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 firstname.lastname@example.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 8th grade 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.
Printable PDFs Common Core Standards Description Related Course Content EXPLORATIONS IN NUMBER THEORY [Printable PDF] CCSS.MATH.PRACTICE.MP7 This introductory number theory set is designed to increase a student’s ability to reason and problem-solve logically. These problems build upon traditional number theory learned in middle school, including primes, composites, evens, odds, factors and multiples. Students apply what they know about relationships between integers to solve these problems as they culminate into the classic scenario of finding patterns among opening and closing doors. Number Theory: Factor Trees DIVISIBILITY RULES [Printable PDF] CCSS.MATH.PRACTICE.MP1 CCSS.MATH.PRACTICE.MP7 While familiarity with divisibility rules can provide time savings and short-cuts for problem solving, it also helps foster a deep understanding of our number system. In these problems, students review divisibility rules, and then apply them to number puzzles. In the culmination, students create a six-digit number that meets a set of criteria. Math Fundamentals: Divisibility CURIOUS CRYPTOGRAMS [Printable PDF] CCSS.MATH.PRACTICE.MP7 Cryptograms are puzzles where capital letters stand in for one or more digits of a number. In these cryptogram puzzles, students combine their knowledge of number patterns and divisibility rules with strategic problem-solving skills. The problems begin with easier puzzles involving divisibility rules such as 4 and 9 before moving onto more complex rules and combinations of several of the rules at once. The culmination cryptogram can be approached in many different ways. Math Fundamentals: Cryptogram Variety Pack PLAYING WITH PATTERNS [Printable PDF] CCSS.MATH.PRACTICE.MP2 CCSS.MATH.PRACTICE.MP8 Finding patterns is at the heart of mathematics. While sometimes these patterns can lead us astray, the ability to recognize and extend patterns is extremely important. In this activity, students bring their intuition about number patterns to the next level by identifying patterns and making predictions with non-linear patterns. In the culmination, students compare and contrast two visually similar patterns. Mathematical Fundamentals: Seeing Patterns
Printable PDFs Common Core Standards Description Related Course Content PERCENTS MEET VARIABLES [Printable PDF] CCSS.MATH.CONTENT.8.EE.B.5 CCSS.MATH.CONTENT.8.EE.C.7 CCSS.MATH.PRACTICE.MP2 It’s time to move beyond typical percent calculations and into the world of percent intuition. In these percent problems, students move beyond simple calculations to complex problems that require applying algebraic reasoning and developing intuition about percents and percent change. The culmination question pushes students to reason through percent change as it pertains to halving and doubling times. Another layer of complexity can be added to this exploration by asking that students prove all answers with variables. Algebra Through Puzzles: Mixing Problems PRICKLY PROPORTIONS [Printable PDF] CCSS.MATH.CONTENT.8.EE.B.5 CCSS.MATH.CONTENT.8.EE.C.7 CCSS.MATH.PRACTICE.MP1 Proportional reasoning is one of the essential building blocks for algebraic thinking and higher-level problem solving. These proportion problems encourage students to think intuitively about proportions and how quantities change in relation to one another. The dairy cow culmination question wraps it all up with one variable-filled proportional reasoning challenge. Algebra Through Puzzles: Proportionality RATIOS MEET ALGEBRA [Printable PDF] CCSS.MATH.CONTENT.8.EE.C.7 CCSS.MATH.PRACTICE.MP1 CCSS.MATH.PRACTICE.MP7 Not just your standard ratio problems, these questions challenge students to extend their understanding of ratios and proportions. Students begin with geometric scenarios and problems that are easily represented pictorially before moving on to mixing problems. In the culmination, students attempt to make sense of fractions and percentages involving multi-part ratios. Throughout this activity, students should be encouraged to apply algebraic strategies to make problem solving easier and more efficient. Algebra Through Puzzles: Mixing Problems
Printable PDFs Common Core Standards Description Related Course Content ARITHMETIC PUZZLES [Printable PDF] CCSS.MATH.PRACTICE.MP1 CCSS.MATH.PRACTICE.MP2 These arithmetic puzzles, ranging from magic squares to magic triangles, are designed to help broaden a student’s problem-solving skills. These methods of strategic thinking lay a strong foundation for algebra. The problems introduce arithmetic puzzles before moving into strategy development. In the culmination problem, students are asked to determine the sum of the three corner values in a magic triangle. Algebra Through Puzzles: Magic Rectangles 1 ARITHMETIC MEETS ALGEBRA [Printable PDF] CCSS.MATH.CONTENT.8.F.B.4 CCSS.MATH.PRACTICE.MP8 Students bring their intuition about number patterns to the next level in this series of problems by identifying patterns, making predictions, and generalizing with variables. Students begin by examining the sum of consecutive odd integers. Then, students identify patterns in sums of consecutive integers and write rules for the patterns they see. Lastly, students dive into ways to generalize patterns they see in pentagonal and hexagonal numbers. Algebra Through Puzzles: The Gauss Trick EXPRESSIONS AND IDENTITIES [Printable PDF] CCSS.MATH.CONTENT.8.EE.C.7 CCSS.MATH.PRACTICE.MP7 Manipulation of algebraic expressions is one of the most basic and important skills in a problem solver's repertoire. Without it, a problem solver would frequently be hopelessly stuck. In this series of problems, students have the opportunity to simplify, manipulate, and compare a wide variety of algebraic expressions. In the more challenging questions, students use variables to prove why numerical patterns that they see will always be true. Mathematical Fundamentals: Always-Sometimes-Never ALGEBRA PUZZLES [Printable PDF] CCSS.MATH.CONTENT.8.EE.C.7 CCSS.MATH.PRACTICE.MP7 Variable equations are one of the most generally applicable problem-solving tools in all of mathematics. In this series of problems, students explore how algebra's symbolic language can be experimented with and manipulated as a problem-solving tool. Students should be encouraged to solve the initial problems using strategic thinking and algebra rather than guess-and-check. The challenge and culmination questions are solved most efficiently and elegantly through the use of algebra. Mathematical Fundamentals: Using Variables
Printable PDFs Common Core Standards Description Related Course Content TILING AND TESSELLATIONS [Printable PDF] CCSS.MATH.CONTENT.8.G.A.1 CCSS.MATH.PRACTICE.MP1 CCSS.MATH.PRACTICE.MP2 This problem set aims to improve students’ understanding of congruence and visual patterns as it explores tiling and tessellations in the infinite plane. Students explore how to tile both regular and irregular 2D shapes. Then, students analyze tiling possibilities with varieties of triangles and pentagons. Finally, students develop a procedure for how to tile any quadrilateral. Outside of the Box Geometry: Regular Tessellations REP-TILES AND SELF-SIMILARITY [Printable PDF] CCSS.MATH.CONTENT.8.G.A.4 CCSS.MATH.CONTENT.8.G.B.7 In this activity, students explore a special kind of tiling of the plane that involves similar figures and scale factors. Rep‑tiles are geometric figures in which 𝑛 copies can be tiled to form a larger, similar figure. Students begin by exploring simple reptiles before examining more complex shapes. The activity culminates with students drawing and then examining characteristics of the right triangle reptile. Outside of the Box Geometry: Reptiles
Printable PDFs Common Core Standards Description Related Course Content VOLUME [Printable PDF] CCSS.MATH.CONTENT.8.G.C.9 This activity explores the volumes of cylinders, cones, spheres, and how the three quantities are related. Students examine how changing the radius or height of these figures impacts volume. They compare relative volumes between the figures. And lastly, in the culmination, students dive into a problem in which they determine volumes of water within nested cylindrical glasses. Mathematical Fundamentals: Volume
Printable PDFs Common Core Standards Description Related Course Content APPLYING THE PYTHAGOREAN THEOREM [Printable PDF] CCSS.MATH.CONTENT.8.G.B.7 These problems deepen students’ understanding of the Pythagorean theorem by presenting a diverse handful of its many applications. The problems begin with determining unknown side lengths in right triangles. Then, students explore various types of right triangle spirals. In the culmination, students use the Pythagorean theorem to compare the areas of figures inscribed in semicircles. Mathematical Fundamentals: The Pythagorean Theorem