# Algebra Two

**Relevant Brilliant Courses (Online Content)**

The four courses below are the foundations for all of the mathematics offered on Brilliant. If you’re an educator with a group of 10 or more students you want to give full access to these courses, contact pricing@brilliant.org to learn more about Brilliant’s discounts for school groups.

**Deep Diving Math Enrichment Problem Sets (Printable Content)**

Too often, school math is all about “racing to finish” instead of diving deep to understand and explore creative, tangential lines of inquiry. These sets were written to inspire deep diving exploration that extends and enriches the core mathematical topics and skills introduced in a standard algebra 2 common core curriculum.

Each of the “Practice-Challenge-Culmination” problem sets listed below takes some foundational skill in the common core curriculum for that grade and, after a few practice problems, extends the concept to more creative challenges, and then to a single, deep-dive question.

#### Contents

## Working with Functions

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentFUNCTIONS INTRODUCTION [Printable PDF] CCSS.MATH.CONTENT.HSF.IF.A.1 CCSS.MATH.CONTENT.HSF.IF.A.2 CCSS.MATH.CONTENT.HSF.BF.A.1 This exploration set aims to deepen students’ understanding of functions. The problem set begins with identifying functions, evaluating functions, and determining domain and range. Then, students dive into a challenging series of writing and evaluating functions. The problems culminate with students creating functions for which 𝑓(𝑥)=−𝑓(−𝑥) and 𝑓(𝑥)=𝑓(−𝑥). (This activity assumes that students have not seen even and odd functions.) Complex Algebra: Functions Warmup COMPOSITION WITH FUNCTIONS [Printable PDF] CCSS.MATH.CONTENT.HSF.BF.A.1.B CCSS.MATH.CONTENT.HSF.BF.A.1.C CCSS.MATH.CONTENT.HSF.BF.B.4.B These puzzle-like questions build upon students’ intuitive understanding of compositions of functions. Students begin by working with single-step compositions before moving on to multi-step compositions. In the culmination, students explore possible compositions for which 𝑓(𝑔(𝑥)) and 𝑔(𝑓(𝑥)) both equal 𝑥. Complex Algebra: Composition

## Polynomials

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentDIFFERENCE OF SQUARES [Printable PDF] CCSS.MATH.CONTENT.HSA.APR.C.4 This set of problems dives deeply into one aspect of polynomial factoring: application of the difference of squares. Students begin by examining a geometric proof of the difference of squares, then apply the identity to some straightforward factoring problems. Lastly, students move on to applying the difference of squares in some more abstract contexts. Algebra Through Puzzles: Difference of Squares POLYNOMIAL FACTORING [Printable PDF] CCSS.MATH.CONTENT.HSA.APR.B.2 CCSS.MATH.CONTENT.HSA.APR.C.4 This set of problems aims to deepen student understanding of factoring and its uses as a problem solving tool. The problems dive into applying factoring intuition and visualizing factoring geometrically before moving into abstract factoring applications. Finally, the problems culminate with a complex combination of factoring, systems of equations, and radicals. Complex Algebra: Finding Roots POLYNOMIALS: ROOTS AND SHAPE [Printable PDF] CCSS.MATH.CONTENT.HSA.APR.B.2 CCSS.MATH.CONTENT.HSA.APR.B.3 This activity begins by identifying roots of polynomials and determining the maximum number of real roots of any polynomial. Then, students build upon their intuition of how factors and roots relate to one another and to the graph of a function. Finally, the culmination asks students to explore numbers of possible solutions when given a factored polynomial in an inequality. Complex Algebra: Graphs of Polynomials ARITHMETIC WITH COMPLEX NUMBERS [Printable PDF] CCSS.MATH.CONTENT.HSN.CN.A.1 CCSS.MATH.CONTENT.HSN.CN.A.2 CCSS.MATH.CONTENT.HSN.CN.A.3 While the thought of imaginary numbers perplexes many people, imaginary numbers are most certainly not imaginary. In this activity, students deepen their understanding of complex numbers as they evaluate and simplify negative square roots and complex number expressions. In the culmination, students strategically combine their algebra skills and knowledge of complex numbers to determine an unknown value. Complex Algebra: It's Not Real

## Rational Functions

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentRATIONAL FUNCTIONS & VERTICAL ASYMPTOTES [Printable PDF] CCSS.MATH.CONTENT.HSF.IF.C.7.D Students begin this activity with a review of how function values change as they approach vertical asymptotes and how to find vertical asymptotes. Then, students move on to comparing the domains and asymptotes of various rational functions. The problems culminate with a challenge to determine the possible number of vertical asymptotes when summing two rational functions. Complex Algebra: Asymptotes RATIONAL FUNCTIONS: ASYMPTOTES [Printable PDF] CCSS.MATH.CONTENT.HSF.IF.C.7.D In this activity, students begin by exploring why some rational functions have horizontal asymptotes and others don’t. Students then transition into identifying and comparing vertical and horizontal asymptotes. Lastly, students explore the minimum and maximum possible number of horizontal asymptotes that a rational function could have. Complex Algebra: Asymptotes

## Radical Functions and Rational Exponents

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentRATIONAL EXPONENTS [Printable PDF] CCSS.MATH.CONTENT.HSN.RN.A.1 CCSS.MATH.CONTENT.HSN.RN.A.2 This activity is designed to deepen students’ understanding of rational exponents and improve their ability to apply exponent rules strategically. Students rewrite, evaluate, and order expressions with rational exponents, solve equations with rational exponents, and determine the validity of statements involving rational exponents. In the culmination, students determine which radical and exponential expressions from a set are equivalent. Algebra Through Puzzles: Fractional Exponents RADICAL EXPRESSIONS & EQUATIONS [Printable PDF] CCSS.MATH.CONTENT.HSA.REI.A.2 CCSS.MATH.CONTENT.HSA.SSE.B.3 This series of problems provides multiple contexts in which students can apply their understanding of radicals. After analyzing, simplifying, and comparing radical expressions, students dive into solving radical equations. Then, students explore two scenarios involving infinite nested radicals. Lastly, in the culmination, students evaluate a long sum of radicals, a problem which can be solved quickly and elegantly with an application of the right simplification tool. Algebra Through Puzzles: Fractional Exponents

## Exponentials and Logarithms

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentEXPONENTIALS [Printable PDF] CCSS.MATH.CONTENT.HSF.LE.A.1 CCSS.MATH.CONTENT.HSF.LE.A.1.A CCSS.MATH.CONTENT.HSF.LE.A.2 This problem set deepens students’ understanding of exponential growth and exponential functions. The early problems address conceptual understanding of exponential growth and how to write basic exponential models. Then, students move on to examine differences between consecutive function values in cases of exponential growth. Finally, students explore decreasing exponential growth and write the function to model this phenomenon when given only the differences between the consecutive function values. Complex Algebra: Exponents Warmup LOGARITHMS [Printable PDF] CCSS.MATH.CONTENT.HSF.BF.B.5 CCSS.MATH.CONTENT.HSF.IF.C.7.E This problem set begins with the evaluation of logarithms before moving into creative applications of the change of base formula. The problems culminate with one challenging equation involving logarithms and several radicals. Note that this entire problem set should be done without a calculator! Complex Algebra: Defining Logarithms LOG SCALES [Printable PDF] CCSS.MATH.CONTENT.HSA.SSE.B.3.C CCSS.MATH.CONTENT.HSF.BF.B.5 CCSS.MATH.CONTENT.HSF.IF.C.7.E In this activity, students examine logarithmic scales and learn how to interpret them. They explore why it's convenient to talk about and compare quantities that grow very quickly, not directly but by their powers. The problems begin by exploring pH scales, before moving into semi-log plots, log-log plots, and finally, decibel scales. Complex Algebra: Log Scales

## Trigonometry

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentSINE AND COSINE [Printable PDF] CCSS.MATH.CONTENT.HSF.TF.A.4 This problem sequence establishes the cofunction identities between the sine and cosine functions. Students begin by examining relationships between points placed on a rotating bicycle wheel. Then, they apply this information to examine relationships between the sine and cosine functions, ultimately establishing the identities and using them to problem-solve. Complex Algebra: The Unit Circle

## Sequences and Series

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentARITHMETIC & GEOMETRIC SEQUENCES [Printable PDF] CCSS.MATH.CONTENT.HSF.BF.A.1.A CCSS.MATH.CONTENT.HSF.BF.A.2 CCSS.MATH.CONTENT.HSF.LE.A.2 This activity is designed to deepen students’ understanding of arithmetic and geometric sequences, culminating in an examination of the number of sides, perimeter, and area of the Koch snowflake. Students begin by analyzing arithmetic sequences and writing rules for geometric sequences. Then, students compare geometric sequences and explore the Sierpinski triangle before diving into the Koch snowflake. Algebra Through Puzzles: Describing Sequences

## Conics

Coming Soon

## Probability and Statistics

Coming Soon