Illustrative Mathematics, Grade 5

In this Grade 5 Illustrative Mathematics lesson (Chapter 1: Finding Volume), students are introduced to volume as the measurable attribute describing the amount of space a solid object takes up. Using connecting cubes, students build and compare three-dimensional objects to discover that different shapes can have the same volume. This lesson lays the conceptual groundwork for standard 5.MD.C.3 before students move on to measuring volume in cubic units and calculating the volume of rectangular prisms.

In this Grade 5 Illustrative Mathematics lesson, students learn to calculate the volume of rectangular prisms by counting unit cubes organized into horizontal or vertical layers. Using connecting cubes and prism images, they practice multiplying the number of layers by the number of cubes in each layer to find total volume, building toward standard 5.MD.C.5. The lesson connects multiplication of three factors to the layered structure of prisms as a systematic strategy for measuring volume.

In this Grade 5 Illustrative Mathematics lesson, students learn to describe rectangular prisms using the terms length, width, and height, and calculate volume by multiplying these three side lengths together. They connect this formula to the earlier method of multiplying the area of the base by the height, reinforcing that any face of a prism can serve as the base. The lesson builds toward standard 5.MD.C.5.b as students move from counting unit cubes to writing and applying volume expressions.

In this Grade 5 Illustrative Mathematics lesson from Chapter 1: Finding Volume, students learn to measure the volume of rectangular prisms using standard cubic units including cubic centimeters, cubic inches, and cubic feet. Students apply the formulas length × width × height and base × height to calculate volume in real-world contexts, such as finding the volume of a moving truck. The lesson also develops students' understanding of how to choose an appropriate unit of measure depending on the size of the object being measured.

In this Grade 5 lesson from Illustrative Mathematics Chapter 1, students learn to find the volume of composite figures made up of two non-overlapping right rectangular prisms by decomposing the shape and adding the volumes of each part. Students practice applying the additive property of volume (5.MD.C.5.c), discovering that the same figure can be split into prisms in different ways yet yield the same total volume. Hands-on work with connecting cubes helps reinforce why the volume of a combined figure equals the sum of its component prisms.

In this Grade 5 Illustrative Mathematics lesson from Chapter 1: Finding Volume, students apply their understanding of volume to solve real-world and mathematical problems using the formulas V = l × w × h and V = B × h. Students practice calculating the volume of rectangular prisms and use their knowledge to reason about composite figures and multi-step problems. This lesson reinforces core volume concepts introduced earlier in the chapter by connecting them to practical contexts.

In this Grade 5 lesson from Illustrative Mathematics Chapter 2, students explore the relationship between division and fractions by working through equal-sharing situations involving sandwiches. Students build on their prior knowledge of whole-number division to understand that dividing, such as 3 sandwiches shared among 4 people, produces a fractional quotient like three-fourths. This lesson lays the foundation for the standard 5.NF.B.3, helping students see that a fraction represents the value of a division expression.

In this Grade 5 lesson from Illustrative Mathematics, students write division equations to represent equal-sharing situations and explain how each part of the equation — dividend, divisor, and fraction — corresponds to the real-world context. Building on the understanding that 1 ÷ b = 1/b, students extend this to recognize that a ÷ b = a/b, connecting the numerator to the quantity being shared and the denominator to the number of sharers. The lesson addresses standard 5.NF.B.3 and develops students' ability to interpret and critique equations involving fractions as quotients.

In this Grade 5 Illustrative Mathematics lesson from Chapter 2, students explore why a ÷ b = a/b, learning to interpret fractions as division of the numerator by the denominator. Using real-world sharing scenarios, they analyze situations where a quotient is greater than, equal to, or less than 1, and practice solving equations where the unknown may be the numerator, denominator, or quotient. This lesson directly addresses standard 5.NF.B.3 and helps students build flexible reasoning about the relationship between division and fractions.

In this Grade 5 lesson from Illustrative Mathematics Chapter 2, students explore the relationship between division and multiplication, learning that dividing by a number produces the same result as multiplying by its reciprocal. Students apply this understanding within the context of fractions as quotients, connecting expressions like a ÷ b to fraction multiplication. This foundational skill prepares students to fluently multiply and divide fractions throughout the chapter.

In this Grade 5 lesson from Illustrative Mathematics Chapter 2, students learn to multiply whole numbers by unit fractions and connect that process to division, discovering that expressions like 4 × 1/3, 1/3 × 4, and 4 ÷ 3 all represent the same value. Students solve real-world story problems, match situations to diagrams and equations, and build fluency with the relationship between fraction multiplication and division as described in standard 5.NF.B.4.a.

In this Grade 5 lesson from Illustrative Mathematics Chapter 2, students calculate the area of rectangles with fractional side lengths less than 1 by applying fraction multiplication. Students connect the area formula (length × width) to multiplying unit fractions and other fractions less than 1, building on their understanding of fractions as quotients. This lesson deepens conceptual understanding of how multiplying two fractions less than 1 produces a product smaller than either factor.

In this Grade 5 Illustrative Mathematics lesson from Chapter 2, students learn to find the area of rectangles when one side length is a fraction greater than 1, such as 11/3 or 14/3. Students write and interpret multiplication expressions like 3 × 11/3 to calculate area, building on prior work with fractions less than 1 and connecting improper fractions to mixed numbers. This lesson addresses standards 5.NF.B.3 and 5.NF.B.4.b, deepening students' understanding of fraction multiplication and its relationship to division.

In this Grade 5 Illustrative Mathematics lesson from Chapter 2, students learn how to decompose rectangles with mixed number side lengths to find area by applying the distributive property and partial products. Students practice multiplying whole numbers by mixed numbers such as 5 × 2½ and use diagrams to represent and calculate areas like 5 yards by 6¼ yards. The lesson builds on prior work with fractional side lengths and addresses standard 5.NF.B.4.b.

In this Grade 5 lesson from Illustrative Mathematics Chapter 2, students apply fraction multiplication to solve area situations involving rectangles with fractional side lengths. Students practice finding the area of a rectangle by multiplying a fraction by a fraction or a whole number by a fraction, connecting the area formula to their understanding of fractions as quotients. The lesson builds fluency with fraction multiplication in real-world geometric contexts.

In this Grade 5 lesson from Illustrative Mathematics Chapter 2, students multiply whole numbers and mixed numbers by applying properties of operations such as the distributive property to decompose and find products. Students practice expressions like 12 × 9⅔ and 3⅝ × 18, using fraction decomposition to break calculations into manageable parts. The lesson builds fluency with multiplying fractions greater than 1 written as mixed numbers, connecting to real-world contexts involving area.

In this Grade 5 lesson from Illustrative Mathematics Chapter 3, students learn to interpret and represent a unit fraction of a unit fraction using diagrams, building toward the concept of multiplying two fractions. Using a real-world context like a pan of macaroni and cheese, students draw and explain area diagrams and tape diagrams to find fractional quantities such as one-fifth of one-half. This lesson aligns with standard 5.NF.B.4.a and prepares students to connect visual representations to multiplication expressions in later lessons.

In this Grade 5 Illustrative Mathematics lesson from Chapter 3: Multiplying and Dividing Fractions, students learn to multiply unit fractions by applying the concept that multiplying by a unit fraction is the same as finding a fractional part of a quantity. Students use visual models and equations to develop fluency with expressions such as 1/2 × 1/3, building foundational understanding of fraction multiplication.

In this Grade 5 lesson from Illustrative Mathematics Chapter 3, students learn to multiply a unit fraction by a non-unit fraction by writing multiplication expressions for shaded rectangular regions in area diagrams. Students explore how the product of the numerators represents the number of shaded pieces and the product of the denominators represents the total pieces in the whole, building on their prior understanding of unit fraction multiplication such as one-fifth times one-third equals one-fifteenth. The lesson uses visual models and estimation to help students make sense of expressions like six-fifths times one-third as six copies of one-fifteenth.

In this Grade 5 lesson from Illustrative Mathematics Chapter 3, students learn to multiply two non-unit fractions by using area diagrams and written expressions. Students discover that the product of two fractions equals the product of the numerators over the product of the denominators, such as finding that 3/6 × 4/5 = (3×4)/(6×5). This builds toward a generalized understanding of fraction multiplication aligned with standard 5.NF.B.4.b.

In this Grade 5 lesson from Illustrative Mathematics Chapter 3, students apply fraction multiplication to solve real-world and mathematical problems involving multiplication of fractions and mixed numbers. Building on prior lessons in the Multiplying and Dividing Fractions unit, students practice setting up and computing fraction multiplication expressions in context. The lesson strengthens fluency with multiplying fractions by fractions and whole numbers through applied problem-solving tasks.

In this optional Grade 5 lesson from Illustrative Mathematics Chapter 3, students apply multiplication of fractions to a real-world design challenge by creating their own flag and calculating the area of fabric needed for each section. Students first explore principles of flag design from the North American Vexillological Association, then use those principles to design an original flag while solving fraction multiplication problems involving area. The lesson directly addresses standard 5.NF.B.6, connecting multiplying fractions to a meaningful, hands-on context.

In this optional Grade 5 lesson from Illustrative Mathematics Chapter 3: Multiplying and Dividing Fractions, students explore the foundational concepts of division, including interpreting division as equal sharing and as finding how many groups. The lesson builds conceptual understanding that prepares students for dividing fractions by connecting division to real-world contexts and visual models.

In this Grade 5 lesson from Illustrative Mathematics Chapter 3, students learn to divide a unit fraction by a whole number, such as finding how much of a pan each person receives when one-half is shared equally among 6 people. Using diagrams and division expressions like 1/2 ÷ 6, students explore how dividing a fractional amount into equal parts connects to standard 5.NF.B.7.a. The lesson builds understanding through real-world context before students later formalize the relationship between multiplication and division of fractions.

In this Grade 5 Illustrative Mathematics lesson from Chapter 3: Multiplying and Dividing Fractions, students learn to divide a whole number by a unit fraction using the context of cutting paper strips into fractional-length pieces. They apply standard 5.NF.B.7.b as they use tape diagrams and equations to represent quotients and explore how the size of the unit fraction affects the result. By the end of the lesson, students can solve problems such as determining how many one-fourth foot pieces can be cut from a 3-foot strip of paper.

In this Grade 5 lesson from Illustrative Mathematics Chapter 3, students apply their understanding of dividing a whole number by a unit fraction and a unit fraction by a whole number to write and solve real-world word problems. Using a card sort activity, students match situations to expressions such as 5 ÷ 1/4 and 1/4 ÷ 5, interpreting what each equation means in context. The lesson builds fluency with fraction division and addresses standards 5.NF.B.7 and 5.NF.B.7.c.

In this Grade 5 lesson from Illustrative Mathematics Chapter 3, students reason about the size of quotients involving unit fractions and whole numbers — such as comparing 10 ÷ ⅓ and 10 ÷ ⅕ — without calculating exact values. Students apply their understanding of how the size of the divisor affects the quotient to assess whether answers are greater than or less than 1 and to check reasonableness. The lesson addresses standard 5.NF.B.7 and builds on prior work dividing unit fractions by whole numbers and whole numbers by unit fractions.

In this Grade 5 lesson from Illustrative Mathematics Chapter 3, students explore the relationship between multiplication and division with fractions, learning to represent the same real-world situation using both a division equation and a multiplication equation. Using contexts like dividing pounds of beef by a unit fraction and sharing a fraction equally between people, students practice writing equations such as 2 ÷ ¼ and its multiplication counterpart. The lesson builds fluency with 5.NF.B.4 and 5.NF.B.7 by connecting diagram representations to symbolic fraction operations.

In this Grade 5 Illustrative Mathematics lesson, students apply their understanding of multiplying and dividing fractions — including unit fractions divided by whole numbers and whole numbers divided by unit fractions — to strategically build expressions with the greatest or smallest possible value. Using a set of given digits, learners analyze how the size of factors, dividends, and divisors affects products and quotients (standards 5.NF.B.4, 5.NF.B.6, 5.NF.B.7). The lesson also develops estimation strategies for products involving fractions and mixed numbers.

In this Grade 5 Illustrative Mathematics lesson from Chapter 4, students estimate and calculate products of multi-digit numbers using place value understanding and properties of operations. Building on their Grade 4 work with partial products, students apply strategies for multiplying numbers such as 15 × 121 and multiples of 10 and 100, then evaluate whether their estimates are too large or too small. The lesson prepares students for the standard algorithm for multi-digit multiplication addressed in 5.NBT.B.5.

In this Grade 5 Illustrative Mathematics lesson, students interpret partial products diagrams to multiply three-digit numbers by two-digit numbers, building on the place value strategies introduced in Grade 4. Using rectangular diagrams, students decompose factors and calculate individual partial products before combining them to find the full product, as in 222 × 14 broken into 2,800 + 280 + 28. This lesson from Chapter 4 lays the groundwork for understanding the standard multiplication algorithm covered later in the chapter.

In this Grade 5 Illustrative Mathematics lesson from Chapter 4, students use the standard algorithm to multiply up to five-digit numbers by one-digit factors, including problems that require composing new units. Students compare the standard algorithm to the partial products method to understand how carry digits represent newly composed tens, hundreds, or thousands. This lesson builds on Grade 4 work with multi-digit multiplication and supports students in developing fluency with the standard algorithm as defined by standard 5.NBT.B.5.

In this Grade 5 lesson from Illustrative Mathematics Chapter 4, students learn to apply the standard algorithm to multiply two- and three-digit numbers by two-digit numbers, including recording the composition of new units. Building on prior work with single-digit multipliers and multiplication without composing, students combine these skills to find products such as 251 × 34 using a structured written method aligned to standard 5.NBT.B.5.

In this Grade 5 Illustrative Mathematics lesson, students build fluency with the standard algorithm for multiplying two-digit and three-digit whole numbers, including problems that require composing multiple new units. Students explore different ways to apply the algorithm and practice placing digits strategically to maximize products, deepening their understanding of place value. The lesson addresses standard 5.NBT.B.5 within Chapter 4 on multi-digit multiplication and division.

In this Grade 5 Illustrative Mathematics lesson from Chapter 4, students apply whole-number multiplication and the standard algorithm to solve real-world problems involving the volume of rectangular prisms. Using birdhouse dimensions as context, they calculate products of multi-digit numbers and explore strategies such as the associative and commutative properties of multiplication. The lesson addresses standards 5.NBT.B.5 and 5.MD.C.5, reinforcing multi-digit multiplication within a measurement and volume context.

In this Grade 5 Illustrative Mathematics lesson, students estimate and solve multi-digit division problems using strategies such as partial quotients and the relationship between multiplication and division. Set in the context of a world record Peruvian folk dance with 4,704 dancers arranged in groups of 8, students practice dividing multi-digit whole numbers in ways that build on their Grade 4 understanding of place value and division. This lesson from Chapter 4 prepares students for more efficient division methods aligned to standard 5.NBT.B.6.

In this Grade 5 lesson from Illustrative Mathematics Chapter 4, students learn to divide multi-digit whole numbers using the partial quotients strategy by applying place value understanding and the relationship between multiplication and division. Students explore how a dividend can be decomposed in different ways to produce partial quotients, comparing approaches that use multiplication expressions versus division expressions to reach the same result. The lesson emphasizes choosing decompositions that can be calculated mentally, building efficiency with the partial quotients algorithm.

In this Grade 5 lesson from Illustrative Mathematics Chapter 4, students practice multi-digit division using the partial quotients strategy to build fluency and accuracy. Learners apply their understanding of place value to break division problems into manageable parts, recording each partial quotient step by step. This lesson reinforces the division concepts developed throughout the chapter's focus on wrapping up multiplication and division with multi-digit numbers.

In this Grade 5 lesson from Illustrative Mathematics Chapter 4, students use the relationship between multiplication and division to find missing side lengths of rectangles and rectangular prisms when given area or volume. They apply multi-digit division (5.NBT.B.6) to solve problems such as determining an unknown width when the area and length are known. The lesson builds on prior knowledge of volume formulas and division strategies developed throughout the unit.

In this Grade 5 Illustrative Mathematics lesson, students apply multi-digit division with mixed number quotients to solve real-world problems inspired by the Guinness World Record for the longest continuous noodle, which measured approximately 10,119 feet. Students practice dividing up to four-digit dividends by two-digit divisors using strategies based on place value and the relationship between multiplication and division, while also interpreting fractional quotients in context. The lesson addresses standards 5.NBT.B.6 and 5.NF.B.3 within Chapter 4's focus on wrapping up multiplication and division with multi-digit numbers.

In this Grade 5 Illustrative Mathematics lesson from Chapter 4, students practice using the partial quotients strategy to divide multi-digit numbers, working through examples such as 243 ÷ 9 by breaking the dividend into manageable chunks. Students learn to subtract successive multiples of the divisor and combine the resulting partial quotients to find the final answer. This optional lesson reinforces division fluency as part of the unit wrapping up multiplication and division with multi-digit numbers.

In this Grade 5 lesson from Illustrative Mathematics Chapter 4, students practice estimating products and quotients of whole numbers using a real-world volume context involving milk consumption. Students apply the associative property of multiplication to build estimates of large quantities, such as calculating the total cubic inches of milk consumed by thousands of students. The lesson connects 5.NBT.B multiplication and division skills with volume measurement concepts from 5.MD.C.5.

In this Grade 5 lesson from Illustrative Mathematics Chapter 4, students apply the standard algorithm for multi-digit multiplication to solve real-world area problems involving the Great Garbage Patch and U.S. states, multiplying two- and three-digit numbers such as 452 × 600. Students also practice estimation to compare the area of states like New Mexico and Wyoming with the size of the Great Garbage Patch, reinforcing standard 5.NBT.B.5. The lesson connects multiplication of whole numbers to measurement concepts, including a review of kilometers and square kilometers.

In this Grade 5 Illustrative Mathematics lesson from Chapter 5: Place Value Patterns and Decimal Operations, students explore the unit of thousandths and its relationship to tenths and hundredths. Students learn to represent 0.001 using hundredths grid diagrams, fractions, and decimal notation, building on their Grade 4 work with tenths and hundredths. The lesson emphasizes the base-ten relationship that one tenth of one hundredth equals one thousandth, aligning with standard 5.NBT.A.1.

In this Grade 5 Illustrative Mathematics lesson, students explore place value relationships among tenths, hundredths, and thousandths by examining how different combinations of decimal weights can balance a scale. Through the "Maintain Your Balance" activities, students discover equivalences such as 10 thousandths equaling 1 hundredth and practice representing three-digit decimal numbers using expanded form and word form. The lesson builds on prior work with decimal notation (5.NBT.A.1, 5.NBT.A.3) to deepen understanding of the multiplicative relationships between adjacent decimal place values.

In this Grade 5 Illustrative Mathematics lesson from Chapter 5: Place Value Patterns and Decimal Operations, students use place value understanding to compare decimals to the thousandths place. Students apply strategies such as place value reasoning, hundredths grids, and number lines to determine which decimal is greater, including finding decimals between values like 5 and 5.01. The lesson addresses standard 5.NBT.A.3.b and builds on students' prior work reading and writing decimals to the thousandths place.

In this Grade 5 Illustrative Mathematics lesson, students use place value understanding to locate and label decimals on number lines, working with tenths, hundredths, and thousandths. Through activities like "Name that Number," students identify which two hundredths a decimal falls between and estimate its precise position, reinforcing the relationship between consecutive place values. This lesson addresses standard 5.NBT.A.3.b and builds fluency with comparing and representing decimals to the thousandths place.

In this Grade 5 Illustrative Mathematics lesson (Chapter 5), students practice ordering decimals to the thousandths place by comparing place value to arrange numbers from least to greatest. Using number lines and place value reasoning, students work through true-or-false decimal inequalities and identify decimal numbers that fall between two given values. The lesson addresses standard 5.NBT.A.3.b and builds fluency in comparing and ordering multi-digit decimals such as 565.4, 556.040, and 565.004.

In this Grade 5 lesson from Illustrative Mathematics Chapter 5, students develop number sense around decimal addition by learning to estimate sums before calculating. Students practice applying place value understanding to judge the reasonableness of decimal addition results, building flexibility with tenths and hundredths. This lesson supports the broader unit on place value patterns and decimal operations by connecting estimation strategies to accurate computation.

In this Grade 5 lesson from Illustrative Mathematics Chapter 5, students learn to add decimals to the hundredths place using the standard algorithm with correct place value alignment. The lesson focuses on a common error in which decimal numbers are right-aligned like whole numbers, causing digits of different place values — such as tenths and hundredths — to be added together incorrectly. Students identify and correct these alignment mistakes, reinforcing that decimal numbers must be lined up by place value, not by their rightmost digit.

In this Grade 5 Illustrative Mathematics lesson from Chapter 5, students learn to subtract decimals to the hundredths place using place value strategies, such as decomposing tenths into hundredths and applying the relationship between addition and subtraction. Students apply these strategies through true-or-false reasoning tasks and hands-on subtraction activities, building on prior work with decimal addition. The lesson develops flexible, sense-making approaches to multi-digit decimal subtraction aligned with standard 5.NBT.B.7.

In this Grade 5 lesson from Illustrative Mathematics Chapter 5, students add and subtract decimals to the hundredths using place value strategies, including adding on, compensation, and the standard algorithm. Students practice choosing and applying the most efficient method for expressions like 75.2 − 4.37 and 236.87 + 5.15, reinforcing standard 5.NBT.B.7. The lesson builds fluency with decimal operations by encouraging students to think strategically rather than relying on a single prescribed approach.

In this Grade 5 Illustrative Mathematics lesson from Chapter 5, students learn to multiply whole numbers by tenths and hundredths, such as finding the value of expressions like 2 × 0.4 and 4 × 0.03. Students apply their understanding of decimal place value and whole number multiplication, using strategies like hundredths grids, tape diagrams, and the distributive property to make sense of decimal products. The lesson builds on prior work with decimal addition and subtraction to help students see how place value patterns extend to multiplication.

In this Grade 5 Illustrative Mathematics lesson from Chapter 5, students use the associative property, distributive property, and compensation strategy to interpret and evaluate multiplication expressions involving whole numbers and decimals. Students practice identifying equivalent expressions for products such as 15 × 0.19 and choosing the most efficient strategy to find the value. The lesson builds fluency with multiplying tenths and hundredths by connecting decimal products to related whole number calculations.

In this Grade 5 Illustrative Mathematics lesson, students use place value understanding to multiply multi-digit decimals, including products of two decimals to the tenths and products of a whole number and a decimal to the hundredths. Students learn to relate decimal products to whole number products, for example recognizing that 2.5 × 6.4 equals (25 × 64) × 0.01 using place value reasoning. The lesson is part of Chapter 5: Place Value Patterns and Decimal Operations and addresses standards 5.NBT.A.1 and 5.NBT.B.7.

In this Grade 5 Illustrative Mathematics lesson from Chapter 5: Place Value Patterns and Decimal Operations, students learn how to divide by decimals by applying place value understanding and patterns. Students explore how dividing by a decimal such as 0.1 or 0.01 relates to multiplying by 10 or 100, building fluency with decimal division. This lesson strengthens students' number sense and prepares them for more complex decimal operations.

In this Grade 5 Illustrative Mathematics lesson from Chapter 5: Place Value Patterns and Decimal Operations, students learn to divide decimals to the hundredths place by whole numbers, such as solving expressions like 0.42 ÷ 5 and 0.8 ÷ 4. Students use place value understanding and hundredths grids to reason through quotients, connecting decimal division to whole number division strategies. The lesson addresses standard 5.NBT.B.7 and builds on prior work dividing whole numbers by decimals.

In this Grade 5 lesson from Illustrative Mathematics Chapter 6, students explore place value patterns by examining how each digit in a decimal represents ten times as much as the digit to its right and one tenth as much as the digit to its left. Students practice expressing these multiplicative relationships using multiplication and division equations with decimals and whole numbers, such as 6 × 0.1 = 0.6 and 600 ÷ 100 = 6. This lesson builds the foundation for working with powers of 10 and measurement conversions covered throughout the chapter.

In this Grade 5 Illustrative Mathematics lesson (Chapter 6, Lesson 2), students learn to convert metric length units — meters to centimeters, millimeters, and kilometers — by multiplying and dividing with powers of 10 (10, 100, and 1,000). Students explore place value patterns, such as how digits shift left when multiplying by a power of 10, connecting decimal multiplication to real-world measurement contexts like track and field. The lesson addresses standards 5.MD.A.1 and 5.NBT.A.2.

In this Grade 5 lesson from Illustrative Mathematics Chapter 6, students solve multi-step problems by converting between metric length units — centimeters, meters, and kilometers — using multiplication and division by powers of 10. Students work with decimal measurements and practice all four operations, choosing whether to convert to a smaller or larger unit to simplify their calculations. The lesson builds on students' understanding of place value and decimal arithmetic developed earlier in the unit.

In this Grade 5 Illustrative Mathematics lesson, students solve multi-step problems involving customary length unit conversions between inches, feet, and yards. Unlike metric conversions, these require multiplying and dividing by numbers such as 12 and 3, so students apply prior skills with whole numbers and fractions to convert strategically in both directions. The lesson is part of Chapter 6 on Place Value Patterns and Decimal Operations and addresses standard 5.MD.A.1.

In this Grade 5 Illustrative Mathematics lesson from Chapter 6, students learn to add and subtract fractions with unlike denominators by finding common denominators, including cases where one denominator is a multiple of the other and cases requiring a new common denominator. Students use fraction strips and number lines to represent equivalent fractions and reason through sums and differences such as 5/6 − 1/3 and 3/4 + 1/2. This lesson addresses standard 5.NF.A.1 and builds on prior knowledge of same-denominator fraction operations from earlier grades.

In this Grade 5 lesson from Illustrative Mathematics Chapter 6, students learn how to find common denominators for fractions with unlike denominators, including cases where multiplying the two denominators gives a valid common denominator. Students practice adding and subtracting fractions such as three-fourths plus seven-eighths and three-fourths minus two-fifths by converting to equivalent fractions with a shared denominator. The lesson addresses standard 5.NF.A.1 and encourages students to choose the common denominator that makes the most sense for each problem.

In this Grade 5 Illustrative Mathematics lesson from Chapter 6: Place Value Patterns and Decimal Operations, students apply their understanding of decimal place value and operations to solve multi-step word problems. Learners practice selecting appropriate strategies to add, subtract, multiply, and divide decimals in real-world contexts. This lesson consolidates key skills from the chapter to build fluency and problem-solving confidence with decimal numbers.

In this Grade 5 Illustrative Mathematics lesson from Chapter 6, students apply and consolidate their skills for adding and subtracting fractions, including fractions with unlike denominators. The lesson brings together key strategies from the chapter to solve multi-step problems involving fraction addition and subtraction. Students practice selecting efficient methods and checking the reasonableness of their answers in real-world and mathematical contexts.

In this Grade 5 Illustrative Mathematics lesson from Chapter 6: Place Value Patterns and Decimal Operations, students practice representing fractions on a line plot by plotting fractional data points and interpreting the distribution. The lesson builds on students' understanding of fractions and number lines as tools for organizing and analyzing real-world measurement data.

Grade 5 students learn to create and interpret line plots displaying fractional measurement data, adding and subtracting fractions with unlike denominators to solve multi-step problems. Using the Info Gap routine with a fruit-picking context, students practice communicating mathematical information to find missing data and answer questions about the line plot. This lesson from Illustrative Mathematics Grade 5, Chapter 6 also connects repeated addition of fractions to multiplication, building on standards 5.MD.B.2, 5.NF.A.2, and 5.NF.B.4.

In this Grade 5 Illustrative Mathematics lesson, students compare the size of a product to the size of one of its factors, focusing on cases where one factor is a whole number and the other is a fraction greater than, equal to, or less than 1. Students apply standard 5.NF.B.5.a by reasoning about whether expressions such as ¼ × 20 or 10/8 × 20 produce results greater than or less than the whole-number factor, using strategies like number lines, diagrams, or fraction reasoning. The lesson builds on prior work with whole number, decimal, and fraction multiplication in Chapter 6 of the Grade 5 curriculum.

In this Grade 5 lesson from Illustrative Mathematics Chapter 6, students learn to compare the size of a product to one of its factors by examining the size of the other factor, applying standard 5.NF.B.5.a. Students practice determining whether multiplying a number by a fraction greater than, equal to, or less than 1 will produce a product that is greater than, equal to, or less than the original number, without calculating the actual product. Number line diagrams are used to visualize and justify these comparisons, including cases where both factors are fractions.

In this Grade 5 Illustrative Mathematics lesson from Chapter 6: Place Value Patterns and Decimal Operations, students learn to compare decimal numbers to the benchmark value of 1, determining whether a given decimal is greater than, less than, or equal to 1. Students apply place value reasoning to make and justify these comparisons using mathematical language and notation.

In this Grade 5 Illustrative Mathematics lesson from Chapter 7, students build familiarity with the structure of the coordinate grid, learning key vocabulary including coordinate grid, axes, horizontal axis, and vertical axis. Students use coordinates to describe the size and location of rectangles on the grid, discovering how numbered axes make it possible to communicate shape locations precisely. This lesson addresses standard 5.G.A.1 and sets the foundation for plotting and interpreting points on the coordinate plane.

In this Grade 5 Illustrative Mathematics lesson (Chapter 7, Lesson 2), students practice plotting and naming points on a coordinate grid, including points on the horizontal and vertical axes where one coordinate equals zero. Students explore how points sharing the same x- or y-coordinate align on a horizontal or vertical line, reinforcing precision in writing coordinate pairs such as (3, 0) and (0, 2). The lesson builds toward standard 5.G.A.1 and deepens understanding of coordinate structure through estimation and reasoning activities.

In this Grade 5 Illustrative Mathematics lesson from Chapter 7, students learn to classify quadrilaterals by analyzing their properties, including parallel sides, right angles, and equal side lengths. Students distinguish between categories such as parallelograms, rectangles, rhombuses, and squares, understanding how these categories relate hierarchically to one another. The lesson builds geometric reasoning skills as students apply precise mathematical vocabulary to sort and describe shapes on the coordinate plane.

In this Grade 5 Illustrative Mathematics lesson, students explore the hierarchy of quadrilaterals by examining how categories such as parallelograms, rectangles, rhombuses, and squares relate to and nest within one another. Students use shared attributes to classify shapes and understand why a square, for example, belongs to multiple categories simultaneously. This lesson builds geometric reasoning skills aligned with the properties and classification of two-dimensional figures.

In this Grade 5 Illustrative Mathematics lesson from Chapter 7, students generate two numerical patterns from rules and form ordered pairs from corresponding terms to plot on a coordinate grid. Students practice identifying relationships between the two patterns, such as recognizing that one set of values is twice the other, and use coordinates to describe the location of each point. This lesson addresses standard 5.OA.B.3 and builds toward coordinate plane skills in 5.G.A.1.

In this Grade 5 Illustrative Mathematics lesson from Chapter 7, students explore how the length and width of rectangles with a fixed perimeter or fixed area relate to each other by plotting coordinate pairs on a grid. Students calculate perimeter using 2l + 2w and area using multiplication, including with fractions and decimals, then graph the results to visually compare how each relationship behaves differently. The lesson addresses standards 5.G.A.2, 5.NBT.B.7, and 5.OA.B.3.