The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. The curriculum is divided into eight units and each unit contains a written End-of-Unit Assessment for individual student completion. The Unit 8 Assessment is an End-of-Course Assessment and it includes problems from across the grade. Examples from End-of-Unit Assessments include: 

• Unit 2, Fractions as Quotients and Fraction Multiplication, End-of-Unit Assessment, Problem 3, “There are 8 ounces of pasta in the package. Jada cooks $\frac{2}{3}$ of the pasta. How many ounces of pasta did Jada cook? A. $2\frac{2}{3}$, B., C. $7\frac{1}{3}$, D, 12.” (5.NF.4a, 5.NF.6)

• Unit 3, Multiplying and Dividing Fractions, End-of-Unit Assessment, Problem 4, “440 meters is $\frac{1}{4}$ of the way around the race track. How far is it around the whole race track? Explain or show your reasoning.” (5.NF.7b, 5.NF.7c)

• Unit 5, Place Value Patterns and Decimal Operations, End-of-Unit Assessment, Problem 3, “What is 1.357 rounded to the nearest hundredth? What about to the nearest tenth? To the nearest whole number? Explain or show your reasoning.” (5.NBT.4)

• Unit 7, Shapes on the Coordinate Plane, End-of-Unit Assessment, Problem 3, “Fill in each blank with the correct word, ‘sometimes,’ ‘always,’ or ‘never.’ a. A parallelogram is ___ a rhombus. A rhombus is ___ a parallelogram. c. A rectangle is ___ a rhombus. d. A quadrilateral with a 35 degree angle is ___ a rectangle.” (5.G.3, 5.G.4)

• Unit 8, Putting It All Together, End-of-Course Assessment, Problem 1, “Select all expressions that represent the volume of this rectangular prism in cubic units. a. $5\times4\times3$, b. $(3\times4)+4$ , c. $5\times(4+3)$ , d. $3\times20$, e. $4\times15$.” A prism is shown next to the problem. (5.MD.5a, 5.MD.5b, 5.OA.A)

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for the materials giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials provide extensive work in Grade 5 as students engage with all CCSSM standards across the grade and a consistent lesson structure, including a Warm Up, one to three Instructional Activities, a Lesson Synthesis, and a Cool-Down. Examples of extensive work include:

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lessons 12 and 16; Unit 3, Multiplying and Dividing Fractions, Lesson 19; and Unit 8, Putting It All Together, Lesson 13 engage students in extensive work with 5.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction). Unit 2, Lesson 12, Decompose Area, Activity 1, students find the area of a rectangle with a whole number side length and a side length that is a mixed number. Student Facing, “1. Noah’s garden is 5 yards by $6\frac{1}{4}$ yards. Draw a diagram of Noah’s garden on the grid. 2. ​Priya’s garden is 6 yards by $5\frac{1}{4}$ yards. Draw a diagram of Priya’s garden on the grid. 3. Whose garden covers a larger area? Be prepared to explain your reasoning.” Unit 2, Lesson 16, Estimate Products, Activity 2, students reason about the value of products by rounding either the whole number or mixed number factors and multiplying. Student Facing, “1. Write a whole number product that is slightly less than, slightly greater than, or about equal to the value of $7\times12\frac{8}{9}$. a slightly less b. slightly greater c. just right. 2. Write a whole number product that is slightly less than, slightly greater than, or about equal to the value of $9\times4\frac{2}{29}$. a. slightly less b. slightly greater c. just right. 3. Without calculating, use the numbers 2, 3, 5, 6, and 7, to complete the expression with a value close to 20. (An equation model for multiplying a whole number by a mixed number is provided.) 4. Explain how you know your expression represents a value close to 20.” Unit 3, Lesson 19, Fraction Games, Warm-up: Estimation Exploration, students develop strategies for finding the product of a fraction and a mixed number. Student Facing “ Record an estimate that is: too high, about right, too low.” Unit 8, Lesson 13, Multiply Fractions Game Day, Activity 1, students practice multiplying fractions. Student Facing, “1. Use the directions to play Fraction Multiplication Compare with your partner. Spin the spinner. Write the number you spun in one of the empty boxes. Once you write a number, you cannot change it. Player two spins and writes the number on their game board. Continue taking turns until all four blank boxes are filled. Multiply your fractions. The player with the greatest product wins. Play again. 2. What strategy do you use to decide where to write the numbers?”

• Unit 5, Place Value Patterns and Decimal Operations, Lessons 11, 12, and 13 engage students in extensive work with 5.NBT.7 (Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used). Lesson 11, Make Sense of Decimal Addition, Activity 2, students use place value understanding to add decimals. Student Facing, “1. Play one round of Target Numbers. Partner A, Start at 0. Roll the number cube. Choose whether to add that number of tenths or hundredths to your starting number. Write an equation to represent the sum. Take turns until you’ve played 6 rounds. Each round, the sum from the previous equations becomes the starting number in the new equation. The partner to get a sum closest to 1 without going over wins. 2. Describe a move that you could have made differently to change the outcome of the game.” Lesson 12, Estimate and Add, Cool-down, students build on knowledge of the standard algorithm for addition from a prior unit. Student Facing, “Find the value of $3.45+21.6$. Explain or show your reasoning.” Lesson 13, Analyze Addition Mistakes, Activity 1, students analyze a common error when using the standard algorithm to add decimals, “‘Solve the first problem on your own.’ 2–3 minutes: independent work time. ‘Now, work on the second problem on your own for a few minutes, and then talk to your partner about it.’” Student Facing, “1. Find the value of $621.45+72.3$. Explain or show your reasoning. 2. Elena and Andre found the value of $621.45+72.3$. Who do you agree with? Explain or show your reasoning.” Work is shown for Elana and Andre, showing that Elena lined digits up incorrectly as she wrote the problem vertically.

• Unit 6, More Decimal and Fraction Operations, Lessons 5 and 7 engage students in extensive work with 5.MD.1 (Convert like measurement units within a given measurement system). Lesson 5, Multi-step Conversion Problems: Metric Length, Activity 2, Student Facing, students convert between meters and kilometers to decide which of two measurements is larger, “1. Use the table to find the total distance Tyler ran during the week. Explain or show your reasoning. A table with columns for day and distance (km) is shown: Monday 8.5, Tuesday 6.25, Wednesday 10.3, Thursday 5.75, Friday 9.25. 2. Use the table to find the total distance Clare ran during the week. Show your reasoning. A table with columns for day and distance (m) is shown. Monday 5,400, Tuesday 7,500, Wednesday 8,250, Thursday 6,750, Friday 7,250. 3. Who ran farther, Clare or Tyler? How much farther? Explain or show your reasoning.” Lesson 7, Multi-step Conversion Problems: Customary Length, Activity 2, students solve multi-step conversion problems using customary length units. Student Facing, “1. A rectangular field is 90 yards long and $42\frac{1}{4}$ yards wide. Priya says that 6 laps around the field is more than a mile. Do you agree with Priya? Explain or show your reasoning. 2. A different rectangular field is $408\frac{1}{2}$ feet long and $240\frac{1}{4}$ feet wide. How many laps around this field would Priya need to run if she wants to run at least 2 miles?”

The materials provide opportunities for all students to engage with the full intent of Grade 5 standards through a consistent lesson structure. According to the IM Teacher Guide, A Typical IM Lesson, “Every warm-up is an instructional routine. The warm-up invites all students to engage in the mathematics of the lesson. After the warm-up, lessons consist of a sequence of one to three instructional activities. The activities are the heart of the mathematical experience and make up the majority of the time spent in class. After the activities for the day, students should take time to synthesize what they have learned. This portion of class should take 5-10 minutes. The cool-down task is to be given to students at the end of the lesson and students are meant to work on the cool-down for about 5 minutes independently.” Examples of meeting the full intent include:

• Unit 1, Finding Volume, Lessons 10 and 12; Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 9; and Unit 8, Putting It All Together, Lesson 6 engage students in the full intent of 5.MD.5 (Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume). Unit 1, Lesson 10, Represent Volume with Expressions, Activity 1, students find volume of figures in different ways. Student Facing “1. Write an expression to represent the volume of the figure in unit cubes. 2. Compare expressions with your partner. a. How are they the same? b. How are they different? 3.If they are the same, try to find another way to represent the volume.” An image of a rectangular prism is provided. Unit 1, Lesson 12, Lots and Lots of Garbage, Activity 1, students find different ways to arrange 60 shipping containers. Student Facing, “1. Find at least 5 different ways to arrange 60 containers. Represent each arrangement with an expression. 2. Create a visual display to show which is the best arrangement for shipping the 3,300 tons of garbage.” Unit 4, Lesson 9, The Birds, Cool-Down, students calculate volume. Student Facing, “To make a birdhouse for a screech owl, the recommended area of the floor is 8 inches by 8 inches and the recommended height is 12 inches to 15 inches. What is the recommended range of volumes for a screech owl birdhouse? Explain or show your thinking.” Unit 8, Lesson 6, Revisit Volume, Activity 1, students revisit the concept that volume is the number of unit cubes required to fill a space without gaps or overlaps. Student Facing, “A company packages 126 sugar cubes in each box. The box is a rectangular prism. 1. What are some possible ways they could pack the cubes? 2. How would you choose to pack the cubes? Explain or show your reasoning. 3. The side lengths of the box are about $1\frac{7}{8}$ inches by $3\frac{3}{4}$ inches by $4\frac{3}{8}$ inches. What can we say about how the sugar cubes are packed?”

• Unit 5, Place Value Patterns and Decimal Operations, Lessons 2, 5, and 7 engage students in the full intent of 5.NBT.3 (Read, write, and compare decimals to thousandths). Lesson 2, Thousandths on Grids and in Words, Activity 2, students consider different ways to name a decimal shown on a hundredths grid. Student Facing, “Several students look at the diagram and describe the shaded region in different ways. Who do you agree with? Why? A. Jada says it’s ‘15 hundredths.’ B. Priya says it’s ‘150 thousandths.’ C. Tyler says it’s ‘15 thousandths.’ D. Diego says it’s ‘1 tenth and 5 hundredths.’ E. Mai says it’s ‘1 tenth and half of a tenth.’” Lesson 5, Compare Decimals, Cool-down, students use place value understanding to compare decimals. Student Facing, “Lin threw the frisbee 5.09 meters. Andre threw the frisbee 5.1 meters. Who threw the frisbee farther? Explain or show your reasoning.” Lesson 7, Round Doubloons, Activity 2, students examine numbers in different situations and decide if they are exact or approximate. Student Facing, “Decide if each quantity is exact or an estimate. Be prepared to explain your reasoning. 1. There are 14 pencils on the desk. 2. The population of Los Angeles is 12,400,000. 3. It's 2.4 miles from the school to the park. 4. The runner finished the race in 19.78 seconds.”

• Unit 7, Shapes on the Coordinate Plane, Lessons 4, 5, 6, and 7 engage students in the full intent of 5.G.4 (Classify two-dimensional figures in a hierarchy based on properties). Lesson 4, Sort Quadrilaterals, Activity 2, students determine appropriate categories as they sort quadrilaterals. Student facing, “Your teacher will give you a set of cards. 1. Sort all of the quadrilateral cards in a way that makes sense to you. Name the categories in your sort. 2. Sort the quadrilateral cards in a different way and name each of the categories in your new sort. Lesson 5, Trapezoids, Warm Up, students share what they know about and how they can represent trapezoids. Student facing, “What do you know about trapezoids?” Lesson 6, Hierarchy of Quadrilaterals, Activity 2, students determine if quadrilaterals are squares, rhombuses, rectangles, or parallelograms. Student facing, “1. Draw 3 different quadrilaterals on the grid, making sure at least one of them is a parallelogram. 2. For each of your quadrilaterals determine if it is a: square, rhombus, rectangle, parallelogram. Explain or show your reasoning. 3. Draw a rhombus that is not a square. Explain or show how you know it is a rhombus but not a square. 4. Draw a rhombus that is a square. Explain or show how you know it is a rhombus and a square. 5. Diego says that it is impossible to draw a square that is not a rhombus. Do you agree with him? Explain or show your reasoning.” Lesson 7, Rectangles and Squares, Activity 1, students deepen their understanding of the quadrilateral hierarchy as they recognize specific attributes. Student facing, “Spread out your shape cards so you and your partner can see all of them. Work together to find a shape that fits each clue. If you don’t think it is possible to find that shape, explain why. You can only use each shape one time. 1. Find a quadrilateral that is not a parallelogram. 2. Find a rhombus that is also a square. 3. Find a rhombus that is not a square. 4. Find a trapezoid that is not a rectangle. 5. Find a rectangle that is not a square. 6. Find a parallelogram that is not a rectangle. 7. Find a square that is not a rectangle.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade. The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade: 

• The approximate number of units devoted to major work of the grade (including assessments and supporting work connected to the major work) is 7 out of 8, approximately 88%.

• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 137 out of 156, approximately 88%. The total number of lessons devoted to major work of the grade includes 129 lessons plus 8 assessments for a total of 137 lessons.

• The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 143.5 out of 164, approximately 88%.

A lesson-level analysis is most representative of the instructional materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each unit. As a result, approximately 88% of the instructional materials focus on major work of the grade.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. 

Materials are designed so supporting standards/clusters are connected to the major standards/ clusters of the grade. These connections are listed for teachers on a document titled “Pacing Guide and Dependency Diagram” found within the Course Guide tab for each unit. Examples of connections include:

• Unit 1, Finding Volume, Lesson 9, Activity 2 connects the supporting work of 5.OA.2 (Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them) to the major work of 5.MD.5c (Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems). Students find the volume of composite figures. Student Facing states, “1. Explain how each expression represents the volume of the figure. Show your thinking. Organize it so it can be followed by others. a. $((2\times3)\times4)+((3\times3)\times2)$. b. $(5\times6)+(3\times4)$. 2. How does each expression represent the volume of the prism? Explain or show your thinking. Organize it so it can be followed by others. a. $(5\times8\times6)+(5\times4\times9)$ cubic inches. b. $(5\times4\times3)+(5\times12\times6)$ cubic inches.”

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 13, Warm-up connects the supporting work of 5.OA.1 (Use parenthesis, brackets, or braces in numerical expressions, and evaluate expressions with these symbols) to the major work of 5.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction). Students multiply a whole number and a fraction as they solve problems with grouping symbols. Student Facing states, “Find the value of each expression mentally. $5\times(7+4)$, $(5\times7)+(5\times4)$, $(5\times7)+(5\times\frac{1}{4})$, $(5\times7)-(5\times\frac{1}{4})$.”

• Unit 6, More Decimal and Fraction Operations, Lesson 14, Activity 1 connects the supporting work of 5.MD.2 (Make a line plot to display a data set of measurements in fractions of a unit [, $\frac{1}{4}$, $\frac{1}{8}$], Use operations on fractions for this grade to solve problems involving information presented in line plots) to the major work of 5.NF.A (Use equivalent fractions as a strategy to add and subtract fractions). Students make a line plot and then analyze the data to solve problems using operations with fractions. A spinner with the fractions $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, $\frac{5}{8}$ is provided. Student Facing states, “1. Play Sums of Fractions with your partner. Take turns with your partner. Spin the spinner twice. Add the two fractions. Record the sum on the line plot. Play the game until you and your partner together have 12 data points. 2. How did you know where to plot the sums of eighths? 3. What is the difference between your highest and lowest number? 4. What do you notice about the data you collected?”

The materials for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. 

Materials are coherent and consistent with the Standards. These connections can be listed for teachers in one or more of the four phases of a typical lesson: warm-up, instructional activities, lesson synthesis, or cool-down. Examples of connections include:

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 16, Activity 2 connects the major work of 5.NBT.B (Perform operations with multi-digit whole numbers and with decimals to hundredths) to the major work of 5.NF.B (Apply and extend previous understandings of multiplication and division to multiply and divide fractions). Students consider the most precise estimate for a fractional length, connecting division to what they know about fractions. Student Facing states, “Han said that each person will get about $25\frac{1}{4}$ feet of noodle. Do you agree with Han? Explain or show your reasoning.” The problem context states that 400 people equally shared a 10,119 foot noodle. Activity Synthesis states, “Display: $25\frac{119}{400}$. ‘What does $25\frac{119}{400}$ mean in this situation?’ (Each person gets 25 feet of the noodle and then the 119 feet leftover would be divided into 400 equal pieces.) Display: $25\frac{1}{4}$25 ’Why is Han's estimate reasonable?’ (Because is $\frac{119}{400}$ really close to $\frac{100}{400}$ and $\frac{100}{400}=\frac{1}{4}$) ‘Do you think they actually measured and cut the noodle into equal pieces when they served it?’ (No, because it would take too long and be too difficult. Yes, because if long noodles represent long life they probably want to serve the noodle soup with sections that are one piece of the original noodle.).”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 21, Warm-Up connects the major work of 5.NBT.A (Understand the place value system) and 5.NBT.B (Perform operations with multi-digit whole numbers and with decimals to hundredths). Students apply what they know about area and multiplication of decimals to a situation where the side length of the rectangle are decimals. Student Facing states, “Central Park is a large park in Manhattan. It is about 3.85 kilometers long and 0.79 km wide. What is the area of Central Park? Record an estimate that is: too low, about right, too high.” Activity Synthesis states, “Invite students to share their estimates. ‘How do you know the area is greater than 2 square kilometers?’ (I know that 3 x 0.7 is 21 tenths or 2.1 and it’s more than that.) ‘How do you know the area is less than 3.2 square kilometers?’ (I know 3.85 is less than 4 and 0.79 is less than 0.8. Then $4\times0.8$ is 32 tenths or 3.2.).”

• Unit 7, Shapes on the Coordinate Plane, Lesson 11, Warm-Up connects the supporting work of 5.OA.B (Analyze patterns and relationships) to the supporting work of 5.G.A (Graph points on the coordinate plane to solve real-world and mathematical problems). Students look for patterns in points plotted on a coordinate grid. Student Facing states, “What do you notice? What do you wonder?” Students see a coordinate grid with points plotted in the first quadrant. Student Response includes, “Students may notice: The points are scattered, There are 4 points labeled A–D, Points B and D are on the same horizontal line, The numbers on the vertical and horizontal axis skip count by two, Some points are not on the vertices of the grid. Students may wonder: What do the points represent? Can we connect the points? If we connect the points, what shape will it make?”

• Unit 8, Putting It All Together, Lesson 8, Activity 1 connects the major work of 5.MD.C (Geometric measurement: Understand concepts of volume and relate volume to multiplication and to addition) to the major work of 5.NBT.B (Perform operations with multi-digit whole numbers and with decimals to hundredths). Students apply what they know about multiplication and division to solve problems involving the volume of the Radio Flyer, a rectangular prism. The Launch states, “Groups of 2. Display: 27 feet long, 13 feet wide, 2 feet deep. ‘These are the approximate dimensions of the actual Radio Flyer. How do they compare to the estimates you made in the previous lesson?’ (We were close for the length and depth but the actual wagon is wider than what we guessed.). ‘Imagine the wagon was being filled with sand. Would you want to buy large bags of sand or small bags of sand? Why?’ (I would want large bags because it would take fewer of them.).” Student Facing states, “The Radio Flyer wagon is 27 feet long, 13 feet wide and 2 feet deep. 1. A 150-pound bag of sand will fill about 9 cubic feet. How many bags of sand will it take to fill the wagon with sand? 2. A 150-pound bag of sand costs about $12. About how much will it cost to fill the wagon with sand? Explain or show your reasoning. 3. How many pounds of sand does the Radio Flyer hold when it is full? Explain or show your reasoning.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. 

Prior and Future connections are identified within materials in the Course Guide, Section Dependency Diagrams which state, “an arrow indicates the prior section that contains content most directly designed to support or build toward the content in the current section.” While future connections are all embedded within the Scope and Sequence, descriptions of prior connections are also found within the Preparation tab for specific lessons, and within the notes for specific parts of lessons. 

Examples of connections to future grades include:

• Course Guide, Scope and Sequence, Unit 3, Multiplying and Dividing Fractions, Section B, Section Learning Goals connect 5.NF.7 (Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions) to the work of interpreting and computing quotients of fractions in 6.NS.1. Lesson Narrative states, “Students may notice that to find $5\div\frac{1}{2}$, they can multiply 5 by 2 because there are 2 halves in each of the 5 wholes. It is not essential, however, that students generalize division of fractions at this point, as they will do so in grade 6.”

• Course Guide, Scope and Sequence, Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section B, Section Learning Goals connect 5.NBT.6 (Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models) to the work of dividing multi-digit numbers using the standard algorithm in 6.NS.2. Lesson Narrative states, “Students see that some decompositions may be more helpful than others for finding whole-number quotients. They use this insight to make sense of algorithms using partial quotients that are more complex. Note that use of the standard algorithm for division is not an expectation in grade 5, but students can begin to develop the conceptual understanding needed to do so. The algorithms using partial quotients seen here are based on place value, which will allow students to make sense of the logic of the standard algorithm they’ll learn in grade 6.”

• Course Guide, Scope and Sequence, Unit 5, Place Value Patterns and Decimal Operations, Unit Learning Goals connect the work of 5.NBT.7 (Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used) to work with operations with decimals in Grade 6. Lesson Narrative states, “Students then apply their understanding of decimals and of whole-number operations to add, subtract, multiply, and divide decimal numbers to the hundredths, using strategies based on place value and the properties of operations. They see that the reasoning strategies and algorithms they used to operate on whole numbers are also applicable to decimals. For example, addition and subtraction can be done by attending to the place value of the digits in the numbers, and multiplication and division can still be understood in terms of equal-size groups. In grade 6, students will build on the work here to reach the expectation to fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.”

Examples of connections to prior knowledge include:

• Unit 1, Finding Volume, Lesson 1, Preparation connects 5.MD.3 (Recognize volume as an attribute of solid figures and understand concepts of volume measurement) to the work with concepts of area from Grades 3 and 4. Lesson Narrative states, “In previous grades, students learned that they can count the number of square tiles that cover a plane shape without gaps or overlaps to find the area of the shape. In this lesson, students explore the concept of volume as they build and compare objects made of cubes. Students learn that objects can have different shapes but still take up the same amount of space and that we call this amount an object’s volume.”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 1, Preparation connects 5.NBT.A (Understand the place value system) to work with decimal fractions from 4.NF.C. Lesson Narrative states, “In grade 4, students studied decimal fractions with denominators 10 and 100. They represented tenths and hundredths with hundredths grids, number lines, and decimal notation. In this lesson students make sense of representations of tenths, hundredths, and thousandths with hundredths grid diagrams, fractions, and decimals. They also see relationships between these values, namely that a tenth of a tenth is a hundredth and a tenth of a hundredth is a thousandth. Students may use informal language to describe the relationship between decimals (for example, to get from 0.01 to .001 you add a zero in front of the one.) This language supports students in sharing their developing understanding. Teachers should ask questions to help students develop more precise language to describe base-ten representations (for example, what does the extra 0 you wrote in .001 represent?). They will have many opportunities to develop this understanding in upcoming lessons.”

• Unit 7, Shapes on the Coordinate Plane, Lesson 4, Warm-up connects 5.G.3 (Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category) to work with classifying two-dimensional shapes from Grade 4. Narrative states, “The purpose of this What Do You Know About ___ is for students to share what they know about and how they can represent quadrilaterals. In previous courses students have drawn and described squares, rectangles, and rhombuses and they will revisit and classify all of these shapes over the next several lessons.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. This is located within IM Curriculum, How to Use These Materials, and the Course Guide, Scope and Sequence. Examples include:

• IM Curriculum, How To Use These Materials, Design Principles, Coherent Progression provides an overview of the design and implementation guidance for the program, “The overarching design structure at each level is as follows: Each unit starts with an invitation to the mathematics. The first few lessons provide an accessible entry point for all students and offer teachers the opportunity to observe students’ prior understandings. Each lesson starts with a warm-up to activate prior knowledge and set up the work of the day. This is followed by instructional activities in which students are introduced to new concepts, procedures, contexts, or representations, or make connections between them. The lesson ends with a synthesis to consolidate understanding and make the learning goals of the lesson explicit, followed by a cool-down to apply what was learned. Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals. In each of the activities, care has been taken to choose contexts and numbers that support the coherent sequence of learning goals in the lesson.”

• Course Guide, Scope and Sequence, provides an overview of content and expectations for the units, “The big ideas in grade 5 include: developing fluency with addition and subtraction of fractions, developing understanding of multiplication and division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions), extending division to two-digit divisors, developing understanding of operations with decimals to hundredths, developing fluency with whole number and decimal operations, and developing understanding of volume.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Several components focus specifically on the content of the lesson. Preparation and lesson narratives within the Warm-up, Activities, and Cool-down provide useful annotations. Examples include:

• Unit 1, Finding Volume, Lesson 4, Activity 1, teachers are provided context to support students when reasoning about the volume of prisms. Narrative states, “This activity continues to develop the idea of decomposing rectangular prisms into layers. Students explicitly multiply the number of cubes in a base layer by the number of layers. Students can use any layer in the prism as the base layer as long as the height is the number of those base layers.” Launch states, “Groups of 2. Display first image from student workbook. ‘What do you know about the volume of this prism? What would you need to find out to find the exact volume of this prism? You are going to work with prisms that are only partially filled in this activity.’ Give students access to connecting cubes.” Activity states, “5 minutes: independent work time. 5 minutes: partner work time. As students work, monitor for: students who notice that prisms A and D and prisms B and C are “the same” but they are sitting on different faces so the layers might be counted in different ways. Students who reason about the partially filled prisms by referring to the cubes in one layer they would see if all of the cubes were shown. Students who recognize that there are several different layers they can use to determine the volume of a prism, all of which result in the same volume.”

• Unit 7, Shapes on the Coordinate Plane, Lesson 10, Lesson Synthesis provides teachers guidance about how to support students to find patterns given two rules, “Today we noticed and explained relationships between patterns. Some of the relationships involved fractions. What relationships did you find between the patterns we studied today? (Sometimes I could multiply each term in one pattern by the same number to get the corresponding number in the other pattern. Sometimes that number was a fraction.) Consider asking students to record their response in a math journal and then share their response with a partner.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject.

Within the Teacher’s Guide, IM Curriculum, About These Materials, there are sections entitled “Further Reading” that consistently link research to pedagogy. There are adult-level explanations, including examples of the more complex grade-level concepts and concepts beyond the grade, so that teachers can improve their own understanding of the content. Professional articles support teachers with learning opportunities about topics such as ensuring mathematical success for all, early understanding of equality, and repeating patterns. Additionally, each lesson provides teachers with a lesson narrative, including adult-level explanations and examples of the more complex grade/course-level concepts. Examples include:

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 2, Preparation, Lesson Narrative states, “In the previous lesson, students explored the relationship between fractions and division by representing situations where some people shared some sandwiches. They used informal language to describe how they knew each person got about the same amount of sandwich. In this lesson, students recognize the relationship between a fraction and a division expression. For example, $\frac{1}{5}=1\div5$. Students interpret $1\div5$ as the amount in one group when a single whole is divided into 5 equal portions. They see that the quantity in that portion is $\frac{1}{5}$ of a whole.”

• IM K-5 Math Teacher Guide, About These Materials, Unit 3, “Why is a negative times a negative a positive? In this blog post, McCallum discusses how the ‘rule’ for multiplying negative numbers is grounded in the distributive property.” 

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 23, Preparation, Lesson Narrative states, “In the previous lesson, students divided whole numbers by one tenth and one hundredth and made generalizations about how to divide any whole number by those units. The purpose of this lesson is for students to extend that work to divide whole numbers by any number of tenths or hundredths (with total value less than 1). Consistent divisors are used in repetition to highlight relationships between the dividends and the quotients (MP8). Students evaluate expressions with larger divisors such as $12\div0.2$ in order to encourage them to use the relationship between multiplication and division. Rather than drawing 12 unit squares and dividing all of them into groups of 2 tenths, students may draw a single whole divided into 2 tenths and then use multiplication.”

• IM K-5 Math Teacher Guide, About These Materials, Unit 7, Making Sense of Distance in the Coordinate Plane, “In this blog post, Richard shares how understanding of the coordinate plane, introduced in grade 5, provides a foundation for conceptual understanding of distance and the Pythagorean Theorem.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Correlation information is present for the mathematics standards addressed throughout the grade level/series and can be found in several places, including the curriculum course guide, within unit resources, and within each lesson. Examples include:

• Grade 5, Course Guide, Lesson Standards includes a table with each grade-level lesson (in columns) and aligned grade-level standards (in rows). Teachers can search any lesson for the grade and identify the standard(s) that are addressed within.

• Grade 5, Course Guide, Lesson Standards, includes all Grade 5 standards and the units and lessons each standard appears in. Teachers can search a standard for the grade and identify the lesson(s) where it appears within materials.

• Unit 1, Finding Volume, Lesson 9, the Core Standards are identified as 5.MD.C.5c and 5.OA.A.2. Lessons contain a consistent structure: a Warm-up that includes Narrative, Launch, Activity, Activity Synthesis; Activity 1, 2, or 3 that includes Narrative, Launch, Activity; an Activity Synthesis; a Lesson Synthesis; and a Cool-down that includes Responding to Student Thinking and Next Day Supports. This provides an additional place to reference standards, and language of the standard, within each lesson.

• Unit 3, Resources, Teacher Guide, outlines standards, learning targets and the lesson where they appear. This is present for all units and allows teachers to identify targeted standards for any lesson.

Each unit includes an overview outlining the content standards addressed within as well as a narrative describing relevant prior and future content connections. Examples include:

• Grade 5, Course Guide, Scope and Sequence, Unit 1: Finding Volume, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “In grade 3, students learned that the area of a two-dimensional figure is the number of square units that cover it without gaps or overlaps. They first found areas by counting squares and began to intuit that area is additive. Later, they recognized the area of a rectangle as a product of its side lengths and found the area of more-complex figures composed of rectangles. Here, students learn that the volume of a solid figure is the number of unit cubes that fill it without gaps or overlaps. First, they measure volume by counting unit cubes and observe its additive nature. They also learn that different solid figures can have the same volume. Next, they shift their focus to right rectangular prisms: building them using unit cubes, analyzing their structure, and finding their volume. They write numerical expressions to represent their reasoning strategies and work with increasingly abstract representations of prisms.”

• Grade 5, Course Guide, Scope and Sequence, Unit 4: Wrapping Up Multiplication and Division with Multi-digit Numbers, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “In this unit, students multiply multi-digit whole numbers using the standard algorithm and begin working toward end-of-grade expectation for fluency. They also find whole-number quotients with up to four-digit dividends and two-digit divisors. In grade 4, students used strategies based on place value and properties of operations to multiply a one-digit whole number and a whole number of up to four digits, and to multiply a pair of two-digit numbers. They decomposed the factors by place value, and used diagrams and algorithms using partial products to record their reasoning. Here, students build on those strategies to make sense of the standard algorithm for multiplication. They recognize that it is also based on place value but records the partial products in a condensed way.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

The IM K-5 Math Teacher Guide, Design Principles, outlines the instructional approaches of the program, “It is our intent to create a problem-based curriculum that fosters the development of mathematics learning communities in classrooms, gives students access to the mathematics through a coherent progression, and provides teachers the opportunity to deepen their knowledge of mathematics, student thinking, and their own teaching practice. In order to design curriculum and professional learning materials that support student and teacher learning, we need to be explicit about the principles that guide our understanding of mathematics teaching and learning. This document outlines how the components of the curriculum are designed to support teaching and learning aligning with this belief.” Examples of the design principles include:

• IM K-5 Math Teacher Guide, Design Principles, All Students are Capable Learners of Mathematics, “All students, each with unique knowledge and needs, enter the mathematics learning community as capable learners of meaningful mathematics. Mathematics instruction that supports students in viewing themselves as capable and competent must leverage and build upon the funds of knowledge they bring to the classroom. In order to do this, instruction must be grounded in equitable structures and practices that provide all students with access to grade-level content and provide teachers with necessary guidance to listen to, learn from, and support each student. The curriculum materials include classroom structures that support students in taking risks, engaging in mathematical discourse, productively struggling through problems, and participating in ways that make their ideas visible. It is through these classroom structures that teachers will have daily opportunities to learn about and leverage their students’ understandings and experiences and how to position each student as a capable learner of mathematics.”

• IM K-5 Teacher Guide, Design Principles, Coherent Progression, “Each unit starts with an invitation to the mathematics. The first few lessons provide an accessible entry point for all students and offer teachers the opportunity to observe students’ prior understandings. Each lesson starts with a warm-up to activate prior knowledge and set up the work of the day. This is followed by instructional activities in which students are introduced to new concepts, procedures, contexts, or representations, or make connections between them. The lesson ends with a synthesis to consolidate understanding and make the learning goals of the lesson explicit, followed by a cool-down to apply what was learned. Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals. In each of the activities, care has been taken to choose contexts and numbers that support the coherent sequence of learning goals in the lesson.” 

• IM K-5 Teacher Guide, Design Principles, Learning Mathematics by Doing Mathematics, “Students learn mathematics by doing mathematics, rather than by watching someone else do mathematics or being told what needs to be done. Doing mathematics can be defined as learning mathematical concepts and procedures while engaging in the mathematical practices—making sense of problems, reasoning abstractly and quantitatively, making arguments and critiquing the reasoning of others, modeling with mathematics, making appropriate use of tools, attending to precision in their use of language, looking for and making use of structure, and expressing regularity in repeated reasoning. By engaging in the mathematical practices with their peers, students have the opportunity to see themselves as mathematical thinkers with worthwhile ideas and perspectives. ‘Students learn mathematics as a result of solving problems. Mathematical ideas are the outcomes of the problem-solving experience rather than the elements that must be taught before problem solving’ (Hiebert et al., 1996). A problem-based instructional framework supports teachers in structuring lessons so students are the ones doing the problem solving to learn the mathematics. The activities and routines are designed to give teachers opportunities to see what students already know and what they can notice and figure out before having concepts and procedures explained to them.”

Research-based strategies are cited and described within the IM Curriculum and can be found in various sections of the IM K-5 Math Teacher Guide. Examples of research-based strategies include:

• IM K-5 Math Teacher Guide, About These Materials, 3–5, Fraction Division Parts 1–4, “In this four-part blog post, McCallum and Umland discuss fraction division. They consider connections between whole-number division and fraction division and how the two interpretations of division play out with fractions with an emphasis on diagrams, including a justification for the rule to invert and multiply. In Part 4, they discuss the limitations of diagrams for solving fraction division problems. Fraction Division Part 1: How do you know when it is division? Fraction Division Part 2: Two interpretations of division Fraction Division Part 3: Why invert and multiply? Fraction Division Part 4: Our final post on this subject (for now). Untangling fractions, ratios, and quotients. In this blog post, McCallum discusses connections and differences between fractions, quotients, and ratios.“

• IM K-5 Math Teacher Guide, Design Principles, Using the 5 Practices for Orchestrating Productive Discussions, “Promoting productive and meaningful conversations between students and teachers is essential to success in a problem-based classroom. The Instructional Routines section of the teacher course guide describes the framework presented in 5 Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011) and points teachers to the book for further reading. In all lessons, teachers are supported in the practices of anticipating, monitoring, and selecting student work to share during whole-group discussions. In lessons in which there are opportunities for students to make connections between representations, strategies, concepts, and procedures, the lesson and activity narratives provide support for teachers to also use the practices of sequencing and connecting, and the lesson is tagged so teachers can easily identify these opportunities. Teachers have opportunities in curriculum workshops and PLCs to practice and reflect on their own enactment of the 5 Practices.”

• IM K-5 Math Teacher Guide, Key Structures in This Course, Student Journal Prompts, “Writing can be a useful catalyst in learning mathematics because it not only supplies students with an opportunity to describe their feelings, thinking, and ideas clearly, but it also serves as a means of communicating with other people (Baxter, Woodward, Olson, & Robyns, 2002; Liedtke & Sales, 2001; NCTM, 2000). NCTM (1989) suggests that writing about mathematics can help students clarify their ideas and develop a deeper understanding of the mathematics at hand.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Course Guide includes a section titled “Required Materials” that includes a breakdown of materials needed for each unit and for each lesson. Additionally, specific lessons outline materials to support the instructional activities and these can be found on the “Preparation” tab in a section called “Required Materials.” Examples include:

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 17, Activity 1, “Colored Paper, Glue, Rulers, Scissors.” Launch states, “Groups of 4. Distribute materials. Make sure each student in the group gets a different color paper.”

• Course Guide, Required Materials for Grade 5, Materials Needed for Unit 3, Lesson 9, teachers need, “Colored pencils or crayons, Paper, Rulers.” 

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 19, Activity 1, Required Materials, “Materials to Copy: Decimal Multiplication Expression Card Sort.” Launch states, “Groups of 2. Distribute one set of pre-cut cards to each group of students.” Activity states, “‘In this activity, you will sort some expressions into categories of your choosing. When you sort the expressions, work with your partner to come up with categories.’ 3 minutes: partner work time.”

• Course Guide, Required Materials for Grade 5, Materials Needed for Unit 8, Lesson 14, teachers need, “Chart paper, Colored pencils, crayons, or markers.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics as suggestions are outlined within each lesson and parts of each lesson. According to the IM K-5 Teacher Guide, Universal Design for Learning and Access for Students with Disabilities, “These materials empower all students with activities that capitalize on their existing strengths and abilities to ensure that all learners can participate meaningfully in rigorous mathematical content. Lessons support a flexible approach to instruction and provide teachers with options for additional support to address the needs of a diverse group of students, positioning all learners as competent, valued contributors. When planning to support access, teachers should consider the strengths and needs of their particular students. The following areas of cognitive functioning are integral to learning mathematics (Addressing Accessibility Project, Brodesky et al., 2002). Conceptual Processing includes perceptual reasoning, problem solving, and metacognition. Language includes auditory and visual language processing and expression. Visual-Spatial Processing includes processing visual information and understanding relation in space of visual mathematical representations and geometric concepts. Organization includes organizational skills, attention, and focus. Memory includes working memory and short-term memory. Attention includes paying attention to details, maintaining focus, and filtering out extraneous information. Social-Emotional Functioning includes interpersonal skills and the cognitive comfort and safety required in order to take risks and make mistakes. Fine-motor Skills include tasks that require small muscle movement and coordination such as manipulating objects (graphing, cutting with scissors, writing).” 

Examples of supports for special populations include: 

• Unit 1, Finding Volume, Lesson 3, Activity 2, Narrative, Access for Students with Disabilities, “Representation: Access for Perception. Invite students to identify correspondences between the visual representation and the prism made of connecting cubes. Make connections between representations visible through gestures or labeled displays. Supports accessibility for: Visual-Spatial Processing, Conceptual Processing.” 

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 3, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were needed to solve the problem. Display the sentence frame, ‘The next time I write division equations, I will pay attention to . . .’ Supports accessibility for: Conceptual Processing, Memory, Language.”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 8, Activity 2, Narrative, Access for Students with Disabilities, “Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy for determining proximity before they begin. Students can speak quietly to themselves, or share with a partner. Supports accessibility for: Organization, Conceptual Processing, Language.”

• Unit 7, Shapes on the Coordinate Plane, Lesson 5, Activity 2, Narrative, Access for Students with Disabilities, “Engagement: Provide Access by Recruiting Interest. Synthesis: Optimize meaning and value. Invite students to share the meaning of a trapezoid and the similarities and differences in the two definitions of a trapezoid with a classmate who missed the lesson. Supports accessibility for: Conceptual Processing; Language.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

While there are no instances where advanced students do more assignments than classmates, materials do provide multiple opportunities for students to investigate grade-level content at a higher level of complexity. These are found in a section titled “Exploration Problems” within lessons where appropriate. According to the IM K-5 Teacher Guide, How To Use The Materials, Exploration Problems, “Each practice problem set also includes exploration questions that provide an opportunity for differentiation for students ready for more of a challenge. There are two types of exploration questions. One type is a hands-on activity directly related to the material of the unit that students can do either in class if they have free time, or at home. The second type of exploration is more open-ended and challenging. These problems go deeper into grade-level mathematics. They are not routine or procedural, and they are not just the same thing again but with harder numbers. Exploration questions are intended to be used on an opt-in basis by students if they finish a main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in exploration problems, and it is not expected that any student works on all of them. Exploration problems may also be good fodder for a Problem of the Week or similar structure.” Examples include:

• Unit 2, Fractions as Quotients and Fraction Multiplication, Section B: Fractions of Whole Numbers, Problem 5, Exploration, “A standard rectangular sheet of paper measures $8\frac{1}{2}$ inches in width and 11 inches in length. How many square inches are there in a sheet of paper? If you get stuck, consider using the grid.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section B: Multi- Digit Division Using Partial Quotients, Problem 7, Exploration, “1. Andre made a noodle that was 102 feet long. The noodle broke into two pieces. One piece was 2 times as long as the other. How long were the two noodles? Explain your reasoning. 2. Priya made a noodle that was 456 feet long. The noodle broke into two pieces. One piece was 5 times as long as the other. How long were the two noodles? Explain your reasoning.”

• Unit 5, Place Value Patterns and Decimal Operations, Section B: Add and Subtract Decimals, Problem 8, Exploration, “Lin is trying to use the digits 1, 3, 4, 2, 5, and 6 to make 2 two-digit decimals whose sum is equal to 1. 1. Explain why Lin can not make 1 by adding together 2 two-digit decimal numbers made with these digits. 2. What is the closest Lin can get to 1? Explain how you know.”

• Unit 7, Shapes on the Coordinate Plane, Section C: Numerical Patterns, Problem 7, Exploration, “Andre starts from 2 and counts by 6s. Clare starts at 1,000 and counts back by 7s. 1. List the first 6 numbers Andre and Clare say. 2. Do Andre and Clare ever say the same number in the same spot on their lists? Explain or show your reasoning.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Guidance is consistently provided for teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the IM K-5 Math Teacher Guide, Mathematical Language Development and Access for English Learners, “In a problem-based mathematics classroom, sense-making and language are interwoven. Mathematics classrooms are language-rich, and therefore language demanding learning environments for every student. The linguistic demands of doing mathematics include reading, writing, speaking, listening, conversing, and representing (Aguirre & Bunch, 2012). Students are expected to say or write mathematical explanations, state assumptions, make conjectures, construct mathematical arguments, and listen to and respond to the ideas of others. In an effort to advance the mathematics and language learning of all students, the materials purposefully engage students in sense-making and using language to negotiate meaning with their peers. To support students who are learning English in their development of language, this curriculum includes instruction devoted to fostering language development alongside mathematics learning, fostering language-rich environments where there is space for all students to participate.” The series provides the following principles that promote mathematical language use and development: 

• “Principle 1. Support sense-making: Scaffold tasks and amplify language so students can make their own meaning. 

• Principle 2. Optimize output: Strengthen opportunities for students to describe their mathematical thinking to others, orally, visually, and in writing. 

• Principle 3. Cultivate conversation: Strengthen opportunities for constructive mathematical conversations. 

• Principle 4. Maximize meta-awareness: Strengthen the meta-connections and distinctions between mathematical ideas, reasoning, and language.” 

The series also provides Mathematical Language Routines in each lesson. According to the IM K-5 Math Teacher Guide, Mathematical Language Development and Access for English Learners, “Mathematical Language Routines (MLRs) are instructional routines that provide structured but adaptable formats for amplifying, assessing, and developing students' language. MLRs are included in select activities in each unit to provide all students with explicit opportunities to develop mathematical and academic language proficiency. These ‘embedded’ MLRs are described in the teacher notes for the lessons in which they appear.” Examples include:

• Unit 3, Multiplying and Dividing Fractions, Lesson 8, Activity 1, Teaching notes, Access for English Learners, “MLR7 Compare and Connect. Invite students to prepare a visual display that shows the strategy they used to calculate the area of different parts of the flag. Encourage students to include details that will help others interpret their thinking. For example, specific language, using different colors, shading, arrows, labels, notes, diagrams, or drawings. Give students time to investigate each other’s work. During the whole-class discussion, ask students, ‘What did the approaches have in common?’, ‘How were they different?’, and ‘Did anyone solve the problem the same way, but would explain it differently?’ Advances: Representing, Conversing.”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 2, Activity 1, Teaching notes, Access for English Learners, “MLR2 Collect and Display. Circulate, listen for, and collect the language students use as they shade and interpret diagrams. On a visible display, record words and phrases such as: ‘fraction,’ ‘part of,’ ‘decimal,’ ‘tenths,’ ‘row,’ ‘hundredths,’ ‘thousandths,’ ‘represents,’ ‘shows.’ Invite students to borrow language from the display as needed, and update it throughout the lesson. Advances: Conversing, Reading.”

• Unit 8, Putting It All Together, Lesson 14, Activity 2, Teaching notes, Access for English Learners, “MLR8 Discussion Supports. At the appropriate time, give students 2–3 minutes to make sure that everyone in their group can explain their Notice and Wonder activity. Invite groups to rehearse what they will say when they share with the whole class. Advances: Speaking, Conversing, Representing.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for providing manipulatives, physical but not virtual, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Suggestions and/or links to manipulatives are consistently included within materials, often in the Launch portion of lessons, to support the understanding of grade-level math concepts. Examples include: 

• Unit 1, Finding Volume, Lesson 2, Activity, 1, references connecting cubes to build understanding of volume. Student Facing states, “Partner A: Build an object using 8–12 cubes and give the object to Partner B. Partner B: Explain how you would count the number of cubes in the object. Partner A: Explain if you would count the cubes in the same way or in a different way. Switch roles and repeat. Which objects were easiest to count? Why?” A picture of a house built out of connecting cubes is shown. Launch states, “Groups of  2. Give 24 connecting cubes to each group. ‘In this activity, you will use unit cubes to build objects and describe how you would measure the volume.’ 10 minutes: partner work time.” Activity states, “As students work, monitor for students who build rectangular prisms to share during the synthesis.”

• Unit 3, Multiplying and Dividing Fractions, Lesson 9, Activity 2, identifies colored pencils or crayons, paper, and rulers to design and analyze a flag, including operations with fractions. Launch states, “Give each student white paper. Use the design principles we discussed in the last activity to make your own flag. As you make the design, think about the meaning of each symbol and color you use.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 7, Activity 1, references a game called Greatest Product and Number Cards (0-10), where students create the greatest product from expressions. Launch states, “Give each group a set of number cards and 2 copies of the blackline master. Remove the cards that show 10 and set them aside. ‘We’re going to play a game called Greatest Product. Let’s read through the directions and play one round together.’ Read through the directions with the class and play a round against the class using the diagram in the student workbook. Display each number card. Think through your choices aloud. Record your move and score for all to see.”

• Unit 6, More Decimal and Fraction Operations, Lesson 3, Activity 1, identifies meter sticks to help students convert meters to centimeters. Launch states, “Give students access to meter sticks. Display image from student workbook. ‘What do you notice? What do you wonder?’ Display additional information about track and field events: The height of a hurdle is 1 meter. The approximate distance between hurdles in 110 meter races is 10 meters. The shortest race in many track competitions is 100 meters. ‘Work with you partner to complete the problems.’”