3rd Grade - Gateway 3

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 3 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 3 include: developing understanding of multiplication and division and strategies for multiplication and division within 100; developing understanding of fractions, especially unit fractions (fractions with numerator 1); developing understanding of the structure of rectangular arrays and of area; and describing and analyzing two-dimensional shapes.”

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Section A, Add Within 1,000, Section Overview, Section Narrative, provides an overview of the content and expectations for the section. “Students begin this section by revisiting the idea of place value, reasoning about different ways to decompose numbers within 1,000, and using familiar strategies from grade 2 to add and subtract within 1,000. From there, they progress toward more abstract addition strategies, but ones that are still based on place value. To support this progression toward algorithms, students use base-ten blocks or diagrams, express numbers in expanded form, and rely on their understanding of properties of operations. For example, here are three ways to add 362+354: (image of addition using base ten blocks, expanded form, and partial sums.) Students look for and make use of structure as they relate the compositions of numbers, expressions, and base-ten blocks or diagrams to find sums and differences (MP7).” 

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, Fractions as Numbers, Lesson 17, Warm-up, provides teachers guidance on how to work with estimation and fractions. Launch, “Groups of 2. Display the image. ‘What is an estimate that’s too high? Too low? About right?’ 1 minute: quiet think time.” Activity, “‘Discuss your thinking with your partner.’ 1 minute: partner discussion. ‘Record responses.’” Activity Synthesis, “Consider asking: Is anyone’s estimate less than _____? Is anyone’s estimate greater than _____? Based on this discussion, does anyone want to revise their estimate?”

• Unit 8, Putting It All Together, Lesson 10, Lesson Synthesis provides teachers guidance on closing the lesson with representations of multiplication and division. “Today we created posters that showed ways to represent division. How does an area diagram show us the relationship between multiplication and division? (It shows that multiplying is like finding the area of a rectangle when the two side lengths are known, and dividing is like finding a side length when we know the area and the other side length.) How does a tape diagram or equal-groups diagram show multiplication and division? (Both show multiplying as a way to find the total when we know the number of groups and how many in each group, and dividing as a way to find either the number of groups or the size of each group when the total is known.) What were some aspects of the posters you saw that helped make the math your classmates used clear for you? (Clear labels on diagrams helped me understand their thinking. Units on their answers. When other students wrote their explanations, it helped me understand their thinking.)”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 3 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 1, Ratio Tables are not Elementary, supports teachers with context for work beyond the grade. “In this blog post, McCallum discusses the difference between multiplication tables and tables of equivalent ratios, highlighting how K–5 arithmetic work prepares students to make sense of these tables.”

• Why is the curriculum designed this way? Further Reading, Unit 5, “Fractions: Units and Equivalence. In this blog post, McCallum discusses equivalent fractions as the same numbers in different units.”

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 16, Design A Carnival, About this Lesson, “This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling. In this lesson, students continue to work with the context of a fair. Students analyze games they might see at a carnival such as a penny toss or marble run and consider what makes a good game. They then create their own games with given materials and integrate mathematical ideas from this unit. Students play the game and consider ways to improve it. When students make choices about quantities and rules, analyze constraints in situations, and adjust their work to meet constraints, they model with mathematics (MP4).”

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 1, About this Lesson, “In previous grades, students sorted shapes into categories based on the attributes of the shape. In this lesson, students revisit this work and learn the terms angle in a shape and right angle in a shape to describe the corners of shapes. This will be helpful in later lessons as students further sort triangles and rectangles by additional attributes. Throughout the lesson, if students have trouble determining if sides have the same length, offer rulers to measure the side lengths.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 3 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 3 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 3 standards and the units and lessons each standard appears in. 

• 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 5, Fractions as Numbers, Lesson 5, the Core Standards are identified as 3.NF.A.3 and 3.NF.A.3.d. 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, Relating Multiplication to Division, Unit Overview, Unit Learning Goals, full Unit Narrative, include an overview of how the math of this module builds from previous work in math. “This unit introduces students to the concept of division and its relationship to multiplication. Previously, students learned that multiplication can be understood in terms of equal-size groups. The expression 5\times2 can represent the total number of objects when there are 5 groups of 2 objects, or when there are 2 groups of 5 objects. Here, students make sense of division also in terms of equal-size groups. For instance, the expression 30\div5 can represent putting 30 objects into 5 equal groups, or putting 30 objects into groups of 5. They see that, in general, dividing can mean finding the size of each group, or finding the number of equal groups.”

• Unit 5, Fractions as Numbers, 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 make sense of fractions as numbers, using various diagrams to represent and reason about fractions, compare their size, and relate them to whole numbers. The denominators of the fractions explored here are limited to 2, 3, 4, 6, and 8. In grade 2, students partitioned circles and rectangles into equal parts and used the language “halves,” “thirds,” and “fourths.” Students begin this unit in a similar way, by reasoning about the size of shaded parts in shapes. Next, they create fraction strips by folding strips of paper into equal parts and later represent the strips as tape diagrams. Using fraction strips and tape diagrams to represent fractions prepare students to think about fractions more abstractly: as lengths and locations on the number line. This work builds on students’ prior experience with representing whole numbers on the number line.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 3 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 1, Ratio Tables are not Elementary. “In this blog post, McCallum discusses the difference between multiplication tables and tables of equivalent ratios, highlighting how K–5 arithmetic work prepares students to make sense of these tables.” Unit 3, “To learn more about the order of operations, see: A world without order (of operations). In this blog post, McCallum describes a world with only parentheses to guide the order of operations and discusses why the conventional order of operations is useful.” Unit 5,  “Fractions: Units and Equivalence. In this blog post, McCallum discusses equivalent fractions as the same numbers in different units.”

• 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 3 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 3 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 a reusable material used in lesson 3.2.1. 60 hexagons and trapezoids, 120 squares and rhombuses, and 240 triangles are needed per 30 students. Cut out shapes from paper or cardstock and Virtual Pattern Blocks are suitable substitutes. 

Grid paper is a consumable material used in lessons 3.2.10, 3.4.15, and 3.4.20. 180 pages (about 2 pages per activity per student) are needed for 30 students. Paper and Virtual Grid Paper are suitable substitutes for the material. Rulers (with whole units) are a reusable material used in lessons 3.2.6 and  3.2.8. 15 rulers per 30 students. A cut out ruler to scale is a suitable substitute for the material.

• Course Overview, Grade Level Resources, Grade 3 Picture Books, contains a “list of suggested picture books to read throughout the curriculum.” Unit 2, Last Stop on Market Street by Matt de la Peña and Christian Robinson is used. Unit 2, City Green by DyAnne DiSalvo-Ryan is used.

• Unit 1, Introducing Multiplication, Lesson 18, Activity 1, Teaching Notes, Materials to gather, “Connecting cubes or counters.” Launch, “Give students access to connecting cubes or counters. Take a minute to represent this situation with an array. You can use drawings or objects.” Activity, “Work with your partner to represent the next three situations with an array. Be prepared to share how you see equal groups in your array. Have students share an array for problems 2–4. Try to show both drawings and arrays made of objects.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 3 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, Introducing Multiplication, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 3, 3.OA.1, “Write a multiplication expression that could represent the number of dots in each drawing.”

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 6, 3.MD.2, “A young humpback whale weighs 835 kg. A young killer whale weighs 143 kg. How much heavier is the humpback whale than the killer whale? Explain or show your reasoning.”

• Unit 8, Putting it All Together, End-of-Course Assessment answer key, denotes standards addressed for each problem. Problem 17, 3.OA.2, “a. Write a division equation for each situation. Use “?” for the unknown quantity. i. There are 35 students in the room. They are seated at 7 tables, with the same number of students at each table. How many students are at each table? ii. There are 35 students in the room. There are 7 students seated at each table. How many tables of students are there? b. How are the situations the same? How are they different?”

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 3, MP2 is found in Unit 5, Lessons 4, 12, 15, and 17. 

• IM K-5 Curriculum Guide, How do you assess progress? Standards For Mathematical Practice, Standards for Mathematical Practices Chart, Grade 3, MP4 is found in Unit 6, Lessons 7, 8, 11, 14, and 16. 

• 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, “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 Imagine Learning Illustrative Mathematics Grade 3 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, Area and Multiplication, End-of-Unit Assessment, develops the full intent of 3.OA.7, fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8\times5=40, one knows 40\div5=8) or properties of operations. Problem 6, “Find the value of each expression. a. 4\times7. b. 3\times9. c. 6\times4. d. 5\times8.”

• Unit 4, Relating Multiplication to Division, End-of-Unit Assessment problems develops the full intent of 3.OA.5, apply properties of operations as strategies to multiply and divide. Problem 6, “Find the value of each expression. Explain or show your reasoning. a. 11\times8, b. 7\times40, c. 5\times13.”

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, End-of-Unit Assessment problems support the full intent of MP6, attend to precision, as students choose objects that weigh about 1 kilogram. Problem 2, “Select 3 items that weigh about 1 kilogram. A. pencil, B. laptop computer, C. pineapple, D. paper clip, E. car, F. dictionary.”

• Unit 8, Putting It All Together, End-of-Course Assessment problems supports the full intent of MP3, construct viable arguments and critique the reasoning of others, as students choose plants for a garden. Problem 15, “Lin's class is designing a garden at school. Their garden is a rectangle that is 8 feet by 12 feet. The table shows how far some different plants need on all sides to grow well. a. Which plant takes up the most amount of space? Which plant takes up the least amount of space? b. Andre wants to plant pumpkins. Lin says that there is not enough room. Do you agree with Lin? Explain or show your reasoning. c. How many lettuce plants can the class fit in the garden? Explain or show your reasoning.”

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 3 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, Relating Multiplication and Division, Section D: Dividing Larger Numbers, Problem 7, Exploration, “What are the different ways you can divide 48 objects into equal groups? 1. Make a list. 2. Write a multiplication or division equation for each different way.” 

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Section C: Problems Involving Time, Problem 5, Exploration, “Priya drew this clock face to show 3:15. Do you think Priya’s clock face is accurate? Explain or show your reasoning.”

• Unit 7, Two-dimensional Shapes and Perimeter, Section B: What is Perimeter?, Problem 6, Exploration, “1. Draw some different shapes that you can find the perimeter of. Then find their perimeters. 2. Can you draw a rectangle whose perimeter is odd? Explain or show your reasoning. 3. Can you draw a pentagon or hexagon (or a figure with even more sides) whose perimeter is odd?”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 3 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 1, Introducing Multiplication, Lesson 14, Activity 1, Teaching Notes, Access for English Learners, “MLR8 Discussion Supports. Invite students to take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “I noticed ___ , so I matched . . .” Encourage students to challenge each other when they disagree. Advances: Conversing, Representing.”

• Unit 4, Relating Multiplication to Division, Lesson 13, 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. “How did the number of chairs show up in each method? Why did the different approaches lead to the same outcome?” 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, Measuring Length, Time, Liquid Volume, and Weight, Lesson 1, Activity 2, Teaching notes, Access for English Learners, “MLR2 Collect and Display. Circulate, listen for and collect the language and numbers students use as they measure objects. On a visible display, record numbers, words and phrases such as: seven half inches, seven halves of an inch, \frac{7}{2}, between 2 and 3 inches, six and a half inches, 6\frac{1}{2}, and less than 5 inches. Invite students to borrow language from the display as needed, and update it throughout the lesson. Advances: Conversing, Reading.”

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 6, Activity 2, Teaching Notes, Access for English Learners, “MLR8 Discussion Supports. Synthesis: To support the transfer of new vocabulary to long-term memory, invite students to chorally repeat these phrases in unison 1–2 times: perimeter and distance around a shape. Advances: Speaking.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 3 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, Area and Multiplication, Lesson 2, Activity 1, students use inch tiles to explore area by creating shapes. Launch, “Groups of 4. Give each group inch tiles. ‘Take some tiles and build your shape.’ Activity, ‘Now work with your group to order the shapes. Be prepared to explain how you ordered the shapes.’”

• Unit 5, Fractions as Numbers, Lesson 18, Activity 1, students create a design using the fraction \frac{1}{2} as a constraint for length. Launch, “Groups of 2. ‘Let’s create a design using the fraction \frac{1}{2}. Take a minute to read the activity statement. Then, turn and talk to your partner about what you are asked to do.’ Give each student a ruler or a straightedge. Provide access to extra paper, in case requested. Activity, ‘Work with your partner to complete the activity. Use a straightedge when you draw lines to connect points.’ 10 minutes: partner work time. Monitor for different strategies and tools students use to partition the sides of the squares, such as: estimating or “eyeballing” the midpoint folding opposite sides of each square in half, copying the side length of each square onto another paper, folding it in half, and using it to mark the midpoint of all four sides using a ruler to measure.”

• Unit 8, Putting It All Together, Lesson 11, Activity 1, students play a game called Race to 1, where students practice dividing until they reach 1. Launch, “‘Let’s look at a sample game. Jada rolled a 3 on her first turn, then rolled 2 a few times afterwards. Talk with your partner about what her next move should be if she rolls 2 on her next turn.’ (She should divide 4 or 6 by 2 because those moves get her really close to one.) Give each group a number cube. Activity, ‘Play Race to 1 with your partner.’”