2nd Grade - Gateway 3

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 2 include: extending understanding of the base-ten number system, building fluency with addition and subtraction, using standard units of measure, and describing and analyzing shapes.”

• Unit 3, Measuring Length, Section A, Metric Measurement, Section Overview, Section Narrative, provides an overview of the content and expectations for the section. “This section introduces two metric units: centimeter and meter. Students use base-ten blocks, which have lengths of 1 centimeter and 10 centimeters, to measure objects in the classroom and to create their own centimeter ruler. Students iterate the 1-centimeter unit Just as they had done with non-standard units in grade 1. Students relate the side length of a centimeter cube to the distance between tick marks on their ruler. They see that each tick mark notes the distance in centimeters from the 0 mark, and that the length units accumulate as they move along the ruler and away from 0. Students then compare the ruler they created to a standard centimeter ruler. They learn the importance of placing the end of an object at 0 and discuss how the numbers on the ruler represent lengths from 0. Students also learn about a longer unit in the metric system, meter, and use it to estimate lengths. They have opportunities to choose measurement tools and to do so strategically (MP5), by considering the lengths of objects being measured. Students also measure the length of longer objects in both centimeters and meters, which prompts them to relate the size of the unit to the measurement. To close the section, students apply their knowledge of measurement to compare the lengths of objects and solve Compare story problems involving lengths within 100, measured in metric units.”

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 5, Numbers to 1,000, Lesson 5, provides teachers guidance on how to represent numbers up to 1,000. Launch, “Display one statement. ‘Give me a signal when you know whether the statement is true and can explain how you know.’ 1 minute: quiet think time.” Activity, “‘Share and record answers and strategy.’ Repeat with each statement.” Activity Synthesis, “What is different about the last equation? (It’s not decomposed into hundreds, tens, and ones. 22 shows some tens and some ones and 10 shows another ten).”

• Unit 7, Adding and Subtracting within 1,000, Lesson 12, Lesson Synthesis provides teachers guidance on ways to decompose 10 to subtract within 1,000. “Today we saw that we can subtract by place with larger numbers, and sometimes a ten is decomposed. How did you know when a ten would be decomposed when you subtracted three-digit numbers? (I could tell when I looked at the ones place and saw I didn't have enough ones to subtract ones from ones.) How was this the same as when you subtracted two-digit numbers? How was it different? (It was just like when we subtracted two-digit numbers. It's different because one of the numbers has hundreds.)”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 4, The Nuances of Understanding a Fraction as a Number supports teachers with context for work beyond the grade. “In this blog post, Gray discusses the role the number line plays in student understanding of fractions as numbers.”

• Why is the curriculum designed this way? Further Reading, Unit 8, What is Multiplication?, “In this blog post, McCallum discusses multiplication beyond repeated addition—as equal groups. The foundation of this understanding is laid in this unit of grade 2.”

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 16, Solve All Kinds of Compare Problems, “The number choices in the Compare problems in this lesson encourage students to use methods based on place value to find the unknown value. Students may look for ways to compose a ten or subtract multiples of ten when finding unknown values within 100. Students will subtract numbers other than multiples of ten within 100 in future lessons. Encourage students to use a tape diagram to make sense of the problem if it is helpful.”

• Unit 5, Numbers to 1,000, Lesson 14, Hundreds of Objects, About this Lesson, “This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling. In this lesson, students build on their previous understandings and experiences with representations of numbers between 100 and 999. Students use their understanding of the base-ten structure of numbers to count and represent quantities of real-world objects (MP7). When students investigate the advantages and disadvantages of different methods of counting a large number of objects and then choose a method to use they critique the reasoning of others and model with mathematics (MP3, MP4).”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 2 standards breakdown, standards are addressed by lesson. Teachers can search a standard in the grade and identify the lesson(s) where it appears within materials.

• Course Guide, Lesson Standards, includes all Grade 2 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 8, Equal Groups, Lesson 10, the Core Standards are identified as 2.NBT.A.2, 2.OA.B.2, 2.OA.C.3, and 2.OA.C.4. Lessons contain a consistent structure that includes a Warm-up with a Narrative, Launch, Activity, Activity Synthesis. An Activity 1, 2, or 3 that includes Narrative, Launch, Activity, Activity Synthesis, Lesson Synthesis. A Cool-down that includes Responding to Student Thinking and Next Day Supports. This provides an additional place to reference standards, and language of the standard, within each lesson.

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

• Unit 5, Numbers to 1,000, 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 extend their knowledge of the units in the base-ten system to include hundreds. In grade 1, students learned that a ten is a unit made up of 10 ones, and two-digit numbers are formed using units of tens and ones. Here, they learn that a hundred is a unit made up of 10 tens, and three-digit numbers are formed using units of hundreds, tens, and ones. To make sense of numbers in different ways and to build flexibility in reasoning with them, students work with a variety of representations: base-ten blocks, base-ten diagrams or drawings, number lines, expressions, and equations.”

• Unit 8, Equal Groups, Unit Learning Goals, full Unit Narrative, include an overview of how the math of this module builds from previous work in math. “In this unit, students develop an understanding of equal groups, building on their experiences with skip-counting and with finding the sums of equal addends. The work here serves as the foundation for multiplication and division in grade 3 and beyond. Students begin by analyzing even and odd numbers of objects. They learn that any even number can be split into 2 equal groups or into groups of 2, with no objects left over. Students use visual patterns to identify whether numbers of objects are even or odd. Next, students learn about rectangular arrays. They describe arrays using mathematical terms (rows and columns). Students see the total number of objects as a sum of the objects in each row and as a sum of the objects in each column, which they express by writing equations with equal addends. They also recognize that there are many ways of seeing the equal groups in an array.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

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

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

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

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

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

• Curriculum Guide, Why is the curriculum designed this way?, Further Reading, Unit 4, “To learn more about the essential nature of the number line (which is introduced in this unit) in mathematics beyond grade 2, see: The Nuances of Understanding a Fraction as a Number. In this blog post, Gray discusses the role the number line plays in student understanding of fractions as numbers. Why is 3–5=3+(-5)? In this blog post, McCallum discusses the use of the number line in introducing negative numbers.” Unit 8, “What is Multiplication? In this blog post, McCallum discusses multiplication beyond repeated addition—as equal groups. The foundation of this understanding is laid in this unit of grade 2.”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 2 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. Pattern blocks are reusable materials used in lessons 2.6.6, 2.6.10, and 2.6.21. Included in the pattern blocks list is a note that 180 triangles and 120 each of other shapes are needed per 30 students. Cut out shapes from paper or cardstock and Virtual Pattern Blocks are suitable substitutes. Inch tiles are a reusable material used in lessons 2.3.8, 2.3.9, and 2.4.1. 360 inch tiles are needed for 30 students. Cut out inch squares from grid paper and Virtual Grid Paper are suitable substitutes for the material. Sticky notes are a consumable material used in lessons 2.2.18, 2.3.14, 2.4.1, and 2.5.14. Sticky notes are needed per 30 students. No suitable substitute is listed.

• Unit 4, Addition and Subtraction on the Number Line, Lesson 4, Activity 1: Compare the Numbers, Teaching Notes, Materials to gather, “Give each group 3 number cubes and 2 counters.” Activity, “You will use the number line you created and work with a partner. Decide with your partner whose number line you will use. As needed, demonstrate the task with a student. I am Partner A. I am going to roll the 3 number cubes and find the sum. Then, I take a counter and place it on the number line to represent the sum. Now it’s my partner's turn. They do the same thing and put their counter on the same number line to represent the sum of their numbers. Then, we decide which number is greater and explain how we know. Last, we use the  <, =, or > symbols to record our comparison.”

• Unit 6, Geometry, Time, and Money, Lesson 7, Activity 1, Teaching Notes, Materials to gather, “Construction paper, Rulers, Scissors.” Launch, “Give each student 3 paper rectangles and access to scissors and rulers. In an earlier lesson, we thought about how shapes could be composed using equal-size smaller shapes. Today, we are going to decompose shapes into equal pieces and name the pieces. Each of you has 3 rectangles. First, cut out each rectangle. Next, fold each rectangle in different ways. You can use a ruler to draw lines first, if it is helpful. You each have 2 pieces. How can you check to see if they are equal? (If you lay them on top of each other, they are the same size.)” Activity, “Have extra paper on hand if students want to try again when making thirds.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 3, Adding and Subtracting within 100, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 3, 2.NBT.5, “Find the number that makes each equation true. Show your thinking using drawings, numbers, or words. a. 23+19=___. b. 75-36= ___.”

• Unit 4, Addition and Subtraction on the Number Line, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 3, 2.MD.6 and 2.NBT.5, “a. Locate and label 43 and 38 on the number line. (number line image with intervals of 5 denoted up to 50.); b. Explain how to use the number line to find the value of 43-38.” Image of a number line with intervals of 5 denoted up to 50 with 43 and 38 plotted.

• Unit 9, Putting it All Together,  End-of-Course Assessment answer key, denotes standards addressed for each problem. Problem 9, 2.NBT.7, “Find the value of each expression. Show your thinking. a. 347+583. b. 612-174.”

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 2, MP1 is found in Unit 3, Lessons 6, 11, and 12. 

• IM K-5 Curriculum Guide How do you assess progress?, Standards For Mathematical Practice, Standards for Mathematical Practices Chart, Grade 2, MP7 is found in Unit 9, Lessons 2, 3, 5, 6, and 9. 

• IM K-5 Curriculum Guide How do you assess progress?, Standards For Mathematical Practice, Standards for Mathematical Practice Student Facing Learning Targets, “MP3 I Can Construct Viable Arguments and Critique the Reasoning of Others. I can explain or show my reasoning in a way that makes sense to others. I can listen to and read the work of others and offer feedback to help clarify or improve the work. I can come up with an idea and explain whether that idea is true.”

• IM K-5 Curriculum Guide, How do you assess progress?, Standards For Mathematical Practice, Standards for Mathematical Practice Student Facing Learning Targets, “MP6 I Can Attend to Precision. I can use units or labels appropriately. I can communicate my reasoning using mathematical vocabulary and symbols. I can explain carefully so that others understand my thinking. I can decide if an answer makes sense for a problem.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 1, Adding, Subtracting, and Working with Data, End-of-Unit Assessment problems support the full intent of 2.OA.2, fluently add and subtract within 20 using mental strategies. Problem 3, “Find the number that makes each equation true. a. 7+___$$=18$$. b. 20-___$$=12$$. c. 9+7=___. d. 9+7=___. e. 19-14=___.”

• Unit 3, Measuring Length, End-of-Unit Assessment problems support the full intent of MP6, attend to precision, as students measure each of the rectangles and determine how much longer one rectangle is than the other. Problem 4, “How many centimeters longer is rectangle A than rectangle B? Explain or show your reasoning. (Images included of 2 rectangles that are different lengths).”

• Unit 6, Geometry, Time, and Money, End-of-Unit Assessment, develops the full intent of 2.G.1, recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Students identify triangles, quadrilaterals, pentagons, hexagons, and cubes. Problem 1, “Draw a quadrilateral with one square corner and two equal sides.” and Problem 2, “Choose the name of the shape. (image of a pentagon) A. Hexagon, B. Triangle, C. Quadrilateral, D. Pentagon”

• Unit 9, Putting It All Together, End-of-Course Assessment supports the full intent of MP2, reason abstractly and quantitatively, as students compare numbers within 1,000. Problem 2, “Fill in each blank with <, =, or > to make the statements true. a. 675 ___ 576 b. 98 ___ 205 c. 500+40+3___$$543$$ d. 675___400+70+1.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 3 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 4, Relating Multiplication to Division, Lesson 3, Activity 3, Narrative, Access for Students with Disabilities, “Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select at least 3 of the 6 problems to complete. Supports accessibility for: Organization, Attention, Social-emotional skills.”

• Unit 5, Fractions as Numbers, Lesson 15, Activity 2, Narrative, Access for Students with Disabilities, “Representation: Access for Perception. To support understanding, begin by demonstrating how to play one round of “Spin to Win.” Supports accessibility for: Memory, Social-Emotional Functioning.”

• Unit 8, Putting It All Together, Lesson 3, Activity 1, Narrative, Access for Students with Disabilities, “Engagement: Develop Effort and Persistence: Chunk this task into more manageable parts. Check in with students to provide feedback and encouragement after each round. Supports accessibility for: Organization, Focus.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 2, Adding and Subtraction Within 100, Section B: Decompose to Subtract, Problem 7, Exploration, “Here is Jada’s method for finding the value of 73-58. 73-60=13. 13+2. 1. Explain why Jada’s method works. 2. Use Jada’s method to find the value of 85-49.”

• Unit 4, Addition and Subtraction on the Number Line, Section A: The Structure of the Number Line, Problem 11, Exploration, “1. Here is a picture of a thermometer. How is the thermometer the same as a number line? How is it different? 2. Here is a picture of a rain gauge. How is the rain gauge the same as a number line? How is it different?”

• Unit 5, Numbers to 1000, Lesson 12, Section A: The Value of Three Digits, Problem 10, Exploration, “1. Can you represent the number 218 without using any hundreds? Explain your reasoning. 2. Can you represent the number 218 without using any tens? Explain your reasoning. 3. Can you represent the number 218 without using any ones? Explain your reasoning.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 4, Addition and Subtraction on the Number Line, Lesson 3, Activity 1, Teaching Notes, Access for English Learners, “MLR2 Collect and Display. Collect the language students use as they work with the number lines and discuss the number patterns. Display words, phrases, and representations such as: number line, distance from zero, in order, interval, spaces, tick mark, point, and pattern. During the synthesis, invite students to suggest ways to update the display: What are some other words or phrases we should include? Invite students to borrow language from the display as needed. Advances: Conversing, Reading.”

• Unit 5, Numbers to 1,000, Lesson 6, Activity 2, Teaching Notes, Access for English Learners, “MLR7 Compare and Connect. Synthesis: After the Gallery Walk, lead a discussion comparing, contrasting, and connecting the different representations of numbers. To amplify student language and illustrate connections, follow along and point to the relevant parts of the displays as students speak. Advances: Representing, Conversing.”

• Unit 6, Geometry, Time, and Money, Lesson 2, Activity 2, Teaching Notes, English Learners, “MLR8 Discussion Supports. Invite students to begin partner interactions by repeating the question, Which attributes match the shape I drew? This gives both students an opportunity to produce language. Advances: Conversing.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 2, Add and Subtract within 100, Lesson 1, Activity 2, students use connecting cubes to compare problems. Launch, “Give students access to towers of ten and loose connecting cubes. Display the image of the cubes. ‘What do you notice? What do you wonder?’ (Lin has more cubes. They have 40 cubes all together. Lin has ten more cubes.) Monitor for students who notice the groups of ten cubes and use this structure to find the total number of cubes or the difference.”

• Unit 8, Equal Groups, Lesson 7, Activity 1, students use counters to create arrays. Launch, “Groups of 2. Give each group 3 sets of counters with 6, 7, and 9. Display A from the warm-up or arrange counters to show: (Image of 4 rows: first and third row have 4 counters, second and fourth row have 2 counters.) ‘The red counters are arranged in rows, but it is not an array. How could we rearrange the counters to make an array like image B?’ (We could move the bottom two counters to the middle row. We could move one from the top row to the next row. We could move 1 from the third row to the bottom row.)” Activity, “‘Arrange each of your sets of counters into an array. Your arrays should have the same number of counters in each row with no extra counters.’ Be prepared to explain how you made an array out of each set. ‘If you have time, try to figure out a different way to make an array out of each set of counters.’”

• Unit 9, Putting It All Together, Lesson 8, Activity 1, students play a game called Heads Up to add and subtract within 100. Launch, “Give students number cards. Activity, ‘We are going to play a game called Heads Up.’ Demonstrate with 2 students. ‘Players A and B pick a card and put it on their foreheads without looking at it. I am Player C. My job is to find the value of the sum and tell my group. Players A and B use the other player’s number and the value of the sum to determine what number is on their head. Finally, each player writes the equation that represents what they did. Demonstrate writing an equation for each of the players. After each round switch roles and play again.’”