Illustrative Mathematics, Grade 6

In this Grade 6 Illustrative Mathematics lesson (Unit 1, Lesson 1), students are introduced to the concept of area by exploring tiling patterns made of rhombuses, trapezoids, and triangles to reason about which shapes cover more of a two-dimensional plane. Students learn that area is the number of square units covering a region without gaps or overlaps, and practice finding area by counting grid squares, decomposing shapes, and comparing regions.

In this Grade 6 Illustrative Mathematics lesson from Unit 1: Area and Surface Area, students explore the defining properties of parallelograms — including parallel opposite sides, equal opposite side lengths, and equal opposite angles — and learn how to calculate their area. Students practice decomposing a parallelogram into a rectangle by rearranging pieces, and apply the base times height formula to find areas of parallelograms with given dimensions.

In this Grade 6 lesson from Illustrative Mathematics Unit 1, students explore the relationship between parallelograms and triangles by decomposing quadrilaterals into two identical triangles using a single diagonal cut. Students discover that any parallelogram can always be split into two congruent triangles, and conversely, two identical triangles can be joined along any of their three sides to form a parallelogram. This special connection between the two shapes builds the foundation for understanding how to calculate the area of any triangle.

In this Grade 6 lesson from Illustrative Mathematics Unit 1, students learn to identify and define polygons by examining their properties, including vertices, sides, and angles. Students distinguish polygons from non-polygons and explore how polygons can be decomposed into triangles and other shapes as a strategy for finding area. This lesson builds the geometric foundation needed for calculating surface area throughout the unit.

In this Grade 6 Illustrative Mathematics lesson from Unit 1: Area and Surface Area, students explore polyhedra by identifying faces, edges, and vertices and learning to distinguish between prisms and pyramids. Students also work with nets — two-dimensional representations that fold into three-dimensional shapes — to build understanding of surface area as the total area covering all faces of a polyhedron. The lesson connects hands-on assembly of nets to key vocabulary including base, face, prism, pyramid, and surface area.

In this Grade 6 lesson from Illustrative Mathematics Unit 1, students learn to identify perfect squares and perfect cubes, calculate the area of squares and volume of cubes, and express these measurements using exponent notation such as s² and s³. Students connect geometric models — including square grids and unit cube constructions — to algebraic representations, building fluency with squared and cubed numbers and the vocabulary of exponents.

In this Grade 6 lesson from Illustrative Mathematics Unit 2, students are introduced to the concept of ratios as associations between two or more quantities. They practice sorting collections into categories and expressing relationships using ratio notation in multiple forms, including "a to b," "a : b," and "for every a of one category, there are b of another." The lesson builds foundational ratio language that students will apply throughout the unit.

In this Grade 6 lesson from Illustrative Mathematics Unit 2, students learn what equivalent ratios are and how to identify them using real-world contexts like recipes and drink mixtures. Through activities involving batches of cookies and powdered drink mix, students discover that ratios such as 5:2, 10:4, and 15:6 are equivalent because they represent the same relationship scaled up by doubling or tripling. Students practice drawing diagrams to represent multiple batches and finding additional equivalent ratios for a given recipe.

In this Grade 6 Illustrative Mathematics lesson from Unit 2, students learn how to use double number line diagrams to represent equivalent ratios. Working with real-world contexts like drink mix recipes and paint mixtures, students practice reading and completing double number lines to find unknown quantities that maintain the same ratio. The lesson builds foundational understanding of ratio equivalence as a multiplicative relationship between two quantities.

In this Grade 6 lesson from Illustrative Mathematics Unit 2, students learn how to represent equivalent ratios using tables and double number line diagrams. They practice scaling ratios up and down to solve real-world problems, such as finding ingredient amounts for different batch sizes of a recipe. Students also develop strategies for using ratio tables to solve problems involving large or non-whole-number values.

In this Grade 6 Illustrative Mathematics lesson from Unit 2, students learn to interpret part-part-whole ratios by recognizing that the two quantities in a ratio can be added together to find a total. Using tape diagrams, students solve problems where a known total must be divided according to a given ratio, such as determining how many students wear sneakers when the sneaker-to-boots ratio is 5:6 in a class of 33. The lesson builds fluency with scaling ratios and applying unit reasoning to real-world mixing and grouping situations.

In this Grade 6 lesson from Illustrative Mathematics Unit 3, students explore units of measurement and apply equivalent ratios to solve real-world constant rate problems, such as calculating how far a climber travels in a given time or how long it takes a crew to complete a task. Using ratio tables and double number lines, students practice finding unknown quantities when a rate — like windows washed per minute or students served per minute — stays the same. The lesson builds foundational skills for understanding rates and proportional reasoning in the context of familiar, concrete situations.

In this Grade 6 Illustrative Mathematics lesson from Unit 3, students build intuition for standard units of measurement by anchoring units of length, volume, and weight or mass to familiar everyday objects, such as comparing 1 millimeter to the thickness of a dime or 1 liter to a reusable water bottle. Students practice sorting units by the attribute they measure and selecting the most appropriate unit for real-world objects ranging from a pencil to a hippopotamus. The lesson also introduces distinguishing between length, area, volume, and weight as distinct measurable attributes.

In this Grade 6 lesson from Illustrative Mathematics Unit 3, students learn how to compare rates — such as speeds and prices — by finding unit rates like meters per minute or dollars per can. Students work through real-world problems involving treadmill workouts and store sales to practice calculating and interpreting unit prices and speeds. The lesson builds understanding of why expressing a ratio as a quantity per 1 unit makes comparing different rates efficient and accurate.

In this Grade 6 Illustrative Mathematics lesson from Unit 3: Rates and Percentages, students learn that a percentage is a rate per 100 and practice identifying and calculating percentages using familiar contexts like U.S. coins and dollar values. Students use double number lines and tape diagrams to represent percentages, including values greater than 100%, and connect percentage notation to equivalent fractions. The lesson builds foundational skills for comparing quantities using percent language across real-world situations.

In this Grade 6 lesson from Illustrative Mathematics Unit 4 (Dividing Fractions), students explore the relationship between the size of the divisor and the size of the quotient, learning to estimate and reason about whether a quotient will be greater than 1, close to 1, or less than 1 without computing. Students practice identifying the dividend, divisor, and quotient and build number sense by comparing expressions such as 30 ÷ ½ and 9 ÷ 10,000 to determine their relative size.

In this Grade 6 Illustrative Mathematics lesson from Unit 4: Dividing Fractions, students explore the two core meanings of fraction division — "how many groups?" and "how many in each group?" — and learn to distinguish between them using diagrams and equations. Students connect fraction division to related multiplication equations, building conceptual understanding before applying procedures.

In this Grade 6 lesson from Illustrative Mathematics Unit 4, students explore the algorithm for dividing fractions by identifying patterns when dividing by unit fractions and non-unit fractions. Using tape diagrams, students discover that dividing by a fraction is equivalent to multiplying by its reciprocal, working through expressions such as 6 ÷ 1/3 and 6 ÷ 2/3 to generalize the rule as a ÷ b/c = a × c/b. The lesson builds conceptual understanding of fraction division before formalizing the procedure.

In this Grade 6 lesson from Illustrative Mathematics Unit 4, students use fraction division to solve real-world comparison problems involving fractional lengths, such as determining how many times as tall one person is compared to another or what fraction of a planned distance a cyclist actually traveled. Students practice dividing mixed numbers by converting them to improper fractions and applying the multiply-by-the-reciprocal algorithm. The lesson builds fluency with writing and interpreting division equations to express multiplicative comparisons between fractional quantities.

In this Grade 6 lesson from Illustrative Mathematics Unit 5, students apply decimal addition, subtraction, multiplication, and division in real-world shopping and budgeting contexts. Using scenarios like concession stand purchases and planning a dinner party with a fixed budget, students practice estimating and calculating exact decimal amounts, rounding, and cost-per-unit reasoning. The lesson builds foundational fluency with decimal operations that will be explored more formally in the lessons ahead.

In this Grade 6 lesson from Illustrative Mathematics Unit 5, students use base-ten block diagrams to represent and add decimal numbers, working with place values including tenths, hundredths, and thousandths. Students explore how the same decimal, such as 0.13 or 0.25, can be represented in multiple equivalent ways using ones, tenths, and hundredths blocks. The lesson builds toward adding decimals like 0.137 + 0.284 by connecting visual diagram-based reasoning to vertical calculation methods.

In this Grade 6 Illustrative Mathematics lesson from Unit 5, students learn how to multiply decimals by expressing them as fractions with powers of ten, such as rewriting 0.7 × 0.09 as (7 × 1/10) × (9 × 1/100) to determine the correct placement of the decimal point in the product. Students explore how multiplying by 0.1, 0.01, and other powers of ten shifts the decimal point and practice applying this reasoning to multi-digit decimal multiplication. The lesson builds fluency with equivalent expressions and connects fraction concepts to decimal arithmetic.

In this Grade 6 Illustrative Mathematics lesson, students learn to divide multi-digit whole numbers using two strategies: base-ten diagrams and the partial quotients method. The partial quotients method involves repeatedly subtracting the divisor times a chosen amount, then adding those partial quotients to find the total. Students compare both approaches and practice applying the partial quotients method to problems such as 1,332 ÷ 9 and 1,032 ÷ 43.

In this Grade 6 Illustrative Mathematics lesson from Unit 6: Expressions and Equations, students learn to represent and solve one-variable equations using tape diagrams and balanced-hanger diagrams. Students practice matching equivalent equations such as 4 + x = 12 and 4 · x = 12 to their corresponding diagrams, reinforcing how different equation forms can describe the same relationship between quantities. By drawing diagrams and reasoning about unknown values, students build foundational skills for solving equations in one variable.

In this Grade 6 Illustrative Mathematics lesson from Unit 6: Expressions and Equations, students learn to write algebraic expressions using variables to represent real-world situations, such as calculating lemonade sales revenue with the expression 0.50c or describing age relationships with expressions like a + 3. Students practice substituting values into expressions and translating word problems involving operations like multiplication, addition, and fractions into variable expressions. The lesson also introduces writing and solving one-variable equations when the value of an expression is known.

In this Grade 6 Illustrative Mathematics lesson from Unit 6: Expressions and Equations, students learn to write and evaluate expressions with exponents, understanding that exponential notation represents repeated multiplication of the same factor. Students practice interpreting expressions such as 4³ and connecting exponential form to expanded form and standard value. This lesson builds the foundational vocabulary and skills needed to work with algebraic expressions and equations throughout the unit.

In this Grade 6 Illustrative Mathematics lesson, students learn to write equations that describe proportional relationships between two quantities, identifying which variable is independent and which is dependent. Using a paint-mixing ratio context, students complete tables of equivalent ratios and then graph the relationships on coordinate grids. The lesson builds core vocabulary and skills for Unit 6 Expressions and Equations, connecting ratio reasoning to algebraic equations and their visual representations.

In this Grade 6 lesson from Illustrative Mathematics Unit 7, students are introduced to negative numbers and absolute value by exploring real-world contexts such as temperatures above and below zero degrees Celsius and elevations above and below sea level. Students learn to represent positive and negative numbers on a vertical number line and interpret the meaning of zero, positive values, and negative values within a given context. The lesson builds foundational understanding of rational numbers using familiar examples like weather data and geographic landmarks such as Death Valley and the Mariana Trench.

In this Grade 6 Illustrative Mathematics lesson from Unit 7: Rational Numbers, students learn to write and graph inequalities using symbols such as < and ≥ to represent real-world constraints, including situations involving strict inequalities and "at least" or "at most" conditions. Students practice matching written scenarios to inequality statements and number line representations, distinguishing between open and closed circles to indicate whether an endpoint value is included. The lesson builds foundational understanding of how variables and inequality notation describe ranges of possible values.

In this Grade 6 lesson from Illustrative Mathematics Unit 7, students extend the coordinate plane to all four quadrants by plotting and identifying ordered pairs with negative x- and y-values. Students practice locating points in Quadrant I, II, III, and IV using coordinates such as (-4, 1) and (-3.5, -3), and explore how the signs of the coordinates determine a point's position relative to the x- and y-axes. The lesson builds fluency with the coordinate plane through activities like graphing on a grid and interpreting coordinates in real-world contexts.

In this Grade 6 Illustrative Mathematics lesson from Unit 7: Rational Numbers, students learn to identify common factors and the greatest common factor (GCF) of two whole numbers by listing all factors and finding the largest shared value. Real-world problems, such as dividing baked goods into equal bags and tiling a bulletin board with squares, help students apply GCF concepts meaningfully. Students also explore common multiples to solve problems involving equal groupings of different items.

In this Grade 6 Illustrative Mathematics lesson from Unit 8, students explore the foundational concepts of data by distinguishing between numerical data and categorical data, interpreting dot plots, and evaluating what makes a valid statistical question. Through a class survey activity, students collect real measurements such as height, travel time, and sleep hours, then classify survey responses by data type and unit of measurement.

In this Grade 6 lesson from Illustrative Mathematics Unit 8, students learn to represent data distributions using dot plots and bar graphs by distinguishing between categorical and numerical data. Students practice organizing frequency tables, constructing dot plots for numerical data sets, and building bar graphs for categorical survey responses. The lesson introduces key vocabulary including frequency and distribution as tools for analyzing and comparing data sets.

In this Grade 6 Illustrative Mathematics lesson from Unit 8: Data Sets and Distributions, students learn how to calculate the mean of a data set by summing all values and dividing by the number of values. They explore two key interpretations of the mean: as a "fair share" that distributes a total equally among all members, and as a balance point on a dot plot where distances of data points to the left and right are equal. Students apply these concepts using real-world contexts such as walk times and quiz scores.

In this Grade 6 Illustrative Mathematics lesson from Unit 8: Data Sets and Distributions, students learn how to find the median of a data set by ordering values and identifying the middle number, including how to average the two middle values when the data set has an even count. Students also compare the median to the mean as measures of center, exploring which better represents a typical value in skewed distributions. The lesson builds conceptual understanding through hands-on activities using real-world contexts like sibling counts and bus travel times.