5th Grade - Gateway 3

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 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 5 include: developing fluency with addition and subtraction of fractions, developing understanding of multiplication and division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions), extending division to two-digit divisors, developing understanding of operations with decimals to hundredths, developing fluency with whole number and decimal operations, and developing understanding of volume.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section A, Multi-Digit Multiplication Using the Standard Algorithm, Section Overview, Section Narrative, provides an overview of the content and expectations for the section. For example, “This section introduces the standard algorithm for multiplication, extending students’ earlier work on multiplication. In grade 4, students used diagrams and partial-products algorithms to find the product of a one-digit number and a number up to four digits, and the product of 2 two-digit numbers. They attended to the role of place value along the way. Students revisit these strategies and representations here, but work with factors with more digits than encountered in grade 4. They make connections between the partial products in diagrams and previous algorithms to the numbers in the standard algorithm. They also learn the notation for recording new place-value units that result from finding partial products. When using the standard algorithm to multiply a two-digit number and a three-digit number, students account for the place value of the digits being multiplied, as they had done before. For example, the 3 in 23 represents 3 ones, so 3\times123 is 369. The 2 in 23, however, represents 2 tens, so the partial product is 2\times10\times123 or 2,460, instead of 2\times123 or 246. The partial products 369 and 2,460 can be seen in a diagram as well. Once students have practiced recording products this way, they learn to multiply factors that require composing new units, such as 264\times38.”

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

• Unit 1, Finding Volume, Lesson 4, Activity 1, teachers are provided context to help students reason about the volume of prisms. Narrative, “This activity continues to develop the idea of decomposing rectangular prisms into layers. Students explicitly multiply the number of cubes in a base layer by the number of layers. Students can use any layer in the prism as the base layer as long as the height is the number of those base layers.” Launch, “Groups of 2. Display first image from the student workbook. ‘What do you know about the volume of this prism? What would you need to find out to find the exact volume of this prism? You are going to work with prisms that are only partially filled in this activity.’ Give students access to connecting cubes.” Activity, “5 minutes: independent work time. 5 minutes: partner work time As students work, monitor for: students who notice that prisms A and D and prisms B and C are “the same” but they are sitting on different faces so the layers might be counted in different ways. Students who reason about the partially filled prisms by referring to the cubes in one layer they would see if all of the cubes were shown. Students who recognize that there are several different layers they can use to determine the volume of a prism, all of which result in the same volume.”

• Unit 7, Shapes on the Coordinate Plane, Lesson 9, About this Lesson provides notes about the lesson for the teacher. “The purpose of this lesson is for students to generate two different numerical patterns and then compare the terms in the two patterns. In this lesson, the patterns are the multiples of given whole numbers, starting with 0, and one of the numbers is a multiple of the other. This means that one of the patterns is contained inside the other. For example, the list of multiples of 9 is contained inside the list of multiples of 3 since every third multiple of 3 is a multiple of 9. Students express relationships within a pattern and between 2 patterns using multiplication and division. Lesson purpose: The purpose of this lesson is for students to generate patterns, given two rules, and identify relationships between corresponding terms in the different patterns. Teacher reflection question: In what ways did you accept students' everyday way of talking as a starting point for joining the math conversation today?” The section also includes lesson overview, learning goals, learning goals (student facing), materials to gather, materials to copy, and required preparation.

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 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, “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.” 

• Why is the curriculum designed this way? Further Reading, Unit 3, “Why is a negative times a negative a positive? supports teachers with context for work beyond the grade. “In this blog post, McCallum discusses how the “rule” for multiplying negative numbers is grounded in the distributive property.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 21, Food Waste Journal, About this Lesson, “This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling. In previous lessons, students made estimates about the volume of recyclable garbage students at their school produce. In this lesson, students make similar estimates and calculations, but now they estimate the weight of food waste produced. In the first activity students estimate the amount of food waste they produce based on average production by individuals in the United States. In the second activity, students are introduced to a food waste journal and make some initial calculations about their food waste. The last activity is optional and can be used after students use the journal to record their food waste for a week. Students share what they notice and compare the amount of food waste they produce to the national average. Students use this sample data to estimate how much food waste they produce monthly and annually. When students recognize the mathematical features of familiar real world situations and use those features to solve problems, they model with mathematics (MP4).”

• Unit 6, More Decimals and Fraction Operations, Lesson 21, Weekend Investigation, About this Lesson, “This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling. In this lesson, students brainstorm and define categories of how to spend time. Then they collect and represent data on a line plot. They analyze and describe the data to tell a story about the time use. When students define categories, choose and ask questions, collect and analyze data, and tell a story about the situation based on data, they model with mathematics (MP4).”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 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 5 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 5 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 6, More Decimal and Fraction Operations, Lesson 5, the Core Standards are identified as 5.MD.A.1 and 5.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 5: Place Value Patterns and Decimal Operations, Unit Overview, 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 expand their knowledge of decimals to read, write, compare, and round decimals to the thousandths. They also extend their understanding of place value and numbers in base ten by performing operations on decimals to the hundredth. In grade 4, students wrote fractions with denominators of 10 and 100 as decimals. They recognized that the notations 0.1 and 110 express the same amount and are both called “one tenth.” They used hundredths grids and number lines to represent and compare tenths and hundredths. Here, students likewise rely on diagrams and their understanding of fractions to make sense of decimals to the thousandths. They see that “one thousandth” refers to the size of one part if a hundredth is partitioned into 10 equal parts, and that its decimal form is 0.001. Diagrams help students visualize the magnitude of each decimal place and compare decimals.”

• Unit 6: More Decimals and Fraction Operations, Unit Overview, full Unit Narrative, include an overview of how the math of this module builds from previous work in math. “In this unit, students deepen their understanding of place-value relationships of numbers in base ten, unit conversion, operations on fractions with unlike denominators, and multiplicative comparison. The work here builds on several important ideas from grade 4. In grade 4, students learned the value of each digit in a whole number is 10 times the value of the same digit in a place to its right. Here, they extend that insight to include decimals to the thousandths. Students recognize that the value of each digit in a place (including decimal places) is \frac{1}{10} the value of the same digit in the place to its left. This idea is highlighted as students perform measurement conversions in metric units. Previously, students learned to convert from a larger unit to a smaller unit. Here, they learn to convert from a smaller unit to a larger unit. They observe how the digits shift when multiplied or divided by a power of 10 and learn to use exponential notation for powers of 10 to represent large numbers.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 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, 3–5, “Fraction Division Parts 1–4. In this four-part blog post, McCallum and Umland discuss fraction division. They consider connections between whole-number division and fraction division and how the two interpretations of division play out with fractions with an emphasis on diagrams, including a justification for the rule to invert and multiply. In Part 4, they discuss the limitations of diagrams for solving fraction division problems. Fraction Division Part 1: How do you know when it is division? Fraction Division Part 2: Two interpretations of division Fraction Division Part 3: Why invert and multiply? Fraction Division Part 4: Our final post on this subject (for now). Untangling fractions, ratios, and quotients. In this blog post, McCallum discusses connections and differences between fractions, quotients, and ratios.”

• 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 5 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 5  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. Meter sticks are a reusable material used in lesson 5.4.19, 5.6.3, and 5.6.5. 15 (one per pair) are needed per 30 students. No suitable substitutes are listed. Number cubes are a reusable material used in lessons 5.5.11, 5.5.14, and 5.8.12. 15 are needed for 30 students. No suitable substitutes for the material are listed. Chart paper is a consumable material used in lessons 5.5.1, 5.5.11, 5.5.14, 5.8.14, and 5.8.15. 30 pages are needed per 30 students. Poster Paper is a suitable substitute for the material.

• Unit 1, Finding Volume, Lesson 7, Activity 1: What are the Units?, Teaching Notes, Materials to gather, “Rulers (centimeters), Rulers (inches), Yardsticks.” Launch, “Write the list of objects (moving truck, freezer, etc…) on a display for all students to see. In this activity we are going to consider using different cubic units of measure to find the volume of different sized objects. There is no right or wrong answer in these questions, but be prepared to explain your choice. Give students access to rulers and yardsticks.”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 12, Activity 2, Teaching Notes, Materials to gather, “Number cubes, Target Numbers Stage 9 Recording Sheet.” Launch, “Give each group 1 number cube. We’re going to play a new stage of the game called Target Numbers. Let’s read through the directions and play one round together. Read through the directions with the class and play a round with the class: Display each roll of the number cube. Think through your choices aloud. Record your move and score for all to see. Now, play the game with your partner.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 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, Multiplying and Dividing Fractions, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 4, 5.NF.7.b and 5.NF.7.c, “440 meters is \frac{1}{4} of the way around the race track. How far is it around the whole race track? Explain or show your reasoning.”

• Unit 5, Place Value Patterns and Decimal Operations, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 3, 5.NBT.4, “What is 1.357 rounded to the nearest hundredth? What about to the nearest tenth? To the nearest whole number? Explain or show your reasoning.”

• Unit 8, Putting it All Together, End-of-Course Assessment  answer key, denotes standards addressed for each problem. Problem 6, 5.NF.2, “There are 8 ounces of pasta in the package. Jada cooks \frac{2}{3} of the pasta. How many ounces of pasta did Jada cook? A. 2\frac{2}{3}, B. 5\frac{1}{3}, C. 7\frac{1}{3}, D. 12.”

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 5, MP3 is found in Unit 5, Lessons 2, 8, 12, and 16. 

• IM K-5 Curriculum Guide, How do you assess progress? Standards For Mathematical Practice, Standards for Mathematical Practices Chart, Grade 5, MP7 is found in Unit 1, Lessons 3, 6, 8, and 10. 

• 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.”

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 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, Finding Volume, End-of-Unit Assessment problems support the full intent of MP4, model with mathematics, as students design a composite prism to meet certain criteria. For example, Problem 7, “Mai's class is designing a garden with two levels and this general shape. The garden should have at least 200 square feet for the plants. The volume should be less than 500 cubic feet. a. Recommend side lengths for the tiered garden that fit the needs of Mai's class. b. Label the diagram to show your choices for the side lengths.”

• Unit 4, Wrapping Up Multiplication and Division With Multi-Digit Numbers, End-of-Unit Assessment, develops the full intent of 5.NBT.5, fluently multiply multi-digit whole numbers using the standard algorithm. Problem 1, “Find the value of each product. Explain or show your reasoning. (both problems are written vertically) a. 213\times54; b. 375\times47.”

• Unit 5, Place Value Patterns and Decimal Operations, End-of-Unit Assessment, develops the full intent of 5.NBT.1, recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and \frac{1}{10} of what it represents in the place to its left. Problem 5, “In which number does the 6 represent \frac{1}{1,000} the value of the 6 in 16.003? A. 3 B. 10.006 C. 16.004 D. 16,003.”

• Unit 6, More Decimal and Fraction Operations, End-of-Unit Assessment problems support the full intent of MP7, look for and make use of structure, as students convert millimeters to kilometers and write numbers in exponential notation and standard notation. Problem 1, “Select all expressions that represent the number of millimeters in a kilometer. A.$$10^3$$ B. 10^5 C. 10^6 D.1,000 E. 100,000 F. 1,000,000. ”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 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 3, Multiplying and Dividing Fractions, Lesson 3, Activity 2, Narrative, Access for Students with Disabilities, “Engagement: Provide Access by Recruiting Interest. Invite students to share a connection between the diagram and something in their own lives that represent the fractional values. Supports accessibility for: Attention, Conceptual Processing.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 19, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Internalize Comprehension.  Activate or supply background knowledge. Provide students with a visual representation of a kilometer for students who are unfamiliar with the distance. Supports accessibility for: Conceptual Processing, Memory, Attention.”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 8, Activity 2, Narrative, Access for Students with Disabilities, “Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy for determining proximity before they begin. Students can speak quietly to themselves, or share with a partner. Supports accessibility for: Organization, Conceptual Processing, Language.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 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 3, Multiplying and Dividing Fraction, Section B: Fraction Division, Problem 9, Exploration, “It takes Earth 1 year to go around the Sun. 1. During the time it takes Earth to go around the Sun, Mercury goes around the Sun about 4 times. How many years does it take Mercury to make 1 full orbit of the Sun? Write an equation showing your answer. 2. During the time it takes Earth to go around the Sun, Saturn goes \frac{1}{29} of the way around the Sun. How many years does it take Saturn to go around the Sun? Write an equation showing your answer.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section C: Let’s Put it to Work, Problem 5, Exploration, “The Pentagon has 5 floors and the Empire State Building has 102 floors. Noah says that the Empire State Building is bigger. Do you agree with Noah? Investigate and justify your answer.”

• Unit 5, Place Value Patterns and Decimal Operations, Section D: Divide Decimals, Problem 7, Exploration, “1. The daily recommended allowance of vitamin C for a 5th grader is 0.05 grams. A vitamin C tablet has 1 gram of vitamin C. How many times the daily recommended allowance of vitamin C is one vitamin C tablet? Use the diagram if it is helpful. 2. A large orange has 0.18 grams of vitamin C. How many times the daily recommended allowance of vitamin C is in a large orange? Use the diagram if it is helpful.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 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 understandings of others’ ideas.” Examples include:

• Unit 1, Finding Volume, Lesson 3, Activity 1, Teaching notes, Access for English Learners, “MLR8 Discussion Supports. During small-group discussion, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “I heard you say . . . .” Original speakers can agree or clarify for their partner. Advances: Listening, Speaking.”

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 11, Activity 1, Teaching notes, Access for English Learners, “MLR8 Discussion Supports. Display sentence frames to support small-group discussion: “I wonder if . . . ”, “____ and ____ are the same because. . . .”, and “____ and ____ are different because ____.  Advances: Conversing, Representing.”

• Unit 4, Wrapping Up Multiplication and Division With Multi-Digit Numbers, Lesson 9, Synthesis, Teaching notes, Access for English Learners, “MLR1 Stronger and Clearer Each Time. Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to What is the possible range of volumes for each type of birdhouse?. Invite listeners to ask questions, to press for details, and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive. Advances: Writing, Speaking, Listening.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 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 5, Place Value Patterns and Decimal Operations, Lesson 17, Activity 2, students use grid paper to find products of whole numbers and some tenths or hundredths. Launch, “Groups of 2. Make hundredths grids available for students. Activity, ‘Take a few minutes to find the value of the expressions in the first problem.’ Student facing, ‘Find the value of each expression. Explain or show your reasoning. a. 3\times0.5 b. 5\times0.3 c. 7\times0.02’”

• Unit 6, More Decimal and Fraction Operations, Lesson 3, Activity 1, students use meter sticks to help them convert meters to centimeters. Launch, “Give students access to meter sticks. Display image from student workbook. ‘What do you notice? What do you wonder?’ Display additional information about track and field events: The height of a hurdle is 1 meter. The approximate distance between hurdles in 110 meter races is 10 meters. The shortest race in many track competitions is 100 meters. ‘Work with your partner to complete the problems.’”

• Unit 7, Shapes on the Coordinate Plane, Lesson 1, Activity 1, students use coordinate grids to communicate and draw shapes.  Launch, “Groups of 2. ‘We are going to play a drawing game. Decide who will be partner A and who will be partner B.’ Student facing, Play three rounds of Draw My Shape using the three sets of cards from your teacher. For each round: Partner A chooses a card—without showing your partner—and describes the shape on the card. Partner B draws the shape as described. Partner A reveals the card and partner B reveals the drawing. Compare the shapes and discuss: ‘What’s the same? What’s different?’ Activity, Circulate, listen for, and collect the language students use to describe the location of each figure on the coordinate grid. Listen for students who: use the grid to determine the side lengths or area of the rectangle. describe the general location of the rectangle. use the numbers on the axes as reference points when describing the rectangle. Record students’ words and phrases on a visual display and update it throughout the lesson.”