The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for assessing grade-level content and if applicable, content from earlier grades. The materials for Grade 5 are divided into eight units, and each unit contains a written End-of-Unit Assessment. Additionally, the Unit 8 Assessment is an End-of-Course Assessment, and it includes problems from the entire grade level. Examples of End-of-Unit Assessments include:

• Unit 1, Finding Volume, End-of-Unit Assessment, Problem 4, students “Find the volume of a rectangular prism with the given side lengths. 1.The length is 2 units, the width is 5 units, and the height is 7 units. 2.The base has an area of 200 square inches and the height is 6 inches.” (5.MD.5)

• Unit 3, Multiplying and Dividing Fractions, End of Unit Assessment, Problem 6, “An apple weighs $\frac{1}{2}$ pound. Diego cuts the apple into 4 equal pieces. How many pounds does each piece of the apple weigh? Explain your reasoning.” (5.NF.7)

• Unit 6, More Decimal and Fraction Operations, End of Unit Assessment, Problem 5, “Han’s backpack weighs $\frac{5}{4}$ as much as Lin’s backpack. Clare’s backpack weighs $\frac{2}{3}$ as much as Lin’s backpack. Whose backpack weighs the most? Whose backpack weighs the least? Explain or show how you know.” (5.NF.5)

• 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)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards. The materials provide extensive work with and opportunities for students to engage in the full intent of Grade 5 standards by including in every lesson a Warm Up, one to three instructional activities, and Lesson Synthesis. Within Grade 5, students engage with all CCSS standards.

Examples of extensive work include:

• Unit 1, Finding Volume, Lesson 4, Using Layers to Determine Volume, Lesson 10, Represent Volume with Expressions, and Unit 2, Fractions as quotients and Fraction Multiplication, Lesson, Lesson 8, Divide to Multiply Non-unit Fractions, and Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 3, Partial Products in Algorithms,   provide students with extensive work with 5.OA.2 (Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them). In Lesson 4, Using Layers to Determine Volume, Activity 2, students interpret multiplication expressions that represent the volume of an illustrated rectangular prism with dimensions 4, 5, 6, “1. Explain or show how the expression  $5\times24$ represents the volume of this rectangular prism., 2. Explain or show how the expression $6\times20$ represents the volume of this rectangular prism., 3. Find a different way to calculate the volume of this rectangular prism. Explain or show your thinking., 4. Write an expression to represent the way you calculated the volume.” In Lesson 10, Activity 1, students write expressions to represent the volume of an illustrated prism that has an L-shaped base, “1. Write an expression to represent the volume of the figure in unit cubes.” In Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 8, Divide to Multiply Non-unit Fractions, students determine if expressions are equivalent during the warm up, “Decide if each statement is true or false. Be prepared to explain your reasoning. $2\times(\frac{1}{3}\times6)=\frac{2}{3}\times6$, $2\times(\frac{1}{3}\times6)=2\times(6\div3)$, $\frac{2}{3}\times6=2\times(\frac{1}{4}\times6)$.” In Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 3, Partial Products in Algorithms, Activity 1, students use different illustrated area models to represent multi-digit multiplication with different expressions, “1. Take turns picking out a set of expressions that are equal to $245\times35$ when added together. Use the diagrams if they are helpful., 2. Explain how you know the sum of your expressions is equal to $245\times35$., 3. What is the value of $245\times35$? Explain or show your reasoning.” In Unit 5, Place Value Patterns and Decimal Operations, Lesson 17, Multiply Decimals and Whole Numbers, Activity 1, students create visual models of different expressions using 100s grids, “Find the value of each expression in a way that makes sense to you. Explain or show your reasoning. Use the grids, if needed., 1. $2times0.7$, 2. $2\times0.08$, 3. $2\times0.78$. In Lesson 23, Activity 2, students relate a multiplication and division expression to a single diagram. The provided diagram is a 100s grid with 2 columns colored blue, followed by 2 columns colored orange, repeated for 10 columns. “2. This is the diagram and explanation Tyler used to justify why $12\div0.2=60$. ‘ There are 5 groups of 0.2 in 1 and there are 12 so that is 12 groups of 5.’ Explain how the expression $12\times(0.1\div0.2)$ relates to Tyler’s reasoning.” In Unit 7, Lesson 12, Represent Problems on the Coordinate Grid, students decide if expressions are equivalent during the warm up, “Decide if each statement is true or false. Be prepared to explain your reasoning., $(2\times10)+(3\times5)=(3\times10)+(1\times5)$, $(3\times25)+(5\times5)=8\times25$, $(4\times25)+(10\times5)=(2\times25)+(10\times10)$.”

• Unit 5, Place Value Patterns and Decimal Operations, Lessons 5, 6, and 9 engage students in extensive work of 5.NBT.3b (Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.) In Lesson 5, Compare Decimals, Activity 1: Faster and Faster, students compare decimals to hundredths place. “1. Diego and Jada were competing to see who could throw the frisbee further. Diego threw the frisbee 5.10 meters. Jada threw the frisbee 5.01 meters. Who threw the frisbee further? Be prepared to explain your thinking. 2. Tyler and Han were competing to see who could swim the length of the pool faster. Tyler swam the length of the pool in 35.15 seconds. Han swam the length of the pool in 35.30 seconds. Who swam the length of the pool faster? Be prepared to explain your thinking.” In Lesson 6, Compare Decimals on a Number Line, Activity 2, Label and Compare Decimals, students accurately label the marks for decimals through the thousandths place on a number line. Students are given 3 number lines to label the thousandths place. “1. Label the tick marks on each number line. 2. Which of the number lines would you use to compare 0.534 and 0.537? Explain or show your reasoning.” In Lesson 9, Ordering Decimals, Warm Up: True or False: Decimal Inequalities, students compare pairs of decimal numbers and explain their reasoning. Students “decide whether each statement is true or false. Be prepared to explain your reasoning. $0.909>0.91$; $4.1<4.100$; $0.99<0.999$$.”

• Unit 6, More Decimal and Fraction Operations, Lesson 5, Multi-step Conversion Problems: Metric Length, students engage with extensive work with 5.MD.1 (Convert like measurement units within a given measurement system). In Activity 2: Who Ran Farther?, Student Task Statements, 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. Column labels include day and distance (km) with the following data: 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. Column labels include day and distance (m) with the following data: 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.” 

Examples of full intent include:

• Unit 5, Place Value Patterns and Decimal Operations, Lessons 7, Round Doubloons, and Lesson 8, Round Decimals, provides the opportunity for students to engage with the full intent of standard 5.NBT.4 (Use place value understanding to round decimals to any place). Lesson 7, Round Doubloons, Activity 1: Gold Doubloons, students round to the nearest tenth and hundredth. “Display the image. “This is a doubloon. What do you notice? What do you wonder?” Problem 4, “Use the number lines to find which hundredth of a gram the doubloon weights are each closest to.” Lesson 8, Round Decimals, Activity 2: Which Number is Closest?, students round a decimal number to the nearest whole number, tenth, and hundredth. Problem 2, “Round 4.158 to the nearest whole number, tenth, and hundredth.” 

• Unit 6, More Decimal and Fraction Operations, Lesson 12, Solve Problems, provides students with opportunities to meet the full intent of 5.NF.2 (Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.) In Activity 1, students are shown a recipe for salad dressing that uses $\frac{3}{4}$ cup of olive oil, and answer questions requiring subtraction of fractions with unlike denominators and assessing the reasonableness of an estimate, “1. Priya has $\frac{2}{3}$ cup of olive oil. She is going to borrow some more from her neighbor. How much olive oil does she need to borrow to have enough to make the dressing?, 2. 1 tablespoon is equal to $\frac{1}{16}$ of a cup. Priya decides that 1 tablespoon of olive oil is close enough to what she needs to borrow from her neighbor. Do you agree with Priya? Explain or show your reasoning.” In the Cool Down, students are required to add fractions with unlike denominators, “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.”

• Unit 7, Shapes on the Coordinate Plane, Lesson 2, Points on the Coordinate Grid, and Lesson 3, Plot More Points, engage students with the full intent of 5.G.1 (Graph points on the coordinate plane to solve real-world and mathematical problems). In Lesson 2, Points on the Coordinate Grid, Activity 2: Plot and Label Points, students write ordered pairs of numbers to represent points in the coordinate plane and to plot points with given coordinates. Student Task Statements, a coordinate plane and three points are provided for students. “1. List the coordinates for each point. 2. Plot points D, E, F on the same grid. D (6,4), E (2,5), F (8,3).” In Lesson 3, Plot More Points, Cool-down, Missing Coordinate,“Here is a grid with some points labeled. Plot and label the points (3,0), (0,2) and (3,2). Explain or show your reasoning.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of the 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 the major work of the grade (including assessments and supporting work connected to 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 major work) is 137 lessons out of 156 lessons, approximately 88%. The total number of lessons include 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 major work) is 143.5 days out of 164 days, approximately 88%.

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

The materials reviewed for Imagine Learning 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 with supporting standards/clusters 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 on the Course Guide tab for each Unit. Teacher Notes also provide the explicit standards listed within the lessons. Examples of connections include:

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 13, Area and Properties of Operations, Warm-up: Number Talk: Parentheses, 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 Task Statements, “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 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 11, Different Partial Quotients, Activity 2: Division Expressions, Student Task Statement, 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.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). Students rewrite division expressions to find their values. “1. Choose a set of expressions that, when added together, is equal to $308\div14$. Not all expressions will be used. 2. Explain to your partner how you know that your cards represent a sum that is equal to $308\div14$. 3. Choose one of the sets of expressions whose sum is equal to $308\div14$ and use it to find the value of $308\div14$.”

• Unit 6, More Decimals and Fraction Operations, Lesson 4, Metric Conversion and Multiplication by Powers of 10, Activity 1: Long Jump, Javelin Throw, and Shot Put connects the supporting work of 5.MD.1 (Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real-world problems) to the major work of 5.NBT.2 (Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10). Student Task Statement, “Below are some results Jackie Joyner-Kersee recorded in different events in 1988. Complete the table. (Table has the following columns with the following information in the rows: Event- long jump, javelin throw, shot put; centimeters- 727, 4,566, and 1,580; meters- (blank). (Students are to convert centimeters to meters)” “What pattern do you notice? $727\div100=7.27$; $4,566\div100=45.66$; $1,580\div100=15.80$”

The materials reviewed for Imagine Learning 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. Examples of connections between major work to major work and/or supporting work to supporting work throughout the materials, when appropriate, include:

• Unit 4, Wrapping Up Multiplication and Division with Multi-digit Numbers, Lesson 16, World’s Record Noodle Soup connects the major work of 5.NF.B (Apply and extend previous understandings of multiplication and division to multiply and divide fractions) to the major work of 5.NBT.B (Perform operations with multi-digit whole numbers and with decimals to hundredths). In Activity 2, Han’s Estimate, students apply understanding of multiplication and division and perform multiplication and division with multi-digit whole and fractions. Lesson Synthesis, “Today, we solved problems about a real life context. We also discussed solutions that were mixed numbers. In what ways did we use division today?”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 21, Multiply More Decimals, Activity 1, connects the major work of 5.NBT.A (Understand The Place Value System) with the major work of 5.NBT.B (Perform Operations With Multi-Digit Whole Numbers And With Decimals To Hundredths). Students explain equivalence among expressions using both decimals and whole numbers. Student Task Statement, “1. Explain or show why each pair of expressions is equivalent., a. $7.2\times5.3$ and $(72\times53)\times0.01$, b. $6.5\times2.8$ and $(65\times28)\div100$, c. $31\times0.44$ and $(31\times44)\times\frac{1}{100}$, 2. Find the value of the products in the previous problem.”

• Unit 7, Shapes on the Coordinate Plane, Lesson 13, Perimeter and Area of Rectangles, Activity 1, 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 draw rectangles on the coordinate plane. Student Task Statement, “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.” 

• Unit 8, Putting it All Together, Lesson 14, Notice and Wonder, Activity 3, Design Your Notice and Wonder, Part 2, connects the major work of 5.NBT (Numbers and Operations In Base Ten) with the major work of 5.NF (Number and Operations - Fractions). Students find meaningful mathematics in images or photographs. Students are encouraged to make connections to a mathematical topic they have become familiar with. “1. Find an image that you find interesting and would encourage your classmates to notice and wonder about a mathematical topic you have learned this year. 2. Fill in the possible things students might notice and wonder about your image.”

The materials reviewed for Imagine Learning 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. 

The Section Dependency Chart explores the Unit sections relating to future grades. The Section Dependency Chart states, “arrow indicates the prior section that contains content most directly designed to support or build toward the content in the current section.” 

Examples of connections to future grades include:

• Unit 3, Multiplying and Dividing Fractions, Section B, Fraction Division, Section summary connects 5.NF.7 (Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.) with work done in grade 6. “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.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-digit Numbers, Section B, Multi-digit Division Using Partial Quotients, connects 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm.), 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.), 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, e.g., by using visual fraction models or equations to represent the problem), 5.OA.2 (Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.) with work done in grade 6. “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.”

• Unit 5, Finding Volume, Section C Narrative: Volume of Solid Figures connects 5.NBT.2 (Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.) and 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm) to work in grade 6. About this Section, “The work reminds students that they can decompose multi-digit factors by place value to find their product, paving the way toward the standard algorithm for multiplication in a later unit.”

Examples of connections to prior knowledge include:

• Unit 1, Finding Volume, Lesson 3, Volumes of Prism Drawings, Warm-up: Number Talk: Multiplication connects 4.MD.5 (Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume) to the work in grade 3, “In grade 3, students learned how to find the area of a rectangle by tiling and found that multiplying the side lengths yields the same result (3.MD.7).”

• Unit 2, Fractions as Quotients and Fractions as Multiplication, Section B: Fractions of Whole Numbers, Section Narrative, “In grade 4, students saw that a non-unit fraction can be expressed as a product of a whole number and a unit fraction, or a whole number and a non-unit fraction with the same denominator. For instance, $\frac{8}{3}$ can be expressed as $8\times\frac{1}{3}$, as $4\times\frac{2}{3}$, or as $2\times\frac{4}{3}$. In the previous section, students interpreted a fraction like $\frac{8}{3}$ as a quotient: $8\div3$. This section allows students to connect these two interpretations of $\frac{8}{3}$ and relate $8\frac{1}{3}$ and $8\div3$.”

• Unit 6, More Decimal and Fraction Operations, Unit Overview, Full Unit Narrative, “In this unit, students deepen their understanding of place-value relationships of numbers in base ten, unit conversion, operations on fractions with unlike denominators, and multiplicative comparison. The work here builds on several important ideas from grade 4. In grade 4, students learned the value of each digit in a whole number is 10 times the value of the same digit in a place to its right. Here, they extend that insight to include decimals to the thousandths. Students recognize that the value of each digit in a place (including decimal places) is $\frac{1}{10}$ the value of the same digit in the place to its left.”

The materials reviewed for Imagine Learning 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. Examples include:

• IM Curriculum, Why is the curriculum designed this way?, Design Principles, Coherent Progression, “To support students in making connections to prior understandings and upcoming grade-level work, it is important for teachers to understand the progressions in the materials. Grade level, unit, lesson, and activity narratives describe decisions about the organization of mathematical ideas, connections to prior and upcoming grade-level work, and the purpose of each lesson and activity. When appropriate, the narratives explain whether a decision about the scope and sequence is required by the standards or a choice made by the authors. The basic architecture of the materials supports all learners through a coherent progression of the mathematics based both on the standards and on research-based learning trajectories. Each activity and lesson is part of a mathematical story that spans units and grade levels. This coherence allows students to view mathematics as a connected set of ideas that makes sense. 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. The invitation to the mathematics is particularly important because it offers students access to the mathematics. It builds on prior knowledge and encourages students to use their own language to make sense of ideas before formal language is introduced, both of which are consistent with the principles of Universal Design for Learning.”

• IM Curriculum, Scope and sequence information, 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.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section A, Multi-Digit Multiplication Using the Standard Algorithm, Section Overview, Section Narrative, provides an overview of the content and expectations for the section. For example, “This section introduces the standard algorithm for multiplication, extending students’ earlier work on multiplication. In grade 4, students used diagrams and partial-products algorithms to find the product of a one-digit number and a number up to four digits, and the product of 2 two-digit numbers. They attended to the role of place value along the way. Students revisit these strategies and representations here, but work with factors with more digits than encountered in grade 4. They make connections between the partial products in diagrams and previous algorithms to the numbers in the standard algorithm. They also learn the notation for recording new place-value units that result from finding partial products. When using the standard algorithm to multiply a two-digit number and a three-digit number, students account for the place value of the digits being multiplied, as they had done before. For example, the 3 in 23 represents 3 ones, so $3\times123$ is 369. The 2 in 23, however, represents 2 tens, so the partial product is $2\times10\times123$ or 2,460, instead of $2\times123$ or 246. The partial products 369 and 2,460 can be seen in a diagram as well. Once students have practiced recording products this way, they learn to multiply factors that require composing new units, such as $264\times38$.”

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, Warm-up, Activities, and Cool-down narratives all provide useful annotations. IM Curriculum, Why is the curriculum designed this way?, Design Principles, Coherent Progressions, “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.” Examples include:

• Unit 1, Finding Volume, Lesson 4, Activity 1, teachers are provided context to help students reason about the volume of prisms. Narrative, “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, “Groups of 2. Display first image from the 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, “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 9, About this Lesson provides notes about the lesson for the teacher. “The purpose of this lesson is for students to generate two different numerical patterns and then compare the terms in the two patterns. In this lesson, the patterns are the multiples of given whole numbers, starting with 0, and one of the numbers is a multiple of the other. This means that one of the patterns is contained inside the other. For example, the list of multiples of 9 is contained inside the list of multiples of 3 since every third multiple of 3 is a multiple of 9. Students express relationships within a pattern and between 2 patterns using multiplication and division. Lesson purpose: The purpose of this lesson is for students to generate patterns, given two rules, and identify relationships between corresponding terms in the different patterns. Teacher reflection question: In what ways did you accept students' everyday way of talking as a starting point for joining the math conversation today?” The section also includes lesson overview, learning goals, learning goals (student facing), materials to gather, materials to copy, and required preparation.

The materials reviewed for Imagine Learning 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, Why is the curriculum designed this way?, 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 current grade so that teachers can improve their own knowledge of the subject. Professional articles support teachers with learning opportunities about topics such as ensuring mathematical success for all, early understanding of equality, and repeating patterns. Examples include:

• Why is the curriculum designed this way? Further Reading, Unit 1, “A world without order (of operations). In this blog post, McCallum describes a world with only parentheses to guide the order of operations and discusses why the conventional order of operations is useful.” 

• Why is the curriculum designed this way? Further Reading, Unit 3, “Why is a negative times a negative a positive? supports teachers with context for work beyond the grade. “In this blog post, McCallum discusses how the “rule” for multiplying negative numbers is grounded in the distributive property.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 21, Food Waste Journal, About this Lesson, “This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling. In previous lessons, students made estimates about the volume of recyclable garbage students at their school produce. In this lesson, students make similar estimates and calculations, but now they estimate the weight of food waste produced. In the first activity students estimate the amount of food waste they produce based on average production by individuals in the United States. In the second activity, students are introduced to a food waste journal and make some initial calculations about their food waste. The last activity is optional and can be used after students use the journal to record their food waste for a week. Students share what they notice and compare the amount of food waste they produce to the national average. Students use this sample data to estimate how much food waste they produce monthly and annually. When students recognize the mathematical features of familiar real world situations and use those features to solve problems, they model with mathematics (MP4).”

• Unit 6, More Decimals and Fraction Operations, Lesson 21, Weekend Investigation, About this Lesson, “This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling. In this lesson, students brainstorm and define categories of how to spend time. Then they collect and represent data on a line plot. They analyze and describe the data to tell a story about the time use. When students define categories, choose and ask questions, collect and analyze data, and tell a story about the situation based on data, they model with mathematics (MP4).”

The materials reviewed for Imagine Learning 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-level resources, Grade 5 standards breakdown, standards are addressed by lesson. Teachers can search for a standard in the grade and identify the lesson(s) where it appears within materials.

• Course Guide, Lesson Standards, includes all Grade 5 standards and the units and lessons each standard appears in. 

• 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.

• Unit 6, More Decimal and Fraction Operations, Lesson 5, the Core Standards are identified as 5.MD.A.1 and 5.NBT.A.2. Lessons contain a consistent structure that includes a Warm-up with a Narrative, Launch, Activity, Activity Synthesis. An Activity 1, 2, or 3 that includes Narrative, Launch, Activity, Activity Synthesis, Lesson Synthesis. 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.

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

• Unit 5: Place Value Patterns and Decimal Operations, Unit Overview, Unit Learning Goals, full Unit Narrative, include an overview of how the math of this module builds from previous work in math. “In this unit, students expand their knowledge of decimals to read, write, compare, and round decimals to the thousandths. They also extend their understanding of place value and numbers in base ten by performing operations on decimals to the hundredth. In grade 4, students wrote fractions with denominators of 10 and 100 as decimals. They recognized that the notations 0.1 and 110 express the same amount and are both called “one tenth.” They used hundredths grids and number lines to represent and compare tenths and hundredths. Here, students likewise rely on diagrams and their understanding of fractions to make sense of decimals to the thousandths. They see that “one thousandth” refers to the size of one part if a hundredth is partitioned into 10 equal parts, and that its decimal form is 0.001. Diagrams help students visualize the magnitude of each decimal place and compare decimals.”

• Unit 6: More Decimals and Fraction Operations, Unit Overview, full Unit Narrative, include an overview of how the math of this module builds from previous work in math. “In this unit, students deepen their understanding of place-value relationships of numbers in base ten, unit conversion, operations on fractions with unlike denominators, and multiplicative comparison. The work here builds on several important ideas from grade 4. In grade 4, students learned the value of each digit in a whole number is 10 times the value of the same digit in a place to its right. Here, they extend that insight to include decimals to the thousandths. Students recognize that the value of each digit in a place (including decimal places) is $\frac{1}{10}$ the value of the same digit in the place to its left. This idea is highlighted as students perform measurement conversions in metric units. Previously, students learned to convert from a larger unit to a smaller unit. Here, they learn to convert from a smaller unit to a larger unit. They observe how the digits shift when multiplied or divided by a power of 10 and learn to use exponential notation for powers of 10 to represent large numbers.”

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

Instructional approaches of the program are described within the Curriculum Guide, Why is the curriculum designed this way? Design Principles. “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 materials through coherent progression, and provides teachers the opportunity to deepen their knowledge of mathematics, student thinking, and their own teaching practice. Examples from the Design Principles include:

• Curriculum Guide, Why is the curriculum designed this way?, Design Principles, includes information about the 11 principles that informed the design of the materials. Balancing Rigor, “There are three aspects of rigor essential to mathematics: conceptual understanding, procedural fluency, and the ability to apply these concepts and skills to mathematical problems with and without real-world contexts. These aspects are developed together and are therefore interconnected in the materials in ways that support student understanding.  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. Warm-ups, practice problems, centers, and other built-in routines help students develop procedural fluency, which develops over time.”

• Curriculum Guide, Why is the curriculum designed this way?, Design Principles, Task Complexity, “Mathematical tasks can be complex in different ways, with the source of complexity varying based on students’ prior understandings, backgrounds, and experiences. In the curriculum, careful attention is given to the complexity of contexts, numbers, and required computation, as well as to students’ potential familiarity with given contexts and representations. To help students navigate possible complexities without losing the intended mathematics, teachers can look to warm-ups and activity launches for built-in preparation, and to teacher-facing narratives for further guidance. In addition to tasks that provide access to the mathematics for all students, the materials provide guidance for teachers on how to ensure that during the tasks, all students are provided the opportunity to engage in the mathematical practices. More details are given below about teacher reflection questions, and other fields in the lesson plans help teachers assure that all students not only have access to the mathematics, but the opportunity to truly engage in the mathematics.”

Research-based strategies within the program are cited and described within the Curriculum Guide, within Why is the curriculum designed this way?. There are four sections in this part of the Curriculum Guide including Design principles, Key Structures, Mathematical Representations, and Further Reading. Examples of research-based strategies include: 

• Curriculum Guide, Why is the curriculum designed this way?, Further Reading, Entire Series, The Number Line: Unifying the Evolving Definition of Number in K–12 Mathematics. “In this article, the authors (Lahme, McLeman, Nakamaye, and Umland) focus their attention on the selection of definitions, notation, and graphical conventions surrounding the development of the real numbers from kindergarten to grade 12, and address the work that students might do in later years.”

• Curriculum Guide, Why is the curriculum designed this way?, Further Reading, 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.”

• Curriculum Guide, Why is the curriculum designed this way?, Design Principles, Instructional Routines, “Instructional routines provide opportunities for all students to engage and contribute to mathematical conversations. Instructional routines are invitational, promote discourse, and are predictable in nature.” They are “enacted in classrooms to structure the relationship between the teacher and the students around content in ways that consistently maintain high expectations of student learning while adapting to the contingencies of particular instructional interactions.” (Kazemi, Franke, & Lampert, 2009)

• Curriculum Guide, Why is the curriculum designed this way?, 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.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Course Overview, Grade-level resources, provides a Materials List intended for teachers to gather materials for each grade level. Additionally, specific lessons include a Teaching Notes section and a Materials List, which include specific lists of instructional materials for lessons. Examples include:

• Course Overview, Grade Level Resources, Grade 5  Materials List, contains a comprehensive chart of all materials needed for the curriculum. It includes the materials used throughout the curriculum, whether they are reusable or consumable, quantity needed, lessons the materials are used in, and suitable substitutes for the materials. Each lesson listed in the chart and any additional virtual materials noted for a lesson are digitally linked in the materials for quick access. Meter sticks are a reusable material used in lesson 5.4.19, 5.6.3, and 5.6.5. 15 (one per pair) are needed per 30 students. No suitable substitutes are listed. Number cubes are a reusable material used in lessons 5.5.11, 5.5.14, and 5.8.12. 15 are needed for 30 students. No suitable substitutes for the material are listed. Chart paper is a consumable material used in lessons 5.5.1, 5.5.11, 5.5.14, 5.8.14, and 5.8.15. 30 pages are needed per 30 students. Poster Paper is a suitable substitute for the material.

• Unit 1, Finding Volume, Lesson 7, Activity 1: What are the Units?, Teaching Notes, Materials to gather, “Rulers (centimeters), Rulers (inches), Yardsticks.” Launch, “Write the list of objects (moving truck, freezer, etc…) on a display for all students to see. In this activity we are going to consider using different cubic units of measure to find the volume of different sized objects. There is no right or wrong answer in these questions, but be prepared to explain your choice. Give students access to rulers and yardsticks.”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 12, Activity 2, Teaching Notes, Materials to gather, “Number cubes, Target Numbers Stage 9 Recording Sheet.” Launch, “Give each group 1 number cube. We’re going to play a new stage of the game called Target Numbers. Let’s read through the directions and play one round together. Read through the directions with the class and play a round with the class: Display each roll of the number cube. Think through your choices aloud. Record your move and score for all to see. Now, play the game with your partner.”

The materials reviewed for Imagine Learning 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. In the Curriculum Guide, How do the materials support all learners? 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 include:

• Unit 3, Multiplying and Dividing Fractions, Lesson 3, Activity 2, Narrative, Access for Students with Disabilities, “Engagement: Provide Access by Recruiting Interest. Invite students to share a connection between the diagram and something in their own lives that represent the fractional values. Supports accessibility for: Attention, Conceptual Processing.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 19, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Internalize Comprehension.  Activate or supply background knowledge. Provide students with a visual representation of a kilometer for students who are unfamiliar with the distance. Supports accessibility for: Conceptual Processing, Memory, Attention.”

• 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.”

The materials reviewed for Imagine Learning 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 do you use the materials?, Practice Problems, 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 that students can do directly related to the material of the unit, 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 3, Multiplying and Dividing Fraction, Section B: Fraction Division, Problem 9, Exploration, “It takes Earth 1 year to go around the Sun. 1. During the time it takes Earth to go around the Sun, Mercury goes around the Sun about 4 times. How many years does it take Mercury to make 1 full orbit of the Sun? Write an equation showing your answer. 2. During the time it takes Earth to go around the Sun, Saturn goes $\frac{1}{29}$ of the way around the Sun. How many years does it take Saturn to go around the Sun? Write an equation showing your answer.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section C: Let’s Put it to Work, Problem 5, Exploration, “The Pentagon has 5 floors and the Empire State Building has 102 floors. Noah says that the Empire State Building is bigger. Do you agree with Noah? Investigate and justify your answer.”

• Unit 5, Place Value Patterns and Decimal Operations, Section D: Divide Decimals, Problem 7, Exploration, “1. The daily recommended allowance of vitamin C for a 5th grader is 0.05 grams. A vitamin C tablet has 1 gram of vitamin C. How many times the daily recommended allowance of vitamin C is one vitamin C tablet? Use the diagram if it is helpful. 2. A large orange has 0.18 grams of vitamin C. How many times the daily recommended allowance of vitamin C is in a large orange? Use the diagram if it is helpful.”

The materials reviewed for Imagine Learning 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 Curriculum Guide, How do the materials support all learners? Mathematical language development, “Embedded within the curriculum are instructional routines and supports to help teachers address the specialized academic language demands when planning and delivering lessons, including the demands of reading, writing, speaking, listening, conversing, and representing in math (Aguirre & Bunch, 2012). While these instructional routines and supports can and should be used to support all students learning mathematics, they are particularly well-suited to meet the needs of linguistically and culturally diverse students who are learning mathematics while simultaneously acquiring English. Mathematical Language Routines (MLR) are also included in each lesson’s Support for English learners, to provide teachers with additional language strategies to meet the individual needs of their students. Teachers can use the suggested MLRs as appropriate to provide students with access to an activity without reducing the mathematical demand of the task. When selecting from these supports, teachers should take into account the language demands of the specific activity and the language needed to engage the content more broadly, in relation to their students’ current ways of using language to communicate ideas as well as their students’ English language proficiency. Using these supports can help maintain student engagement in mathematical discourse and ensure that struggle remains productive. All of the supports are designed to be used as needed, and use should fade out as students develop understanding and fluency with the English language.” The series provides 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 (MLR) in each lesson. Curriculum Guide, How do the materials support all learners? Mathematical language development, “A mathematical language routine is a structured but adaptable format for amplifying, assessing, and developing students' language. The MLRs were selected because they are effective and practical for simultaneously learning mathematical practices, content, and language. These routines can be adapted and incorporated across lessons in each unit to fit the mathematical work wherever there are productive opportunities to support students in using and improving their English and disciplinary language use. These routines facilitate attention to student language in ways that support in-the-moment teacher, peer, and self-assessment. The feedback enabled by these routines will help students revise and refine not only the way they organize and communicate their own ideas, but also ask questions to clarify their understandings of others’ ideas.” Examples include:

• Unit 1, Finding Volume, Lesson 3, Activity 1, Teaching notes, Access for English Learners, “MLR8 Discussion Supports. During small-group discussion, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “I heard you say . . . .” Original speakers can agree or clarify for their partner. Advances: Listening, Speaking.”

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 11, Activity 1, Teaching notes, Access for English Learners, “MLR8 Discussion Supports. Display sentence frames to support small-group discussion: “I wonder if . . . ”, “____ and ____ are the same because. . . .”, and “____ and ____ are different because ____.  Advances: Conversing, Representing.”

• Unit 4, Wrapping Up Multiplication and Division With Multi-Digit Numbers, Lesson 9, Synthesis, Teaching notes, Access for English Learners, “MLR1 Stronger and Clearer Each Time. Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to What is the possible range of volumes for each type of birdhouse?. Invite listeners to ask questions, to press for details, and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive. Advances: Writing, Speaking, Listening.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The materials consistently include suggestions and/or links, within the lesson notes, for virtual and physical manipulatives that support the understanding of grade-level math concepts. Examples include: 

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 17, Activity 2, students use grid paper to find products of whole numbers and some tenths or hundredths. Launch, “Groups of 2. Make hundredths grids available for students. Activity, ‘Take a few minutes to find the value of the expressions in the first problem.’ Student facing, ‘Find the value of each expression. Explain or show your reasoning. a. $3\times0.5$ b. $5\times0.3$ c. $7\times0.02$’”

• Unit 6, More Decimal and Fraction Operations, Lesson 3, Activity 1, students use meter sticks to help them convert meters to centimeters. Launch, “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 your partner to complete the problems.’”

• Unit 7, Shapes on the Coordinate Plane, Lesson 1, Activity 1, students use coordinate grids to communicate and draw shapes.  Launch, “Groups of 2. ‘We are going to play a drawing game. Decide who will be partner A and who will be partner B.’ Student facing, Play three rounds of Draw My Shape using the three sets of cards from your teacher. For each round: Partner A chooses a card—without showing your partner—and describes the shape on the card. Partner B draws the shape as described. Partner A reveals the card and partner B reveals the drawing. Compare the shapes and discuss: ‘What’s the same? What’s different?’ Activity, Circulate, listen for, and collect the language students use to describe the location of each figure on the coordinate grid. Listen for students who: use the grid to determine the side lengths or area of the rectangle. describe the general location of the rectangle. use the numbers on the axes as reference points when describing the rectangle. Record students’ words and phrases on a visual display and update it throughout the lesson.”