# 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*

*Exercises*

Develop probability distributions: Theoretical probabilities |

*Videos*

**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 |