## 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
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## Report for Kindergarten

### Overall Summary

The materials reviewed for Kendall Hunt's Illustrative Mathematics Kindergarten 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.

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

### Focus & Coherence

The materials reviewed for Kendall Hunt's Illustrative Mathematics Kindergarten 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 Kindergarten 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 Kindergarten 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 an End-of-Unit Assessment. While Unit 1 includes an End-of-Unit Assessment as an Interview, all other units include a written assessment for individual student completion. Additionally, 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, Numbers 1-10, End-of-Unit Assessment, Problem 4, “a. Circle the number that is more. 4, 6. b. Circle the number that is less. 8, 5.” (K.CC.7)

• Unit 5, Composing and Decomposing Numbers to 10, End-of-Unit Assessment, Problem 3, “Mai has a train of 7 connecting cubes. She snaps the train into two pieces. Show 1 way to snap the cubes. Show a different way to snap the cubes.” A picture of seven snap cubes is shown. (K.OA.3)

• Unit 6, Numbers 0 - 20, End-of-Unit Assessment, Problem 1, ”Draw 17 dots. Use the 10-frame if it helps you.” An image of a ten frame is provided. (K.NBT.1)

• Unit 7, Solid Shapes All Around Us, End-of-Unit Assessment, Problem 3, “Consider the ball and box your teacher has displayed. How are the shapes the same? How are they different? Show your thinking with drawings or words.” (K.G.4)

• Unit 8, Putting it All Together, End-of-Course Assessment, Problem 4, “a. How many dots are there?  b. How many triangles are there?  c. How many counters are there?” There is a picture of 11 dots, 14 triangles arranged in a circle, and 17 counters on ten frames. (K.CC.5)

##### 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 Kindergarten 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 Kindergarten as students engage with all CCSSM standards within a consistent daily lesson structure, including a Warm Up, one to three Instructional Activities, a Lesson Synthesis, and 15-25 minutes for Centers. Examples of extensive work include:

• Unit 1, Math in Our World, Lesson 12; Unit 3, Flat Shapes all Around Us, Lesson 4; Unit 4, Understanding Addition and Subtraction, Lesson 12; Unit 5, Composing and Decomposing Numbers to 10 Lesson 9; Unit 6, Numbers 0-20, Lesson 11; and Unit 7, Solid Shapes all Around Us, Lesson 9 engage students in extensive work with K.CC.1 (Count to 100 by ones and by tens). Unit 1, Lesson 12, How Many Are There (Part 1), Activity 2, students work on the verbal count sequence to 10, “‘Let’s count to 10 all together.’ Count to 10 all together. ‘Let’s count to 10 and clap our hands when we say each number.’ Count to 10 and clap all together. ‘Let’s count to 10 and touch the table when we say each number.’ Count to 10 and touch the table all together. ‘Let’s count to 10 and put up 1 finger when we say each number.’ Count to 10 and put up each finger all together.” Throughout Kindergarten, students work up to counting to 100. Unit 3, Lesson 4, Describe, Compare and Sort Shapes, Warm Up: Choral Count, students continue the counting up to 30, “Count to 30 together. Record as students count. Count to 30 1–2 times. Point to the numbers as students count.” Unit 4, Lesson 12, Compare Addition and Subtraction Story Problems, students and the teacher count to 40 together. Warm Up: Choral Count, “Count to 40 together. Record as students count. Count to 40 1–2 times. Point to the numbers as students count.” As students progress to Unit 5, they count up to 70, count to 90 in Unit 6, and count to 100 in Unit 7. Unit 5, Lesson 9, All of the Story Problems, Warm Up: Choral Count, “Let’s count to 70. Count to 70 1–2 times as a class.” Unit 6, Lesson 11, Count Images (Part 1), Warm Up: Choral Count, “Let’s count to 90. Count to 90 1–2 times as a class.” Unit 7, Lesson 9, Compare Capacity, Warm Up: Choral Count, “Display numbers from 1 to 100. ‘Let’s count to 100.’ Point to the numbers as students count to 100.”

• Unit 2, Numbers 1-10, Lessons 5, 8, and 10 engage students in extensive work with K.CC.6 (Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies. Include groups with up to ten objects). Lesson 5, Make Groups of More, Fewer, or the Same, Warm-up: How Many Do You See?, students recognize quantities represented on fingers, without having to count, “How many do you see? How do you see them?” Lesson 8, Compare Matching Images, Activity 1, students compare groups of images that are lined up and decide which group has more or fewer items. Directions state, “Groups of 2, Display the image from the student book. ‘What do you notice? What do you wonder? (There are people and apples. There are 6 people. How many apples are there? Are there enough apples for each person to get one?) Have you ever helped to set the table for a meal or pass out a snack? What did you do?’” Lesson 10, Find More or Fewer, Cool-down: Unit 2, Section B Checkpoint, students use the structure of 5 (in 5-frames or fingers) to count on from 5 to tell how many, “Use ‘more,’ ‘fewer,’ and ‘the same number’ to describe comparisons.”

The materials provide opportunities for all students to engage with the full intent of Kindergarten 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.” In Kindergarten, most lessons do not include cool-downs like those common in other grades, “During these lessons, checkpoints are used to formatively assess understanding. Since activities are shorter, each lesson includes 15-25 minutes of time for centers.” Examples of meeting the full intent include:

• Unit 4, Understanding Addition and Subtraction, Lesson 14, and Unit 5, Composing and Decomposing Numbers to 10, Lesson 4 engage students with the full intent of K.CC.2 (Count forward beginning from a given number within the known sequence). In Unit 4, Lesson 14, Expressions and Story Problems, Warm-up: Choral Count, students count on from a given number, “‘Let’s count to 10.’ Count to 10. ‘Now start at the number 3 and count to 10.’ Count on from 3 to 10. Repeat 3–4 times starting with other numbers within 10.” In Unit 5, Lesson 4, Find All the Ways, Warm-up: Choral Count, students count on when given a number, “‘Let’s count to 60.’ Count to 60. ‘Now, start at the number 9 and count to 20.’ Count on from 9 to 20. Repeat 3–4 times starting with other numbers within 10.”

#### Criterion 1.2: Coherence

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

The materials reviewed for Kendall Hunt's Illustrative Mathematics Kindergarten 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 Kindergarten 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 6 out of 8, approximately 75%.

• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 123 out of 145, approximately 85%. The total number of lessons devoted to major work of the grade includes 115 lessons plus 8 assessments for a total of 123 lessons.

• The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 131 out of 153, approximately 86%.

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 85% of the instructional materials focus on major work of the grade.

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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 Kindergarten 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, Math in Our World, Lesson 4, Activity 1 connects the supporting work of K.MD.3 (Classify objects into given categories; count the numbers of objects in each category and sort the categories by count) to the major work of K.CC.3 (Understand the relationship between numbers and quantities; connect counting to cardinality). Students are introduced to geoblocks and explore, sort, and count the geoblocks. The activity states, “10 minutes: partner work time, ‘Share with your partner one thing you did or made with the blocks.’ 2 minutes: partner discussion. Sample responses: Students use geoblocks to build towers, buildings, and other things. Students sort the geoblocks by shape. Students use comparison language like more, bigger, or smaller when discussing their creations. Students use shape names to describe the blocks.”

• Unit 3, Flat Shapes All Around Us, Lesson 5, Activity 1 connects the supporting work of K.G.5 (Model shapes in the world by building shapes from components) to the major work of K.CC.5 (Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects) and to the major work of K.CC.3 (Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 with 0 representing a count of no objects). Students identify examples of circles and triangles. The activity states, “‘Choose 1 triangle that you colored in. Tell your partner 1 thing that you know about that shape.’ 30 seconds: quiet think time. 30 seconds: partner discussion. ‘Write a number to show how many triangles you colored. Write a number to show how many circles you colored.’ 1 minute: independent work time. ‘Did you color more triangles or more circles? How do you know?’”

##### 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 Kindergarten 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, Understanding Addition and Subtraction, Lesson 4, Warm Up connects the major work of K.CC.B (Count to tell the number of objects) to the major work of K.OA.A (Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from). Students count two groups of numbers to find a total. The activity states, “‘How many students would rather be a bird? How do you know?’ Share responses. Demonstrate or invite students to demonstrate counting. ‘How many students would rather be a fish? How do you know?’ Share responses. Demonstrate or invite students to demonstrate counting.”

• Unit 6, Numbers 0–20, Lesson 5, Activity 1 connects the major work of K.CC.B (Count to tell the number of objects) to the major work of K.NBT.A (Work with numbers 11-19 to gain foundations for place value). Students count to answer “how many” questions about images displayed on fingers. The Launch states, “Groups of 2. Display the student page. ‘Let’s practice reading numbers.’ Point to and read each written number. Invite students to chorally repeat each written number 1–2 times. ‘Now, figure out how many fingers there are. Draw a line from the fingers to the number that shows how many there are.’” Student Facing shows questions 1-5 with a pair of hands with fingers corresponding to numbers 13 to 19.

• Unit 7, Solid Shapes All Around Us, Lesson 11, Activity 2 connects the supporting work of K.G.B (Analyze, compare, create, and compose shapes) to the supporting work of K.MD.B (Classify objects and count the number of objects In each category). Students determine defining characteristics for sorting solid shapes into groups. The Launch states, “Give each group of students a collection of at least 6-8 solid shapes. ‘Work with your partner to sort the shapes into two groups. Write a number to show how many shapes are in each group.’” The activity states, “‘Think of a name for each group of shapes that describes why you put those shapes together. You can write the name above each group. Pair up with another group. Explain to them which shapes you put together and why. Sort your shapes in a different way.’ Monitor for students who sort the shapes based on attributes, as described in the activity narrative.”

• Unit 8, Putting It All Together, Lesson 2, Activity 1 connects the major work of K.CC.A (Know number names and the count sequence) to the major work of K.CC.B (Count to tell the number of objects). Students count collections of up to 20 objects and represent their count with drawings and numbers. The Launch states, “Give each student a collection of objects and access to 10-frames.” Student Facing states, “How many objects are in your collection? Show your thinking using drawings, numbers, or words ___.”

##### 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 Kindergarten 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.” Connections are further described within the Unit Learning Goals embedded in the Scope and Sequence, within the Preparation tab for specific lessons, and within the notes for specific parts of lessons.

Examples of connections to future grades include:

• Unit 3, Flat Shapes All Around Us, Lesson 3, Preparation connects K.G.4 (Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts and other attributes) to work defining attributes of shapes in Grade 1. Lesson Narrative states, “Students look at pictures of objects in the environment as well as common flat shapes. They describe and compare shapes. This lesson is an opportunity to see what attributes of shapes students notice and attend to. Students notice and describe both defining (number of sides and corners, flat or straight sides) and non-defining (size, color, orientation) attributes of shapes. This allows teachers to see the vocabulary students use to describe shapes (MP6). In grade 1 students will distinguish between these defining and non-defining attributes of shapes.”

• Unit 6, Numbers 0-20, Lesson 7, Preparation connects K.CC.5 (Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects) to work with relating counting to addition and subtraction in 1.OA.5. Lesson Narrative states, “Students write a number to represent a quantity greater than 10 for the first time. Students use full 10-frames and some more to identify and create numbers 11–19. Students may count all of the dots or counters to determine the teen number, or they may count on from 10. Counting on to determine the total is not an expectation in kindergarten.”

• Course Guide, Scope and sequence, Unit 7, Solid Shapes All Around Us, Unit Learning Goals connect K.G.5 (Model shapes in the world by building shapes from components and drawing shapes) and K.G.6 (Compose simple shapes to form larger shapes) to the work of creating composite shapes in Grade 1. Lesson Narrative states, “Students use their own language to describe attributes of solid shapes as they identify, sort, compare, and build them, while also learning the names for cubes, cones, spheres, and cylinders. The work here prepares students to identify defining attributes of shapes and to use flat and solid shapes to create composite shapes in grade 1.”

Examples of connections to prior knowledge include:

• Course Guide, Scope and Sequence, Unit 1, Math in Our World, Unit Learning Goals connect K.CC.A (Know number names and the count sequence), K.CC.B (Count to tell the number of objects), and K.G.A (Identify and describe shapes) to previous work with counting. Lesson Narrative states, “Students enter kindergarten with a range of counting experiences, concepts, and skills. This unit is designed to be accessible to all learners regardless of their prior experience. To that end, no counting is required for students to engage in the activities in the first three sections, though students may choose to count. Students also have opportunities to work with math tools and topics related to geometry, measurement, and data through a variety of centers.”

• Unit 2, Numbers 1–10, Lesson 4, Preparation connects K.CC.6 (Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group) to previous work comparing quantities in Kindergarten. Lesson Narrative states, “In a previous lesson, students identified groups that had more or fewer objects than a given group. The number of objects in the groups made it easy to compare the groups visually. For example, students could tell by looking that a group of two cubes was fewer than a group of nine cubes. In this lesson, students compare groups of objects that are closer in quantity. Students also practice using the words fewer, more, and the same in sentences that compare quantities (MP6). For example, students hear and repeat statements such as, “There are fewer red counters than yellow counters.”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 2, Preparation connects K.OA.3 (Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from) to work composing shapes in Kindergarten Unit 3. Lesson Narrative states, “In a previous unit, students made designs with pattern blocks and counted how many of each pattern block they used. In this lesson, students make and share a design with the same total number of pattern blocks but different numbers of individual pattern blocks.”

##### 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 Kindergarten 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 K, there are 153 days of instruction including:

• 137 lesson days

• 16 unit assessment days

There are eight units in Grade K and, within those units, there are between 13 and 22 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. In kindergarten, most lessons do not include cool-downs. During these lessons, checkpoints are used to formatively assess understanding of the lesson. Since activities are shorter, each lesson includes 15–25 minutes of time for centers.” There is a Preparation tab for lessons, including specific guidance and time allocations for each phase of a lesson.

In Grade K, each lesson is composed of:

• 10 minutes Warm-up

• 10-25 minutes (each) for one to three Instructional Activities

• 10 minutes Lesson Synthesis

• 0-5 minutes Cool-down

• 15-25 minutes Centers

### Rigor & the Mathematical Practices

The materials reviewed for Kendall Hunt's Illustrative Mathematics Kindergarten 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 Kindergarten 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 Kindergarten 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 2, Numbers 1-10, Lesson 9, Warm-up, students develop conceptual understanding as they identify groups that have more, less, or the same number as a given group of images. Activity states, “‘How can we figure out how many students like apples better?’ 30 seconds: quiet think time. 30 seconds: partner discussion. Share responses. Demonstrate or invite students to demonstrate counting. ‘How many students like apples better? How can we figure out how many students like bananas better?’ 30 seconds: quiet think time. 30 seconds: partner discussion. Share responses. Demonstrate or invite students to demonstrate counting. ‘How many students like bananas better?’” (K.CC.B)

• Unit 3, Flat Shapes Around Us, Lesson 1, Activity 2, students develop conceptual understanding as they use informal language to describe shapes and share what they know about different shapes. An image of a Backgammon game is shown. Launch states, “‘What games do you play with your family?’ 30 seconds: quiet think time. 30 seconds: partner discussion. Share responses. ‘Backgammon is a popular game in many different countries, such as Iraq, Lebanon, Egypt, and Syria. Lots of people play backgammon in our country, too. Have you ever played this game or a game like this? Tell your partner about a shape you see in the backgammon game. Take turns describing the shapes you see in the picture with your partner.’ 30 seconds: quiet think time.” (K.G.4)

• Unit 8, Putting It All Together, Lesson 2, Warm-up, students develop conceptual understanding of 10 as they subitize or use grouping strategies to describe the images they see. Dot images are provided and Student Facing states, “How many do you see? How do you see them?” Activity Synthesis states, “‘How is the 10-frame helpful when figuring out how many dots there are?”’ (I know that there are 10 dots on the 10-frame and 10 and 5 is 15. I start counting at 10 and count the rest of the dots.)” (K.NBT.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, Math In Our World, Lesson 12, Activity 1, students demonstrate conceptual understanding as they count collections of objects and say one number for each object. Activity states, “Give each student a bag of objects. Give students access to 5-frames and a counting mat. ‘Figure out how many objects are in your collection. Use the tools if they are helpful.’ 2 minutes: independent work time. ‘Switch collections with a partner. Figure out how many objects are in your new collection.’ 2 minutes: independent work time. Monitor for students who say one number for each object.” (K.CC.4a)

• Unit 4, Understanding Addition and Subtraction, Lesson 16, Cool-down, students demonstrate conceptual understanding as they find the value of and represent an expression. Student Facing states, “Find the value of the expression 1+4. Show your thinking using objects, drawings, numbers, or words.” (K.OA.1)

• Unit 7, Solid Shapes All Around Us, Lesson 14, Activity 2, students demonstrate conceptual understanding as they build and describe figures composed of solid shapes. Launch states, “Groups of 2. Give students access to solid shapes and geoblocks. ‘Choose who will build first. The first partner will use the solid shapes to build something. Watch as your partner builds.’ 2 minutes: independent work time. ‘Use the solid shapes to build the same thing as your partner.’ 1 minute: independent work time. Repeat the steps above, with students switching roles.” Activity Synthesis states, “Invite students to share how they changed their building using positional words and names of shapes.” (K.G.1, K.G.6)

• Unit 8, Putting It All Together, Lesson 21, Activity 2, students demonstrate conceptual understanding as they compose and decompose numbers 11–19. Launch states, “Give students access to connecting cubes or two-color counters, 10-frames, and bead tools. Display the student book. ‘Kiran wrote equations to show the total number of students and how many students sat at the table and how many sat on the rug, but he didn’t finish the equations. Finish filling in each equation. You can use connecting cubes or two-color counters if they are helpful.’” Student Facing states, “$$17=10+$$___. 19=___$$+9$$. 10+___$$=14$$. ___$$+2=12$$. 11=___$$+1$$. 15=10+___.” (K.NBT.1)

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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 Kindergarten 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 2, Numbers 1-10, Lesson 6, Warm-Up, students develop procedural skill and fluency as they recognize quantities represented with fingers. Launch states, “Groups of 2. ‘How many do you see? How do you see them?’ Display 4 fingers.” Activity states, “‘Discuss your thinking with your partner.’ 30 seconds: partner discussion. Share responses. Repeat with 8 fingers and 10 fingers.” An image of two hands is shown with one hand showing four fingers and one hand balled in a fist. (K.CC)

• Unit 6, Numbers 0- 20, Lesson 3, Activity 3, students develop fluency with addition and subtraction within 5 as they find the number that makes 5 when added to a given number. Launch states, “Groups of 2. Give each student a set of cards, a recording sheet, and access to two-color counters, 5-frames, and 10-frames. ‘We’re going to learn a center called Find the Pair. Put your cards in a pile in the middle of the table. You and your partner will both draw 5 cards. Keep your cards hidden from your partner.’ Demonstrate drawing 5 cards. Invite a student to act as the partner and draw 5 cards. ‘I am going to look at my cards. I need to choose 1 card and figure out which number I need to make 5 with the card.’ Display a card with the number 4. ‘My card says 4. What card do I need to go with it to make 5? (1) I need a 1 card. I’m going to ask my partner if they have a 1 card. If my partner has a 1 card, they will give it to me. I will put the 4 card and 1 card down as a match and write an expression. If I have a 4 card and a 1 card, what expression should I write?’ ($$4+1$$ or 1+4).” (K.OA.5)

• Unit 8, Putting It All Together, Lesson 1, Warm-Up, students develop procedural skills and fluency as they practice counting and finding patterns in the count. Launch states, “‘Count by 1, starting at 57.’ Record as students count. Stop counting and recording at 77.” (K.CC.2, K.CC.4c)

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, Understanding Addition and Subtraction, Lesson 17, Activity 1, students demonstrate fluency as they find the value of addition expressions with +0 and +1. Launch states, “‘Mai has a tower with 3 cubes. Mai wants to add 0 cubes to the tower. What should Mai do?’ (Nothing. When you add 0, you don’t add anything.) 30 seconds: quiet think time. Share responses. Give each group of students a copy of the blackline master and a connecting cube. Give students access to connecting cubes and two-color counters. ‘Take turns with your partner. Roll the cube to figure out if you need to add 0 or 1. Fill in the expression. Find the value of the expression and write the number on the line. You can use objects or drawings if they are helpful.’” (K.OA.5)

• Unit 7, Solid Shapes All Around Us, Lesson 6, Activity 3, students demonstrate procedural skill and fluency as they use addition and subtraction within 5. Launch states, “Groups of 2. Give each group of students a cup, 5 two-color counters, and 2 copies of the blackline master. ‘We are going to learn a new way to do the Shake and Spill center. It is called Shake and Spill, Cover. Let’s play a round together.’ ‘I am going to put 3 counters in the cup and shake them up. Before I spill the counters, you will close your eyes so I can cover all the yellow counters with the cup. Then you will open your eyes and figure out how many counters are under the cup.’ Put 3 counters in a cup and shake them up. ‘Close your eyes.’ Spill the counters and cover 1 yellow counter. Leave 2 red counters on the table. ‘Open your eyes. Look at the counters on the table. How many counters are under the cup? How do you know?’ (One because there are 2 on the table and 2 and 1 more makes 3.) 30 seconds: partner discussion. Share responses. Pick up the cup showing the 1 counter that was covered. ‘Now we fill in the recording sheet. We had 3 counters total. Then we fill in the expression that matches the parts we broke 3 into. There were 2 counters outside the cup and 1 counter in the cup.’ Demonstrate completing the recording sheet. ‘Take turns with your partner spilling and covering the yellow counters. On each turn you can decide to use 3, 4, or 5 counters. Make sure you and your partner agree on how many total counters you are using before you shake, spill, and cover.’”(K.OA.5)

• Unit 8, Putting It All Together, Lesson 12, Cool-Down: Unit 8, Section C Checkpoint, students demonstrate their fluency as they use strategies to find sums and differences. Student Response states, “Students count all to find the sum. Students use their knowledge of the count sequence to find certain sums. Students know certain sums. Students represent all, then cross off or remove to find the difference. Students use their knowledge of the count sequence to find certain differences. Students know certain differences.” (K.OA.5)

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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 Kindergarten 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, Flat Shapes All Around Us, Lesson 4, Activity 2, students consider and describe attributes of shapes and sort shapes into categories. Launch states, “Give each group a set of shape cards. ‘In the last activity we sorted our objects into groups. We put the objects together based on something that was the same about them.’ Display a couple of shape cards. ‘You and your partner will sort the shape cards into two groups. You can decide how to sort the shapes. Put each shape in one of your groups. Talk to your partner about why each shape fits in the group.’” Activity states, “Monitor for groups that sort the shapes in different ways. ‘Write a number to show how many shapes are in your groups.’ 2 minutes: independent work time. ‘Which group has more shapes? How do you know?’” (K.G.4, K.MD.3)

• Unit 4, Understanding Addition and Subtraction, Lesson 11, Activity 1, students draw a picture to represent and solve a story problem. Student Facing states, “There were 7 kids playing soccer in the park. 3 of the kids left to go play on the swings. How many kids are playing soccer in the park now?” There is an image of four kids playing soccer. Activity states, “Monitor for students who draw pictures with details to represent the story. Monitor for students who use symbols such as circles.” (K.OA.2)

• Unit 8, Putting It All Together, Lesson 18, Cool-down, students solve a real-world problem by composing and decomposing within 10. Student Facing states, “There are 10 birds on the wire. Some of the birds are red. The rest of the birds are blue. How many of the birds are red? Then how many of the birds are blue? Show your thinking using objects, drawings, words, or numbers. Find more than 1 solution to the problem.” (K.OA.2, K.OA.3)

Examples of non-routine applications of the math include:

• Unit 2, Numbers 1- 10, Lesson 5, Activity 1, students make groups that have more, fewer, or the same number of objects as another group. Launch states, ”Give each group a mat and access to collections of between 2–9 objects and connecting cubes. ‘Choose a group of objects and place them in the box at the top of the mat. Use cubes to make a new group of objects for each box below. Make a group that has fewer objects, a group that has the same number of objects, and a group that has more objects. Discuss with your partner how you know each group has more, fewer, or the same number of objects.’” (K.CC.6)

• Unit 7, Story Problems about Shapes, Lesson 5, Activity 2, students connect the action in the story to the meaning of the addition and subtraction signs. Activity states, “Reread the first story problem. ‘Show your thinking using objects, drawings, numbers, or words.’ 2 minutes: independent work time. 2 minutes: partner discussion. ‘Lin began writing this equation but didn’t finish it. Finish her equation to show what happened in the story problem.’ 2 minutes: independent work time. Repeat the steps with the second story problem. Display 9-3=___ for students to complete the equation.” Student Facing states,”1. Andre put together 4 pattern blocks to make a shape. Then Andre put 4 more pattern blocks on the shape. How many pattern blocks are in Andre’s shape? ___ equation: 8=___$$+$$___ 2. Elena used 9 pattern blocks to make a train. Then she took 3 of the pattern blocks off of the train and put them back in the bucket. How many pattern blocks are in Elena's train now? ___ equation 9-3=___.” (K.OA.1, K.OA.2)

• Unit 8, Putting It All Together, Lesson 3, Activity 1, students use their knowledge of the count sequence to solve Add To, Result Unknown and Take From, Result Unknown story problems where one is added or taken away. Launch states, “Give students access to connecting cubes and 10-Frames. ‘Today you are going to solve two story problems about people on a bus.’” Student Facing states, “1. There were 7 people on the bus. Then 1 more person got on the bus. How many people are on the bus now? Show your thinking using objects, drawings, numbers, or words. 2. There were 10 people on the bus. Then 1 person got off the bus. How many people are on the bus now? Show your thinking using objects, drawings, numbers, or words.” (K.CC.2, K.CC.4, K.OA.2)

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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 Kindergarten 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 4, Understanding Addition and Subtraction, Lesson 4, Activity 1, students extend their conceptual understanding as they use objects to represent addition. Launch states, “Give each student at least 10 counters and access to 5-frames. ‘Count out 3 counters. You can use a 5-frame if it is helpful.’ 30 seconds: independent work time. ‘Now add 4 more counters.’ 30 seconds: independent work time. ‘How many counters are there altogether?’ Write ‘There are 7 counters altogether.’ ‘This sentence now says There are 7 counters altogether. We are going to continue to work on problems where we add more counters to the group we started with. Let’s add and find the total number of counters.’” (K.CC.5, K.OA.1)

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 1, Activity 1, students develop procedural skill and fluency as they decompose 6 into two parts using connecting cubes. Activity states, “‘You have 6 cubes. Put some of the cubes in your hand and some on your desk.’ 30 seconds: independent work time. ‘Tell your partner how many cubes are in your hand. Show them the cubes. Tell your partner how many cubes are on your desk. Show them the cubes. Tell your partner how many cubes you have altogether.’” (K.OA.3, K.OA.5)

• Unit 8, Putting It All Together, Lesson 11, Activity 1, students apply their understanding of addition and subtraction as they represent and solve a story problem about their school community. Launch states, “Give each student a piece of chart paper and access to connecting cubes or two-color counters and crayons. ‘Tell your partner the story problem that you came up with yesterday. Today you are going to make a poster to show your story problem. Solve the story problem. Show your thinking using drawings, numbers, or words.’” Activity states, “10 minutes: independent work time. ‘If you have time, you may want to show different ways to solve the problem using pictures, numbers, words, or symbols.’ 10 minutes: independent work time.” (K.OA.1, K.OA.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, Number 1-10, Lesson 2, Activity 2, students develop conceptual understanding alongside procedural skill and fluency as they recognize that the arrangement of a group of objects does not change the number of objects. Launch states, “‘We are going to learn a new center called Shake and Spill. Let's play a round together. Choose who will go first and start with all of the counters in the cup. Shake the cup and spill the counters on the table.’ 30 seconds: partner work time. ‘Take turns figuring out how many counters there are. When you know how many counters there are, tell your partner and see if you both agree.’ 1 minute: partner work time. ‘Put the counters back into the cup, shake them and spill them again. Take turns figuring out how many counters there are and share with your partner.’ 1 minute: partner work time. ‘Now you can take turns playing with your partner. Take some of the counters out of the cup and put them away so that you are using a different number of counters this time. Remember to spill the counters, figure out how many there are, spill the counters again, and figure out how many there are.’” (K.CC.4, K.CC.5)

• Unit 4, Understanding Addition and Subtraction, Lesson 14, Activity 1, students develop conceptual understanding alongside application as they explain how a subtraction expression represents a story problem. Launch states, “Give students access to connecting cubes or two-color counters. Read and display the task statement. ‘Tell your partner what happened in the story.’ 30 seconds: quiet think time. 1 minute: partner discussion. Monitor for students who accurately retell the story. Choose at least one student to share with the class. Write the expression 10 - 6. ‘How does this expression show what happens in the story problem?’” Student Facing states, “There were 10 people riding bikes in the park. Then 6 of the people stopped riding to have lunch. How many people are riding bikes now?” (K.OA.1, K.OA.2)

#### 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 Kindergarten meet expectations for practice-content connections. The materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).

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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 Kindergarten 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, Math In Our World, Lesson 11, Activity 1, students think of different ways to represent a story. Student Facing states, “4 little speckled frogs sat on a speckled log, eating the most delicious bugs. Yum! Yum! 1 jumped into the pool, where it was nice and cool. Now there are 3 green speckled frogs. Glub! Glub!” An image of a green frog is shown. Narrative states, “Acting out gives students opportunities to make sense of a context (MP1). Monitor for suggestions of acting out the story with concrete objects such as cubes, fingers, or students, as well as representing the story with pictures.”

• Unit 2, Numbers 1-10, Lesson 4, Activity 1, students identify a group of objects that has more. Student Facing states, “Priya and her family are sitting down at the table for dinner. There are 4 people sitting at the table. There are 6 spoons. Are there enough spoons for each person to get one?” Launch states, “Groups of 2. Give each group of students access to connecting cubes and two-color counters. ‘We have been learning about different tools that we use at home and in our classroom. What kind of tools do you use when you eat at home?’ (Spoons, forks, chopsticks, plates, bowls, napkins, cups, straws). 30 seconds: quiet think time. 1 minute: partner discussion. Share and record responses. ‘We use many different tools when we eat.’ Display and read the story. ‘What is the story about?’ (A family eating dinner, Priya’s family, spoons for dinner). 30 seconds: quiet think time. Share responses. Read the story again. ‘How can you act out this story?’ (We can pretend we are sitting at the table and pretend to hand out spoons. We can use the cubes to show the people and the counters to show the spoons. We can draw a picture.). 30 seconds: quiet think time. 1 minute: partner discussion. Share responses.” Activity states, “‘Act out the story with your partner.’ 3 minutes: partner work time. ‘Are there more people or spoons? How do you know?’ (There are more spoons than people. Each person gets one spoon and then there are some more spoons.). 2 minutes: partner work time. Monitor for students who matched one spoon to each person to see if there were enough spoons and which there was more of.” Narrative states, “The context of family mealtimes that is introduced in this activity will be revisited throughout the unit. Acting it out gives students an opportunity to make sense of a context (MP1).”

• Unit 7, Solid Shapes All Around Us, Lesson 5, Warm-up, students reason about a problem context involving quantities within ten. Student Facing states, “What do you notice? What do you wonder? Elena used 9 pattern blocks to make a train. Then she took 3 of the pattern blocks off of the train and put them back in the bucket.” Narrative states, “This warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (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 1, Math in Our World, Lesson 5, Warm-up, students reason about tools and consider different representations. Launch states, “Groups of 2. Display the image.” Student Facing states, “What do you notice? What do you wonder?” Narrative states, “The purpose of this warm-up is for students to consider how different tools can be used to represent the same thing. When students describe how each object represents a house and make connections between the objects, they show their ability to reason abstractly and quantitatively (MP2).”

• Unit 3, Flat Shapes All Around Us, Lesson 10, Activity 2, students use pattern blocks to fill in simple puzzles. Student Facing shows an image of a pattern puzzle, with different pattern blocks shown to make the figure. Launch states, “Groups of 2. Give each group of students pattern blocks. ‘In the last activity we put together pattern blocks to make quilts. We can also use pattern blocks to make things that we see in real life. Close your eyes and think about something that you see at home or in your community. Use the pattern blocks to make what you see.’ 2 minutes: independent work time. ‘Tell your partner about what you made and why.’ 2 minutes: partner discussion. Share responses. ‘Each puzzle looks like something we see in real life. Use the pattern blocks to fill in each puzzle. Write a number to show how many of each pattern block you used. Ask your partner a question about each puzzle using the word “fewer”.’ Activity states, “6 minutes: partner work time.” Narrative states, “When students make connections between the pattern blocks and the shape outlines in the puzzle, they show their ability to reason abstractly and quantitatively (MP2).”

• Unit 6, Numbers 0-20, Lesson 11, Activity 2, students show that numbers 11-19 consist of 10 ones and and some more ones as they color images to match expressions. Student Facing states, “1. Color the squares to show 10+2. 10+2=____. An image of 12 squares is shown. 2. Color the triangles to show 10+8. 10+8=____. An image of 18 triangles is shown. 3. Color the hexagons to show 10+4. 10+4=____. An image of 14 hexagons is shown. 4. Color the circles to show 10+9. 10+9+____. Two images are shown: a rectangular shape with 10 circles and 9 circles.” Launch states, “Groups of 2. Give each student access to at least two different colored crayons. ‘Color the shapes to show each expression. Then complete the equation to show how many shapes there are altogether.’” Activity states, “2 minutes: independent work time. 3 minutes: partner work time. Monitor for students who count on from 10.” Narrative states, “Because students are coloring in the shapes to show 10+____, students may count on from 10 to determine the total number of shapes. It is important that students connect their equations to the corresponding representations (MP2).”

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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 Kindergarten 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 1, Math in Our World, Lesson 13, Warm-up, students construct viable arguments as they count collections of objects, focused on keeping track of which objects have been counted. Launch states, “‘We need to figure out how many of us are here. How can we make sure that we count each person one time?’ 30 seconds: quiet think time. Share responses. Monitor for students who suggest a way to organize the students, such as having all of the students line up.” Activity states, “Count the students using two of the methods suggested by students. ‘How many of us are here today?’” Narrative states, “As students share answers to questions such as ‘How can we figure out how many of us are here?’ and ‘Did I count the students correctly?’ they are beginning to construct viable arguments and attend to precision (MP3, MP6).”

• Unit 3, Flat Shapes Around Us, Lesson 1, Warm-up, students construct arguments as they compare four different images and analyze the characteristics or attributes of the images. Launch states, “Display the image. ’What is the same and what is different about the teddy bears?’ 30 seconds: quiet think time. 30 seconds: partner discussion. Share responses.” Activity states, “Which one doesn’t belong? Display the image. 30 seconds: quiet think time. ‘Tell your partner which teddy bear doesn’t belong and why.’ 30 seconds: partner discussion. Student Facing states, “Which one doesn’t belong?” Images of teddy bears are provided. Narrative states, “In this warm-up, students only work with three images of teddy bears. By the end of the section, students will compare four images of shapes. Emphasize to students that there is no right answer to the question and that it is important to explain their choice. Listen to how students create an argument and use or revise their language to make their argument clear to others (MP3).”

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 3, Flats Shapes All Around Us, Lesson 5, Activity 2, students construct arguments and begin to critique the reasoning of others as they distinguish triangles from other shapes. Launch states, “Groups of 4, Give each group a set of cards. ‘Work with your group to sort the shapes into 2 groups. Put the shapes that are triangles on the left side of your page. Put the shapes that are not triangles on the right side of your page. When you place a shape, tell your group why you think the shape belongs in that group.’” Activity states, “4 minutes: small-group work time. Monitor for students who discuss attributes of triangles when sorting. ‘Write a number to show how many shapes are in each group.’ 1 minute: independent work time. ‘Walk around to see how the other groups organized their shapes. Did they organize them the same way that your group did?’ 6 minutes: work time.” Student Facing states, “Let’s put the shapes into 2 groups.Triangle, Not a Triangle” Activity Synthesis states, “Display cards O, K, and G next to each other. ‘Noah says that the shape in the middle is not a triangle because it is pointing down and triangles have to point up. Do you agree with Noah? Why or why not?’”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 1, students construct arguments and begin to critique the reasoning of others as they decompose numbers into two groups. Launch states, “Groups of 2. Give students access to connecting cubes. ‘Diego and Lin also put some cubes in their hands and some on their desks. Diego has 3 in his hand and 1 on his desk. He says he has 4 cubes altogether. Lin has 2 in her hand and 2 on her desk. She also says she has 4 cubes total. Can they both have 4 cubes altogether?’” Activity states, “3 minutes: partner discussion. Monitor for students who count the groups to determine that both students have 4 even though they are broken into different parts.” Activity Synthesis states, “Do Diego and Lin both have 4 cubes? Invite previously identified students to share. ‘What parts did Diego break 4 into? (3 and 1)’ Write 3+1. ‘What parts did Lin break 4 into?’ (2 and 2) Write 2+2. ‘Diego and Lin showed us that we can break numbers apart in different ways.’”

• Unit 8, Putting It All Together, Lesson 7, Cool-down, students critique the work of others as they use numbers to create a number book using objects in their environment. Student Facing states, “Choose 1 object in our classroom. Create a number book page about the object. Include a number, a drawing, and letters, a word, or words.” Narrative states, “Students share their work with a partner, receive feedback, and then improve their work (MP3).”

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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 Kindergarten 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 3, Flat Shapes All Around Us, Lesson 14, Activity 2, students put shapes together to form larger shapes, Launch states, “Give each student a sheet of construction or white paper. Give students access to cut out shapes, glue, crayons, colored pencils, and markers. ‘We noticed that artists use shapes in different ways to create art. Some artists make patterns and designs. Some put shapes together to form people or animals. Now you are going to make your own piece of artwork using shapes. You can use any of these materials. Think about how you can draw or put shapes together to make larger shapes.’” Activity states, “8 minutes: independent work time.” Narrative states, “When students recognize mathematical features of objects in the real world, they model with mathematics (MP4).”

• Unit 7, Solid Shapes All Around Us, Lesson 3, Cool-down, Section A Checkpoint, students ask and answer mathematical questions about shapes composed of pattern blocks. The teacher observes to capture evidence of student thinking on the checkpoint checklist. Student Response states, “Count all to determine the total. Use objects, drawings, or equations to represent a story problem.” Lesson Narrative states, “In this lesson, students create a shape out of pattern blocks and brainstorm questions that they could ask about other students’ shapes. Students create and solve story problems about shapes made out of pattern blocks (MP2, MP4).”

• Unit 8, Putting It All Together, Lesson 8, Activity 2, students recognize different ways math is all around them in their community. Student Facing states, “Find something that you can count. Find 2 objects that you can compare the weight of. Find something that you know how many there are without counting. Find something that there are 5 of. Find 2 groups of objects that make 10 objects altogether. Find a group of objects that you could use to fill in a 10-frame. Find something that you could make using solid shapes. Find 2 groups of objects that you can compare the number of. Find something that has a number on it. Find 2 objects that you can compare the length of.” Launch states, “Give students access to 10-frames, geoblocks, and solid shapes. ‘Work with your partner to find an object or objects that goes with each prompt.’” Activity states, “‘Find something that you can count.’ 30 seconds: quiet think time. 2 minutes: partner work time. ‘Now count what you found.’ 1 minute: partner work time. Repeat the steps with the rest of the prompts.” Narrative states, “When students identify objects in the classroom that fit different constraints they are taking an important step toward modeling with mathematics (MP4).”

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 1, Math In Our World, Lesson 13, Activity 1, students count collections of objects and the focus is saying one number for each object. Launch states, “Today you’re going to count another collection of objects. As you’re working, think about how to make sure you count each object.” Activity states, “Give each student a bag of objects. Give students access to 5-frames and a counting mat. ‘Figure out how many objects are in your collection.’ 2 minutes: independent work time. ‘Switch collections with a partner. Figure out how many objects are in your new collection.’ 2 minutes: independent work time. Monitor for students who have a method of keeping track of which objects have been counted, such as moving and counting or lining up the objects and counting them in order.” Narrative states, “Students use appropriate tools strategically as they choose which tools help them count their collections (MP5).”

• Unit 2, Numbers 1–10, Lesson 21, Activity 1, students compare numbers in a way that makes sense to them, Student Facing states, “Circle the number that is more. 1. 5, 8, 2. 9, 4,” Launch states, “Give students access to connecting cubes or counters. ‘Work with your partner to figure out which number is more. Circle the number that is more.’” Activity states, “5 minutes: partner work time. Monitor for students who create representations of the numbers using cubes or a drawing and use these representations to compare. Monitor for students who counted to figure out which number is more.” Narrative states, “Students can use physical objects or make drawings to represent each number (MP5).”

• Unit 4, Understanding Addition and Subtraction, Lesson 8, Cool-down, Section B checkpoint, students represent and solve story problems using a strategy that makes sense to them. Teachers observe and capture evidence of student thinking on the checkpoint checklist. Student Response states, “Accurately retell a story problem in their own words. Understand the action in a story problem and act it out or demonstrate it with objects or drawings. Use objects or drawings to represent a story problem.” Narrative states, “Students may use objects, math tools, or drawings to represent and solve the story problem (MP5).”

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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 Kindergarten 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, Math in Our World, Lesson 4, Warm-up, students use specialized language to describe shapes. Narrative states, “The purpose of this activity is to elicit ideas students have about geoblocks. This allows teachers to see what language students use to describe shapes (MP6). There is no need to introduce formal geometric language at this point since this will happen in a later unit.” Launch states, “Groups of 2. Give each student a few geoblocks and display a collection of geoblocks or the image in the student book. ‘What do you notice?’ 30 seconds: quiet think time.” Activity states, “‘Tell your partner what you noticed.’ 1 minute: partner discussion. Share and record responses. ‘What do you wonder?’ 1 minute: quiet think time. ‘Tell your partner what you wondered.’ 30 seconds: quiet think time. 1 minute: partner discussion. Share and record responses.” Activity Synthesis states, “These are called geoblocks. What is one thing that you think you could do or make with the geoblocks?”

• Unit 1, Math In Our World, Lesson 8, Activity 2, students attend to precision as they recognize, name, and match groups with the same number of images. Launch states, “Groups of 2. Display the image from the student book. ‘When I point to each group, show your partner with your fingers and tell your partner how many things there are.’ Point to the ducks. 30 seconds: partner work time. Repeat the steps with the cats and dogs. ‘Which groups have the same number of things? How do you know?’ (There are 3 ducks and 3 dogs. They are both 3.) 30 seconds: quiet think time. Share responses. Display or write “3”. ‘There are 3 ducks and 3 dogs. They both have the same number of things.’” Activity states, “Give each group of students a set of cards. ‘Work with your partner to match the cards that have the same number of things. Explain to your partner how you know.’ 4 minutes: partner work time.”  Narrative states, “When students say that two cards match because they have the same number of objects, they attend to precision in their language (MP6).”

• Unit 2, Numbers 1- 10, Lesson 6, Cool Down, students attend to precision as they compare groups of objects and describe their comparisons using “more,” “fewer,” and “the same number.” Student Facing states, “Compare the number of objects in groups. Use ‘more,’ ‘fewer,’ and ‘the same number’ to describe comparisons. Make groups with more, fewer, or the same number of objects than a given group.” Narrative states, “In making comparisons, students have a reason to use language precisely (MP6).”

• Unit 3, Flat Shapes All Around Us, Lesson 9, Warm-up, students attend to precision as they compare attributes of shapes to determine which one does not belong. Launch 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.” Narrative states, “This warm-up prompts students to carefully analyze and compare the attributes of 4 shapes. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the words students know and how they talk about attributes of shapes.” Activity Synthesis states, “Display the image of the square. ‘Noah said that this shape doesn’t belong because it is not a rectangle. What do you think?’ 30 seconds: quiet think time. Share responses. ‘A square is a special kind of rectangle.’”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 5, Cool-down, students use grade appropriate math terms to restate and represent story problems. Student Facing states, “Accurately retell a story problem in your own words. Use objects or drawings to represent a story problem. Explain how objects or drawings represent a story problem. Use labels, colors, numbers, or other methods to represent the two groups in a story problem.” Narrative states, “Students are encouraged to use clear and precise language to explain how their representation shows the story problem (MP6).”

• Unit 7, Solid Shapes Around Us, Lesson 13, Activity 1, students use specific mathematical language to describe the solid shapes. Launch states, “Groups of 2. Give students access to solid shapes. ‘Choose 2 solid shapes.’ 30 seconds: independent work time.” Activity states, “‘We are going to go for a walk. Your job is to look for objects that look like your solid shapes. Tell your partner about the shapes you find.’ 10 minutes: shape walk. Monitor for students who use positional words to describe the location of shapes. ‘Tell your partner about your favorite object. Where did you see it?’ 30 seconds: quiet think time. 1 minute: partner discussion. Share responses.” Narrative states, “The purpose of this activity is for students to identify and describe solid shapes in their environment (MP4, MP6).” Activity Synthesis states, “Invite students who used positional words to describe the location of shapes to share. ‘____ saw a round light bulb below the lamp shade. It looked like a sphere. _____ saw a book on the bookshelf. It looked like a box.’ Display image: ‘Which shape does this clock look like?’ (Students say “cylinder” or hold up a cylinder.) Display image: ‘Which shape does this party hat look like? (Students say “cone” or pick up a cone.) In the next activity, we are going to use clay to make shapes that show the objects we saw.’”

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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 Kindergarten 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 2, Numbers 1–10, Lesson 8, Activity 2, students look for and make use of structure while they compare groups of images. Launch states, “Groups of 2. Display the student page. ‘How are these pictures different from the ones we worked with in the first activity?’ (There are different pictures. The pictures aren’t matched up.)” Activity states, “‘Are there enough cartons of milk for each student? How do you know?’ 30 seconds: quiet think time. 30 seconds: partner discussion. ‘Are there more students or cartons of milk? How do you know?’ 30 seconds: independent work time. 30 seconds: partner discussion. ‘There are more cartons of milk than students. How many students are there? How many cartons of milk are there?’ 1 minute: independent work time. ‘8 cartons of milk is more than 7 students.’ Repeat the steps with each group of images. Switch between asking students ‘Are there more _____ or _____?’ and ‘Are there fewer _____ or _____?’ Monitor for students who draw lines to match each image. Narrative states, “Matching the images helps students relate the comparisons to the situation they just worked with where the images were already matched (MP7).”

• Unit 4, Understanding Addition and Subtraction, Lesson 15, Cool-down: Unit 4, Section C Checkpoint, students look for and make use of structure while they connect expressions to drawings. Student Facing states, ”Lesson observations. Explain how an expression connects to a drawing or story problem. Fill in an expression to represent a drawing.” Activity 1 Narrative states, “The purpose of this activity is for students to match drawings to expressions. Students use the structure of the dots to decide whether they represent an addition or subtraction expression and then identify that expression (MP2, MP7).” The cool-down also assesses students’ ability to connect expressions to drawings. Teacher Instructions state, “For this Checkpoint Assessment, a full checklist for observation of students can be found in the Assessments for this unit. The content assessed is listed below for reference. Relate addition and subtraction expressions to story problems. Explain how an expression connects to a drawing or story problem. Fill in an expression to represent a drawing. Find the value of addition and subtraction expressions within 5. Use fingers, objects, or drawings to find the value of an expression. Count all to determine the total when 0 or 1 are added. Use knowledge of the count sequence to determine the total when 1 is added.”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 13, Warm-up, students look for and make use of structure while they subitize or use grouping strategies to describe the images they see. Student Facing states, “How many do you see? How do you see them?” Activity states, “Display image. ‘Discuss your thinking with your partner.’ 1 minute: partner discussion. Record responses. Repeat for each image.” Activity Synthesis states, “Display the hands showing 8 fingers. ‘How many fingers are up? (8) How many fingers need to go up so there are 10 fingers?’ (2). Repeat the steps with the rest of the images.” Narrative states, “When students think about quantities in relation to 5 and 10, they look for and make use of structure (MP7).”

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 1, Math in Our World, Lesson 11, Cool-down, students use repeated reasoning using one-to-one correspondence as they make groups of objects. Student Facing states, “Lesson observations.” Sample Student Responses include, “Say the count sequence to 10. Say one number for each object. Answer how many without counting again. Recognize and name groups of 1, 2, or 3 objects or images without counting. Recognize and name groups of 4 objects or images without counting. Show quantities on fingers.Identify groups with the same number of objects (for groups of up to 4 objects).” Lesson Narrative states, “As students notice that when you get enough of an object for each student to have one, the number of students and the number of objects are the same, they look for and express regularity in repeated reasoning (MP8).”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 12, Activity 2, students use repeated reasoning to find how many counters are needed to fill a 10-frame. Launch states, “Groups of 2. Give students access to two-color counters. ‘Figure out how many counters are needed to fill each 10-frame. Write a number to show how many counters are needed to fill it. Circle the equation that shows the number of counters in the 10-frame and the number of counters needed to fill the 10-frame.’” Student Facing states, “1. (A ten frame with 7 counters is shown)$$10=7+3$$, 10=8+2, 10=5+5 2. (A ten frame with 4 counters is shown) 10=8+2, 10=1+9, 10=4+6 3. (A ten frame with 9 counters is shown) 10=9+1, 10=5+5, 10=7+3 4. (A ten frame with 3 counters is shown) 10=5+5, 10=3+7, 10=2+8 5. (A ten frame with 5 counters is shown) 10=9+1, 10=6+4, 10=5+5 6. (A ten frame with 2 counters is shown) 10=1+9, 10=2+8, 10=4+6.” Narrative states, ”With repeated experience composing 10 in many ways, students may begin to know the combinations to make 10 (MP8).”

• Unit 6, Numbers 0–20, Lesson 4, Warm-up, students count collections of objects and understand that the number of objects in a collection stays the same, regardless of how they are arranged, Student Facing states, “What do you notice? What do you wonder?” Launch states, “Groups of 2. Display the image. ‘What do you notice? What do you wonder?’ 1 minute: quiet think time.” Activity states, “‘Discuss your thinking with your partner.’ 1 minute: partner discussion. Share and record responses.” Activity Synthesis states, “Which arrangements do you think would be easiest to count? Why? (The lined up dots would be easy to count. I could count one line and then the other line.).” Lesson Narrative states, “Students will count the same collection of objects in different arrangements to build this conservation of number, which develops through experience over time. While developing conservation of number, students may need to recount the objects each time they are rearranged. With repeated practice, some students may know that the number of objects is the same without recounting (MP8).”

### Usability

The materials reviewed for Kendall Hunt's Illustrative Mathematics Kindergarten 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 Kindergarten 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.

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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 Kindergarten 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 kindergarten include: representing and comparing whole numbers, initially with sets of objects; understanding and applying addition and subtraction; and describing shapes and space. More time in kindergarten is devoted to numbers than to other topics. In these materials, particularly in units that focus on addition and subtraction, teachers will find terms that refer to problem types, such as Add To, Take From, Put Together or Take Apart, Compare, Result Unknown, and so on. These problem types are based on common addition and subtraction situations, as outlined in Table 1 of the Mathematics Glossary section of the Common Core State Standards.”

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 2, Numbers 1–10, Lesson 2, Activity 1, teachers are provided context to support students in understanding that the arrangement of objects does not change the total. Narrative states, “Students grab a handful of connecting cubes and count to see how many they have. They then rearrange the connecting cubes using a 5-frame and discover that although the connecting cubes are arranged differently, the number of connecting cubes stays the same. This understanding develops over time with repeated experience working with quantities in many different arrangements. Students may continue to recount the objects in this and future lessons until they understand and are confident that the number of objects remains the same when they are rearranged.” Launch states, “Groups of 2. Give each group of students connecting cubes. ‘We are going to play a game with our connecting cubes and 5-frame. One person will grab a handful of connecting cubes and figure out and tell their partner how many there are. Then the other partner will organize the connecting cubes using the 5-frame, and figure out and tell their partner how many there are. Take turns playing with your partner.’” Activity states, “5 minutes: partner work time. Monitor for students who notice that the number of objects is the same after they are rearranged.”

• Unit 8, Putting It All Together, Lesson 18, Lesson Synthesis provides guidance around strategies for composing and decomposing within 10, “Display the chart with solutions to the story problem. ‘Tyler and Priya recorded the different ways that the pigeons could be in the fountain and on the bench. What do you notice? What patterns do you see?’ (There are lots of ways to make 10. The numbers in one column are counting up and the numbers in the other column are counting down. I see that there are 7 and 3 and 3 and 7.)”

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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 Kindergarten 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:

• IM K-5 Math Teacher Guide, About These Materials, Unit 1, When is a number line not a number line?, “In this blog post, McCallum shares why the number line is introduced in grade 2 in IM K–5 Math, emphasizing the importance of foundational counting skills.”

• Unit 4, Understanding Addition and Subtraction, Lesson 10, Preparation, Lesson Narrative states, “In a previous lesson, students solved Add To and Take From, Result Unknown story problems and explained how both objects and drawings represented the story. In this lesson, students solve story problems and compare how different drawings represent the story. Students interpret both drawings that correctly and incorrectly represent the story problem, as well as unorganized and organized drawings. While students are not expected to produce a drawing to represent and solve a story problem in this lesson, students make sense of various drawings, which will help them be prepared to create drawings in a future lesson. The purpose of the lesson synthesis is for students to discuss how it can be easier to see what happens in the story problem in an organized drawing.”

• IM K-5 Math Teacher Guide, About These Materials, Kindergarten, Unit 7, What is a Measurable Attribute?, “In this blog post, Umland wonders about what counts as a measurable attribute and discusses how this interesting and important mathematical idea begins to develop in kindergarten.”

• Unit 7, Solid Shapes All Around Us, Lesson 1, Preparation, Lesson Narrative states, “In previous units, students put together pattern blocks to form larger shapes and filled in puzzles. They counted groups of up to 20 objects and images and wrote numbers to record their count. Students use only 1 kind of pattern block to fill in puzzles and eventually create given shapes without outlines provided, which requires students to think informally about the attributes of shapes. Students need to change the orientation of the pattern blocks and align the sides of the pattern blocks. Students may be able to visualize how to turn or flip the shape to fill a particular space or may need to use trial and error. In both activities, students count and write a number to record how many pattern blocks they used.”

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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 Kindergarten 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 K, 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) addressed within.

• Grade K, Course Guide, Lesson Standards, includes all Kindergarten 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, 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 3, Flat Shapes All Around Us, Lesson 5, the Core Standards are identified as K.G.B.4 and K.MD.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; and a Lesson Synthesis.

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 K, Course Guide, Scope and Sequence, Unit 7, Solid Shapes All Around Us, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “In this unit, students explore solid shapes while reinforcing their knowledge of counting, number writing and comparison, and flat shapes. They compose figures with pattern blocks and continue to count up to 20 objects, write and compare numbers, and solve story problems. In an earlier unit, students investigated two-dimensional shapes. They named shapes (circle, triangle, rectangle, and square) and described the ways the shapes are different. Students used pattern blocks to build larger shapes and used positional words (above, below, next to, beside) along the way. Here, students distinguish between flat and solid shapes before focusing on solid shapes. They consider the weight and capacity of solid objects and identify solid shapes around them. Geoblocks, connecting cubes, and everyday objects are used throughout the unit. Standard geoblock sets do not include cylinders, spheres, and cones. When these shapes are required, ‘solid shapes’ are indicated as required materials. If solid shapes are not available, students can work with everyday items that represent each shape. Students use their own language to describe attributes of solid shapes as they identify, sort, compare, and build them, while also learning the names for cubes, cones, spheres, and cylinders.”

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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 Kindergarten 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 K, Unit 1, Math in Our World, Family Support Materials, “In this unit, students recognize numbers and quantities in their world. Section A: Exploring Our Tools, In this section, students discuss what it looks like to do math in their classrooms. They work with the math tools they will use during math activities and centers throughout the year. Students have the opportunity for free exploration in order to think of mathematical purposes for the tools. Students are encouraged to use their own language to describe their work, as well as listen to the ideas of others in the class, which positions students as mathematicians who have interesting and worthwhile ideas to share. Images of connecting cubes, pattern blocks, geoblocks, and 5-frames are shown. Section B: Recognizing Quantities, In this section, students continue to explore math in their classrooms, focusing on small groups of objects or images. Students may begin to see dot images in arrangements that allow them to know how many without counting such as these.” Images of arrangements of dots are shown.

• For Families, Grade K, Unit 3, Flat Shapes All Around Us, Family Support Materials, Try it at home!, “Near the end of the unit, ask your student to go on a scavenger hunt to find shapes around the home or in places you visit often. Questions that may be helpful as they work: Can you find a square, a rectangle, a triangle, and a circle? Find two shapes that are the same. What is the same about these shapes? What is different?”

• For Families, Grade K, Unit 8, Putting It All Together, Family Support Materials, “In this unit, students put together their understanding from throughout the year to cap off major work and fluency goals of the grade. Section A: Counting and Comparing, In this section, students count and compare collections of up to 20 objects. Students focus on the count sequence up to 20 and use their knowledge of the count sequence to determine one more or one less than a given quantity or number. Section B: Math in Our School, In this section, students explore and describe the math that they see in their environment. Students participate in multiple activities that allow them to notice, record, ask questions, and tell stories about math in their community. Students record quantities that they see in their school by making their own number book. Then students ask and answer their own mathematical questions about their school, such as ‘how many tiles are there from the office to the cafeteria?’ or ‘are there more doors or more windows in the library?’ Finally students create, share, and solve story problems about their school environment and community. While the school building is used as a context, the activities in this section can be adapted for students to do in the community or at home.”

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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 Kindergarten 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 Certified, Blog, In the Beginning: Unit 1 in Kindergarten, Alex Clayton, Exploring our math tools, “Unit 1, Section A is titled Exploring Our Math Tools, but really it is all about exploring our new math community as well! During this section, we are introduced to many of the tools that will be used throughout the year: connecting cubes, two-color counters, pattern blocks, 5-frames, and geoblocks. Students get to explore these materials freely. They discover how the materials work and try out their own ideas, before they are ever asked to use them for a specific mathematical purpose.Equally important, students get to practice sharing their ideas (‘What do you want to make or do with the connecting cubes?’) and listening to the ideas of others. These are some of the first steps in building a mathematical community where everyone has valuable mathematical ideas. We learn from each other.”

• 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.”

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

The materials reviewed for Kendall Hunt's Illustrative Mathematics Kindergarten 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 or on the “Lesson” tab in a section called “Required Materials.” Examples include:

• Course Guide, Required Materials for Kindergarten, Materials Needed for Unit 1, Lesson 5, teachers need, “Connecting cubes, Materials from previous centers, Pattern blocks, Connecting Cubes Stage 2 Cards (groups of 2), Pattern Blocks Stage 2 Mat (groups of 2).”

• Course Guide, Required Materials for Kindergarten, Materials Needed for Unit 5, Lesson 10, teachers need, “Glue, Materials from previous centers, Scissors, Two-color counters, Numbers on Fingers and 10-frames (groups of 1), 5-Frames to Cut Out (groups of 1).”

• Unit 4, Understanding Addition and Subtraction, Lesson 9, Activity 1, Required Materials, “Connecting cubes or two-color counters, makers.” Launch states, “Groups of 2. Give students access to two-color counters, connecting cubes, and markers. Read and display the task statement. ‘Tell your partner what happened in the story.’ 30 seconds: quiet think time. 1 minute: partner discussion. Monitor for students who accurately retell the story. Choose at least one student to share with the class. Reread the task statement. ‘Show your thinking using drawings, numbers, words, or objects.’”

##### 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 Kindergarten 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 Kindergarten 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 3, Flat Shapes All Around Us, End-of-Unit Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 3, K.MD.2, “a. Circle the rectangle that is longer. b. Circle the rectangle that is shorter.” 3a shows red and blue horizontal rectangles. 3b shows red and blue vertical rectangles.

• Unit 5, Composing and Decomposing Numbers to 10, End-of-Unit Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 4, K.OA.2, “Diego has 3 toy cars on the floor. He has 5 more toy cars on his bed. How many toy cars does Diego have altogether? Show your thinking using drawings, numbers, or words.”

• Unit 8, Putting it All Together, End-of-Course Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 6, K.NBT.1, “Write numbers to make each equation true. a, 10+6=___. b. 3+10=___. c. ___ + ___$$=13$$. d. ___ + ___$$=17$$.”

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 K, MP2 is found in Unit 1, Lessons 5, 7, 14, and 15.

• IM K-5 Math Teacher Guide, How to Use These Materials, Standard for Mathematical Practices Chart, Grade K, MP6 is found in Unit 4, Lessons 2, 3, 7, and 10.

• 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, “MP4 I Can Model with Mathematics. I can wonder about what mathematics is involved in a situation. I can come up with mathematical questions that can be asked about a situation. I can identify what questions can be answered based on data I have. I can identify information I need to know and don’t need to know to answer a question. I can collect data or explain how it could be collected. I can model a situation using a representation such as a drawing, equation, line plot, picture graph, bar graph, or a building made of blocks. I can think about the real-world implications of my model.”

##### 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 Kindergarten 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 2, Numbers 1–10, End-of-Unit Assessment, Problem 5, “Write the missing numbers. 0, 1, ___, ___, ___, ___, ___, 7.” The Assessment Teacher Guide states, “Students use their knowledge of the count sequence to write the missing numbers. As with other problems on the assessment, students may know how to say the count sequence but may draw the incorrect numbers. Students who are struggling with counting in the correct order will likely have difficulty with most of the questions on this assessment. Read the task statement aloud.” The answer key aligns this problem to K.CC.3.

• Unit 7, Solid Shapes All Around Us, End-of-Unit Assessment, Problem 1, “a. How many squares are in the puzzle?___. b. How many triangles are in the puzzle?___. c. How many pattern blocks are in the puzzle?___.“ The Assessment Teacher Guides states, “Students identify and count different pattern block shapes in a puzzle. They also count all of the pattern blocks in the puzzle. Since they cannot move the pieces, they will need to count carefully in order to count each shape once and only once.” The answer key aligns this problem to  K.CC.5 and K.G.2.

• Unit 8, Putting It All Together, End-of-Course Assessment, Problem 7, “There are 8 crabs on the beach. Then 5 of the crabs go into the ocean. How many crabs are on the beach now? Show your thinking using drawings, numbers, or words.” The Assessment Teacher Guide states, “Students solve a Take Away, Result Unknown story problem. They may draw a picture as in the provided solution or they may write an equation or explain their reasoning in words.” The answer key aligns this problem to K.OA.1 and K.OA.2.

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 7, Solid Shapes All Around Us, Lesson 2, Cool-down, Student Facing states, “Circle the shape that is filled with more pattern blocks.” Responding to Student Thinking states, “Students circle the penguin with fewer pattern blocks.” Next Day Supports states, “During the launch of the first activity in the next lesson, have two students share shapes that they created with pattern blocks. Invite students to share methods for comparing the number of pattern blocks in each shape.” This problem aligns to K.CC.C.

##### 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 Kindergarten 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 3, Flat Shapes All Around Us, End-of-Unit Assessment develops the full intent of K.G.2 (Correctly name shapes regardless of their orientations or overall size). For example, Problem 1 states, “Color the 3 rectangles.” Students are provided images of five shapes.

• Unit 4, Understanding Addition and Subtraction, End-of-Unit Assessment, supports the full intent of MP4 (Model with mathematics) as students represent an addition problem with drawings, numbers, words or objects. For example, Problem 2 states, “There are 3 stickers on the book. Then Jada puts 2 more stickers on the book. How many stickers are on the book now? Show your thinking using drawings, numbers, words, or objects.”

• Unit 6, Numbers 0-20, End-of-Unit Assessment develops the full intent of K.NBT.1 (Compose and decompose numbers from 11 to 19 into ten ones and some further ones and record each composition or decomposition by drawing or equation; understand that these numbers are composed of ten ones, and one, two, three, four, five, six, seven, eight, or nine ones). For example, Problem 3 states, “Circle the 2 images that make 14 dots together.”

• Unit 8, Putting It All Together, End-of-Course Assessment supports the full intent of MP3 (Construct viable arguments and critique the reasoning of others) as students reason about subtraction within 10. For example, Problem 9b states, “Han has 7 flowers. He gives Elena 1 flower. How many flowers does Han have now? Show your thinking using drawings, numbers, or words.”

##### 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 Kindergarten 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, How to Assess Progress, Summative Assessment Opportunity, “In K–2, the assessment may be read aloud to students, as needed.”

• 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 Kindergarten 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 Kindergarten 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 2, Numbers 1–10, Lesson 18, Activity 2, Narrative, Access for Students with Disabilities, “Engagement: Develop Effort and Persistence. Students might benefit from counting the first tower that was built to determine how many cubes they need to create a tower that is 1 fewer or 1 more. Invite students to count in sequence the number of cubes and remind them to stop at the number that is 1 less or 1 more. Supports accessibility for: Memory, Attention, Organization.

• Unit 3, Flat Shapes All Around Us, Lesson 5, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Access for Perception. Synthesis: Students might need extra support determining that the oval and the pizza slice shapes are not circles or triangles. Hold up a triangle and circle next to the shapes to visually show that the oval and pizza slice shapes do not match the triangle and circle. Supports accessibility for: Visual-Spatial Processing.

• Unit 4, Understanding Addition and Subtraction, Lesson 2, Activity 2, Narrative, Access for Students with Disabilities, “Action and Expression: Develop Expression and Communication. Some students may benefit from using 5-frames to help count the number of green and red apples. Give students access to 5-frames and counters to represent the apples in each problem. Invite students to use the 5-frames to figure out how many apples there are altogether. Supports accessibility for: Organization, Conceptual Processing.

• Unit 8, Putting It All Together, Lesson 5, Activity 2, Narrative, Access for Students with Disabilities, “Engagement: Internalize Self-Regulation. Provide students an opportunity to self-assess and reflect on the number clue and if that number clue matches the number they will stand by. For example, students can choral count together to check that the number 9 is 1 less than 10. Supports accessibility for: Memory, Conceptual Processing.

##### 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 Kindergarten 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, Numbers 1–10, Section C: Connect Quantities and Numbers, Problem 6, Exploration, “Han says he sees 5. Lin says she sees 4. Tyler says he sees 3. Explain or show how Han, Lin, and Tyler can all be correct.”

• Unit 4, Understanding Addition and Subtraction, Section A: Count to Add and Subtract, Problem 7, Exploration, “Pick a number from the list to put in the blank space. 2, 7, 6, 3. Then try the problem you made. Count out 8 counters. Take away ___ counters. How many counters are left? After you try the problem you made, try it again with a different number in the blank space. Do you think your answer will be the same or different? Explain.”

• Unit 6, Numbers 0–20, Section B: 10 Ones and Some More, Problem 8, Exploration, “1. Arrange 18 dots in a way that helps you see there are 18. 2. Arrange 18 dots in a way that makes it hard to see how many there are. 3. Explain why you chose your arrangements. Try again with other numbers up to 19.”

• Unit 7, Solid Shapes All Around Us, Section B: Describe, Compare, and Create Solid Shapes, Problem 4, Exploration, “1. Can you find an object in the classroom that fits the description? I am not flat. I am heavy. You can see some rectangles on me. Can you find more than one object? 2. Can you find an object in the classroom that fits the description? I am flat. I have lots of colors and different shapes. I have some rectangles. Can you find more than one object?”

##### 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 Kindergarten 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 2, Numbers 1-10, Lesson 17, Warm-up, students show different ways to represent a mathematical quantity. Launch states, “Groups of 2. Display and read the story. ‘What is the story about?’ 30 seconds: quiet think time. Share responses.” Activity states, “Read the story again. ‘How can you show the plates?’ 30 seconds: quiet think time. ‘Discuss your thinking with your partner.’ 1 minute: partner discussion. Share responses.” Student Facing states, “Han is helping his grandfather set the table for dinner. Han puts 8 plates on the table.”

• Unit 4, Understanding Addition and Subtraction, Lesson 10, Activity 3, students use a game to count dots and match the dice that they roll. Launch states, “Give each group of students 2 connecting cubes, two-color counters, and a dot mat. Give each student a gameboard. ‘We are going to learn a new way to do the Bingo center. It is called Bingo, Add and Cover.’ Display a dot mat 1-5. ‘I’m going to roll 2 cubes onto the mat. Then I need to figure out how many dots I have altogether.’ Demonstrate rolling 2 cubes onto the mat. ‘How can I figure out how many dots I have altogether?’ (You can count all of the dots. You can just see that there are 2 and 2, which is 4.) Display gameboard. ‘I have 4 dots altogether. Now I need to cover all of the squares on my gameboard that also have 4 things. Which squares should I cover?’ Take turns rolling the cubes onto the mat, then each person covers the squares on their mat. The game ends when someone has 4 counters in a row.”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 13, Activity 1, students work to determine how many more fingers are needed to make 10. Launch states, “Groups of 2. Give each group of students a connecting cube, a number mat, and a recording sheet. ‘We are going to learn a new way to do the Math Fingers center. It is called Math Fingers, Make 10.’  Display and roll a connecting cube onto the number mat. ‘I rolled 7, so I am going to hold up 7 fingers. Now my partner needs to figure out how many more fingers I need to put up to show 10 fingers. How many more fingers do I need to hold up to make 10?’ (3.) 30 seconds: quiet think time. Share responses. Display the recording sheet. ‘Now we need to fill in an equation to show how many fingers are up and how many more fingers are needed to make 10. How should I fill in an equation?’ (7+3) 30 seconds: quiet think time. Share responses. ‘Take turns with your partner rolling to find a number and showing that number with your fingers. Your partner figures out how many more fingers are needed to make 10. You both fill in an equation to show how many fingers are up and how many more fingers are needed to make 10.’”

• Center, Find the Value of Expressions (K), Stage 1: Color the Total or Difference, students find the value of the expression card. Narrative states, “One partner chooses an expression card. The other partner finds the value of the expression. When both partners agree, they both color in that number on the recording sheet. All expressions have values within 10.”

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

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

The materials reviewed for Kendall Hunt's Illustrative Mathematics Kindergarten 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 1, Math in Our World, Lesson 7, Activity 1, students work with a partner as they recognize quantities without counting. Launch states, “Groups of 2. Display the first image. ‘How many do you see? How do you see them?’ 30 seconds: quiet think time. Activity states, “‘Use your fingers to show your partner how many dots you see.’ 30 seconds: partner work time. ‘Tell your partner how many dots you see and how you see them.’ 1 minute: partner discussion. Record responses. Repeat for the second image.”

• Unit 2, Numbers 1-10, Lesson 12, Activity 1, students work in groups of four to match a number with the bag that holds that number of objects. Launch states, “Groups of 4. Give each group of students 4 bags. Write or display the number 8. ‘Find the bag that has 8 objects.’” Activity states, “2 minutes: small-group work time. Repeat the steps with the numbers 5, 7, and 9.”

• Unit 3, Flat Shapes All Around Us, Lesson 15, Activity 1, students work in groups to use shapes to compose animals. Launch states, “Groups of 4. Give each group of students a set of shape stamps and a paper plate with black paint. ‘You will use these stamps to make an animal. I am going to make a cat. What shapes should I use? (Circle for the head, triangles for the ears, rectangles for the legs and tail.) If I want to make a circle for the cat’s head, which stamp should I use?’ Invite students to point to the correct shape. Demonstrate dipping the stamp into the paint and pressing it on the paper. ‘Take turns using the shape stamps with your group. You only need a little bit of paint for each stamp.’” Activity states, “‘Which animal do you want to make? What shapes will you use to make the animal?’ 30 seconds: quiet think time. 10 minutes: small-group work time.”

• Unit 8, Putting It All Together, Lesson 21, Activity 1, students work with a partner to compose and decompose numbers between 11 and 19. Launch states, “Groups of 2. Give each group of students access to collections of 11–19 objects. ‘We are going to pretend that the objects in the collection are students. The students will either sit at the table or on the rug.’ Display the image. ‘All of the students want to sit at the table. How many of the students can sit at the table?’ (10 students) 1 minute: independent work time. ‘10 students can sit at the table. The students who do not fit at the table will sit on the rug. Work with your partner to figure out how many students will sit at the table, how many will sit on the rug, and how many students there are altogether. Fill in an equation for each bag of objects.’”

##### 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 Kindergarten 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 1, Math in Our World, Lesson 15, Activity 1, Teaching notes, Access for English Learners, “MLR8 Discussion Supports. Provide multiple opportunities for verbal output. Invite students to chorally repeat each count in unison. Advances: Listening, Speaking.”

• Unit 3, Flat Shapes All Around Us, Lesson 14, Activity 1, Teaching Notes, Access for English Learners, “MLR7 Compare and Connect. Synthesis: To amplify student language and illustrate connections, follow along and point to the relevant parts of the images as students compare how they are alike and different. Advances: Representing, Conversing.”

• Unit 4, Understanding Addition and Subtraction, Lesson 13, Activity 1, Teaching notes, Access for English Learners, “MLR2 Collect and Display. Circulate, listen for and collect the language students use as they create story problems. On a visible display, record words and phrases such as: ‘more,’ ‘joined,’ ‘went away,’ ‘take away,’ and ‘less.’ Review the language on the display, then ask, “Which of these words tell you the story is about addition?” and “Which of these words tell you the story is about subtraction?” Advances: Representing, Listening.”

##### 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 Kindergarten 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 Kiran, Mai, Elena, and Han 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 Kindergarten 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.

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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 Kindergarten 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 1, Math in Our World, Lesson 6, Activity 2 introduces the use of picture books, representing a variety of cultures, to be used throughout the school year. Narrative states, “The purpose of this activity is for students to recognize and name quantities in picture books. If students have not heard the story this year, read the book aloud to students as a part of the launch. Students may notice and wonder many things about the page in the book, especially after hearing the story. This should be encouraged and recorded as students are making sense of the context. If students do not mention the groups of objects displayed on the page, ask them ‘What things on the page remind you of things we have been doing in math class?’ to encourage them to mathematize the situation (MP4). This prepares students to see and analyze quantities so that they can use mathematics to describe their world. This is stage 1 of the Picture Books center. Students continue working with picture books throughout this unit in centers.” Some examples of picture books include: Grandma’s Purse by Vanessa Brantlet-Newton, My Heart Fills with Happiness by Monique Gray Smith, Pablo’s Tree by Pat Mora, Saturday by Oge Mora, There is a Bird on Your Head by Mo Willems, Last Stop on Market Street by Matt de la Peña, Miss Bindergarten Gets Ready for Kindergarten by Joseph Slate, Big Red Lollipop by Rukhsana Khan, Count on Me by Miguel Tanco, and The Girl with the Parrot on Her Head by Daisy Hirst.

• Unit 3, Flat Shapes All Around Us, Lesson 10, Warm-up, students reference a quilt, woven by a women’s group in the South. Narrative states, “The purpose of this warm-up is to elicit the idea that shapes can be combined to make patterns and pictures, which will be useful when students put together pattern blocks to make shapes in a later activity. While students may notice and wonder many things about these images, the shapes in the design of the quilt are the important discussion points. The images in this warm-up are of quilts made by a group of women in Gee’s Bend, Alabama. Consider reading the book, ‘Stitchin’ and Pullin: A Gee’s Bend Quilt’ by Patricia McKissack and showing students more examples of quilts as a part of the Notice and Wonder activity. Examples of quilts from the book that are made of different shapes than the one shown in the student workbook will give students the opportunity to notice and wonder different things.”

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

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

• Unit 1, Math in Our World, Lesson 2, Warm-up, “The purpose of this activity is to elicit ideas students have about pattern blocks. This allows teachers to see the vocabulary students use to describe shapes (MP6). There is no need to introduce formal geometric language at this point since this will happen in a later unit.”

• Unit 3, Flat Shapes All Around Us, Lesson 3, Activity 2, Access for Students with Disabilities, “Representation: Access for Perception. Students might benefit from using gestures to connect the meaning of words describing shapes with the attributes they see. Invite students to mimic gestures during the launch when using words like round, point, square, flat, etc.”

• Unit 6, Numbers 0–20, Lesson 5, Activity 3, Access for Students with Disabilities, “Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words.”

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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 Kindergarten 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, Math in Our World, Lesson 5, Activity 1, identifies two-color counters and 5-frames as strategies for students to engage in the math of the lesson. Launch states, “Give each student a 5-frame. ‘As you explore the two-color counters, you will also explore a new tool called a 5-frame.’ Display the 5-frame. ‘Why do you think we call this a 5-frame?’ (Because it has five spaces or squares in it.) Share responses. Give each group of students a container of two-color counters. ‘Let’s explore two-color counters and 5-frames.’”

• Unit 2, Numbers 1–10, Lesson 14, Activity 1, references the use of cards and counters to count out objects and match the quantity to a number. Launch states, “Groups of 2. Give each group of students a set of number cards and counters. ‘What is your favorite pizza topping?’ Display the student book and a number card. ‘If my partner showed me this card, how many pizza toppings should I add to my pizza?’ 30 seconds: quiet think time. 1 minute: partner discussion. Share responses.”

• Unit 3, Flat Shapes All Around Us, Lesson 11, Activity 1, identifies pattern blocks for use in identifying shapes that are the same, regardless of orientation. Launch states, “Groups of 2. Give students pattern blocks. Display the student book. ‘What do you notice? What do you wonder?’ (There are lots of different pattern blocks. I wonder why they are all missing a piece.) 30 seconds: quiet think time. 30 seconds: partner discussion. Share responses. ‘Figure out which pattern block is missing from each puzzle. Tell your partner how you know.’”

• Unit 4, Understanding Addition and Subtraction, Lesson 6, Activity 3, describes the use of connecting cubes and a number mat to support understanding of subtraction problems. Launch states, “Give each group of students 10 connecting cubes and a number mat. ‘We are going to learn a center called Subtraction Towers.’ Display a connecting cube tower with 7 cubes. ‘How many cubes are in the tower? If I have to subtract, or take away, 3 cubes from my tower, what should I do?’ (Break off 3 cubes, take off 1 cube at a time as you count.) One partner uses up 5-10 cubes to build a tower. Then the other partner rolls to figure out how many cubes to take away, or subtract, from the tower.”

#### 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 Kindergarten 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.

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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 Kindergarten 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.

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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 Kindergarten 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.

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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 Kindergarten 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.

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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 Kindergarten 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

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### Overall Summary

###### Alignment
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###### Usability
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