Course Updates for Pre-Calculus in July 2021

We are excited to share that we've made significant enhancements to our Pre-Calculus course. One consequence of adding this new content is that it may impact learners’ Mastery percentages.

When we add new content and remove old content, Mastery percentages often change because the number of available skills changes. For example, if you mastered 90 out of 100 skills in a course, and we added 10 new skills, your mastery skill count would then be 90 out of 110. This change would change your Mastery percentage from 90% to 82%.

With this specific course update, we expect that Mastery percentages will change by a maximum of 52%. Most people will see a much smaller Mastery percentage change.

You can see exactly what content items we’ve added and removed below. Happy learning! 

Content Added

Articles

Adding & subtracting rational expressions (advanced)

Addition rule for probability (basic)

Estimating limit values from graphs

Intermediate value theorem review

Intro to adding & subtracting rational expressions

Intro to rational expressions

Laws of sines and cosines review

Limits intro

Reducing rational expressions to lowest terms

Strategy in finding limits

Using tables to approximate limit values

Exercises

Add & subtract rational expressions

Add & subtract rational expressions: unlike denominators

Classify discontinuities

Complex plane and polar form

Conclusions from direct substitution (finding limits)

Connecting limits and graphical behavior

Continuity and common functions

Continuity at a point (algebraic)

Continuity at a point (graphical)

Continuity over an interval

Creating tables for approximating limits

Develop probability distributions: Empirical probabilities

Develop probability distributions: Theoretical probabilities

Direct substitution with limits that don't exist

End behavior of rational functions

Estimating limit values from graphs

Estimating limits from tables

Find expected payoffs

General triangle word problems

Graph probability distributions

Graphically multiply complex numbers

Graphs of rational functions

Infinite limits: algebraic

Infinite limits: graphical

Interpret expected value

Interpret probabilities of compound events

Interpret solutions of trigonometric equations in context

Inverse functions: graphs and tables

Justification with the intermediate value theorem

Limits at infinity of quotients

Limits at infinity of quotients with square roots

Limits at infinity: graphical

Limits by direct substitution

Limits by factoring

Limits intro

Limits of combined functions: products and quotients

Limits of combined functions: sums and differences

Limits of composite functions

Limits of piecewise functions

Limits of trigonometric functions

Limits using conjugates

Limits using trig identities

Mean (expected value) of a discrete random variable

Multiply & divide rational expressions

Multiply & divide rational expressions (basic)

Next steps after indeterminate form (finding limits)

One-sided limits from graphs

One-sided limits from tables

Probability with discrete random variables

Probability with general multiplication rule

Rational function points of discontinuity

Removable discontinuities

Simplify rational expressions: common binomial factors

Simplify rational expressions: common monomial factors

Solve triangles using the law of cosines

Solve triangles using the law of sines

Squeeze theorem

Strategy in finding limits

Trig identities from reflections and rotations

Trig values of special angles

Trig values of π/6, π/4, and π/3

Two-way tables, Venn diagrams, and probability

Use matrices to manipulate data

Use matrices to represent data

Use matrices to solve systems of equations

Use matrices to transform the plane

Use probabilities to make fair decisions

Using the intermediate value theorem

Videos

Adding & subtracting rational expressions: like denominators

Adding rational expression: unlike denominators

Adding vectors in magnitude and direction form

Analyzing structure word problem: pet store (1 of 2)

Analyzing structure word problem: pet store (2 of 2)

Analyzing unbounded limits: mixed function

Analyzing unbounded limits: rational function

Approximating limits using tables

Composing 3x3 matrices

Connecting limits and graphical behavior

Constructing a probability distribution for random variable

Continuity at a point

Continuity over an interval

Cosine equation algebraic solution set

Cosine equation solution set in an interval

Cosine, sine and tangent of π/6 and π/3

Discontinuities of rational functions

Dividing rational expressions

End behavior of rational functions

Estimating limit values from graphs

Estimating limits from tables

Finding area of figure after transformation using determinant

Functions continuous at specific x-values

Functions continuous on all real numbers

Functions with same limit at infinity

Graphing rational functions according to asymptotes

Graphs of rational functions: horizontal asymptote

Graphs of rational functions: vertical asymptotes

Graphs of rational functions: y-intercept

Graphs of rational functions: zeros

Infinite limits and asymptotes

Intermediate value theorem

Interpreting determinants in terms of area

Interpreting solutions of trigonometric equations

Interpreting statements about vectors

Intro to adding rational expressions with unlike denominators

Intro to determinant notation and computation

Intro to rational expression simplification

Introduction to infinite limits

Introduction to limits at infinity

Introduction to vector components

Inverse matrices and matrix equations

Inverse matrix introduction

Invertible matrices and determinants

Invertible matrices and transformations

Justification with the intermediate value theorem: equation

Justification with the intermediate value theorem: table

Least common multiple of polynomials

Limit of (1-cos(x))/x as x approaches 0

Limit of sin(x)/x as x approaches 0

Limit properties

Limits at infinity of quotients (Part 1)

Limits at infinity of quotients (Part 2)

Limits at infinity of quotients with square roots (even power)

Limits at infinity of quotients with square roots (odd power)

Limits by direct substitution

Limits by factoring

Limits by rationalizing

Limits intro

Limits of combined functions

Limits of combined functions: piecewise functions

Limits of composite functions: external limit doesn't exist

Limits of composite functions: internal limit doesn't exist

Limits of piecewise functions

Limits of piecewise functions: absolute value

Limits of trigonometric functions

Matrices as transformations of the plane

Mean (expected value) of a discrete random variable

Meaningfully composing functions

Multiplying & dividing rational expressions: monomials

Multiplying rational expressions

One-sided limits from graphs

One-sided limits from graphs: asymptote

One-sided limits from tables

Parallelogram rule for vector addition

Probability with discrete random variable example

Proof of the law of cosines

Proof of the law of sines

Proof of the tangent angle sum and difference identities

Proof: Matrix determinant gives area of image of unit square under mapping

Rational equations word problem: combined rates

Rational equations word problem: combined rates (example 2)

Rational equations word problem: eliminating solutions

Reading inverse values from a graph

Reading inverse values from a table

Reasoning about unknown variables

Reasoning about unknown variables: divisibility

Removing discontinuities (factoring)

Removing discontinuities (rationalization)

Representing quantities with vectors

Representing systems of any number of equations with matrices

Representing systems of equations with matrices

Sine equation algebraic solution set

Solving for a side with the law of cosines

Solving for a side with the law of sines

Solving for an angle with the law of cosines

Solving for an angle with the law of sines

Squeeze theorem intro

Strategy in finding limits

Structure in rational expression

Subtracting rational expressions

Subtracting rational expressions: factored denominators

Subtracting rational expressions: unlike denominators

Subtracting vectors with parallelogram rule

Theorem for limits of composite functions

Theorem for limits of composite functions: when conditions aren't met

Trig limit using Pythagorean identity

Trig limit using double angle identity

Trig values of π/4

Trig word problem: stars

Types of discontinuities

Unbounded limits

Undefined limits by direct substitution

Using matrices to manipulate data: Game show

Using matrices to manipulate data: Pet store

Using matrices to represent data: Networks

Using matrices to represent data: Payoffs

Using matrices to transform a 4D vector

Using matrices to transform the plane: Composing matrices

Using matrices to transform the plane: Mapping a vector

Using specific values to test for inverses

Using the tangent angle addition identity

Valid discrete probability distribution examples

Vector word problem: resultant force

Vector word problem: resultant velocity

Verifying inverse functions from tables

Worked example: Continuity at a point (graphical)

Worked example: point where a function is continuous

Worked example: point where a function isn't continuous

Worked example: using the intermediate value theorem

Working with matrices as transformations of the plane

Content Removed

Articles

Adding and subtracting polynomials with two variables review

Circle equation review

Complex number operations review

Intro to complex numbers

Intro to the imaginary numbers

Matrix row operations

Multiplying complex numbers

Parabola focus & directrix review

Powers of the imaginary unit

Recognizing vectors practice

Representing linear systems with matrices

Solving equations graphically

Vector forms review

Vector operations review

Exercises

Add & subtract complex numbers

Add & subtract polynomials: find the error

Add & subtract polynomials: two variables

Add & subtract polynomials: two variables (intro)

Add vectors: magnitude & direction to component

Analyze scalar multiplication

Classify complex numbers

Combined vector operations

Dependent probability

Determinant of a 2x2 matrix

Determine inverse matrices

Divide polynomials by monomials (with remainders)

Divide polynomials with remainders

Equation of a parabola from focus & directrix

Explicit formulas for arithmetic sequences

Explicit formulas for geometric sequences

Extend arithmetic sequences

Extend geometric sequences

Extend geometric sequences: negatives & fractions

Features of a circle from its expanded equation

Features of a circle from its graph

Features of a circle from its standard equation

Graph a circle from its expanded equation

Graph a circle from its features

Graph a circle from its standard equation

Independent probability

Interpret equations graphically

Matrix dimensions

Matrix elements

Matrix equations: addition & subtraction

Matrix equations: scalar multiplication

Matrix row operations

Multiply complex numbers

Multiply complex numbers (basic)

Parts of complex numbers

Permutations & combinations

Powers of the imaginary unit

Probabilities of compound events

Recursive formulas for arithmetic sequences

Recursive formulas for geometric sequences

Represent linear systems with matrices

Sequences word problems

Simple probability

Simplify roots of negative numbers

Solve equations graphically

Transform polygons using matrices

Unit vectors

Use arithmetic sequence formulas

Use geometric sequence formulas

Write standard equation of a circle

Videos

"At least one" probability with coin flipping

Adding complex numbers

Adding polynomials: two variables (intro)

Adding vectors algebraically & graphically

Adding vectors in magnitude & direction form (1 of 2)

Adding vectors in magnitude & direction form (2 of 2)

Birthday probability problem

Classifying complex numbers

Combined vector operations

Common tangent of circle & hyperbola (1 of 5)

Common tangent of circle & hyperbola (2 of 5)

Common tangent of circle & hyperbola (3 of 5)

Common tangent of circle & hyperbola (4 of 5)

Common tangent of circle & hyperbola (5 of 5)

Compound events example with tree diagram

Conic section from expanded equation: circle & parabola

Conic section from expanded equation: ellipse

Conic section from expanded equation: hyperbola

Converting recursive & explicit forms of geometric sequences

Dependent probability example

Dependent probability: coins

Determinant of a 2x2 matrix

Determining invertible matrices

Die rolling probability

Die rolling probability with independent events

Divide polynomials by monomials (with remainders)

Dividing complex numbers: polar & exponential form

Dividing polynomials with remainders

Equation of a parabola from focus & directrix

Equivalent vectors

Exactly three heads in five flips

Example: Combinatorics and probability

Explicit & recursive formulas for geometric sequences

Explicit formulas for arithmetic sequences

Features of a circle from its expanded equation

Features of a circle from its graph

Features of a circle from its standard equation

Finding an error in polynomial subtraction

Focus & directrix of a parabola from equation

Free-throw probability

Generalizing with binomial coefficients (bit advanced)

Getting exactly two heads (combinatorics)

Graphing a circle from its standard equation

Graphing circles from features

Independent & dependent probability

Interpreting equations graphically

Interpreting equations graphically (example 2)

Intersection of circle & hyperbola

Intro to arithmetic sequences

Intro to complex numbers

Intro to focus & directrix

Intro to geometric sequences

Intro to geometric sequences (advanced)

Intro to matrix inverses

Intro to the imaginary numbers

Intro to theoretical probability

Matrix word problem: prices

Matrix word problem: vector combination

Multiplying complex numbers

Polynomials review

Powers of complex numbers

Powers of the imaginary unit

Probability & combinations (2 of 2)

Probability with counting outcomes

Probability without equally likely events

Recognizing vectors

Representing a line tangent to a hyperbola

Representing linear systems with matrix equations

Sequences intro

Sequences word problem: growth pattern

Simple probability: non-blue marble

Simple probability: yellow marble

Simplifying roots of negative numbers

Solving equations graphically (1 of 2)

Solving equations graphically (2 of 2)

Solving linear systems with matrices

Subtracting complex numbers

Subtracting polynomials: two variables

Subtracting polynomials: two variables (intro)

The Monty Hall problem

Three-pointer vs free-throw probability

Transforming polygons using matrices

Transforming vectors using matrices

Unit vectors intro

Vectors word problem: tug of war

Visual representation of transformation from matrix

Worked example: Scaling unit vectors

Worked example: finding unit vector with given direction

Writing standard equation of a circle

i as the principal root of -1