## Kendall Hunt’s Illustrative Mathematics

##### v1.5
###### Usability
Our Review Process

Title ISBN Edition Publisher Year
Kendal Hunt's Illustrative Mathematics Grade 1 978-1-7924-6275-7 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 1 978-1-7924-6289-4 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 4 978-1-7924-6278-8 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 4 978-1-7924-6292-4 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 3 978-1-7924-6277-1 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 3 978-1-7924-6291-7 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Kindergarten 978-1-7924-6274-0 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Kindergarten 978-1-7924-6287-0 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 5 978-1-7924-6279-5 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 5 978-1-7924-6293-1 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 2 978-1-7924-6276-4 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 2 978-1-7924-6290-0 2021 Kendall Hunt Publishing Company 2021
Showing:

### Overall Summary

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and meet expectations for practice-content connections.

###### Alignment
Meets Expectations
###### Usability
Meets Expectations

### Focus & Coherence

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for focus and coherence. For focus, the materials assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards. For coherence, the materials are coherent and consistent with the CCSSM.

##### Gateway 1
Meets Expectations

#### Criterion 1.1: Focus

Materials assess grade-level content and give all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for focus as they assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards.

##### Indicator {{'1a' | indicatorName}}

Materials assess the grade-level content and, if applicable, content from earlier grades.

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.$$5\frac{1}{3}$$, 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)

##### Indicator {{'1b' | indicatorName}}

Materials give all students extensive work with grade-level problems to meet the full intent of grade-level standards.

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 “$$28\times2\frac{8}{9}$$ 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 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.”

#### Criterion 1.2: Coherence

Each grade’s materials are coherent and consistent with the Standards.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for coherence. The materials: address the major clusters of the grade, have supporting content connected to major work, make connections between clusters and domains, and have content from prior and future grades connected to grade-level work.

##### Indicator {{'1c' | indicatorName}}

When implemented as designed, the majority of the materials address the major clusters of each grade.

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.

##### Indicator {{'1d' | indicatorName}}

Supporting content enhances focus and coherence simultaneously by engaging students in the 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 [$$\frac{1}{2}$$, \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?”

##### Indicator {{'1e' | indicatorName}}

Materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

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.” ##### Indicator {{'1f' | indicatorName}} 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. 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.” ##### Indicator {{'1g' | indicatorName}} In order to foster coherence between grades, materials can be completed within a regular school year with little to no modification. The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 foster coherence between grades and can be completed within a regular school year with little to no modification. According to the IM K-5 Teacher Guide, About These Materials, “Each grade level contains 8 or 9 units. Units contain between 8 and 28 lesson plans. Each unit, depending on the grade level, has pre-unit practice problems in the first section, checkpoints or checklists after each section, and an end-of-unit assessment. In addition to lessons and assessments, units have aligned center activities to support the unit content and ongoing procedural fluency. The time estimates in these materials refer to instructional time. Each lesson plan is designed to fit within a class period that is at least 60 minutes long. Some units contain optional lessons and some lessons contain optional activities that provide additional student practice for teachers to use at their discretion.” In Grade 5, there are 164 days of instruction including: • 148 lesson days • 16 unit assessment days There are eight units in Grade 5 and, within those units, there are between 12 and 26 lessons. According to the IM K-5 Teacher Guide, A Typical IM Lesson, “A typical lesson has four phases: 1. a warm-up 2. one or more instructional activities 3. the lesson synthesis 4. a cool-down.” There is a Preparation tab for lessons, including specific guidance and time allocations for each phase of a lesson. In Grade 5, each lesson is composed of: • 10 minutes Warm-up • 10-25 minutes (each) for one to three Instructional Activities • 10 minutes Lesson Synthesis • 5 minutes Cool-down ###### Overview of Gateway 2 ### Rigor & the Mathematical Practices The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for rigor and balance and practice-content connections. The materials help students develop procedural skills, fluency, and application. The materials also make meaningful connections between the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs). ##### Gateway 2 Meets Expectations #### Criterion 2.1: Rigor and Balance Materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications. The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for rigor. The materials develop conceptual understanding of key mathematical concepts, give attention throughout the year to procedural skill and fluency, and spend sufficient time working with engaging applications of mathematics. There is a balance of the three aspects of rigor within the grade. ##### Indicator {{'2a' | indicatorName}} Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Materials develop conceptual understanding throughout the grade level. According to IM K-5 Math Teacher Guide, Design Principles, conceptual understanding is a part of the design of the materials. Balancing Rigor states, “Opportunities to connect new representations and language to prior learning support students in building conceptual understanding. Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations.” Additionally, Purposeful Representations states, “Across lessons and units, students are systematically introduced to representations and encouraged to use representations that make sense to them. As their learning progresses, students are given opportunities to make connections between different representations and the concepts and procedures they represent.” Examples include: • Unit 1, Finding Volume, Lesson 1, Activity 1, students develop conceptual understanding of volume as they recognize that objects with the same volume take up the same amount of space. Students are given different pictures of pattern block formations. Student Facing states, “1. Which is bigger? Explain or show your reasoning. 2. Which is bigger? Explain or show your reasoning. 3. What does it mean for an object to be ‘bigger’?” (5.MD.3) • Unit 5, Place Value Patterns and Decimal Operations, Lesson 5, Warm-up, students develop conceptual understanding as they use place value understanding to compare decimals to the thousandths place. Student Facing states, “Decide if each statement is true or false. Be prepared to explain your reasoning. 7.06=7.006, 7.06=7.060, 7.06=7.600.” (5.NBT.3) • Unit 8, Putting It All Together, Lesson 10, Activity 1, students develop conceptual understanding as they practice adding fractions with unlike denominators and reason about how the size of the numerators and denominators impact the value of a fraction. Student Facing states, “Use the directions to play Greatest Sum with a partner. 1. Spin the spinner. 2. Each player writes the number that was spun in an empty box for Round 1. Be sure your partner cannot see your paper. 3. Once a number is written down, it cannot be changed. 4. Continue spinning and writing numbers in the empty boxes until all 4 boxes have been filled. 5. Find the sum. 6. The person with the greater sum wins the round. 7. After all 4 rounds, the player who won the most rounds wins the game. 8. If there is a tie, players add the sums from all 4 rounds and the highest total sum wins the game. Total sum of all 4 rounds: ___.” Activity Synthesis states, “‘What strategies were helpful as you played Greatest Sum?’ (I tried to make fractions that have a larger numerator than denominator so they would be greater than one. I tried to make sure the ones and twos were in the denominator and put bigger numbers in the numerator.) ‘How did you add your fractions?’ (My denominators were 1, 2, 3, and 4 so I used 12 as a common denominator for all of them.)” (5.NF.1) According to IM K-5 Math Teacher Guide, materials were designed to include opportunities for students to independently demonstrate conceptual understanding, when appropriate. Design Principles, Coherent Progress states, “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.” A Typical IM Lesson states, “The cool-down task is to be given to students at the end of the lesson. Students are meant to work on the cool-down for about 5 minutes independently and turn it in. The cool-down serves as a brief formative assessment to determine whether students understood the lesson. Students’ responses to the cool-down can be used to make adjustments to further instruction.” Examples include: • Unit 1, Finding Volume, Lesson 2, Cool-down, students demonstrate conceptual understanding of volume when they use their understanding of volume as the amount of unit cubes that fill a space. Students see a picture of a rectangular prism and Student Facing states, “Find the volume of the rectangular prism. Explain or show your reasoning.” (5.MD.4) • Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 12, Activity 2, students demonstrate conceptual understanding as they deepen their understanding of an algorithm that uses partial quotients. Students are provided three division problems and Student Facing states, “Use Elena’s strategy to complete the following problems: 492\div12, 630\div15, 364\div14.” (5.NBT.6) • Unit 6, More Decimal and Fraction Operations, Lesson 20, Activity 2, students demonstrate conceptual understanding as they compare a product of fractions to one of the factors. Student Facing states, “Andre says: When you multiply any fraction by a number less than 1, the product will be less than the fraction. When you multiply any fraction by a number greater than 1, the product will be greater than the fraction. Each partner choose one of the statements and describe why it is true. You may want to include details such as notes, diagrams, and drawings to help others understand your thinking.” (5.NF.5) ##### Indicator {{'2b' | indicatorName}} Materials give attention throughout the year to individual standards that set an expectation for procedural skill and fluency. The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. According to IM Curriculum, Design Principles, Balancing Rigor, “Warm-ups, practice problems, centers, and other built-in routines help students develop procedural fluency, which develops over time.” Examples include: • Unit 4, Wrapping Up Multiplication and Division with Multi–Digit Numbers, Lesson 4, Warm-Up, students develop procedural skill and fluency as they notice the patterns in calculations within the number talk, leading towards the standard algorithm. Student Facing states, “Find the value of each product mentally. 3\times3, 3\times20, 3\times600, 3\times623.” Activity Synthesis states, “‘How is the last product related to the first three?’ (It is the sum of the first three.) ‘Did the first three calculations help you find the last product?’ (Yes, I was able to add them together to find.)” (5.NBT.5) • Unit 6, More Decimal and Fraction Operations, Lesson 9, Warm-up, students develop procedural skill and fluency with adding and subtracting fractions with different denominators. Student Facing states, “Decide if each statement is true or false. Be prepared to explain your reasoning. \frac{1}{4}+\frac{2}{4}=\frac{3}{4}, \frac{1}{2}+\frac{1}{4}=\frac{2}{4}, \frac{3}{4}+\frac{1}{2}=\frac{2}{4}.” (5.NF.1) • Unit 8, Putting It All Together, Lesson 2, Activity 1, students develop procedural skill and fluency as they find mistakes when they multiply large numbers. Launch states, “Display or write for all to see. 650\times27. Display each number in a different corner of the room: 14,000, 18,000, 13,000, 19,000. ‘When I say go, stand in the corner with the number that you think is the most reasonable estimate for 650\times27. Be prepared to explain your reasoning.’ 1 minute: quiet think time. Ask a representative from each corner to explain their reasoning. ‘Does anyone want to switch corners?’ Ask a student who switched corners to explain their reasoning. ‘Now you are going to find this product and analyze some work.’” Student Facing states, “1. Find the value of the product. 650\times27. 2. Below is Kiran’s work finding the value of the product 650\times27. Is his answer reasonable? Explain your reasoning. 3. What parts of the work do you agree with? Be prepared to explain your reasoning. 4. What parts of the work do you disagree with? Be prepared to explain your reasoning. 5. Look at your solution to problem 1. Is there anything you want to revise? Be prepared to explain.” (5.NBT.5) According to IM K-5 Math Teacher Guide, materials were designed to include opportunities for students to independently demonstrate procedural skill and fluency, when appropriate. Design Principles, Coherent Progress states, “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.” A Typical IM Lesson states, “The cool-down task is to be given to students at the end of the lesson. Students are meant to work on the cool-down for about 5 minutes independently and turn it in. The cool-down serves as a brief formative assessment to determine whether students understood the lesson. Students’ responses to the cool-down can be used to make adjustments to further instruction.” Examples include: • Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 5, Cool-down, students demonstrate fluency when multiplying a multi-digit number. Student Facing states, “Use the standard algorithm to find the value of 203\times23.” (5.NBT.5) • Unit 6, More Decimal and Fraction Operations, Lesson 1, Cool-down, students demonstrate procedural skill and fluency as they use place value patterns when multiplying and dividing whole numbers and numbers in decimal form. Student Facing states, “Fill in the blank to make each equation true. 1. 0.06\times10=___. 2. 60=___$$\times0.6$$. 3. ___$$= 6\div100$$.” (5.NBT.A) • Unit 8, Putting It All Together, Lesson 1, Activity 2, students demonstrate procedural skill and fluency as they practice using the standard algorithm to find products. Student Facing states, “1. Use the digits 7, 3, 2, and 5 to make the greatest product.” Launch states, “Groups of 2. Display: 7, 3, 2, 5. ‘Using only these digits, what multiplication expressions could we write?’ (, , , .) 1 minute: quiet think time. Record answers for all to see. ‘Which of these expressions do you think would make the greatest product? Be prepared to explain your reasoning.’ (I think the three-digit by one-digit expression would make the greatest product because you can put the 7 in the hundreds place.) ‘Use the digits 7, 3, 2, and 5 to make the greatest product.’ 5-7 minutes: work time.” (5.NBT.5) ##### Indicator {{'2c' | indicatorName}} Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. According to IM Curriculum, Design Principles, Balancing Rigor, “Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations.” Multiple routine and non-routine applications of the mathematics are included throughout the grade level and these single- and multi-step application problems are included within Activities or Cool-downs. Students have the opportunity to engage with applications of math both with support from the teacher and independently. According to IM K-5 Math Teacher Guide, materials were designed to include opportunities for students to independently demonstrate application of grade-level mathematics, when appropriate. Design Principles, Coherent Progress states, “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.” A Typical IM Lesson states, “The cool-down task is to be given to students at the end of the lesson. Students are meant to work on the cool-down for about 5 minutes independently and turn it in. The cool-down serves as a brief formative assessment to determine whether students understood the lesson. Students’ responses to the cool-down can be used to make adjustments to further instruction.” Examples of routine applications of the math include: • Unit 3, Multiplying and Dividing Fractions, Lesson 18, Activity 1, students work with real-world problems involving multiplication and division of fractions. Student facing states, “1. Diego’s dad is making hamburgers for the picnic. There are 2 pounds of beef in the package. Each burger uses \frac{1}{4} pound. How many burgers can be made with the beef in the package? a. Draw a diagram to represent the situation. b. Write a division equation to represent the situation. c. Write a multiplication equation to represent the situation. 2. Diego and Clare are going to equally share \frac{1}{4} pound of potato salad. How many pounds of potato salad will each person get? a. Draw a diagram to represent the situation. b. Write a division equation to represent the situation. c. Write a multiplication equation to represent the situation.” (5.NF.4, 5.NF.6, 5.NF.7) • Unit 3, Multiplying and Dividing Fractions, Lesson 15, Cool-Down, students solve problems involving division of whole numbers and unit fractions. Student facing states, “Match each expression to a situation. Answer each question. 5\div\frac{1}{4}, \frac{1}{4}\div5 a. Han cut 5 feet of ribbon into pieces that are \frac{1}{4} foot long. How many pieces are there? b. Han cut a \frac{1}{4} foot long piece of ribbon into 5 equal pieces. How long is each piece?” (5.NF.7c) • Unit 6, More Decimal and Fraction Operations, Lesson 12, Cool-Down, students solve real-world problems that involve adding and subtracting fractions with unlike denominators. Student facing states, “Priya hiked 1\frac{2}{3} miles. Diego hiked \frac{1}{2} mile. How much farther did Priya hike than Diego? Explain or show your reasoning. 2. On Monday, Andre hiked \frac{3}{4} mile in the morning and 1\frac{1}{3} miles in the afternoon. How far did Andre hike on Monday? Explain or show your reasoning.” (5.NF.1, 5.NF.2) Examples of non-routine applications of the math include: • Unit 1, Finding Volume, Lesson 5, Activity 3, students interpret equations in order to match information given about rectangular prisms. Activity states, “2 minutes: quiet think time. 4 minutes: partner work time. Monitor for students who: use informal language, such as layers, use the terms length, width, height, and base in their questions.” Student facing states, “This is the base of a rectangular prism that has a height of 5 cubes. These are answers to questions about the prism. Read each answer and determine what question it is answering about the prism. 1. 3 is the answer. What is the question? 2. 5 is the answer. What is the question? 3. 3\times4=12. The answer is 12. What is the question? 4. 12\times5=60. The answer is 60 cubes. What is the question? 5. 3 by 4 by 5 is the answer. What is the question?” (5.MD.5b) • Unit 5, Place Value Patterns and Decimal Operations, Lesson 26, Activity 2, students solve a real-world problem including operations with numbers in decimal form. Student Facing states, “Price list from the publisher: type of book, price. boxed sets & collections24.95. comic books $2.60. science books$8.00. chapter books $9.99. history books$14.49. audiobooks $20.00. activity books$4.50. reference books $12.00. Spanish language books$6.00. biographies $6.05. Plan a book fair: 1. Choose 3–5 types of books you want to order. 2. Decide on the mark-up price for each type of book you chose. 3. Estimate the amount of money your school will raise as a profit with your book sale. Record an estimate that is: too low, about right, too high. 4. Show or explain your reasoning for the estimate. Include the assumptions you made.” (5.NBT.7) • Unit 7, Shapes on the Coordinate Plane, Lesson 13, Activity 1, students plot points that represent the length and width of a rectangle with a given perimeter. Activity states, “2 minutes: independent think time. 5 minutes: partner work time.” Student Facing states, “1. Jada drew a rectangle with a perimeter of 12 centimeters. What could the length and width of Jada’s rectangle be? Use the table to record your answer. 2. Plot the length and width of each rectangle on the coordinate grid. 3. If Jada drew a square, how long and wide was it? 4. If Jada’s rectangle was 2.5 cm long, how wide was it? Plot this point on the coordinate grid. 5. If Jada’s rectangle was 3.25 cm long, how wide was it? Plot this point on the coordinate grid.” (5.G.2, 5.NBT.7, 5.OA.3) ##### Indicator {{'2d' | indicatorName}} The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. All three aspects of rigor are present independently throughout the grade. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include: • Unit 1, Finding Volume, Lesson 6, Cool-down, students demonstrate conceptual understanding as they use their understanding of volume to identify and explain the correct expression. Student Facing states, “1. Which of these expressions does not represent the volume of the rectangular prism in cubic units? Explain or show your reasoning. 2. Choose one of the expressions from above and explain why it represents the volume of the prism in cubic units.” (5.MD.5b) • Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 5, Activity 2, students develop procedural skill and fluency as they use the standard algorithm to multiply three-digit numbers by two-digit numbers. Student Facing states, “Use the standard algorithm to find the value of each expression. 1. 202\times12, 2. 122\times33, 3. 321\times24. 4. Diego found the value of 301\times24 Here is his work. Why doesn’t Diego’s answer make sense?” The answer shown is 1,806. (5.NBT.5) • Unit 7, Shapes on the Coordinate Plane, Lesson 12, Activity 1, students apply their understanding of the coordinate plane as they interpret data about a series of coin flips. Student facing states, “Han and Jada flipped a penny several times and counted how many times it came up heads and how many times it came up tails. Their results are plotted on the graph. 1. How many heads did Jada get? How many tails did Jada get? Explain or show how you know. 2. How many heads did Han get? How many tails did Han get? Explain or show how you know. 3. Flip the coin 10 times and record how many heads and tails you get. Plot the point on the coordinate grid that represents your coin flips. 4. Show your partner the point you plotted on the coordinate grid. Look at your partner's coordinate grid. How many heads did your partner flip? How many tails did your partner flip? Explain or show your reasoning. 5. Do any of the points you plotted lie on the horizontal axis? What would a point on the horizontal axis mean in this situation? 6. If time allows, toss the coin 10 more times and record your results and your partner’s results on the coordinate grid.” (5.G.2) Multiple aspects of rigor are engaged simultaneously throughout the materials in order to develop students’ mathematical understanding of a single unit of study or topic. Examples include: • Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 10, Cool-down, students use conceptual understanding and procedural fluency as they compute the area of rectangles when there is one non-unit fractional side length and one whole number side length. Student Facing states, “1. Write a multiplication expression to represent the area of the shaded region. 2. Find the area of the shaded region.” An image shows a rectangle with a length of 5 and a width of \frac{3}{4}. (5.NF.4) • Unit 6, More Decimal and Fraction Operations, Lesson 12, Activity 2, students develop all three aspects of rigor, conceptual understanding, procedural skill and fluency, and application, as they solve multi-step problems involving the addition and subtraction of fractions. Student Facing states, “1. Choose a problem to solve. Problem A: Jada is baking protein bars for a hike. She adds \frac{1}{2} cup of walnuts and then decides to add another \frac{1}{3} cup. How many cups of walnuts has she added altogether? If the recipe requires 1\frac{2}{3} cups of walnuts, how many more cups of walnuts does Jada need to add? Explain or show your reasoning. Problem B: Kiran and Jada hiked 1\frac{1}2} miles and took a rest. Then they hiked another \frac{4}{10} mile before stopping for lunch. How many miles have they hiked so far? If the trail they are hiking is a total of 2\frac{1}{2} miles, how much farther do they have to hike? Explain or show your reasoning. 2. Discuss the problems and solutions with your partner. What is the same about your strategies and solutions? What is different? 3. Revise your work if necessary.” (5.NF.2) • Unit 7, Shapes on the Coordinate Plane, Lesson 6, Activity 1, students use conceptual understanding and application to construct quadrilaterals and explain their attributes. Student Facing states, “1. Build a square with your toothpicks. How do you know it is a square? 2. Use the same four toothpicks to build this shape. (a parallelogram is shown) What stayed the same? What changed? 3. Build a rectangle with six toothpicks. How do you know it is a rectangle? 4. Use the same six toothpicks to build this shape. (a parallelogram is shown) What stayed the same? What changed?” (5.G.3) #### Criterion 2.2: Math Practices Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs). The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for practice-content connections. The materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs). ##### Indicator {{'2e' | indicatorName}} Materials support the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives). MP1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with support of the teacher and independently throughout the units. Examples include: • Unit 1, Finding Volume, Lesson 5, Warm-up, students reason about the attributes of a rectangular prism. Narrative states, “The purpose of this warm-up is for students to notice that each face of a prism can be the base, which will be useful when students use a base of a prism to find the prism’s volume in a later activity. While students may notice and wonder many things about these images, the relationship between the images of the prism and the images of the rectangles are the important discussion points.” Student Facing states, “What do you notice? What do you wonder?” Three rectangular prisms and three rectangles are shown. • Unit 4, Wrapping up Multiplication and Division with Multi-Digit Numbers, Lesson 10, Cool- Down, students make sense of multi-digit division problems. Preparation, Lesson Narrative states, “In this lesson, students explore a context to make sense of division with multi-digit numbers (MP1). This builds on work students did in grade 4 where they divided with up to 4-digit dividends and single-digit divisors. Students used place value understanding, the relationship between multiplication and division and partial quotients to divide. The work in this lesson gives teachers an opportunity to see how students apply their prior understanding, including multiplying multi-digit numbers in the last section. In future lessons, students work toward using more efficient methods to divide multi-digit numbers, including partial quotients.” Student Facing states, “A different group of 4,632 dancers make groups of 8. 1. Write a division expression to represent the situation. 2. How many groups of 8 will there be? Explain or show your thinking.” • Unit 8, Putting It All Together, Lesson 8, Activity 2, students make sense of problems as they reason about multiplication and division. Student Facing states, “The Radio Flyer wagon is 27 feet long 13 feet wide and 2 feet deep. The wagon is being used to deliver 4,000 boxes that each have the side lengths 2 feet by 2 feet by 2 feet. How many trips will the wagon have to make? Explain or show your reasoning.” Narrative states, “The purpose of this activity is for students to solve another problem about the Radio Flyer using multiplication and division. Instead of filling the wagon with sand, they consider filling the wagon with boxes and determine how many boxes will fill the wagon. Unlike with the sand, the boxes do not fill the wagon completely and the number of boxes that do fit is not a divisor of the total number of boxes. Accounting for these considerations will be the focus of the synthesis. When students account for these constraints of the situation, they persevere in solving the problem (MP1).” MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with support of the teacher and independently throughout the units. Examples include: • Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 7, Activity 2, students reason abstractly and quantitatively as they match expressions and diagrams. Student Facing states, “Han, Lin, Kiran, and Jada together ran a 3 mile relay race. (They each ran the same distance.) 1. ”Find the expressions and diagrams that match this situation. Be prepared to explain your reasoning. 2. How far did each person run?” Narrative states, “Students reason abstractly and quantitatively (MP2) when they relate the story to the diagrams and expressions. All of the diagrams and expressions involve the same set of numbers so students need to carefully analyze the numbers in the story, the diagrams, and the expressions in order to choose the correct matches.” • Unit 6, More Decimal and Fraction Operations, Lesson 14, Activity 2, students reason about information presented in a line plot. Student Facing states, “1. Here are the weights of some eggs, in ounces. Use them to make a line plot. 1\frac{7}{8}, 2\frac{1}{2}, 2\frac{3}{8}, 1\frac{3}{4}, 2\frac{1}{4}, 2\frac{4}{8}, 2\frac{1}{8}, 1\frac{7}{8}, 2\frac{1}{4}, 1\frac{6}{8}, 2\frac{1}{8}, 1\frac{7}{8} 2. Jada said that \frac{1}{4} of the eggs weigh 1\frac{7}{8} ounces. Do you agree? Explain or show your reasoning. 3. How much heavier is the heaviest egg than the lightest egg? Explain or show your reasoning.” Narrative states, “The purpose of this activity is for students to use measurement data to make a line plot and then solve problems about the data presented in the line plot (MP2).” • Unit 7, Shapes on the Coordinate Plane, Lesson 10, Cool-down, students think abstractly as they determine rules for given patterns. Lesson Narrative states, “In this lesson students continue to generate two patterns and observe relationships between their corresponding terms. Most of the relationships are more complex in this lesson, involving either multiplication by a fractional amount or both multiplication and addition or subtraction. Students begin to express the relationships between patterns using equations (MP2).” Student Facing states, “1. Jada and Priya are creating rules for patterns. Follow each rule to complete the patterns. Jada’s rule: start with 0 and add 3. Priya’s rule: start with 0 and add 4. 2. Kiran says that when Jada’s number is 45, Priya’s corresponding number will be 90. Do you agree? Why or why not?” ##### Indicator {{'2f' | indicatorName}} Materials support the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with MP3 across the year and it is often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives). According to the Course Guide, Instructional Routines, Other Instructional Routines, 5 Practices, “Lessons that include this routine are designed to allow students to solve problems in ways that make sense to them. During the activity, students engage in a problem in meaningful ways and teachers monitor to uncover and nurture conceptual understandings. During the activity synthesis, students collectively reveal multiple approaches to a problem and make connections between these approaches (MP3).” Students construct viable arguments, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include: • Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 5, Cool-down, students construct viable arguments as they write division expressions and equations that represent real world situations. Student Facing states, “Explain why 8\div5=\frac{8}{5}.” Preparation, Lesson Narrative states, “In this lesson, students generalize their understanding that a fraction can be interpreted as division of the numerator by the denominator. They interpret situations where a certain amount of pounds of blueberries is shared with a certain number of people when the pounds of blueberries each person gets is equal to 1, greater than 1, and less than 1. Then, they construct arguments about why an equation would make sense for any numerator and for any denominator.” • Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 1, Activity 1, students construct a viable argument as they reason about appropriate estimates for multi-digit multiplication calculations. Narrative states, “The purpose of this activity is for students to make a reasonable estimate for a given product. In addition to estimating the product, students also decide whether the estimate is too large or too small. In the activity synthesis, students consider how far their estimate is from the actual product. In the next activity, students will evaluate the expressions using a strategy of their choice. Students choose between several different possible estimates and justify their choice before they calculate the product (MP3).” Activity states, “5–7 minutes: independent work time. 2–3 minutes: partner discussion. Monitor for students who: relate the given expression to each proposed answer by rounding or changing one or both factors, estimate by rounding the factors, use benchmark numbers, use place value reasoning or the properties of operations to explain why their estimate is reasonable.” Student Facing states, “Which estimate for the product 18\times149 is most reasonable? Explain or show your reasoning. A. 2,000 B. 4,000 C. 3,000 D. 1,500 2. Are any of the estimates unreasonable? Explain or show your reasoning. 3. Do you think the actual product will be more or less than your estimate? Explain or show your reasoning.” • Unit 8, Putting It All Together, Lesson 5, Activity 1, students construct an argument and critique the reasoning of others as they defend a strategy to solve a division problem. Narrative states, “The purpose of this activity is for students to revisit the partial quotients method to find whole number quotients. Students compare their strategy with Elena's strategy and reason about the similarities and differences using their understanding of place value. They may use estimation to identify that Elena's answer is not reasonable while they may also use parts of their own calculation to identify Elena's error (MP3).” Activity states, “‘Work with your partner to complete the second, third, and fourth problems.’ 5–7 minutes: partner work time. ‘Now you will have a chance to revisit your work from the first problem.’ 1–2 minutes: independent work time. Monitor for students who: revised their original solution, used different partial quotients.” Student Facing states, “1. Find the value of the quotient. 6773\div13$ 2. Here is how Elena found the quotient. Is her answer reasonable? (Students see work done by Elena using partial quotient strategy.) Explain or show your reasoning. 3. What parts of the work do you agree with? Be prepared to explain your reasoning. 4. What parts of the work do you disagree with? Be prepared to explain your reasoning. 5. Look at your solution to problem 1. Is there anything you want to revise? Be prepared to explain.”

Students critique the reasoning of others, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Finding Volume, Lesson 7, Activity 1, students construct arguments and critique the reasoning of others as they reason about the volume measurements of different items. Narrative states, “The purpose of this activity is for students to consider how the size of an object impacts the unit we use to measure the volume of that object. Since this is the students’ first experience with these cubic units of measure, it may be helpful for them to see the actual length of a centimeter, inch and foot. Have rulers or cubes available to provide extra support to visualize the size of the cubic units of measure. Because there are no mathematically correct or incorrect answers, this activity provides a rich opportunity for students to discuss and defend different points of view (MP3).” Activity states, “2 minutes: independent work time. 5 minutes: partner discussion. As students work, monitor for students who discuss how big or small the object is when choosing the size of the unit of measure. Ask these students to share during the synthesis. If students finish early, ask them to find other objects they would measure the volume of using the different cubic units of measure. If the objects are in the room, they could estimate and check their estimates.” Student Facing states, “For each object, choose the cubic unit you would use to measure the volume: cubic centimeter, cubic inch, or cubic foot.” A table is included with the following objects: the volume of a moving truck, the volume of a freezer, the volume of a juice box, the volume of a classroom, the volume of a dumpster, and the volume of a lunch box.

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 3, Activity 1, students critique the reasoning of others when working with fraction division. Narrative states, “The last problem provides an opportunity for students to think critically about a proposed solution to a problem (MP3). Different ways to think about the proposed solution include: estimation: with 3 friends sharing 2 liters, each friend gets less than 1 liter thinking about the meaning of the numerator (how many liters are being shared) and denominator (how many people are sharing the water).” Activity states, “5 minutes: independent work time. 5 minutes: partner discussion. As students work, monitor for students who: draw a diagram to determine the amount of water each dancer drinks if 3 dancers share 2 liters of water, revise their solution for how much water each dancer gets after explaining why Mai’s answer doesn’t make sense.”  Student Facing states, “Three dancers share 2 liters of water. How much water does each dancer get? Write a division equation to represent the situation. Mai said that each dancer gets \frac{3}{2} of a liter of water because 3 divided into 2 equal groups is  \frac{3}{2}. Do you agree with Mai? Show or explain your reasoning.”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 15, Activity 3, students construct a viable argument and critique the reasoning of others as they analyze a common error when using the standard algorithm to subtract decimals. Student Facing states, “1. Find the value of 622.35-71.4 Explain or show your reasoning. 2. Elena and Andre found the value of 622.35-71.4. Who do you agree with? Explain or show your reasoning.” Narrative states, “When students identify and correct Elena's error they construct viable arguments and critique the reasoning of others (MP3).”

##### Indicator {{'2g' | indicatorName}}

Materials support the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities to solve real-world problems, model situations with appropriate representations, and describe what they do with the model and how it relates to the problem. Students model with mathematics as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 17, Activity 1, students model with mathematics as they use the context of art to multiply fractions. Lesson Narrative states, “When students make decisions and choices, analyze contextual objects with mathematical ideas, and translate a mathematical answer back into the context of a situation, they model with mathematics (MP4).” Student Facing states, “1. Use the colored paper and scissors to cut identical rectangles. Make sure the measurement of one side of the rectangle is a whole number and the other is a fraction greater than one. 2. What is the area of one of your rectangles? Show your reasoning. 3. Use the rectangles from your group to make a group mosaic by arranging some of the different colored rectangles on a blank piece of paper.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 20, Cool-down, students model with math when they make estimates and solve complex problems. Lesson Narrative states, “Throughout the lesson, students make estimates and simplifying assumptions in order to answer complex mathematical questions (MP4).” Student facing states, “1. A different shipping container is 40 feet long, 9 feet wide, and 8 feet tall. a. What is the volume of this container? Explain or show your thinking. b. A school makes 24 cubic feet of recyclable plastic each day. How many days does it take the school to fill this container? Explain or show your thinking.”

• Unit 8, Putting it All Together, Lesson 7, Activity 1, students model with math by selecting the appropriate unit of measure as well as what the estimate means within the real-world situation. Narrative states, “Choosing an appropriate unit of measure for an estimation and understanding how that choice affects both the calculations and the meaning of the estimate are important aspects of applying mathematics to solve real world problems (MP4).” Activity states, “3–5 minutes: quiet work time. 5 minutes: partner discussion time. Monitor for students who: notice that the wagon has a rectangular prism shape, roughly, and recognize that we need to know the side lengths of the wagon in order to make a reasonable estimate about its volume, use references, such as the size and number of people in the wagon, to help estimate the wagon’s volume, choose different units of length and volume for their estimates.”  Student Facing states, “1. What measurements would you take of the wagon to accurately estimate its volume? 2. What units would you use to measure the wagon? Explain your reasoning. 3. Record an estimate for the volume of the wagon that is: too low, about right, too high. 4. What can you use in the picture to refine your estimate?”

MP5 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities to identify and use a variety of tools or strategies, working with the support of the teacher and independently, throughout the units to support their understanding of grade-level math. Examples include:

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 14, Activity 1, students use appropriate tools strategically to subtract decimals. Student Facing states, “Find the value of 2.26-1.32. Explain or show your reasoning.” Lesson Narrative states, “Students should be encouraged to use whatever strategies make sense to them, including using place value understanding and the relationship between addition and subtraction. Strategies students may use include using hundredths grids (MP5), using place value and writing equations.”

• Unit 6, More Decimal and Fraction Operations, Lesson 8, Cool-down, students use an appropriate strategy as a tool to find solutions to problems involving addition and subtraction of fractions and then explain their strategy. Student facing states, “Find the value of each expression. Explain or show your reasoning. 1. \frac{5}{6}-\frac{1}{3}. 2. \frac{3}{4}+\frac{1}{2}.” Lesson Narrative states, “Students find the sums and differences in a way that makes sense to them. The denominators of the fractions used in this lesson are familiar from grade 3, inviting students to use a variety of different familiar representations.”

##### Indicator {{'2h' | indicatorName}}

Materials attend to the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have many opportunities to attend to precision and the specialized language of math, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Finding Volume, Lesson 5, Activity 2, students use precise language as they complete a table showing multiplication expressions for the volume of prisms. Student Facing states, “Here is a base of a rectangular prism. 1. Fill out the table for the volumes of rectangular prisms with this base and different heights.” Lesson Narrative states, “Students may still use informal language, such as layers, to describe the prisms and find their volume. During the lesson synthesis, connect their informal language to the more formal math language of length, width, height, and area of the base.”

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 2, Activity 2, students attend to precision of language when connecting mathematical representations for a real world problem. Narrative states, “This sorting task gives students opportunities to analyze and connect representations, situations, and expressions (MP2, MP7). As students work, encourage them to refine their descriptions of how the diagrams represent the situations and expressions using more precise language and mathematical terms (MP6).” Activity states, “‘This set of cards includes diagrams, expressions, and situations. Match each diagram to a situation and an expression. Some situations and expressions will match more than one diagram. Work with your partner to justify your choices. Then, answer the questions in your workbook.’ 5–8 minutes: partner work time. Monitor for students who: notice that the number of large rectangles in the picture and the dividend in the expressions represent the number of sandwiches, notice that the number of pieces in each whole and the divisor in the expressions represent the number of people sharing the sandwiches.” Student Facing states, “Your teacher will give you a set of cards. Match each representation with a situation and expression. Some situations and expressions will have more than one matching representation. Choose one set of matched cards. 1. Show or explain how the diagram(s) and expression represent the number of sandwiches being shared. 2. Show or explain how the diagram(s) and expression represent the number of people sharing the sandwiches. 3. How much sandwich does each person get in the situation?”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 6, Cool-down, students attend to precision when they apply the standard algorithm for multiplication. Student Facing states, “Use the standard algorithm to find the product 251\times34.” Lesson Narrative states, “Because these calculations have new units composed in almost every place value, students will need to locate and use the composed units carefully. It gives students a reason to attend to the features of their calculation and to use language precisely (MP6).”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 7, Warm-up, students attend to precision when working with weight measurements. Narrative states, “The weights on the scale total 12.32 ounces, but the scale reads 12.3 ounces. There are different possible explanations for this discrepancy. For example, the scale might be inaccurate. Or the scale might only give readings in tenths of an ounce. In the discussion, students consider the idea that the value shown on the scale is not always exact. It may just show the closest value that it is capable of reading, which is the nearest tenth of an ounce in this case (MP6).” Student Facing states, “What do you notice? What do you wonder?” Activity Synthesis states, “What do you notice about the weights on the scale and the reading of the scale? (They aren’t the same. The weights are 12.32 ounces and the scale says 12.3 ounces.) Why do you think the scale and the weights don’t agree? (The scale could be wrong.) What if the scale only shows tenths of an ounce, and it can’t show hundredths of an ounce? (The value is still not accurate but it’s the best the scale can do.) In today’s lesson we will look at scales that show different numbers of decimals and see how that influences what they show.”

• Unit 6, More Decimal and Fraction Operations, Lesson 5, Cool-down, students attend to precision when they compare two measurements. Student Facing states, “Jada ran 15.25 kilometers. Han ran 8,500 meters. Who ran farther? How much farther? Explain or show your reasoning.” Narrative states, “This gives students an opportunity to think about which units are most helpful for communicating a distance (MP6).”

• Unit 7, Shapes on the Coordinate Plane, Lesson 5, Cool-down, students use accurate mathematical language to classify quadrilaterals as trapezoids. Student Facing states, “1. When is a quadrilateral also a trapezoid?  2. Which of the following shapes are trapezoids? Show or explain your reasoning.” Student Response sample states, “1. A quadrilateral is a trapezoid if it has at least one pair of opposite sides that are parallel. 2. All of the shapes except D are trapezoids because they have at least one pair of opposite sides that are parallel.”

##### Indicator {{'2i' | indicatorName}}

Materials support the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities to look for, describe, and make use of patterns within problem-solving as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Finding Volume, Lesson 3, Cool-down, students look for and make use of structure while they consider the layered structure of a prism to find its volume. Lesson Narrative states, “In previous lessons, students built objects, including rectangular prisms, with unit cubes and counted the number of cubes. In this lesson, students continue to count the number of unit cubes needed to build a rectangular prism, but now they are presented with images of prisms instead of the objects themselves. To encourage students to develop a systematic way to count the cubes, they are shown prisms made from larger numbers of cubes. As students use horizontal or vertical layers to measure the volume, they make use of the layered structure of prisms (MP7).” Student Facing states, “Jada’s prism has 4 layers and each layer has 9 cubes. 1. Circle the prism that is Jada's. 2. Find the volume of Jada’s prism. Explain or show your reasoning.” Students see four prisms with two layers of 12, four layers of 9, three layers of 9, and three layers of 8.

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 2, Warm-up, students look for and make use of structure as they use a hundreds grid to estimate a shaded region. Narrative states, “When students reflect about how the hundredths grid could help refine their estimate, they observe the value and power of its structure (MP7).” Launch states, “Groups of 2. Display the image. ‘What is an estimate that’s too high? Too low? About right?’ 1 minute: quiet think time.” Student Facing states, “How much of the square is shaded?” Activity Synthesis states, “Why is estimating the shaded region more difficult without the gridlines of a hundredths grid? (The gridlines show me the tenths and hundredths. Without that, I can only guess or estimate.)”

• Unit 7, Shapes on the Coordinate Plane, Lesson 3, Activity 1, students look for and make use of structure as they plot points on the coordinate grid. Student Facing states, “Partner A. 1. Estimate the location of each point. A(5,1) B(5,2) C(5,3) D(5,4). 2. Plot and label the points on the coordinate grid. 3. What do the points have in common? 4. Plot the point with coordinates (5,0) on the coordinate grid. Partner B. 1. Estimate the location of each point. A(4,3) B(5,3) C(6,3) D(7,3). 2. Plot and label the points on the coordinate grid. 3. What do the points have in common? 4. Plot the point with coordinates (0,3) on the coordinate grid.” Lesson Narrative states, “The purpose of this activity is for students to plot several points with the same vertical or horizontal coordinate and observe that they lie on a horizontal or vertical line respectively (MP7). Students also plot points on the axes for the first time. Before plotting the points on a grid with grid lines, students first estimate the location of the points. This encourages them to think about the coordinates as distances (from the vertical axis for the first coordinate and from the horizontal axis for the second coordinate).” Activity Synthesis states, “Ask previously identified students to share their thinking. ‘What can we say about a set of points when they share the same first coordinate?’ (They will be on the same vertical line.) Display image from student solution showing points with first coordinate 5. ‘How did you know where to put the point with coordinates (5,0)?’ (I put it on the horizontal axis. I went over 5 but did not go up at all.) ‘What happens when a set of points share the same second coordinate?’ (They will be on the same horizontal line.) Display image from student solution showing points with second coordinate 3. ‘What does the zero in (0,3) tell us?’ (It means the point will be on line zero of the horizontal axis, which is the vertical axis.) (0,0) is an important point because it's where we start when we plot a point on the coordinate grid. Find (0,0) on the grid you have been working with.”

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities to use repeated reasoning in order to make generalizations and build a deeper understanding of grade-level math concepts as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 13, Activity 2, students use repeated reasoning as they use partial quotients to divide up to four-digit dividends by a two-digit divisor. Student Facing states, “Estimate the value of each quotient. Then, use an algorithm using partial quotients to find the value. 1. A reasonable estimate for 612\div34 is: ___. 2. A reasonable estimate for 529\div23 is: ___. 3. A reasonable estimate for 1,044\div29 is: ___.” Narrative states, “Before finding the quotient, students estimate the value of the quotient which both helps students decide which partial quotients to use and helps them evaluate the reasonableness of their solution (MP8).” Activity Synthesis states, “Ask 2–3 students to share their work for the same problem that shows different partial quotients. ‘How can you make sure that the whole number quotient you got at the end is reasonable?’ (It should be close to my estimate. I can multiply the quotient and divisor and that should give me the dividend.) If students pair and share with other partners, ask, ‘How did explaining your work to others help you today?’ or ‘What did someone say today that helped you in your understanding of division?’ (I learned that it’s ok to take more steps because I was comfortable with the multiples I used.)”

• Unit 6, More Decimal and Fraction Operations, Lesson 2, Warm-up, students use repeated reasoning as they analyze a diagram and make connections to exponents. Narrative states, “When students analyze the diagram and determine how many segments there are of each length, they are observing and making use of the repeated structure of ten segments joining at the different vertices (MP7, MP8).” Launch states, “Groups of 2. ‘How many do you see? How do you see them?’ Display the image. 1 minute: quiet think time.” Student Facing states, “How many do you see? How do you see them?” Activity Synthesis states, “Invite students to share their estimates for how many of the smallest line segments there are in the diagram. ‘How can you find out exactly how many there are?’ (I can count the number of long segments and then the number of medium size segments on one long segment and then the number of tiny segments on one medium size one. Then I multiply those numbers.) Invite students to count and then display the expression: 10\times10\times10. ‘How does the expression relate to the diagram?’ (It’s the total number of tiny segments.) ‘Another way to write 10\times10\times10 is 10^310^3. This is called a power of ten. The number 3 tells us how many factors of 10 there are, or how many times we multiply 10 to get the number.’”

• Unit 7, Shapes on the Coordinate Plane, Lesson 9, Cool-down, students use repeated reasoning as they generate patterns, given two rules, and identify relationships between corresponding terms. Student Facing states, “1. List the first 10 numbers for these 2 patterns. Jada’s rule: Start with 0 and keep adding 5. Priya’s rule: Start with 0 and keep adding 10. 2. What number will be in Priya’s pattern when Jada’s pattern has 100? 3. What relationship do you notice between corresponding numbers in the two patterns?” Activity 2 Narrative states, “When students find and explain patterns related rules and relationships, they look for and express regularity in repeated reasoning (MP8).” Students have the opportunity to use this repeated reasoning in the Cool-down as well.

### Usability

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for Usability. The materials meet expectations for Criterion 1, Teacher Supports; partially meet expectations for Criterion 2, Assessment; and meet expectations for Criterion 3, Student Supports.

##### Gateway 3
Meets Expectations

#### Criterion 3.1: Teacher Supports

The program includes opportunities for teachers to effectively plan and utilize materials with integrity and to further develop their own understanding of the content.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for Teacher Supports. The materials: provide teacher guidance with useful annotations and suggestions for enacting the student and ancillary materials; contain 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; include standards correlation information that explains the role of the standards in the context of the overall series; provide explanations of the instructional approaches of the program and identification of the research-based strategies; and provide a comprehensive list of supplies needed to support instructional activities.

##### Indicator {{'3a' | indicatorName}}

Materials provide 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.

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.”

##### Indicator {{'3b' | indicatorName}}

Materials contain adult-level explanations and examples of the more complex grade-level/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

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.”

##### Indicator {{'3c' | indicatorName}}

Materials include standards correlation information that explains the role of the standards in the context of the overall series.

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.”

##### Indicator {{'3d' | indicatorName}}

Materials provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

Each unit has corresponding Family Support Materials, in English and Spanish, that provide a variety of supports for families. Each unit includes a narrative for stakeholders, describing what students will learn within each section. Additionally, Try it at home! includes suggestions for at home activities and questions families can ask, all geared towards supporting the mathematical ideas in the unit. Examples include:

• For Families, Grade 5, Unit 2, Fractions as Quotients and Fraction Multiplication, Family Support Materials, Try it at home!, “Near the end of the unit, ask your student the following questions: 1. Write as many expressions as you can that represent this diagram: \frac{3}{5} and 4 is shown in the diagram. 2. What is the area of the following rectangle? Questions that may be helpful as they work: How are the two problems similar? How are they different? How does your expression represent the diagram? How did you break up the rectangle to help you solve for the entire area? What are the side lengths of the rectangle?”

• For Families, Grade 5, Unit 3, Multiplying and Dividing Fractions, Family Support Materials, “In this unit, students use area concepts to represent and solve problems involving the multiplication of two fractions, and generalize that when they multiply two fractions, they need to multiply the two numerators and the two denominators to find their product. They also reason about the relationship between multiplication and division to divide a whole number by a unit fraction and a unit fraction by a whole number. Section A: Fraction Multiplication. In this section, students build on their knowledge of fraction multiplication developed in the previous unit by using area concepts to understand the multiplication of a fraction times a fraction. Students draw diagrams to represent the fractional area. For example, students learn that the diagrams below can represent the situation ‘Kiran eats macaroni and cheese from a pan that is \frac{1}{3} full. He eats \frac{1}{4} of the remaining macaroni and cheese in the pan. How much of the whole pan did Kiran eat?’ Students extend this conceptual understanding to multiply all types of fractions including fractions greater than 1 (for example, \frac{7}{4}). In each case, the students relate this multiplication to finding the area of a rectangle with fractions as side lengths. As the lessons progress, they notice that they can multiply the two numerators and the two denominators to find their product. This reasoning holds true for fractions greater than 1.”

• For Families, Grade 5, Unit 7, Shapes on the Coordinate Plane, Family Support Materials, “In this unit, students are introduced to the structure of the coordinate grid, and the convention and notation of coordinates to name points. They classify triangles and quadrilaterals in a hierarchy based on properties of side length and angle measure. In their work with numerical patterns, students generate two different numerical patterns, and identify relationships between the corresponding terms in the patterns. Section A: The Coordinate Plane. In this section, students explore the coordinate grid. They recognize that a point is located where two lines intersect. They describe points on the grid based on the numbers on the horizontal and vertical axes. For example, the point shown is located at (7,3).”

##### Indicator {{'3e' | indicatorName}}

Materials provide explanations of the instructional approaches of the program and identification of the research-based strategies.

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.”

##### Indicator {{'3f' | indicatorName}}

Materials provide a comprehensive list of supplies needed to support instructional activities.

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.”

##### Indicator {{'3g' | indicatorName}}

This is not an assessed indicator in Mathematics.

##### Indicator {{'3h' | indicatorName}}

This is not an assessed indicator in Mathematics.

#### Criterion 3.2: Assessment

The program includes a system of assessments identifying how materials provide tools, guidance, and support for teachers to collect, interpret, and act on data about student progress towards the standards.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 partially meet expectations for Assessment. The materials identify the standards and the mathematical practices assessed in formal assessments. The materials provide multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance but do not provide suggestions for follow-up. The materials include opportunities for students to demonstrate the full intent of grade-level standards and mathematical practices across the series.

##### Indicator {{'3i' | indicatorName}}

Assessment information is included in the materials to indicate which standards are assessed.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for having assessment information included in the materials to indicate which standards are assessed.

End-of-Unit Assessments and the End-of-Course Assessments consistently and accurately identify grade-level content standards. Content standards can be found in each Unit Assessment Teacher Guide. Examples from formal assessments include:

• Unit 1, Finding Volume, End-of-Unit Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 4, 5.MD.5b, “Find the volume of a rectangular prism with the given side lengths. a. The length is 2 units, the width is 5 units, and the height is 7 units. b. The base has an area of 200 square inches and the height is 6 inches.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, End-of-Unit Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 4, 5.NBT.6, “Find the value of 1,530\div34. Explain or show your reasoning.”

• Unit 8, Putting it All Together, End-of-Course Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 6, 5.NF.2, “Jada’s math book is \frac{5}{16} of an inch thick. Her science book is \frac{1}{4} of an inch thick. a. Which book is thicker? How much thicker? Explain or show your reasoning. b. How thick are the math and science books together?”

Guidance is provided within materials for assessing progress of the Mathematical Practices. According to IM K-5 Math Teacher Guide, How to Use These Materials, “Because using the mathematical practices is part of a process for engaging with mathematical content, we suggest assessing the Mathematical Practices formatively. For example, if you notice that most students do not use appropriate tools strategically (MP5), plan in future lessons to select and highlight work from students who have chosen different tools.” For each grade, there is a chart outlining a handful of lessons in each unit that showcase certain mathematical practices. There is also guidance provided for tracking progress against “I can” statements aligned to each practice, “Since the Mathematical Practices in action can take many forms, a list of learning targets for each Mathematical Practice is provided to support teachers and students in recognizing when engagement with a particular Mathematical Practice is happening. The intent of the list is not that students check off every item on the list. Rather, the ‘I can’ statements are examples of the types of actions students could do if they are engaging with a particular Mathematical Practice.” Examples include:

• IM K-5 Math Teacher Guide, How to Use These Materials, Standards for Mathematical Practices Chart, Grade 5, MP2 is found in Unit 3, Lessons 1, 4, 11, 14, and 18.

• IM K-5 Math Teacher Guide, How to Use These Materials, Standard for Mathematical Practices Chart, Grade 5, MP6 is found in Unit 5, Lessons 6, 7, 13, 16, and 25.

• IM K-5 Math Teacher Guide, How to Use These Materials, Standards for Mathematical Practice Student Facing Learning Targets, “MP1 I Can Make Sense of Problems and Persevere in Solving Them. I can ask questions to make sure I understand the problem. I can say the problem in my own words. I can keep working when things aren’t going well and try again. I can show at least one try to figure out or solve the problem. I can check that my solution makes sense.”

• IM K-5 Math Teacher Guide, How to Use These Materials, Standards for Mathematical Practice Student Facing Learning Targets, “MP5 I Can Use Appropriate Tools Strategically. I can choose a tool that will help me make sense of a problem. These tools might include counters, base-ten blocks, tiles, a protractor, ruler, patty paper, graph, table, or external resources. I can use tools to help explain my thinking. I know how to use a variety of math tools to solve a problem.”

##### Indicator {{'3j' | indicatorName}}

Assessment system provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 partially meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Each End-of-Unit Assessment and End-of-Course Assessment provides guidance to teachers for interpreting student performance, with an answer key and standard alignment. According to the Teacher Guide, Summative Assessments, “All summative assessment problems include a complete solution and standard alignment. Multiple choice and multiple response problems often include a reason for each potential error a student might make.” Examples from the assessment system include:

• Unit 1, Finding Volume, End-of-Unit Assessment, Problem 1, “Find the volume of the figure. Explain or show your reasoning. A: 3\times4\times5, B: 3+4+5, C: 20+20+20, D: 15\times15\times15\times15, E: 5\times12.” A picture of a prism is shown. The Assessment Teacher Guide states, “Students find the volume of a figure. No strategy is suggested but students will likely cut the figure into two rectangular prisms and add the volumes of those prisms. But they may decompose the figure in any way that allows them to count the total number of cubes that make the figure.” The answer key aligns this problem to 5.MD.5c.

• Unit 2, Fractions as Quotients and Fraction Multiplication, End-of-Unit Assessment, Problem 5, “A hiking trail is 7 miles long. Han hikes \frac{2}{3} of the trail and then stops for water. Jada hikes \frac{1}{2} of the trail and then stops for water. a. How many miles did Han hike before stopping for water? Explain or show your reasoning. b. How many miles did Jada hike before stopping for water? Explain or show your reasoning.” The Assessment Teacher Guide states, “Students multiply a whole number by a fraction to solve a story problem. No representation for the problem is requested so students may draw a tape diagram (or discrete diagram), or an area diagram, or they may reason about the quantities without a picture.” The answer key aligns this problem to 5.NF.4a and 5.NF.6.

• Unit 8, Putting It All Together, End-of-Course Assessment, Problem 17, “Clare read that a bath can use between 120 and 200 liters of water. She wants to check if this is reasonable and makes some measurements of her bathtub. A table chart shows the length of the bathtub 1.5 meters and width of the bathtub 0.6 meters. Clare estimates that the depth of water in her bath is about 20 cm. a. What is the area of the base of the bathtub in square meters? Explain your reasoning.b. What is the area of the base of the bathtub in square centimeters? c. What is the volume of the water in Clare’s bath in cubic centimeters? d. There are 1,000 cubic centimeters in 1 liter. Does the volume of water in Clare’s bath agree with what she read? Explain your reasoning.” The Assessment Teacher Guide states, “Students multiply decimal and whole numbers to find a volume. They also perform two unit conversions, each of which requires either multiplying or dividing by a power of 10, giving students an opportunity to use what they have learned about place value. Implicit in the problem is that Clare’s bathtub is generally shaped like a rectangular prism. The teacher may wish to highlight this natural modeling assumption. This problem can also be made more hands-on by asking students to take measurements of a bathtub. The numbers students are likely to get with their own measurements will not be as nice as the ones provided in the problem, making the arithmetic more difficult and possibly beyond the standards but it will make the problem more meaningful and they might be instructed to round their numbers to facilitate further calculations.” The answer key aligns this problem to 5.MD.1, 5.MD.5, 5.NBT.A, and 5.NBT.7.

While assessments provide guidance to teachers for interpreting student performance, suggestions for following-up with students are either minimal or absent. Cool-Downs, at the end of each lesson, include some suggestions. According to IM Curriculum, Cool-Downs, “The cool-down (also known as an exit slip or exit ticket) is to be given to students at the end of the lesson. This activity serves as a brief check-in to determine whether students understood the main concepts of that lesson. Teachers can use this as a formative assessment to plan further instruction. When appropriate, guidance for unfinished learning, evidenced by the cool-down, is provided in two categories: next-day support and prior-unit support. This guidance is meant to provide teachers ways in which to continue grade-level content while also giving students the additional support they may need.” An example includes:

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 1, Cool-down, Student Facing states, “1. Draw a diagram to show how much sandwich each person will get. 3 sandwiches are equally shared by 4 people. Explain or show how you know that each person gets the same amount of sandwich.” Responding to Student Thinking states, “Students do not draw a diagram that shows equal shares.” Next Day Supports states, “During Activity 1, encourage students to draw a diagram to represent each situation in the table and explain where they see the number of people sharing the sandwich in each diagram.” This problem aligns to 5.NF.3.

##### Indicator {{'3k' | indicatorName}}

Assessments include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

Formative assessment opportunities include some end of lesson cool-downs, interviews, and Checkpoint Assessments for each section. Summative assessments include End-of-Unit Assessments and the End-of-Course Assessment. Assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types, including multiple choice, multiple response, short answer, restricted constructed response, and extended response. Examples from summative assessments include:

• Unit 1, Finding Volume, End-of-Unit Assessment supports the full intent of MP4 (Model with mathematics) as students design a composite prism to meet certain criteria. For example, Problem 7 states, “Mai's class is designing a garden with two levels and this general shape. The garden should have at least 200 square feet for the plants. The volume should be less than 500 cubic feet. a. Recommend side lengths for the tiered garden that fit the needs of Mai's class. b. Label the diagram to show your choices for the side lengths.”

• Unit 2, Fractions as Quotients and Fraction Multiplication, End-of-Unit Assessment develops the full intent of 5.NF.3 (Interpret a fraction as division of the numerator by the denominator (\frac{a}{b}=a\div b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers), 5.NF.4a (Interpret the product (\frac{a}{b})\times q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations $$a\times q\div b.)$$, and 5.NF.6 (Solve real world problems involving multiplication of fractions and mixed numbers). For example, Problem 7 states, “A farm is rectangular in shape. It is 2 km long and 3 km wide. a. What is the area of the farm? Explain or show your reasoning. b. The farm is divided into 5 equal parts. Corn is grown in one of the parts. Draw a diagram to show where the corn is grown. c. What is the area of the part of the farm where corn is grown? Explain or show your reasoning.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, End-of-Unit-Assessment develops the full intent of 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). For example, Problem 5 states, “The area of a rectangular yard is 5,063 square feet and its length is 61 feet. What is its width? Explain or show your reasoning.”

• Unit 8, Putting It All Together, End-of-Course Assessment supports the full intent of MP7 (Look for and make use of structure) as students identify different expressions that have the value one million. For example, Problem 7 states, “Select all expressions that have the value one million. A. 10^3B. 10^6 C. 10^7 D. 1,000,000 E. 10\times10\times10\times10\times10\times10. F. 100\times100 G. 100\times100\times100.”

##### Indicator {{'3l' | indicatorName}}

Assessments offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment. These suggestions are provided within the Teacher Guide in a section called “Universal Design for Learning and Access for Students with Disabilities.” As such, they are included at the program level and not specific to each assessment.

Examples of accommodations include:

• IM K-5 Teacher Guide, UDL Strategies to Enhance Access states, “Present content using multiple modalities: Act it out, think aloud, use gestures, use a picture, show a video, demonstrate with objects or manipulatives. Annotate displays with specific language, different colors, shading, arrows, labels, notes, diagrams, or drawings. Provide appropriate reading accommodations. Highlight connections between representations to make patterns and properties explicit. Present problems or contexts in multiple ways, with diagrams, drawings, pictures, media, tables, graphs, or other mathematical representations. Use translations, descriptions, movement, and images to support unfamiliar words or phrases.”

• IM K-5 Teacher Guide, UDL Strategies to Enhance Access states, “It is important for teachers to understand that students with visual impairments are likely to need help accessing images in lesson activities and assessments, and prepare appropriate accommodations. Be aware that mathematical diagrams are provided as scalable vector graphics (SVG format), because this format can be magnified without loss of resolution. Accessibility experts who reviewed this curriculum recommended that students who would benefit should have access to a Braille version of the curriculum materials, because a verbal description of many of the complex mathematical diagrams would be inadequate for supporting their learning. All diagrams are provided in the SVG file type so that they can be rendered in Braille format.”

• IM K-5 Teacher Guide, UDL Strategies to Enhance Access states, “Develop Expression and Communication, Offer flexibility and choice with the ways students demonstrate and communicate their understanding. Invite students to explain their thinking verbally or nonverbally with manipulatives, drawings, diagrams. Support fluency with graduated levels of support or practice. Apply and gradually release scaffolds to support independent learning. Support discourse with sentence frames or visible language displays.”

#### Criterion 3.3: Student Supports

The program includes materials designed for each child’s regular and active participation in grade-level/grade-band/series content.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for Student Supports. The materials provide: strategies and supports for students in special populations and for students who read, write, and/or speak in a language other than English to support their regular and active participation in learning grade-level mathematics; multiple extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity; and manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

##### Indicator {{'3m' | indicatorName}}

Materials provide strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

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.”

##### Indicator {{'3n' | indicatorName}}

Materials provide extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

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.”

##### Indicator {{'3o' | indicatorName}}

Materials provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 provide various approaches to learning tasks over time and variety in how students are expected to demonstrate their learning, but do not provide opportunities for students to monitor their learning.

Students engage with problem-solving in a variety of ways within each lesson: Warm-up, Instructional Activities, Cool-down, and Centers. According to the IM K-5 Teacher Guide, A Typical IM 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. An activity can serve one or more of many purposes. Provide experience with a new context. Introduce a new concept and associated language. Introduce a new representation. Formalize a definition of a term for an idea previously encountered informally. Identify and resolve common mistakes and misconceptions that people make. Practice using mathematical language. Work toward mastery of a concept or procedure. Provide an opportunity to apply mathematics to a modeling or other application problem. The purpose of each activity is described in its narrative. Read more about how activities serve these different purposes in the section on design principles.” Examples of varied approaches include:

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 14, Warm-up, students analyze four different strategies for a division problem and determine which one does not belong. Activity states, “Groups of 2. Display the image. ‘Pick one that doesn’t belong. Be ready to share why it doesn’t belong.’ 1 minute: quiet think time.”  Student Facing states, “$$1312\div82$$ (students show 4 different ways to solve it).”

• Unit 7, Shapes on the Coordinate Plane, Lesson 8, Activity 2, students sort triangles based on attributes. Activity states, “5 minutes: independent work time. 5 minutes: small-group work time. MLR7 Compare and Connect ‘Create a visual display that shows your thinking about the problems. You may want to include details such as notes, diagrams, or drawings to help others understand your thinking.’ 2–5 minutes: independent or group work. 5–7 minutes: gallery walk.” Student Facing states, “Sort the triangle cards from the previous activity in a way that makes sense to you. Describe how you sorted the cards. Now sort out the triangles with a 90 degree angle. For these triangles, write statements about each category. All of the triangles with a 90 degree angle… Some of the triangles with a 90 degree angle… None of the triangles with a 90 degree angle…”

• Unit 8, Putting It All Together, Lesson 11, Activity 1, students practice subtracting fractions with unlike denominators. Student Facing states, “Use the directions to play Greatest Difference with a partner. 1. Spin the spinner. 2. Each player writes the number that was spun in an empty box for Round 1. Be sure your partner cannot see your paper. 3. Once a number is written down, it cannot be changed. 4. Continue spinning and writing numbers in the empty boxes until all 4 boxes have been filled. 5. Find the difference. 6. The person with the greatest difference wins the round. 7. After all 4 rounds, the player who won the most rounds, wins the game. 8. If there is a tie, players add the differences from all 4 rounds and the highest total wins the game.”

• Center, Number Puzzles: Multiplication and Division (4-5), Stage 1: Two-Digit Factors, students use the numbers 0-9 to create multiplication equations. Narrative states, “Students use the digits 0–9 to make multiplication equations with two-digit factors true. Each digit may only be used one time.”

##### Indicator {{'3p' | indicatorName}}

Materials provide opportunities for teachers to use a variety of grouping strategies.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 provide opportunities for teachers to use a variety of grouping strategies. Suggestions are consistently provided for teachers within the facilitation notes of lesson activities and include guidance for a variety of groupings, including whole group, small group, pairs, or individual. Examples include:

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 21, Activity 3, students share their completed journals with a partner and discuss noticings. Launch states, “Groups of 2. ‘Share your food waste journal with a partner. As you look over your journals, what do you notice? What is the same? What is different?’” Activity states, “10 minutes: independent work time. 2 minutes: partner discussion.”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 25, Activity 1, students work with a partner as they divide decimal numbers by 0.1 and 0.01. Launch states, “Groups of 2.” Activity states, “5 minutes: independent work time. 5 minutes: partner work time. Monitor for students who: Describe how Jada's diagram shows the value of 1.6\div0.1 as 16. Describe how Jada's diagram also represents the expression 160\div10.”

• Unit 7, Shapes on the Coordinate Plane, Lesson 8, Activity 2, students work in groups of two or four to sort triangles in a way that makes sense to them. Launch states, “Groups of 2 or 4 (if doing a gallery walk).” Activity states, “5 minutes: independent work time. 5 minutes: small- group work time. MLR7 Compare and Connect ‘Create a visual display that shows your thinking about the problems. You may want to include details such as notes, diagrams, or drawings to help others understand your thinking.’ 2–5 minutes: independent or group work. 5–7 minutes: gallery walk.”

• Unit 8, Putting It All Together, Lesson 5, Activity 2, students work in partnerships to practice using partial quotients. Launch states, “Groups of 2, then 4. ‘You and your partner will each find a quotient independently. After you’re done, discuss your work with your partner.’” Activity states, “3–5 minutes: independent work time. 1–3 minutes: partner discussion. ‘Now, find another group of 2 and compare your work. How is it the same? How is it different?’”

##### Indicator {{'3q' | indicatorName}}

Materials provide 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.

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.

##### Indicator {{'3r' | indicatorName}}

Materials provide a balance of images or information about people, representing various demographic and physical characteristics.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 provide a balance of images or information about people, representing various demographic and physical characteristics.

Images of characters are included in the student facing materials when they connect to the problem tasks. These images represent different races and portray people from many ethnicities in a positive, respectful manner, with no demographic bias for who achieves success based on the grade-level mathematics and problem circumstances. Names include multi-cultural references such as Han, Mai, Lin, and Priya and problem settings vary from rural, to urban, and international locations. Additionally, lessons include a variety of problem contexts to interest students of various demographic and personal characteristics.

##### Indicator {{'3s' | indicatorName}}

Materials provide guidance to encourage teachers to draw upon student home language to facilitate learning.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 partially provide guidance to encourage teachers to draw upon student home language to facilitate learning.

The student materials are available in Spanish. Directions for teachers are in English with prompts for students available in Spanish. The student materials including warm ups, activities, cool-downs, centers, and assessments are in Spanish for students.

The IM K-5 Teacher Guide includes a section titled “Mathematical Language Development and Access for English Learners” which outlines the program’s approach towards language development in conjunction with the problem-based approach to learning mathematics, which includes the regular use of Mathematical Language Routines, “The MLRs included in this curriculum were selected because they simultaneously support students’ learning of mathematical practices, content, and language. They are particularly well-suited to meet the needs of linguistically and culturally diverse students who are learning mathematics while simultaneously acquiring English.” While Mathematical Language Routines (MLRs) are regularly embedded within lessons and support mathematical language development, they do not include specific suggestions for drawing on a student’s home language.

##### Indicator {{'3t' | indicatorName}}

Materials provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 provide some guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.

Spanish materials are consistently accessible for a variety of stakeholders within the Family Support Materials for each unit. Within lessons, cultural connections are included within the context of problem solving, picture book centers, or games. Examples include:

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 17, students study and then create mosaics using multiplication of fractions. Lesson Narrative states, “This lesson is optional because it does not address any new mathematical content standards. This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling. In previous lessons, students found products of whole numbers and fractions, including fractions greater than 1. In this lesson, they apply what they learned about multiplying whole numbers and fractions to make mosaic art pieces out of rectangles and use area to determine how much it costs to recreate the mosaic with hard material like stone, tile, and glass. Throughout the activity, students make sense of problems and persevere in solving them (MP1). In the first activity, students create rectangles from colored paper. Each rectangle has a side that is a fraction greater than 1 and a side that is a whole number. Students multiply whole numbers by fractions to find the area of one rectangle and then find the combined area of all of their rectangles. In the second activity, students exchange their different sized and colored rectangles and make a mosaic. They analyze and compare their mosaics by area. Finally in the synthesis, students sort selected mosaics from different groups. For example, they sort from smallest to largest area covered.”

• Unit 3, Multiplying and Dividing Fractions, Lesson 9, Warm-up, students prepare to create their own flag later in the lesson. The warm-up has students look at two flags, Botswana and Taiwan, to make noticings and wonderings about the designs used on both flags. Narrative states, “The purpose of this warm-up is for students to discuss the meaning and intention behind flag design, which will be useful when students design their own flag in a later activity. While students may notice and wonder many things about these images, the colors, symbols, and the shapes used in the flag are the important discussion points. In the synthesis, students consider questions to ask the designers of the flag. As an extension to this warm up, students can further explore these questions to learn more about the flag.” Launch states, “Groups of 2. Display the image. ‘What do you notice? What do you wonder?’ 1 minute: quiet think time.”  Student response states, “Students may notice: The first image has rectangles. The second image has rectangles and something that looks like the sun. They look like flags. The first image uses the colors blue and white and the second one uses red, blue, and white. Students may wonder: What country are these flags from? What do the colors represent? What does the sun represent?” Activity synthesis states, “‘The first image is the flag of Botswana.’ If needed, show students where Botswana is on the map. ‘The second image is the flag of Taiwan.’ If needed, show students where Taiwan is on the map. ‘What are some questions you might ask the designer of these flags?’ (What do the different colors represent? Why is there a sun on the second flag?)”

##### Indicator {{'3u' | indicatorName}}

Materials provide supports for different reading levels to ensure accessibility for students.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 provide some supports for different reading levels to ensure accessibility for students.

• Unit 1, Fractions as Numbers, Lesson 7, Activity 2, Access for Students with Disabilities, “Representation: Access for Perception. Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems, and other text-based content. Supports accessibility for: Language, Conceptual Processing.”

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 1, Activity 2, Narrative states, “This activity uses MLR2 Collect and Display. Advances: Reading, Writing, Activity states, “MLR2 Collect and Display, Circulate, listen for, and collect the language students use to describe how they know each person gets the same amount. Listen for these words and phrases: divide, same, equal, fair, size of the piece, number of pieces, and one third of one half. Record students’ words, phrases, and expressions on a visual display and update it throughout the lesson.”

• Unit 7, Shapes on the Coordinate Plane, Lesson 13, Activity 1, Access for Students with Disabilities, “Representation: Access for Perception. Read tasks aloud. Students who both listen to and read the information will benefit from extra processing time. Supports accessibility for: Language Conceptual Processing.

##### Indicator {{'3v' | indicatorName}}

Manipulatives, both virtual and physical, are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

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.’”

#### Criterion 3.4: Intentional Design

The program includes a visual design that is engaging and references or integrates digital technology, when applicable, with guidance for teachers.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 do not integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level standards. The materials do not include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other. The materials have a visual design that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic. The materials do not provide teacher guidance for the use of embedded technology to support and enhance student learning.

##### Indicator {{'3w' | indicatorName}}

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 do not integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.

According to the IM K-5 Teacher Guide, About These Materials, “Teachers can access the teacher materials either in print or in a browser as a digital PDF. When possible, lesson materials should be projected so all students can see them.” While this format is provided, the materials are not interactive.

According to the IM K-5 Teacher Guide, Key Structures in This Course, “Across lessons and units, students are systematically introduced to representations and encouraged to use representations that make sense to them. As their learning progresses, students are given opportunities to make connections between different representations and the concepts and procedures they represent. Over time, they will see and understand more efficient methods of representing and solving problems, which support the development of procedural fluency. In general, more concrete representations are introduced before those that are more abstract.” While physical manipulatives are referenced throughout lessons and across the materials, they are not virtual or interactive.

##### Indicator {{'3x' | indicatorName}}

Materials include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 do not include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

According to IM K-5 Teacher Guide, Key Structures in this Course, “Classroom environments that foster a sense of community that allows students to express their mathematical ideas—together with norms that expect students to communicate their mathematical thinking to their peers and teacher, both orally and in writing, using the language of mathematics—positively affect participation and engagement among all students(Principles to Action, NCTM).” While the materials embed opportunities for mathematical community building through student task structures, discourse opportunities, and journal and reflection prompts, these opportunities do not reference digital technology.

##### Indicator {{'3y' | indicatorName}}

The visual design (whether in print or digital) supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 have a visual design (whether in print or digital) that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

There is a consistent design within units and lessons that supports student understanding of the mathematics. According to the IM K-5 Teacher Guide, Design Principles, “Each unit, lesson, and activity has the same overarching design structure: the learning begins with an invitation to the mathematics, is followed by a deep study of concepts and procedures, and concludes with an opportunity to consolidate understanding of mathematical ideas.” Examples from materials include:

• Each lesson follows a common format with the following components: Warm-up, one to three Activities, Lesson Synthesis, and Cool-Down, when included in lessons. The consistent structure includes a layout that is user-friendly as each component is included in order from top to bottom on the page.

• Student materials, in printed consumable format, include appropriate font size, amount and placement of directions, and space on the page for students to show their mathematical thinking.

• Teacher digital format is easy to navigate and engaging. There is ample space in the printable Student Task Statements, Assessment PDFs, and workbooks for students to capture calculations and write answers.

##### Indicator {{'3z' | indicatorName}}

Materials provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 do not provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

While the IM K-5 Teacher Guide provides guidance for teachers about using the “Launch, Work Synthesize” structure of each lesson, including guidance for Warm-ups, Activities, and Cool-Downs, there is no embedded technology.

## Report Overview

### Summary of Alignment & Usability for Kendall Hunt’s Illustrative Mathematics | Math

#### Math K-2

The materials reviewed for Illustrative Mathematics Kendall Hunt Grades K-2 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and practice-content connections. In Gateway 3, the materials meet expectations for usability including Teacher Supports and Student Supports; the materials partially meet expectations for Assessment.

##### Kindergarten
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations

#### Math 3-5

The materials reviewed for Illustrative Mathematics Kendall Hunt Grades 3-5 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and practice-content connections. In Gateway 3, the materials meet expectations for usability including Teacher Supports and Student Supports; the materials partially meet expectations for Assessment.

###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations

## Report for {{ report.grade.shortname }}

### Overall Summary

###### Alignment
{{ report.alignment.label }}
###### Usability
{{ report.usability.label }}

### {{ gateway.title }}

##### Gateway {{ gateway.number }}
{{ gateway.status.label }}