3rd-5th Grade - Gateway 3

The materials reviewed for Kiddom IM® v.360 Grade 3 through 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. 

The Course Overview contains sections titled What’s in an IM Lesson, Key Structures in This Course, Scope and Sequence, and Pacing Guides and Dependency Diagrams, which provide instructional guidance related to the use of student and ancillary materials. Examples include:

• Course Overview, Key Structures in This Course, 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 the opportunity to observe students’ prior understandings. Each lesson starts with a Warm-up to activate students’ prior knowledge and to 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 Synthesis at the end consolidates understanding and makes the learning goals of the lesson explicit. In the Cool-down that follows, students apply what they learned. Each activity starts with a Launch that gives all students access to the task. Independent work time follows, allowing them to grapple with problems individually before working in small groups. In the Activity Synthesis at the end, students consolidate their learning by making connections between their work and the mathematical goals. Each activity includes carefully chosen contexts and numbers that support the coherent sequence of learning goals in the lesson.” 

• Course Overview, Key Structures in This Course, Principles of IM Curriculum Design, Productive Discussions states, "Promoting productive and meaningful conversations between students and teachers is essential to success in a problem-based classroom. To facilitate these conversations, the IM curriculum incorporates the framework presented in 5 Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011). The 5 Practices are: anticipating, monitoring, selecting, sequencing, and making connections between students’ responses. All IM lessons support the practices of anticipating, monitoring, and selecting students’ work to share during whole-group discussions. In lessons where students make connections between representations, strategies, concepts, and procedures, the Lesson Narrative and the Activity Narrative support the practices of sequencing and connecting. The lesson is also tagged so that these opportunities are easily identifiable. For additional opportunities to connect students’ work, look for activities tagged with MLR7 Compare and Connect. Similar to the 5 Practices routine, MLR7 supports the practices of monitoring, selecting, and making connections. In curriculum workshops and PLCs, rehearse and reflect on enacting the 5 Practices."

• Course Overview, What’s in an IM Lesson, Narratives Tell The Story states, “The story of each grade is told in eight or nine units. Each unit has a narrative that describes the mathematical work that will unfold in that unit. Each lesson and each activity in the unit also has a narrative. The Lesson Narrative explains: The mathematical content of the lesson and its place in the learning sequence. The meaning of any new terms introduced in the lesson. How the mathematical practices come into play, as appropriate. The Activity Narrative explains: The mathematical purpose of the activity and its place in the learning sequence. What students are doing during the activity. What to look for, while students are working on an activity, to orchestrate an effective Activity Synthesis. Connections to the mathematical practices, when appropriate.”

• Course Overview, Scope and Sequence, lists each of the eight units, explains connections to prior learning, describes the progression of learning throughout the unit, and integrates new terminology/vocabulary throughout each unit.

• Course Overview, Pacing Guides, and Dependency Diagrams, show the interconnectedness between lessons and units within each grade and across all grades.

• Resources from Kiddom, Grade 3 Glossary, provides a visual glossary for teachers that includes both definitions and illustrations.

Materials include sufficient annotations and suggestions presented within the context of the specific learning objectives. 

An example in Grade 3 includes:

• Unit 5, Fractions as Numbers, Lesson 17, Warm-up, provides teachers guidance on how to work with estimation and fractions. Launch states, “Groups of 2. Display the image. ‘What is an estimate that’s too high? Too low? About right?’ 1 minute: quiet think time. Activity: ‘Discuss your thinking with your partner.’ 1 minute: partner discussion. Record responses. Activity Synthesis, Consider asking: ‘Is anyone’s estimate less than ¼ inch? Is anyone’s estimate greater than ½ inch?’ ‘Based on this discussion, does anyone want to revise their estimate?’”

An example in Grade 4 includes:

• 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 measurements of different angles.’ Display: ‘Here are some angles whose measurements we tried to find: Angle P, 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 of P make a right angle, which is 90°, then dividing 90° by 2 gives us the measure of P.) ‘Which unknown angle can we find by multiplication?’ (The angle made up of four 30° angles has a measurement of 430.) ‘Which unknown angle can we find by subtracting one angle from another? (Angle S: We can subtract 30° from 180° and divide by 2 to find the measure of S, which is 75°.) ‘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 105°)’”

An example in Grade 5 includes:

• Unit 1, Finding Volume, Lesson 4, Activity 1, teachers are provided context to help students reason 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 the first image from the student book. ‘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. Monitor for students who: Notice that Prisms A and D and Prisms B and C are ‘the same’ but sitting on different faces, so the layers might be counted in different ways. Reason about the partially filled prisms by referring to the cubes in one layer they would see if all of the cubes were shown. 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.”

The materials reviewed for Kiddom IM® v.360 Grade 3 through 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 knowledge of the subject. 

The Further Reading section, located within the Course Overview, connects research to pedagogy. It includes explanations and examples of both grade-level and above-grade-level content to support teacher understanding. Additionally, Unit Overviews, Lesson Narratives, and Activity Synthesis sections throughout the lessons provide similar support, offering explanations and examples of how grade-level concepts connect to content in other grade levels.

 Examples in Grade 3 include:

• Course Overview, Further Reading, “Unit 1, Ratio Tables are not Elementary”, supports teachers with context for work beyond the grade. “In this blog post, McCallum discusses the difference between multiplication tables and tables of equivalent ratios, highlighting how K–5 arithmetic work prepares students to make sense of these tables.”

• Unit 6, Measuring Length, Time, Lesson 16, Activity 1, Narrative, “The purpose of this activity is for students to use the provided materials to design their own game. Students decide the rules and objectives of the game. After playing the game at least once, students revise their design to include two of the following elements: measuring elapsed time, measuring distance, multiplication or division within 100, addition or subtraction within 1,000. If there is time, a pair of students from each group can swap with another group at different points of this activity so they have an opportunity to play a different game.”

Examples in Grade 4 include:

• Course Overview, Further Reading, “Unit 7, Making Peace with the Basics of Trigonometry” supports teachers with context for work beyond the grade. “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 6, Multiplying and Dividing Multi-digit Numbers, Lesson 25, Lesson Narrative, “In the final lesson of the unit, students apply their knowledge of numbers in base-ten and their estimation and computation skills to solve problems about languages and populations in the United States. The census data used here prompts students to work with large numbers and to interpret them carefully.”

Examples in Grade 5 include:

• Course Overview, 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 6, More Decimals and Fraction Operations, Lesson 21, Lesson Narrative, “This lesson is optional because it does not address any new mathematical content standards. This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling (MP4). 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 use of time.”

The materials reviewed for Kiddom IM® v.360 Grade 3 through Grade 5 meet expectations for including a year-long scope and sequence with standards correlation information.

The Course Overview includes multiple components that support planning and understanding of the program’s structure and standards alignment. 

Examples in Grade 3 include:

• The Scope and Sequence section narratively outlines unit content, prior knowledge, future learning, and terminology. The materials state: “The big ideas in IM Grade 3 include: developing understanding of multiplication and division, and strategies for multiplication and division within 100; developing understanding of fractions, especially unit fractions (fractions with numerator 1); developing understanding of the structure of rectangular arrays and of area; and describing and analyzing two-dimensional shapes.”

• Lessons by Standard, which provides a table that shows each content standard for the grade level and the lessons in which it appears. For example, 3.OA.A.1 is addressed in Unit 1, Lessons 9, 10, 11, 12, 13, 14, 16, 17, 18, and 19; Unit 2, Lesson 1; and Unit 8, Lesson 13.

• Standards by Lesson, which provides a table listing the standards covered within each lesson. For example, Unit 8, Lesson 9 addresses 3.MD.C.7; 3.OA.B.5; and 3.OA.C.7.

• Standards for Mathematical Practice, mapping practice standards (MPs) to lessons. For example, Unit 4, Lesson 4 integrates MP2, and MP6.

Examples in Grade 4 include:

• The Scope and Sequence section narratively outlines unit content, prior knowledge, future learning, and terminology. The materials state: “The big ideas in IM 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.”

• Lessons by Standard, which provides a table that shows each content standard for the grade level and the lessons in which it appears. For example, 4.NBT.B.4 is addressed in Unit 4, Lessons 10, 18, 19, 20, 21, 22, and 23; Unit 6, Lessons 24 and 25; and Unit 9, Lessons 4 and 9.

• Standards by Lesson provides a table listing the standards covered within each lesson. For example, Unit 5, Lesson 2 addresses 4.OA.A.1 and 4.OA.A.2.

• Standards for Mathematical Practice, mapping practice standards (MPs) to lessons. For example, Unit 3, Lesson 6 integrates MP2, and MP3.

Examples in Grade 5 include:

• The Scope and Sequence section narratively outlines unit content, prior knowledge, future learning, and terminology. The materials state: “The big ideas in IM Grade 5 include: developing fluency with addition and subtraction of fractions, and 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 understanding of division to two-digit divisors; developing understanding of operations with decimals to hundredths, and developing fluency with whole- number and decimal operations; and developing understanding of volume.”

• Lessons by Standard, which provides a table that shows each content standard for the grade level and the lessons in which it appears. For example, 5.NF.B.5 is addressed in Unit 6, Lesson 20.

• Standards by Lesson provides a table listing the standards covered within each lesson. For example, Unit 3, Lesson 9 addresses 5.NF.B.4.a and 5.NF.B.6.

• Standards for Mathematical Practice, mapping practice standards (MPs) to lessons. For example, Unit 5, Lesson 14 integrates MP5, and MP8.

In addition, Pacing Guides and Dependency Diagrams within the Course Overview outline the number of lessons and suggested teaching days per unit, supporting year-long planning and implementation. Each lesson includes references to the standards addressed and often notes how the lesson builds on prior learning.

The materials reviewed for Kiddom IM® v.360 Grade 3 through Grade 5 meet expectations for explaining the program’s instructional approaches, identify research-based strategies, and explain the role of the standards. Examples include:

• Course Overview, Problem-Based Teaching and Learning, “Illustrative Mathematics is 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 offers teachers the opportunity to deepen their knowledge of mathematics, students’ thinking, and their own teaching practice. The curriculum and the professional-learning materials support students’ and teachers’ learning, respectively. This document defines the principles that guide IM’s approach to mathematics teaching and learning. It then outlines how each component of the curriculum supports teaching and learning, based on these principles.”

• Course Overview, Problem-Based Teaching and Learning, Learning Mathematics By Doing Mathematics, “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 problem solvers learning the mathematics.”

• Course Overview, Key Structures in This Course, Coherent Progression, “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. Activities and lessons are parts 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.”

• Course Overview, What’s in an IM Lesson, Instructional Routines, Instructional routines (IRs) in the materials are designed to promote student engagement in mathematical conversations through predictable, discourse-driven structures. Instructional Routines state, “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). A small, intentionally selected set of IRs is used throughout the curriculum to support consistent implementation and reduce cognitive load for teachers. Each routine is aligned to specific unit, lesson, or activity learning goals and is intended to support student access to mathematics by requiring them to think and communicate mathematically. Routines are identified by name within activities, and professional learning includes classroom videos and opportunities for educators to observe, practice, and reflect on their use.

The materials reviewed for Kiddom IM® v.360 Grade 3 through Grade 5 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Course Overview includes a comprehensive Required Materials list detailing all materials needed for the grade-level content. Each unit also provides a list titled Materials Needed, which outlines the materials required throughout the unit. At the lesson level, the Required Materials section specifies what is needed for that specific lesson; if no materials are required, this section will indicate that by being left blank or stating "none."

An example in Grade 3 includes: 

• Unit 5, Fractions as Numbers, Materials Needed, Materials to Gather, “Base-ten blocks, Colored pencils, Colored pencils, crayons, or markers, Materials for creating a display, Materials from a previous lesson, Number cubes, Paper clips, Rulers or straightedges, Scissors, String.” Materials Needed, Lesson 5, “Materials to Gather,  Scissors: Activity 2, Materials to Copy, Card Sort Number Lines Cards: Activity 1, Fold and Label the Number Line Handout: Activity 2.”

An example in Grade 4 includes: 

• Unit 2, Fraction Equivalence and Comparison, Lesson 17, Activity 1, Required Materials, “Markers, Paper clips, Tape (painter’s or masking), Paper.”

An example in Grade 5 includes: 

• Unit 6, Shapes on the Coordinate Grid, Lesson 14, Activity 1, Required Materials, “Paper clips, Pencils.”

The materials reviewed for Kiddom IM® v.360 Grade 3 through Grade 5 meet expectations for providing consistent opportunities to determine student learning throughout the school year. The assessment system provides sufficient teacher guidance for evaluating student performance and determining instructional next steps. 

Each End-of-Unit Assessment and the End-of-Course Assessment includes answer keys and standards alignment to support teachers in interpreting student understanding. According to the Course Overview and Assessment Guidance, “All summative-assessment problems include a complete solution and standards alignment. Multiple-choice and multiple-response problems often include a reason for each potential error that a student might make. Unlike formative assessments, problems on summative assessments generally do not prescribe a method of solution.” 

An example in Grade 3 includes:

• Unit 5, Fractions as Numbers, End-of-Unit Assessment, Question 4 states, “Which fraction is equivalent to \frac{9}{3}? A: \frac{1}{3}, B: \frac{10}{4}, C: \frac{6}{2}, D: \frac{8}{2}.” The Narrative for Question 4 states, “Students identify a fraction that is equivalent to a whole number expressed as a fraction. While they are not directly asked to write \frac{9}{3} as a whole number, the most likely reasoning is to identify that this is 3 wholes and so is \frac{6}{2}. Students may select A if they confuse the meanings of the numerator and the denominator, and think of the fraction \frac{1}{3}. They may select B if they add 1 to both the numerator and the denominator. They may select D if they subtract 1 from both the numerator and the denominator.” The answer key aligns the item to standards 3.NF.3a and 3.NF.3c, and includes sample representations to support teacher interpretation of student strategies.

An example in Grade 4 includes:

• Unit 3, Extending Operations to Fractions, End-of-Unit Assessment, Question 5 states, “The line plot shows the lengths of some colored pencils. (There is an image of a line graph showing colored pencil lengths in inches.) Part A. What is the difference in length between the longest pencil and the shortest pencil shown in this line plot? Show your reasoning. Part B. How many pencils measure 4\frac{1}{2}inches or more? Part C. Two more colored pencils measure 2\frac{1}{4} inches and 5\frac{1}{8} inches. Plot these measurements on the line plot.” The Narrative for Question 5 states, “Students interpret the measurement data on the line plot to answer questions and use the data to subtract fractions. For the first question, students may use the numbers on the line plot to help find the difference, or they may reason more abstractly as in the provided solution.” The answer key aligns the item to standard 4.MD.4, and includes sample representations to support teacher interpretation of student strategies.

An example in Grade 5 includes:

• Unit 2, Fractions as Quotients and Fraction Multiplication, End-of-Unit Assessment, Question 5 states, “A hiking trail is 7 miles long. Han hikes \frac{1}{3} of the trail and then stops for water. Jada hikes \frac{2}{3} of the trail and then stops for water. Part A. How many miles did Han hike before stopping for water? Part B. How many miles did Jada hike before stopping for water?” The Narrative for Question 5 states, “Students multiply a whole number by a fraction to solve a story problem. No representation for the problem is requested so students may draw a tape diagram (or discrete diagram), or an area diagram, or they may reason about the quantities without a picture. Students may use the strategy of doubling the distance they found for Han because they know \frac{2}{3} is twice as long as \frac{1}{3}. If students calculate Han's distance incorrectly before doubling it to find Jada's distance, their answer for Jada's distance should still count as correct.” The answer key aligns the item to standards 5.NF.4a and 5.NF.6, and includes sample representations to support teacher interpretation of student strategies.

The materials also include guidance for determining next instructional steps, integrated into both formative and summative assessment opportunities. Most lessons conclude with a Cool-down task designed to assess student thinking in relation to the lesson’s learning goal. The Course Overview section titled Key Structures in This Course and Authentic Use of Contexts and Suggested Launch Adaptations: Response to Student Thinking includes the following description:

• “The materials offer guidance to support students in meeting the learning goals. This guidance falls into one of two categories, Next-Day Support or Prior-Unit Support, based on anticipated student responses. This guidance offers ways to continue teaching grade-level content, with appropriate and aligned practice and support for students. These suggestions range from providing students with more concrete representations in the next day’s lesson to recommending a section from a prior unit, with activities that directly connect to the concepts in the lesson.”

In addition to this formative support, the materials provide teachers with structured guidance following summative assessments. The End-of-Unit Assessment Guidance describes how teachers might observe patterns of student understanding and offers suggestions for addressing unfinished learning alongside upcoming grade-level instruction: “The End-of-Unit Assessment Guidance includes example observations of students’ unfinished learning and strategies for support in the Next-Unit Support. The guidance is organized around evidence for understanding and mastery of the grade-level content standards. Rather than provide item-by-item analysis, the observations encourage analyzing multiple items (when appropriate) to look for evidence of what students understand about the standards. The Next-Unit Support offers ideas for how to address any unfinished learning alongside upcoming grade-level work or before the concept is needed for upcoming grade-level work. These supports include suggestions for questions to ask during activities, representations to use, centers to encourage, and ways to incorporate the End-of-Unit Assessment as an additional learning opportunity. When needed, supports also include ways to revisit activities (for example, a Card Sort) in new ways to build on what students already know and focus on both unfinished and new learning.” 

For example, in Grade 3: 

• Unit 2, Area and Multiplication, End-of-Unit Assessment, Question 5, Responding To Student Thinking states, “Observation: Students show they understand that area can be found by decomposing the figure, but only find parts of the area or use values that do not match the lengths of the sides. Next Unit Support: As they play Stage 2 of Rectangle Rumble in the next unit, invite students to compose rectilinear figures from the rectangles that share sides on the gameboards. Ask them to identify the areas of the rectilinear figures they create. Before they multiply with greater numbers in Unit 4 Section C, consider inviting selected students to use inch squares to recreate the problems in the assessment (or related practice problems). Encourage students to physically decompose the area into non-overlapping rectangles and use what they notice to identify the area of the rectilinear figure. Observation: Students find the area of a rectangle in which the unit squares are visible, but do not yet show they can find the area of a rectangle by multiplying the side lengths. Students show they understand area can be found by performing an operation with the values of the side lengths, but do not consistently use multiplication. Next Unit Support: As students play Stage 2 of Rectangle Rumble in the next unit, invite them to connect the factors to the side lengths of the rectangles they draw. Prompt students to draw a rectangle with the same dimensions, using inches as the unit for each side. Ask them how they could find the area in square inches.”

For example, in Grade 4: 

• Unit 1, Factors and Multiplies, End-of-Unit Assessment, Question 1, Responding To Student Thinking states, “Observation: Student responses show they may be confusing the meaning of factor and multiple or prime and composite. Next Unit Support: Throughout the course, students will apply the idea of factors and multiples. In the next unit as students play Mystery Number, Stages 3  and 4, ask them to reflect on how they used the given vocabulary (which includes prime, composite, factor, and multiple) to create clues and to guess their partner’s number. Observation: Students list only some of the factor pairs for a given number. Next Unit Support: Before students use the language of multiples and factors to find equivalent fractions in Section B of the next unit, provide opportunities for students to continue to associate side lengths of rectangles (factors 1–5) for a given area as they play Can You Build It, Stage 2. Ask how they know that they have found all the possible rectangles. Observation: Students show they may understand factor, multiple, prime, and composite, but may not yet be fluent with their multiplication facts. For example, they say 27 is prime because they do not determine that 3\times 9=27. Next Unit Support: Before students work with multiplying numerators and denominators ( 2, 3, 4, 5, 6, 8 ) to generate equivalent fractions in Section B of the next unit, invite them to play Capture Squares, Stage 7. Ask students to reflect on which facts they know and which they need some support.”

For example, in Grade 5:

• Unit 7, Shapes on a Coordinate Grid, End-of-Unit Assessment, Question 4, Responding To Student Thinking states, “Observation: Students show they may understand the attributes of different categories of shapes, but may not yet understand which categories are subcategories of others or which attributes distinguish subcategories of larger categories. Next Unit Support: During the next unit, consider inviting selected students to revisit the Quadrilaterals Grade 5 Card Sort. Ask students to find shapes that fit certain attributes and explain how they know to help them make sense of the problems on the assessment. For example, find a rhombus that is not a square, and explain how you know it is a rhombus but not a square. Alternatively, ask students to find all the shapes that are parallelograms and ask them if all of the parallelograms they found are rhombuses.  Encourage students to use the class chart that shows the hierarchy of quadrilaterals to help them find the shapes.”

The materials reviewed for Kiddom IM® v.360 Grade 3 through Grade 5 meet expectations for providing strategies and support for students in special populations to work with grade-level content and meet or exceed grade-level standards, which support their regular and active participation in learning. Examples include:

• Course Overview, Advancing Mathematical Language and Access For Multilingual Learners, Mathematical Language Development And Access For Multilingual Learners states, “To support students who are learning English in their development of language, this curriculum includes instruction devoted to advancing language development alongside mathematics learning, and fostering language-rich environments in which there is space for all students to participate.” Mathematical Language Routines states, “Mathematical Language Routines (MLRs) are instructional routines that provide structured but adaptable formats for amplifying, assessing, and developing students' language. The MLRs included in this curriculum were selected because they simultaneously support students’ learning of mathematical practices, content, and language. They are particularly well-suited to meet the needs of linguistically and culturally diverse students who are learning mathematics while concurrently acquiring English.” MLRs are included in select activities of each unit, and are described in the Teacher Guide for the lessons in which they appear within the Activity Narrative, Supporting Multilingual Learners.

• Course Overview, Universal Design for Learning and Access for Students with Disabilities, Access For Students With Disabilities states, “Supplemental instructional strategies, included in Access for Students with Disabilities of each lesson, increase access, reduce barriers and maximize learning. Each support is aligned to the Universal Design for Learning Guidelines (udlguidelines.cast.org), and based on one of the three principles of UDL, providing alternative means of engagement, representation, or action and expression. These supports offer additional ways to adjust the learning environment so that students can access activities, engage in content, and communicate their understanding. Supports are tagged, with the areas of cognitive functioning they are designed to address, to help identify and select appropriate supports for students. Designed to facilitate access to Tier 1 instruction by capitalizing on students’ strengths to address obstacles related to cognitive functions or challenges, these strategies and supports are appropriate for any student who needs additional support to access rigorous, grade-level content. Use these lesson-specific supports, as needed, to help students succeed with a specific activity, without reducing the mathematical demands of the task. Phase them out as students gain understanding and fluency. Use a UDL approach and students’ IEPs, their strengths, and their challenges to ensure access. When students may benefit from alternative means of access or support, draw on ideas from the tables below or visit udlguidelines.cast.org for more information.”

• Course Overview, Universal Design for Learning and Access for Students with Disabilities, Accessibility For Students With Visual Impairments states, “For students with visual impairments, accessibility features are built into the materials: 1. A palette of colors distinguishable to people with the most common types of color blindness. 2. Tasks and problems are designed so that success does not depend on the ability to distinguish between colors. 3. Mathematical diagrams, presented in scalable vector graphic (SVG) format, can be magnified, without loss of resolution, and rendered in Braille. 4. Where possible, text associated with images is not part of the image file, but rather included as an image caption accessible to screen readers. 5. Alt text on all images makes interpretation easier for users accessing the materials, with a screen reader. All images in the curriculum have alt text: a very short indication of the image’s contents, so that the screen reader doesn’t skip over as if nothing is there. Some images have a longer description to help students’ with visual impairments recreate the image in their mind. Understand that students with visual impairments likely will need help accessing images in lesson activities and assessments. Prepare appropriate accommodations. Accessibility experts, who reviewed this curriculum, recommended that eligible students have access to a Braille version of the curriculum materials, because a verbal description of many of the complex mathematical diagrams is inadequate to support their learning.”

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

• Unit 7 in Grade 4, Angles and Angle Measurement, Lesson 9, Activity 2, 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. Advances: reading, writing.”

• Unit 5 in Grade 5, Place Value Patterns and Decimal Operations, Lesson 8, Activity 2, 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 Kiddom IM® v.360 Mathematics Grade 3 through Grade 5 meet expectations for regularly providing extensions and/or opportunities for advanced students to engage with grade-level mathematics at greater depth. Examples include:

• Course Overview, What's in an IM Lesson?, Practice Problems, Exploration Problems states, “Each section has two or more exploration practice problems that offer differentiation for students ready for a greater challenge. There are two types of exploration problems. One type is a hands-on activity directly related to the material of the unit that students complete either in class, if they have free time, or at home. The second type of exploration problem is more open-ended and challenging. These problems go deeper into grade-level mathematics. They are not routine or procedural, and they are not ‘the same thing again but with harder numbers.” While there are no instances where advanced students do more assignments than their classmates, materials do provide multiple opportunities for students to investigate grade-level content at greater depth.

• Course Overview, Key Structures in This Course, Authentic Use of Contexts and Suggested Launch Adaptations, Advancing Student Thinking states, “This section offers look-fors and questions to support students as they engage in an activity. Effective teaching requires supporting students as they work on challenging tasks, without taking over the process of thinking for them (Stein, Smith, Henningsen, & Silver, 2000). As teachers monitor during the course of an activity, they gain insight into what students know and are able to do. Based on these insights, the Advancing Student Thinking section provides questions that advance students’ understanding of mathematical concepts, strategies, or connections between representations.” Respond to Student Thinking states, “Most lessons end with a Cool-down to formatively assess students’ thinking in relation to the learning goal of the day. The materials offer guidance to support students in meeting the learning goals. This guidance falls into one of two categories, Next-Day Support or Prior-Unit Support, based on anticipated student responses. This guidance offers ways to continue teaching grade-level content, with appropriate and aligned practice and support for students. These suggestions range from providing students with more concrete representations in the next day’s lesson to recommending a section from a prior unit, with activities that directly connect to the concepts in the lesson.”

• In Grade 3, Unit 7, Two-dimensional Shapes and Perimeter, Section B: What is Perimeter?, Section B Practice Problems, Exploration 6, “Part A: Draw some different shapes that you can find the perimeter of. Then find their perimeters. Part B: Can you draw a rectangle whose perimeter is an odd number of units? Explain or show your reasoning. A. Yes, you can draw a rectangle whose perimeter is an odd number of units. B. No, you can't draw a rectangle whose perimeter is an odd number of units. Part C: Can you draw a pentagon or hexagon (or a figure with even more sides) whose perimeter is an odd number of units? A. Yes, you can draw a pentagon or hexagon (or a figure with even more sides) whose perimeter is an odd number of units. B. No, you can't draw a pentagon or hexagon (or a figure with even more sides) whose perimeter is an odd number of units.”

• In Grade 4, Unit 2, Fraction Equivalence and Comparison, Section C: Fraction Comparison, Section C Practice Problems, Exploration 7, “Jada lists these fractions that are all equivalent to 1⁄2 : 2⁄4, 3⁄6, 4⁄8, 5⁄10 She notices that each time the numerator increases by 1, the denominator increases by 2. Will this pattern continue? A. Yes, the pattern will continue. B. No, the pattern will not continue.”

• In Grade 5, Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section C: Let’s Put It to Work, Section C Practice Problems, Exploration 5, “The Pentagon has 5 floors. The Empire State Building has 102 floors. Noah says that the Empire State Building is larger. Do you agree? Explain how you know.”

The materials reviewed for Kiddom IM® v.360 Grade 3 through Grade 5 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they support and, when appropriate, are connected to written methods.

Teachers can assign digital manipulatives to individual students or the entire class based on student need, allowing for personalized learning experiences.

Course Overview, Key Structures in This Course, Purposeful Representations states, “Curriculum representations, and the grade levels at which they are used, are determined by their usefulness for particular mathematical learning goals. More concrete representations are introduced before those that are more abstract. For example, in IM Kindergarten, students begin by counting and moving objects—before they represent these objects in 5- and 10-frames—to lay the foundation for understanding the base-ten system. In later grades, these familiar representations are extended so that as students encounter greater numbers, they use place-value diagrams and more symbolic methods, such as equations, to represent their understanding. When appropriate, the reasoning behind the selection of certain representations in the materials is made explicit.” Manipulatives are referenced within lessons as appropriate to support concept development. 

An example in Grade 3 includes:

• Unit 2, Area and Multiplication, Lesson 2, Activity 1, students make figures out of square tiles and order the figures from smallest to largest. Launch states, “Groups of 4. Give each group inch tiles. ‘Take some tiles and build a shape.’ 2 minutes: independent work time.” Activity states, “Now work with your group to order the figures. Be prepared to explain how you ordered the figures’ 5 minutes: group work time, Monitor for groups who order by: The amount of space the figure takes up on the table. The length of the figure. The number of tiles used to create the figure.” 

An example in Grade 4 includes:

• Unit 3, Extending Operations to Fractions, Lesson 2, Activity 1, students interpret multiplication expressions and diagrams as the number of groups and the amount in each group, and match representations of the same quantity. Launch states, “Groups of 2. Give each group a set of cards. Activity states, “‘This set of cards include expressions and diagrams. Match the expressions to the diagrams. Work with your partner to explain your reasoning. Some expressions will not have a matching diagram.’ 5 minutes: partner work time. Monitor for students who reason about the number of groups and the amount in each group as they match. Pause for a discussion. Invite students to share their matches and their reasoning. Highlight reasoning that clearly connects one factor in the expression to the number of groups and the other factor to the size of each group. ‘Now you will complete an unfinished diagram for 7\times \frac{1}{8}, and then draw a new diagram for an expression without a match.’ 5 minutes: independent work time.”

An example in Grade 5 includes:

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 17, Activity 2, students find products of a whole number and some tenths or hundredths using a hundredths grid or a strategy that made sense to them. Launch states, “Groups of 2. Make hundredths grids available for students. Activity states, ‘Take a few minutes to find the value of the expressions in the first problem.’ 1-2 minutes: quiet think time. 5 minutes: partner work time. Question 1, Find the value of each expression. Explain or show your reasoning. Part A: 3\times 0.5.”