4th Grade - Gateway 3

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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 4 include: developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.”

• Unit 3, Extending Operations to Fractions, Section B, Addition and Subtraction of Fractions, Section Overview, Section Narrative, provides an overview of the content and expectations for the section. “In this section, students learn to add and subtract fractions by decomposing them into sums of smaller fractions, writing equivalent fractions, and using number lines to support their reasoning. Students begin by thinking about a fraction as a sum of unit fractions with the same denominator and then as a sum of other smaller fractions. They represent different ways to decompose a fraction by drawing “jumps” on number lines and writing different equations. Working with number lines helps students see that a fraction greater than 1 can be decomposed into a whole number and a fraction, and then be expressed as a mixed number. This can in turn help us add and subtract fractions with the same denominator. For example, to find the value of 3−\frac{2}{5}, it helps to first decompose the 3 into 2+\frac{5}{5}, and then subtract \frac{2}{5} from the \frac{5}{5}. Later in the section, students organize fractional length measurements ($$\frac{1}{2}$$, \frac{1}{4}, and \frac{1}{8} inch) on line plots. They apply their ability to interpret line plots and to add and subtract fractions to solve problems about measurement data.”

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, Multiplicative Comparison and Measurement, Lesson 7, Warm-up, provides teachers guidance on how to work with meters and centimeters in regards to size. Launch, “Groups of 2. Display the image. ‘What do you notice? What do you wonder?’ 1 minute: quiet think time.” Activity, “‘Discuss your thinking with your partner.’ 1 minute: partner discussion. ‘Share and record responses.’” Activity Synthesis, “Consider sharing that the large insect is a stick insect. (The longest species ever found measured more than 60 cm.) The small insect is a green potato bug. ‘If each unit in the ruler is 1 centimeter, about how long is the potato bug? (1 cm) What about the stick insect? (About 16 cm with the antennae, about 12 cm otherwise.)’”

• Unit 7, Angles and Angle Measurement, Lesson 13, Lesson Synthesis provides teachers guidance on how to help students find unknown measurements by composing or decomposing known measurements. “‘Today we used different operations to find the measurement of different angles.’ Display: ‘Here are some angles whose measurements we tried to find: anglep, angle s, and some angles composed of smaller angles. We used different operations to find the unknown measurements. Which of these angles can we find by using division? (Angle p: If we know that 2 copies ofpmake a right angle, which is 90\degree, then dividing 90\degree by 2 gives us the measure of .) Which unknown angle can we find by multiplication?’ (The angle made up of four 30° angles has a measurement of 4\times30.) ‘Which unknown angle can we find by subtracting one angle from another? (Angle s: We can subtract 30\degree from 180\degree and divide by 2 to find the measure of s, which is 75\degree.) Which unknown angle can we find by adding known angles? (Once we know the measure of angle s, we can find the last angle: 15+75+15, which is 150\degree.)’”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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, Fractions: Units and Equivalence, supports teachers with context for work beyond the grade. “In this blog post, McCallum discusses equivalent fractions as the same numbers in different units.”

• Why is the curriculum designed this way? Further Reading, Unit 7, Making Peace with the Basics of Trigonometry, “In this blog post, Phillips highlights how student exploration in trigonometry allows them to see that trigonometric ratios come from measuring real triangles, fostering conceptual understanding. This blog is included in this unit as an example of how concepts of angle come into play in mathematics beyond elementary school.”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 18, About this Lesson, “This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling. In this lesson, students write true and false statements involving multiplicative comparisons and unit conversions. They have an opportunity to choose the animals to compare and which facts to use. Then, students determine which of their classmates’ statements are true and which are false. At the end of this activity, they have an opportunity to revise their earlier statements to make them clearer or stronger. As students discuss and justify their decisions, they share a mathematical claim and the thinking behind it (MP3).”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 25, Paper Flower Decorations, 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 prior understanding and experiences with creating and analyzing patterns to solve multi-step problems in a real-world context. In the first activity, students make different types of paper flowers. In the second activity, they consider patterns and solve problems involving paper flower garlands. In the third activity, students think of their own pattern and multi-step problems inspired by their process of making paper flowers. When students ask and answer questions that arise from a given situation, use mathematical features of an object to solve a problem, make choices, analyze real-world situations with mathematical ideas, interpret a mathematical answer in context, and decide if an answer makes sense in the situation, they model with mathematics (MP4).”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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 4 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 4 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 5, From Hundredths to Hundred-thousands, Lesson 5, the Core Standards are identified as 4.NBT.A.2. 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 2, Fraction Equivalence and Comparison, 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 grade 3, students partitioned shapes into parts with equal area and expressed the area of each part as a unit fraction. They learned that any unit fraction \frac{1}{b} results from a whole partitioned into b equal parts. They used unit fractions to build non-unit fractions, including fractions greater than 1, and represent them on fraction strips and tape diagrams. The denominators of these fractions were limited to 2, 3, 4, 6, and 8. Students also worked with fractions on a number line, establishing the idea of fractions as numbers and equivalent fractions as the same point on the number line.  Here, students follow a similar progression of representations. They use fraction strips, tape diagrams, and number lines to make sense of the size of fractions, generate equivalent fractions, and compare and order fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.”

• Unit 7, Angles and Angle Measurement, 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 earlier grades, students learned about two-dimensional shapes and their attributes, which they described informally early on but with increasing precision over time. Here, students formalize their intuitive knowledge about geometric features and draw them. They identify and define some building blocks of geometry (points, lines, rays, and line segments), and develop concepts and language to more precisely describe and reason about other geometric figures. Students analyze cases where lines intersect and where they don’t, as in the case of parallel lines. They learn that an angle as a figure composed of two rays that share an endpoint. Later, students compare the size of angles and consider ways to quantify it. They learn that angles can be measured in terms of the amount of turn one ray makes relative to another ray that shares the same vertex.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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 2, “Fractions: Units and Equivalence. In this blog post, McCallum discusses equivalent fractions as the same numbers in different units.” Unit 7, ”Making Peace with the Basics of Trigonometry. In this blog post, Phillips highlights how student exploration in trigonometry allows them to see that trigonometric ratios come from measuring real triangles, fostering conceptual understanding. This blog is included in this unit as an example of how concepts of angle come into play in mathematics beyond elementary school.”

• 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 4 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 4 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. Base 10 Blocks are a reusable material used in lesson 4.4.6, 4.4.8, 4.6.16, 4.6.17, and 4.6.18. 60 hexagons and trapezoids, 15 thousands, 300 hundreds, 600 tens, 600 ones are needed per 30 students. Paper cut-outs or Virtual Base-ten blocks are suitable substitutes.  Paper clips are a reusable material used in lessons 4.2.17 and 4.5.9. 150 paper clips are needed for 30 students. No suitable substitutes for the material are listed. Grid paper is a consumable material used in lessons 4.1.1, 4.1.2, 4.1.3, 4.4.18, 4.4.19, 4.4.20, 4.4.21, 4.4.22, and (4.6.3). 15 are needed per 30 students. Paper or Virtual Grid Paper are suitable substitutes for the material.

• Unit 2, Fraction Equivalence and Comparison, Lesson 17, Activity 1: Paper Clip Tossing Game, Teaching Notes, Materials to gather, “Markers, Paper, Paper clips, Tape (painter's or masking).” Launch, “Give each group strips of paper, markers, paper clips. Work with your group to play a version of the paper-clip tossing game. To play the game, you will toss some paper clips and use fractions to label where they land. First, we’ll make the game board. Fold your paper strip in half and then in half again. Carefully tape it down to your workspace (desk or floor can work) and label the benchmark fractions.”

• Unit 7, Angles and Angle Measurement, Lesson 12, Activity 3, Teaching Notes, Materials to gather, “Pattern blocks, Protractors.” Launch, “Give students access to protractors and pattern blocks.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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 4, From Hundredths to Hundred-thousands, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 3, 4.NBT.1, “The distance between New York City and Boston is 225 miles. The distance between New York City and Salt Lake City is 10 times as far. How many miles is it between New York City and Salt Lake City? Explain or show your reasoning.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 5, 4.NBT.6, “Find the value of each quotient. Explain or show your reasoning. a. 714\div6. b. 3,626\div7.”

• Unit 9, Putting it All Together, End-of-Course Assessment answer key, denotes standards addressed for each problem. Problem 5, 4.NBT.5, 4.OA.2, “Andre ran 1,270 meters. Clare ran 3 times as far as Andre. How many meters did Clare run? Explain or show your reasoning.”

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 4, MP1 is found in Unit 3, Lessons 11, 14, 15, and 18. 

• IM K-5 Curriculum Guide How do you assess progress? Standards For Mathematical Practice, Standards for Mathematical Practices Chart, Grade 4, MP8 is found in Unit 2, Lessons 3, 5, 8, and 11. 

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

• IM K-5 Curriculum Guide, How do you assess progress? Standards For Mathematical Practice, Standards for Mathematical Practice Student Facing Learning Targets, “MP4 I Can Model with Mathematics. I can wonder about what mathematics is involved in a situation. I can come up with mathematical questions that can be asked about a situation. I can identify what questions can be answered based on data I have. I can identify information I need to know and don’t need to know to answer a question. I can collect data or explain how it could be collected. I can model a situation using a representation such as a drawing, equation, line plot, picture graph, bar graph, or a building made of blocks. I can think about the real-world implications of my model.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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, Fraction Equivalence and Comparison, End-of-Unit Assessment problems support the full intent of MP1, make sense of problems and persevere in solving them, as students compare fractions to benchmarks \frac{1}{2} and 1. Problem 2, “Select all fractions that are greater than \frac{1}{2} but less than 1. A. \frac{4}{5} B. \frac{1}{3} C. \frac{5}{4} D. \frac{4}{7} E. \frac{5}{10}.”

• Unit 4, From Hundredths to Hundred-Thousands, End-of-Unit Assessment develops the full intent of 4.NBT.4, fluently add and subtract multi-digit whole numbers using the standard algorithm. Problem 5, “Find the sum or difference. (Both problems are written vertically.) a. 324,567+34,762; b. 827,419-80,125.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, End-of-Unit Assessment develops the full intent of 4.OA.3, solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted, represent these problems using equations with a letter standing for the unknown quantity, and assess the reasonableness of answers using mental computation and estimation strategies including rounding. Problem 7, “Jada’s family is getting soil for a garden. The garden will be 160 square feet. Each bag of soil covers 6 square feet. How many bags of soil does Jada’s family need? Explain or show your reasoning.”

• Unit 9, Putting It All Together, End-of-Course Assessment problems supports the full intent of MP4, model with mathematics, as students find a whole number multiple of a fraction and a difference of a whole number and a fraction. Problem 13, “a. Mai’s house is \frac{5}{8} mile from school. She walked to school all 5 days of the week. How many miles did Mai walk altogether from home to school? Explain or show your reasoning. b. Mai wants to walk 6 miles total for the week. How much farther does she need to walk? Explain or show your reasoning.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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 7, Angles and Angle Measurement, Lesson 9, Activity 2, Narrative, Access for Students with Disabilities, “Action and Expression: Internalize Executive Functions. Invite students to estimate the size of the angle before finding each precise measurement. Offer the sentence frame: “This angle will be greater than _____ and less than _____. It will be closer to _____.” Supports accessibility for: Conceptual Processing, Visual-Spatial Processing, Attention”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 2, Activity 1, Narrative, Access for Students with Disabilities, “Action and Expression: Internalize Executive Functions. Invite students to discuss the steps they might take to complete the task. For example, students may decide to look at one triangle at a time and decide which attributes it has. Alternatively, they may decide to look at each triangle through the lens of one attribute at a time. Supports accessibility for: Organization.”

• Unit 9, Putting It All Together, Lesson 6, Activity 1, Narrative, Access for Students with Disabilities, “Engagement: Provide Access by Recruiting Interest. Provide choice. Tell students they will be finding the value of 7,465\div5, and that there are four unfinished strategies to look at. Invite students to choose whether they want to solve it in their own way or look at the unfinished strategies first. Supports accessibility for: Organization, Attention, Social-Emotional Functioning.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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, Fraction Equivalence and Comparison, Section C: Fraction Comparison, Problem 7, Exploration, “Jada lists these fractions that are all equivalent to \frac{1}{2}: \frac{2}{4}, \frac{3}{6}, \frac{4}{8}, \frac{5}{10}. She notices that each time the numerator increases by 1 and the denominator increases by 2. Will the pattern Jada notices continue? Explain your reasoning.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Section A: Features of Patterns, Problem 11, Exploration, “Tyler draws this picture and writes the equation 1+3+5=9. 1. How do you think the equation relates to the picture? 2. Tyler keeps drawing circles to make larger squares. How many new circles does he need to draw to make a 4-by-4 square, and then a 5-by-5 square? 3. What pattern do you notice in the number of circles Tyler adds each time? 4. Why do you think the number of circles is increasing that way?”

• Unit 7, Angles and Angle Measurement, Section A: Points, Lines, Segments, Rays, and Angles, Problem 8, Exploration, “Here is a riddle. Can you solve it? “I am a capital letter made of more than 1 segment with no curved parts. I have no perpendicular segments or parallel segments. What letter could I be?’”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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, Fraction Equivalence and Comparison, Lesson 13, Synthesis, Teaching Notes, Access for English Learners, “MLR7 Compare and Connect. Synthesis: After each strategy has been presented, lead a whole-class discussion comparing, contrasting, and connecting the different approaches. Ask, “Did anyone solve the problem the same way, but would explain it differently?” and “Why did the different approaches lead to the same outcome?” Advances: Representing, Conversing.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 4, Activity 1, Teaching Notes, Access for English Learners, “MLR8 Discussion Supports. Use multimodal examples to show the patterns of both columns. Use verbal descriptions along with gestures, drawings, or concrete objects to show the connection between the multiples of 9 and 10. Advances: Listening, Representing.”

• Unit 7, Angles and Angle Measurement, Lesson 10, Activity 1, Teaching Notes, Access for English Learners, “MLR8 Discussion Supports. Synthesis: For each strategy that is shared using the protractor, invite students to turn to a partner and restate what they heard using precise mathematical language. Advances: Listening, Speaking.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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, Extending Operations to Fractions, Lesson 2, Activity 1, students use their understanding of equivalent fractions and decimals by sorting a set of cards by their value. Launch, “Groups of 2. Give each group a set of cards from the blackline master. Activity, ‘Work with your partner to match each expression to a diagram that represents the same equal-group situation and the same amount. Be prepared to explain how you know the two representations belong together.’”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 7, Activity 1, students use scissors, tape, and centimeter grid paper to help them visualize a meter. Launch, “Give each group 2–3 sheets of centimeter grid paper, 2–3 pairs of scissors, and some tape. ‘Each grid on the paper is 1 centimeter long. Work with your group to cut the centimeter grid paper into strips and then join them to make a strip of paper that is exactly 1 meter long.’”

• Unit 7, Angles and Angle Measurement, Lesson 8, Activity 2, students construct a protractor- like tool that has benchmark angles. Launch, “Groups of 2 to 4. Give each student one paper half-circle and access to rulers or straightedges. ‘Your sheet of paper is in the shape of half a circle. It shows a ray on the bottom right and two angles ($$120\degree$$ and 180\degree) measured from the ray. We see the 120\degree label. Where is the 120\degree angle? Where are the two rays that make this angle?’ 1 minute: quiet think time for the first problem 1 minute: group discussion ‘Where do you think the second ray of a 90\degree angle would be?’ (Between 0 and 120, but closer to 120.)”