Complete Lesson List
Mr. Math web lessons are a free online resource that outlines every topic from Pre-Algebra through Advanced Calculus, utilizing the revolutionary "DNA of Math". Think of it as the world's largest and best math textbook - the likes of which have never before been possible.
Lesson content is still a very long way from 100% complete, but several lessons are published and more all the time! Below you can browse lessons, use the search box, or filter by topic tags with the checkboxes.
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- Integers and the Real Number Line
Pre-Algebra $\rightarrow$ Integers, Operations, and Expressions $\rightarrow$ Working with Integers
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Priority: High
Lesson Description
This lesson introduces the concept of integers, and some common properties and methods we use when working with them, such as positive or negative, absolute value, and relative order.
Learning Objectives
- Know what the word integer means
- Learn how the number line works
- Learn how to compare integers
- Learn what absolute value is
- Adding and Subtracting Integers
Pre-Algebra $\rightarrow$ Integers, Operations, and Expressions $\rightarrow$ Working with Integers
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Priority: VIP Knowledge
Lesson Description
Any two integers can be added together or subtracted from one another. This lesson demonstrates how to combine any two integers, whether using addition or subtraction, and whether each number is positive or negative.
Learning Objectives
- Learn how to add positive and negative integers using a number line
- Learn how to subtract positive and negative integers using a number line
- Be able to add and subtract positive and negative numbers fluently, without a number line
- Setup and answer word problems that ask for a sum or difference
- Variables and Variable Expressions
Pre-Algebra $\rightarrow$ Integers, Operations, and Expressions $\rightarrow$ Using Variables and Expressions
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Priority: High
Lesson Description
Numbers are the essence of math, but variables are what makes algebra algebra. To start, we'll define what variables are and introduce some ways that we use them. Then we'll take a look at what variable expressions are, how to evaluate them given a value of the variable, and why they are different from equations.
Learning Objectives
- Define what variables are and why we use them
- Know how to represent an unknown quantity with a variable
- Introduce the idea of figuring out what value the variable must be, aka solving for a variable
- Know what an expression is
- Learn how to evaluate an expression using given values for the variables
- Understand the difference between an expression and an equation
- Combining Like Terms
Pre-Algebra $\rightarrow$ Integers, Operations, and Expressions $\rightarrow$ Using Variables and Expressions
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Priority: VIP Knowledge
Lesson Description
This lesson shows you how to tell whether or not variable expressions can be combined by adding or subtracting, which, when possible, we refer to the expressions as "like terms". After that, we will practice both adding and subtracting like terms with both positive and negative coefficients.
Learning Objectives
- Understand what the phrase "like terms" means
- Tell the difference between pairs of terms that can and cannot be combined
- Practice combining like terms
- Turning Word Expressions Into Math Expressions
Pre-Algebra $\rightarrow$ Integers, Operations, and Expressions $\rightarrow$ Using Variables and Expressions
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Priority: High
Lesson Description
Often times when word problems describe a situation that we must work with, it is our job to write down an expression or equation to work with. This means we need to be experts at picking up on key clue words and knowing what math operations are implied. This lesson shows us how to master the art of writing down math based on words.
Learning Objectives
- Know how to use numbers to represent real life situations
- Learn how variable expressions can be written from word descriptions
- Identify key words that dictate which operation is appropriate (addition, subtraction, multiplication, division)
- Sets and Real Numbers
Pre-Algebra $\rightarrow$ Properties of Real Numbers $\rightarrow$ Properties for All Numbers
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Priority: Normal
Lesson Description
Though we have been working with numbers for some time now, here we seek to more concisely define real numbers, with a mind for sets of numbers. Mostly, we will examine sets, what they mean, and what properties of them that we study.
Learning Objectives
- Define and understand what sets are, what they mean, and what we usually do with them
- Know how to find the union or intersection of two sets
- Understand what subsets are and how they relate to their parent set
- Define real numbers and understand major common subsets of real numbers, such as rational numbers and irrational numbers
- Rational vs. Irrational Numbers
Pre-Algebra $\rightarrow$ Properties of Real Numbers $\rightarrow$ Properties for All Numbers
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Priority: Normal
Lesson Description
This lesson will help you see and understand the difference between a rational and irrational number. You will also learn about important properties of each.
Learning Objectives
- Concisely define what makes a number rational or not
- Know when a decimal is rational or irrational based on pattern
- Gain familiarity with common irrational numbers and where they come from
- Place Value (Ones to Millions)
Pre-Algebra $\rightarrow$ Properties of Real Numbers $\rightarrow$ Properties for All Numbers
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Priority: High
Lesson Description
For most students, place value up to the millions place is very likely review from grammar school material. In order to make sure we are all on the same page, this lesson reviews place value of each digit from ones to millions.
Learning Objectives
- Define and understand what place value refers to
- Know what each place value represents from ones to millions
- Understand and translate between number form and word form of a number
- Rounding and Estimation
Pre-Algebra $\rightarrow$ Properties of Real Numbers $\rightarrow$ Properties for All Numbers
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Priority: Normal
Lesson Description
Another topic that is likely review from several years prior, rounding is an important concept that manifests itself in many ways in math. While many students probably know how to round, this lesson spells it out so that going forward we know we've covered what's needed.
Learning Objectives
- Learn how to round to the nearest 1, 10, 100, etc.
- Be able to round to the nearest specified decimal place
- Gain confidence about using judgement in which place value to round to
- Basic Exponent Rules
Pre-Algebra $\rightarrow$ Properties of Real Numbers $\rightarrow$ Properties for All Numbers
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Priority: VIP Knowledge
Lesson Description
Earlier in this course, you learned a bit about exponents. Exponents will be really important for us in algebra and even calculus and beyond! Since they're so important, here we will review what we know, add another trick or two to our toolkit, and master it all with some practice.
Learning Objectives
- Review the exponent rules for same base multiplication and powers raised to powers
- Learn a new exponent rule about same base division
- Understand the meaning of exponents of 1 and 0
- Apply exponent knowledge to variables as well
- Know the trick for quickly raising 10 to any power
- Revised Order of Operations
Pre-Algebra $\rightarrow$ Properties of Real Numbers $\rightarrow$ Properties for All Numbers
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Priority: Normal
Lesson Description
By now we've seen and used order or operations in our algebra work. With this lesson, we seek to better understand exactly how the classic "PEMDAS" works, with emphasis on MD and AS as single operations, and working with nested groups within groups, as well as more complex exponent applications.
Learning Objectives
- Fully review what we already know about PEMDAS
- Learn why and how MD refer to the same operation, and why and how AS also refer to the same operation
- Successfully sort through several levels of grouping symbols
- Divisibility and Prime Numbers
Pre-Algebra $\rightarrow$ Properties of Real Numbers $\rightarrow$ Properties for Whole Numbers Only
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Priority: High
Lesson Description
Formally, this lesson defines what it means when we say a number is "divisible" by another number. We also look at what it means to say a number is prime.
Learning Objectives
- When dividing two numbers, know the difference between observing divisibility and not observing divisibility
- Understand what we mean when we say a number is prime
- Greatest Common Factor (GCF)
Pre-Algebra $\rightarrow$ Properties of Real Numbers $\rightarrow$ Properties for Whole Numbers Only
Web Lesson Coming Soon! »
Priority: VIP Knowledge
Lesson Description
Finding the Greatest Common Factor of two or more terms is an incredibly useful skill from Pre-Algebra to Calculus and beyond. Here, we'll see first what GCF means, before moving on to methods of finding the GCF of two or more integers. In future courses we will revisit GCF for variable expressions as we learn more about variables and unknown quantities.
Learning Objectives
- Understand what Greatest Common Factor means
- Learn how to find the GCF of two numbers
- Learn how to find the GCF of more than two numbers
- Understanding Square Roots
Pre-Algebra $\rightarrow$ Properties of Real Numbers $\rightarrow$ Properties for Whole Numbers Only
Web Lesson Coming Soon! »
Priority: High
Lesson Description
In our first Mister Math official look into what square roots are all about, we'll understand what perfect square integers are, and learn how to find their square roots. We'll also see that square roots of other integers are irrational, not integers. Note that if you need to learn how to simplify square roots of integers, this is covered in an upcoming Algebra One lesson.
Learning Objectives
- Define and understand what perfect square integers are
- Reconcile the inverse operations of squaring and square rooting
- Learn that square roots of integers are either integers or irrational, and know when each case happens
- Do NOT (yet) learn how to simplify square roots. This is an upcoming lesson in Algebra One!
- Place Value Again (tenths to millionths)
Pre-Algebra $\rightarrow$ Decimals and Fractions $\rightarrow$ Working with Decimals
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Priority: Normal
Lesson Description
The first time we talked about place value, we were studying how large numbers can become, up to the millions place value. Now that it's time to learn about decimals, we will study how exact a number can get using decimal place value. This lesson also covers the skill of writing the decimal number from the words, and vice versa, even thought we aren't often asked to do this.
Learning Objectives
- Identify each digit's place value in a decimal, up to the millionths place
- Know how to translate back and forth between number form and word form, for decimals with up to six decimal digits
- Ordering Decimals
Pre-Algebra $\rightarrow$ Decimals and Fractions $\rightarrow$ Working with Decimals
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Priority: Normal
Lesson Description
If we are given decimals that are close together, or may even look physically similar by coincidence, we need to be able to know how to put them in order from least to greatest. This lesson shows us how!
Learning Objectives
- Approximate the location of decimals on a number line
- Order a set of given decimal values from least to greatest
- Properties of Fractions and Simplifying
Pre-Algebra $\rightarrow$ Decimals and Fractions $\rightarrow$ Numerical Fractions
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Priority: VIP Knowledge
Lesson Description
We've seen fractions in early grade math, but we're about to start using them a whole lot in Algebra and beyond. This first Mister Math lesson on fractions will remind us of the essential properties we need to have mastered for fractions.
Learning Objectives
- Recall the structure and meaning of fractions
- Fitting in fractions on the number line
- Simplifying numeric fractions to lowest terms
- Identifying and obtaining equivalent fractions
- Identifying and obtaining the reciprocal of a fraction
- Comparing Fractions
Pre-Algebra $\rightarrow$ Decimals and Fractions $\rightarrow$ Numerical Fractions
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Priority: Normal
Lesson Description
If two fractions are not equal, then one is larger than the other. When comparing two fractions, we usually need to analyze before it's clear which fraction is larger. This lesson outlines the best practices for knowing which fraction is larger than the other in this case.
Learning Objectives
- Understand why comparing fractions is more complicated than comparing integers
- Learn how to compare two fractions and know which one is larger than the other - equal denominators
- Learn how to compare two fractions and know which one is larger than the other - different denominators
- Adding and Subtracting Fractions and Mixed Numbers
Pre-Algebra $\rightarrow$ Decimals and Fractions $\rightarrow$ Numerical Fractions
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Priority: VIP Knowledge
Lesson Description
Adding and subtracting fractions is not very different than adding and subtracting integers. The only major difference is that it's only possible to combine fractions simply when the denominators are the same. Here we will look at the method of adding and subtracting fractions when the denominator is the same, and then look at how to proceed when the denominators are not the same. We'll also look at how to add and subtract mixed numbers.
Learning Objectives
- Know how to add and subtract fractions when the denominators are equal
- Know how to add and subtract fractions when the denominators are not equal
- Know the most efficient way to add and subtract mixed numbers
- Equating Fractions and Decimals
Pre-Algebra $\rightarrow$ Decimals and Fractions $\rightarrow$ Using Decimals and Fractions
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Priority: Normal
Lesson Description
Any fraction can be represented in decimal form, and any terminating decimal or repeating non-terminal decimal can be converted to fraction form. Beside familiarizing ourselves with the difference between terminating decimals and repeating non-terminating decimals, this lesson focuses on the necessary and common techniques used to turn terminating decimals into fractions, and vice versa. Note: turning non-terminating repeating decimals into fractions will be covered in the next lesson.
Learning Objectives
- Categorize decimals as terminating or non-terminating (repeating), and know how to notate a repeating decimal
- Know how to turn terminating decimals into fractions
- Understand why a special process is required for turning repeating non-terminating decimals into fractions (this process will be covered two lessons from now!)
- Turn fractions into decimals via long division for fractions with common number denominators that will not cause non-terminating decimals
- Know which decimal answers result from working with 1/2, 1/3, 1/4, 1/5, and 1/6
- Convert common fraction and accompanying decimal representations as required, using mental math only
- Repeating Decimals
Pre-Algebra $\rightarrow$ Decimals and Fractions $\rightarrow$ Using Decimals and Fractions
Web Lesson Coming Soon! »
Priority: Normal
Lesson Description
Intentionally, we have only thus far worked with converting decimals that have finite place value into fractions. Repeating decimals are naturally occurring rational numbers, but they are just a little harder to work with. This lesson shows you how to turn any repeating decimal into a fraction, which, beside something a teacher will expect you to know how to do, will make life easier if we have to perform further arithmetic.
Learning Objectives
- Understand why repeating decimals exist
- Practice turning fractions into decimals when the denominator will cause a repeating decimal to appear
- Turn typical repeating decimals into fractions
- Turn more complicated repeating decimals into fractions
- Comparing Rational Numbers
Pre-Algebra $\rightarrow$ Decimals and Fractions $\rightarrow$ Using Decimals and Fractions
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Priority: Normal
Lesson Description
Any two real numbers compare to one another with two outcomes: they are equal, or one is larger than the other. Focusing on rational numbers, and using our fraction and decimal skills, we see a few ways to look at a set of numbers and determine which number is larger or smaller than the other numbers.
Learning Objectives
- Review what we already know about comparing fractions to fractions or decimals to decimals
- Learn best practices of which numbers to convert, when working with a mixed bag of decimals and fractions
- Combining Decimals and Fractions
Pre-Algebra $\rightarrow$ Decimals and Fractions $\rightarrow$ Using Decimals and Fractions
Web Lesson Coming Soon! »
Priority: Optional
Lesson Description
Though an uncommon skill, some teachers will include quizzes and tests on your ability to add and subtract mixed quantities of decimals and fractions. When do you convert one to the other or vice versa? This lesson gives you the right advice to make it as efficient as possible.
Learning Objectives
- Learn how to add and subtract mixed quantities of decimals and fractions
- Properties of Equations
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Manipulating Equations
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Priority: High
Lesson Description
Even though you've probably seen the classic "=" before, it's time to take a closer look at exactly what defines an equation, and how equations work.
Learning Objectives
- See the definition of what an equation is
- Know the classifications of equations (open vs closed, true vs false)
- Define exactly what we mean when we refer to the "solution" of an equation
- Solving One-Step Integer Equations
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Manipulating Equations
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Priority: High
Lesson Description
In this introductory equation solving lesson, you'll learn how to solve the two different kinds of one-step linear equations.
Learning Objectives
- Recognize the two types of one-step equations
- Solve addition and subtraction one-step equations
- Solve multiplication and division one-step equations
- Solving Two-Step Integer Equations
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Manipulating Equations
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Priority: High
Lesson Description
Two-step equations are probably the most common type of equation we work with throughout Algebra. In this lesson we build upon what we learned in the last lesson, and master solving two-step linear equations.
Learning Objectives
- Recognize the general form of a two-step equation
- Solve two step-equations and show the right work
- Solving Multi-Step Integer Equations
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Manipulating Equations
Web Lesson Coming Soon! »
Priority: VIP Knowledge
Lesson Description
Up to now, we've solved equations that involved at most two steps. Now we'll learn how to solve equations that require several steps, using our newly gleaned knowledge of combining like terms.
Learning Objectives
- Combine like terms at every chance to make the equation simpler
- Use the distributive property when required to clean up equations
- Move multiple terms to and from each side of the equation to get all variable terms on one side only
- Solving Equations with Fractions
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Manipulating Equations
Web Lesson Coming Soon! »
Priority: Normal
Lesson Description
Using everything we just learned about fractions and what we learned much earlier on solving equations, we will see how to solve single-step, two-step, and multi-step equations that have fractions in them.
Learning Objectives
- Solve equations with fraction coefficients and/or fraction constants
- Know the best approach for equations that have mixed numbers
- Solving Equations with Decimals
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Manipulating Equations
Web Lesson Coming Soon! »
Priority: Normal
Lesson Description
Now that we've studied how to work with decimals, we will apply what we know to solving single-step, two-step, and multistep equations that have decimal coefficients.
Learning Objectives
- Solve equations with decimal coefficients and/or decimal constants
- Learn common techniques to make these equations more like integer equations that we recently learned about
- Strategies for Solving Fraction and Decimal Equations
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Manipulating Equations
Web Lesson Coming Soon! »
Priority: Optional
Lesson Description
This lesson shows you a few techniques for making it easier to solve fraction and decimal equations. No new concepts are taught here, just some mastery, tips, and practice.
Learning Objectives
- Know to take care of distributive property expressions first
- Clear fractions using the LCD to make things easier
- Clear decimals by multiplying the entire equation by powers of 10
- Linear Absolute Value Equations
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Manipulating Equations
Web Lesson Coming Soon! »
Priority: Normal
Lesson Description
Equations that include an absolute value expression are solvable only when we isolate the absolute value and consider both cases. We'll learn the right way to do that here, after first remember important facts and know-how about the absolute value operator.
Learning Objectives
- Review what absolute value operators are and how they work
- Solve absolute value equations by splitting the problem into two cases
- Learn how to properly prepare the equations to split into two cases, and why it cannot necessarily be done as the first step
- Know when and how to check for extraneous solutions
- Properties of Inequalities
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Working with Inequalities
Web Lesson Coming Soon! »
Priority: High
Lesson Description
The first Mister Math lesson on inequalities focuses on what they are, and their properties. There are many things we already know how to do by translating what we already know about equations.
Learning Objectives
- Understand what inequalities mean and why they are aptly named
- Learn the similarities and differences between working with inequalities and working with equations
- Determine endpoint inclusion or exclusion, based on wording
- Graphing Solutions to Inequalities
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Working with Inequalities
Web Lesson Coming Soon! »
Priority: Normal
Lesson Description
We just learned that solutions to inequalities are ranges of values, not one specific value. We are often required to graph the solution range on a number line. This lesson covers how to do just that.
Learning Objectives
- Learn best practices for graphing inequality solution ranges on a number line
- Maintain cognizance of including or excluding the endpoints
- Know what teachers usually mark points off for not doing or forgetting
- Solving Inequalities with Decimals and Fractions
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Working with Inequalities
Web Lesson Coming Soon! »
Priority: Normal
Lesson Description
The difference between solving inequalities with integers and inequalities with fraction or decimal coefficients is similar to what we saw in the world of working with equations. This lesson gives us the chance to practice our newly gained knowledge of how to work with and solve inequalities, while also giving us the chance to revisit the nuances of working with non-integer coefficients.
Learning Objectives
- Continue to reinforce the mechanics of solving inequalities, including sign switching behavior
- Recall tips and tricks for making it easier to work with non-integer coefficients
- Practice solving multi-step inequalities that contain decimals and fractions
- Compound Linear Inequalities
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Working with Inequalities
Web Lesson Coming Soon! »
Priority: Normal
Lesson Description
Because the solution to an inequality problem is a range of values, it is possible and common to solve for solution ranges that satisfy two inequalities at the same time. This lesson helps you determine the difference between AND compound inequalities and OR compound inequalities, as well as what the solution ranges look like in each case.
Learning Objectives
- Understand what a compound inequality is and why it is different then a standard, single inequality
- Recognize the two types of "AND" compound inequalities
- Recognize the "OR" type of compound inequality
- Be able to graph compound inequalities on a number line
- Practice solving general compound inequalities by categorizing the problem and working with it accordingly
- Solving Absolute Value Inequalities
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Working with Inequalities
Web Lesson Coming Soon! »
Priority: High
Lesson Description
Solving absolute value inequalities requires a very similar approach to solving absolute value equations, which we looked at recently. Here we will look at how to solve these inequalities and graph their solution on the number line.
Learning Objectives
- Solve absolute value inequalities by splitting the problem into two cases
- Give appropriate answers to inequalities using the words "and" or "or"
- Graph the solutions to absolute value inequalities on the real number line
- Know how to check for extraneous solutions
- Using Formulas and Solving for a Variable
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Using Equations and Inequalities
Web Lesson Coming Soon! »
Priority: High
Lesson Description
In math and science applications, when we are working with formulas, we typically know the value of all the variables except one of them. Since we may need to isolate any one of the variables, this lesson focuses on how to start with any formula and isolate any variable.
Learning Objectives
- Be able to fluidly move variables back and forth to each side of a formula equation
- Isolate a given variable in a formula when the variable appears once
- Isolate a given variable in a formula when the variable appears more than once
- Employ the same skills with inequalities
- One Variable Linear Word Problems
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Using Equations and Inequalities
Web Lesson Coming Soon! »
Priority: Normal
Lesson Description
Congrats! You're an equation solving pro by now! It's time for that classic favorite, the one that everyone knows and loves - word problems! You will learn how to make an equation from the described situation, and then solve said equation.
Learning Objectives
- Solve word problems with a single missing quantity using multi-step equations
- Master the approach to solving word problems
- Understand situations and setup your own equation
- Decimal and Fraction Word Problems
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Using Equations and Inequalities
Web Lesson Coming Soon! »
Priority: Normal
Lesson Description
Similar to what we've seen in the past for word problems, we need to set up expressions and equations based on the wording of a question. This time, we will be working with word problems that require solving equations that involve decimals and fractions.
Learning Objectives
- Continue practicing setting up word problems
- Use the word problem step approach to make sure the question is answered
- Practice solving word problems that describe situations involving decimals and fractions
- Problem Solving with Inequalities
Pre-Algebra $\rightarrow$ Equations and Inequalities $\rightarrow$ Using Equations and Inequalities
Web Lesson Coming Soon! »
Priority: Normal
Lesson Description
Linear situations where we need the answer to be bigger or smaller than something else implies working with inequalities. Here we will look at some word problems that require setting up and then solving an inequality, as well as interpreting the results in context.
Learning Objectives
- Look for key words that clue us into the need to use inequalities
- Set up an inequality based on the wording in a problem
- Solve inequalities that you set up, and interpret your answer to ensure it makes sense in context of the problem
- Solving Percent Equations and Word Problems
Pre-Algebra $\rightarrow$ Percents, Ratios, and Proportions $\rightarrow$ Percentages
Web Lesson Coming Soon! »
Priority: High
Lesson Description
Solving equations that involve percents are challenging without the right perspective. This lesson demonstrates the common ways a percent equation is structured, and gives you the tools to set up your own percent equations correctly no matter what the missing quantity is...100% of the time (sorry).
Learning Objectives
- See and know the different missing quantities (the percent, the base number, or the result) that we may need to solve for in a percent equation
- Understand how to solve percent equations using the decimal form definition of a percent
- Understand how to solve percent equations using the fraction form definition of a percent
- Solve percent word problems by turning key words directly into math symbols, yielding a solvable equation
- Estimating Percentages Mentally
Pre-Algebra $\rightarrow$ Percents, Ratios, and Proportions $\rightarrow$ Percentages
Web Lesson Coming Soon! »
Priority: Optional
Lesson Description
This light optional lesson gives you the tools to make quick and dirty estimates of quantities that involve percents. As a real world skill, this ranks highly for answering quick questions that require only an estimate.
Learning Objectives
- Use mental math rounding techniques to get estimates that involve percents
- Learn the "1% estimation" method for getting a ballpark estimate
- Percent Change
Pre-Algebra $\rightarrow$ Percents, Ratios, and Proportions $\rightarrow$ Percentages
Web Lesson Coming Soon! »
Priority: High
Lesson Description
If a pair of shoes increases in price by a dollar, is that "a lot"? It depends on whether the shoes originally cost 5 dollars or 500 dollars! This very important lesson shows you how to measure change as a percent - a skill that is actually useful IRL!
Learning Objectives
- Define percent change formulaically
- Understand why the original value matters to the answer, not just the new discounted or marked up value
- Use problem context to determine if we are performing percent increase or percent decrease
- Solve problems involving known percent change but unknown values
- Working with Ratios
Pre-Algebra $\rightarrow$ Percents, Ratios, and Proportions $\rightarrow$ Ratios and Proportions
Web Lesson Coming Soon! »
Priority: Normal
Lesson Description
This lesson introduces the concept of ratios, and how they are similar yet different from fractions. We will understand exactly what they are, and examine some properties and concepts specific to working with ratios. Then we'll see how ratios are used to solve problems.
Learning Objectives
- Understand what ratios are and why they are different from fractions, even though they are related
- Learn the three major notations we see for expressing ratios
- Know how to convert ratios to equivalent ratios, similar to what we can do with fractions
- Work with three part ratios
- How to solve the missing quantity when either one of the parts is missing or the total is missing
- Set up and solve ratio based problems and equations, from word problems
- Working with Rates and Proportions
Pre-Algebra $\rightarrow$ Percents, Ratios, and Proportions $\rightarrow$ Ratios and Proportions
Web Lesson Coming Soon! »
Priority: High
Lesson Description
In addition to working with ratios, proportionality also applies when a task is being done at a certain rate. This lesson shows us how to use proportions to answer rate based word problems, where we will measure in units that use "per" (e.g. Miles per hour, words per minute, gallons per dollar, etc.).
Learning Objectives
- Use unit based ratios and turn ratios into single rates
- Define what a proportion is and how to solve for a missing quantity in a proportion using "cross multiplication"
- Know how to identify when a word problem implies a proportion
- Set up and solve rate based word problems
- Scale Models
Pre-Algebra $\rightarrow$ Percents, Ratios, and Proportions $\rightarrow$ Ratios and Proportions
Web Lesson Coming Soon! »
Priority: Optional
Lesson Description
Using proportions, we can calculate the size of an object when we know the size of another object that is bigger or smaller, but otherwise identical. We'll see how to do that in the context of word problems.
Learning Objectives
- Know what it means when we says two shapes are similar
- Use ratios or proportions to calculate unknown quantities
- Apply principles of proportionality to word problems
- Pre-Algebra Exponent Recap
Algebra One $\rightarrow$ Polynomials $\rightarrow$ Advanced Exponent Properties
Web Lesson Coming Soon! »
Priority: Normal
Lesson Description
This lesson serves to remind students of the three exponent rules learned in the Pre-Algebra course. Each is incredibly important and used frequently in algebra and beyond. This lesson is one of the rare ones that is comprised completely of review of past material, but is useful for putting all exponent rules in one place.
Learning Objectives
- Review three prior learned exponent rules
- Review what raising to the power of 1 or 0 means, and why
- Negative Exponents
Algebra One $\rightarrow$ Polynomials $\rightarrow$ Advanced Exponent Properties
Web Lesson Coming Soon! »
Priority: VIP Knowledge
Lesson Description
We've worked with exponents before, but in this lesson we will see what negative exponents are (and what they are not!).
Learning Objectives
- Define and understand what negative exponents are
- Evaluate expressions that use negative exponents
- Re-write expressions that use negative exponents using only positive exponents
- Scientific Notation
Algebra One $\rightarrow$ Polynomials $\rightarrow$ Advanced Exponent Properties
Web Lesson Coming Soon! »
Priority: Normal
Lesson Description
Very large numbers and very small numbers can be tedious to write out in their traditional forms. Scientific Notation solves this problem, as well as helps us articulate what level of accuracy we are certain about (e.g. we can estimate the distance from Earth to the sun to the nearest million miles, but not the nearest mile - you would not likely believe someone who insisted that it is exactly 93,254,803 miles away).
Learning Objectives
- Understand what scientific notation is and why it is needed
- Turn very large and very small numbers into scientific notation form
- Quickly turn scientific notation form numbers back into standard numbers
- Exponents and Fractions
Algebra One $\rightarrow$ Polynomials $\rightarrow$ Advanced Exponent Properties
Web Lesson Coming Soon! »
Priority: High
Lesson Description
When it comes to exponents, we've learned a lot so far, but have really only used exponents on whole numbers. Sometimes we have to raise a fraction to a power. This lesson shows us how to do just that.
Learning Objectives
- Review exponent rules up to this point
- Know what to do when exponents are applied to an entire fraction
- Know what to do when fractions have numerators and denominators with the same exponent
- Raising Products to Powers
Algebra One $\rightarrow$ Polynomials $\rightarrow$ Advanced Exponent Properties
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Priority: High
Lesson Description
After a quick review of all the exponent rules we have learned up to this point, this lesson adds on one final exponent rule for the course, involving what happens when you raise a product collectively to a power. This lesson also quickly shows why this rule cannot apply to a sum.
Learning Objectives
- Know what to do when simplifying an entire product that is raised to a power
- Be able to reverse this process and group terms together with a single common exponent
- Defining Monomials
Algebra One $\rightarrow$ Polynomials $\rightarrow$ Working with Monomials
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Priority: High
Lesson Description
Though by now we are used to working with variables, we need to better define objects called "monomials", which act like standalone units in many ways. This lesson examines the structure of monomials, as well as some basic properties about them.
Learning Objectives
- Understand exactly what a monomial is
- Apply concepts that we already know about coefficients and like terms to monomials
- Adding and Subtracting Monomials
Algebra One $\rightarrow$ Polynomials $\rightarrow$ Working with Monomials
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Lesson Description
Similar to working with single variable like terms we've already seen, adding and subtracting monomials is a matter of knowing whether or not two monomials are compatible, and if so, combining their coefficients properly. This lesson covers the know-how on adding and subtracting monomial expressions.
Learning Objectives
- Be aware when two monomials can and cannot be combined
- Practice combining monomials with addition and subtraction
- Dividing and Simplifying Monomials
Algebra One $\rightarrow$ Polynomials $\rightarrow$ Working with Monomials
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Priority: VIP Knowledge
Lesson Description
In this VIP Knowledge lesson, we'll learn how to comprehensively simplify fractions composed of monomials, where coefficients and variables each may reduce by canceling common factors. We will use exponent rules that we recently learned to reduce variable expressions to simplest form.
Learning Objectives
- Recall what simplifying a fraction entails by refreshing with numerical examples
- Observe the connection between dividing variables and the exponent same-base division subtraction rule
- When dividing two monomials, go through each variable systematically and produce a final, simplified answer
- GCF and LCM of Monomials
Algebra One $\rightarrow$ Polynomials $\rightarrow$ Working with Monomials
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Priority: High
Lesson Description
In Pre-Algebra we learned a bit about how to find the Greatest Common Factor or the Least Common Multiple of two or more numbers. Each process was unique and had its best practices. Here, we will learn the same conceptual techniques to carry these processes out for monomials rather than integers, and understand any differences in finding GCF and LCM between working with monomials and working with numbers.
Learning Objectives
- Recall what factors and multiples are for integers
- Understand what constitutes a factor or a multiple for a monomial
- Recall how to find the GCF of two integers
- Learn how to identify the GCF of two or more monomials
- Recall how to find the LCM of two integers
- Learn how to find the LCM of two or more monomials
- Understanding Polynomials
Algebra One $\rightarrow$ Polynomials $\rightarrow$ Working with Polynomials
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Priority: High
Lesson Description
Generally a polynomial is formed when we add monomials together. Since we work with polynomials a whole lot in Algebra, Calculus and, believe it or not, the real world, it makes sense to start by first defining exactly what they are, and what kinds of polynomials we typically work with.
Learning Objectives
- Define polynomials by their structure
- Gain familiarity about the types of polynomials you are likely to analyze and work on.
- Adding and Subtracting Polynomials
Algebra One $\rightarrow$ Polynomials $\rightarrow$ Working with Polynomials
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Priority: Normal
Lesson Description
Hopefully it won't be too surprising to learn that adding or subtracting polynomials is going to look very similar to what we did when we studied adding and subtracting monomials. We'll see some common tips and traps to keep in mind to make life easier.
Learning Objectives
- Learn how to add polynomials keeping a "like terms" approach in mind
- Using the distributive property properly when subtracting an entire polynomial
- Polynomials Divided by Monomials
Algebra One $\rightarrow$ Polynomials $\rightarrow$ Working with Polynomials
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Priority: Normal
Lesson Description
Now that we are familiar with many different arithmetic operations involving polynomials, this lesson will show us how to divide a polynomial by a monomial.
Learning Objectives
- Learn how to simplify if possible when dividing a polynomial by a monomial
- Verify your results of dividing by multiplying and getting back the original answer
- NOT learning how to perform polynomial long division - that is a separate lesson in Algebra Two!
- Polynomial Word Problems
Algebra One $\rightarrow$ Polynomials $\rightarrow$ Working with Polynomials
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Priority: Optional
Lesson Description
In this lesson, you will practice setting up a polynomial arithmetic equation from the description in a word problem. Since we haven't yet practiced solving complicated polynomials, this lesson focuses on problem setup, and only occasionally solving the problem.
Learning Objectives
- Learn how to set up word problems that involve polynomial multiplication, addition, or subtraction
- Solve such word problems, but for this lesson, only if the problem reduces to a multistep equation that we've already learned how to solve
- Factoring Polynomials Using GCF
Algebra One $\rightarrow$ Polynomials $\rightarrow$ Getting Started with Factoring
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Priority: High
Lesson Description
Using GCF know-how when it comes to monomials, we can seek out the GCF of a polynomial and factor it out. This is a skill we will use once in a while, but when we use it, it's a life saver.
Learning Objectives
- Factoring out a GCF from one-variable polynomials
- Factoring By Grouping
Algebra One $\rightarrow$ Polynomials $\rightarrow$ Getting Started with Factoring
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Priority: Normal
Lesson Description
First we'll take GCF factoring to the next level by learning how to factor out entire expressions from a larger one. Then, leveraging what we have learned about factoring out entire expressions, we will look at how to turn the sum of four objects into a product with a technique called Factoring by Grouping.
Learning Objectives
- Examine the distributive property backward to understand how to factor out an entire expression
- Learn and practice the Factoring by Grouping technique
- Factoring Out Negatives
Algebra One $\rightarrow$ Polynomials $\rightarrow$ Getting Started with Factoring
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Priority: High
Lesson Description
While this will not be the most common factoring we do, it will sometimes be necessary to factor out a negative from an expression. We will see how to accomplish this, and look at a few reasons why we may want to be able to.
Learning Objectives
- Quadratic Trinomial Factoring Basics
Algebra One $\rightarrow$ Intro to Quadratics $\rightarrow$ Quadratic Factoring Forms
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Priority: VIP Knowledge
Lesson Description
Of all the factoring you will do in your math career, this skill may be the one you do the most of. Here we will factor quadratic trinomials into the product of two binomials, which is often referred to as "reverse FOIL".
Learning Objectives
- Learn the method behind the trinomial factoring process, which is often referred to as "reverse FOIL"
- Know how to factor quadratic trinomials with a leading coefficient of 1 ($x^2 + bx + c$)
- Discern when these trinomials can and cannot be factored
- Advanced Trinomial Factoring
Algebra One $\rightarrow$ Intro to Quadratics $\rightarrow$ Quadratic Factoring Forms
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Priority: Normal
Lesson Description
If you've mastered the basics from the last lesson, you're ready to really turn up the heat on your trinomial factoring skills. This lesson has you practice more factoring but in cases when the "a" term has a coefficient. It also covers a special two-variable case.
Learning Objectives
- Continue to master the trinomial factoring method
- Completely factor trinomials that have a GCF present
- Know how to factor quadratic trinomials with any leading coefficient ($ax^2 + bx + c$)
- Discern when these trinomials can and cannot be factored
- Apply the method to the special two variable case $ax^2 + bxy + cy^2$
- Difference of Squares
Algebra One $\rightarrow$ Intro to Quadratics $\rightarrow$ Quadratic Factoring Forms
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Priority: VIP Knowledge
Lesson Description
A binomial that consists of one squared object subtracted from another squared object is a very special form that factors into a product of two binomials. Beside classic trinomial factoring, this form, called the Difference of Squares, is one of the most popular forms we encounter through our entire math career that we are expected to know how to factor.
Learning Objectives
- Derive the Difference of Squares factoring form from what we know about FOIL
- Be able to use the factoring method forward and backward
- Utilize the Difference of Squares method when a GCF is present in both objects
- Utilize the Difference of Squares method when one of the squared objects is a perfect square trinomial
- Choosing a Factoring Method
Algebra One $\rightarrow$ Intro to Quadratics $\rightarrow$ Quadratic Factoring Forms
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Priority: Normal
Lesson Description
In more advanced math courses, factoring isn't something you're tested on, it's something you're expected to use to your advantage. You need to be well-familiarized with your factoring options. The focus of this lesson is how to proceed generally when you won't know which approach you are supposed to use.
Learning Objectives
- Develop independence for choosing how to proceed with factoring in situations where you are not told what method to use
- Understand general guidelines for factoring and why certain factoring techniques make sense to try before certain other techniques
- Solving Quadratics Using Square Roots
Algebra One $\rightarrow$ Intro to Quadratics $\rightarrow$ Solving Quadratic Equations
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Priority: Normal
Lesson Description
This first lesson on solving quadratic equations shows us just how straight forward it can sometimes be. Equipped with only the rules we know about working with equations, we can solve vertex form quadratics by acting equally on both sides of the equation.
Learning Objectives
- See how to solve vertex form quadratics using algebra equation rules
- Ensure that you always have two solutions, even if they are the same number or imaginary
- Recall prior knowledge of square roots to understand why taking the square root of a squared object means that the result could be positive or negative
- Solving Quadratics with Factoring
Algebra One $\rightarrow$ Intro to Quadratics $\rightarrow$ Solving Quadratic Equations
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Priority: High
Lesson Description
The next way we can solve quadratic equations is to use trinomial factoring. We learned and practiced this in Algebra One, but here, after we review what we know, we'll add in some GCF factoring and tricks for working with fraction coefficients.
Learning Objectives
- Use what we know about how to factor quadratics to solve equations
- Review solving quadratic equations using the "reverse FOIL" trinomial factoring method
- Recall and use the zero product property to solve equations
- The Quadratic Formula
Algebra One $\rightarrow$ Intro to Quadratics $\rightarrow$ Solving Quadratic Equations
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Priority: VIP Knowledge
Lesson Description
Solving quadratic equations using the techniques we've learned so far are great, but they are not certain. They only work under the right conditions. Here we will take our first look at the quadratic formula, which is the only fail safe method for solving a quadratic equation. It is also the fastest method for quadratics with annoying coefficients, since it is direct.
Learning Objectives
- Love and memorize the quadratic formula
- Use the discriminant to identify what kind of roots a quadratic equation has
- Practice using the formula on quadratic equations that yield integer results, and relate those results to factoring the quadratic
- Practice using the formula to get more complicated (non-integer) results
- Completing the Square
Algebra One $\rightarrow$ Intro to Quadratics $\rightarrow$ Solving Quadratic Equations
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Priority: High
Lesson Description
This technique for solving quadratics is often puzzling to students, since it involves some strange equation manipulation. But this technique has a few important uses in the future, and particularly is a topic that gets easier with practice!
Learning Objectives
- Learn how to solve a quadratic equation by completing the square
- Practice the simple case of this technique with quadratics that have a leading coefficient of 1
- Practice the general case of this technique with quadratics that have leading coefficients other than 1
- Know how to handle all steps of the process in cases when fraction coefficients are present in the initial problem
- Defining Rational Expressions
Algebra One $\rightarrow$ Rational Expressions $\rightarrow$ Working with Linear Rational Expressions
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Priority: High
Lesson Description
This first lesson on rational expressions sets up some basic ideas and defines some common properties about rational expressions that will help us take our work further in the upcoming lessons.
Learning Objectives
- Understanding what someone means when they refer to an expression as being "rational"
- Understanding what rational equations look like with linear factors in the numerator and/or the denominator
- Know when expressions are equivalent even if we do not know the value of the variable
- Adding and Subtracting Basic Rational Expressions
Algebra One $\rightarrow$ Rational Expressions $\rightarrow$ Working with Linear Rational Expressions
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Priority: High
Lesson Description
Rational expressions, like numeric fractions, can only be added or subtracted when both expressions have the same denominator. Getting rational expressions to have the same denominator is slightly more involved than getting numeric fractions to have the same denominator, so first we'll see some best practices to make that process as smooth as possible. Then, we'll be able to practice the overall process of adding and subtracting rational expressions, start to finish.
Learning Objectives
- Understand what needs to be done to prepare rational expressions to be combined with addition or subtraction
- Understand what a least common denominator means for fractions with variable expressions
- Know the method for finding the least common denominator, not just any common denominator
- Identify the LCD of two or more rational expressions
- Apply the multiplicative identity to each turn each rational expression into an equivalent fraction with the least common denominator
- Combine rational expressions properly once they have the same least common denominator
- Variation
Algebra One $\rightarrow$ Rational Expressions $\rightarrow$ Problem Solving with Rational Terms
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Lesson Description
Relationships between quantities that are proportional are said to have variation with one another. This can happen in a few ways, in fact. This lesson qualifies the three categories of variation relationships, and shows you how to quantify specific variation relationships. With a specific relationship defined, we can also solve application problems.
Learning Objectives
- Understand what variation means in general
- Know the differences between direct, inverse, and joint variation relationships
- Understand how the variation constant defines the variation relationship
- Use the relationship to solve problems
- Unit Analysis and Conversions
Algebra One $\rightarrow$ Rational Expressions $\rightarrow$ Problem Solving with Rational Terms
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Priority: Normal
Lesson Description
Any time we are given one quantity in one unit and asked to find an answer in a different unit, we can set up a unit analysis chain. This lesson shows us how to do that, converting from one unit to another one step at a time to solve problems that are otherwise too many steps to process at once.
Learning Objectives
- Identify when unit analysis is applicable to a problem
- Practice setting up unit analysis problems as a string of fractions multiplied together
- Linear Problem Solving with Rational Expressions
Algebra One $\rightarrow$ Rational Expressions $\rightarrow$ Problem Solving with Rational Terms
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Priority: Normal
Lesson Description
Problem solving for complex relationships between quantities that have fraction relationships is a topic that most Algebra students see along their journey. Fortunately, there are only a few common types, so we can master it with some targeted instruction and structured practice.
Learning Objectives
- Be familiar with several common categories of word problems: mixtures, combined work, distance/rate/time, and others
- Set up word problems from scratch with confidence based on the category or type of problem, and then solve the problem
- Learn why "mixture" problems tempt you with false logic, and how to set them up properly
- Understand the general principle of setting up a collaborative work equation, and solve for any unknown variable in a collaborative work situation
- Understand what the DERT formula is and how it works, both for single trips and multi-part trips
- Perfect Square Integers
Algebra One $\rightarrow$ Radical Expressions and Roots $\rightarrow$ Exploring Numeric Square Roots
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Priority: Normal
Lesson Description
Square roots are a common operation that we probably didn't need much before now. Here we'll make sure we understand exactly what roots are, and that when a square root of a number is rational, it is because the number was what we call a "perfect square". In addition to helping us better understand what square roots are, this lesson will help us better understand where perfect square integers come from, and how we can quickly generate a list of them.
Learning Objectives
- Recall what we know so far about square roots from Pre-Algebra
- Remember what rational vs irrational numbers are, and know when square roots are one or the other
- Know what perfect square integers are
- Be able to quickly generate and identify perfect square integers
- Be familiar with the square roots of the common perfect square integers
- Estimating Square Roots
Algebra One $\rightarrow$ Radical Expressions and Roots $\rightarrow$ Exploring Numeric Square Roots
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Priority: Optional
Lesson Description
While perfect square integers have square roots that are also integers, the number of perfect square integers compared to the number of integers that are not perfect squares is very small - most integers do not have clean square roots. However, it is important that you can estimate the square root of any integer with or without a calculator, with at least some measure of reasonability.
Learning Objectives
- Know how to estimate square roots without a calculator when the answer is not an integer
- Be able to reasonably estimate the square root of an integer
- See approximation methods for computing square roots
- Advanced Difference of Squares
Algebra One $\rightarrow$ Radical Expressions and Roots $\rightarrow$ Using Roots
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Priority: Normal
Lesson Description
Most of the time when we need to use Difference of Squares, it will be the cases we already looked at in the first Difference of Squares lesson. However, occasionally we need to factor slightly more complicated variable expressions using the Difference of Squares technique.
Learning Objectives
- Confirm the knowledge we already have about the Difference of Squares factoring technique
- Take the square root of multivariable monomials to use the Difference of Squares factoring method
- Understand complex forms of a Difference of Squares such as perfect square trinomials
- Basic Square Root Equations
Algebra One $\rightarrow$ Radical Expressions and Roots $\rightarrow$ Using Roots
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Priority: High
Lesson Description
One of the things we can use root properties for is solving equations that have root expressions in them. In this lesson we'll do just that - we'll practice isolating x in an equation when x is part of a root expression.
Learning Objectives
- Solve equations that contain one single variable root expression and no other variable terms
- Solve equations that contain one variable root expression and also have other variable terms
- Check for extraneous solutions when solving root equations
- The Cartesian Coordinate Plane
Algebra One $\rightarrow$ Linear Graphing $\rightarrow$ The Coordinate Plane and Line Equations
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Priority: VIP Knowledge
Lesson Description
In this first Mister Math lesson on the Coordinate Plane, we'll get up to speed on all the important basics that you may or may not have seen before high school, such as naming points and quadrants, and some vocabulary.
Learning Objectives
- Know how to navigate the Coordinate Plane
- Define important vocab used to label qualities about the Plane
- Learn how to plot ordered pairs on the plane
- Equations With Two variables
Algebra One $\rightarrow$ Linear Graphing $\rightarrow$ The Coordinate Plane and Line Equations
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Priority: Normal
Lesson Description
The key purpose of the coordinate plane is to study relationships between two variables. Before we start graphing two-variable equations (in the next lesson), we need to be better familiar with the properties of two-variable equations, in terms of what they mean and how to work with them, and understand the similarities and differences to regular one-variable equations we've worked with up to this point.
Learning Objectives
- Learn what a two-variable equation is
- Understand what the solution set to a two-variable equation is, and what it means
- Know when a two-variable relationship is categorized as linear
- Find and/or verify a solution to a two-variable equation
- Preview the visual interpretation of a two-variable equation
- Getting Started with Graphing Lines
Algebra One $\rightarrow$ Linear Graphing $\rightarrow$ The Coordinate Plane and Line Equations
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Priority: High
Lesson Description
Because an open equation with two variables has infinitely many solutions, it also has a (usually) smooth graph on the Coordinate Plane that allows us to quickly see which specific ordered pairs are solutions. This lesson focuses on how to get information about the graph from the equation, and vice versa.
Learning Objectives
- Clarify the relationship between a two-variable equation and its visual representation on the plane
- Learn basic how-to for graphing a two-variable relationship in the Coordinate Plane
- Define $x$ and $y$ intercepts of a relationship based on the graph
- Define $x$ and $y$ intercepts of a relationship based on the equation
- Horizontal and Vertical Lines
Algebra One $\rightarrow$ Linear Graphing $\rightarrow$ The Coordinate Plane and Line Equations
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Priority: High
Lesson Description
Two special cases of straight lines in the plane occur when one of the variables, $x$ or $y$, is omitted from the equation. This lesson helps us understand what happens and why, when only one of $x$ or $y$ is present.
Learning Objectives
- Know the general form of straight, horizontal lines
- Know the general form of straight, vertical lines
- Correctly describe the slope of horizontal and vertical lines
- Slope
Algebra One $\rightarrow$ Linear Graphing $\rightarrow$ Graphing Linear Equations
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Priority: VIP Knowledge
Lesson Description
Linear two-variable equations always graph as a straight line, and every straight line has a fixed slope. This slope, or steepness, is the first thing we want to understand as we start to learn how to graph linear equations.
Learning Objectives
- Recall what makes an equation linear vs non-linear
- Recognize that all linear equations have fixed slope
- Understand how we measure and identity slope
- Given the graph of a line, determine the slope of the line visually by counting
- Given two ordered pairs, determine the slope between them with algebra
- Slope - Intercept Form of a Line
Algebra One $\rightarrow$ Linear Graphing $\rightarrow$ Graphing Linear Equations
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Priority: VIP Knowledge
Lesson Description
Of the three linear forms we will need to become friends with, this is often the most commonly used. This lesson focuses on how to work with linear equations in Slope-Intercept form, and how to get linear equations to be in this form if they are not given to you that way.
Learning Objectives
- Define the slope-intercept form of a linear equation
- Practice manipulating two-variable $x$-$y$ equations into this form when it is not given to you that way
- Using equations in this form to quickly graph the line without having to plot random points
- Point - Slope Form of a Line
Algebra One $\rightarrow$ Linear Graphing $\rightarrow$ Graphing Linear Equations
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Priority: High
Lesson Description
The second linear form we need to know is incredibly helpful in the right situations. Here we will understand what Point-Slope linear form is, as well as when we want to use it.
Learning Objectives
- Define the point-slope form of a linear equation
- Know when this form is and is not the right form to use to make your job easiest
- Graphing a line from an equation in point-slope form
- Writing the point-slope equation from given information
- Standard Form of a Line
Algebra One $\rightarrow$ Linear Graphing $\rightarrow$ Graphing Linear Equations
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Priority: Normal
Lesson Description
The third linear form we must be familiar with is called Standard Form, and is useful for finding both $x$ and $y$ intercepts quickly. It will also be useful in the future when studying simultaneous linear equations (aka Systems of Equations).
Learning Objectives
- Define Standard Form for a linear relationship
- Understand the advantages of using this form, as well as the disadvantages
- Know the best way to graph a line that is given to you in Standard Form
- Line Equations in General
Algebra One $\rightarrow$ Linear Graphing $\rightarrow$ Graphing Linear Equations
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Priority: Normal
Lesson Description
Each of the three linear forms has its perks and pitfalls. It is your job not only to learn the material in each of the three preceding lessons, but also gain insight as to which forms are advantageous in which situations, which we will practice here. You also will become masterful at the essential skill of converting back and forth among the three forms, which will help you greatly on tests and quizzes.
Learning Objectives
- Know all three linear forms, and master knowing when each is best for your task
- Practice being able to change any form to any other form
- Solidify the similarities and differences among the forms when you are asked to graph a linear equation, so that you can graph lines no matter what form you are given
- Two Variable Linear Inequalities
Algebra One $\rightarrow$ Linear Graphing $\rightarrow$ Graphing Inequalities and Absolute Value Equations
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Priority: Normal
Lesson Description
Graphing linear inequalities is incredibly similar to graphing linear equalities (equations). This lesson outlines the process you should follow, along with the small details that are specific to inequalities that we wouldn't have seen with equations.
Learning Objectives
- Interpret solutions to linear inequalities
- Compare solution sets of linear equations and linear inequalities
- Learn the graphing process for linear inequalities
- Two Variable Linear Absolute Value Equations
Algebra One $\rightarrow$ Linear Graphing $\rightarrow$ Graphing Inequalities and Absolute Value Equations
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Priority: High
Lesson Description
Linear functions that include an absolute value operator will graph in a particular shape that is closely related to a straight line. This lesson shows us how to work with and become familiar with functions of the form $f(x) = m|x-h| + k$.
Learning Objectives
- Recall the conceptual interpretation of absolute value as a measure of positive distance
- Graph linear absolute value functions in the coordinate plane
- Learn how to define linear absolute value functions using piecewise functions instead of with absolute value bars
- Solve for unknown function input $x$ by setting up an equation and solving it in two cases, both similar to techniques learned in Algebra One and using the piecewise function approach
- Two Variable Absolute Value Inequalities
Algebra One $\rightarrow$ Linear Graphing $\rightarrow$ Graphing Inequalities and Absolute Value Equations
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Priority: High
Lesson Description
Instead of solving absolute value inequalities algebraically, we can use them in forms $y \geq m|x-h| + k$ and look at it as a two variable graph, similar to what we did in Algebra One with lines in the plane. This lesson focuses on how to work with absolute value inequalities with two variables by using what we already know about linear absolute value functions.
Learning Objectives
- Remember what absolute value functions tell us about the shape of two variable absolute value equations
- Apply knowledge from graphing linear inequalities to understand how to shade the solution space
- Draw connections between two variable absolute value inequalities and one variable absolute value inequalities
- Parallel and Perpendicular Lines
Algebra One $\rightarrow$ Linear Graphing $\rightarrow$ Analyzing Linear Graphs
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Priority: High
Lesson Description
Lines that are parallel or perpendicular to one another have a special slope connection. Whether you need to check if two lines are parallel or perpendicular, or construct two lines that are, here we'll see what the slope relationships are, and how to proceed.
Learning Objectives
- Know what lines must have in common when they are parallel
- Understand the relationship between two lines that are perpendicular
- Use the equations of two lines and determine whether they are parallel, perpendicular, or neither
- Find the equation of a line through a specific point that is parallel or perpendicular to a given line
- The Midpoint Formula
Algebra One $\rightarrow$ Linear Graphing $\rightarrow$ Analyzing Linear Graphs
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Priority: Normal
Lesson Description
In both Algebra and Geometry, sometimes we need to identify the midpoint location between two points, when working in the Coordinate Plane. Here we'll see how to do this using a simple formula approach.
Learning Objectives
- Learn the midpoint formula and understand where it comes from
- Practice finding the midpoint coordinate of two given ordered pairs
- The Distance Formula
Algebra One $\rightarrow$ Linear Graphing $\rightarrow$ Analyzing Linear Graphs
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Priority: Normal
Lesson Description
In the coordinate plane, the distance between two points can be quickly measured by borrowing something familiar from geometry. This lesson will show the approach that many teachers demand of memorizing the formula, but more importantly, we will explore the Distance Formula in a way that will help you remember and use it without relying on memorization.
Learning Objectives
- Learn the Distance Formula and understand its derivation
- Practice plugging in coordinates into the memorized formula
- Find the same answer more easily by using right triangle properties
- Estimating Lines of Best Fit
Algebra One $\rightarrow$ Linear Graphing $\rightarrow$ Analyzing Linear Graphs
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Priority: Optional
Lesson Description
Given data points, we may seek to estimate a relationship using a linear trend line, often referred to as a "line of best fit". While computer software commonly takes in your data points and spits out an exact best fit line, we are often called upon to estimate the trend using approximation and judgement. Here we will practice choosing a best fit line for data, as well as using it to answer questions.
Learning Objectives
- Choose a decent estimate of the line of best fit based solely on visual guessing
- Subsequently measure the slope and y-intercept of the best fit line, thus yielding a slope-intercept form line estimate
- Use the best estimate line to answer theoretical and estimation questions about the data
- Understand the effect that outlier data has on both the best estimate line and its predictive power
- Intro to Linear Systems and Solutions from Graphs
Algebra One $\rightarrow$ Linear Systems $\rightarrow$ Systems of Two Equations
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Priority: High
Lesson Description
We will define linear systems as simultaneous equations, understand what that means, and how to obtain the solution to them using graphing techniques.
Learning Objectives
- Know the definition of a system of equations
- Understand what the solution to a system of equations means, and the nature of solutions of linear systems
- Learn how to check whether an ordered pair is or is not a solution to a system of equations
- Use what we already know about graphing to solve linear equation systems
- Solving Linear Systems Algebraically
Algebra One $\rightarrow$ Linear Systems $\rightarrow$ Systems of Two Equations
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Priority: VIP Knowledge
Lesson Description
The graphing approach is a great way to understand and start out with linear systems, but graphing is not always feasible. Most commonly, solving systems requires an algebra approach. This lesson covers the two ways we can solve linear systems without graphing: Substitution and Elimination.
Learning Objectives
- Solve linear systems using Substitution technique
- Manipulate systems so that the Elimination technique can be employed, and practice using it
- Understand what situations you should choose one technique over the other for, in terms of making your job much easier
- Two Variable Linear Word Problems
Algebra One $\rightarrow$ Linear Systems $\rightarrow$ Systems of Two Equations
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Priority: High
Lesson Description
For some reason, teachers tend to really like turning linear systems into word problems. This lesson will help you continue practicing solving linear systems, adding the step of also needing to write down the system from a worded description.
Learning Objectives
- Define an approach to solving linear system word problems using what you already know about problems solving
- Choose your own variables and setup a system of equations based on the description in a word problem
- Continue practicing solving linear systems of equations
- Graphing Two Linear Inequalities
Algebra One $\rightarrow$ Linear Systems $\rightarrow$ Systems of Inequalities
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Priority: Normal
Lesson Description
Graphing two lines in a system was a fast way to get the answer visually. This is now the preferred way to proceed when graphing two linear inequalities, except now we'll have entire regions of solutions, not just the point of intersection.
Learning Objectives
- Understand why we will never (hopefully) solve systems of inequalities using algebra techniques like Substitution or Elimination
- Plug in to check whether or not a provided answer is a solution to the system of inequalities
- Review how to graph each linear inequality, including the dashed or solid line, and the correct side to shade
- From this process, visually identify the region of the plan that has solutions in it
- Graphing More Than Two Linear Inequalities
Algebra One $\rightarrow$ Linear Systems $\rightarrow$ Systems of Inequalities
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Priority: Normal
Lesson Description
Systems of equations are very commonly only studied when there are two variables and two equations. If there are two variables and three equations, there is either no solution, or one of the equations was not needed at all. With inequalities, however, we can solve two variable systems with many simultaneous inequalities. Because the answers are shaded regions, and each inequality is needed even if we have three or four. We will solve these types of inequality systems in this lesson.
Learning Objectives
- Graph each inequality in the two-variable inequality system to find the shared region, with three, four, or more inequalities
- Verify that a suggested solution is indeed a solution to the inequality system algebraically or with a graph
- Attempt and fail to organically solve these systems algebraically, and bow down to the ease of the graphing approach
- Linear Programming
Algebra One $\rightarrow$ Linear Systems $\rightarrow$ Systems of Inequalities
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Priority: Optional
Lesson Description
Though this is a real life business topic that many courses cover, its real world use is left in the dust thanks to advances in computing. Nevertheless, many courses cover this topic, and understanding how linear programming works will give you a cursory idea of how computers would be programmed to solve optimization problems iteratively.
Learning Objectives
- Understand the problem that linear programming seeks to solve
- Apply the linear programming method and obtain a best answer
- Compare the best answer to other feasible answers and understand why the best one is indeed best
- Intro to Graphing in Three-Space
Algebra One $\rightarrow$ Linear Systems $\rightarrow$ Intro to Systems of Three Variables
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Priority: Optional
Lesson Description
We will use this concept very rarely, but there is a three dimensional plotting space similar to the 2 dimensional plane we usually use. This lesson explores how it works, and how to plot points in it. Understanding three space will help us understand solution sets when working with systems of three variables in the next lesson, but probably won't be super helpful again until advanced calculus.
Learning Objectives
- Familiarize with the 3D Cartesian space and draw similarities to the already familiar 2D Cartesian plane
- Orient the new third axis the correct way
- Plotting points in three-space
- Substitution With Three Variables
Algebra One $\rightarrow$ Linear Systems $\rightarrow$ Intro to Systems of Three Variables
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Priority: Optional
Lesson Description
Though we won't really see this again until Calculus (usually), a quick extension of solving two equations with two variables is solving three equations with three variables. This lesson employs the same substitution strategy we used for systems of two equations. Honestly though, this subject is rarely covered.
Learning Objectives
- Solving systems of three equations with three unknowns
- Understand the scenarios that can happen, analogous to knowing the scenarios of systems of two equations (one solution, no solution, many solutions, etc.)
- Intro to Functions
Algebra Two $\rightarrow$ Functions $\rightarrow$ Function Basics
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Priority: VIP Knowledge
Lesson Description
This is the first proper Mister Math lesson on functions. We'll see what they are, how they work, and how to determine whether or not a relationship can be called a function.
Learning Objectives
- Define and understand what a relation is
- Define and understand what a function is
- Determine whether or not a relationship is a function by looking at sets of input and output
- Function Notation
Algebra Two $\rightarrow$ Functions $\rightarrow$ Function Basics
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Priority: VIP Knowledge
Lesson Description
In this short VIP Knowledge lesson, we seek to understand the standard function notation that mathematicians use.
Learning Objectives
- Read and understand standard function notation
- Interpret worded math instruction and write it using function notation
- Evaluating Functions
Algebra Two $\rightarrow$ Functions $\rightarrow$ Function Basics
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Priority: VIP Knowledge
Lesson Description
Functions are fancy, special relationships, but they aren't very helpful in math unless we can use them. This lesson first shows us how to evaluate a function at a given input value, and then shows us how to figure out what the input value was if all we know is the output.
Learning Objectives
- Understand how functions behave like input/output processes.
- Write and interpret input / output lists
- Evaluate functions at a specific value
- Solve for an unknown input given the output
- Input an algebraic expression into a function
- Properties of Graphs of Functions
Algebra Two $\rightarrow$ Functions $\rightarrow$ Function Basics
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Priority: High
Lesson Description
Every function has a graph, though some are simple and some are complex. This lesson helps us understand the why and how of graphing functions, what it means to say a function is continuous, and what asymptotes are and how to graph them. We can also use a graph to determine whether or not we have a function or a relation, using what we call "The Vertical Line Test".
Learning Objectives
- Know how and why functions can be graphed on the coordinate plane
- Learn important characteristics that a function's graph can have
- See what asymptotes are and how to denote them on a graph
- Learn what the vertical line test is and how it works
- Understand why the vertical line test is a fail-safe method
- Become a champ at getting a function's graph with a graphing calculator
- Piecewise Functions
Algebra Two $\rightarrow$ Functions $\rightarrow$ Function Basics
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Priority: High
Lesson Description
While we most commonly work with one function definition for any input value, it is possible to use several function definitions one at a time in different intervals. Here we will see how that is done and how to work with functions of this type.
Learning Objectives
- Define what a piecewise function is and the notation we use for them
- Learn how to work with, evaluate, and graph piecewise functions
- Determine an unknown piecewise function based on its graph
- Defining Domain and Range
Algebra Two $\rightarrow$ Functions $\rightarrow$ Function Basics
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Priority: VIP Knowledge
Lesson Description
In this first lesson about domain and range, we will learn some basic techniques for finding the domain and range of functions based on clues from either the function definition or the function graph.
Learning Objectives
- Learn what each domain and range means
- Practice finding domain and range of a function by inspecting the function definition
- Practice finding domain and range of a function by inspecting the graph of the function
- Basic Function Transformations - Translations
Algebra Two $\rightarrow$ Functions $\rightarrow$ Function Basics
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Priority: High
Lesson Description
If a function is translated, its original shape remains intact but its graph shifts up, down, left, or right. This lesson shows us how to draw relationships between two functions that are otherwise the same, but translated, or shifted over from one another. We will continue using translations throughout the entirety of Pre-Calculus, including trigonometry.
Learning Objectives
- Identify translations based on graphs only
- Use function notation to understand how to translate a function up, down, left, or right
- Algebraically find the new translated function from the original function, given the desired translation
- Straight Lines as Functions
Algebra Two $\rightarrow$ Functions $\rightarrow$ Linear Functions
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Priority: Normal
Lesson Description
Already we've seen various function definitions, with all kinds of curvy and crazy graphs. This lesson focuses on the function type that graphs as a straight line, putting together past knowledge you have about two variable linear equations and newer knowledge you've learned about functions.
Learning Objectives
- Know how to identify whether or not a function is linear
- Understand why slope-intercept form is the only easy way to translate between the perspective of a two variable linear equation and the perspective of a function
- See why this simple function case exhibits similar behavior to lines in the place that we studied in Algebra One, both algebraically and graphically
- Modeling with Linear Functions
Algebra Two $\rightarrow$ Functions $\rightarrow$ Linear Functions
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Priority: Normal
Lesson Description
Any situation in which two quantities are related by a simple slope is linear. This lesson focuses on how to create and use linear functions that are specific to a given situation. We will be able to not only solve word problems, but also interpret results in context of the situation.
Learning Objectives
- Generate a linear function based on tables, graphs, or descriptions
- Interpret a linear function in context of a real-life scenario, including reasonable limits on domain
- Solve word problems using linear functions, where the function was either given to us or created by us
- Defining Complex Numbers
Algebra Two $\rightarrow$ Complex Numbers $\rightarrow$ Using Complex Numbers
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Priority: Normal
Lesson Description
Real numbers and imaginary numbers are not like terms and cannot be combined, but their sum comprises a type of number we call a "Complex Number". These are very useful for polynomial analysis and other situations where imaginary numbers are meaningful, such as advanced Physics applications.
Learning Objectives
- Define what complex numbers are
- Understand that each real numbers and imaginary numbers are subsets of the set of complex numbers
- Know the concepts of norm and conjugate as applicable to complex numbers
- Plotting Complex Numbers
Algebra Two $\rightarrow$ Complex Numbers $\rightarrow$ Using Complex Numbers
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Priority: Optional
Lesson Description
While we often use the Cartesian Coordinate Plane to plot relationships between two real variables, we can use a modified version to plot complex numbers as points. We will see how to do that here, and the useful properties of complex numbers that we are able to visualize when we do so.
Learning Objectives
- See and understand how the Complex Number Plane works
- Learn how to plot a complex number on the plane
- Visualize the norm and conjugate of a complex number by using the complex plane
- Factoring Quadratics Overview
Algebra Two $\rightarrow$ Using Quadratic Relationships $\rightarrow$ Quadratic Functions
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Priority: Optional
Lesson Description
This pure review lesson revisits all the techniques for factoring quadratics that we learned in Algebra One, including Trinomial Factoring (aka Reverse FOIL), GCF factoring, Difference of Squares, and Perfect Square Trinomials. With the opportunity to look at them all in one place, we can make sure we're solid on each technique before moving forward with new quadratics concepts in the following lessons.
Learning Objectives
- Review factoring techniques from Algebra One
- Recall factoring techniques including Trinomial Factoring ("reverse FOIL"), GCF factoring, difference of squares, and perfect square trinomials
- Properties of Quadratic Functions
Algebra Two $\rightarrow$ Using Quadratic Relationships $\rightarrow$ Quadratic Functions
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Priority: VIP Knowledge
Lesson Description
In Algebra One we learned a little bit about factoring and solving quadratic equations. In Algebra Two we're going to study quadratics in detail. To start, this lesson will define quadratics in perspective of functions, and review some common characteristics all quadratics share, both graphically and algebraically. Some important concepts we'll go over include the meaning of the intercepts on the graph, how to find the coordinates of the vertex, and how to tell if a given number is a root of the quadratic.
Learning Objectives
- Define a quadratic function
- Know and recognize basic characteristics common to all quadratics
- See how and why the shape of graphs of quadratic functions is a parabola
- Find the coordinates of the parabola vertex and the y-intercept
- Find how many roots a quadratic has and what kind based on the graph
- Verify that a given number is a root of a quadratic function
- Forms of Expressing a Quadratic Function
Algebra Two $\rightarrow$ Using Quadratic Relationships $\rightarrow$ Quadratic Functions
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Priority: Normal
Lesson Description
Though we most commonly are presented with quadratics in what we call "standard form" ($ax^2 + bx + c$), there are two other forms that we can manipulate a quadratic into. This is very useful in the right situation, as each form has special properties. This lesson familiarizes us with the three forms (Note: a later lesson is dedicated to changing back and forth between the three forms).
Learning Objectives
- Recognize the three forms that a quadratic might be arranged in
- Know the advantages each form has
- Graphing Quadratic Relationships
Algebra Two $\rightarrow$ Using Quadratic Relationships $\rightarrow$ Quadratic Functions
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Priority: High
Lesson Description
When asked to graph a quadratic from scratch, we can use the common properties that all quadratics share that we recently learned to make the process structured. We will zone in on how to graph quadratic functions in three ways - one for each of the three quadratic forms we now know.
Learning Objectives
- Learn the common ideas to graphing a quadratic
- Know how to graph a quadratic function that is presented in standard form
- Know how to graph a quadratic function that is presented in factored form
- Know how to graph a quadratic function that is presented in vertex form
- Problem Solving with Quadratic Factoring
Algebra Two $\rightarrow$ Using Quadratic Relationships $\rightarrow$ Applications of Quadratics
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Priority: Normal
Lesson Description
We will continue what we learned in the last lesson to solve equations that require quadratic factoring. Here, however, we will need to give answers in context of a real-world problem, and sometimes we will have to build the equation to solve instead of being supplied with the equation to solve.
Learning Objectives
- Review techniques for solving word problems
- Setup equations for number based, geometry, and real world word problems, and then solve the problem
- Quadratic Solutions via Graphing Calculator
Algebra Two $\rightarrow$ Using Quadratic Relationships $\rightarrow$ Applications of Quadratics
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Priority: Normal
Lesson Description
Sometimes it will be sufficient to solve a quadratic equation with a graphing calculator, even though we usually only get answers as decimal approximations. This lesson will show you the basics of using the TI-83/84 functionality, as well as a few other common graphing calculator interfaces such as the Casio graphing calculator.
Learning Objectives
- Learn how to solve any quadratic equation quickly using a graphing calculator
- Learn how to find the vertex of any quadratic using a graphing calculator
- Problem Solving with Quadratic Functions
Algebra Two $\rightarrow$ Using Quadratic Relationships $\rightarrow$ Applications of Quadratics
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Priority: Normal
Lesson Description
Now that you're an expert on solving quadratic equations, let's look at real world situations that are modeled with quadratic behavior. You will practice setting up your own equation and then solving it.
Learning Objectives
- Set up a quadratic equation from a word problem
- Know to restrict the domain subjectively for realistic possible values of $x$ in context of the problem
- Become familiar with common situations that have quadratic behavior
- Equations in Quadratic Form
Algebra Two $\rightarrow$ Using Quadratic Relationships $\rightarrow$ Applications of Quadratics
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Priority: Normal
Lesson Description
Looks at how to solve equations that are of the general quadratic form (variable squared + variable + constant) but not necessarily $ax^2 + bx + c$.
Learning Objectives
- Recognize a general quadratic form, not just the typical $ax^2 + bx + c$
- Apply factoring techniques that we already know to these special quadratics
- Solve general quadratic form equations using the quadratic formula.
- Systems of One Variable Quadratic Equations
Algebra Two $\rightarrow$ Using Quadratic Relationships $\rightarrow$ Applications of Quadratics
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Priority: Optional
Lesson Description
In Algebra One we learned how to solve systems of linear equations, which was conceptually equivalent to finding the place that two straight lines intersect. Here we will do the same thing but with quadratics, so that we are conceptually looking for the places where two parabolas intersect.
Learning Objectives
- Solve systems of quadratic equations visually
- Solve systems of quadratics equations algebraically
- Know when we do and do not need a calculator for this
- Quadratic Inequalities
Algebra Two $\rightarrow$ Using Quadratic Relationships $\rightarrow$ Applications of Quadratics
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Priority: Normal
Lesson Description
When you have an inequality that involves a quadratic expression, we can still use our quadratic knowledge but we have to modify our thinking slightly. We'll see how to methodically get the solutions to quadratic inequalities.
Learning Objectives
- Learn how to solve quadratic inequalities algebraically
- Learn how to graph quadratic inequalities as a two-variable relationship
- Learn how to solve quadratic inequalities with a graph
- Polynomial Properties Overview
Algebra Two $\rightarrow$ Polynomial Functions $\rightarrow$ Analyzing Polynomials
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Priority: Normal
Lesson Description
This lesson starts with a quick overview of everything we know about polynomials up to this point, including what they are, what they are not, and important vocab that we should know. We'll also re-visit the GCF and Factor-by-Grouping factoring techniques.
Learning Objectives
- Be able to identify whether a given object is or is not a polynomial
- Re-visit important vocal terms such as degree, coefficient, leading term, etc.
- Discern the degree of a polynomial for single variable and multi-variable polynomials
- Review the GCF factoring and Factoring by Grouping techniques
- Sums and Differences of Cubes
Algebra Two $\rightarrow$ Polynomial Functions $\rightarrow$ Analyzing Polynomials
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Priority: Normal
Lesson Description
Similar to the Difference of Squares formula we already know, the Sum and Difference of Cubes formula provides a successful factoring approach to expressions of the form $a^3 \pm b^3$.
Learning Objectives
- Know and (probably) memorize the sum of cubes and difference of cubes formulas
- Practice using the formulas on simple cases of sums and difference of cubes
- Use GCF factoring and other prior techniques for complex expressions of sums and differences that have one or more variables
- Solving Polynomials with Roots and the Factor Theorem
Algebra Two $\rightarrow$ Polynomial Functions $\rightarrow$ Analyzing Polynomials
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Priority: High
Lesson Description
Factoring and using the zero product property is one of the most common ways we solved quadratic equations in the past. We'll extend our prior knowledge to learn how we can use the same property to solve general polynomials, and understand the connection between the roots of a polynomial (the numbers that make the polynomial zero) and the factors of a polynomial (the expressions that divide evenly into the polynomial).
Learning Objectives
- Recall what the zero-product property of equations is and how it works
- Understand what factors of polynomials are by definition
- Use the zero-product property to solve polynomials that are already factored
- Understand and use the Factor Theorem for polynomials
- Define what a root of a polynomial is and what root multiplicity means
- Use basic factoring techniques to factor polynomials of degree 3 and higher, and then solve the equation using the zero-product property
- Solving Polynomial Inequalities
Algebra Two $\rightarrow$ Polynomial Functions $\rightarrow$ Analyzing Polynomials
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Priority: Optional
Lesson Description
This lesson expands on a basic principle we used when we looked at quadratic inequalities. We will use the zeros of the polynomial and sign analysis to find the solution ranges of polynomial inequalities.
Learning Objectives
- Use polynomial zeroes and sign analysis to solve polynomial inequalities
- Polynomial Long and Synthetic Division
Algebra Two $\rightarrow$ Polynomial Functions $\rightarrow$ Analyzing Polynomials
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Priority: Normal
Lesson Description
First we'll look at long division - that's right, a throwback to grade school, compadre. But instead of numbers, we will divide polynomials by polynomials. We will also learn a shortcut method of polynomial division called "synthetic division" that saves a ton of time and writing, and understand when we can and cannot use it.
Learning Objectives
- Remember how to divide polynomials by monomials
- Use the long division technique for division of any polynomial by any other polynomial
- Learn how to setup polynomial division using shorthand notation that we call "Synthetic Long Division"
- Practice correctly using this method as an alternative to long-hand polynomial long division notation
- Know that long-form polynomial long division is required when dividing by non-linear factors
- The Remainder Theorem
Algebra Two $\rightarrow$ Polynomial Functions $\rightarrow$ Analyzing Polynomials
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Priority: Normal
Lesson Description
There exists an important connection between the result we get when we plug a specific value $c$ in for $x$ in any polynomial $P(x)$, and the result we get when we divide the polynomial by $(x-c)$. The Remainder Theorem tells us what that connection is, and in this lesson, we'll not only see when this information is useful, but also how teachers typically structure quiz questions on this topic.
Learning Objectives
- Understand and use the Remainder Theorem for polynomials
- Decide when long division is an easier way to solve a problem than plugging in values, and vice versa
- Polynomial Graphs and End Behavior
Algebra Two $\rightarrow$ Polynomial Functions $\rightarrow$ Advanced Polynomial Properties
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Priority: High
Lesson Description
Here we will look more closely at the relationship between characteristics of polynomials and what those characteristics mean when the polynomial is graphed, including degree, leading coefficient, and root multiplicity. We will also discuss "end behavior" and what we can know about it with and without a graph to look at.
Learning Objectives
- Understand common properties that all polynomial graphs share
- See how properties that are specific to the degree of the polynomial affect the shape of its graph
- Understand how we identify and describe "end behavior" of polynomials
- Understand the effect of root multiplicity on the graph of the polynomial
- Be able to sketch a function based on its algebraic function form, either factored or not factored
- Theorems for Roots of Polynomials
Algebra Two $\rightarrow$ Polynomial Functions $\rightarrow$ Advanced Polynomial Properties
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Priority: Normal
Lesson Description
There are several important theorems about polynomials that help us find all the roots of a polynomial, including the ones that are easy to miss. This lesson shows us the most important theorems, and we'll see a few more in the next lesson.
Learning Objectives
- Use the Fundamental Theorem of Algebra to determine the number of roots of a polynomial
- Understand and use the rational roots (aka rational zeroes) theorem
- Specify a minimum number of real roots for odd degree polynomials
- Advanced Theorems for Roots of Polynomials
Algebra Two $\rightarrow$ Polynomial Functions $\rightarrow$ Advanced Polynomial Properties
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Priority: Normal
Lesson Description
Though they are used infrequently, there are a few theorems about properties of polynomial roots that we are expected to know. This lesson looks at a handful of these theorems, and how they can be used.
Learning Objectives
- Learn two theorems about roots that always come in conjugate pairs
- Understand Descartes's Rule of Signs as well as when and how to use it
- Discern the pattern between the coefficients of the polynomial and either the sum or product of its roots
- Solving Advanced Polynomial Equations
Algebra Two $\rightarrow$ Polynomial Functions $\rightarrow$ Advanced Polynomial Properties
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Priority: Normal
Lesson Description
This lesson combines the knowledge from the last two lessons, and reviews a few facts that we already knew about polynomials so that we can solve higher degree polynomial equations by hand if needed.
Learning Objectives
- Use the theorems from the prior two lessons to solve higher degree polynomials by hand
- Break a polynomial of degree n down completely so that all $n$ roots are identified
- Finding a Polynomial from Its Roots
Algebra Two $\rightarrow$ Polynomial Functions $\rightarrow$ Advanced Polynomial Properties
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Priority: Normal
Lesson Description
Working backward to find a polynomial solely based on knowing its roots is a great way to recap the wealth of information you now have regarding working with polynomials. Many courses test this skill explicitly as well.
Learning Objectives
- Understand why knowing only the roots of a polynomial is not sufficient enough to know exactly what the polynomial is
- Utilize the pattern between the coefficients of the polynomial and either the sum or product of its roots
- Leverage relationships among the roots to specify the exact polynomial
- Using Technology with Polynomials
Algebra Two $\rightarrow$ Polynomial Functions $\rightarrow$ Advanced Polynomial Properties
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Priority: Optional
Lesson Description
Some of the common things we study about polynomials can be handled in a flash with your graphing calculator. Sometimes the calculator cannot get you the exact, correct answer you need, but rather a decimal approximation - but sometimes that's all we need. This lesson shows you how to accomplish several tasks related to our study of polynomials, including solving a polynomial equation, finding the roots of a polynomial, and finding places where the polynomial has a relative max or min value.
Learning Objectives
- Learn how to use your TI or graphing calculator to solve a polynomial equation
- Learn how to use your TI or graphing calculator to find the roots of a polynomial
- Learn how to use your TI or graphing calculator to find a relative max or min point of the polynomials
- The Complete List of Exponent Rules
Algebra Two $\rightarrow$ Radical and Rational Relationships $\rightarrow$ Advanced Radicals and Roots
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Priority: VIP Knowledge
Lesson Description
This VIP Knowledge lesson starts with the exponent rules that we've learned about so far, and adds a couple more on to the list. We must be absolute exponent rule masters, both for the entirety of this Mr. Math Section, and for almost every part of our future math career!
Learning Objectives
- Pull together every exponent rule we've ever learned
- Square Root Overview for Advanced Algebra
Algebra Two $\rightarrow$ Radical and Rational Relationships $\rightarrow$ Advanced Radicals and Roots
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Priority: Normal
Lesson Description
Before we directly relate square root operations with exponents, we must review and re-master taking square roots, of both integers and variable expressions, as well as simplifying down in both cases.
Learning Objectives
- Remaster everything we know about taking square roots and what square roots are
- Practice simplifying square root expressions involving integers only
- Practice simplifying square root expression involving variables and general monomials
- Fractions and Rationalizing with n-th Roots
Algebra Two $\rightarrow$ Radical and Rational Relationships $\rightarrow$ Working with Advanced Roots
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Priority: Normal
Lesson Description
This lesson focuses on applying n-th roots to fractions and rationalizing when the denominator is an n-th root, by using similar mechanics and ideas as the ones we saw in our study of applying square roots to fractions in Algebra One.
Learning Objectives
- Learn how to take the n-th root of a fraction
- Learn how to rationalize simple n-th root denominators
- Get tips for special complex n-th root denominators
- Fraction Exponents
Algebra Two $\rightarrow$ Radical and Rational Relationships $\rightarrow$ Working with Advanced Roots
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Priority: VIP Knowledge
Lesson Description
Now that we can take any root of an object and/or raise an object to any power, we introduce a very important exponent operation that lets you do both at once. This lesson introduces fraction exponents, which are a more consistent way to express a combination of roots and powers. For example, squaring a variable and then taking the third root can be done all at once, and the result is more easily obtained when we use fraction exponents. We will also see how to solve very simple equations that involve fraction exponents.
Learning Objectives
- Understand why taking an n-th root is the same as raising to the $1/n$ power
- Interpret fraction exponents of the form $a/b$ and know best practices for using them
- Learn how to solve basic equations with a given variable raised to the $a/b$ power
- Using and Graphing Radical Functions
Algebra Two $\rightarrow$ Radical and Rational Relationships $\rightarrow$ Working with Advanced Roots
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Priority: High
Lesson Description
Having learned more about roots recently, we can study functions that have root expressions in them, and get to know and love the properties that such functions have.
Learning Objectives
- Know the general properties of radical functions, including domain and range
- Graph radical functions and know common graph properties
- Power Functions
Algebra Two $\rightarrow$ Radical and Rational Relationships $\rightarrow$ Working with Advanced Roots
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Priority: Optional
Lesson Description
Although we have thoroughly studied polynomial functions and their behavior, there is an "in-between" type of function that has terms that look similar to polynomial ones, but without integer exponents. These Power Functions have similar properties but have a few unique differences.
Learning Objectives
- Learn what makes a general Power Function similar to and different from a polynomial function
- Study the graphical properties that are unique to Power Functions
- Solving Any Root Equation
Algebra Two $\rightarrow$ Radical and Rational Relationships $\rightarrow$ Working with Advanced Roots
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Priority: High
Lesson Description
This lesson is a very comprehensive exploration into how to solve equations that have a root or radical expression in them. We'll start with square roots, and then use the rational expressions knowledge we have from the last lesson to solve radicals that are not square roots, both for single variable radicals as well as roots of entire expressions.
Learning Objectives
- Review solving equations with just one radical from Algebra One
- Practice the recently learned "exponent neutralizing" method of solving for variables raised to fraction exponent powers
- Learn how to solve for $x$ when the equation has two square root expressions
- Know that results often yield extraneous solutions and that all solutions of radical equations must be verified
- Common Denominators for Rational Expressions
Algebra Two $\rightarrow$ Radical and Rational Relationships $\rightarrow$ Rational Expressions
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Priority: Normal
Lesson Description
To prepare ourselves to combine rational expressions, we will need to zone in on taking a set of rational expressions and identifying what their least common denominator is, as well as whichever extra factors each fraction will need to be turned into one with the least common denominator.
Learning Objectives
- Revisit LCD and LCM for variable expressions
- Identify the simplest common denominator of two fractions
- Determine the missing factors that each original denominator needs to turn into the common denominator
- Convert given rational expressions to have a common denominator
- Adding and Subtracting Rational Expressions
Algebra Two $\rightarrow$ Radical and Rational Relationships $\rightarrow$ Rational Expressions
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Priority: Normal
Lesson Description
Rational expressions can be added and subtracted, just like numeric fractions. In fact, the same rule applies - the denominators must be the same before you can. In this lesson we'll see how to get rational expressions to have the same denominator, and then we'll add or subtract them.
Learning Objectives
- Find the simplest common denominator in an addition or subtraction problem
- Know why finding the "not simplest" common denominator will create a bad situation
- Add or subtract expressions that have common denominators and simplify the answers completely
- Solving Rational Equations
Algebra Two $\rightarrow$ Radical and Rational Relationships $\rightarrow$ Using Rational Expressions
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Priority: High
Lesson Description
With what we now know about rational expressions, we are ready to solve equations that contain them. This lesson looks at how to solve each of the three main flavors of these equations - purely numeric denominators, variable denominators with equations equal to zero, and variable denominators with equations equal to anything.
Learning Objectives
- Get familiar with categorizing a given equation as rational or not
- Learn the three major types of rational equations we solve in Algebra
- Review least common denominator requirement and why other common denominators are not good enough
- Understand why extraneous solutions often occur in rational equations, and consequently why we must always check solutions for validity
- Rational Inequalities
Algebra Two $\rightarrow$ Radical and Rational Relationships $\rightarrow$ Using Rational Expressions
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Priority: Optional
Lesson Description
If a rational expression appears in an inequality with variables in the denominator, the complete solution set might not be obvious. Here we learn how to systematically solve the inequality thoroughly in such cases.
Learning Objectives
- Learn and practice a strategy for solving inequalities that contain rational expressions
- Recall and use the sign analysis technique with intervals on a number line
- Using and Graphing Rational Functions
Algebra Two $\rightarrow$ Radical and Rational Relationships $\rightarrow$ Using Rational Expressions
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Priority: VIP Knowledge
Lesson Description
This lesson examines rational functions and their behavior, first in the case where the denominator is a linear factor, and then generally where each the numerator and denominator can be any polynomial. After this lesson, we will be able to identify several key properties for any given rational functions, including asymptotes, intercepts, and general graph shape.
Learning Objectives
- Analyze one-variable rational functions that contain a rational expression with a linear denominator
- Understand the three cases of this function form when the denominator is of the form $ax+b$
- Examine the three cases of general rational functions where both the numerator and denominator are polynomials
- Discern the domain and range of rational functions
- Determine which types of rational functions do and do not have zeros
- Find vertical, horizontal, and slant asymptotes, when they exist
- Understand the common properties and shapes of the graphs of rational functions
- Learn how to graph these functions by hand, as well as identify a function based on its graph
- What Matrices Are
Algebra Two $\rightarrow$ Matrices $\rightarrow$ Matrices and Arithmetic
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Priority: Normal
Lesson Description
Matrices are a somewhat oddball topic. Whether or not you have ever seen them before now, this lesson will help us understand exactly what matrices are, and see some important similarities and differences between matrices and numbers.
Learning Objectives
- Better understand what matrices are as objects
- Relate important similar concepts between numbers and matrices, such as the multiplicative identity and zero product property
- Learn what scalars are and how they operate on matrices multiplicatively
- Basic Matrix Arithmetic
Algebra Two $\rightarrow$ Matrices $\rightarrow$ Matrices and Arithmetic
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Priority: Normal
Lesson Description
This lesson will outline how to add or subtract two matrices, as well as how to multiply a matrix by a scalar.
Learning Objectives
- Learn which matrices can and cannot be operated on with addition and subtraction
- Practice performing matrix addition and subtraction
- Apply multiplicative scalars to matrices
- Solving Systems of Equations with Matrices
Algebra Two $\rightarrow$ Matrices $\rightarrow$ Solving Systems of Equations with Matrices
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Priority: VIP Knowledge
Lesson Description
This valuable lesson will ultimately show you a commonly tested process - how to solve systems of equations with matrices. First we will turn a system into a special matrix. Then we will use row operations to manipulate the matrix until it is in a form that gives us the solution to the system.
Learning Objectives
- Learn how to translate a two or three variable system of equations into an augmented matrix
- Learn how to manipulate augmented matrices with row operations
- Master the method of Gauss-Jordan Elimination to turn an augmented matrix into what we call "Row Echelon Form"
- Turn Row Echelon matrices into identify form matrices so that the solution is plainly obtained
- Matrix Determinants
Algebra Two $\rightarrow$ Matrices $\rightarrow$ Solving Systems of Equations with Matrices
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Priority: High
Lesson Description
Matrices have a special number associated with them called a determinant, which is useful for a few upcoming concepts and shortcuts. For now, we will learn how to find the determinant of a matrix, as well as which kinds of matrices do and do not have a determinant.
Learning Objectives
- Define what the determinant of a matrix is
- Understand which types of matrices do and do not have determinants
- Learn how to manually calculate the determinant of smaller matrices
- Learn how to theoretically determine the determinant of larger matrices
- Inverse Matrices
Algebra Two $\rightarrow$ Matrices $\rightarrow$ Solving Systems of Equations with Matrices
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Priority: Normal
Lesson Description
Just like numbers have inverses (which we called reciprocals), so too do matrices. But just like multiplication of matrices is more complex than multiplication of numbers, so too are matrix inverses. This lesson defines what an inverse matrix is, and how we can use them in a practical way.
Learning Objectives
- Understand what an inverse matrix is, and how they are like the matrix world equivalent to reciprocals in the world of numbers
- Find the inverse matrix of a 2x2 matrix explicitly using a formula
- Solve two variable systems of equations instantly without row operations by applying inverse matrices
- Solving Systems with Cramer's Rule
Algebra Two $\rightarrow$ Matrices $\rightarrow$ Solving Systems of Equations with Matrices
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Priority: Normal
Lesson Description
If you are familiar and comfortable with finding matrix determinants, you can utilize Cramer's rule for solving two or three variable systems of equations, which is essentially a massive shortcut. In this lesson we will see how Cramer's rule works and practice using it.
Learning Objectives
- See and understand how Cramer's Rule works
- Practice solving systems of equations with matrices via Cramer's Rule for 2x2 or 3x3 (two or three variable, respectively) matrix systems.
- A Review of Function Basics
Pre-Calculus $\rightarrow$ Advanced Function Analysis $\rightarrow$ Properties of Functions
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Priority: High
Lesson Description
Calculus is almost entirely about working with functions, so to begin a Calculus readiness course like Pre-Calculus, we'll start by making sure we're up to speed on what we are already supposed to know.
Learning Objectives
- Review what functions are and how they work
- Recap the knowledge we gained in Algebra Two about function notation, and domain and range
- Function Composition
Pre-Calculus $\rightarrow$ Advanced Function Analysis $\rightarrow$ Properties of Functions
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Priority: VIP Knowledge
Lesson Description
Function composition is the "Inception" of functions - a function within a function. At first glance, function composition confuses many students, just as the movie equivalent might. But once you get familiar with what it means, you'll be able to work with these the same way you work with standalone functions.
Learning Objectives
- Understand conceptually how function composition works
- Navigate function notation for function composition, using what you already know about notation
- Work with variable expression inputs and/or work backwards to an unknown input from a known output
- Be able to break apart a complex function into a series of function compositions of simpler functions
- Even and Odd Functions
Pre-Calculus $\rightarrow$ Advanced Function Analysis $\rightarrow$ Properties of Functions
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Priority: Normal
Lesson Description
Whether or not your teacher is going to test you on this property of functions outright, in Calculus it is occasionally necessary to know whether a function is odd, even, or neither. This lesson will help you understand exactly what that means, as well as the best tips and shortcuts for knowing whether or not a function is indeed odd, even, or neither.
Learning Objectives
- Know what is means to say a function is even or odd, algebraically
- Understand the implications that being odd or even has on a function's graph
- Learn shortcuts to determine whether a polynomial function is even, odd, or neither
- See and remember which other common functions we know are even or odd
- One-to-one Functions
Pre-Calculus $\rightarrow$ Advanced Function Analysis $\rightarrow$ Properties of Functions
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Priority: Normal
Lesson Description
One-to-one functions are slightly easier to study than many-to-one functions, and having this quality (or not having this quality) makes the difference between whether we can or cannot work with it in the next lesson. So while this isn't the most earth-shattering function property, it is worth knowing and understanding - especially in the context of studying function inverses in the next lesson.
Learning Objectives
- Define conceptually what it means for a function to be "one-to-one"
- Be able to identify one-to-one functions from a function's graph using visual clues
- Understand what will be true about a function definition algebraically if it is one-to-one
- Inverse Functions
Pre-Calculus $\rightarrow$ Advanced Function Analysis $\rightarrow$ Properties of Functions
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Priority: High
Lesson Description
Two functions are inverses of each other if they "undo" one another. This lesson will define what this means using function language, and help us identify what types of functions do and do not have inverses.
Learning Objectives
- See how we write function inverses using function notation
- Be able to tell whether or not a function has an inverse based on its graph
- Find function inverses algebraically by swapping $x$ and $y$
- Make a connection between the domain of a function and the range of that function's inverse
- Know the relationship between the graph of a function and the graph of its inverse
- Misc Function Characteristics
Pre-Calculus $\rightarrow$ Advanced Function Analysis $\rightarrow$ Properties of Functions
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Priority: Optional
Lesson Description
This lesson covers the function characteristics of continuous, smooth, and monotone slope, and recaps what you recently learned about one-to-one and even/odd functions by looking at all of these properties simultaneously.
Learning Objectives
- Learn what the terms "monotone increasing" and "monotone decreasing" refer to for functions
- Review the function property of "continuous" and add on a stronger property called "smooth"
- Think about functions of y instead of our typical functions of x
- Periodic Functions
Pre-Calculus $\rightarrow$ Advanced Function Analysis $\rightarrow$ Properties of Functions
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Priority: Optional
Lesson Description
While many functions we study extend left and right forever, periodic ones are special in that they repeat over and over, and do not approach very large or small values for very large or small $x$. This lesson will show you all you should know about studying periodic functions (at least, until we get to trigonometry).
Learning Objectives
- Define what periodic functions are
- Understand both graphical and algebraic techniques for finding the "wavelength" or "period"
- Characterize periodic functions as even, odd, or neither
- Arithmetic of Functions
Pre-Calculus $\rightarrow$ Advanced Function Analysis $\rightarrow$ Working with Functions
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Priority: Normal
Lesson Description
Just like variables, functions can be added, subtracted, multiplied, or divided with one another. This lesson shows you how to do that, and looks at what we might expect for results.
Learning Objectives
- Learn to add, subtract, multiply, and divide two functions similar to how you would work with two variables
- Understand why these operations create new functions (not relations) and how the resulting functions are similar and different to their comprising pieces
- See shortcuts for evaluating arithmetic expressions involving entire functions
- Set-Builder and Interval Notation
Pre-Calculus $\rightarrow$ Advanced Function Analysis $\rightarrow$ Working with Functions
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Priority: Normal
Lesson Description
The classic way to describe a range of numbers is to use an inequality. However, there are other ways to express a range of numbers, and not only do we have to learn them because our teachers will use it, but they are also sometimes easier and less work to use.
Learning Objectives
- Review what we know up to this point about describing ranges and sets of numbers
- Learn interval notation to describe a range of real numbers
- Learn set-builder notation to describe a set of numbers
- Understand the motivation for these notations and the advantages each has
- Function Translations
Pre-Calculus $\rightarrow$ Advanced Function Analysis $\rightarrow$ Working with Functions
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Priority: High
Lesson Description
If a function is translated, its original shape remains intact but its graph shifts up, down, left, or right. This lesson shows us how to draw relationships between two functions that are otherwise the same, but translated, or shifted over from one another. We will continue using translations throughout the entirety of Pre-Calculus, including trigonometry.
Learning Objectives
- Identify translations based on graphs only
- Use function notation to understand how to translate a function up, down, left, or right
- Algebraically find the new translated function from the original function, given the desired translation
- Advanced Function Transformations
Pre-Calculus $\rightarrow$ Advanced Function Analysis $\rightarrow$ Working with Functions
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Priority: Normal
Lesson Description
In Algebra Two we learned how to algebraically shift functions by moving them left, right, up, and down. This lesson will introduce ways to manipulate functions by stretching, compressing, or reflecting. We'll also see how to manage several simultaneous transformations.
Learning Objectives
- Review translations (aka sliding) type transformations that we learned in Algebra Two
- Understand how stretch-type transformations visually change the graph of a function
- Know how to apply stretch and compression transformations to a function algebraically
- Apply translation and stretch / compression transformations simultaneously
- Learn how to reflect a function over the $x$ or $y$ axis
- Learn how to reflect a function over any horizontal or vertical line
- Finding Domain and Range
Pre-Calculus $\rightarrow$ Advanced Function Analysis $\rightarrow$ Working with Functions
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Priority: High
Lesson Description
Up to this point, the only ways we have been able to find domain or range have been by inspection or by examining graphs. This lesson will go beyond what we already know about domain and range, giving us a better understanding and a definite approach to finding them.
Learning Objectives
- Review what Domain and Range each is and what we already know about finding them from inspection
- Compile the authoritative list of what to consider when seeking the domain of a function
- Define what to do algebraically to determine the domain of a function
- Find the range of a function using function inverse techniques
- Determine the domain of composite and piecewise functions
- Identify general patterns of domain and range based on function family
- Multivariable Functions
Pre-Calculus $\rightarrow$ Advanced Function Analysis $\rightarrow$ Working with Functions
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Priority: Optional
Lesson Description
It is possible (and common in the world of computer programming) for a function to require more than one input. This lesson gives a quick yet useful overview at how multivariable functions operate conceptually.
Learning Objectives
- Understand how functions work when they require more than one input
- Use common multivariable functions
- Real Number Exponents
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Understanding Variable Exponents
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Priority: Normal
Lesson Description
Building up to the next several lessons on exponential functions and relationships, we need to understand that we can let any real number be an exponent, from integer and fraction exponents that we have already seen, to irrational numbers like $\sqrt{2}$ and $\pi$.
Learning Objectives
- Quickly recap all previously learned exponent rules
- Make connections between rational exponents, which we've seen, and terminating decimal exponents, which we haven't yet seen
- Understand how irrational exponents work and what they mean, even though they are not as intuitive
- Relationships With Variable Exponents
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Understanding Variable Exponents
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Priority: High
Lesson Description
For the first time, we will begin to understand relationships that involve constants raised to variable exponents instead of the other way around (e.g. $3^x$ instead of $x^3$). We will examine the behavior of these relationships using integer values for $x$ before seeing what happens when we allow exponent variable $x$ to be non-integer values. Finally, we will look at exponential growth versus exponential decay, and begin to work with basic growth and decay models.
Learning Objectives
- Introduce the concept of having the unknown variable in an equation be the exponent
- See and understand the general patterns associated with relationships that involve a variable as the exponent, using x-y tables
- Further justify evaluating expressions when the exponent is an irrational number
- Define the concepts of exponential growth and exponential decay, and know how to tell which one describes a given relationship
- Investigate basic growth and decay models that use rate of growth or decay $r$ and initial quantity $a$
- Be able to set up and interpret a simple growth or decay model in the context of a word problem
- Basic Exponential Equations
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Understanding Variable Exponents
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Priority: High
Lesson Description
Exponential equations are equations in which the unknown variable is an exponent. There are two ways to solve these: either using integer and fraction knowledge or using logarithms. While we will study logarithms very soon, this lesson shows us how to solve exponential equations using only what we know about integers and fractions. We'll also understand when we can use this technique, and when we instead need to rely on logarithms.
Learning Objectives
- Solve for variable exponents in an equation using only what we know about integers and fractions
- Learn the "same base" trick for solving certain equations that involve variable exponents, and know when the trick can and can't be used
- Understand that when the "same base" trick cannot be used, that there is necessarily no rational solution (solution requires forthcoming logarithm techniques)
- Solve special kinds of exponential equations that are quadratic equations in disguise
- Exponential Functions
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Understanding Variable Exponents
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Priority: VIP Knowledge
Lesson Description
This lesson will put together several ideas that we saw in the prior three lessons, defining exponential relationships in the context of looking at a function. Not only will be employ techniques for solving for a variable like we saw in the prior lesson, but we will also define general properties of an exponential function based on the coefficients in the typical exponential form we see.
Learning Objectives
- Apply recently acquired knowledge about exponential relationships to the input/output behavior we are familiar with for functions
- Know and understand the form of a basic exponential function, $f(x) = ab^x$
- Describe common properties of exponential functions, and continue to understand the difference between growth and decay
- Interpret the coefficients of an exponential function, and what happens when the coefficient is positive vs negative
- Using growth or decay and a positive or negative coefficient, learn and understand the four major categories that an exponential function can fall into: positive growth, positive decay, negative growth, and negative decay
- See the general shape of exponential graphs, and how that shape changes in each of the four exponential categories
- Graphing Exponential Functions
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Understanding Variable Exponents
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Priority: High
Lesson Description
Using what we now know about how exponential functions work, we can understand the general shape of their graphs, and the common properties that the graphs of all exponential functions share.
Learning Objectives
- Learn how to graph specific exponential functions by plotting a few points and eyeballing it
- Know general properties of the graph of an exponential function, as well as what points are commonly on the graph
- Be able to graph an exponential function quickly by applying transformations to its parent function, visually
- Analyzing Exponential Transformations
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Understanding Variable Exponents
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Priority: Normal
Lesson Description
In the last lesson, we learned how to graph exponential functions, including some basic transformations such as shifting and scaling. Here, we will practice working with transformations algebraically, and learn specific implications for exponentials that certain transformations have.
Learning Objectives
- Apply what we know about algebraic function transformations to exponential functions
- Be able to look at an exponential written function definition and know quickly what its parent function is and the transformations that it has
- Know how to start with a parent function in written form and obtain a new transformed function in simplest form
- Understand how reflection transformations are connected to the exponential properties of growth vs decay, and positive vs negative
- Working With the Number e
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Understanding Variable Exponents
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Priority: Normal
Lesson Description
This lesson defines the number $e$ - one of the most important numbers in math. It is what we call the "natural exponential base". Like $\pi$, it is an irrational number that comes from a special definition.
Learning Objectives
- See the limit definition of the number $e$, but more importantly understand what it represents conceptually
- Set up basic continuous growth models with given information
- Given an exponential model using $e$, find the equivalent model that uses a base of $1+r$ and vice versa
- Logarithms
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Understanding Logarithms
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Priority: VIP Knowledge
Lesson Description
This important lesson introduces logarithms: what they are, and their relationship to what you already know about exponentials.
Learning Objectives
- Know he definition of a logarithm forward and backward (literally)
- Learn which two log bases are most common to use and why
- Evaluate simple logarithm expressions without a calculator using integer relationships
- Turn simple equations with logarithms into exponential equations that we know how to solve
- Use rounding and estimation techniques to approximate the numerical value of a logarithm
- Logarithmic Functions
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Understanding Logarithms
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Priority: High
Lesson Description
Now that we know what logarithms are, we will think about them in the context of functions to further understand their nature and their relationship to exponential functions.
Learning Objectives
- Examine functions of the form $f(x) = log_b (x)$ and understand their properties
- Use the definition of logarithms to show that logarithmic functions are inverses of exponential functions and vice versa
- Understand and be able to find the domain and range of any logarithmic functions
- Graphs of Log Functions
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Understanding Logarithms
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Priority: Normal
Lesson Description
As we often do after learning about a new function type, let's see what graphs of logarithmic relationships look like, and how they relate to other graphs that we've already seen.
Learning Objectives
- See what shape and pattern to expect from the graph of a logarithmic function
- Compare graphs of logarithmic and exponential functions, keeping in mind their inverse relationship to one another
- Further validate the domain and range patterns of logarithmic functions with graphs
- Apply typical function transformations to log functions, and connect the transformation's visual effect to the changes in the function definition, and vice versa
- Logarithm Rules
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Understanding Logarithms
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Priority: High
Lesson Description
Because logarithms really represent exponents, the rules that apply to them look oddly similar to things we know about working with exponents. The three rules in this lesson may look unusual at first for the same reason, but we use them often, because they allow us to manipulate logarithms to make it easier to solve certain problems.
Learning Objectives
- Learn the three important logarithm manipulation rules
- Practice using them backward and forward (literally), since we need to use the rules backward just as often as we do forward
- Expanding and Condensing Logarithmic Expressions
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Understanding Logarithms
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Priority: High
Lesson Description
Using the logarithmic rules we recently learned, we can now operate on logarithms by either taking a single logarithm of a complex object and breaking it up into simpler logarithms, or conversely, taking several simple logarithms and combining them into a single logarithm. These opposite processes each have their place in your math future, each making certain complex tasks easier.
Learning Objectives
- Apply multiple logarithm manipulation rules at the same time
- Begin with a single logarithm of a complicated term and end with sums and differences of several much simpler logarithms (expanding)
- Begin with a string of sums and differences of logarithms and end with a single logarithm by using logarithm manipulation formulas (condensing)
- Log Change of Base Formula
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Understanding Logarithms
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Priority: Normal
Lesson Description
It is possible, but not intuitive, to turn a logarithm of base b into a logarithm of some other base. This formula is particularly important if you are working with only variable, or if you need a decimal approximation and only have an older calculator that does not allow you to determine the base (many calculators only have input functions for base 10 and base $e$). This short lesson demonstrates how to change a log expression into an equivalent expression with a different base.
Learning Objectives
- Learn the logarithm change of base formula
- Understand the motivation for such a formula, and why most calculators only have natural and common logarithm functionality
- Practice using the formula both with hand written problems and with calculator based problems
- Solving Exponential Equations Using Logarithms
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Exponential and Logarithmic Equations
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Priority: VIP Knowledge
Lesson Description
Exponential equations can be definitely solved using logarithm manipulation, regardless of whether or not the equation has a "nice" integer solution. This lesson shows you how to solve equations with unknown variable exponents, using logarithms.
Learning Objectives
- Learn how to solve equations that contain unknown variable exponents using logarithms
- If asked for a decimal approximation, get an answer from your calculator after solving for the variable using logarithms
- Understand the difference between exponential equations we solve in this lesson that require logarithms and same-base exponential equations we solved recently without logarithms
- Identify the types of exponential equations that can and cannot be solved with algebra (even with logarithms)
- Know how to use a graphing calculator to find the solutions to ones that cannot be solved any other way
- Solving Logarithmic Equations
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Exponential and Logarithmic Equations
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Priority: High
Lesson Description
This brief lesson shows you how to solve equations that have logarithms on one or both sides of the equal sign, as well as other equations with logarithms of unknown bases, by using the exponent definition of logarithms.
Learning Objectives
- Know how to solve equations that have logarithms on one side of the equation
- Know how to solve equations that have logarithms on both sides of the equation
- Learn how to solve equations that have logarithms with unknown base $x$
- Understand why to check for extraneous solutions, and know how to check
- Know which types of logarithmic equations cannot be solved explicitly, and therefore must be solved with a computer
- Using Common Exponential Models
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Exponential and Logarithmic Equations
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Priority: Normal
Lesson Description
Recall from much earlier lesson on variable exponent relationships that exponentials come in two flavors - growth and decay. In this lesson, we will examine specific scenarios of both growth and decay models and how to solve for unknown quantities with these models. We will also see some examples of exponential relationships in the real world.
Learning Objectives
- Learn how to set up commonly used exponential growth and decay models (but not money accumulation with interest - that is the topic of the next lesson!)
- Learn how to solve for the missing quantity in all kinds of exponential models, including initial amount, growth rate, and time
- Simple, Compound, and Continuous Interest
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Exponential and Logarithmic Equations
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Priority: Normal
Lesson Description
Now that we can manipulate equations with logarithms, we can look at all three common ways that interest on money may accrue. This lesson will show us how to set up interest models, and solve for the unknown missing quantity, which may be initial amount, interest rate, or time.
Learning Objectives
- Learn how each interest accrual method works (simple, compound, or continuous interest)
- Practice setting up the correct equation to model out a given money growth situation
- Be able to solve for the missing quantity whether that is the initial amount, the interest rate, or the amount of time, regardless of which of the there interest methods is being used
- Advanced Modeling With Exponential Functions
Pre-Calculus $\rightarrow$ Exponential Relationships $\rightarrow$ Exponential and Logarithmic Equations
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Priority: Optional
Lesson Description
The complexity of exponential modeling can go far beyond the typical growth and decay situations we have already seen, such as half-life or multiplying bacteria. Here we will look at more complex real life situations that use exponentials to model advanced behavior.
Learning Objectives
- Get acquainted with situations that use exponentials in a non-traditional way to model situations, such as Newton's Law of Cooling or Carrying Capacity Population Growth
- Interpret the coefficients and horizontal asymptotes in advanced exponential models
- Solve for any missing unknown in advanced exponential models
- Right Triangle Trigonometry
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Right Triangles
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Priority: VIP Knowledge
Lesson Description
The three major trigonometry ratios exist in any right triangle - the sine, cosine, and tangent. While we will look at these big three a lot throughout the course of trigonometry, this first lesson helps us understand what the ratios mean and where they come from.
Learning Objectives
- Define the sine, cosine, and tangent ratios for a given non-right angle in a right triangle (aka SOH-CAH-TOA)
- Using the Pythagorean Theorem and the definition of sine, cosine, and tangent either to answer questions about unknown triangle side lengths or to find trig ratio values
- Solving a Right Triangle
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Right Triangles
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Priority: VIP Knowledge
Lesson Description
With very little initial information, you can find all three sides and all three angles of a right triangle, particularly with the help of trig ratios that we just learned about. In this lesson we'll see the common ways in which we'll be asked to do this.
Learning Objectives
- Know what it means to "solve a triangle"
- Use the Pythagorean Theorem and basic properties of triangles, in conjunction with trig ratios, to find all missing values of a right triangle
- Learn how to use trig ratios with your calculator to solve for missing sides in a right triangle
- Learn how to use your calculator's "inverse" trig ratio functionality to solve for missing angles in a right triangle
- Special Right Triangles
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Right Triangles
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Priority: High
Lesson Description
For right triangles with special angles, we can skip the trig ratio magic and use some standard ratios. Not only is this a faster approach, but it is also commonly prevalent on tests like the SAT.
Learning Objectives
- Learn the two special right triangles, and the side ratios that accompany each
- Given any one of the three sides of these special triangles, be able to find the other two
- Applied Right Triangle Problems
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Right Triangles
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Priority: Normal
Lesson Description
Using the very same skills we recently learned to solve a right triangle, we can set up and solve word problems that describe right triangles with unknown sides or angles. We'll look at problems involving angles of elevation and depression, including complicated diagrams.
Learning Objectives
- Independently set up geometry word problems for situations that measure angles of elevation and depression
- Set up other types of geometry word problems that are described by a right triangle with unknown sides or angles
- Solve complex applied right triangle word problems using a combination of Pythagorean Theorem, trig ratios, and algebra
- Measuring Angles in Degrees
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Angles and Angle Measurement
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Priority: Normal
Lesson Description
At this point in your math career, you've probably measured angles using degrees in the past. This first lesson on angle measurement will reinforce what we already know about degrees, and also discuss a couple of common ways to work with partial degrees.
Learning Objectives
- Specifically define / recall what exactly degrees are
- Define partial degrees as fractions of degrees in a system called DMS (degrees minutes seconds)
- Learn how to convert partial degrees between our natural decimal system and DMS measure
- Measuring Angles in the Coordinate Plane
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Angles and Angle Measurement
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Priority: High
Lesson Description
Having re-remembered all the important facts from the last lesson about measuring angles with degrees, we take our first look at a very common way of working with angles - by putting them in the coordinate plane. Then we can open up angles at any measure we want, and look at the properties that result, such as which quadrant the angle ends up in.
Learning Objectives
- Look at the standard way in which we measure angles in the coordinate plane
- Understand why and how negative angles are possible, and interpret how they are different from positive angles
- Understand why and how angles can be larger than 360 degrees (one full circle rotation)
- Know what co-terminal angles are and how to tell if two angles are co-terminal
- Radian Angle Measure
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Angles and Angle Measurement
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Priority: High
Lesson Description
For much of the ride ahead through Trigonometry (and very much so in Calculus), we aren't able to use degrees as a way to measure angles. This lesson introduces the very important alternative, dimension-less way to measure angles, measured in units we call radians.
Learning Objectives
- Learn the definition of radians and understand why it is a measure without dimension
- Translate between common degree measure angles and their equivalent radian measure
- Revisit measuring angles in the coordinate plane but this time using radians
- Become an expert on measuring angles with radians instead of degrees - you'll need it for the future!
- Angular and Linear Speed of Rotation
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Angles and Angle Measurement
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Priority: Normal
Lesson Description
This lesson, which is both popular and potentially confusing, looks at the major difference between how fast a wheel is rotating and how fast a wheel is actually moving (which depends entirely on the size of the wheel). We'll see lots of examples and practice to understand how teachers test you on this.
Learning Objectives
- Understand the difference between angular speed and linear speed
- Using basic circle properties, be able to convert angular speed to linear speed and vice versa
- Determine the speed relationships between two differently sized spinning connected wheels, such as on a bicycle
- Arc Length and Sector Area
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Angles and Angle Measurement
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Priority: Normal
Lesson Description
When we measure angles in radians, we can quickly calculate the arc length and area of partial circles, commonly referred to as sectors. This lesson shows you how to measure each in radians, and then how to do it similarly with degrees.
Learning Objectives
- Understand exactly what a sector is and how it is defined
- Know the difference between sector arc length and sector perimeter, and how to find either when using radian measure
- Learn how to find the area of a sector measured in radians
- See the analogous formulas to use when working with degree measure
- The Sine and Cosine Functions
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Trigonometry Functions
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Priority: VIP Knowledge
Lesson Description
Trigonometric ratios are introduced to us in a way that has a nice visual interpretation with right triangles, but the reality is that we can take the sine or cosine of any angle - not just acute angles. This lesson explores this idea by looking at a definition of sine and cosine in the coordinate plane.
Learning Objectives
- Look at sine and cosine with triangles in the coordinate plane
- Define sine and cosine values based on the actual coordinates of points in the plane
- With the coordinate plane definitions of sine and cosine, find the sine and cosine of any angle in any quadrant, not just acute angles
- Create functions of the sine and cosine ratios with the angle you are measuring at as the input
- Understand basic function properties that $y=\sin(x)$ and $y=\cos(x)$ each have
- The Unit Circle
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Trigonometry Functions
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Priority: VIP Knowledge
Lesson Description
Because of the way we define sine and cosine with coordinates in the plane, it turns out that looking at the coordinates on a unit length circle will give us a great reference table for values that we will need to memorize, or at least know how to quickly look up without a calculator.
Learning Objectives
- See and understand what the unit circle is in the coordinate plane
- Learn how to use the unit circle to obtain the sine and cosine values for the most common angle measures
- The Tangent Function
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Trigonometry Functions
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Priority: High
Lesson Description
Much of what we just learned about the sine and cosine functions apply to working with the tangent trig ratio as well - particularly creating a coordinate based definition in the coordinate plane. This lesson works through the content we just saw for sine and cosine, but for the tangent trig ratio.
Learning Objectives
- Define the tangent trig ratio using coordinates in the plane
- Build the important relationship between sine, cosine, and tangent, that tan = sin / cos
- Define the tangent function similar to the sine and cosine Functions, where the input is the angle and the output is the trig ratio
- Be able to find the tan of any angle in any quadrant, not just acute angles
- Calculate the tangent of an unknown angle given information about the point on the plane that the angle passes through
- Calculate the tangent of an unknown angle given which quadrant the angle lies in, and the value of the sine or cosine of that angle
- Reference Angles
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Trigonometry Functions
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Priority: Normal
Lesson Description
Understanding and being able to identify reference angles is a major step in the general process of evaluating trig functions. This lesson focuses on getting you to understand what reference angles are, and to become masterful at finding them.
Learning Objectives
- For angles oriented in the coordinate plane, understand what reference angles are
- Practice finding the reference angle of angles in any quadrant
- Learn a big shortcut for finding reference angles when measuring angles with radians
- Evaluating Trig Functions
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Trigonometry Functions
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Priority: VIP Knowledge
Lesson Description
One of the most important and common tasks we are required to perform in trigonometry (and again in Calculus) is to evaluate a trig function at a specific value. For common angles we are often expected to do this without a calculator. This lesson teaches you how to do just that, and provides a three-step approach to make sure you handle it the right way every time.
Learning Objectives
- Review recent lesson material all together to understand which trig functions in which quadrants are positive or negative
- Leveraging recent lesson material, evaluate sine, cosine, and tangent at common angles in all four quadrants without a calculator
- Learn the Mister Math three-step approach to correctly evaluate sine, cosine, and tangent for common angles
- Determine the sine, cosine, and tangent of an unknown angle knowing only which point on the unit circle it passes through
- Better your understanding of what you will and will not be expected to do without the aide of a calculator
- All Six Trig Functions
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Trigonometry Functions
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Priority: High
Lesson Description
So far we have looked at sine, cosine, and tangent. Additionally, the reciprocal of each of these creates a new trig function, and because of their use and commonality, we give them their own names and study their properties. We will learn these new three trig function names and coordinate plane definitions of each.
Learning Objectives
- Learn how to obtain three new trig functions (cotangent, secant, cosecant) by reciprocating each of the three original trig functions, and which new function comes from which old function
- Define the cotangent, secant, and cosecant functions using points in the plane ($x$, $y$, and $r$), like we did before for sine, cosine, and tangent
- Learn basic identities that show how all 6 trig functions can be easily defined using only sine and cosine
- Learn how to evaluate the three new trig functions at common angles, analogous to how we learned to do it for sine, cosine, and tangent
- Learn how to evaluate the three new trig functions at non-common angles with a calculator, since most calculators do not have buttons for them
- Learn how to find the values of all six trig functions of an unknown angle by drawing a right triangle, given information about one of the function values and which quadrant the angle falls in
- Inverse Trig Functions
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Trigonometry Functions
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Priority: High
Lesson Description
Like many typical algebra functions, we can find an inverse function for trig functions such that we input a trig ratio value into the inverse trig function and it gives the original angle as output (as opposed to normal trig functions that take an angle as the input and give a ratio value as the output). This lesson will look at the definitions and properties of these inverse trig functions, before showcasing their applications and common types of exam questions.
Learning Objectives
- Understand what reverse trig functions are in relation to trig functions
- Understand the inherent difficulty of defining a perfect inverse function due to the periodic nature of normal trig functions, and learn how we mitigate that issue
- Know whether a question is asking you for a specific answer versus a list of all possible answers
- Based on knowledge of common angle trig values, know when we can and cannot answer a questions without a calculator
- Use the domain of inverse trig functions to answer questions that involve both inverse and normal trig functions together
- Draw triangles in the coordinate plane to find either exact values or generic relationships between inverse trig functions and normal trig functions
- Trig Function Relationships
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Using and Graphing Trig Functions
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Priority: Normal
Lesson Description
This lesson demonstrates relationships that exist among trig functions, first by reviewing a few fundamental inverse relationships that we've already seen, and then introducing ones based on negatives and angle shifts.
Learning Objectives
- Review relationships we already know about trig functions, such as the reciprocal relationship between sine and cosecant, for example
- Explore other relationships such as negative argument (e.g. $sin(-x) = -sin(x)$) and shifted argument relationships (e.g. $sin(x) = cos(\pi/2-x)$).
- Solving Basic Trigonometric Equations
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Using and Graphing Trig Functions
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Priority: Normal
Lesson Description
Now that we know more about how to work with inverse trig functions, we can solve fairly basic equations for unknown x when x is the argument of a trigonometric function, such as solving $2\sin(x) - 1 = 0$. Note that an upcoming lesson is dedicated to solving similar types of equations, but focusing on equations with more complex expressions present.
Learning Objectives
- Similar to solving for a variable, learn to isolate the trig function in the equation
- Properly apply the inverse trig function operation to both sides of an equation, once one side of the equation contains an isolated trig function
- Pay attention to the directions - they are often the only indicator as to whether your answer should be a finite or infinite list
- As with recent lessons, know the difference between questions that can and cannot be solved without a calculator
- Trigonometric Graphs of Sine and Cosine
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Using and Graphing Trig Functions
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Priority: High
Lesson Description
Functions that are transformed versions of $y = \sin(x)$ and $y = \cos(x)$ are wave-shaped patterns that we refer to as sinusoidal. This lesson, which is very commonly covered on exams, shows us how to graph functions of this form, and conversely write down functions of this form based on a graph. We'll also see some application word problems.
Learning Objectives
- Learn the general properties of the graphs of sine and cosine functions, both of which are referred to as sinusoidal
- Leverage our knowledge of function transformations to see how we might account for vertical and horizontal stretching, and vertical and horizontal shifting, relative to the base functions $y = \sin(x)$ and $y = \cos(x)$
- Practice writing down functions based on graphs by examining which transformations appear on the graph, relative to the base function
- Use the Mister Math four-step approach to graphing written sinusoidal functions by accounting for any transformations in the order that makes it easiest for you
- Work practice problems and word problems that require working with and manipulating sinusoidal functions
- Graphs of Other Trigonometric Functions
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Using and Graphing Trig Functions
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Priority: Optional
Lesson Description
When it comes to graphing trig functions, the general focus is definitely on the sine and cosine forms we saw in the last lesson. However, many teachers will quickly cover the graphs of the other four trig functions, and so, this lesson will show you their forms and properties, leveraging some of the work we accomplished in the last lesson when possible.
Learning Objectives
- Learn how to graph basic forms of $y = \tan(x)$ and $y = \cot(x)$, without transformations
- Discuss how transformations can be implemented in tan and cot graphs, although this is not commonly studied
- Graph basic and transformed versions of functions of the form $y = \sec(x)$ and $y = \csc(x)$ by first graphing their reciprocal functions using what we recently learned about graphing sinusoids
- Solving Advanced Trigonometric Equations
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Advanced Trigonometry
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Priority: Normal
Lesson Description
In a prior lesson, we saw how to solve basic trigonometric equations where the unknown variable to solve for was the angle measured in radians - but only how to solve equations with a single trig function in them. Now we'll turn our attention to more complicated trig equations that require either trig identities or advanced ideas to solve.
Learning Objectives
- Learn to solve any equation that involves trig functions, regardless of complexity, using tricks and knowledge of trig identities
- Categorize equations by what type of expressions are present, and become familiar with common categories and their solution processes
- Give appropriate answers based on the problem set instructions (finite vs infinite lists of solutions)
- Know when to check for extraneous solutions
- The Law of Sines
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Solving Scalene Triangles
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Priority: Normal
Lesson Description
We've learned a lot about solving right triangles, but some of the trig knowledge we acquired will also help us solve scalene (non-right) triangles. In this lesson we'll see how the Law of Sines works on any triangle, and when we can and can't use it.
Learning Objectives
- Learn what the Law of Sines states, and understand when and how to use it
- Learn which situations have one solution vs multiple solutions, because most teachers mark it wrong if you only give one
- Set up word problems from scratch and apply the Law of Sines to answer questions
- The Law of Cosines
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Solving Scalene Triangles
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Priority: Normal
Lesson Description
Similar to the Law of Sines, the Law of Cosines is a tool for working with scalene triangles. It is slightly more complicated but works well for situations in which the Law of Sines fails to work. This lesson will teach us when this is the right approach, and how it works.
Learning Objectives
- Learn the Law of Cosines conceptually, and how it may be rewritten three ways - one for each side of the triangle
- Know the right situations to use Law of Cosines versus the preferred Law of Sines method, since Law of Sines is faster and less complicated
- Recognize when there is potentially more than one solution, in the same ambiguous case we saw with Law of Sines (neither LoS or LoC removes the ambiguity)
- Set up triangle word problems and solve them using either Law of Sines or Law of Cosines - whichever is most efficient
- The Area of a Triangle Using Sin
Pre-Calculus $\rightarrow$ Trigonometry $\rightarrow$ Solving Scalene Triangles
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Priority: Optional
Lesson Description
In geometry we had to do a lot of work to find the area of a scalene triangle, because the formula $A = 1/2 \, bh$ required that the base and height be perpendicular. With a quick formula involving the sine function, we can find the area of any triangle.
Learning Objectives
- Learn and practice the formula for the area of any triangle using two sides and the angle between them
- What a Conic Section Is
Pre-Calculus $\rightarrow$ Conic Sections and Analytic Geometry $\rightarrow$ Conic Sections, Circles, and Ellipses
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Priority: Normal
Lesson Description
We've learned a lot about parabolas through our study of quadratics, but equations of parabolas, circles, ellipses, and hyperbolas are all related in a special way. This lesson is our first look at what conic sections are.
Learning Objectives
- Learn the four main categories of conic sections (circles, ellipses, parabolas, and hyperbolas), and why they are all related and referred to as "conic"
- See the generic algebraic equation form of a conic section, and understand that the equation can represent any of the four categories depending solely on the coefficients
- Get a cursory look at the coefficient patterns that produce each category, before we look at each category with its own upcoming lesson
- The Equation of a Circle
Pre-Calculus $\rightarrow$ Conic Sections and Analytic Geometry $\rightarrow$ Conic Sections, Circles, and Ellipses
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Priority: VIP Knowledge
Lesson Description
One of the most common and useful quadratic forms is the circle. This lesson will examine conic section equation circles, and what we need to know about both the quadratic equation and the accompanying graph in the coordinate plane.
Learning Objectives
- Understand the quadratic form conic equation of a circle, and its derivation via the Distance Formula
- Use the Completing the Square technique to rewrite a generic quadratic form equation into the standard circle form
- Learn how to graph circles based on the equation, and conversely, how to obtain an equation based on a graph
- Equations and Properties of Ellipses
Pre-Calculus $\rightarrow$ Conic Sections and Analytic Geometry $\rightarrow$ Conic Sections, Circles, and Ellipses
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Priority: Normal
Lesson Description
Though ellipses are visually similar to circles, their oblique shape without a fixed radius is not as easy to define as a circle. This lesson explores the definitions, equations, and properties of ellipses, and what we need to know about working with them.
Learning Objectives
- Learn the quadratic equation forms that represent ellipses, and how they are similar to and different from circles
- Define vertices and axes for ellipses
- Define focus points and focal distance in an ellipse
- Define the directrix line associated with an ellipse
- Interpret an ellipse's measure of eccentricity
- Practice graphing ellipses based on the quadratic equations, and conversely finding a equation that matches the graph
- The Conic Equation of a Parabola
Pre-Calculus $\rightarrow$ Conic Sections and Analytic Geometry $\rightarrow$ Parabolas and Hyperbolas
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Priority: Normal
Lesson Description
Parabolas that we study in the context of conic sections look and behave similarly to the ones we studied during out work on quadratic functions in algebra. However, the form and attributes of conic section parabolas that we will look at is really like taking on a new point of view about something we are already familiar with. Here we will work with a specific equation form of parabolas, and quantify attributes such as its axis, focus, vertex, and directrix line.
Learning Objectives
- Learn the standard form for conic section parabolas, for both vertically and horizontally oriented parabolas
- For a parabola with vertex $(h,k)$, know what its focal point and directrix line are, formulaically
- Understand a geometric interpretation of what the focus point is of a parabola
- The Equations of Hyperbolas
Pre-Calculus $\rightarrow$ Conic Sections and Analytic Geometry $\rightarrow$ Parabolas and Hyperbolas
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Priority: Normal
Lesson Description
Hyperbolas are perhaps the least familiar conic section, because we work with them the least. This lesson will describe exactly what they are, and show you what you are supposed to know about them.
Learning Objectives
- Learn about the general shape and orientation of horizontal and vertical hyperbolas
- Find the equation of the hyperbola and the equations of its asymptotes from basic given information (such as its center, its focus, etc.)
- Graph hyperbolas quickly by putting them in standard conic section equation form and utilizing the vertex and asymptotes
- Classifying Conic Sections
Pre-Calculus $\rightarrow$ Conic Sections and Analytic Geometry $\rightarrow$ Analytic Geometry
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Priority: Normal
Lesson Description
At this point, we've seen all the types of conic sections. This short lesson focuses on how to look at equations in conic form and figure out which type of conic section you're dealing with, based on the coefficients.
Learning Objectives
- Identify conic sections using a determinant method similar to the quadratic formula, for standard form conic equations
- Recall and be able to categorize conic sections by their literal definition (e.g. which one is the set of points such that the sum of distances from any point to the two focus points is constant)
- Nonlinear Systems of Equations
Pre-Calculus $\rightarrow$ Conic Sections and Analytic Geometry $\rightarrow$ Analytic Geometry
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Priority: Normal
Lesson Description
The solutions to nonlinear systems can be a topic so complicated that it wouldn't belong in this course. However, if we restrict our discussion to polynomials of degree 2 or less, we'll see some manageable processes, and familiar conic section graphs.
Learning Objectives
- See common techniques for solving systems of equations that involve quadratics
- Understand the inherent differences from solving linear systems
- Know why we graphically interpret solutions as intersections, just like we did for linear systems
- Solve these systems with a graphing calculator and know the intricacies of doing that
- Conic Rotations
Pre-Calculus $\rightarrow$ Conic Sections and Analytic Geometry $\rightarrow$ Analytic Geometry
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Priority: Optional
Lesson Description
Usually, conic sections are naturally oriented along either the $x$ or $y$ axis. If we want to create ones that are rotated at an angle, can use trig functions to achieve this. This lesson will show you how to transform conic sections via rotation.
Learning Objectives
- Understand which conic sections can be rotated and which ones cannot
- Learn the formulaic approach for applying a rotational transformation to a conic section
- The Polar Axis System
Pre-Calculus $\rightarrow$ Miscellaneous and Calculus Prep Topics $\rightarrow$ Polar Coordinates
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Priority: High
Lesson Description
For some topics, it will be more convenient (and often required) to measure the location of points in the plane using a different approach than the $(x,y)$ system we're used to using. This lesson introduces the Polar Axis coordinate measurement system, as well as its relationship to the usual rectangular system.
Learning Objectives
- Describe the location of points in the plane using the polar axis system $(r,\theta)$ instead of rectangular $(x,y)$ system we usually use
- Define relationships among $x$, $y$, $r$, and $\theta$.
- Be able to swiftly convert between rectangular and polar coordinates
- Equations and Functions in Polar Coordinates
Pre-Calculus $\rightarrow$ Miscellaneous and Calculus Prep Topics $\rightarrow$ Polar Coordinates
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Priority: High
Lesson Description
In general functions are operations that require an input and compute an output. This lesson introduces functions of the form $r(\theta)$ instead of the usual $f(x)$ that we are accustomed to. Here we input some angle $\theta$ into the function and it gives distance from the origin as output.
Learning Objectives
- Define polar functions conceptually and understand what the input / output means
- See what kind of unique graphs are possible using polar equations and functions
- Graph simple polar functions by hand using common angles
- Convert equations from polar to rectangular coordinates, if possible
- Learn how to graph polar functions with your graphing calculator
- CIS and De Moivre's Theorem
Pre-Calculus $\rightarrow$ Miscellaneous and Calculus Prep Topics $\rightarrow$ Polar Coordinates
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Priority: Normal
Lesson Description
When we studied radicals and polynomials, the Fundamental Theorem of Algebra dictated that there are $n$ $n$-th roots for any number, thought they do not all have to be real. Using polar complex "cis" form that we learned in the last lesson, here we will learn how to find all the roots of any real or complex number.
Learning Objectives
- Continue building familiarity with the cis polar form which was introduced in the last lesson
- Learn De Moivre's Theorem, which has to do with raising a complex number to an exponent
- Use De Moivre's Theorem to find all $n$ of the $n$th-roots of a complex number
- Since real numbers are a subset of complex numbers, see how to use De Moivre's Theorem to find all $n$ of the $n$th-roots of a real number
- Classic Polar Functions
Pre-Calculus $\rightarrow$ Miscellaneous and Calculus Prep Topics $\rightarrow$ Polar Coordinates
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Priority: Normal
Lesson Description
Continuing on with our study of polar functions, this lesson examines some common polar function forms that create graphs of spirals, roses, and cardioids, among other things. Beside the function form that each has, we will also strive to understand some common properties they share as well.
Learning Objectives
- Become familiar with functions that graph as spirals, circles, cardioids, lemniscates, and roses
- For each form, understand how the coefficients being positive or negative affect the graph
- For each form that involves a trig function, know the graphical differences between using sine vs cosine in otherwise identical polar functions
- Vectors as Arrows
Pre-Calculus $\rightarrow$ Miscellaneous and Calculus Prep Topics $\rightarrow$ Vectors
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Priority: High
Lesson Description
The concept of vectors is an incredibly useful one for applied science. In this introductory lesson on them, we will look at what they are visually and how to combine two vectors.
Learning Objectives
- Define vectors as objects, similar to numbers but also having a direction property
- Visually represent a vector as an arrow, the length of which dictates the magnitude
- Learn the end-to-end method of adding and subtracting vectors
- Know the outcome of multiplying a vector by a real number, which we call a scalar
- Vector Properties and Operations
Pre-Calculus $\rightarrow$ Miscellaneous and Calculus Prep Topics $\rightarrow$ Vectors
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Priority: High
Lesson Description
Vectors may be given in the form of magnitude and direction or in the form of $x$ and $y$ components. This lesson focuses on the common things we do when working with vectors in component form.
Learning Objectives
- Examine vectors in component form and understand the difference between this and examining vectors with magnitude and direction
- Learn how to add, subtract, and scale vectors when using component form
- Calculate the norm and / or direction of a vector based on its components
- Define unit length vectors and learn how to find them
- Calculate the angle between two component form vectors using the law of cosines
- Vector Application Problems
Pre-Calculus $\rightarrow$ Miscellaneous and Calculus Prep Topics $\rightarrow$ Vectors
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Priority: Normal
Lesson Description
Using the concepts of magnitude and direction together, we'll see some common motion and force word problems, and develop an approach to vector based word problems in general.
Learning Objectives
- Use a combination of algebra and graphing to set up vector based word problems
- Represent moving objects and / or forces on a diagram using vectors in the form of arrows with both a magnitude and direction
- Combine vectors acting on the same object by adding their components and finding the resultant vector in word problem situations
- General Recursive Formulas
Pre-Calculus $\rightarrow$ Miscellaneous and Calculus Prep Topics $\rightarrow$ Intro to Sequences and Series
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Priority: Normal
Lesson Description
To kick off this section on the study of formula defined patterns, we will examine recursive relationships, where each number in the pattern is defined in terms of the number that precedes it.
Learning Objectives
- Understand conceptually what recursive formulas do, and why we need a given first term
- Practice generating the $n$-th term in a pattern using a recursive formula
- Use what we know about functions, inverses, and solving equations to move backward and discern a previous term using recursive formulas
- Understanding Sequences
Pre-Calculus $\rightarrow$ Miscellaneous and Calculus Prep Topics $\rightarrow$ Intro to Sequences and Series
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Priority: Normal
Lesson Description
Sequences are similar to recursive formulas, except in a sequence, instead of using the prior term to define the next term, we will use $n$ to define the $n$-th term (where $n$ is literally representative of which term number you're looking at).
Learning Objectives
- Understand exactly what sequences are and how they are both similar to and different from recursive relationships
- Label sequences as finite or infinite based on whether or not we want the list to terminate
- Identify common patterns and be able to write a term-generating rule using $n$, as to be able to find any term in a sequence
- Finite Differences
Pre-Calculus $\rightarrow$ Miscellaneous and Calculus Prep Topics $\rightarrow$ Intro to Sequences and Series
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Priority: Optional
Lesson Description
If a list of numbers can be obtained by a polynomial sequence-generating term, there will be a special relationship between the differences between the terms and the degree of that polynomial. This lesson looks at the topic of finite differences, which not only yields a way to generate a polynomial for sequence generating, but also similarly can be used to fit a polynomial of degree $n-1$ to $n$ data points, and find out what that polynomials is.
Learning Objectives
- Understand what finite differences are and how to calculate them
- Know the relationship between the number of non-constant finite differences and the degree of the polynomial that will generate the list
- Leverage this polynomial relationship among $n$ terms to be able to find a polynomial to fit through $n$ ordered pairs in the coordinate plane
- Series and Sigma Notation
Pre-Calculus $\rightarrow$ Miscellaneous and Calculus Prep Topics $\rightarrow$ Intro to Sequences and Series
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Priority: Normal
Lesson Description
When you add up the terms of a sequence, we call it a series. This lesson introduces us to the notation used in series, and the common things we do with them.
Learning Objectives
- Define series in terms of sequences and understand the conceptual difference between the two
- Learn the notation and vocal commonly associated with studying series
- Apply linear operator principles to series, such as factoring out a constant from the series, or that the series of a sum is the sum of each series
- Develop an understanding of the difference between a finite and infinite series
- Learn (but hopefully not memorize) formulas for special types of finite series
- Exploring Infinite Series
Pre-Calculus $\rightarrow$ Miscellaneous and Calculus Prep Topics $\rightarrow$ Intro to Sequences and Series
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Priority: Normal
Lesson Description
Series, like sequences, can be finite or infinite, even though series means to sum terms together. If the terms have certain properties, it turns out that the sum of an the infinite list of terms is exactly equal to a specific number. Here we will understand when an infinite series does and does not add up to a finite result.
Learning Objectives
- Learn which types of series will and will not have finite sums when there are an infinite number of terms
- Understand what is meant when we say that a particular infinite series is divergent
- Functions in Review
Calculus $\rightarrow$ Functions and Limits $\rightarrow$ Function Review
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Priority: Optional
Lesson Description
Since Calculus is almost entirely about function analysis, this first lesson makes sure you're up to speed on all the major properties and mechanics you should already know about functions before starting Calculus.
Learning Objectives
- Recap the most important properties of functions from prior courses
- Review Domain, Range, and Function Inverses
- Review basic how-to regarding graphing functions
- Transformations of Functions Review
Calculus $\rightarrow$ Functions and Limits $\rightarrow$ Function Review
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Priority: Optional
Lesson Description
Algebraically manipulating functions to translate or scale them is another skill we must brush up on. This lesson covers concepts from Algebra Two and Pre-Calculus, putting all function transformation knowledge in one place.
Learning Objectives
- Recall the mechanics involved with function translations both horizontally and vertically, sometimes referred to as "slides" or "shifts"
- Use multiplication and division to transform a function with stretching or compressing
- Combining Functions Review
Calculus $\rightarrow$ Functions and Limits $\rightarrow$ Function Review
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Priority: Normal
Lesson Description
This lesson reviews combining functions both by arithmetic (e.g. $h(x) = f(x) + g(x)$) and function composition (e.g. $h(x) = f(g(x))$).
Learning Objectives
- Review how to apply arithmetic to functions as if they were variables
- Thoroughly refresh on what function composition is all about, and being able to identify the "inner" and "outer" function
- Be able to deconstruct a complex function into a series of compositions of simpler functions
- Important Types of Functions
Calculus $\rightarrow$ Functions and Limits $\rightarrow$ Function Review
Web Lesson Coming Soon! »
Priority: Normal
Lesson Description
Overall knowledge and familiarity of functions is crucial to success in Calculus. You'll be able to recognize families of functions and the characteristics of each, either by looking at the graph, or by seeing the function.
Learning Objectives
- Know how to categorize functions, and each category's characteristics
- Be able to identify and categorize very commonly used functions visually
- Discern certain numerical characteristic about common function categories
- Defining Limits and Finding Graph-Based Limits
Calculus $\rightarrow$ Functions and Limits $\rightarrow$ Limits
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Priority: Normal
Lesson Description
In this inaugural discussion of limits, we seek to understand what limits are, and be able to find limits based on graphs.
Learning Objectives
- Understand what limits are and know their notation
- Evaluating finite limits based on graphs
- Know the difference between one-sided and two-sided limits
- Define two-sided limits based on both one-sided limits
- Evaluate limits computationally using a calculator
- Evaluating Limits with Limit Laws
Calculus $\rightarrow$ Functions and Limits $\rightarrow$ Limits
Web Lesson Coming Soon! »
Priority: High
Lesson Description
As we come to be more familiar with limits, we want to treat them like algebra operations. This lesson starts with some detail about how limits are used purely in an algebra sense (without relying on a graph), as well as how to evaluate them
Learning Objectives
- Recognize the technical definition of a one-sided limit
- Know laws governing limits of sums and differences, products, quotients, and constants
- Find limits using substitution and/or limit laws
- Determine when a limit requires further analysis to evaluate, versus when it does not exist
- Squeeze Theorem, End Behavior, and Limits at Infinity
Calculus $\rightarrow$ Functions and Limits $\rightarrow$ Limits
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Priority: Normal
Lesson Description
In Algebra Two we studied the end behavior of polynomials. Here, using the concepts of infinity and limits, we can look at end behavior for any function. Specifically we will focus on rational functions. We will also examine what the "squeeze theorem" guarantees for well-behaved functions.
Learning Objectives
- Use the squeeze theorem to evaluate otherwise indeterminate limits
- Recall what we already know about polynomial end behavior
- Understanding what limits mean as x goes to positive or negative infinity
- Apply infinite limits to functions we are already familiar with such as exponential or logarithmic functions
- Focus on determining end behavior of rational functions
- Three Important Limits
Calculus $\rightarrow$ Functions and Limits $\rightarrow$ Limits
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Priority: Optional
Lesson Description
This isolated topic is worth a quick look, since both AP exams and college professors often put limits of these forms on exams. The focus of this lesson is solely to expose you to what these limits are, and what they correctly evaluate to so that you can answer exam questions if they should appear.
Learning Objectives
- Learn three specific limits that AP exams and college professors often use for exams
- Focus on what the result of each limit is, not the derivation or justification, and be able to give the correct answer when asked about limits in these special forms
- Continuity
Calculus $\rightarrow$ Functions and Limits $\rightarrow$ Continuity
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Priority: Normal
Lesson Description
We've talked in the past in Algebra Two and Pre-Calculus about a loose definition of what it means for a function to be continuous. Now we'll talk about a rigorous definition, using limits.
Learning Objectives
- Understand continuity at a point or over an interval conceptually and visually based on graphs
- Define continuity of a function at a point using limits
- Define continuity of a function over an interval using limits
- Classify types of discontinuities, either based on graphs or based on function definitions
- The Continuity Value Theorems
Calculus $\rightarrow$ Functions and Limits $\rightarrow$ Continuity
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Priority: Optional
Lesson Description
When a function is known to be continuous. there are two value theorems that tell us about where the roots of the function are, or about the min or max point over a closed interval.
Learning Objectives
- Learn and understand the intermediate value theorem
- Learn and understand the extreme value theorem
- The Slope Problem
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ The Concept of the Derivative
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Priority: Normal
Lesson Description
One of the major calculus tasks is to solve "the slope problem". This lesson seeks to define the issue we are addressing when we study differential calculus, and introduces us to some estimation techniques that will eventually evolve into calculus principles.
Learning Objectives
- Define and understand the slope problem of Calculus
- Examine ways we can estimate the instantaneous slope of a nonlinear function
- Look at how and why we might want to interpret the slope of a function in a real life application
- Average Rate of Change
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ The Concept of the Derivative
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Priority: Normal
Lesson Description
In order to further study the instantaneous slope of a function, it will be helpful to first study the average slope of a function over a specified interval. This lesson defines and works with this concept, which is commonly referred to as Average Rate of Change.
Learning Objectives
- Define the Average Rate of Change over an interval
- Understand why AROC only makes sense to examine over a fixed, finite interval
- Learn how AROC can be used to approximate instantaneous slope
- Briefly look at kinematics graphs and interpret AROC in context of a displacement or velocity graph
- The Derivative as a Limit
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ The Concept of the Derivative
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Priority: High
Lesson Description
Using what we know about the slope formula, AROC, and limits, we are ready to define a function's exact instantaneous slope at a point using a limit.
Learning Objectives
- Know what the derivative of a function is and what it means
- Define the derivative using the difference quotient
- Use the difference quotient and limits to determine a function's instantaneous slope at a point
- The Derivative as a Function
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ The Concept of the Derivative
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Priority: High
Lesson Description
The limit approach to instantaneous slope from the last lesson got us exact slopes at a specific points. This lesson instead will show us how to obtain a function's "slope function" which will yield us the slope of the function at any point without having to repeatedly use the limit definition.
Learning Objectives
- Obtain a function's "slope function" using derivatives
- Use the difference quotient to get the "slope function" of a function for basic polynomial, radical, and rational functions
- Understand why the generic slope function is better than finding specific slopes
- Derivative Notation
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ The Concept of the Derivative
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Priority: Normal
Lesson Description
There are two ways to write derivatives using math symbols. A derivative is a derivative, but while each way means the same thing, some derivative applications are easier to communicate with one versus the other. This lesson will explain what each way looks like and help you understand why we might favor one over the other depending on the situation.
Learning Objectives
- See both ways we commonly notate derivatives, the "prime" and the "differential" notations
- Understand why each notation has unique applications
- Know the proper way to use each notation
- Derivatives of Sums and Constants
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ The Concept of the Derivative
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Priority: High
Lesson Description
Not coincidentally similar to limits, the derivative of a sum or difference of functions can be handled by taking things one piece at a time. This lesson outlines this for us with practice, and also looks at the derivative of a function that is multiplied by a constant.
Learning Objectives
- Learn how the derivative of a function is affected if the function is multiplied by a constant
- Learn why the derivative of a constant on its own is zero
- Understand why and how we can take the derivative of sums and differences by handling one term at a time
- Polynomial Derivatives: The Power Rule
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ Derivatives of Common Functions
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Priority: VIP Knowledge
Lesson Description
By far the most common type, polynomial derivatives have a simplistic, formulaic approach that we can use instead of performing the long hand limit definition method of derivatives. We will learn the massive time-saving formula approach in this lesson.
Learning Objectives
- Learn the Power Rule for quickly taking the derivative of any single variable monomial
- Take the derivative of entire polynomials by applying the power rule one term at a time
- Apply the Power Rule for variables raised to fraction exponent powers
- See and practice using the power rule correctly when the variable is in the denominator
- Derivatives of Trig Functions
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ Derivatives of Common Functions
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Lesson Description
Here we will learn the knowledge and tricks to take the derivative of sin, cos, and the other four trig functions. Note: this lesson covers the basics. If you are looking for more challenging practice problems involving trig functions, check out the upcoming lessons on Differentiation Techniques.
Learning Objectives
- Learn how to take the derivative of any of the six trig functions
- Use derivatives of trig functions alongside derivative rules we've already seen
- Derivatives of Exponentials
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ Derivatives of Common Functions
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Priority: VIP Knowledge
Lesson Description
Here we will learn how to take the derivative of exponential terms, starting with the foundational natural exponentiation function $e^x$. After studying the derivative pattern of this vital function, we will see how to take the derivative of similar functions, including exponentials with bases other than $e$ as well as exponentials with constant multipliers on the variable, such as $e^{5x}$.
Learning Objectives
- Learn how to take the derivative of $e^x$
- Modify these new facts to take the derivative of things other than base $e$
- Know the rule for the derivative of $e^{kx}$ where $k$ is a constant
- Derivatives of Logarithms
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ Derivatives of Common Functions
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Priority: High
Lesson Description
The derivative of the natural logarithm function is incredibly useful for both semesters of Calculus. This lessons will start by showing us how to take the derivative of this fundamental function, before also showing us how to leverage old-school log rules for taking derivatives of complicated log expressions.
Learning Objectives
- Learn how to take the derivative of $\ln (x)$
- Modify the derivative of the natural log to be able to take the derivative of any base log
- Extend our knowledge to know the derivative of an expression of the form $\log_b (ax + c)$ where $a$, $b$, and $c$ are constant coefficients
- Leverage logarithm manipulation rules to find derivatives of otherwise complicated log expressions
- Derivatives of Inverse Trig Functions
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ Derivatives of Common Functions
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Priority: Normal
Lesson Description
We will learn how to take the derivative of the inverse trig functions we studied in trigonometry. While this lesson is very commonly tested, it is almost entirely a matter of memorization. You will only see this again infrequently in the future, but if you do, many teachers expect you to remember it.
Learning Objectives
- Know the derivatives of arcsin(x), arccos(x), arctan(x)
- See (but perhaps not memorize) the derivatives of arccot(x), arcsec(x), and arccsc(x)
- Understand similarities and relationships among these derivatives to aide memorization
- Learn a $u$ substitution formula to take the derivative of any inverse trig function when the input is not merely $x$ (previewing the Chain Rule)
- Differentiating with Hyperbolic Trig Functions
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ Derivatives of Common Functions
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Priority: Optional
Lesson Description
Hyperbolic trig functions are similar to the classic SOH-CAH-TOA trig functions, but they have different meanings and properties. Though they are not commonly used, some professors expect you to know what they are, and how to take the derivatives of both the functions themselves and their inverse functions. We'll start with what inverse trig functions are and what you're expected to know about them, and then lay down some formulas that you may or may not be asked to memorize depending on your professor's level of insanity.
Learning Objectives
- Know the derivatives of sinh(x), cosh(x), and tanh(x)
- Higher Derivatives
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ Differentiation Techinques
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Priority: High
Lesson Description
Knowing derivatives is one thing, but what about the derivative of the derivative? What does that describe about a function? What about the derivative or the derivative of the derivative? This lesson formally discusses higher derivatives and in which situations we care about their use and interpretation.
Learning Objectives
- Learn what compound derivatives mean and how we usually name and notate them
- See (but not usually use) the limit definition of the second derivative
- Understand important patterns and relationships between the function and its derivative, second derivative, third derivative, etc.
- The Product and Quotient Rules
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ Differentiation Techinques
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Priority: High
Lesson Description
Unlike limit laws, it is not true that the derivative of a product is the product of the derivatives. The derivative of a product has a pattern based approach, though, and we will learn and practice it here, before immediately moving on to how to take the derivative of a quotient.
Learning Objectives
- Learn the product rule for differentiating a product of two pieces when you know the derivative of each piece
- Learn the quotient rule for differentiating the quotient of two terms when you know the derivative of each term
- Know expert tips and tricks to avoid very common pitfalls for using these techniques mistake-free
- Because the quotient rule is more involved, learn how we can often turn a quotient problem into a multiplication problem, and then proceed with the product rule
- The Chain Rule
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ Differentiation Techinques
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Lesson Description
The Chain Rule is incredibly important to derivatives because it gives us the ability to take the derivative of almost anything. This lesson will teach us how the Chain Rule works, and then get us exposed to the common ways in which we use it practically.
Learning Objectives
- See and understand the concept of how the Chain Rule works
- Apply the Chain Rule one step at a time by thinking about function composition
- Practice seeing and using the chain rule with a level of automaticity
- Implicit Differentiation
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ Differentiation Techinques
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Priority: High
Lesson Description
In some ways, regular differentiation is just a subset of implicit differentiation. Implicit can also be viewed as an extension of the chain rule. This lesson shows you what it's all about, but most importantly, gives you the know-how to get the right answer for exams and future applications.
Learning Objectives
- Understand implicit differentiation as an extension of the chain rule
- Understand how implicit differentiation gives us generic results for unknown functions of $x$
- Recognize when we do and do not necessarily need to use this approach
- Logarithmic Differentiation
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ Differentiation Techinques
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Priority: Optional
Lesson Description
When working with functions of $x$ that have $x$ as both a variable exponent and part of the base expression (e.g. $f(x) = x^x$, or $g(x) = [\cos(x)]^{x^2}$), the normal rules of differentiation do not work. In this case we can get the correct derivative expression by using a technique called Logarithmic Differentiation.
Learning Objectives
- Learn the step by step method of Logarithmic Differentiation
- Recognize when Logarithmic Differentiation is required to find a derivative
- Understand what types of complicated functions are optionally (but much more easily) differentiated using Logarithmic Differentiation
- Find The Derivative of Anything
Calculus $\rightarrow$ Discovering Derivatives $\rightarrow$ Differentiation Techinques
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Priority: High
Lesson Description
This lesson represents one of the major payouts of Calculus, both practically for your future and for exams. Here you will practice putting together all that you have learned, often applying concepts simultaneously, so that you can take the derivative of any one variable function that you could ever dream of.
Learning Objectives
- Master knowing how to differentiate anything
- Write the right amount of scratch work so that you never make a mistake
- Understand when and why each required rule is needed, and in what order, for problems that require many steps.
- Understand when the Chain Rule is needed, with and without needing another simultaneous rule such as the Product Rule or Quotient Rule
- Learn to manipulate complicated log functions to simpler form before taking their derivative
- Differentiability
Calculus $\rightarrow$ Applications of Derivatives $\rightarrow$ Function Analysis with Derivatives
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Priority: Normal
Lesson Description
Not all functions have well-defined slope functions. This lesson explores the relationship between whether a function has a well-defined slope function and the characteristics of that function.
Learning Objectives
- Understand what "differentiable" means and how it is similar and different from "continuous"
- See visual cues from graphs that help us define and understand differentiability
- Define differentiability using a limit definition
- Relative Extrema and Intervals of Increase
Calculus $\rightarrow$ Applications of Derivatives $\rightarrow$ Function Analysis with Derivatives
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Priority: High
Lesson Description
Critical points are places where a function can possibly have its slope change from positive to negative or vice versa. Using your new knowledge of derivatives, we will learn how to find the location of a critical point, as well as figuring out whether each critical point is a relative max, relative min, or neither, without having to rely on a graph.
Learning Objectives
- Use derivatives to identify the critical points of a function.
- Know the two things we look for when identifying critical points
- Use derivatives to identify points that are potential relative maximum and minimum values of a function
- Decide whether a critical point is a max, min, or neither by using the second derivative test
- Use sign analysis on the number line to determine whether a critical point is a max or min
- Find and concisely state a function's intervals of increase and decrease
- Inflection Points and Concavity
Calculus $\rightarrow$ Applications of Derivatives $\rightarrow$ Function Analysis with Derivatives
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Priority: High
Lesson Description
Similar to the last lesson, we can use derivatives to identify special points on a function. This time, instead of looking for max or min points, we will look for points where the function concavity changes. We will also seek to concisely express the intervals of concavity for a function, employing the aide of sign analysis, as we did in the last lesson.
Learning Objectives
- Define the concavity of a function
- Understand what inflection points are
- Learn how to identify inflection points
- Use sign analysis on the number line to determine whether a potential inflection point is truly an inflection point
- Find and concisely state a function's intervals of concavity
- Interpreting Graphs of Derivatives
Calculus $\rightarrow$ Applications of Derivatives $\rightarrow$ Function Analysis with Derivatives
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Priority: Normal
Lesson Description
Without knowing exactly what a function is, we can identify key relationships between it and its derivatives, using only graphs. Here we will understand what those relationships are, and find an even more concise result when working with polynomial functions.
Learning Objectives
- Identify visual relationships between the graph of a function and the graph of that function's derivative
- For polynomial functions, find a specific relationship between the shape of a function and the shape of its first derivative as well as higher derivatives
- Absolute Max and Min Values
Calculus $\rightarrow$ Applications of Derivatives $\rightarrow$ Function Analysis with Derivatives
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Priority: Normal
Lesson Description
In this lesson, you will learn how to find the absolute largest or smallest function value on a closed interval, using only Calculus techniques.
Learning Objectives
- Use derivatives and critical points to determine the maximum and minimum points on a closed interval
- Know how and why to consider the endpoints of the interval
- Tell whether or not a function has a global max or min using end behavior
- Curve Sketching
Calculus $\rightarrow$ Applications of Derivatives $\rightarrow$ Function Analysis with Derivatives
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Priority: Normal
Lesson Description
This lesson allows you to draw an unknown function using only clues about its behavior, with both specific points and derivative information.
Learning Objectives
- Without explicitly defining a function, sketch a potential graph of it based on clues about it and its derivatives
- Understand why there is no one right sketch for these questions, but also what can happen to make a sketch be incorrect
- Mean Value Theorem and Rolle's Theorem
Calculus $\rightarrow$ Applications of Derivatives $\rightarrow$ Function Analysis with Derivatives
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Priority: Optional
Lesson Description
The Mean Value Theorem (MVT) is the kind of thing that makes sense intuitively and is tested in fairly predictable ways, but is not very practical for real world applications. With that in mind, this lesson will make sure you understand the theorem and be able to answer typical exam questions based on it.
Learning Objectives
- Learn what the Mean Value Theorem is and how you're expected to use it
- Understand the geometric interpretation of the Mean Value Theorem
- Tangent Line Equations
Calculus $\rightarrow$ Applications of Derivatives $\rightarrow$ Other Applications of Derivatives
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Priority: High
Lesson Description
On quizzes and exams, we are commonly asked to find not only the instantaneous slope at a point on a function, but also the linear equation of the straight line at that point. More advanced questions ask for equations of lines that are tangent to the function but pass through a point that is not on the function. Each question has its own short solution process, and this lesson will cover both cases.
Learning Objectives
- Know how to find the tangent lines of a function at a point
- Use the same technique to find the equation of a normal line at a point
- Learn a guaranteed-to-work approach for finding equations of tangent lines that go through a specified point
- Optimization
Calculus $\rightarrow$ Applications of Derivatives $\rightarrow$ Other Applications of Derivatives
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Priority: High
Lesson Description
One of the most common and useful applications of derivatives is finding the optimal solution to a problem - the one that maximizes or minimizes some quantity. This lesson will show you the common techniques used to do this, as well as the common ways this topic shows up on exams.
Learning Objectives
- Understand why calculus is the most efficient way to find the max or min of something
- Master the general technique for solving an optimization problem
- Practice setting up optimization problems from scratch by relating several unknown quantities with a single variable
- Equations of Motion (Derivatives Only)
Calculus $\rightarrow$ Applications of Derivatives $\rightarrow$ Other Applications of Derivatives
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Priority: Normal
Lesson Description
Given the equation that describes an object's position at a given time, we can discern other measures of its motion using derivatives. This lesson will showcase the common ways in which you can expect professors and AP exams to test this topic.
Learning Objectives
- Learn the Calculus-based relationships between acceleration, velocity, and location based on derivatives
- Use graphs to understand relationships between acceleration, velocity, and position
- Differentiate to find velocity from position, or acceleration from velocity
- Marginality Applications
Calculus $\rightarrow$ Applications of Derivatives $\rightarrow$ Other Applications of Derivatives
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Priority: Optional
Lesson Description
This topic, which is fairly specific to business operations, will show you how to analyze and quantify profitability of producing $q$ number of goods based on how much it revenue and cost is involved, where revenue and cost each may vary based on $q$.
Learning Objectives
- Examine the idea of producing goods to make a profit by defining profit, revenue, and cost as functions of $q$, the quantity of the good produced
- Interpret the give revenue function $R(q)$ for $q$ number of goods produced, or create $R(q)$ from a price function if $R(q)$ is not given explicitly
- Interpret the cost function $C(q)$, which is almost always given explicitly
- Find and interpret the marginal cost and marginal revenue functions using differentiation, and know how these marginal functions may be used to make business decisions
- Know how to calculate the ideal quantity of goods to produce to maximize the profit
- L'Hopital's Rule
Calculus $\rightarrow$ Applications of Derivatives $\rightarrow$ Other Applications of Derivatives
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Priority: Normal
Lesson Description
If you are evaluating a limit which yields an indeterminate result (such as 1/0, 0/0, $1^{\infty}$, $0^0$, etc.), there is a good chance L'Hopital's Rule will give you the answer quickly. This lesson will teach you more specifically when to employ the rule, and, of course, how to use it properly.
Learning Objectives
- Review and master being able to recognize all indeterminate forms
- Understand when and why L'Hopital's rule is required
- Know how to use L'Hopital's Rule
- The Derivative of a Function's Inverse
Calculus $\rightarrow$ Applications of Derivatives $\rightarrow$ Other Applications of Derivatives
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Priority: Optional
Lesson Description
Though a fairly isolated topic, the derivative of a function's inverse has a formulaic approach that allows you to find the answer without finding the function inverse function. This lesson shows how this works.
Learning Objectives
- Understand the motivation for wanting to use a formula approach for this situation rather than explicitly finding the inverse
- Learn the formulaic approach to talking the derivative of a function's inverse
- Linearization and Newton's Method
Calculus $\rightarrow$ Applications of Derivatives $\rightarrow$ Other Applications of Derivatives
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Priority: Optional
Lesson Description
Linearization and Newton's Method are two quick commonly studied derivative applications. Both of these methods are what we call "numerical methods" because rather than finding an exact solution, these methods can be used to find very precise (but not exact) decimal solutions.
Learning Objectives
- Learn the method of linearization, which approximates function values using the derivative
- Use Newton's Method to iteratively find the roots of equations that have no algebraic exact solution
- Intro to Integration
Calculus $\rightarrow$ Understanding Integration $\rightarrow$ The Calculus Area Problem
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Priority: High
Lesson Description
To get us started with integration calculus, we will first understand the problem we are trying to solve, and some basic properties about the area enclosed between a function and the $x$ axis.
Learning Objectives
- Understand the "area problem" that integration calculus seeks to solve, similar to how differential calculus solves the slope problem
- Know the difference between positive and negative net area
- Rectangular Approximation
Calculus $\rightarrow$ Understanding Integration $\rightarrow$ The Calculus Area Problem
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Priority: Normal
Lesson Description
As a stepping stone for the processes that answer "The Area Problem" exactly and definitively, we will examine a common technique used to approximate the area under a curve
Learning Objectives
- Learn the method of approximating area under a curve using a finite number of rectangles
- Understand the three ways the rectangles could be oriented in an approximation: left, right, and midpoint
- Use function concavity and the number of rectangles used to draw conclusions about the accuracy of the approximation
- Other Area Approximations
Calculus $\rightarrow$ Understanding Integration $\rightarrow$ The Calculus Area Problem
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Priority: Normal
Lesson Description
While rectangle approximations yield reasonable estimates, there are two similar methods which are slightly more accurate at the expense of being slightly more complicated. Here we will look at everything we need to know about the Trapezoidal Rule and Simpson's Rule.
Learning Objectives
- Get a first look at the trapezoidal approximation approach by seeing how it works visually
- Learn the trapezoidal approximation approach by learning the formula
- Learn and use the formula for Simpson's approach
- For both approximations, learn how to quantify the error bounds in estimates as compared to the true value
- Riemann Sums
Calculus $\rightarrow$ Understanding Integration $\rightarrow$ The Calculus Area Problem
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Priority: Normal
Lesson Description
The concept of Riemann Sums is to integrals what the difference quotient was for derivatives - it's the "long way" of getting the exact answer we seek. Using limits and finite sum formulas, we'll be able to get an exact answer for the area under a curve.
Learning Objectives
- Understand the concept of Riemann Sums as an extension of the rectangle approximation method
- Practice setting up infinite sums with limits that represent the exact area under a function from starting point $a$ to ending point $b$
- Use limits, summation laws, and finite sum formulas to compute Riemann Sums
- Integration and Antiderivatives
Calculus $\rightarrow$ Understanding Integration $\rightarrow$ Antiderivatives
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Lesson Description
One of the tools we will need to find specific areas is the process of finding a function's antiderivative. This first lesson will conceptually familiarize us with what they are and what we need to know about them.
Learning Objectives
- Learn what antiderivatives are and their relation ship to functions
- Understand the relationship between antiderivatives and derivatives and why they are very very nearly (but not quite) inverse operations
- Similar to derivatives, know the linear operator rules for working with sums, differences, and constant coefficients
- Learn what the process of indefinite integration means
- The Definite Integral
Calculus $\rightarrow$ Understanding Integration $\rightarrow$ Antiderivatives
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Lesson Description
This lesson more precisely defines the operation of finding the area under a curve between two $x$ values. We'll look at properties of definite integrals, and relate definite integrals to antiderivatives using a very important theorem.
Learning Objectives
- Learn what a definite integral represents conceptually and visually
- Split or combine integrals of a function using given information and integration limits
- Know what happens when you switch the limits of integration and why
- Relate functions and their Antiderivatives using the a Fundamental Theorem of Calculus
- Use the FTC to evaluate definite integrals
- The Fundamental Theorem of Calculus
Calculus $\rightarrow$ Understanding Integration $\rightarrow$ Antiderivatives
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Lesson Description
The definite integral of a known function does not require a graph or computer, but rather can be expressed algebraically using the antiderivatives and the Fundamental Theorem of Calculus. This lesson shows us how to execute on definite integrals of known functions, which we will continue practicing as we learn more about specific function families' antiderivative processes.
Learning Objectives
- Relate functions and the area under their curve using Antiderivatives and the Fundamental Theorem of Calculus
- Use the FTC to evaluate definite integrals
- Understand Part 2 of the FTC, involving simultaneous derivatives and integrals
- The Power Rule for Antiderivatives
Calculus $\rightarrow$ Understanding Integration $\rightarrow$ Antiderivatives
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Priority: VIP Knowledge
Lesson Description
We recently learned that integration involves finding antiderivatives, so we need to know how to find them for any given function type. Here we will start with polynomials, using a Power Rule for integration similar to the one that exists for derivatives.
Learning Objectives
- Learn the Power Rule for polynomial antiderivatives
- Be able to specifically state why it behaves like an inverse to the Power Rule for derivatives
- Practice taking definite and indefinite integrals of polynomials
- Antiderivatives of Exponentials
Calculus $\rightarrow$ Understanding Integration $\rightarrow$ Antiderivatives
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Priority: High
Lesson Description
Leveraging the derivative knowledge we have about exponentials, we will understand how to find antiderivatives of them as well. We'll start with the base case antiderivative of $e^x$, before working on antiderivatives of both $e^{kx}$ and $a^{kx}$, where $a$ and $k$ are real number constants.
Learning Objectives
- Using what we know about derivatives, understand how to find the antiderivative of terms of the form $e^{kx}$
- Learn how to integrate exponential terms when the base is not $e$
- Practice taking definite and indefinite integrals of exponential functions
- Antiderivatives Requiring Logarithms
Calculus $\rightarrow$ Understanding Integration $\rightarrow$ Antiderivatives
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Priority: Normal
Lesson Description
There are a few cases where logarithms get thrown in the mix during our study of integration. First we'll see how to use logarithms to integrate $x^{-1}$, since the Power Rule cannot give a definitive answer for this term. Then, we'll look at the antiderivatives of logarithm expressions on their own.
Learning Objectives
- See why the Power Rule fails for integrating $x^{-1}$ and find its antiderivative by recalling the function that has a derivative of $x^{-1}$
- Practice taking definite and indefinite integrals of logarithmic terms and $x^{-1}$
- Learn the best way to deal with integrating logarithm terms, first with natural logarithms and then with any-base logarithms
- Trigonometric Antiderivatives
Calculus $\rightarrow$ Understanding Integration $\rightarrow$ Antiderivatives
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Priority: Normal
Lesson Description
Here we will see some trig function antiderivatives that will align with what we already know about trig function derivatives. We'll also look at the antiderivative of each of the six trig functions, though some of them are not common or useful due to their complexity.
Learning Objectives
- Leverage our knowledge of derivatives to instantly know six specific antiderivatives
- See (but hopefully not memorize or even use) four other trig function antiderivatives
- Integrals with Absolute Value
Calculus $\rightarrow$ Integration Techniques $\rightarrow$ Common Methods of Integration
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Priority: Normal
Lesson Description
Whenever expressions containing absolute value appear in an integral, a minor but important step is needed to make sure that the result of integration is correct. This lesson will show you what's needed for both indefinite and definite integration.
Learning Objectives
- Learn the principle behind splitting integrals of absolute value based on the definition of absolute value
- For indefinite integrals, understand the need and process for representing results as piecewise functions
- For definite integrals, understand the net-area reasons and implications of splitting the integral into two integrals
- Basic U-Substitutions
Calculus $\rightarrow$ Integration Techniques $\rightarrow$ Common Methods of Integration
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Lesson Description
The method of U Substitutions is a very common technique in integration. This lesson shows you how to do it, and when the method is and is not applicable. We will learn the mechanics to apply this method for both indefinite and definite integrals.
Learning Objectives
- Learn the integration method of U Substitutions, and when it does and does not work
- Specific to U Substitutions, learn the Mr. Math method to write scratch work to avoid calculation errors
- Practice integrating indefinitely using the U Substitution method
- Learn the minute yet important differences between indefinite and definite integrals when it comes to the U Substitution approach
- Advanced U-Substitutions
Calculus $\rightarrow$ Integration Techniques $\rightarrow$ Common Methods of Integration
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Lesson Description
While a vast majority of the cases you'll need U substitution for will closely resemble the problems we saw in the last lesson, there are specific cases where U substitution is used in unintuitive ways. This lesson will demonstrate these less common uses for U substitutions.
Learning Objectives
- Reinforce what we learned about the mechanics of U substitution in the previous lesson
- Utilize the substitution method for untraditional situations such as manipulated linear expressions
- Learn how to apply U substitution for integrals involving rational functions with the variable in two places
- Integration By Parts
Calculus $\rightarrow$ Integration Techniques $\rightarrow$ Common Methods of Integration
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Lesson Description
Another technique we commonly use to solve more complex integration problems is the method of Integration By Parts (IBP). First we'll focus on indefinite integration to learn how to use the formula both on its own and recursively, and then we'll see what changes for definite integrals.
Learning Objectives
- Memorize and use integration by parts formula
- Recognize when this is the correct approach
- Use Integration By Parts formula recursively
- Apply the tabular approach to integrating certain common recursive forms
- After gaining mastery of the process, learn how to use IBP correctly with definite integration
- Partial Fraction Decomposition
Calculus $\rightarrow$ Integration Techniques $\rightarrow$ Advanced Integration Techniques
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Lesson Description
Though this concept is based entirely in algebra, splitting complex fractions into pieces doesn't really have a convenient use beside integrating. We'll learn the fraction decomposition technique, and then apply it while integrating.
Learning Objectives
- Algebraically reduce fractions using partial fraction decomposition
- Understand when this algebra technique is useful for integration, and also when it cannot be applied
- Practice applying this technique for both indefinite and definite integrals
- Apply the technique for linear factors of multiplicity 1 (all levels)
- Apply the technique for linear factors with multiplicity, and for irreducible quadratic factors (BC and College level only)
- Trigonometric Substitution
Calculus $\rightarrow$ Integration Techniques $\rightarrow$ Advanced Integration Techniques
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Lesson Description
When integrating a function that contains one of three special root expressions, there is a trig function substitution that works when a U Sub fails. This lesson will teach you which expressions this method applies to, and then how to execute the method.
Learning Objectives
- Learn the $x$-substitution method with trig functions for integrating special root expressions
- Learn the three common root expressions this technique is good for, and which trig functions are the right fit in each case
- When these expressions are present, know to first check to see if a U Sub will suffice (it's less work)
- Practice this method using both indefinite and definite integration
- Integration by Division
Calculus $\rightarrow$ Integration Techniques $\rightarrow$ Advanced Integration Techniques
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Lesson Description
Integrating rational expressions is among one of the greater puzzle challenges of integral calculus, because similar looking situations may require vastly different approaches. This lesson combines long division with knowledge from recent lessons and gives us a better overview of how to handle any feasible rational expression integral.
Learning Objectives
- See the common ways in which monomial division masks the fact that integration could be done with the Power Rule
- Classify rational expressions using properties of each the numerator and denominator to help choose an integration method
- See and get familiar with common patterns so that you will be able to classify problems quickly for test day
- Advanced Trig Integrals
Calculus $\rightarrow$ Integration Techniques $\rightarrow$ Advanced Integration Techniques
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Priority: Optional
Lesson Description
Through the use of iterative Integration By Parts, it is possible to develop formulaic approaches to integrals that involve trig functions raised to large powers. We'll look at these types of integrals and how they might appear on exams, though they are uncommon.
Learning Objectives
- Become familiar with the forms of trig function expressions that we will solve using a formula based approach
- Learn the methods and formulas for integrating these types of problems
- Improper Integrals
Calculus $\rightarrow$ Integration Techniques $\rightarrow$ Advanced Integration Techniques
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Lesson Description
Normal integrals are performed on smooth, continuous intervals. If an integral is performed where either one of the limits is a point where the function is undefined, or the interval between the limits contains a singularity, it is improper. This lesson shows you how to evaluate improper integrals, if possible.
Learning Objectives
- Know the definition of an improper integral, and how to recognize one
- For each of the two types of improper integrals, know how to proceed with trying to evaluate them
- Choosing an Integration Approach
Calculus $\rightarrow$ Integration Techniques $\rightarrow$ Advanced Integration Techniques
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Priority: High
Lesson Description
This wrap-up integration lesson takes an important step back to think about how we should approach general problems. Often on tests, we know what method to use, because we are being tested on one or two methods at a time. If a totally random integral is throw at you, it's helpful to know what to try first and what to look for. This lesson develops your expertise in the "art" of knowing how to efficiently evaluate any integral they throw at you.
Learning Objectives
- Have a general approach and "pecking order" of methods ready if you're going to integrate a random problem
- See common integrals that students find confusing because the required method is not immediately clear
- Area Between Two Curves
Calculus $\rightarrow$ Applications of Integration $\rightarrow$ Graphical Applications of Integrals
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Lesson Description
By now we know that definite integration yields the area under a function's graph. This lesson outlines a simple way to use definite integration to find the area enclosed between two functions.
Learning Objectives
- Learn how to find the area enclosed between two functions that do not intersect
- Learn how to find the area enclosed between two functions that intersect
- Find the area between curves for functions of $y$ (College Calc and BC only)
- Volume of Revolution by Disks or Washers
Calculus $\rightarrow$ Applications of Integration $\rightarrow$ Graphical Applications of Integrals
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Priority: Normal
Lesson Description
Rotating a finite area in the coordinate plane around the $x$ or $y$ axis in three dimensions creates a volume - and using the technique explained in this lesson, we can measure that volume. First we'll better visualize and understand the 3-d volume that we get from revolution of area, and then we'll learn how to calculate it.
Learning Objectives
- Understand conceptually what is meant by creating a volume of revolution
- Learn the disk method of calculating the volume of revolution created by a single function.
- Learn the similar "washer" method for calculating the volume of revolution created by the area bound between two functions
- Be able to apply this technique for functions of $x$ and for functions of $y$ depending on the axis of rotation
- Know which situations this method works best for
- Volume of Revolution by Cylindrical Shells
Calculus $\rightarrow$ Applications of Integration $\rightarrow$ Graphical Applications of Integrals
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Lesson Description
The disk / washer method we just learned works well for most functions, but a second, entirely different method is cleaner and more efficient for certain situations. This lesson first goes over this second method (cylindrical shells), as well as how and when this method is the preferred approach.
Learning Objectives
- Understand the "Russian Doll" concept of integrating to obtain volume
- Learn and practice applying the method of cylindrical shells for functions of $x$
- Learn and practice applying the method of cylindrical shells for functions of $y$
- Understand which method is preferable (washers vs shells) based on the problem
- Arc Length of a Function
Calculus $\rightarrow$ Applications of Integration $\rightarrow$ Graphical Applications of Integrals
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Priority: Normal
Lesson Description
Although it is a less common need, Calculus also allows us to find the arc length of a function between two points. This lesson will show you where the integral formula for function arc length comes from, and how to use it.
Learning Objectives
- Derive and understand the integration formula to find the arc length of a function
- Memorize and use the formula
- Understand which types of functions do and do not lend themselves well to this operation
- Surface of Revolution
Calculus $\rightarrow$ Applications of Integration $\rightarrow$ Graphical Applications of Integrals
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Priority: Normal
Lesson Description
A direct application of the function arc length formula that we learned in the prior lesson is the ability to calculate the surface area of a solid of revolution. This is conceptually similar to volume of revolution concepts we recently learned, but instead yields the surface area.
Learning Objectives
- Use the arc length formula to derive the formula for the surface area of a volume of revolution
- Learn tricks for memorizing the formula, if you are required to memorize it
- 3D Volumes via Cross Sections
Calculus $\rightarrow$ Applications of Integration $\rightarrow$ Graphical Applications of Integrals
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Priority: Optional
Lesson Description
Volumes of revolution are not the only way to obtain volume using single variable integration. Volumes that are defined by perpendicular cross sections over a region can be calculated when those cross sectional shapes are related to the shape of the region. While not all professors cover this potentially confusing and hard-to-picture topic, many do!
Learning Objectives
- Understand what kinds of volumes are described this way
- Draw the appropriate picture in the coordinate plane to aide your work, even thought the actual solid is nearly impossible to draw
- Be able to set up and evaluate the integral that gives the required volume
- Unknown Function Initial Conditions
Calculus $\rightarrow$ Applications of Integration $\rightarrow$ Integration Applications to Functions
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Lesson Description
When we integrate indefinitely, our answer contains an unknown constant (the infamous "plus c"). However, if we are also given more information about the answer, we can solve for the value of that coefficient in that case. This is called initial conditions, and this lesson shows how to work with this type of information.
Learning Objectives
- Learn how to replace the random "+C" integration constant with a specified constant in cases where enough information is provided to do so
- See the similarities and differences between applying initial conditions to $f(x)$ and $F(x)$ versus to $f'(x)$ and $f(x)$
- Average Value of a Function
Calculus $\rightarrow$ Applications of Integration $\rightarrow$ Integration Applications to Functions
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Lesson Description
This lesson describes the methodology for finding the average value of a function over an interval, and helps us interpret what the average value represents.
Learning Objectives
- Understand what the "average function value" means conceptually
- See the formula for average function value and understand where it comes from
- Practice applying the average function value formula
- Integrals as Accumulation Over Time
Calculus $\rightarrow$ Applications of Integration $\rightarrow$ Integration Applications to Functions
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Lesson Description
As we saw with the last lesson on equations of motion, differentiation and integration can be used to relate quantities of net change over time. This lesson will use the same ideas as the last lesson but in the context of unit analysis and non-motion applications.
Learning Objectives
- Use derivatives and Integrals depending on the situation to translate between rate functions and net amounts
- Apply basic curve fitting techniques to data and integrate the result to find accumulated amounts
- Practice using this concept for problems in which you must create your own rate function from a description
- Equations of Motion (Derivatives and Integrals)
Calculus $\rightarrow$ Applications of Integration $\rightarrow$ Integration Applications to Functions
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Lesson Description
Using both differentiation and integration, we can examine the physics relationships between location, velocity, and acceleration. This lesson will explore the relationships among all three, both graphically and algebraically.
Learning Objectives
- Learn the Calculus-based relationships between acceleration, velocity, and location
- Use graphs to understand relationships between acceleration, velocity, and position
- Differentiate to find velocity from position, or acceleration from velocity
- Integrate to find velocity from acceleration, or position from velocity
- Solve initial value or initial condition problems
- First Look at Differential Equations
Calculus $\rightarrow$ Applications of Integration $\rightarrow$ Intro to Differential Equations
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Lesson Description
Entire college courses are dedicated to studying differential equations, but we're often asked to understand them at a basic level in a Calculus course (including AP courses). This lesson will introduce us to the idea of what a differential equation is, and how we can prove or disprove proposed solutions to such equations.
Learning Objectives
- Define and understand what a differential equation is
- Know how to verify whether or not a proposed solution of a differential equation is indeed a solution
- Separable Differential Equations
Calculus $\rightarrow$ Applications of Integration $\rightarrow$ Intro to Differential Equations
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Lesson Description
Solving differential equations is generally left to its own course, but there are two types of differential equations that we are often expected to be able to solve in a Calculus course. The first of these is called Separable Differential Equations, because we will be able to separate variables on each side of the equation and integrate. This lesson will outline the specifics of how this process works.
Learning Objectives
- Understand what makes a differential equation "separable"
- Be able to identify differential equations as separable or not separable
- Be able to setup and solve separable differential equations
- The Logistic Differential Equation
Calculus $\rightarrow$ Applications of Integration $\rightarrow$ Intro to Differential Equations
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Lesson Description
The second of the two types of differential equations that we are often expected to know how to solve in a Calculus course as opposed to a proper differential equations course is the Logistic Differential Equation, which can model realistic population growth with a given carrying capacity.
Learning Objectives
- Recognize when you are presented with a differential equation that is in the Logistic form
- Be able to write your own Logistic differential equation from a word problem or description
- Know how to organically solve a Logistic differential equation using algebra and calculus manipulation techniques
- Know how to solve a Logistic differential equation based on the prescribed solution formula
- Infinite Sequences and Series Overview
Calculus $\rightarrow$ Sequences, Series, and their Applications $\rightarrow$ Infinite Sequences and Series
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Lesson Description
Commonly in a Pre-Calculus course, students will have studied sequences and series to an extent that leaves them familiar with arithmetic and geometric sequences. This lesson recaps the material that you would be expected to know going into a Calculus course that covers series analysis - whether or not you learned it in a prior course.
Learning Objectives
- Understand that sequences are infinitely long lists, and understand how they are defined
- Recall the common arithmetic and geometric sequence types
- Learn how to work with generic sequences that have terms which are defined relative to their term number
- Define series in terms of sequence language
- Recall how to calculate partial sums of arithmetic and geometric series
- Understand what infinite series are, even if it is difficult to understand their result
- Derive and understand how and when infinite geometric series converge to a finite result
- Common Infinite Series
Calculus $\rightarrow$ Sequences, Series, and their Applications $\rightarrow$ Infinite Sequences and Series
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Lesson Description
From Pre-Calculus knowledge, which we recalled in the last lesson, we have some light familiarity with infinite geometric sequences. This lesson introduces some other common types of infinite series, and the properties of each type, so that by the end of this lesson, we'll have a strong understanding of what infinite series tend to look like.
Learning Objectives
- Quickly revisit the infinite geometric series, which, until now, is the only one we have been familiar with
- Further understand what convergence and divergence means for an infinite series
- Define the alternating and p-series common series types
- Examine series that we call "telescopic" and learn techniques for computing the finite sum of these types of series
- Basic Convergence Tests
Calculus $\rightarrow$ Sequences, Series, and their Applications $\rightarrow$ Infinite Sequences and Series
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Lesson Description
One of the major tasks we are charged with when we study series is determining whether a given infinite series converges or diverges (i.e. has a finite sum or not). This lesson will introduce a few common ways to determine whether or not a series converges.
Learning Objectives
- Define and understand how The Divergence Test works, and what the results of it specifically mean
- Learn the Comparison Test and Limit Comparison Test and when it is useful
- Practice using these tests in situations that would otherwise be difficult to analyze with other (forthcoming) convergence tests
- The Integral Test
Calculus $\rightarrow$ Sequences, Series, and their Applications $\rightarrow$ Infinite Sequences and Series
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Priority: Normal
Lesson Description
When the infinite series we are analyzing can be defined using continuous functions such as polynomial, exponential, or logarithm functions, we can apply the Integral Test. This is a quick and decisive way to identify convergence or divergence of such series, using an improper integral.
Learning Objectives
- Learn how the Integral Test works and know how to interpret its results
- Gain familiarity with which types of series this test can apply to
- Know when this test is a better choice than some of the tests we have learned up to this point, and vice versa
- The Ratio Test
Calculus $\rightarrow$ Sequences, Series, and their Applications $\rightarrow$ Advanced Series and Convergence
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Priority: High
Lesson Description
The Ratio Test is yet another means of analyzing whether an infinite series converges to a finite sum. This test is also useful for looking at Taylor Series, which is a near-future topic.
Learning Objectives
- Understand what the Ratio Test says and how to use it
- Know the types of series that this test excels at examining
- Understand the limitations of this test and what to do if the test is inconclusive
- The Root Test
Calculus $\rightarrow$ Sequences, Series, and their Applications $\rightarrow$ Advanced Series and Convergence
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Priority: Normal
Lesson Description
The Root Test is a convergence test that is not widely applicable, but very useful in certain situations. In some ways, this makes it easier to know when this is the best test to use, once we practice working some problems. This lesson discusses what it is and when it is particularly useful.
Learning Objectives
- Understand what the Root Test says and how to use it
- Know the types of series that this test excels at examining
- Understand the limitations of this test and what to do if the test is inconclusive
- Alternating Series Convergence
Calculus $\rightarrow$ Sequences, Series, and their Applications $\rightarrow$ Advanced Series and Convergence
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Priority: Normal
Lesson Description
Series with sequential terms that alternate between being positive and negative are referred to simply as alternating series. Because of this behavior, the criteria for determining whether or not a series converges is slightly different for these types of series, as it is "easier" for them to be convergent in a sense (in fact some converging alternating series would diverge if all the terms were instead positive). This lesson will help us understand the approach we should always take when working with alternating series.
Learning Objectives
- Recall from an earlier lesson exactly what we mean when we refer to a series as "alternating"
- Learn the two-part criteria for determining the convergence of an alternating series
- Define convergence of an alternating series as either conditional or absolute, depending on how its behavior would change if all the terms were instead positive
- Power Series
Calculus $\rightarrow$ Sequences, Series, and their Applications $\rightarrow$ Polynomial Approximations with Series
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Lesson Description
Up until now, the series we have examined have been defined without variables. In other words, while we could describe the pattern of the terms by using $n$, the term number, there was no free variable that could take on any value. Power series are infinite series of the form $\Sigma a_n (x-a)^n$, and whether or not they converge may depend on our choice of $x$. This lesson introduces Power Series, and ways to examine their convergence.
Learning Objectives
- Understand what Power Series are and the difference that it makes to have a variable in the series
- Learn how to determine which values of $x$ cause a Power Series to converge or diverge
- Learn the two ways that we are asked to describe the range of $x$ values that allow convergence, as a radius of convergence and as an interval of convergence
- Know how to validate the behavior at the interval end points, and determine whether or not those endpoints allow absolute or conditional convergence
- Taylor Polynomials
Calculus $\rightarrow$ Sequences, Series, and their Applications $\rightarrow$ Polynomial Approximations with Series
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Lesson Description
Using derivatives and Power Series, we can approximate any continuous function using a polynomial. The more terms we include, the more like the original function the polynomial will look. This phenomenon, known as Taylor Polynomials, is the subject of this lesson.
Learning Objectives
- See visually how any function can be approximated by a polynomial, centered at any $x$ value you choose
- Understand how our polynomial approximation becomes more and more like the function it's trying to replicate as we include more and more terms
- Define the "order" of a Taylor polynomial based on the degree of polynomial we want as a result
- Maclaurin Series
Calculus $\rightarrow$ Sequences, Series, and their Applications $\rightarrow$ Polynomial Approximations with Series
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Priority: Normal
Lesson Description
Taylor polynomial approximations can be centered around any $x$ value, but when they are centered at $x=0$, we call the resulting infinite Taylor series a Maclaurin Series. These series have special properties, which we will study in this lesson.
Learning Objectives
- Define a Maclaurin Series as a special instance of a Taylor Series
- Know (and possible memorize) the Maclaurin series for common functions
- Derive Maclaurin series for derivatives, integrals, and transformations of common functions by applying those operations term-wise to the corresponding series
- What is Data?
Prob and Stats $\rightarrow$ Understanding Data $\rightarrow$ Data Basics
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Priority: Normal
Lesson Description
The basis of all statistics is data - whether we want to understand data we have, or predict outcomes for future data. This introductory lesson will help us get a high level understanding of what data is, and some characteristics data can have.
Learning Objectives
- Define data in the context of statistics
- Understand the distinction between categorical and quantitative data
- Broadly define the goals of statistics in terms of data
- Learn important vocabulary such as quantitative, qualitative, bias, and outliers
- Displaying Categorical Data
Prob and Stats $\rightarrow$ Understanding Data $\rightarrow$ Data Basics
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Priority: High
Lesson Description
While we can't do much in the way of calculation to categorical data, looking at it can be extremely useful. This lesson will show us ways to summarize and examine categorical data, so that we can better interpret any patterns we observe.
Learning Objectives
- Understand the inherent similarities and differences between displaying categorical data and displaying quantitative data
- Be able to create frequency tables, bar charts, and pie charts using reasonable increments
- Identify ways in which these tables and charts can be presented in a misleading or inappropriate way
- Two Way Tables
Prob and Stats $\rightarrow$ Understanding Data $\rightarrow$ Data Basics
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Priority: High
Lesson Description
A common practice in surveys is to categorize responses into multiple categories - for example we may want to split a cohort of survey participants by gender as well as political affiliation. Instead of showing these splits in two separate tables, statisticians often create subgrouping in a two-way table, so that we can see all at once how the cohort splits into group combinations (e.g. male democrat, female democrat, male republican, female republican, etc.).
Learning Objectives
- Learn what two-way tables are and understand their structure
- Identify subtotals and category totals in a two-way table
- Learn what a marginal distribution is
- See an introduction to conditional distributions
- Percentiles and Range
Prob and Stats $\rightarrow$ Understanding Data $\rightarrow$ Data Basics
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Lesson Description
When we have numeric data, we can perform calculations on it - for example, you're probably familiar with what it means to look at the average of a data set. Beside average, we will learn about other ways in which we can describe the pattern of how data is distributed in a given data set.
Learning Objectives
- Precisely define percentiles and understand how they can be used to describe a data set
- Learn common data set descriptors such as median, and range
- Learn what a data set's five number summary is
- Define IQR and its use for identifying outliers
- Displaying Quantitative Data
Prob and Stats $\rightarrow$ Understanding Data $\rightarrow$ Data Basics
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Priority: VIP Knowledge
Lesson Description
Quantitative data are collections of numeric measures, and as such, we have many familiar options for visualizing a quantitative data set. This lesson will formally define dot plots, histograms, box plots, and other common graphical means of showing numeric data.
Learning Objectives
- Continue to understand what outliers are, and how we may choose to visualize them
- Visualize a Five Number Summary with box plots
- Use Dot Plots, Bar Graphs, and Histograms to visualize relative frequency of data points
- Use tabular methods such as the Stem-and-Leaf plot to concisely write organized lists of data points
- Understand visual what skewed and bimodal data sets look like
- Measures of Center and Spread
Prob and Stats $\rightarrow$ Understanding Data $\rightarrow$ Data Basics
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Priority: VIP Knowledge
Lesson Description
One of the most common ways we summarize and understand a data set is to examine the center and spread of it. Both measures make comparisons between data sets easier and relevant, and as such, understanding these concepts and having the skills to calculate these measure will be a frequent ask in any intro or AP course, both with and without relaying on technology.
Learning Objectives
- Formally define mean
- Define the standard deviation as a measure of data spread, and learn its intrinsic meaning
- Revisit median and IQR as measures of center and spread, respectively
- Compare standard deviation and IQR, and know which measure is more useful in which situations
- Learn how to practically calculate these measures based on the way in which data is given to us
- Use box plots and visual representations of data to find or estimate these measures if we are not given the actual data
- Skewness
Prob and Stats $\rightarrow$ Understanding Data $\rightarrow$ Data Basics
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Priority: Normal
Lesson Description
As we have seen throughout our introductory exposure to visualizations of data sets, not all data looks the same. We tend to expect data to look symmetric (with good reason as we will learn in the future) and when it doesn't, we say it has the quality of being skewed. While there aren't many common numeric measurements to determine the extent of the presence of skewness in data sets, there are a few facts that students are expected to know and understand about skewed distributions of data.
Learning Objectives
- Visually define skewness and what it means to say a data set is skewed one way or another
- Learn about the necessary relationship between the median and the mean in a skewed distribution
- Bivariate Data
Prob and Stats $\rightarrow$ Understanding Data $\rightarrow$ Data Basics
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Priority: High
Lesson Description
Bivariate data is a fancy word for identifying two things that are studied together because they may or may not be numerically related. We see this very often in statistics and as well as science courses. This lesson will help us understand what situations call for us to look at bivariate data, and the ways in which we can visualize both the data itself and the potential relationship between the two variables.
Learning Objectives
- Understand what bivariate data is and how it is different from data we've seen up to this point
- Learn how to visualize bivariate data with scatterplots
- Learn introductory ideas about lines of fit and the potential relationship between the two variables
- Sampling Surveys
Prob and Stats $\rightarrow$ Understanding Data $\rightarrow$ Gathering Data
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Priority: High
Lesson Description
Obtaining data from a relatively small sample rather than from every single member of a population is the cornerstone of many of the goals of statistics. The results of a sample are only credible if the sample was surveyed following appropriate guidelines, such as choosing sample members randomly and making sure that the sample will be representative of the whole population, on average. This lesson helps us become familiar with several common sampling methods, and why we might choose one method over another depending on the situation and concerns we have in a specific scenario.
Learning Objectives
- Understand the purpose of surveying and its role in the goal of statistics
- Know common methods of collecting survey responses
- Be able to distinguish similar sampling methods from one another
- Experiments and Studies
Prob and Stats $\rightarrow$ Understanding Data $\rightarrow$ Gathering Data
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Priority: Normal
Lesson Description
To learn about a population from a selected sample, we may or may not seek to control behavior or environment for the subjects to isolate the effect of the interest of the study. For example, researchers testing a weight loss drug may wish to test it on human subjects who are subjected to prescribed diets to eliminate the chance that subjects are eating vastly differently on or off the drug (then the diet may be affecting the results, not the drug). Controlled experiments are distinguishable from retrospective or prospective observations for their level of control on subjects, but there are a number of situations that make it unclear which is which. This lesson will help us hone the ability to tell them apart, and help us understand when we should consider doing one or the other based on the situation.
Learning Objectives
- Specifically define each what an experiment and what an observational study is
- Understand the major differences between experiments and observational studies
- Know how professors and exams typically test your knowledge of these ideas
- Bias and Error
Prob and Stats $\rightarrow$ Understanding Data $\rightarrow$ Gathering Data
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Priority: Normal
Lesson Description
Even when a statistician conducts a survey with the best intentions, the results have the potential to contain significant bias. Being aware of what types of bias can exist is the first step toward avoiding it in data collection, including types of bias that can happen unintentionally. This lesson discusses these factors and their causes, and, in light of the AP curriculum, showcases the ways in which exams test a student's ability to tell various biases apart.
Learning Objectives
- Understand what bias is and why it can exist
- Learn about the types of bias and their causes
- Be able to tell types of bias apart from one another
- Learn about other types of measuring error that can occur in data collection and surveying
- Venn Diagrams and Counting Basics
Prob and Stats $\rightarrow$ Probability Basics $\rightarrow$ Combinatorics
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Priority: VIP Knowledge
Lesson Description
One important principle of combinatorics is knowing how to account for 100% of objects in a problem, when two or more categories can be true at once (e.g. in a student group there are 5 sophomores and 7 basketball players - we could be talking about as few as 7 individuals or as many as 12 depending on the overlap in categories). Here we will learn techniques for counting appropriately based on given information.
Learning Objectives
- Learn the Inclusion / Exclusion Principle for ensuring we do not double count for overlapping categories
- Utilize Venn Diagrams to account for each category in question
- Understand why it is important to save space for those which are not in any category
- The Fundamental Counting Principle
Prob and Stats $\rightarrow$ Probability Basics $\rightarrow$ Combinatorics
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Priority: High
Lesson Description
When you have to make several independent choices, there is a multiplicative principle known as the Fundamental Counting Principle which can help us quickly enumerate the total ways you can make choices. This lesson helps us understand what the FCP tells us, and will help us understand when and how to use it.
Learning Objectives
- Define the multiplication principle for finding the total number of combinations in a situation where multiple decisions are in play
- Introduce a shortcut for the number of ways to order $n$ objects
- Factorials
Prob and Stats $\rightarrow$ Probability Basics $\rightarrow$ Combinatorics
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Priority: VIP Knowledge
Lesson Description
Along your math journey, you may have briefly encountered numbers with an exclamation point after them. These are called factorials, and represent a specific chain of multiplication that is used to count the number of arrangements of a list of objects. Beside formally defining factorials, this lesson will also help you better understand what their use is and the types of situations that require us to use them.
Learning Objectives
- Learn what factorials are mechanically, and what they represent
- Understand the specific situations that can be counted with factorials
- Know how to calculate larger factorials with technology
- Combinations and Permutations
Prob and Stats $\rightarrow$ Probability Basics $\rightarrow$ Combinatorics
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Priority: High
Lesson Description
Counting the number of ways a decision can be made will sometimes involves selecting multiple objects from a group. This lesson introduces factorials, combinations, and permutations, which are all used together for counting possible selections from groups, depending on how many are selected, and whether or not the order of selection matters in the context of the problem.
Learning Objectives
- Use factorials to define the formula for permutations of $r$ objects chosen from a pool of $n$ possibilities, where the order of the selection matters
- Practice setting up and executing word problems that involve factorials, combinations, permutations, or a combination of the three, and knowing when each is needed based on the problem
- Know how to count the ways to order objects with repeating elements
- Be fluent with how to use technology to obtain combinations and permutations results
- Definition of Probability
Prob and Stats $\rightarrow$ Probability Basics $\rightarrow$ Basic Probability Concepts
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Priority: VIP Knowledge
Lesson Description
Probability is a deep and important subject matter in math that helps us quantify the unknown. This introductory probability lesson defines probability and other closely related concepts, such as events, sample spaces, and random variables.
Learning Objectives
- Understand how probability is generally defined in terms of sample spaces, events, and outcomes
- Learn what the complement of a probability is, and when it is useful to use
- Know why probability is theoretical and what that means
- Random Variables
Prob and Stats $\rightarrow$ Probability Basics $\rightarrow$ Basic Probability Concepts
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Priority: High
Lesson Description
In algebra, we often use variables to represent an unknown quantity. In the study of probability, we use random variables to represent unknown future outcomes - and while, unlike algebra variables, we cannot solve for a definitive value of a random variable, we can measure likelihoods and expected outcomes of the unknown future occurrence that it represents. This lesson will introduce us to these ideas, and why we can about something that we can't actually know with certainty.
Learning Objectives
- Define and understand what Random Variables are, and their nature
- Learn the difference between discrete and continuous random variables
- Understand the conceptual connection between probability and random variables
- Learn why we should care about random variables, even though they are something we can't specifically value like we can with algebra variables
- Discrete Probability Distributions
Prob and Stats $\rightarrow$ Probability Basics $\rightarrow$ Basic Probability Concepts
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Priority: VIP Knowledge
Lesson Description
As we become more familiar with probability, it becomes necessary to work with specific types of probability distributions. This lesson introduces discrete distributions (separate countable outcomes), and specifies several properties that they must have to be legitimate. We'll also see both tabular and function forms of defining the potential outcomes and their respective probabilities.
Learning Objectives
- Understand what a probability distribution is and what it means
- Be able to differentiate discrete outcomes from continuous ones
- Know what properties are unique to discrete distributions
- Understand how discrete distributions may have a finite or infinite number of outcomes
- Expected Values and Variance
Prob and Stats $\rightarrow$ Probability Basics $\rightarrow$ Basic Probability Concepts
View Web Lesson »
Priority: VIP Knowledge
Lesson Description
While we do not know what the outcome will be of something that hasn't happened yet, if we know the probabilities of what could happen, we can calculate the expectation of what will happen. This is one of the most powerful uses of probability, but also one with a commonly misunderstood interpretation. This lesson will first ensure that we understand what expected value, variance, and standard deviation are and what they mean, before learning how we calculate them in various situations.
Learning Objectives
- Define expected value in terms of practical interpretation
- Find the expected value, standard deviation, and variance of an empirical distribution
- Find the expected value and expected squared value of a theoretical probability distribution
- Calculate the standard deviation and variance of a theoretical distribution
- Combining Random Variables
Prob and Stats $\rightarrow$ Probability Basics $\rightarrow$ Basic Probability Concepts
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Priority: High
Lesson Description
When studying probability, it is sometimes (but not often) required that we study a random variable that is derived from another random variable (such as $Y = 2X$ or $W = X + Y$). This could happen because the situation requires it, or simply because we're asked outright to do this. The "parent" random variable has an expected value and variance that is related to the underlying random variables from which it is derived. This lesson will help us understand the formulas that govern these relationships, and also familiarize us with the types of word problems that require us to use these methods.
Learning Objectives
- Know how to find the expected value of a random variable when it is scaled up or down by a constant multiplier
- Know how to find the expected value of a sum of random variables
- Understand the difference between the variance of a scaled random variable and the variance of a sum of random variables
- Be able to combine both techniques (scaling and summing) to find the expected value and variance of $aX + bY$ where $a$ and $b$ are constants and $X$ and $Y$ are random variables
- See word problems that require us to do this and be able to recognize this need going forward (though it is not frequently needed)
- Independent Events
Prob and Stats $\rightarrow$ Probability Basics $\rightarrow$ Multiple Event Probability
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Priority: VIP Knowledge
Lesson Description
Frequently, we are asked to calculate the probability that multiple events happen sequentially. Whatever the case may be, if the events are independent, there is a relatively simple way to calculate the total probability that multiple events happen one after another, and this lesson makes sure we know what that way is.
Learning Objectives
- Further digest and understand what it means for two events to be independent
- Know how to find the probability that two independent events both happen
- Know how to find the probability that one or another of two independent events happen
- Drawing without Replacement
Prob and Stats $\rightarrow$ Probability Basics $\rightarrow$ Multiple Event Probability
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Priority: Normal
Lesson Description
By now, we've likely encountered at least a few word problems that ask us about probabilities for drawing random balls from an urn. However, for the first time in the MM curriculum, we will consider probability calculations for drawing successively without putting previous draws back, which makes successive draws "dependent" on the prior draws. While the next lesson dives deeper into the concept of dependent probability, this lesson will get you familiarized with the idea of drawing "without replacement".
Learning Objectives
- Understand the concept of "drawing" one-at-a-time from a finite number of objects
- Identify specifically why dependent probabilities are present for multiple "draws" when we do not put previously drawn objects back in play
- Be able to calculate dependent probabilities based on specific situations of drawing without replacement
- Dependent and Conditional Probability
Prob and Stats $\rightarrow$ Probability Basics $\rightarrow$ Multiple Event Probability
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Priority: High
Lesson Description
Drawing without replacement is a great introductory dives into the inner workings of dependent probability, but this lesson shows us the explicit formula that is used in cases where we know two events are not independent. We will look at the dependent probability formula, relate it to what we already know about independent events, and practice solving for missing quantities using this dependent probability formula.
Learning Objectives
- Continue to understand what dependent probabilities are
- Explicitly learn the formula for calculating dependent probabilities
- Relate the formula to the simpler case when both events were independent
- Practice setting up the three-part equation and solving for whichever of the three pieces is missing
- Probability Using Combinatorics
Prob and Stats $\rightarrow$ Probability Basics $\rightarrow$ Multiple Event Probability
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Priority: Normal
Lesson Description
When multiple decisions need to be made, the Fundamental Counting Principle tells us to multiply the number of options together. When each calculation is a combination or permutation, our job is to digest and calculate each decision using the appropriate approach.
Learning Objectives
- When presented with multiple decisions to make, identify each choice as combinations or permutations
- Be able to write probabilities that involve all possible combinations in the denominator and all wanted combinations in the numerator
- Continue to develop expertise at telling apart combinations and permutations word problems
- Using Decision Trees
Prob and Stats $\rightarrow$ Probability Basics $\rightarrow$ Multiple Event Probability
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Priority: High
Lesson Description
When a series of decisions can be made via several yes / no scenarios, a decision tree is a helpful visual aide to help identify not only all possible combinations of outcomes, but also the probability associated with each outcome. This lesson touches upon what decision trees look like, and how to construct one of your own for any yes/no or success / failure sequence of consecutive decisions.
Learning Objectives
- Be able to identify word problems and other descriptions as binomial, and therefore able to be represented via decision trees
- Recall what binomial decisions imply and why this is a necessary prerequisite for each level of decision in a decision tree
- Create and use binomial trees to answer questions about the probability of any final outcome
- Use binomial trees to calculate dependent probabilities
- The Binomial Distribution
Prob and Stats $\rightarrow$ Common Probability Distributions $\rightarrow$ Common Discrete Distributions
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Priority: VIP Knowledge
Lesson Description
Due to its simplicity and far-reaching applicability, the Binomial Distribution is one of the most common discrete distributions. It quantifies how likely it is to have a certain number of successes in $n$ attempts, given a fixed per-trial success probability. E.g. this distribution can tell you how likely it is that you got 10 out of 20 SAT questions correct if you were guessing blindly. This lesson will introduce both the concept of the binomial distribution as well as the formulas you will be expected to retain for quizzes and tests on this distribution.
Learning Objectives
- Know the specific circumstances that define a binomial distribution
- Understand what probability the binomial distribution measures
- Know and memorize the formulas for the mean and the standard deviation of a binomial distribution
- Master the difference between the "pdf" and the "cdf"
- Be able to get fast answers from your TI-84 Calculator for binomial problems
- The Geometric Distribution
Prob and Stats $\rightarrow$ Common Probability Distributions $\rightarrow$ Common Discrete Distributions
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Priority: Normal
Lesson Description
The geometric distribution is another success / failure based probability distribution, though it is not always studied in introductory courses. This lesson will help us understand what this distribution measures, and how it is similar to and different from the binomial distribution.
Learning Objectives
- Learn and understand what the geometric distribution measures
- Be able to tell the difference between word problems that require the binomial distribution versus those that require the geometric distribution
- Practice and master the calculations and formulas of the geometric distribution
- The Poisson Distribution
Prob and Stats $\rightarrow$ Common Probability Distributions $\rightarrow$ Common Discrete Distributions
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Priority: Normal
Lesson Description
The Poisson Distribution is a common discrete distribution that measures the number of occurrences of an event, often measured in a rate per unit time. This lesson will show you what you really need to know about working with random variables that are distributed following a Poisson distribution - namely how to get specific probabilities, average values, and the standard deviation.
Learning Objectives
- Learn how the Poisson distribution works
- Understand its nature as both a discrete and infinite distribution
- Know how to formulaically obtain probabilities using the Poisson distribution
- Know how to calculate the expected value and standard deviation of a Poisson random variable
- Use common probability ideas to solve word problems using Poisson random variables
- Continuous Probability Distributions
Prob and Stats $\rightarrow$ Common Probability Distributions $\rightarrow$ Common Continuous Distributions
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Priority: High
Lesson Description
When you measure something for which partial values make sense (e.g. a person's height in inches), it makes sense to work with probability distributions that are not restricted to integer outcomes like discrete distributions tend to be. Continuous distributions allow us to measure such quantities as time or distance that do not need to be restricted to a countable number of outcomes. This lesson will help us understand how probability applies to such continuous distributions and gain experience telling continuous situations apart from discrete ones.
Learning Objectives
- Translate important properties of probability to a continuous setting
- Learn the differences and similarities of what the pdf and cdf mean in the continuous setting versus the discrete setting
- Understand the difference between the types of things continuous distributions measure and the types of things discrete distributions measure
- The Standard Normal Distribution
Prob and Stats $\rightarrow$ Common Probability Distributions $\rightarrow$ Common Continuous Distributions
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Priority: VIP Knowledge
Lesson Description
Normal distributions have a very important place in probability and statistics, due to their commonplace applicability to the real world. This first lesson on the Normal distribution will help us understand what properties it has and the ways in which we measure probability with it.
Learning Objectives
- Learn the properties and characteristic of the Normal distribution
- Know how to obtain probabilities from a normal distribution
- Understand what a "z-score" is and know how to obtain it from tables and calculators
- Memorize the amount of area that is captured within 1, 2, and 3 standard deviations of the mean
- Normal Distributions
Prob and Stats $\rightarrow$ Common Probability Distributions $\rightarrow$ Common Continuous Distributions
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Priority: VIP Knowledge
Lesson Description
Up to this point, we have become acquainted with what the normal distribution is and how we can use it to measure probability. Now, we will focus on applying it to generic situations where the mean and standard deviations can be any numbers. We'll be using the associated formulas both forward and backward to answer questions in common scenarios and word problems.
Learning Objectives
- Learn how to standardize and un-standardize a Normal distribution with any given mean and standard deviation
- Use the properties of the normal distribution to calculate probability that a random selection falls within a specified range (via calculator)
- Given a desired probability or percentile, find the upper or lower range bounds that match the description (via calculator)
- Perform normal distribution calculations by hand if your class requires you to do so
- Be able to work with linear combinations of two differently distributed normal variables, such as the sum of $X$ and $Y$
- The Central Limit Theorem
Prob and Stats $\rightarrow$ Common Probability Distributions $\rightarrow$ Common Continuous Distributions
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Priority: Normal
Lesson Description
While its proof may be beyond the scope of introductory statistics (not to mention the attention span of students), it is true that large quantities of random occurrences tend towards a normal distribution. The Central Limit Theorem specifies when this happens and what it means for us. This lesson will also make sure you have an understanding of the two ways we typically implement the CLT.
Learning Objectives
- Know what the Central Limit Theorem says and when it applies
- Understand the CLT as a tendency rather than an exact calculation
- Use the theorem for approximating very large binomial distributions
- Use the theorem for measuring averages of non-Normal continuous random variables
- The Uniform Distribution
Prob and Stats $\rightarrow$ Common Probability Distributions $\rightarrow$ Common Continuous Distributions
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Priority: Normal
Lesson Description
While the Normal Distribution is by far the most common and widely applicable continuous distribution, the Uniform Distribution is also on the syllabus for AP and intro college courses due to its simplistic nature and common applicability. This lesson will define the Uniform Distribution both in terms of technical properties and conceptual application, so that it will be clear when this distribution must be used even for problems that don't explicitly use the word uniform.
Learning Objectives
- Define the Uniform Distribution in terms of its constant density function
- Understand the Uniform Distribution conceptually as an equal probability for any outcome
- Know which key words in a word problem dictate that you must use the Uniform Distribution
- Understanding Hypothesis Tests
Prob and Stats $\rightarrow$ Inference Testing $\rightarrow$ Intro to Inference Testing
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Priority: VIP Knowledge
Lesson Description
Hands down, one of the most powerful tools in Statistics is inference testing, which gives us mathematical evidence of a hypothesis. This method is used in real world medical and science applications every single day. This introductory lesson will help us understand how hypothesis tests generally work, and how to use them appropriately.
Learning Objectives
- Learn why hypothesis tests exist and what they are intended to do
- Define the Null Hypothesis and the Alternative Hypothesis
- Understand the concept of statistical significance
- Learn how test statistics are usually calculated and how they are used in hypothesis tests
- Know what the p-value result is and what it means
- Begin to understand the difference between measuring single populations and measuring the difference between two populations
- Understanding Confidence Intervals
Prob and Stats $\rightarrow$ Inference Testing $\rightarrow$ Intro to Inference Testing
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Priority: VIP Knowledge
Lesson Description
Our best guess for an unknown future outcome is a specific number that we commonly refer to as a "point estimate." However, since it's a guess and has inherent uncertainty, we often provide our estimates in the form of a range of values. This lesson will introduce us to the concept of confidence intervals, and how we may use and interpret them.
Learning Objectives
- Define what a confidence interval is and what is seeks to accomplish
- Know and recognize the general form of a confidence interval
- Know what a confidence interval means, and just as importantly, what it doesn't mean
- Begin to understand the difference between measuring single populations and measuring the difference between two populations
- One Proportion Hypothesis Tests
Prob and Stats $\rightarrow$ Inference Testing $\rightarrow$ Tests for Proportions
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Priority: High
Lesson Description
The first type of inference test we'll study is probably the most common. After understanding the premise of proportions tests, we will walk through the proper way to conduct a test and draw conclusions from it, step by step. Finally, we will load up with lots of practice problems to see what we are expected to do on an exam.
Learning Objectives
- Understand the premise of a proportions hypothesis test
- Know what assumptions have to be stated to make the test valid and check to make sure they are true
- Know the formula for the test statistic for one-proportions null hypotheses and know how to use it
- Know the formula for creating a confidence interval for one-proportions inference and how to interpret it
- Learn how to get instant results via TI calculator, if you are allowed to use one
- Practice correctly answering questions from described scenarios or word problems
- One Proportion Confidence Intervals
Prob and Stats $\rightarrow$ Inference Testing $\rightarrow$ Tests for Proportions
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Priority: High
Lesson Description
Here we will apply what we've learned about confidence intervals to the concept of true population proportions. We'll learn and practice the type of questions we will be expected to know when it comes to confidence intervals, including the correct way to interpret their meaning in the context of proportions.
Learning Objectives
- Know how to obtain a confidence interval that contains the true population proportion with a desired level of confidence
- Be able to state the appropriate interpretation of a proportion confidence interval
- Memorize the formula to calculate the margin of error
- Know how to efficiently use a calculator to get a confidence interval for a proportion
- Hypothesis Tests for Differences in Proportions
Prob and Stats $\rightarrow$ Inference Testing $\rightarrow$ Tests for Proportions
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Priority: High
Lesson Description
Using a similar setup as we saw in the last lesson, this lesson will demonstrate how to determine whether there is a statistically significant difference between two proportions. First we will understand the assumptions that have to be true for this test to be valid, then we will see and practice it.
Learning Objectives
- Reinforce the ideas surrounding proportions hypothesis tests
- Know what assumptions have to be stated to make the test valid and check to make sure they are true
- Learn what is similar and different between one proportions hypothesis tests and two proportions hypothesis tests
- Understand why the null hypothesis is virtually always the same for these tests
- Know how to choose the correct alternative hypothesis
- Know how to obtain and interpret a confidence interval for two proportions tests
- Correctly use the TI calculator to conduct the test, if you are allowed to do so
- Put it all together and master proportions hypothesis tests via extensive practice
- Two Proportion Confidence Intervals
Prob and Stats $\rightarrow$ Inference Testing $\rightarrow$ Tests for Proportions
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Priority: Normal
Lesson Description
Here we will apply what we've learned about proportions and confidence intervals in the context of measuring the difference between two proportions. The concept will remain the same in a broad sense, including the general meaning and interpretation of the result, but the standard error and particular calculations will be different than what we saw recently when looking at confidence intervals for single proportions.
Learning Objectives
- Continue to understand the concept of measuring the difference between proportions instead of measuring a single proportion
- Know what is similar and what is different about measuring the difference between two proportions instead of measuring a single proportion
- Be able to calculate the standard error and margin of error from memory when given the right information
- Know how to efficiently use a calculator to get a confidence interval for the difference between two proportions
- Hypothesis Tests for Means - Known Sigma
Prob and Stats $\rightarrow$ Inference Testing $\rightarrow$ Tests for Sample Means
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Priority: High
Lesson Description
Hypothesis testing can be used to draw conclusions about a population's unknown average value based on a sample, similar to how we used hypothesis testing to draw conclusions about the proportion of a population based on a sample (in recent lessons). This lesson will showcase the conceptual process and goals of a sample mean hypothesis test, and show us the process and formulas involved in executing one and interpreting the results. We will also learn how to leverage the TI calculator capabilities, before finally putting it all to practice with some typical exam questions.
Learning Objectives
- Digest the idea of making inferences on a population based on the mean of a sample
- Know what assumptions have to be stated to make the test valid and check to make sure they are true
- Know how to set up appropriate null and alternative hypotheses
- Learn the formulas and process for executing and drawing conclusions on a population means hypothesis test when sigma is known
- Be able to leverage the TI calculator to perform the test, and correctly interpret the results
- Practice typical population means exam questions
- Hypothesis Tests for Means - Unknown Sigma
Prob and Stats $\rightarrow$ Inference Testing $\rightarrow$ Tests for Sample Means
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Priority: Normal
Lesson Description
When conducting hypothesis tests for population means based on a sample, we are not always equipped with the underlying distribution of the random variable we are trying to measure. If we do not know the variance of the underlying population, which we often refer to as sigma, then this extra level of uncertainty must be adjusted for by using a different distribution called the $t$ distribution, which is similar to the normal distribution but also changes gradually based on the sample size. Overall the process will be very similar to what we just learned in the last lesson, but we'll look at the minor difference and understand what it is we're expected to do.
Learning Objectives
- Know how to tell apart unknown sigma situations from known sigma situations and understand why they are different
- Know what assumptions have to be stated to make the test valid and check to make sure they are true
- Learn about the $t$ distribution and what its degrees of freedom represent
- Learn the formulas and process for executing and drawing conclusions on a population means hypothesis test when sigma is unknown
- Be able to leverage the TI calculator to perform the test, and correctly interpret the results
- Be able to read a $t$ distribution table if a calculator is not allowed
- Cement your ability to choose the right hypothesis test process by practicing common exam questions
- Confidence Intervals for Population Means
Prob and Stats $\rightarrow$ Inference Testing $\rightarrow$ Tests for Sample Means
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Priority: High
Lesson Description
We will once again revisit the concept of confidence intervals, this time with the perspective of measuring the average value of a population. We'll learn the formulas and practice the common question types associated with single population mean confidence intervals.
Learning Objectives
- Know the formulas and interpretations of population mean confidence intervals
- Be able to modify the formula based on whether or not sigma is known
- Know how to use a calculator to obtain the correct confidence interval depending on the situation
- Hypothesis Tests for Differences in Means
Prob and Stats $\rightarrow$ Inference Testing $\rightarrow$ Tests for Sample Means
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Priority: High
Lesson Description
We need to know how to use the hypothesis test process to make inferences about the difference between the means of two populations based on a sample from each. Parallel to one versus two proportions test, we need only adapt what we know about single sample mean inference tests to understand two sample mean inference tests. In addition to executing these tests by hand we will also learn the nuances of using technology to run the test, and know how to interpret the results. Finally we will have the chance to practice on exam style questions.
Learning Objectives
- Understand the difference in objectives between one sample tests and two sample tests
- Correctly set up the null and alternative hypothesis for two sample $Z$ or $t$ tests depending on whether sigma is known
- Use and memorize the formulas for sample mean, standard error, and test statistic
- Compare the technical and approximate approaches to degrees of freedom for a two sample $t$ test and know what teachers expect you to do
- Know how to use the test statistic to make a decision about the null hypothesis
- Use the TI Calculator to obtain or validate test results
- Practice thoroughly on realistic exam questions
- Confidence Intervals for Differences in Means
Prob and Stats $\rightarrow$ Inference Testing $\rightarrow$ Tests for Sample Means
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Priority: Normal
Lesson Description
In this final lesson on confidence intervals, we will apply all we know to the situation of measuring the difference between the means of two proportions. Depending on whether or not we know the sigma of the underlying population, we will set up a CI with either the normal distribution or the $t$ distribution. As always, we'll practice common problems and questions for this type of confidence interval.
Learning Objectives
- Continue to reinforce your understanding of the difference between a single population confidence interval and a two population difference confidence interval
- Know the formulas for confidence intervals of two means
- Know how to use the calculator to get an appropriate confidence interval based on the situation
- Matched Pair t Tests
Prob and Stats $\rightarrow$ Inference Testing $\rightarrow$ Tests for Sample Means
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Priority: Normal
Lesson Description
Measuring the difference between two quantities of the same population is not exactly the same as measuring the difference between measuring the difference between one quantity of two different populations. The latter case is a two sample test, which we learned in the last lesson, while the former case requires us to use a matched or paired difference one sample t test. This lesson will help us see how this case works, and understand exactly when we should and should not use it.
Learning Objectives
- Know the scenario and assumptions required to use a matched pair t test
- Understand when a matched pair t test is applicable, and why a two sample means test cannot be simultaneously applicable
- Adapt the t test formulas you already know to work for this scenario
- Construct, conduct, and interpret matched pair hypothesis tests
- Use a calculator appropriately when this scenario applies
- See common problems and know the typical word problem language that will alert you that you have a matched pair scenario
- Chi Squared Goodness of Fit
Prob and Stats $\rightarrow$ Inference Testing $\rightarrow$ Inferences for Counts
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Priority: High
Lesson Description
Goodness of fit testing stands out from other types of hypothesis tests we've seen before because it seeks to determine whether or not we observe counts or relative frequencies in the same proportions that we would expect to. In other words, unlike previous tests that consider a single statistic such as a proportion of responses or a sample average, this test looks at an entire set of categories and quantifies whether the response counts among all possible categories match closely enough to what would have been expected. This lesson will help us digest this idea, and show several common types of practice problems that we'll see on this topic.
Learning Objectives
- Understand what the Goodness of Fit test seeks to measure and how to set up the hypothesis
- Know what types of situations this test is applicable for
- Learn how to conduct this hypothesis test using a similar approach to tests we've already learned about
- Learn the TI calculator approach to obtaining the result of the hypothesis test
- Learn the manual table-lookup by-hand approach to finding Chi-Squared hypothesis test results
- Practice the two common situations that we are asked to apply this test in, both uniform and non-uniform expected distributions
- Chi Squared Independence
Prob and Stats $\rightarrow$ Inference Testing $\rightarrow$ Inferences for Counts
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Priority: Normal
Lesson Description
The Chi-Square statistic that is used for goodness of fit tests can also be used to determine independence in multi-category surveys. This lesson will show you how to use the same Chi-Squared statistic table to check for independence among two categories of labels.
Learning Objectives
- Understand how to use the Chi-Squared test statistic to test for independence among categories of response counts
- Be able to use the matrix functionality of the TI calculator to answer Chi-Squared independence hypothesis test questions
- Be able to answer Chi-Squared independence hypothesis test questions manually using a Chi-Squared statistic table
- Chi Squared Homogeneity
Prob and Stats $\rightarrow$ Inference Testing $\rightarrow$ Inferences for Counts
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Priority: Normal
Lesson Description
The final of the three Chi-Squared test statistic uses will allow us to evaluate hypothesis tests of whether two samples come from the same population. The mechanics are nearly identical to the prior lesson on category independence, but the interpretation is slightly different, and we're expected to understand that difference.
Learning Objectives
- Understand how to use the Chi-Squared test statistic to test for homogeneity among two samples
- Be able to use the matrix functionality of the TI calculator to answer Chi-Squared homogeneity hypothesis test questions
- Be able to answer Chi-Squared homogeneity hypothesis test questions manually using a Chi-Squared statistic table
- Which Hypothesis Test Should I Use?
Prob and Stats $\rightarrow$ Inference Testing $\rightarrow$ Inferences for Counts
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Priority: High
Lesson Description
Having learned about several types of hypothesis tests, it is important to reflect on them as a whole to understand which one is appropriate for which situation. This is an important perspective to obtain, and one we can only really consider once we've studied each one. This lesson will show you how to take all that you have learned up to this point about hypothesis tests and confidently be able to see which one is the correct choice in any given situation. Confusing exam questions will purposefully try to make it unclear, but there is always a firm reason why one must be used over another.
Learning Objectives
- Reflect on the several hypothesis tests that we have seen, as a bundle
- Use your knowledge of each hypothesis test to understand what makes it unique when compared to another hypothesis test
- Practice distinguishing tests correctly in common ways that exams ask you to do so
- Correlation
Prob and Stats $\rightarrow$ Regression Processes $\rightarrow$ Linear Regression
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Priority: High
Lesson Description
To begin to understand correlation, this lesson introduces us to the idea of visually and mathematically identifying how strongly two variables are related based on a data sample. We will explore loose visual cues from the scatterplot, as well as numeric quantifiers such as the correlation coefficient and the "R squared" value.
Learning Objectives
- Understand relationship strength of bivariate variables based on a combination of tightness and direction
- Conceptually define the correlation coefficient and the "R squared" measure of relationship strength
- Mathematically understand what kinds of R Squared and correlation coefficient numbers correspond to strong, moderate, or weak relationships
- Understanding Linear Regression
Prob and Stats $\rightarrow$ Regression Processes $\rightarrow$ Linear Regression
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Priority: VIP Knowledge
Lesson Description
Building off of the introductory material we saw in the previous lesson, this lesson will continue to explore exactly what regression means and what it seeks to do, in both a conceptual and mathematical sense. We will define such mathematical quantities as the regression line equation, residuals and correlation, and the technical hypothesis definition of regression significance.
Learning Objectives
- Further explore the nature of examining a bivariate relationship
- Formally define the hypothesis test that is used to determine statistical significance
- Define what residuals are for each observation
- Know how to use the TI to obtain the correlation coefficient, R Squared result, and the equation of the regression line
- Begin to understand how the regression line should be interpreted
- Using Regression Models Predictively
Prob and Stats $\rightarrow$ Regression Processes $\rightarrow$ Linear Regression
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Priority: High
Lesson Description
After gaining understanding about regression and its basics, this lesson focuses on one of its most common applications - predictive analytics. We can use the line of best fit as a predictor for future expectation to the extent that we trust the credibility of the line of best fit, and this lesson discusses this concept as well as the mechanics of how to go about interpreting what predictive regression questions actually want you to do.
Learning Objectives
- Use the line of best fit to solve for predicted values of future occurrences
- Know what conditions make it inappropriate to predict using the regression line
- State and interpret confidence intervals for predictions
- State and interpret predictive intervals and know how they are different from confidence intervals
Pre-Algebra
Integers, Operations, and Expressions - Working with Integers
Integers, Operations, and Expressions - Using Variables and Expressions
Properties of Real Numbers - Properties for All Numbers
Properties of Real Numbers - Properties for Whole Numbers Only
Decimals and Fractions - Working with Decimals
Decimals and Fractions - Numerical Fractions
Decimals and Fractions - Using Decimals and Fractions
Equations and Inequalities - Manipulating Equations
Equations and Inequalities - Working with Inequalities
Equations and Inequalities - Using Equations and Inequalities
Percents, Ratios, and Proportions - Percentages
Percents, Ratios, and Proportions - Ratios and Proportions
Algebra One
Polynomials - Advanced Exponent Properties
Polynomials - Working with Monomials
Polynomials - Working with Polynomials
Polynomials - Getting Started with Factoring
Intro to Quadratics - Quadratic Factoring Forms
Intro to Quadratics - Solving Quadratic Equations
Rational Expressions - Working with Linear Rational Expressions
Rational Expressions - Problem Solving with Rational Terms
Radical Expressions and Roots - Exploring Numeric Square Roots
Radical Expressions and Roots - Simplifying Radical Expressions
Radical Expressions and Roots - Using Roots
Linear Graphing - The Coordinate Plane and Line Equations
Linear Graphing - Graphing Linear Equations
Linear Graphing - Graphing Inequalities and Absolute Value Equations
Linear Graphing - Analyzing Linear Graphs
Linear Systems - Systems of Two Equations
Linear Systems - Systems of Inequalities
Linear Systems - Intro to Systems of Three Variables
Algebra Two
Functions - Function Basics
Functions - Linear Functions
Complex Numbers - The Imaginary Number
Complex Numbers - Using Complex Numbers
Using Quadratic Relationships - Quadratic Functions
Using Quadratic Relationships - Applications of Quadratics
Polynomial Functions - Analyzing Polynomials
Polynomial Functions - Advanced Polynomial Properties
Radical and Rational Relationships - Advanced Radicals and Roots
Radical and Rational Relationships - Working with Advanced Roots
Radical and Rational Relationships - Rational Expressions
Radical and Rational Relationships - Using Rational Expressions
Matrices - Matrices and Arithmetic
Matrices - Solving Systems of Equations with Matrices
Pre-Calculus
Advanced Function Analysis - Properties of Functions
Advanced Function Analysis - Working with Functions
Exponential Relationships - Understanding Variable Exponents
Exponential Relationships - Understanding Logarithms
Exponential Relationships - Exponential and Logarithmic Equations
Trigonometry - Right Triangles
Trigonometry - Angles and Angle Measurement
Trigonometry - Trigonometry Functions
Trigonometry - Using and Graphing Trig Functions
Trigonometry - Advanced Trigonometry
Trigonometry - Solving Scalene Triangles
Conic Sections and Analytic Geometry - Conic Sections, Circles, and Ellipses
Conic Sections and Analytic Geometry - Parabolas and Hyperbolas
Conic Sections and Analytic Geometry - Analytic Geometry
Miscellaneous and Calculus Prep Topics - Polar Coordinates
Miscellaneous and Calculus Prep Topics - Vectors
Miscellaneous and Calculus Prep Topics - Intro to Sequences and Series
Calculus
Functions and Limits - Function Review
Functions and Limits - Limits
Functions and Limits - Continuity
Discovering Derivatives - The Concept of the Derivative
Discovering Derivatives - Derivatives of Common Functions
Discovering Derivatives - Differentiation Techinques
Applications of Derivatives - Function Analysis with Derivatives
Applications of Derivatives - Other Applications of Derivatives
Understanding Integration - The Calculus Area Problem
Understanding Integration - Antiderivatives
Integration Techniques - Common Methods of Integration
Integration Techniques - Advanced Integration Techniques
Applications of Integration - Graphical Applications of Integrals
Applications of Integration - Integration Applications to Functions
Applications of Integration - Intro to Differential Equations
Sequences, Series, and their Applications - Infinite Sequences and Series
Sequences, Series, and their Applications - Advanced Series and Convergence
Sequences, Series, and their Applications - Polynomial Approximations with Series
Prob and Stats
Understanding Data - Data Basics
Understanding Data - Gathering Data
Probability Basics - Combinatorics
Probability Basics - Basic Probability Concepts
Probability Basics - Multiple Event Probability
Common Probability Distributions - Common Discrete Distributions
Common Probability Distributions - Common Continuous Distributions
Inference Testing - Intro to Inference Testing
Inference Testing - Tests for Proportions
Inference Testing - Tests for Sample Means
Inference Testing - Inferences for Counts
Regression Processes - Linear Regression
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