# Algebra One

**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 the algebra 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.

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## Exploring Patterns

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentDESCRIBING PATTERNS [Printable PDF] CCSS.MATH.CONTENT.HSA.SSE.A.1 CCSS.MATH.CONTENT.HSA.CED.A.1 CCSS.MATH.CONTENT.HSA.REI.B.3 This set of problems dives deeply into analyzing patterns and making predictions. The problems require students to interpret numerical and visual patterns, use a variety of strategies for describing the patterns, and write algebraic rules. Students begin with linear patterns before moving to more challenging ones. In the culmination, students use variables to describe the number of colors needed to complete a complex tile mosaic. Algebra Through Puzzles: Describing Sequences FIND THE FORMULA [Printable PDF] CCSS.MATH.CONTENT.HSA.SSE.A.1 CCSS.MATH.CONTENT.HSA.SSE.B.3 CCSS.MATH.CONTENT.HSA.CED.A.1 CCSS.MATH.CONTENT.HSA.REI.B.3 This examination of patterns begins with students matching algebraic descriptions with visual patterns. Then, students move on to writing formulas for numerical patterns. Lastly, students move through a series of problems that help them to derive the algebraic formula for the sum of a set of consecutive integers. Algebra Through Puzzles: Describing Sequences

## Algebraic Reasoning

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentALWAYS-SOMETIMES-NEVER [Printable PDF] CCSS.MATH.CONTENT.HSA.SSE.A.2 CCSS.MATH.CONTENT.HSA.SSE.B.3 In this activity, students explore the importance of considering all possible scenarios when determining whether a mathematical statement is true or false. From simple to challenging examples, students determine if the algebraic equations and inequalities are always, sometimes, or never true. In every problem, students are encouraged to justify their reasoning. Mathematical Fundamentals: Always-Sometimes-Never ALGEBRAIC ARITHMETIC [Printable PDF] CCSS.MATH.CONTENT.HSA.SSE.A.2 CCSS.MATH.CONTENT.HSA.SSE.B.3 These problems begin by asking students to use arithmetic properties to write, simplify, evaluate and compare algebraic expressions. Then, students move on to analyzing someone else’s work for mistakes and factoring monomials from polynomials. While most problems could be solved with some long and drawn out arithmetic, they are solved far more elegantly and efficiently with the properties. Mathematical Fundamentals: Using Variables ALGEBRA RIDDLES [Printable PDF] CCSS.MATH.CONTENT.HSA.CED.A.1 CCSS.MATH.CONTENT.HSA.REI.A.1 CCSS.MATH.CONTENT.HSA.REI.B.3 Variable equations are one of the most generally applicable problem-solving tools in all of mathematics. In this series of problems, students explore ways in which algebra's symbolic language can be manipulated as a problem-solving tool. While the early problems can be solved without algebra, later problems are solved much more easily and elegantly with algebra, and the culmination question requires an algebraic proof. Mathematical Fundamentals: Using Variables MAGIC PERIMETERS [Printable PDF] CCSS.MATH.CONTENT.HSA.SSE.A.1 CCSS.MATH.PRACTICE.MP8 CCSS.MATH.PRACTICE.MP7 This deep dive into magic number puzzles allows students to put their logic and reasoning skills to the test. It is possible to solve these puzzles by brute force, but especially as they get more complex, it's better to use algebra. In particular, it's useful to set up equations relating the sum of the corner numbers to the magic sum to determine which magic sums are and are not possible. This will allow students to provethat certain results are true rather than exhaustively searching through all possible arrangements.Algebra Through Puzzles: Magic Perimeters CALCDOKU [Printable PDF] CCSS.MATH.PRACTICE.MP8 Calcdoku puzzles involve logical thinking about arithmetic relationships between numbers. These puzzles are simultaneously a review of basic arithmetic and a preview of algebra — using given information to determine unknown quantities. These problems first walk students through the basic logic and strategy of solving Calcdoku puzzles and then give them the opportunity to solve these puzzles from scratch. Algebra Through Puzzles: Learn Calcdoku CAN IT BE SOLVED? [Printable PDF] CCSS.MATH.CONTENT.HSA.SSE.A.1 CCSS.MATH.CONTENT.HSA.SSE.B.3 CCSS.MATH.CONTENT.HSA.CED.A.1 CCSS.MATH.CONTENT.HSA.REI.B.3 Sometimes 𝑥 is unknown, and sometimes it’s unknowable. In these algebra puzzles involving combination locks, students must determine which is which. When do they have enough information to figure out the combination, when do they need more information, and when is the information they have logically inconsistent? These problems serve as a gentle introduction to systems of equations. Algebra Through Puzzles: Combination Locks

## Systems of Equations

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentARRAY PUZZLES [Printable PDF] CCSS.MATH.CONTENT.HSA.REI.C.5 CCSS.MATH.CONTENT.HSA.REI.C.6 CCSS.MATH.CONTENT.HSA.CED.A.2 In these array puzzles, logic meets algebra. Students have the opportunity to solve the visual puzzles, or determine if a solution is even possible, by applying logic and writing and solving systems of equations. The culmination question asks students to determine the maximum possible number of objects that can be used for a given array size. Algebra Through Puzzles: Array Puzzles MAKE IT BALANCE [Printable PDF] CCSS.MATH.CONTENT.HSA.REI.C.5 CCSS.MATH.CONTENT.HSA.REI.C.6 CCSS.MATH.CONTENT.HSA.CED.A.2 In these balance puzzles, logic meets algebra. Students work through systems of equations that are presented pictorially using balances. The problems require a variety of problem-solving strategies as students explore pictorial ways to represent elimination, substitution, and solving with fractions. In the culmination, students work through a complex, multi-level balance challenge. Algebra Through Puzzles: Balancing Scales SYSTEMS PROBLEM SOLVING [Printable PDF] CCSS.MATH.CONTENT.HSA.REI.C.5 CCSS.MATH.CONTENT.HSA.REI.C.6 CCSS.MATH.CONTENT.HSA.CED.A.2 In these problems students apply their understanding of what systems of equations are and what their solutions, or lack thereof, represent. Rather than simply solving the systems, students answer conceptual questions about them. After examining several cases of systems with two variables, students apply their intuition to multi-variable systems, including the culmination question that involves five variables. Algebra Through Puzzles: Problem Solving

## Modeling with Functions

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentRATES OF CHANGE & SLOPE [Printable PDF] CCSS.MATH.CONTENT.HSF.IF.B.4 CCSS.MATH.CONTENT.HSF.IF.B.6 CCSS.MATH.CONTENT.HSF.LE.B.5 In this activity, students dive into an exploration of slope. Students begin by exploring some conceptual properties of slope, and then move on to compare rates and slopes. Finally, students explore an application of average rate of change in a non-linear relationship and build intuition around how average rate of change relates to instantaneous rate of change. Mathematical Fundamentals: Slope EQUATIONS OF LINES [Printable PDF] CCSS.MATH.CONTENT.HSA.CED.A.2 CCSS.MATH.CONTENT.HSA.REI.D.10 CCSS.MATH.CONTENT.HSF.LE.A.2 This problem set aims to deepen students’ understanding of equations of lines, slopes, intercepts, and how they’re all connected. Students begin by answering conceptual questions about line characteristics. Then, students explore reflected lines, parallel lines, and perpendicular lines. Finally, students examine how the Kelvin, Celsius, and Fahrenheit temperature scales relate to one another. Mathematical Fundamentals: Slope

## Quadratics

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentZERO PRODUCT PROPERTY [Printable PDF] CCSS.MATH.CONTENT.HSA.APR.B.3 CCSS.MATH.CONTENT.HSA.SSE.B.3 This activity is designed to deepen students’ understanding of the zero product property. Students begin by exploring various basic scenarios where the property comes into play. Then, students extend their intuition as they reason through how the property relates to graphs of parabolas. In the culmination, students extend their understanding of the zero product property by applying a bit of logic to solve a quadratic inequality. Algebra Through Puzzles: Difference of Squares FACTORING QUADRATICS [Printable PDF] CCSS.MATH.CONTENT.HSA.SSE.A.2 CCSS.MATH.CONTENT.HSA.SSE.B.3.A This activity aims to broaden students’ experience manipulating and factoring quadratics. After exploring variations on common factoring scenarios, students dive into applications of factorization of quadratics. In the culmination, students use a dice scenario to determine the possible number of quadratics that are factorable into binomials using integers. Through this activity, they see the rarity of factoring with integers. Algebra Through Puzzles: Difference of Squares

## Exponents, Radicals, and Exponential Models

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentMANIPULATING EXPONENTS [Printable PDF] CCSS.MATH.CONTENT.HSA.SSE.A.2 CCSS.MATH.CONTENT.HSA.SSE.B.3 This activity aims to deepen students’ understanding of exponents and how exponential expressions can be manipulated. In this problem set, the manipulations focus on common bases and/or negative exponents. Over the course of this activity, students evaluate, compare, analyze, and order exponential expressions. In the culmination, students determine how to order digits within an exponential expression to produce the largest and smallest possible values. Algebra Through Puzzles: Same or Similar Bases

## Inequalities

Coming Soon

## Proportions and Rational Equations

Coming Soon

## Statistics and Probability

Coming Soon