Reveal Math, Course 3

In this Grade 8 lesson from Reveal Math, Course 3, students learn how to use integer exponents to express repeated multiplication of rational numbers as powers, identifying the base and exponent in expressions involving integers, fractions, and negative numbers. The lesson covers writing both numerical and algebraic products in exponential form, including how parentheses affect the value of expressions with negative bases such as the difference between (-3)⁴ and -3⁴. Students also practice evaluating powers using the order of operations.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 1: Exponents and Scientific Notation), students learn how to multiply and divide monomials using the Product of Powers and Quotient of Powers properties. They practice adding exponents when multiplying powers with the same base and subtracting exponents when dividing, applying these Laws of Exponents to both numerical and algebraic expressions.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 1), students learn to simplify expressions involving monomials raised to powers by applying the Power of a Power Property and the Power of a Product Property. They practice multiplying exponents when raising a power to another power, such as simplifying (k⁷)⁵ to k³⁵, and extending this to multi-factor expressions like (−2m⁷n⁶)⁵. The lesson builds foundational skills with integer exponents needed for working with scientific notation throughout the module.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 1: Exponents and Scientific Notation), students learn to apply the Zero Exponent Rule and the definition of negative exponents to simplify expressions involving zero and negative integer exponents. They practice converting between negative exponents and fractions using the rule x⁻ⁿ = 1/xⁿ, and apply the Product of Powers and Quotient of Powers properties to simplify expressions with negative exponents. The lesson also connects these concepts to real-world measurement by expressing small decimals as powers of 10.

In this Grade 8 lesson from Reveal Math, Course 3, students learn to write very large and very small numbers using scientific notation, expressing them in the form a × 10ⁿ where 1 ≤ a < 10. The lesson covers converting between scientific notation and standard form by moving the decimal point based on the sign and value of the exponent. Students also learn to interpret calculator E-notation, such as recognizing that 4E-7 represents 4 × 10⁻⁷.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 1: Exponents and Scientific Notation), students learn how to multiply, divide, add, and subtract numbers written in scientific notation. The lesson applies the Product of Powers Property, Quotient of Powers Property, and Distributive Property to solve real-world problems involving very large and very small numbers. Students also practice converting results into proper scientific notation form.

In this Grade 8 lesson from Reveal Math Course 3, Module 2, students learn to identify and classify terminating and repeating decimals as forms of rational numbers, using bar notation to represent repeating digits. Students practice converting fractions and mixed numbers to their decimal forms and determining whether each decimal terminates or repeats. The lesson also introduces the relationship between natural numbers, whole numbers, integers, and rational numbers within the broader real number system.

In this Grade 8 lesson from Reveal Math, Course 3, students learn how to find square roots and cube roots, including principal, negative, and both square roots of whole numbers, decimals, and fractions. Students explore key vocabulary such as perfect squares, perfect cubes, radical signs, and inverse operations, while discovering why negative numbers have no rational square root. The lesson also guides students through using square and cube roots as inverse operations to solve equations of the form x² = p and x³ = p.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 2: Real Numbers), students learn to identify irrational numbers and classify real numbers by determining which sets — natural numbers, whole numbers, integers, rational numbers, or irrational numbers — a given number belongs to. Students practice distinguishing between rational numbers, which have terminating or repeating decimal expansions, and irrational numbers like pi and non-perfect-square roots, which have non-terminating, non-repeating decimals. The lesson uses a Venn diagram and worked examples to build fluency with classifying numbers such as repeating decimals, square roots, and negative integers within the real number system.

In this Grade 8 lesson from Reveal Math, Course 3, Module 2: Real Numbers, students learn how to estimate irrational numbers — including square roots and cube roots of non-perfect squares and non-perfect cubes — by locating them between consecutive integers or tenths on a number line. Students practice squaring interval endpoints to narrow down approximations and use truncating to express decimal expansions more precisely. The lesson builds fluency with inequality notation and the approximation symbol as tools for working with irrational values like √83 and ∛320.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 2: Real Numbers), students learn to compare and order real numbers — including irrational numbers like square roots and pi — by using rational approximations and decimal expansions. Students practice applying comparison symbols and plotting values such as fractions, mixed numbers, square roots, and negative irrational numbers on a number line to determine their relative order.

In this Grade 8 lesson from Reveal Math, Course 3, students learn to solve equations with variables on each side by applying the properties of equality — including the Addition, Subtraction, Division, and Multiplication Properties — to isolate the variable. The lesson also covers equations with rational coefficients, teaching students two methods: solving directly with fractions or multiplying by the LCD to eliminate fractions first. Students practice verifying solutions by substituting values back into the original equation.

In this Grade 8 lesson from Reveal Math, Course 3, students learn to write and solve linear equations in one variable with rational coefficients where variables appear on each side of the equation. Using real-world contexts such as comparing gym membership costs and car rental rates, students apply the properties of equality — including the Subtraction Property and Division Property — to isolate the variable and find the solution. The lesson builds fluency in translating word problems into algebraic equations of the form ax + b = cx + d and verifying solutions by substitution.

In this Grade 8 lesson from Reveal Math, Course 3, students learn to solve multi-step linear equations with rational coefficients by applying the Distributive Property to expand expressions with grouping symbols, combining like terms, and using the Properties of Equality. The lesson covers equations involving integers, decimals, and fractions across three worked examples, with students verifying solutions by substituting back into the original equation.

In this Grade 8 lesson from Reveal Math, Course 3, Module 3, students learn to write and solve multi-step linear equations with rational coefficients by applying the Distributive Property and combining like terms. Using real-world contexts such as geometry problems and field trip costs, students translate word problems into equations with variables on both sides and solve them step by step. The lesson builds key algebraic skills for setting up and solving equations like 4(t + 9.50) = 8 + 5(t + 3) and verifying solutions.

In this Grade 8 lesson from Reveal Math, Course 3, students learn how to determine whether a linear equation in one variable has one solution, no solution, or infinitely many solutions by simplifying both sides and comparing coefficients and constants. The lesson covers applying the Distributive Property and combining like terms to recognize when a simplified equation yields a contradiction like 32 = 12 (no solution) or an identity like -8 = -8 (infinitely many solutions). Students also practice constructing equations that produce a specific number of solutions by analyzing the relationship between coefficients and constants on each side.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 4), students learn to graph and compare proportional relationships using tables, equations, and coordinate graphs while identifying the constant rate of change and slope. The lesson connects unit rate to slope by showing that in any proportional linear relationship, the slope equals the unit rate, calculated as the change in one quantity divided by the change in another between any two points on the line. Students practice interpreting slope in real-world contexts, such as weekly savings and unit conversions, building foundational skills in linear relationships ahead of broader work with linear equations.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 4), students learn how to identify and calculate the slope of a line as the ratio of rise to run, interpreting it as a constant rate of change. The lesson covers finding slope from a graph, a table, and real-world contexts, with examples involving positive and negative slopes. Students practice using the formula slope = rise/run to analyze linear relationships between quantities such as cost, water loss, and bank account balances.

In this Grade 8 lesson from Reveal Math, Course 3, students learn how to use slope triangles and the properties of similar figures to prove that the slope of a line is constant between any two points. By identifying similar right triangles along a line and showing that the ratios of their corresponding sides — rise over run — are proportional, students connect geometric reasoning to the concept of slope. The lesson builds key vocabulary including similar figures, corresponding parts, and slope triangles within Module 4's focus on linear relationships.

In this Grade 8 lesson from Reveal Math, Course 3, Module 4, students learn how to derive the direct variation equation y = mx from the slope formula and identify the constant of variation as the slope, unit rate, and constant of proportionality. Students practice writing direct variation equations from graphs, words, and tables to model proportional linear relationships that pass through the origin. Real-world contexts such as gymnastics lesson costs and the speed of sound help students interpret the meaning of the constant of variation in practical situations.

In this Grade 8 lesson from Reveal Math, Course 3, students learn to identify and apply slope-intercept form (y = mx + b) by recognizing slope and y-intercept in linear equations. The lesson covers how to derive y = mx + b from the slope formula and write equations from graphs, tables, and verbal descriptions of real-world situations. Students also explore how the y-intercept represents an initial value and the slope represents a rate of change in nonproportional linear relationships.

In this Grade 8 lesson from Reveal Math, Course 3, students learn to graph linear equations by interpreting the slope and y-intercept from slope-intercept form (y = mx + b). The lesson covers plotting lines using the y-intercept as a starting point and applying the slope to locate a second point, as well as graphing special cases including horizontal lines (y = b, with slope 0) and vertical lines (x = a, with undefined slope). Real-world contexts, such as gecko growth and novel writing, help students connect the rate of change and initial value to their graphs.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 5), students learn to determine whether a relation is a function by checking whether each input is assigned exactly one output. Using mapping diagrams, tables, and the vertical line test, they practice identifying functions and distinguishing them from non-functions across multiple representations. Key vocabulary includes relation, input, output, function, and vertical line test.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 5: Functions), students learn how to create function tables by substituting input values into function rules such as y = 4x − 1 and y = 770p to find corresponding outputs. They also explore how to select appropriate input values for real-world contexts and use ordered pairs from function tables to graph linear functions on a coordinate plane.

In this Grade 8 lesson from Reveal Math, Course 3, Module 5, students learn how to compare two functions represented in different forms — such as tables, graphs, equations, and verbal descriptions — by analyzing their rates of change and initial values. Students practice determining which function increases faster or has a greater output for a given input using real-world contexts like high-speed train distances and cell phone billing plans.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 5: Functions), students learn to identify nonlinear functions by analyzing rate of change, graph shape, and equation structure. They practice distinguishing linear from nonlinear functions using graphs, tables, and real-world examples such as the area of a square (A = s²) and the volume of a cube (s³). By the end of the lesson, students can explain why a function is nonlinear when its rate of change is not constant and its graph is not a straight line.

In this Grade 8 lesson from Reveal Math, Course 3, Module 5, students learn to recognize, analyze, and sketch qualitative graphs — graphs that show relationships between two quantities without specific numerical values on the axes. Students practice identifying rates of change, including increasing at a constant rate, decreasing at a varied rate, and no change, by interpreting real-world scenarios such as water levels and bouncing tennis balls. By the end of the lesson, students can translate verbal descriptions of situations into accurate qualitative graph sketches and determine whether those graphs are linear or nonlinear.

In this Grade 8 lesson from Reveal Math Course 3, Module 6, students learn how to solve systems of linear equations by graphing, identifying the point of intersection as the solution. The lesson covers writing equations in slope-intercept form, verifying solutions algebraically, and recognizing systems with one solution versus no solution when lines are parallel. Key vocabulary includes system of equations and solution, with practice using real coordinate plane examples.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 6), students learn how to determine whether a system of linear equations has no solution, one solution, or infinitely many solutions by comparing slopes and y-intercepts in slope-intercept form. Students analyze cases where lines are parallel, intersecting, or identical, and practice rewriting equations into slope-intercept form before making comparisons. The lesson builds fluency with systems of equations through worked examples and real-world coordinate geometry problems.

In this Grade 8 lesson from Reveal Math Course 3, Module 6, students learn how to solve systems of linear equations using the substitution method, including cases where one or both equations must first be rewritten in terms of a single variable. The lesson walks through the step-by-step process of replacing a variable with an equivalent algebraic expression, solving for the remaining variable, and checking solutions as ordered pairs. Students also explore systems that yield zero or infinitely many solutions through this algebraic approach.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 6), students learn how to solve systems of linear equations using elimination, an algebraic method that removes one variable by adding or subtracting equations with opposite or equal coefficients. Students practice both elimination by addition, when coefficients are opposites, and elimination by subtraction, when coefficients are the same, then substitute back to find the solution as an ordered pair. This lesson builds core algebra skills needed for working with systems of equations throughout Course 3.

In this Grade 8 lesson from Reveal Math, Course 3, Module 6, students learn how to translate real-world scenarios into systems of linear equations and solve them using graphing, substitution, or elimination. The lesson walks through identifying variables, writing algebraic equations from verbal models, and choosing the most efficient solution method for a given system. Students practice interpreting solutions in context, such as finding unknown costs or determining when two pricing plans are equal.

In this Grade 8 lesson from Reveal Math, Course 3, Module 7, students learn how to identify and apply angle relationships formed when a transversal intersects parallel lines, including alternate interior angles, alternate exterior angles, and corresponding angles. Students practice classifying angle pairs and using these relationships to find the measures of missing angles.

In this Grade 8 lesson from Reveal Math, Course 3, students learn how to find missing interior angle measures using the Triangle Angle Sum Theorem, which states that the three interior angles of any triangle always add up to 180 degrees. The lesson also introduces exterior angles and remote interior angles, teaching students how these angle relationships connect. Students practice applying these concepts through equations and ratios to solve for unknown angle measures in real-world and geometric contexts.

In this Grade 8 lesson from Reveal Math, Course 3, students learn to apply the Pythagorean Theorem (a² + b² = c²) to find the missing side lengths of right triangles, including both legs and the hypotenuse. The lesson covers key vocabulary such as legs and hypotenuse, and walks through real-world problems involving ladders, flagpole wires, and airplane paths using square roots to solve for unknown measurements. Students also extend the theorem to three-dimensional contexts, rounding irrational results to the nearest tenth.

In this Grade 8 lesson from Reveal Math, Course 3, students learn how to apply the converse of the Pythagorean Theorem to determine whether a triangle is a right triangle by testing whether its side lengths satisfy the equation a² + b² = c². Through real-world examples such as carpentry framing and rectangular patios, students practice substituting given side lengths into the equation and evaluating whether the relationship holds true. This lesson builds directly on the Pythagorean Theorem as part of Module 7: Triangles and the Pythagorean Theorem.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 7), students learn how to find the distance between two points on a coordinate plane by applying the Pythagorean Theorem. They practice forming a right triangle between two points, identifying the horizontal and vertical leg lengths from the coordinates, and solving for the hypotenuse using the equation a² + b² = c². The lesson also extends this skill to real-world map problems where students calculate and compare distances using a given unit scale.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 8: Transformations), students learn how to translate figures on the coordinate plane by sliding a preimage to a new position without turning it. Students practice graphing translated images of triangles and writing the coordinates of the image vertices. The lesson also introduces coordinate notation, expressed as (x, y) → (x + a, y + b), to describe horizontal and vertical translations algebraically.

In this Grade 8 lesson from Reveal Math Course 3, Module 8, students learn how to perform and describe rotations as a type of rigid transformation, including 90°, 180°, and 270° clockwise and counterclockwise rotations about a vertex or the origin. Students use coordinate notation rules such as (x, y) → (y, −x) for a 90° clockwise rotation about the origin to find the coordinates of a rotated image. The lesson also introduces key vocabulary including center of rotation and rotation, and reinforces that rotations preserve congruence between the preimage and image.

In this Grade 8 lesson from Reveal Math, Course 3, students learn how to perform and describe dilations on the coordinate plane, including how to use scale factor to determine whether a figure is enlarged, reduced, or unchanged. Students practice using coordinate notation in the form (x, y) → (kx, ky) to find image coordinates and graph dilations with the origin as the center of dilation. Key vocabulary covered includes dilation, scale factor, and center of dilation.

In this Grade 8 lesson from Reveal Math, Course 3, students learn how to apply properties of similar triangles to solve indirect measurement problems, including shadow reckoning and finding distances that cannot be measured directly. Students set up and solve proportions using corresponding sides of similar triangles to calculate unknown heights and lengths in real-world scenarios such as statues, trees, and distances across lakes or ravines. This lesson is part of Module 9: Congruence and Similarity and builds on students' understanding of similar polygons and proportional reasoning.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 10), students learn how to calculate the volume of a cylinder using the formula V = πr²h, applying it to problems where either the radius or diameter is given. Students practice expressing volume in terms of π as an exact answer and as a decimal approximation rounded to the nearest tenth. The lesson also extends to multi-step real-world problems, such as finding the weight of a cylindrical object by multiplying its calculated volume by a given unit weight.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 10: Volume), students learn how to calculate the volume of a cone using the formula V = ⅓πr²h, building on the relationship that a cone's volume is one-third that of a cylinder with the same base and height. Students practice applying the formula given a cone's radius or diameter and height, including real-world problems such as finding the volume of a cone-shaped paper cup. The lesson also challenges students to compare the volumes of cylindrical and conical containers to solve cost-based problems.

In this Grade 8 lesson from Reveal Math, Course 3, students learn to calculate the volume of spheres and hemispheres using the formulas V = 4/3πr³ and V = 2/3πr³. The lesson covers working with both radius and diameter as starting values, expressing answers in terms of π or as decimals, and applying the sphere volume formula to real-world problems such as inflating a volleyball or finding the volume of a spherical stone.

In this Grade 8 lesson from Reveal Math, Course 3 (Module 10), students learn how to use the volume formulas for cylinders, cones, and spheres to solve for missing dimensions such as radius or height when the volume and one other measurement are known. Using the Division Property of Equality and operations like taking square roots and cube roots, students isolate the unknown variable in each formula. The lesson reinforces algebraic reasoning in a geometric context through worked examples and real-world applications.

In this Grade 8 lesson from Reveal Math, Course 3, students learn to construct scatter plots using bivariate data by graphing ordered pairs on a coordinate plane with appropriately scaled axes. Students then practice interpreting scatter plots by identifying positive, negative, or no association and classifying patterns as linear or nonlinear. The lesson also introduces key vocabulary including cluster and outlier, helping students describe real-world data relationships with precision.

In this Grade 8 lesson from Reveal Math, Course 3, students learn how to write equations in slope-intercept form for lines of fit drawn on scatter plots, using the slope formula and y-intercept to model real-world data. Students practice interpreting the slope and y-intercept in context and use their equations to make conjectures about values not present in the original data set. The lesson builds on prior scatter plot work in Module 11 and emphasizes practical prediction skills with linear equations.

In this Grade 8 lesson from Reveal Math, Course 3, students learn how to construct and interpret two-way tables by organizing survey data into rows and columns representing two categories. They practice calculating joint and marginal frequencies, using row and column totals to find missing values through addition and subtraction. The lesson also introduces relative frequencies as a tool for analyzing relationships between categorical data sets.

In this Grade 8 lesson from Reveal Math, Course 3, students learn how to use row and column relative frequencies in two-way tables to determine whether an association exists between two categorical variables. By comparing relative frequencies across groups, such as age and news source or gender and sport preference, students can identify and interpret patterns of association. The lesson builds data analysis skills central to Module 11's focus on scatter plots and two-way tables.