5th Grade - Gateway 3

The materials reviewed for Kendall Hunt's 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. This is located within IM Curriculum, How to Use These Materials, and the Course Guide, Scope and Sequence. Examples include:

• IM Curriculum, How To Use These Materials, Design Principles, Coherent Progression provides an overview of the design and implementation guidance for the program, “The overarching design structure at each level is as follows: 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.”

• Course Guide, Scope and Sequence, 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.”

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 within the Warm-up, Activities, and Cool-down provide useful annotations. Examples include:

• Unit 1, Finding Volume, Lesson 4, Activity 1, teachers are provided context to support students when reasoning about the volume of prisms. Narrative states, “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 states, “Groups of 2. Display first image from 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 states, “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 10, Lesson Synthesis provides teachers guidance about how to support students to find patterns given two rules, “Today we noticed and explained relationships between patterns. Some of the relationships involved fractions. What relationships did you find between the patterns we studied today? (Sometimes I could multiply each term in one pattern by the same number to get the corresponding number in the other pattern. Sometimes that number was a fraction.) Consider asking students to record their response in a math journal and then share their response with a partner.”

The materials reviewed for Kendall Hunt's 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, About These Materials, 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 grade, so that teachers can improve their own understanding of the content. Professional articles support teachers with learning opportunities about topics such as ensuring mathematical success for all, early understanding of equality, and repeating patterns. Additionally, each lesson provides teachers with a lesson narrative, including adult-level explanations and examples of the more complex grade/course-level concepts. Examples include:

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 2, Preparation, Lesson Narrative states, “In the previous lesson, students explored the relationship between fractions and division by representing situations where some people shared some sandwiches. They used informal language to describe how they knew each person got about the same amount of sandwich. In this lesson, students recognize the relationship between a fraction and a division expression. For example, \frac{1}{5}=1\div5. Students interpret 1\div5 as the amount in one group when a single whole is divided into 5 equal portions. They see that the quantity in that portion is \frac{1}{5} of a whole.”

• IM K-5 Math Teacher Guide, About These Materials, Unit 3, “Why is a negative times a negative a positive? In this blog post, McCallum discusses how the ‘rule’ for multiplying negative numbers is grounded in the distributive property.” 

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 23, Preparation, Lesson Narrative states, “In the previous lesson, students divided whole numbers by one tenth and one hundredth and made generalizations about how to divide any whole number by those units. The purpose of this lesson is for students to extend that work to divide whole numbers by any number of tenths or hundredths (with total value less than 1). Consistent divisors are used in repetition to highlight relationships between the dividends and the quotients (MP8). Students evaluate expressions with larger divisors such as 12\div0.2 in order to encourage them to use the relationship between multiplication and division. Rather than drawing 12 unit squares and dividing all of them into groups of 2 tenths, students may draw a single whole divided into 2 tenths and then use multiplication.”

• IM K-5 Math Teacher Guide, About These Materials, Unit 7, Making Sense of Distance in the Coordinate Plane, “In this blog post, Richard shares how understanding of the coordinate plane, introduced in grade 5, provides a foundation for conceptual understanding of distance and the Pythagorean Theorem.”

The materials reviewed for Kendall Hunt's 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 5, Course Guide, Lesson Standards includes a table with each grade-level lesson (in columns) and aligned grade-level standards (in rows). Teachers can search any lesson for the grade and identify the standard(s) that are addressed within.

• Grade 5, Course Guide, Lesson Standards, includes all Grade 5 standards and the units and lessons each standard appears in. Teachers can search a standard for the grade and identify the lesson(s) where it appears within materials.

• Unit 1, Finding Volume, Lesson 9, the Core Standards are identified as 5.MD.C.5c and 5.OA.A.2. Lessons contain a consistent structure: a Warm-up that includes Narrative, Launch, Activity, Activity Synthesis; Activity 1, 2, or 3 that includes Narrative, Launch, Activity; an Activity Synthesis; a Lesson Synthesis; and 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.

• 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.

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

• Grade 5, Course Guide, Scope and Sequence, Unit 1: Finding Volume, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “In grade 3, students learned that the area of a two-dimensional figure is the number of square units that cover it without gaps or overlaps. They first found areas by counting squares and began to intuit that area is additive. Later, they recognized the area of a rectangle as a product of its side lengths and found the area of more-complex figures composed of rectangles. Here, students learn that the volume of a solid figure is the number of unit cubes that fill it without gaps or overlaps. First, they measure volume by counting unit cubes and observe its additive nature. They also learn that different solid figures can have the same volume. Next, they shift their focus to right rectangular prisms: building them using unit cubes, analyzing their structure, and finding their volume. They write numerical expressions to represent their reasoning strategies and work with increasingly abstract representations of prisms.”

• Grade 5, Course Guide, Scope and Sequence, Unit 4: Wrapping Up Multiplication and Division with Multi-digit Numbers, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “In this unit, students multiply multi-digit whole numbers using the standard algorithm and begin working toward end-of-grade expectation for fluency. They also find whole-number quotients with up to four-digit dividends and two-digit divisors. In grade 4, students used strategies based on place value and properties of operations to multiply a one-digit whole number and a whole number of up to four digits, and to multiply a pair of two-digit numbers. They decomposed the factors by place value, and used diagrams and algorithms using partial products to record their reasoning. Here, students build on those strategies to make sense of the standard algorithm for multiplication. They recognize that it is also based on place value but records the partial products in a condensed way.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

The IM K-5 Math Teacher Guide, Design Principles, outlines the instructional approaches of the program, “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 mathematics through a coherent progression, and provides teachers the opportunity to deepen their knowledge of mathematics, student thinking, and their own teaching practice. In order to design curriculum and professional learning materials that support student and teacher learning, we need to be explicit about the principles that guide our understanding of mathematics teaching and learning. This document outlines how the components of the curriculum are designed to support teaching and learning aligning with this belief.” Examples of the design principles include:

• IM K-5 Math Teacher Guide, Design Principles, All Students are Capable Learners of Mathematics, “All students, each with unique knowledge and needs, enter the mathematics learning community as capable learners of meaningful mathematics. Mathematics instruction that supports students in viewing themselves as capable and competent must leverage and build upon the funds of knowledge they bring to the classroom. In order to do this, instruction must be grounded in equitable structures and practices that provide all students with access to grade-level content and provide teachers with necessary guidance to listen to, learn from, and support each student. The curriculum materials include classroom structures that support students in taking risks, engaging in mathematical discourse, productively struggling through problems, and participating in ways that make their ideas visible. It is through these classroom structures that teachers will have daily opportunities to learn about and leverage their students’ understandings and experiences and how to position each student as a capable learner of mathematics.”

• IM K-5 Teacher Guide, Design Principles, Coherent Progression, “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.” 

• IM K-5 Teacher Guide, Design Principles, Learning Mathematics by Doing Mathematics, “Students learn mathematics by doing mathematics, rather than by watching someone else do mathematics or being told what needs to be done. Doing mathematics can be defined as learning mathematical concepts and procedures while engaging in the mathematical practices—making sense of problems, reasoning abstractly and quantitatively, making arguments and critiquing the reasoning of others, modeling with mathematics, making appropriate use of tools, attending to precision in their use of language, looking for and making use of structure, and expressing regularity in repeated reasoning. By engaging in the mathematical practices with their peers, students have the opportunity to see themselves as mathematical thinkers with worthwhile ideas and perspectives. ‘Students learn mathematics as a result of solving problems. Mathematical ideas are the outcomes of the problem-solving experience rather than the elements that must be taught before problem solving’ (Hiebert et al., 1996). A problem-based instructional framework supports teachers in structuring lessons so students are the ones doing the problem solving to learn the mathematics. The activities and routines are designed to give teachers opportunities to see what students already know and what they can notice and figure out before having concepts and procedures explained to them.”

Research-based strategies are cited and described within the IM Curriculum and can be found in various sections of the IM K-5 Math Teacher Guide. Examples of research-based strategies include:

• IM K-5 Math Teacher Guide, About These Materials, 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.“

• IM K-5 Math Teacher Guide, 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.”

• IM K-5 Math Teacher Guide, Key Structures in This Course, Student Journal Prompts, “Writing can be a useful catalyst in learning mathematics because it not only supplies students with an opportunity to describe their feelings, thinking, and ideas clearly, but it also serves as a means of communicating with other people (Baxter, Woodward, Olson, & Robyns, 2002; Liedtke & Sales, 2001; NCTM, 2000). NCTM (1989) suggests that writing about mathematics can help students clarify their ideas and develop a deeper understanding of the mathematics at hand.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Course Guide includes a section titled “Required Materials” that includes a breakdown of materials needed for each unit and for each lesson. Additionally, specific lessons outline materials to support the instructional activities and these can be found on the “Preparation” tab in a section called “Required Materials.” Examples include:

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 17, Activity 1, “Colored Paper, Glue, Rulers, Scissors.” Launch states, “Groups of 4. Distribute materials. Make sure each student in the group gets a different color paper.”

• Course Guide, Required Materials for Grade 5, Materials Needed for Unit 3, Lesson 9, teachers need, “Colored pencils or crayons, Paper, Rulers.” 

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 19, Activity 1, Required Materials, “Materials to Copy: Decimal Multiplication Expression Card Sort.” Launch states, “Groups of 2. Distribute one set of pre-cut cards to each group of students.” Activity states, “‘In this activity, you will sort some expressions into categories of your choosing. When you sort the expressions, work with your partner to come up with categories.’ 3 minutes: partner work time.”

• Course Guide, Required Materials for Grade 5, Materials Needed for Unit 8, Lesson 14, teachers need, “Chart paper, Colored pencils, crayons, or markers.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for having assessment information included in the materials to indicate which standards are assessed. 

End-of-Unit Assessments and the End-of-Course Assessments consistently and accurately identify grade-level content standards. Content standards can be found in each Unit Assessment Teacher Guide. Examples from formal assessments include:

• Unit 1, Finding Volume, End-of-Unit Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 4, 5.MD.5b, “Find the volume of a rectangular prism with the given side lengths. a. The length is 2 units, the width is 5 units, and the height is 7 units. b. The base has an area of 200 square inches and the height is 6 inches.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, End-of-Unit Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 4, 5.NBT.6, “Find the value of 1,530\div34. Explain or show your reasoning.”

• Unit 8, Putting it All Together, End-of-Course Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 6, 5.NF.2, “Jada’s math book is \frac{5}{16} of an inch thick. Her science book is \frac{1}{4} of an inch thick. a. Which book is thicker? How much thicker? Explain or show your reasoning. b. How thick are the math and science books together?”

Guidance is provided within materials for assessing progress of the Mathematical Practices. According to IM K-5 Math Teacher Guide, How to Use These Materials, “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 Math Teacher Guide, How to Use These Materials, Standards for Mathematical Practices Chart, Grade 5, MP2 is found in Unit 3, Lessons 1, 4, 11, 14, and 18.

• IM K-5 Math Teacher Guide, How to Use These Materials, Standard for Mathematical Practices Chart, Grade 5, MP6 is found in Unit 5, Lessons 6, 7, 13, 16, and 25. 

• IM K-5 Math Teacher Guide, How to Use These Materials, 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 Math Teacher Guide, How to Use These Materials, 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 Kendall Hunt's 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 some end of lesson cool-downs, interviews, and Checkpoint Assessments for each section. 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 from summative assessments include:

• Unit 1, Finding Volume, End-of-Unit Assessment supports the full intent of MP4 (Model with mathematics) as students design a composite prism to meet certain criteria. For example, Problem 7 states, “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 2, Fractions as Quotients and Fraction Multiplication, End-of-Unit Assessment develops the full intent of 5.NF.3 (Interpret a fraction as division of the numerator by the denominator (\frac{a}{b}=a\div b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers), 5.NF.4a (Interpret the product (\frac{a}{b})\times q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations $$a\times q\div b.)$$, and 5.NF.6 (Solve real world problems involving multiplication of fractions and mixed numbers). For example, Problem 7 states, “A farm is rectangular in shape. It is 2 km long and 3 km wide. a. What is the area of the farm? Explain or show your reasoning. b. The farm is divided into 5 equal parts. Corn is grown in one of the parts. Draw a diagram to show where the corn is grown. c. What is the area of the part of the farm where corn is grown? Explain or show your reasoning.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, End-of-Unit-Assessment develops the full intent of 5.NBT.6 (Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models). For example, Problem 5 states, “The area of a rectangular yard is 5,063 square feet and its length is 61 feet. What is its width? Explain or show your reasoning.”

• Unit 8, Putting It All Together, End-of-Course Assessment supports the full intent of MP7 (Look for and make use of structure) as students identify different expressions that have the value one million. For example, Problem 7 states, “Select all expressions that have the value one million. A. 10^3B. 10^6 C. 10^7 D. 1,000,000 E. 10\times10\times10\times10\times10\times10. F. 100\times100 G. 100\times100\times100.”

The materials reviewed for Kendall Hunt's 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 as suggestions are outlined within each lesson and parts of each lesson. According to the IM K-5 Teacher Guide, Universal Design for Learning and 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 of supports for special populations include: 

• Unit 1, Finding Volume, Lesson 3, Activity 2, Narrative, Access for Students with Disabilities, “Representation: Access for Perception. Invite students to identify correspondences between the visual representation and the prism made of connecting cubes. Make connections between representations visible through gestures or labeled displays. Supports accessibility for: Visual-Spatial Processing, Conceptual Processing.” 

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 3, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were needed to solve the problem. Display the sentence frame, ‘The next time I write division equations, I will pay attention to . . .’ Supports accessibility for: Conceptual Processing, Memory, Language.”

• 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.”

• Unit 7, Shapes on the Coordinate Plane, Lesson 5, Activity 2, Narrative, Access for Students with Disabilities, “Engagement: Provide Access by Recruiting Interest. Synthesis: Optimize meaning and value. Invite students to share the meaning of a trapezoid and the similarities and differences in the two definitions of a trapezoid with a classmate who missed the lesson. Supports accessibility for: Conceptual Processing; Language.”

The materials reviewed for Kendall Hunt's 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 To Use The Materials, 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 directly related to the material of the unit that students can do 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, Fractions as Quotients and Fraction Multiplication, Section B: Fractions of Whole Numbers, Problem 5, Exploration, “A standard rectangular sheet of paper measures 8\frac{1}{2} inches in width and 11 inches in length. How many square inches are there in a sheet of paper? If you get stuck, consider using the grid.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section B: Multi- Digit Division Using Partial Quotients, Problem 7, Exploration, “1. Andre made a noodle that was 102 feet long. The noodle broke into two pieces. One piece was 2 times as long as the other. How long were the two noodles? Explain your reasoning. 2. Priya made a noodle that was 456 feet long. The noodle broke into two pieces. One piece was 5 times as long as the other. How long were the two noodles? Explain your reasoning.”

• Unit 5, Place Value Patterns and Decimal Operations, Section B: Add and Subtract Decimals, Problem 8, Exploration, “Lin is trying to use the digits 1, 3, 4, 2, 5, and 6 to make 2 two-digit decimals whose sum is equal to 1. 1. Explain why Lin can not make 1 by adding together 2 two-digit decimal numbers made with these digits. 2. What is the closest Lin can get to 1? Explain how you know.”

• Unit 7, Shapes on the Coordinate Plane, Section C: Numerical Patterns, Problem 7, Exploration, “Andre starts from 2 and counts by 6s. Clare starts at 1,000 and counts back by 7s. 1. List the first 6 numbers Andre and Clare say. 2. Do Andre and Clare ever say the same number in the same spot on their lists? Explain or show your reasoning.”

The materials reviewed for Kendall Hunt's 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 IM K-5 Math Teacher Guide, Mathematical Language Development and Access for English Learners, “In a problem-based mathematics classroom, sense-making and language are interwoven. Mathematics classrooms are language-rich, and therefore language demanding learning environments for every student. The linguistic demands of doing mathematics include reading, writing, speaking, listening, conversing, and representing (Aguirre & Bunch, 2012). Students are expected to say or write mathematical explanations, state assumptions, make conjectures, construct mathematical arguments, and listen to and respond to the ideas of others. In an effort to advance the mathematics and language learning of all students, the materials purposefully engage students in sense-making and using language to negotiate meaning with their peers. To support students who are learning English in their development of language, this curriculum includes instruction devoted to fostering language development alongside mathematics learning, fostering language-rich environments where there is space for all students to participate.” The series provides the following 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 in each lesson. According to the IM K-5 Math Teacher Guide, Mathematical Language Development and Access for English Learners, “Mathematical Language Routines (MLRs) are instructional routines that provide structured but adaptable formats for amplifying, assessing, and developing students' language. MLRs are included in select activities in each unit to provide all students with explicit opportunities to develop mathematical and academic language proficiency. These ‘embedded’ MLRs are described in the teacher notes for the lessons in which they appear.” Examples include:

• Unit 3, Multiplying and Dividing Fractions, Lesson 8, Activity 1, Teaching notes, Access for English Learners, “MLR7 Compare and Connect. Invite students to prepare a visual display that shows the strategy they used to calculate the area of different parts of the flag. Encourage students to include details that will help others interpret their thinking. For example, specific language, using different colors, shading, arrows, labels, notes, diagrams, or drawings. Give students time to investigate each other’s work. During the whole-class discussion, ask students, ‘What did the approaches have in common?’, ‘How were they different?’, and ‘Did anyone solve the problem the same way, but would explain it differently?’ Advances: Representing, Conversing.”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 2, Activity 1, Teaching notes, Access for English Learners, “MLR2 Collect and Display. Circulate, listen for, and collect the language students use as they shade and interpret diagrams. On a visible display, record words and phrases such as: ‘fraction,’ ‘part of,’ ‘decimal,’ ‘tenths,’ ‘row,’ ‘hundredths,’ ‘thousandths,’ ‘represents,’ ‘shows.’ Invite students to borrow language from the display as needed, and update it throughout the lesson. Advances: Conversing, Reading.”

• Unit 8, Putting It All Together, Lesson 14, Activity 2, Teaching notes, Access for English Learners, “MLR8 Discussion Supports. At the appropriate time, give students 2–3 minutes to make sure that everyone in their group can explain their Notice and Wonder activity. Invite groups to rehearse what they will say when they share with the whole class. Advances: Speaking, Conversing, Representing.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for providing manipulatives, physical but not virtual, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Suggestions and/or links to manipulatives are consistently included within materials, often in the Launch portion of lessons, to support the understanding of grade-level math concepts. Examples include: 

• Unit 1, Finding Volume, Lesson 2, Activity, 1, references connecting cubes to build understanding of volume. Student Facing states, “Partner A: Build an object using 8–12 cubes and give the object to Partner B. Partner B: Explain how you would count the number of cubes in the object. Partner A: Explain if you would count the cubes in the same way or in a different way. Switch roles and repeat. Which objects were easiest to count? Why?” A picture of a house built out of connecting cubes is shown. Launch states, “Groups of  2. Give 24 connecting cubes to each group. ‘In this activity, you will use unit cubes to build objects and describe how you would measure the volume.’ 10 minutes: partner work time.” Activity states, “As students work, monitor for students who build rectangular prisms to share during the synthesis.”

• Unit 3, Multiplying and Dividing Fractions, Lesson 9, Activity 2, identifies colored pencils or crayons, paper, and rulers to design and analyze a flag, including operations with fractions. Launch states, “Give each student white paper. Use the design principles we discussed in the last activity to make your own flag. As you make the design, think about the meaning of each symbol and color you use.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 7, Activity 1, references a game called Greatest Product and Number Cards (0-10), where students create the greatest product from expressions. Launch states, “Give each group a set of number cards and 2 copies of the blackline master. Remove the cards that show 10 and set them aside. ‘We’re going to play a game called Greatest Product. Let’s read through the directions and play one round together.’ Read through the directions with the class and play a round against the class using the diagram in the student workbook. Display each number card. Think through your choices aloud. Record your move and score for all to see.”

• Unit 6, More Decimal and Fraction Operations, Lesson 3, Activity 1, identifies meter sticks to help students convert meters to centimeters. Launch states, “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 you partner to complete the problems.’”