Kindergarten - Gateway 3

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.)”

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.”

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 4, Understanding Addition and Subtraction, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “In this unit, students develop their understanding of addition and subtraction as they represent and solve story problems. Previously, students built their counting skills and represented quantities in a group with their fingers, objects, drawings, and numbers. Here, they relate counting to the result of two actions: putting objects together or taking objects away. Students enact addition by counting the total number of objects in two groups, and subtraction by counting what remains after some objects are taken away. (The word ‘total’ is used here instead of ‘sum’ to reduce potential confusion with the word ‘some’ or part of a whole.) Students then make sense of stories without questions and later solve story problems of two types—Add To, Result Unknown and Take From, Result Unknown. Students represent the problems in different ways, by acting them out, drawing, using numbers, or using objects. Connecting cubes should be accessible in all lessons for students who wish to use them, including for cool-downs. All story problems should be read aloud by the teacher, multiple times if needed. Students are also introduced to expressions, a symbolic way to represent addition and subtraction. Initially, the teacher records the process of adding and subtracting using words such as ‘5 and 3’ or ‘4 take away 1.’ Later, students see that ‘5 and 3’ and ‘4 take away 1’ can be expressed by 5+3 and 4+1 , respectively. They learn that these expressions are read ‘5 plus 3’ and ‘4 minus 1.’ (Students are not expected to read expressions out loud or to use precise language at this point.) Later in the section, students connect expressions to pictures and story problems. They find the value of addition and subtraction expressions within 10. In a future unit, students will compose and decompose numbers up to 10 and solve other types of addition and subtraction problems.”

• 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.” 

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.”

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).” 

• Unit 3, Flat Shapes All Around Us, Lesson 3, Activity 3, Required Materials, “Counters, Materials from previous centers, Which One Stage 1 Gameboard.” Launch states, “We are going to learn a center called Which One. Let’s play one round together. Pick a shape on the board to be your mystery shape. I’ve chosen a shape that is on the board. Your job is to ask me questions that will help you figure out which shape I chose. You can only ask questions that I can answer with a ‘yes‘ or a ‘no.’ For example, you cannot ask, How many sides does your shape have? But you can ask, Does your shape have more than 3 sides? After each question, ask students to share which shapes they can rule out based on the question and place a counter on those shapes. When students feel ready to guess your shape, invite students to guess the shape, asking them to explain why they think it’s your shape. Take turns choosing a mystery shape and asking questions with your partner.” Activity states, “Now you can choose another center. You can also continue playing Which One. Display the center choices in the student book. Invite students to work at the center of their choice.”

• 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.’”

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.”

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.”

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.”

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?”

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.”

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.”