1st Grade - Gateway 3

The materials reviewed for Imagine Learning 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. Examples include:

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

• IM Curriculum, Scope and sequence information, provides an overview of content and expectations for the units. “The big ideas in grade 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.”

• Unit 4, Numbers to 99, Section C, Compare Numbers to 99, Section Overview, Section Narrative, provides an overview of the content and expectations for the section. “In this section, students use their understanding of the base-ten structure to compare and order numbers to 99. They notice that if a two-digit number has more tens it will be greater than another number with fewer tens, no matter how many ones there are. They then generalize this insight to compare numbers based on the digits. The < and > symbols are introduced here. Before using the symbols to write true comparison statements, students gain familiarity by reading and interpreting statements with these symbols. They have opportunities to work with the symbols throughout the section. The lesson activities intentionally use mathematical language to support students in recalling how to read or write the symbols. For example, initially students are encouraged to notice that the side of the symbol with the greater amount of space between the top and the bottom segments faces the greater number. Avoid using non-mathematical or imaginative language that may distract from the focus of the unit and delay fluency with reading and writing the symbols.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Several components focus specifically on the content of the lesson. Preparation and Lesson Narratives, Warm-up, Activities, and Cool-down Narratives all provide useful annotations. IM Curriculum, Why is the curriculum designed this way?, Design Principles, Coherent Progressions, “Each unit starts with an invitation to the mathematics. The first few lessons provide an accessible entry point for all students and offer teachers the opportunity to observe students’ prior understandings. Each lesson starts with a warm-up to activate prior knowledge and set up the work of the day. This is followed by instructional activities in which students are introduced to new concepts, procedures, contexts, or representations, or make connections between them. The lesson ends with a synthesis to consolidate understanding and make the learning goals of the lesson explicit, followed by a cool-down to apply what was learned. Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals. In each of the activities, care has been taken to choose contexts and numbers that support the coherent sequence of learning goals in the lesson.” Examples include:

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 4, Activity 1, teachers are provided context as they help students subtract. Narrative, “In this stage, students subtract one or two from a number within 10. Some students may count back and some may count all then count back or remove 1 or 2 then count the remaining objects. Provide access to 10-frames and counters and encourage students to use them only if needed.” Launch, “Groups of 2. 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.’” Activity, “10 minutes: partner work time. As students work, consider asking: ‘How did you subtract? How did you decide whether to subtract 1 or 2?’ Monitor for students who: represent the number, remove 1 or 2, count all that are left, represent the number, remove 1 or 2, know how many are left without recounting, count back 1 or 2, use the counting sequence to find the difference.”

• Unit 7, Geometry and Time, Lesson 3, Lesson Synthesis provides teachers guidance on ways to sort and describe two-dimensional shapes. “Display Card A. ‘Today we looked at flat shapes and described them in different ways in order to sort shapes. How might you describe this shape? (There are three sides that are the same. There are three corners. It is a triangle.) Display Card Q. How might you describe this shape?” (There are four sides. Three of the sides are the same length and the bottom side is long.)’ Continue with shape cards U and C as time allows.”

The materials reviewed for Imagine Learning 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, Why is the curriculum designed this way?, there are sections entitled “Further Reading” that consistently link research to pedagogy. There are adult-level explanations including examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject. Professional articles support teachers with learning opportunities about topics such as ensuring mathematical success for all, early understanding of equality, and repeating patterns. Examples include:

• Why is the curriculum designed this way? Further Reading, Unit 2, Representing Subtraction of Signed Numbers: Can You Spot the Difference?, supports teachers with context for work beyond the grade. “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.”

• Why is the curriculum designed this way? Further Reading, Unit 4, Rethinking Instruction for Lasting Understanding: An Example. “In this blog post, Nowak uses the progression of inequalities as an example of how to build reliable mathematical understanding.”

• Unit 2, Addition and Subtraction Story Problems, Lesson 4, Result or Change Unknown, About this Lesson, “Since this lesson includes all three of the problem types introduced to the students at this point, students need to pay close attention to each problem to determine the action in the story and the question that is being asked. This lesson provides an opportunity to assess student progress on making sense of different types of story problems, the methods they use to solve, and the equations they write to match the problems.”

• Unit 5, Adding Within 100, Lesson 14, Food Drive, About this Lesson, “This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling. In previous lessons, students found the value of sums within 100 using methods based on place value and the properties of operations, including adding tens and tens and ones and ones, and adding on by place. In this lesson, students apply these methods to make sense of and solve real-world problems within 100. Students may use base-ten representations or equations to represent their thinking. In the warm-up, they are introduced to a food drive context. In the first activity, they solve problems which involve combining quantities of collected cans in various ways. In the second activity, students make choices about which numbers to combine based on their values and the constraints of the problem. Students may use trial and error to reach the target value. This gives them an opportunity to persevere in problem solving (MP1). When students make and articulate mathematical choices and adhere to mathematical constraints, they model with mathematics (MP4).”

The materials reviewed for Imagine Learning 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-level resources, Grade 1 standards breakdown, standards are addressed by lesson. Teachers can search for a standard in the grade and identify the lesson(s) where it appears within materials.

• Course Guide, Lesson Standards, includes all Grade 1 standards and the units and lessons each standard appears in. 

• 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, Adding and Subtracting Within 20, Lesson 11, the Core Standards are identified as MP8, MP1, 1.OA.D.8, 1.OA.D.7, 1.OA.C.6, and 1.OA.A.1. Lessons contain a consistent structure that includes a Warm-up with a Narrative, Launch, Activity, Activity Synthesis. An Activity 1, 2, or 3 that includes Narrative, Launch, Activity, Activity Synthesis, Lesson Synthesis. A Cool-down that includes Responding to Student Thinking and Next Day Supports. This provides an additional place to reference standards, and language of the standard, within each lesson.

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

• Unit 4, Numbers to 99, Unit Overview, Unit Learning Goals, full Unit Narrative, includes an overview of how the math of this module builds from previous work in math. “This unit develops students’ understanding of the structure of numbers in base ten, allowing them to see that the two digits of a two-digit number represent how many tens and ones there are. Previously, students counted forward by one and ten within 100 in the Choral Counting routine. They learned that 10 ones make a unit called a ten and that a teen number is a ten and some ones. Here, as they count and group quantities, students generalize the structure of two-digit numbers in terms of the number of tens and ones. This understanding enables students to transition from counting by one to counting by ten and then counting on. For example, to count to 73, they may count 7 tens and count on—71, 72, 73.”

• Unit 7, Geometry and Time, Unit Overview, Unit Learning Goals, full Unit Narrative, 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). Here, students extend those experiences as they work with shape cards, pattern blocks, geoblocks, and solid shapes. They develop increasingly precise vocabulary as they use defining attributes (“squares have four equal length sides”) rather than non-defining attributes (“the square is blue”) to describe why a specific shape belongs to a given category. Students should, however, focus on manipulating, comparing, and composing shapes and using their own language, rather than learning the formal definitions of shapes.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

Instructional approaches of the program are described within the Curriculum Guide, Why is the curriculum designed this way? Design Principles. “It is our intent to create a problem-based curriculum that fosters the development of mathematics learning communities in classrooms, gives students access to the materials through coherent progression, and provides teachers the opportunity to deepen their knowledge of mathematics, student thinking, and their own teaching practice. Examples from the Design Principles include:

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

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

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

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

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

• Curriculum Guide, Why is the curriculum designed this way?, Further Reading, 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.” Unit 3, “Russell, S.J., Schifter D., & Bastable, V. (2011). 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, Rethinking Instruction for Lasting Understanding: An Example. In this blog post, Nowak uses the progression of inequalities as an example of how to build reliable mathematical understanding.

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

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

• Course Overview, Grade Level Resources, Grade 1 Materials List, contains a comprehensive chart of all materials needed for the curriculum. It includes the materials used throughout the curriculum, whether they are reusable or consumable, quantity needed, lessons the materials are used in, and suitable substitutes for the materials. Each lesson listed in the chart and any additional virtual materials noted for a lesson are digitally linked in the materials for quick access.10 frames are a reusable material used in lessons 1.2.1, 1.2.2, 1.2.3, 1.2.4, 1.2.6, 1.2.9, (1.2.10), 1.2.11, 1.2.12, 1.2.13, 1.2.15, (1.2.16), 1.2.18, 1.2.19, 1.2.20…. 30 10 frames per 30 students. Drawn frames on a piece of paper are suitable substitutes. Centimeter cubes are a reusable material used in lessons 1.2.16, (1.2.21), (1.3.7), (1.3.14), 1.3.17, (1.3.21), 1.3.22, 1.3.23, (1.3.27), 1.4.7, 1.4.9, 1.4.10, (1.4.12), (1.4.13), 1.4.16, (1.4.22), 1.5.6, (1.5.8), and (1.5.13). 960 centimeter cubes are needed for 30 students. No suitable substitutes for the material are listed. Number cards 0-10 are a reusable material used in lessons 2.2.18, 2.3.14, 2.4.1, and 2.5.14. 15 are needed per 30 students. A spinner with numbers 0-10 cut from paper with a paperclip is a suitable substitute. 

• Course Overview, Grade Level Resources, Grade 1 Picture Books, contains a “list of suggested picture books to read throughout the curriculum.” Unit 3, The Sky Painter by Margarita Engle is used. Unit 6, Sadako and the Thousand Paper Cranes by Eleanor Coer is used.

• Unit 5, Adding Within 100, Lesson 2, Activity 3, Teaching Notes, Materials to gather, “Paper clips, Two-color counters, Five in a Row Addition and Subtraction Stage 5 Gameboard.” Launch, “Give each group two paper clips, a gameboard, and two-color counters. We are going to learn a new way to play Five in a Row. Display the gameboard. The first player chooses one number from each row to add together. They place a paperclip on each number. Demonstrate putting a paperclip on a one-digit number and on a two-digit number. Then that player finds the sum of the numbers and puts a counter on the sum on the gameboard. Demonstrate finding the sum of the two numbers and placing the counter on the gameboard.  The next player only moves one of the paper clips to a new number. Then they find the sum of their two numbers and cover it with a counter on the gameboard. Continue taking turns moving one paper clip and covering numbers on the gameboard until someone gets five counters in a row. They are the winner.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 meet expectations for having assessment information included in the materials to indicate which standards are assessed. The materials consistently identify the content standards assessed for formal assessments, and the materials provide guidance, including the identification of specific lessons, as to how the mathematical practices can be assessed across the series.

End-of-Unit Assessments and End-of-Course Assessments consistently and accurately identify grade-level content standards within each End-of-Unit Assessment answer key. Examples from formal assessments include:

• Unit 1, Adding, Subtracting, and Working with Data, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 2, 1.OA.6, “Find the value of each expression. a. 3+6. b. 7-5. c. 10-6.”

• Unit 6, Length Measurements Within 120 Units, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 3, 1.MD.1, “The straw is longer than the pencil. The noodle is shorter than the pencil. Circle 2 true statements. A. The straw is longer than the noodle. B.The straw is shorter than the noodle. C. The noodle is longer than the straw. D. The noodle is shorter than the straw.”

• Unit 8, Putting it All Together, End-of-Course Assessment answer key, 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 Curriculum Guide, How do you assess progress?, Standards For Mathematical Practice, “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 Curriculum Guide, How do you assess progress?, Standards For Mathematical Practice, Standards for Mathematical Practices Chart, Grade 1, MP2 is found in Unit 4, Lessons 4, 8, 17, and 19. 

• IM K-5 Curriculum Guide, How do you assess progress?, Standards For Mathematical Practice, Standards for Mathematical Practices Chart, Grade 1, MP6 is found in Unit 7, Lessons 3, 4, 5, 6, 9 and 10. 

• IM K-5 Curriculum Guide, How do you assess progress?, Standards For Mathematical Practice, 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 Curriculum Guide, How do you assess progress?, Standards For Mathematical Practice, Standards for Mathematical Practice Student Facing Learning Targets, “MP8 I Can Look for and Express Regularity in Repeated Reasoning. I can identify and describe patterns and things that repeat. I can notice what changes and what stays the same when working with shapes, diagrams, or finding the value of expressions. I can use patterns to come up with a general rule.”

The materials reviewed for Imagine Learning 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 instructional tasks, practice problems, and checklists in each section of each unit. 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 of summative assessment items include:

• Unit 2, Addition and Subtraction Story Problems, End-of-Unit Assessment problems support the full intent of MP7, look for and make use of structure, as students choose equations which match an add to, change unknown story problem. Problem 4, “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 6, Length Measurements Within 120 Units, End-of-Unit Assessment, develops the full intent of 1.MD.1, order three objects by length; compare the lengths of two objects indirectly by using a third object. Problem 1, Three images of different size rectangles are provided. “a. Write a sentence comparing the length of Rectangle A and the length of Rectangle B. b. Write a sentence comparing the length of Rectangle A and the length of Rectangle C.”

• Unit 7, Geometry and Time, End-of-Unit Assessment, develops the full intent of 1.G.1, distinguish between defining attributes versus non-defining attributes; build and draw shapes to possess defining attributes. Problem 1, “Circle the 3 shapes that are triangles. (5 images of shapes that include defining and non-defining attributes),” and Problem 2, “a. Draw a square. Label it with an S. b. Draw a rectangle. Label it with an R. c. Draw a triangle. Label it with a T.”

• 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 use two different methods to subtract from a teen number. Problem 8, “a. Kiran says “15-11 is 4. I counted 14, 13, 12, 11. That’s 4.” Explain why Kiran is correct. b. Elena says “15-11 is 4. I counted 12, 13, 14, 15 to get 15 so that’s 4.” Explain why Elena is correct.”

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

Examples include:

• Unit 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 6, Length Measurements Within 120 Units, Lesson 15, Activity 2, Narrative, Access for Students with Disabilities, “Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words. Keep a display of directions visible throughout the activity. Supports accessibility for: Memory, Organization.”

• Unit 8, Putting It All Together, Lesson 4, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Access for Perception. Provide appropriate reading accommodations and supports to ensure student access to word problems and other text-based content. Supports accessibility for: Language, Visual-Spatial Processing, Attention.”

The materials reviewed for Imagine Learning 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 do you use the materials?, Practice Problems, Exploration Problems, “Each practice problem set also includes exploration questions that provide an opportunity for differentiation for students ready for more of a challenge. There are two types of exploration questions. One type is a hands-on activity that students can do directly related to the material of the unit, either in class if they have free time, or at home. The second type of exploration is more open-ended and challenging. These problems go deeper into grade-level mathematics. They are not routine or procedural, and they are not just “the same thing again but with harder numbers. Exploration questions are intended to be used on an opt-in basis by students if they finish a main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in exploration problems, and it is not expected that any student works on all of them. Exploration problems may also be good fodder for a Problem of the Week or similar structure.” 

Examples include:

• Unit 4, Numbers to 99, Section A: Units of Ten, Problem 8, Exploration, “You can use towers of 10 cubes to help you with these questions. 1. Noah has 70 cubes in towers of 10. He gave some towers of 10 to Clare. Then he gave some towers of 10 to Andre. Now Noah has no cubes left. What is one way Noah could have done this? Show your thinking using drawings, numbers, or words. Write equations to represent the problem. 2. What is another way Noah could have done this? Show your thinking using drawings, numbers, or words. Write equations to represent the problem. 3. Diego has 10 cubes in a tower. Elena gave him some more towers of 10. Then Mai gave him some more towers of 10. Now Diego has 60 cubes in towers of 10. What is one way this could have happened? Show your thinking using drawings, numbers, or words. Write equations to represent the problem. 4. What is another way this could have happened? Show your thinking using drawings, numbers, or words. Write equations to represent the problem.”

• Unit 6, Length Measurements Within 120 Units, Section B: Measure by Iterating up to 120 Length Units, Problem 6, Exploration, “Priya and Noah want to measure their classroom in steps. Priya takes 28 steps to cross the room and Noah takes 26 steps. 1. How could Priya and Noah get different measurements? 2. Measure the length of your classroom in steps.”

• Unit 7, Geometry and Time, Section A: Flat and Solid Shapes, Problem 12, Exploration, “1. What is the smallest number of pattern blocks you can use to fill in the puzzle? 2. What is the largest number of pattern blocks you can use to fill in the puzzle? 3. Can you fill in the puzzle using exactly 12 pattern blocks?”

The materials reviewed for Imagine Learning 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 Curriculum Guide, How do the materials support all learners? Mathematical language development, “Embedded within the curriculum are instructional routines and supports to help teachers address the specialized academic language demands when planning and delivering lessons, including the demands of reading, writing, speaking, listening, conversing, and representing in math (Aguirre & Bunch, 2012). While these instructional routines and supports can and should be used to support all students learning mathematics, they are particularly well-suited to meet the needs of linguistically and culturally diverse students who are learning mathematics while simultaneously acquiring English. Mathematical Language Routines (MLR) are also included in each lesson’s Support for English learners, to provide teachers with additional language strategies to meet the individual needs of their students. Teachers can use the suggested MLRs as appropriate to provide students with access to an activity without reducing the mathematical demand of the task. When selecting from these supports, teachers should take into account the language demands of the specific activity and the language needed to engage the content more broadly, in relation to their students’ current ways of using language to communicate ideas as well as their students’ English language proficiency. Using these supports can help maintain student engagement in mathematical discourse and ensure that struggle remains productive. All of the supports are designed to be used as needed, and use should fade out as students develop understanding and fluency with the English language.” The series provides principles that promote mathematical language use and development: 

• Principle 1. Support sense-making: Scaffold tasks and amplify language so students can make their own meaning. 

• Principle 2. Optimize output: Strengthen opportunities for students to describe their mathematical thinking to others, orally, visually, and in writing. 

• Principle 3. Cultivate conversation: Strengthen opportunities for constructive mathematical conversations. 

• Principle 4. Maximize meta-awareness: Strengthen the meta-connections and distinctions between mathematical ideas, reasoning, and language. 

The series also provides Mathematical Language Routines (MLR) in each lesson. Curriculum Guide, How do the materials support all learners? Mathematical language development, “A mathematical language routine is a structured but adaptable format for amplifying, assessing, and developing students' language. The MLRs were selected because they are effective and practical for simultaneously learning mathematical practices, content, and language. These routines can be adapted and incorporated across lessons in each unit to fit the mathematical work wherever there are productive opportunities to support students in using and improving their English and disciplinary language use. These routines facilitate attention to student language in ways that support in-the-moment teacher, peer, and self-assessment. The feedback enabled by these routines will help students revise and refine not only the way they organize and communicate their own ideas, but also ask questions to clarify their understanding of others’ ideas.” Examples include:

• Unit 2, Addition and Subtraction Story Problems, Lesson 3, Synthesis, Teaching Notes, Access for English Learners, “MLR7 Compare and Connect. Synthesis: After all methods have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, How were the different methods the same? How were they different?” Advances: Representing, Conversing.”

• Unit 3, Adding and Subtracting Within 20, Lesson 2, Activity 1, Teaching Notes, Access for English Learners, “MLR6 Three Reads. Keep books or devices closed. To launch this activity, display only the problem stem, without revealing the question. “We are going to read this story problem three times. After the 1st Read: Tell your partner what happened in the story. After the 2nd Read: What are all the things we can count in this story? Reveal the question. After the 3rd Read: What are different ways we can solve this problem? Advances: Reading, Representing.”

• Unit 7, Geometry and Time, Lesson 2, Activity 1, Teaching Notes, Access for English Learners, “MLR8 Discussion Supports. Display sentence frames to support small-group discussion: I made a… and The shapes I used were… Advances: Speaking, Conversing.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

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

• Unit 4, Numbers to 99, Lesson 20, Activity 1, students use connecting cubes in towers of 10 and singles to make given numbers with different combinations of tens and ones. Launch, “Groups of 2. Give each group access to connecting cubes in towers of 10 and singles. Activity, ‘Today’s challenge is to find as many ways as you can to make 94 using tens and ones. You can use cubes if they will help you. Each way you make 94 should have a different number of tens.’” 

• Unit 6, Length Measurements Within 120 Units, Lesson 4, Activity 2, students use Target Numbers, Five in a Row and Get Your Numbers in Order to add and subtraction of two digit numbers. Launch, “‘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.”

• Unit 7, Geometry and Time, Lesson 7, Activity 1, students use pattern blocks to compose two-dimensional shapes into larger shapes in different ways. Launch, “Give students pattern blocks and the flat shape puzzles. Activity, ‘Use the pattern blocks to fill the outline in different ways. Each time, record how you filled in the shape with pictures, numbers, or words.’”