1st Grade - Gateway 3

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

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

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

• Course Guide, Scope and Sequence, provides an overview of content and expectations for the units, “The big ideas in grade 1 include: developing understanding of addition, subtraction, and strategies for addition and subtraction within 20; developing understanding of whole-number relationships and place value, including grouping in tens and ones; developing understanding of linear measurement and measuring lengths as iterating length units; and reasoning about attributes of, and composing and decomposing geometric shapes. 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 3, Adding and Subtracting Within 20, Lesson 17, Activity 1, teachers are provided guidance to support students in transitioning from using 10 frames to writing mathematical equations. Narrative states, “The purpose of this activity is for students to find sums when one addend is nine. Students represent sums on the 10-frame to encourage them to use the structure of a ten. During the launch, the teacher demonstrates playing a round of the game. It is important to let students discover patterns as they play the game. For example, when finding the sum of 9+5, some students may represent each addend on a separate 10-frame and count to find the sum. Other students may use the associative property and move one counter from the five, and add it to the nine to make a ten. (A picture of a 10-frame is provided.) Students may generalize that when they take one from an addend to make 10, the sum has one less one than that addend. When students build this understanding, they may no longer need to show their thinking on the 10-frame and can just write an equation. By repeatedly making the ten by taking one from an addend, students may see and use the structure of ten to add on (MP7, MP8).”

• Unit 4, Numbers to 99, Lesson 7, Warm-up, provides teachers guidance about how two-digit numbers are composed of tens and ones. 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, “‘The numbers in Set B are called two-digit numbers.’ Display 89. ‘This is one number, the number eighty-nine. This number has two digits, an 8 and a 9. In the number 89, the 8 tells us how many tens are in the number and the 9 tells us how many ones are in the number. Today you will work on making two-digit numbers.’”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 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 2, The Power of Small Ideas, “In this blog post, McCallum discusses, among other ideas, the use of a letter to represent a number. The foundation of this idea is introduced in this unit when students first represent an unknown with an empty box.” Representing Subtraction of Signed Numbers: Can You Spot the Difference?, “In this blog post, Anderson and Drawdy discuss how counting on to find the difference plays a foundational role in understanding subtraction with negative numbers on the number line in middle school.” 

• IM K-5 Math Teacher Guide, About These Materials, Unit 3, Connection to a book by Russell, S.J., Schifter D., & Bastable, V. (2011), supports teachers with context for work beyond the grade, “Connecting Arithmetic to Algebra: Strategies for Building Algebraic Thinking in the Elementary Grades. Heinemann. This book explains how generalizing the basic operations, rather than focusing on isolated computations, strengthens students’ fluency and understanding which helps prepare them for the transition from arithmetic to algebra. Chapter 1, Generalizing in Arithmetic, is available as a free sample from the publisher.”

• Unit 4, Numbers to 99, Lesson 1, Preparation, Lesson Narrative states, “In the previous unit, students learned that a ten is a unit made up of 10 ones. Students learned that teen numbers are made up of 1 ten and some more ones, using 10-frames, drawings, and expressions $$(10+n)$$. In kindergarten, students learned the counting sequence by ones and tens up to 100. The purpose of this lesson is for teachers to formatively assess how students count objects up to 60 through two counting activities. In the first activity, students count objects and represent how many in a way that makes sense to them, then compare the ways they counted. In the second activity, students count bags of different quantities that are multiples of 10, and begin to make sense of grouping objects into tens. Suggested objects include pennies, paper clips, buttons, connecting cubes, inch tiles, counters, or any other objects around the classroom. Students should also be given access to cups, paper plates and double 10-frames to help them organize their collections if they would like.”

• Unit 6, Length Measurements Within 120 Units, Lesson 5, Preparation, Lesson Narrative states, “In previous lessons, students ordered a set of three objects by length. Students also compared lengths of objects indirectly by using a third object. The purpose of this lesson is for students to describe lengths of objects in terms of connecting cubes. Students measure by using connecting cube towers because the units are lined up without gaps or overlaps, a concept they will explore in future lessons. In the first activity, students use connecting cube towers to measure the length of different animals. Students build towers that are exactly the same length as the animals and make a comparison statement (‘The grasshopper is the same length as a tower of 7 cubes’). In the second activity, students use connecting cube towers to measure the length of more animals and describe the length as ‘___ cubes long.’ Even though the side-length of the cube is the unit, it’s appropriate for students to describe length in terms of ‘x cubes long.’ This transition in language helps students understand that the length of objects can be described as a number of length units (MP6). In this lesson, the length unit is the length of a single connecting cube.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 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 1, Course Guide, Lesson Standards includes a table with each grade-level lesson (in columns) and aligned grade-level standards (in rows). Teachers can search any lesson for the grade and identify the standard(s) that are addressed within.

• Grade 1, Course Guide, Lesson Standards, includes all Grade 1 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 3, Resources, Teacher Guide, outlines standards, learning targets and the lesson where they appear. This is present for all units and allows teachers to identify targeted standards for any lesson.

• Unit 6, Length Measurements Within 120 Units, Lesson 3, the Core Standard is identified as 1.MD.A.1. Lessons contain a consistent structure: a Warm-up that includes Narrative, Launch, Activity, Activity Synthesis; Activity 1, 2, or 3 that includes Narrative, Launch, Activity; an Activity Synthesis; a Lesson Synthesis; and a Cool-down that includes Responding to Student Thinking and Next Day Supports. This provides an additional place to reference standards, and language of the standard, within each lesson.

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 1, Course Guide, Scope and Sequence, Unit 2: Addition and Subtraction Story Problems, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “In kindergarten, students solved a limited number of types of story problems within 10 (Add To/Take From, Result Unknown, and Put Together/Take Apart, Total Unknown, and Both Addends Unknown). They represented their thinking using objects, fingers, mental images, and drawings. Students saw equations and may have used them to represent their thinking, but were not required to do so. Here, students encounter most of the problem types introduced in grade 1: Add to/Take From, Change Unknown, Put Together/Take Apart, Unknowns in All Positions, and Compare, Difference Unknown. The numbers are kept within 10 so students can focus on interpreting each problem and the relationship between counting and addition and subtraction. This also allows students to continue developing fluency with addition and subtraction within 10.”

• Grade 1, Course Guide, Scope and Sequence, Unit 7: Geometry and Time, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “In this unit, students focus on geometry and time. They expand their knowledge of two- and three-dimensional shapes, partition shapes into halves and fourths, and tell time to the hour and half of an hour. Center activities and warm-ups continue to enable students to solidify their work with adding and subtracting within 20 and adding within 100. In kindergarten, students learned about flat and solid shapes. They named, described, built, and compared shapes. They learned the names of some flat shapes (triangle, circle, square, and rectangle) and some solid shapes (cube, sphere, cylinder, and cone).”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 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, The Story of Grade 1, Brianne Durst, Deep Dive into New Learning in Unit 4 (Numbers to 99) and Unit 5 (Adding within 100), “In Unit 4, students use what they have learned about teen numbers and the unit of ten to generalize the structure of two-digit numbers, relating the two digits to the number of tens and ones. They interpret and use multiple representations of numbers up to 99, such as connecting cubes, base-ten diagrams, words, and expressions. Connecting cubes in towers of 10 and singles are used throughout grade 1, rather than base-ten blocks, so that units of ten can be physically composed and decomposed with the cubes. Although students work physically with connecting cubes, they interpret base-ten diagrams, recognizing the diagram as a simplified image of the connecting cubes. This helps students make sense of a more efficient way of drawing diagrams to match their connecting cubes. As students develop their understanding of place value and work with each of these representations, they are able to compare any two-digit numbers by comparing the number of tens, and, if needed, the number of ones.”

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

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

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

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

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 4, Activity 1, Required Materials, “10-frames, Materials from a previous activity, Number cards 0–10, Two-color counters.” Launch states, “Give each group a set of number cards, a game board, two-color counters, and access to 10-frames. We are going to learn a new way to play, Five in a Row. Last time we played, we added one or two to the number on our card. This time, you will take turns flipping over a card and choosing whether to subtract one or two from the number. Then put a counter on the number on the game board. The first person to get five counters in a row wins. Remember, your counters can be in a row across, up and down, or diagonally.” 

• Course Guide, Required Materials for Grade 1, Materials Needed for Unit 3, Lesson 4, teachers need, “10-frames, Crayons, Cups, Materials from previous centers, Two-color counters, Shake and Spill Stage 3 Recording Sheet Grade 1 (groups of 1).” 

• Unit 4, Numbers to 99, Lesson 7, Activity 1, Required Materials, “Connecting cubes in towers of 10 and singles, Number cards 0-10, Materials to copy (Make It, Two-Digit Numbers Recording Sheet Number, Drawing, Words).” Launch states, “Groups of 2. Give each group a set of number cards, connecting cubes in towers of 10 and singles, and recording sheets. Ask students to take out the cards with 10 on them. ‘We are going to play a game called Make It. You will work with your partner to make a two-digit number and represent the number in different ways.’ Display two number cards and the recording sheet. ‘First, one partner picks two number cards and makes a two-digit number. I picked a [3] and a [5]. What two-digit numbers can I make?’ (35 or 53). Demonstrate writing one of the numbers on the recording sheet. ‘Now both partners say the number. Then, the partner who made the number watches the other partner build the number with connecting cubes. Make sure you both agree on how to build the number. Then both partners complete the recording sheet with a drawing and the number of tens and ones.”

• Course Guide, Required Materials for Grade 1, Materials Needed for Unit 7, Lesson 1, teachers need, “Bags (brown paper), Geoblocks, Materials from a previous activity, Solid shapes.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 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, Adding and Subtracting Within 20, End-of-Unit Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 5, 1.OA.4 and 1.OA.6, “Clare says that 16-7 must be 9 because 9+7=16. Do you agree with Clare? Show your thinking using drawings, numbers, or words.”

• Unit 4, Numbers to 99, End-of-Unit Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 2, 1.NBT.2, “Circle the 2 expressions that are equal to 53. A. 3+50. B. 30+5. C. 40+10. D. 50+3. E. 5+3.”

• Unit 8, Putting it All Together, End-of-Course Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 5, 1.OA.1, “Jada’s bracelet has 12 beads. 7 of the beads are green and the rest are pink. How many pink beads are on Jada’s bracelet? Show your thinking using drawings, numbers, or words.”

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 1, MP1 is found in Unit 3, Lessons 5, 11, 12, 15, and 20.

• IM K-5 Math Teacher Guide, How to Use These Materials, Standard for Mathematical Practices Chart, Grade 1, MP7 is found in Unit 5, Lessons 3, 5, 7, 9, 10, and 12. 

• IM K-5 Math Teacher Guide, How to Use These Materials, Standards for Mathematical Practice Student Facing Learning Targets, “MP2 I Can Reason Abstractly and Quantitatively. I can think about and show numbers in many ways. I can identify the things that can be counted in a problem. I can think about what the numbers in a problem mean and how to use them to solve the problem. I can make connections between real-world situations and objects, diagrams, numbers, expressions, or equations.”

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

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

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

• Unit 1, Adding, Subtracting, and Working with Data, End-of-Unit Assessment supports the full intent of MP1 (Make sense of problems and persevere in solving them) as students count tallies from a table. For example, Problem 3 states, “The table shows the different shapes on Jada’s desk. How many squares are on Jada’s desk? ___ How many shapes are on Jada’s desk?___ .” An image of a table showing a triangle, circle, and square with tally marks for each is shown. 

• Unit 2, Addition and Subtraction Story Problems, End-of-Unit Assessment supports the full intent of MP7 (Look for and make use of structure) as students choose equations to represent a story problem. For example, Problem 4 states, “Mai drew 2 stars in her notebook. Then she drew some hearts. Now there are 8 shapes altogether. How many hearts did Mai draw in her notebook? Circle 2 equations that match the story. A. ___$$-8=2$$. B. 2+___$$=8$$. C. 8-2=___. D. ___$$+2=10$$. E. 2+8=___.”

• Unit 5, Adding Within 100, End-of-Unit Assessment develops the full intent of 1.NBT.4 (Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten). For example, Problem 3 states, “Find the value of each sum. Show your thinking using drawings, numbers, or words. a. 74+5 b. 45+9, c. 23+48.”

• Unit 6, Length Measurements Within 120 Units, End-of-Unit Assessment develops the full intent of 1.OA.1 (Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions). For example, Problem 6 states, “There were some students on the bus. 7 students got off at the bus stop. Now there are 6 students on the bus. a. Write an equation that matches the story. Use a ? for the unknown number. How does the equation match the story? Show your thinking using drawings, numbers, or words. b. How many students were on the bus before the stop? Show your thinking using drawings, numbers, words, or equations.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

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

Examples of supports for special populations include: 

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 9, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Access for Perception. Students with color blindness will benefit from verbal emphasis, gestures, or labeled displays to distinguish between colors of connecting cubes. Supports accessibility for: Visual-Spatial Processing.”

• Unit 2, Addition and Subtraction Story Problems, Lesson 11, Activity 2, Narrative, Access for Students with Disabilities, “Engagement: Provide Access by Recruiting Interest. Provide choice and autonomy. In addition to connecting cubes, provide access to red, yellow, and blue crayons or colored pencils they can use to represent and solve the story problems. Supports accessibility for: Visual-Spatial Processing, Conceptual Processing.”

• Unit 5, Adding Within 100, Lesson 4, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Develop Language and Symbols. Make connections between the representations visible. For example, ask students to identify correspondences between the visual representation and the expression 37+26. Supports accessibility for: Visual-Spatial Processing, Conceptual Processing.”

• Unit 7, Geometry and Time, Lesson 9, Activity 3, Narrative, Access for Students with Disabilities, “Action and Expression: Develop Expression and Communication. Give students access to a straight edge or ruler. Supports accessibility for: Fine Motor Skills, Visual-Spatial Processing.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 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 1, Adding, Subtracting, and Working with Data, Section C: What Does the Data Tell Us?, Problem 7, Exploration, “Gather data at home or school and make a display of the data. Ask a math question about the data. Trade displays and questions with a partner and answer your partner’s question.”

• Unit 2, Addition and Subtraction Story Problems, Section A: Add To and Take From Story Problems, Problem 8, Exploration, “Choose one of the equations. 1. 5+___$$=8$$. 2. 8-3=___. 3. 3+___$$=8$$. 4. 5+3=___. Write a story problem that the equation matches. Trade with a partner and decide which equation matches your partner’s story.”

• Unit 5, Adding Within 100, Section B: Make a Ten: Add One- and Two-digit Numbers, Problem 7, Exploration, “Priya is playing the game Target Numbers. Priya starts at 25 and picks these six cards: 1, 2, 3, 5, 6, 8. She chooses whether to add that many tens or ones for each card. What is the highest score she can get without going over 95? Use equations to show your thinking.”

• Unit 7, Geometry and Time, Section C: Tell Time in Hours and Half Hours, Problem 4, Exploration, “Show the time during the day when you might do each of these things. 1. wake up in the morning, 2. go to school, 3. have a snack, 4. go for recess, 5. have lunch.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 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 2, Addition and Subtraction Story Problems, Lesson 9, Activity 1, Teaching notes, Access for English Learners, “MLR8 Discussion Supports. Some students may benefit from the opportunity to act out the scenario. Listen for and clarify any questions about the context of each problem. Advances: Speaking, Representing.”

• Unit 5, Adding Within 100, Lesson 1, Activity 1, Teaching notes, Access for English Learners, “MLR2 Collect and Display. Circulate, listen for and collect the language students use as they talk with their partners. On a visible display, record words and phrases such as: tens, ones, sum, equation, starting number, secret number. Invite students to borrow language from the display as needed, and update it throughout the lesson. Advances: Conversing, Speaking, Listening.”

• Unit 6, Length Measurements within 120 Units, Lesson 12, Activity 1, Launch, “MLR6 Three Reads, Display only the problem stem, without revealing the question. ‘We are going to read this problem three times.’ 1st Read: ‘Priya and Han are comparing the lengths of their friendship bracelets. Han’s bracelet is 14 cubes long. The length of Priya’s bracelet is 4 cubes fewer than Han’s bracelet. What is this story about?’ 1 minute: partner discussion. Listen for and clarify any questions about the context. 2nd Read: ‘Priya and Han are comparing the lengths of their friendship bracelets. Han’s bracelet is 14 cubes long. The length of Priya’s bracelet is 4 cubes fewer than Han’s bracelet. What can be counted or measured?’ (The length of Priya's bracelet. The length of Han's bracelet. The difference in length between the two bracelets.) 30 seconds: quiet think time. 1 minute: partner discussion. Share and record all quantities. 3rd Read: Read the entire problem, including the question, aloud. ‘What are different ways we can solve this problem?’ (use connecting cubes to represent the bracelets, draw a picture, think about the numbers), 30 seconds: quiet think time. 1–2 minutes: partner discussion.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 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, Adding, Subtracting, and Working with Data, Lesson 8, Activity 1, describes the use of shape cards and a three-column table to help students sort shapes into categories. Launch states, “Give each group a set of shape cards and access to copies of the three-column table. ‘Look at all of your shape cards. Take a minute to look over the cards by yourself first and think about how you would sort them.’” Activity states, “Work with your partner to sort the cards into three categories in any way that you want. You do not need to use all of the cards.”

• Unit 3, Adding and Subtracting Within 20, Lesson 21, Activity 1, identifies number cards, 10-frames, connecting cubes or two-color counters as strategies for students to add numbers. Launch states, “Give each group a set of number cards, two recording sheets, and access to double 10-frames and connecting cubes or two-color counters. ‘We are going to learn a game called How Close? Add to 20. Let's play the first round together. First we take out any card that has the number 10. We will not use those cards for the game.’ Display 5 cards. ‘I can choose two or three of these cards to add to get as close to 20 as I can. What cards should I choose? I write an equation with the numbers I chose and the sum of the numbers.’ Demonstrate writing the equation on the recording sheet. ‘The person who gets a sum closer to 20 gets a point for the round. Then you each get more cards so you always have five cards to choose from. Play again. The first person to get 10 points wins.’”

• Unit 5, Adding Within 100, Lesson 10, Activity 2, identifies connecting cubes in towers of 10 and singles to support understanding of the associative and commutative properties when adding 2 two-digit numbers. Launch states, “Groups of 2. Give students access to connecting cubes in towers of 10 and singles. Read the first problem. 4 minutes: partner work time. ‘What is the difference between how you solved  28+56 and 27+44.’ (For 28+56, I added the tens first, then the ones. For 27+44 I added the ones first, then the tens.) 1 minute: partner discussion. Share responses.”

• Unit 6, Length Measurements Within 120 Units, Lesson 4, Activity 2, references number cubes and game recording handouts to support understanding during centers. Launch states, “‘Now you will choose from centers we have already learned.’ Display the center choices in the student book. Target Numbers, ‘On your turn: Start at 55. Roll the number cube. Add that number to your starting number and write an equation to represent the sum. Take turns until you’ve played 6 rounds. Each round, the sum from the previous equations is the starting number in the new equation. The partner to get a sum closest to 95 without going over wins.’”