Illustrative Mathematics, Grade 8

In this Grade 8 Illustrative Mathematics lesson, students explore how figures move in a plane by learning the three basic rigid transformations: translation (sliding a figure without turning it), rotation (turning a figure around a center point clockwise or counterclockwise), and reflection (placing a figure on the opposite side of a reflection line). Through activities like describing dance-like movements of shapes and drawing sequential positions, students build precise vocabulary for these transformations, including key terms such as image, corresponding points, and vertex. This lesson opens Chapter 1: Rigid Transformations and Congruence and lays the foundation for understanding how figures can change position while preserving their shape and size.

In this Grade 8 lesson from Illustrative Mathematics Chapter 1, students explore the properties of rigid transformations — translations, rotations, and reflections — and discover that these moves preserve side lengths and angle measures. By translating polygons, rotating triangles, and reflecting pentagons, students verify that corresponding sides and corresponding angles in the original figure and its image always have equal measurements. This hands-on investigation builds the foundation for understanding congruence through the concept that rigid transformations produce figures identical in size and shape.

In this Grade 8 lesson from Illustrative Mathematics Chapter 1, students learn the formal definition of congruence — that two figures are congruent if one can be mapped exactly onto the other using a sequence of rigid transformations such as translations, rotations, and reflections. Students practice identifying congruent shapes, comparing rectangles by area and perimeter, and using geometry tools to determine which figures are truly congruent. The lesson connects the new vocabulary term "congruent" to the rigid transformation concepts students have already been exploring in the unit.

In this Grade 8 lesson from Illustrative Mathematics, students learn how alternate interior angles are formed when a transversal crosses two parallel lines and why these angle pairs are always congruent. Using 180° rotational symmetry around the midpoint between two intersections, students build a logical framework connecting vertical angles, supplementary angles, and alternate interior angles to determine all eight angle measures created by a transversal from just one known angle. The lesson is part of Chapter 1: Rigid Transformations and Congruence, reinforcing how geometric transformations explain angle relationships.

In this Grade 8 Illustrative Mathematics lesson, students explore the concept of scaled copies and scale factors by analyzing rectangles created from an 8½ by 11-inch sheet of paper. Students identify which rectangles are scaled copies by comparing side length ratios and discovering that aligned scaled rectangles share a common diagonal. This lesson builds the foundation for understanding dilations and similarity in Chapter 2.

In this Grade 8 Illustrative Mathematics lesson from Chapter 2, students learn that two figures are similar if one can be mapped onto the other using a sequence of translations, rotations, reflections, and dilations. Students practice identifying and applying these similarity transformations to triangles and polygons, including choosing appropriate scale factors and centers of dilation. The lesson builds a precise, transformation-based definition of similarity that prepares students for deeper work with slope and proportional reasoning.

In this Grade 8 lesson from Illustrative Mathematics Chapter 2, students learn what slope means by using slope triangles — right triangles whose hypotenuse lies on a line — and discover that the slope equals the vertical side length divided by the horizontal side length. Students explore why all slope triangles on the same line produce the same quotient by connecting the concept to similar triangles. They practice calculating and comparing slopes such as 2/3, 1/2, and 3/2 across multiple lines and coordinate grid activities.

In this Grade 8 Illustrative Mathematics lesson, students explore proportional relationships by interpreting and comparing graphs of constant-speed motion, identifying the constant of proportionality, and writing equations for lines through the origin. Students practice scaling and labeling coordinate axes to extract meaningful information, such as comparing speeds and predicting distances over time. The lesson builds foundational skills for understanding linear relationships in Chapter 3.

In this Grade 8 lesson from Illustrative Mathematics Chapter 3, students learn to identify and represent linear relationships by connecting slope to rate of change. Using real-world contexts like stacking cups and babysitting earnings, students discover that the rate of change in a linear relationship is constant and equal to the slope of its graph. The lesson also distinguishes linear relationships from proportional relationships, noting that not all linear graphs pass through the origin.

In this Grade 8 Illustrative Mathematics lesson from Chapter 3: Linear Relationships, students learn how to calculate the slope of a line using two coordinate points by finding the change in y-values divided by the change in x-values. The lesson reinforces why the order of subtraction must stay consistent across both coordinates to produce the correct sign for the slope. Students also connect the coordinate formula to slope triangles and practice applying it to lines with negative slopes.

In this Grade 8 Illustrative Mathematics lesson, students learn what it means for an ordered pair (x, y) to be a solution to a linear equation with two variables by substituting values and determining whether both sides of the equation are equal. Using real-world contexts like buying apples and oranges, students write two-variable equations, identify multiple solutions, and plot them on a coordinate plane. They discover that all solutions to a linear equation form a straight line, while non-solutions fall off the line.

In this Grade 8 lesson from Illustrative Mathematics Chapter 4, students solve multi-step number puzzles by working backward and representing problems using verbal reasoning, diagrams, and linear equations such as 2x + 4 = 18. Students practice translating word problems into algebraic equations and compare solution strategies with partners. The lesson builds foundational skills for writing and solving increasingly complex linear equations throughout the chapter.

In this Grade 8 Illustrative Mathematics lesson from Chapter 4, students use balanced hanger diagrams as a visual model for linear equations in one variable, learning that adding or subtracting the same amount from both sides maintains equality. Students write and solve equations such as a + 2b = 5b by reasoning about unknown weights assigned to shapes like squares, triangles, and circles. The lesson builds foundational understanding of the properties of equality that underpin solving linear equations.

In this Grade 8 Illustrative Mathematics lesson, students are introduced to systems of linear equations and learn to identify a solution as an ordered pair that satisfies both equations simultaneously. Students explore what it means for a point to lie on one line, both lines, or neither, building the conceptual foundation for solving systems. This lesson is part of Chapter 4: Linear Equations and Linear Systems.

In this Grade 8 Illustrative Mathematics lesson from Chapter 5, students explore input-output rules and learn the definition of a function as a rule that assigns exactly one output to each allowable input. Through activities like completing tables and identifying "black box" rules, students practice applying operations such as addition, multiplication, and division to determine outputs from given inputs. The lesson also addresses why certain inputs are not allowable, using division by zero as a key example of an undefined output.

In this Grade 8 lesson from Illustrative Mathematics Chapter 5, students learn how to represent functions using equations and function diagrams, writing rules that express the output as a function of the input. Students practice identifying independent and dependent variables across real-world contexts such as circle circumference, distance-rate-time, and coin value problems. They also explore how a two-variable equation like 0.1d + 0.25q = 12.5 can be rearranged to express either variable as a function of the other.

In this Grade 8 Illustrative Mathematics lesson, students explore linear functions by identifying rates of change and initial values in equations of the form y = mx + b, where m represents the rate of change and b represents the starting value. Students work with real-world contexts — including filling and draining water tanks, savings accounts, and unit conversions — to determine whether a linear function is increasing or decreasing based on the sign of the slope. They also practice comparing two linear functions represented in different forms, such as an equation and a graph, to draw conclusions about rates of change and initial values.

In this Grade 8 Illustrative Mathematics lesson from Chapter 5: Functions and Volume, students explore how the height of water in a cylinder changes as a function of volume, examining how radius affects the slope of the height-versus-volume graph. Students use data tables and graphs to analyze and interpret linear relationships between volume and height for cylinders with different dimensions. The lesson builds understanding of how container shape determines the behavior of height-volume functions by comparing cylinders with the same height but different radii.

In this Grade 8 Illustrative Mathematics lesson, students explore how changing a single dimension — such as the height of a cylinder or the edge length of a rectangular prism — affects volume through proportional relationships. Students write and graph equations relating volume to one variable for prisms, cylinders, and cones, then analyze how doubling or halving that dimension scales the volume by the same factor. The lesson reinforces that these volume relationships are linear functions and proportional relationships.

In this Grade 8 Illustrative Mathematics lesson from Chapter 6, students explore how organizing bivariate data into ordered tables and scatter plots makes patterns and associations easier to identify. Using real measurements such as triangle side lengths and perimeters, students practice sorting data, estimating unknown values based on observed trends, and matching data tables to their corresponding scatter plots by labeling axes.

In this Grade 8 Illustrative Mathematics lesson from Chapter 6, students explore associations in numerical data by analyzing scatter plots and linear models to predict one variable from another, such as diamond price from weight or foot width from foot length. Students practice evaluating how well a linear equation fits real data by comparing predicted values to actual values and identifying cases where the model overpredicts or underpredicts. The lesson also introduces the concept of outliers as data points that fall far from the overall trend in a scatter plot.

In this Grade 8 Illustrative Mathematics lesson, students explore associations in categorical data by interpreting two-way tables of frequencies and relative frequencies, and matching them to corresponding bar graphs and segmented bar graphs. Students practice calculating relative frequencies, labeling graphical representations, and using these tools to determine whether an association exists between two categorical variables, such as age and cell phone use. The lesson builds foundational data literacy skills aligned with Chapter 6: Associations in Data.

In Grade 8 Illustrative Mathematics, Lesson 7.1 introduces students to exponents as a notation for repeated multiplication, exploring concepts such as exponential expressions, base, and exponent through doubling and halving patterns. Students write and evaluate expressions like 2 to the 28th power and one-half to the 6th power, connecting exponent notation to real-world growth and decay scenarios. This lesson builds the foundational skills needed for the chapter's focus on scientific notation and properties of exponents.

In this Grade 8 Illustrative Mathematics lesson from Chapter 7: Exponents and Scientific Notation, students discover and apply the product of powers rule, learning that multiplying powers of 10 is equivalent to adding their exponents (for example, 10⁵ · 10² = 10⁷). Students build this understanding by expanding expressions in a table, then generalize the rule as 10ⁿ · 10ᵐ = 10⁽ⁿ⁺ᵐ⁾. The lesson also uses the product rule to explore the meaning of the zero exponent, establishing why 10⁰ equals 1.

In this Grade 8 lesson from Illustrative Mathematics Chapter 7, students learn to represent very small numbers expressed as multiples of negative powers of 10 on a number line, building foundational skills in scientific notation. Using real-world measurements such as the radius of an electron and the mass of a proton, students practice converting decimals like 0.0000000000003 cm into the form a × 10ⁿ and plotting them accurately on scaled number lines. The lesson also develops students' ability to compare and estimate the relative size of quantities written in scientific notation.

In this Grade 8 lesson from Illustrative Mathematics Chapter 8, students explore the relationship between the side lengths and areas of squares, including rotated squares on a grid. They practice finding areas by decomposing figures into simpler shapes — subtracting the areas of surrounding triangles from a bounding square — and work backward from a given area to determine side length. The lesson builds foundational skills for the Pythagorean Theorem by helping students recognize that some square areas, like 73 square units, fall between perfect squares and have side lengths that are not whole numbers.

In this Grade 8 lesson from Illustrative Mathematics, students explore the Pythagorean Theorem by comparing side lengths of right and non-right triangles to discover that a² + b² = c² holds specifically for right triangles. Students practice identifying the hypotenuse and legs, then apply the theorem to find missing side lengths in various triangles on a coordinate grid. The lesson builds conceptual understanding of why the relationship between the squares of the legs and hypotenuse is a defining property of right triangles.

In this Grade 8 Illustrative Mathematics lesson from Chapter 8: Pythagorean Theorem and Irrational Numbers, students explore the relationship between the side lengths and volumes of cubes, building understanding of how to find an unknown side length when the volume is known. The lesson lays the groundwork for cube roots and irrational numbers by asking students to reason about perfect and non-perfect cube volumes. This connects directly to the chapter's broader focus on number types and geometric measurement.

In this Grade 8 lesson from Illustrative Mathematics Chapter 8, students explore how rational numbers can be expressed as both fractions and decimals, including finite decimal expansions and repeating decimals. Using long division and zoomed number lines, students convert fractions such as 2/11 and 3/8 into their decimal representations and learn to use bar notation to indicate repeating digits. The lesson builds understanding of how decimal expansions can be used to accurately locate rational numbers on a number line.