Illustrative Mathematics, Grade 7

In this Grade 7 Illustrative Mathematics lesson from Chapter 1: Scale Drawings, students learn what a scaled copy is — a figure where every length from the original is multiplied by the same number. Through activities comparing portraits, drawings of the letter F, and polygon card matching, students identify scaled copies and examine how corresponding side lengths relate. This lesson builds foundational understanding of proportional scaling as preparation for deeper work with scale factors and scale drawings.

In this Grade 7 Illustrative Mathematics lesson from Chapter 1: Scale Drawings, students learn what a scale drawing is and how to interpret a scale to convert measurements between a drawing and real-world dimensions. Using a basketball court and silhouettes of famous tall structures, students practice measuring distances on scale drawings and calculating actual lengths using a given scale such as 1 centimeter representing 2 meters. The lesson establishes that every length in a scale drawing is reduced or enlarged by the same scale factor, making scale drawings accurate two-dimensional representations of real objects and places.

In this Grade 7 lesson from Illustrative Mathematics Chapter 1: Scale Drawings, students apply scale factors to solve real-world problems involving maps and floor plans. They practice calculating actual distances and dimensions from scaled representations using proportional reasoning. This lesson reinforces the relationship between scale drawings and the objects they represent through hands-on problem-solving tasks.

In this Grade 7 lesson from Illustrative Mathematics Chapter 2, students review equivalent ratios using double number lines, real-world mixture recipes, and geometric figures to identify proportional relationships. Through hands-on activities, they practice determining whether two ratios are equivalent by checking if the same scale factor connects corresponding values. This lesson builds the foundation for understanding proportional relationships as sets of equivalent ratios throughout the chapter.

In this Grade 7 lesson from Illustrative Mathematics Chapter 2, students learn how to write equations that represent proportional relationships, using the constant of proportionality to express one quantity in terms of another. Working through real-world contexts like rice servings, flight distance, and paint mixing, students practice identifying the constant of proportionality and writing equations in the form y = kx. The lesson builds students' understanding of how tables, equations, and proportional reasoning connect across different problem situations.

In this Grade 7 lesson from Illustrative Mathematics Chapter 2, students learn to distinguish proportional relationships from nonproportional relationships by analyzing tables, calculating unit rates, and checking for a constant quotient between two quantities. Students work with real-world contexts such as park entrance fees and running laps to determine whether a relationship can be represented by an equation of the form y = kx. The lesson emphasizes that a constant quotient suggests a proportional relationship, while varying quotients confirm a nonproportional one.

In this Grade 7 Illustrative Mathematics lesson, students learn to represent proportional relationships using graphs on the coordinate plane, identifying that such graphs form a straight line passing through the origin (0, 0). Students plot ordered pairs from tables, match tables to graphs, and distinguish proportional relationships from non-proportional ones based on these visual characteristics. The lesson is part of Chapter 2: Introducing Proportional Relationships.

In this Grade 7 lesson from Illustrative Mathematics Chapter 3: Measuring Circles, students explore how to determine whether two measured quantities have a proportional relationship by plotting values on a coordinate plane and examining whether the points fall close to a line through the origin. Using squares as a hands-on context, students measure diagonal length alongside both perimeter and area, then compare how each relationship behaves graphically and numerically. The lesson builds foundational skills for recognizing and modeling proportional relationships from real measurement data.

In this Grade 7 lesson from Illustrative Mathematics Chapter 3, students practice estimating the areas of irregular and complex shapes by approximating them with simpler polygons such as rectangles and triangles. Students apply strategies like enclosing a shape in a rectangle and subtracting corner areas, using real-world examples including a house floor plan, the state of Nevada, and Lake Tahoe. This builds foundational spatial reasoning skills needed for more precise area calculations later in the chapter.

In this Grade 7 lesson from Illustrative Mathematics Chapter 4, students explore proportional relationships that involve fractions and decimals by working with scaled copies of the U.S. flag. Using real dimensions such as 7⅝ feet by 5⅜ feet, students apply scale factors, equivalent ratios, and unit rates to find missing dimensions of scaled figures. The lesson also introduces percentage as a rate per 100, setting the foundation for upcoming work with proportional relationships and percentages.

In this Grade 7 lesson from Illustrative Mathematics Chapter 4, students explore percent increase and decrease, learning how to calculate and interpret changes in quantities as a percentage of the original value. The lesson builds proportional reasoning skills by connecting percent change to real-world contexts. This content supports students in applying percentages flexibly across problems involving growth, reduction, and comparison.

In this Grade 7 lesson from Illustrative Mathematics Chapter 4, students apply percentages to calculate sales tax and tips in real-world contexts. They practice finding tax rates from given data, computing sales tax using fractional percentages such as 7.5%, and determining total cost as a percent increase of the original price. Students also work backwards from a known tax amount to find an original subtotal, building fluency with proportional reasoning.

In this Grade 7 lesson from Illustrative Mathematics Chapter 5, students review signed numbers by interpreting negative numbers in real-world contexts such as temperature and elevation above or below sea level. Students practice reading thermometers, comparing positive and negative values on a number line, and ordering rational numbers from least to greatest. The lesson also introduces absolute value as a way to describe a number's distance from zero and reinforces the concept of opposites.

In this Grade 7 lesson from Illustrative Mathematics Chapter 5, students practice adding and subtracting rational numbers by modeling temperature changes as positive and negative integers on a number line. They learn to represent signed numbers using directional arrows and apply tip-to-tail addition to write addition equations such as 3 + (-7) = -4. Real-world temperature scenarios involving cities like Houston and Minneapolis reinforce how positive and negative values describe increases and decreases in context.

In this Grade 7 Illustrative Mathematics lesson, students use signed numbers to represent position, speed, and direction on a number line, learning how the concept of velocity combines magnitude with sign to indicate direction of movement. Through activities involving distance-rate-time relationships, students discover the rule that a negative number multiplied by a positive number produces a negative number. This lesson from Chapter 5: Rational Number Arithmetic builds foundational understanding of multiplying rational numbers in real-world motion contexts.

In this Grade 7 Illustrative Mathematics lesson from Chapter 5, students practice applying all four operations — addition, subtraction, multiplication, and division — to rational numbers, including fractions, decimals, and negative numbers. They evaluate and compare algebraic expressions such as -a, -4b, a ÷ -b, and b³ for given values, and explore how sums, differences, products, and quotients of rational numbers can be written in multiple equivalent forms. The lesson builds fluency with signed number arithmetic and interpreting numerical and algebraic expressions involving rational numbers.

In this Grade 7 Illustrative Mathematics lesson from Chapter 5, students practice solving one-step equations with negative numbers and rational number coefficients by applying additive inverses and multiplicative inverses (reciprocals) to isolate variables. Students work through equations such as -½x = ¼ and -2/9t = -12, learning to rewrite subtraction as adding the opposite and division as multiplying by the reciprocal. Real-world word problems involving elevation changes and trip costs reinforce how to set up and solve equations when values include negative integers, decimals, and fractions.

In this Grade 7 Illustrative Mathematics lesson from Chapter 6, students explore real-world situations involving two-step equations of the forms px+q=r and p(x+q)=r, learning to distinguish these from simpler proportional relationships. Through contexts like entrance fees and pizza delivery costs, students recognize how a fixed fee combined with a per-unit rate creates non-proportional relationships that require multi-operation equations to model. This lesson builds the foundation for writing and solving equations with more than one operation throughout the unit.

In this Grade 7 Illustrative Mathematics lesson, students learn to solve two-step equations of the form px + q = r by connecting balanced hanger diagrams to algebraic reasoning. Using hangers as visual models, they practice applying inverse operations — subtracting a constant from each side and then multiplying each side by a unit fraction — to isolate an unknown variable. The lesson builds foundational skills for writing and solving equations with rational number coefficients drawn from real-world contexts.

In this Grade 7 lesson from Illustrative Mathematics Chapter 6, students learn what it means to write and solve inequalities, including how a solution to an inequality is any value that makes the inequality true. Using a real-world fundraiser context, students explore why an inequality like 100 + 12n ≥ 500 can have many solutions, unlike a single-answer equation. The lesson also reinforces writing equations from tape diagrams as a bridge to understanding algebraic expressions and inequalities.

In this Grade 7 lesson from Illustrative Mathematics, Chapter 6, students learn to rewrite subtraction expressions as equivalent addition expressions by replacing subtracted terms with their additive inverses. They apply the commutative and associative properties to rearrange terms and simplify calculations, then extend this understanding by using the distributive property to expand expressions with subtraction and negative coefficients. Practice problems reinforce converting between subtraction and addition forms and distributing a factor across multi-term expressions involving signed numbers.

In this Grade 7 Illustrative Mathematics lesson, students explore fundamental angle relationships by identifying and measuring right angles, straight angles, and adjacent angles. Using pattern blocks and protractors, students discover that a right angle measures 90°, a straight angle measures 180°, and a full rotation measures 360°, while also practicing how to correctly read a protractor. This lesson from Chapter 7: Angles, Triangles, and Prisms builds the vocabulary and reasoning skills students need to work with more complex geometric figures.

In this Grade 7 Illustrative Mathematics lesson, students explore how given side lengths determine whether a polygon can be built, discovering that the same four side lengths can produce many different quadrilaterals while certain combinations — like 3 in, 4 in, and 9 in — cannot form a triangle at all. Students construct triangles and quadrilaterals using specific side lengths and examine whether a unique shape is produced or multiple figures are possible. This hands-on investigation builds the foundation for understanding the conditions required to construct valid polygons.

In this Grade 7 lesson from Illustrative Mathematics Chapter 7, students explore cross sections by slicing three-dimensional figures such as rectangular prisms, square pyramids, and cubes with horizontal, vertical, and diagonal planes. Students identify and describe the two-dimensional shapes — including rectangles, squares, and triangles — that result from different cuts through the same solid. The lesson builds spatial reasoning and introduces key vocabulary including cross section, prism, and pyramid in the context of solid geometry.

In this Grade 7 Illustrative Mathematics lesson from Chapter 8: Probability and Sampling, students explore the probability of single-step events by using experimental data to make predictions about future outcomes. Through hands-on activities like drawing colored blocks from a bag and analyzing real-world scenarios, students learn to distinguish between situations where probability can be determined from known conditions versus situations where past observations must guide estimates. The lesson builds foundational understanding of experimental probability and how repeated trials produce data that can predict the likelihood of future events.

In this Grade 7 lesson from Illustrative Mathematics Chapter 8, students learn how to identify and organize sample spaces for multi-step experiments using three methods: lists, tables, and tree diagrams. The lesson applies these strategies to real-world scenarios like choosing meals or clothing combinations, and introduces the multiplication principle to count total outcomes. Students then use sample spaces to calculate probabilities of specific outcomes in multi-step events.

In this Grade 7 Illustrative Mathematics lesson, students learn how to compare two groups of data using measures of center such as mean and median alongside measures of variability such as mean absolute deviation (MAD) and interquartile range. Using real-world contexts like team heights and song lengths, students practice creating and interpreting dot plots to determine whether differences between groups are meaningful. The lesson builds statistical reasoning skills by showing how a single summary statistic must be considered alongside spread to draw valid comparisons.

In this Grade 7 lesson from Illustrative Mathematics Chapter 8, students use random samples to estimate population measures of center, practicing when to choose mean versus median based on data distribution. They calculate mean absolute deviation (MAD) for real-world data sets — including TV viewer ages and movie ratings — and use MAD to evaluate how variability in a population affects the reliability of a sample estimate. Students apply these skills to draw conclusions and justify answers in context, such as determining whether a film qualifies for an award based on sampled review scores.