K-2nd Grade - Gateway 1
Back to K-2nd Grade Overview
Note on review tool versions
See the series overview page to confirm the review tool version used to create this report.
- Our current review tool version is 2.0. Learn more
- Reports conducted using earlier review tools (v1.0 and v1.5) contain valuable insights but may not fully align with our current instructional priorities. Read our guide to using earlier reports and review tools
Loading navigation...
Focus and Coherence
| Score | |
|---|---|
Gateway 1 - Meets Expectations | 100% |
Criterion 1.1: Focus | 12 / 12 |
Criterion 1.2: Coherence | 8 / 8 |
The materials reviewed for Illustrative Mathematics® v.360, Kindergarten through Grade 2 meet expectations for focus and coherence. For focus, the materials assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards. For coherence, the materials are coherent and consistent with the CCSSM.
Criterion 1.1: Focus
Information on Multilingual Learner (MLL) Supports in This Criterion
For some indicators in this criterion, we also display evidence and scores for pair MLL indicators.
While MLL indicators are scored, these scores are reported separately from core content scores. MLL scores do not currently impact core content scores at any level—whether indicator, criterion, gateway, or series.
To view all MLL evidence and scores for this grade band or grade level, select the "Multilingual Learner Supports" view from the left navigation panel.
Materials assess grade-level content and give all students extensive work with grade-level problems to meet the full intent of grade-level standards.
The materials reviewed for Illustrative Mathematics® v.360, Kindergarten through Grade 2 meet expectations for focus. They assess grade-level content, clearly identify the content standards and mathematical practices assessed in formal assessments, offer opportunities for students to demonstrate the full intent of grade-level standards and mathematical practices across the series, and provide all students with extensive work on grade-level problems to support mastery of grade-level expectations.
Indicator 1a
Materials assess the grade-level content and, if applicable, content from earlier grades.
The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.
The materials for Kindergarten are divided into eight units, each containing an End-of-Unit Assessment. Unique to Kindergarten, Unit 1 End-of-Unit Assessment includes an interview assessment. All other units include an End-of-Unit written assessment. The Unit 8 Assessment is an End-of-Course Assessment, including problems from the entire grade level. An example of an End-of-Unit Assessment in Kindergarten includes:
Unit 4, Understanding Addition and Subtraction, End-of-Unit Assessment, Problem 5, ‘Write the value of each expression. 1. 2 + 3 =___, 2. 8 - 1 =___, 3. 3 + 0 =___.” (K.OA.1)
The materials for Grade 1 are divided into eight units, each containing an End-of-Unit Assessment. The Unit 8 Assessment is an End-of-Course Assessment, including problems from the entire grade level. An example of an End-of-Unit Assessment in Grade 1 includes:
Unit 2, Addition and Subtraction Story Problems, End-of-Unit Assessment, Problem 5, ”Kiran plays Shake and Spill with 7 counters. 2 counters are red. How many counters are yellow? Circle 2 equations that match the story. A. 7 - 2 =____, B. 7 + 2 =____, C. 9 - 7 =____, D. 7 = 2 + ____, E. 2 + 5 = ____” (1.OA.1, 1.OA.4,1.OA.7)
The materials for Grade 2 are divided into nine units, each containing an End-of-Unit Assessment. The Unit 9 Assessment is an End-of-Course Assessment, including problems from the entire grade level. An example of an End-of-Unit Assessment in Grade 2 includes:
Unit 6, Geometry, Time, and Money, End-of-Unit Assessment, Problem 4, “1. Split the circle into 4 equal parts. 2. Is 4 fourths of a circle the whole circle? How do you know?” (2.G.3)
Indicator 1b
Assessment information is included in the materials to indicate which standards are assessed.
The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for having assessment information included in the materials to indicate which standards are assessed.
Formal assessments, including End-of-Unit and End-of-Course Assessments, consistently align with grade-level content standards.
An example from Kindergarten includes:
Unit 6 End-of-Unit Assessment answer key specifies the standards addressed for each problem, such as Problem 1, which aligns with K.CC.5 and K.NBT.1: “Draw 17 dots. Use the 10-frame if it helps you.” The materials also provide guidance for assessing Mathematical Practices (MPs).
An example from Grade 1 includes:
Unit 6 End-of-Unit Assessment answer key specifies the standards addressed for each problem, such as Problem 3, which aligns with 1.MD.1: “The noodle is shorter than the pencil. Circle 2 true statements. A. The straw is longer than the noodle. B. The straw is shorter than the noodle. C. The noodle is longer than the straw. D. The noodle is shorter than the straw.” The materials also provide guidance for assessing Mathematical Practices (MPs).
An example from Grade 2 includes:
Unit 2 End-of-Unit Assessment answer key specifies the standards addressed for each problem, such as Problem 3, which aligns with 2.NBT.5: “Find the number that makes each equation true. Show your thinking using drawings, numbers, or words. 1. 23+19=___. 2. 75-36= ___.” The materials also provide guidance for assessing Mathematical Practices (MPs).
According to the Course Guides for Kindergarten through Grade 2, 9. Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. Some instructional routines are generally associated with certain MPs. For example, the Card Sort routine often asks students to reason abstractly and quantitatively (MP2), and to look for and make use of structure (MP7). The Estimation Exploration offers students opportunities to share a mathematical claim and the thinking behind it (MP3), and to make an estimate or a range of reasonable answers, with incomplete information, which is a part of modeling with mathematics (MP4). The unit-level Mathematical Practice Chart is meant to highlight a handful of lessons in each unit that showcase certain MPs.” Examples include:
“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, a ruler, patty paper, a graph, a table, or external resources. I can use tools to help explain my thinking. I can use a variety of math tools to solve a problem.”
“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 I’m having a hard time, and try again. I can show that I try to figure out or solve the problem at least once. I can check that my solution makes sense.”
“MP3: I Can Construct Viable Arguments and Critique the Reasoning of Others. I can explain or show my reasoning in a way that makes sense to others. I can listen to and read the work of others and offer feedback to help clarify or improve their work. I can explain my reason for why something is true.”
Indicator 1c
Assessments include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.
The materials reviewed for Illustrative Mathematics® v.360 Grade Kindergarten through Grade 2 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 checkpoints 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.
An example of a summative assessment item in Kindergarten includes:
Unit 2, Number 1-10, End-of-Unit Assessment develops the full intent of K.CC.6, identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies. Problem 3, “1. Circle the group that has more things. (2 images: a 5 frame with dots in each box and 1 added dot outside of the frame and an image of two hands with 8 fingers raised.) 2. Circle the group that has fewer things. (2 images: a straight row of 7 black dots and a circular configuration of 9 black dots).”
An example of a summative assessment item in Grade 1 includes:
Unit 8, Putting It All Together, End-of-Course Assessment develops the full intent of MP3, construct viable arguments and critique the reasoning of others, as students use two different methods to subtract from a teen number. Problem 8, “1. Kiran says “$$15-11$$ is 4. I counted 14, 13, 12, 11. That’s 4.” Explain why Kiran is correct. 2. Elena says “$$15-11$$ is 4. I counted 12, 13, 14, 15 to get 15 so that’s 4.” Explain why Elena is correct.”
An example of a summative assessment item in Grade 2 includes:
Unit 1, Adding, Subtracting, and Working with Data, End-of-Unit Assessment develops the full intent of 2.OA.2, fluently add and subtract within 20 using mental strategies. Problem 3, “Find the number that makes each equation true. 1. 7+___$$=18$$. 2. 20-___$$=12$$. 3. 9+7=___. 4. 19-14=___.”
Indicator 1d
Materials give all students extensive work with grade-level problems to meet the full intent of grade-level standards.
The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.
The materials provide extensive work in Kindergarten through Grade 2 by including in every lesson a Warm-Up, one to three instructional Activities, and Lesson Synthesis. Within Kindergarten through Grade 2, students engage with all CCSS standards.
An example of extensive work in Kindergarten includes:
Unit 2, Numbers 1-10, Lessons 1, 15, and 16 engage students in extensive work with K.CC.5 (Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects.). In Lesson 1, Activity 1, Launch, students recognize and name numbers represented on fingers. Directions include, “Groups of 2, Hold up 3 fingers. ‘How many fingers am I holding up?’ 30 seconds: quiet think time, Share responses. Repeat with 6 fingers. ‘Take turns playing with your partner. One partner holds up some fingers. The other partner figures out how many fingers they are holding up.’” In Lesson 15, Lesson Synthesis, students practice matching a number to the number of items it represents. “Write the number 5 and display the 2 cards. ‘Mai matched the fingers and the dots in a circle to the number 5. Diego says that the fingers and dots can’t both show 5 because the dots are in a circle and the fingers are in a line. What do you think?’ (There are 5 fingers and 5 dots, so it doesn’t matter how they are arranged.) ‘There are 5 fingers and 5 dots, so they can both go with the number 5. There are many different ways to show numbers. We can use ‘5’ to describe many groups that look different, but all have the same number of things.’” In Lesson 16, Lesson Synthesis, students draw or group objects to match a number. “‘Today we counted out objects to show numbers. What other things could we do to show the number 9?’ (We could show 9 fingers. We could draw a picture with 9 things.) ‘Let’s practice counting to 20.’ Demonstrate counting to 20. Count to 20 as a class 1-2 times.”
An example of extensive work in Grade 1 includes:
Unit 3, Adding and Subtracting Within 20, Lesson 3 and Lesson 16, engages students in extensive work with 1.OA.3 (Apply properties of operations as strategies to add and subtract.) In Lesson 3, Activity 1, students sort addition expressions by their value. “Give each group 1 set of addition expression cards. Or 1 student in each group can retrieve their cards from a previous lesson. ‘Today we are going to sort the cards into groups with the same value.’ Display an addition expression card, such as 2+5. ‘I know the value of this sum is 7. It is a sum that I just know. I will start a pile for sums of 7. This set of cards show addition expressions. Sort the cards into groups with the same value. Work with your partner to explain why the cards go together. Make sure that each partner has a chance to find the value before you place the card in a group. If you and your partner disagree, work together to find the value of the sum.’” Activity 2, students determine whether equations are true or false. Student Task Statement, “Determine whether each equation is true or false. Be ready to explain your reasoning. 1.) 4+2=2+4. 2.) 3+6=6+4. 3.) 5+3=1+7. 4.) 6+4=5+3. 5.) 6+2=8+1.” Lesson 16, Warm-up, Related Expressions, students use strategies and understandings students have for adding on to. Teacher Guide, “Display one expression. ‘Give me a signal when you have an answer and can explain how you got it.’” Student Task Statements, “Find the value of each expression mentally. 7+10, 7+2+8, 10+9, 4+9+6.”
An example of extensive work in Grade 2 includes:
Unit 4, Addition and Subtraction on the Number Line, Lessons 2 and 3; Unit 5, Numbers to 1,000, Lesson 1; Unit 6, Geometry, Time, and Money, Lesson 12; and Unit 7, Addition and Subtraction Within 1,000, Lesson 1, students engage with extensive work with 2.NBT.2 (Count within 1000; skip-count by 5s, 10s, and 100s)., as students choral count from 0-100 by 5s. In Lesson 2, Warm Up, “‘Count by 5, starting at 0.’ Record as students count. Stop counting and recording at 100.” In Lesson 3, Cool-down, Problem 1, students are provided a number line that begins at 15, includes large tick marks in intervals of 5 and small tick marks in intervals of 1. The numbers 15, 20 and 45 are included on the number line, and there are blanks below the other multiples of 5. “‘Complete each number line by filling in the missing labels with the number the tick mark represents.’ b, Locate and label 37 on the number line.” Problem 2 is similar, but includes blanks only below multiples of ten. Students are asked to locate and label 35 on the number line. In Unit 5, Lesson 1, Warm Up, “‘Count by 1, starting at 90.’Record as students count. Record 10 numbers in each row. Then start a new row directly below. Stop counting and recording at 120.” In Unit 6, Geometry, Time, and Money, Lesson 12, Warm Up, students count by 5s on a clock. The Warm Up includes an image of an analog clock with the labels 5, 10, 15, etc. shown outside the clock adjacent to 1, 2, 3, etc. In Unit 7, Add and Subtract within 1,000, Lesson 1, students count by 100s. In Activity 2, students complete number lines. “‘1. Fill in the missing numbers. Does this number line show counting on by tens or counting on by hundreds?’ The problem includes an image of a number line with labels at 502, 702, and 902, and blanks at 602, 802. Problems 2 and 3 are similar, but include different images on the number lines. On the Cool Down, students count by 100. Problem 2, “Complete the list of numbers to show counting on by 100. 552, ___, ___, 852, 952. Explain how you know your list shows counting on by 100 and not counting on by 10.”
The materials provide opportunities for students to engage in the full intent of the standards in Kindergarten through Grade 2 by including in every lesson a Warm-Up, one to three instructional Activities, and Lesson Synthesis. In Kindergarten through Grade 2, students engage with all CCSS standards.
An example in Kindergarten includes:
Unit 5, Composing and Decomposing Numbers to 10, Lessons 12 and 13 engage students with the full intent of K.OA.4 (For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation). In Lesson 12, Activity 2, students are asked to find how many counters are needed to fill a 10-frame. “Figure out how many counters are needed to fill each 10-frame. Write a number to show how many counters are needed to fill it. Circle the equation that shows the number of counters in the 10-frame and the number of counters needed to fill the 10-frame.” In Lesson 13, Activity 1, students are asked to show a number on their fingers and determine how many fingers are needed to make 10. Students fill in an equation to represent each composition and decomposition of 10. “I rolled 7, so I am going to hold up 7 fingers. Now my partner needs to figure out how many more fingers I need to put up to show 10 fingers. How many more fingers do I need to hold up to make 10? Now we need to fill in an equation to show how many fingers are up and how many more fingers are needed to make 10. How should I fill in an equation? Take turns with your partner rolling to find a number and showing that number with your fingers. Your partner figures out how many more fingers are needed to make 10. You both fill in an equation to show how many fingers are up and how many more fingers are needed to make 10.”
An example in Grade 1 includes:
Unit 6, Length Measurements Within 120 Units, Lesson 1 and Lesson 2, students engage with the full intent of 1.MD.1 (Order three objects by length; compare the lengths of two objects indirectly by using a third object.) In Lesson 1, Activity 2, students order 3 objects from shortest to longest and longest to shortest. Students are provided a collection of objects, “Give each group of 10-12 objects to measure.” Students use these objects to respond to the following task, “1. Pick 3 objects. Put the objects in order from shortest to longest. Trace or draw the objects in that order. 2. Pick 3 new objects. Put them in order from longest to shortest. Write the names of the objects in that order.” In Lesson 2, Activity 2, students use string to measure objects that cannot be lined up end to end to make comparisons about their length, “‘We saw that sometimes we can compare length without lining up the objects. Now, you are going to compare the length of a side of your desk to the length of one of the legs of your desk.’ Display the image. ‘This image shows which side we will be measuring. Trace the length of the side you will measure with your finger. Why is it important that everybody knows which side of the desk we should measure? Does it matter which leg of the desk you measure?’ (One side is longer than the other, so we need to make sure we are measuring the same thing. All the legs are the same length, so it shouldn’t matter which one we measure.)” Students complete, “‘Compare the length of the side of your desk and the length of one of the legs of your desk using the string. Then show how you know one is longer using words or drawings.’” Section A Practice Problems also provide students with an opportunity to compare the length of objects. Problem 4 shows an image of different length rectangles labeled A, B, and C. “Use any tool you would like to compare the length of the rectangles. List the rectangles from longest to shortest.” Problem 5 has students compare the length of objects indirectly, “Look at the top and side of your workbook. Which is longer? Show your thinking using drawings or words.”
An example in Grade 2 includes:
Unit 3, Measuring Length, Lesson 4, Lesson 5, Lesson 8, and Lesson 9 meet the full intent of 2.MD.3 (Estimate lengths using units of inches, feet, centimeters, and meters.) Lesson 4, Activity 1, students make estimates of an object’s length in centimeters, beginning with a notebook. The guidance instructs the teacher to show the notebook next to a 10 cm tool, “‘Let’s look at another image of the object.’ Display the image or hold a folder next to a 10-centimeter tool.” After launching the activity, the teacher instructs students to estimate the length of different objects, “‘Now look at the objects I gave each group and think about how long they are. Record your estimates on the recording sheet on your own. When you and your partner finish, compare your estimates and explain why you think they are ‘about right.’” In Activity 2, students measure the objects and compare their estimates to the measurements. In Lesson 5, students estimate the length of an object in meters during the Cool-down, “Noah held a gecko at the zoo. The length of the gecko fit in his hands. He measured it and said it was about 13 meters long. Do you think his measurement is correct? Why or why not?” In Lesson 8, Activity 2, students estimate the length of the sides of different shapes that are pictured in the materials, “1. Here is a rectangle. Estimate the length of the long side of the rectangle. Use inches. Estimate: ___ in. Measure the long side of the rectangle. Use inches. Actual length: ___ in.” Problem 2 follows the same structure but includes a square, while Problem 3 is about a triangle. Finally, in Lesson 9, Activity 2, students estimate lengths in feet, “Estimate the length of objects around the room. Decide if you will measure in inches or feet. Circle the unit” The Lesson 9 Cool-down also provides an opportunity to estimate length in feet, “Tyler told Han that a great white shark is about 16 inches long, but Han disagrees. Han believes it would be about 16 feet long. Who do you agree with? Explain.”
Indicator 1d.MLL
Materials assess the grade-level content and, if applicable, content from earlier grades.
The instructional materials reviewed for Grades K-2 of Illustrative Mathematics® v.360 meet the criteria of providing support for Multilingual Learners’ (MLLs’) full and complete participation in extensive work with grade-level problems to meet the full intent of grade-level standards.
At the lesson level, the materials provide consistent, embedded strategies and scaffolds that enable MLLs to access and engage with rigorous, grade-level mathematical content. The materials intentionally designed these supports to develop both language and content knowledge through structured routines and opportunities for discourse across all four language domains—listening, speaking, reading, and writing. The Course Guide, 3. What’s in an IM Lesson describes the problem-based lesson design, which begins with a Warm-Up, then engages students with one to three instructional Activities, and ends with a Lesson Synthesis and Cool-Down formative assessment opportunity. The Course Guide, 4. Advancing Mathematical Language and Access for English Learners outlines how this lesson design centers the unique language needs of MLLs by embedding Stanford University’s four design principles: Support Sense-Making, Optimize Output, Cultivate Conversation, and Maximize Meta-Awareness. The materials feature multimodal instruction, which creates accessible entry points and structured opportunities for disciplinary language usage alongside mathematics learning. Additionally, the materials describe the language and mathematics goals in the following features: Unit Goals, Section Goals, Lesson Narrative, Lesson Purpose, and Learning Goals (both teacher- and student-facing). The Course Guide, 2. Problem-Based Teaching and Learning states, “Good instruction starts with explicit learning goals… Without a clear understanding of the learning objectives, activities in the classroom, implemented haphazardly, have little impact on advanced students’ understanding.” This is especially pertinent in English language development; language development research states that MLLs understanding clear, explicit learning goals helps to facilitate their language development by setting an authentic purpose for using language.
The materials consistently employ adapted versions of the Mathematical Language Routines (MLRs) by Stanford University UL/SCALE, which designed the MLRs to support the simultaneous development of mathematical practices, content, and language. The materials reference MLRs in two ways: in the lesson facilitation or as an additional suggestion in notes titled Access for English Language Learners.
The materials feature all eight of Stanford University UL/SCALE’s MLRs:
MLR1 Stronger and Clearer Each Time: Students construct a verbal or written response to a math problem, then verbally share their response with a partner to get feedback from the listener to improve the response, and revise their original response based on the feedback they received.
MLR2 Collect and Display: Students access their own and others’ mathematical ideas as the teacher scribes the language, strategies, and concepts students use during partner, small group, or whole-class discussions using written words, diagrams, and pictures.
MLR3 Critique, Correct, Clarify: Students rewrite a math response from an example that is incorrect, incomplete, or otherwise ambiguous.
MLR4 Information Gap: In a group, each student has different parts of a mathematical situation, and they work together to piece together that information orally or visually to bridge the gap between the parameters of the situation and a question to solve a mathematical problem.
MLR5 Co-Craft Questions: Students examine a problem stem, a graph, a video, an image, or a list of interesting facts and author a mathematical question that might be asked about the situation. With partners or as a class, they compare questions before the teacher reveals the mathematical question of the task as designed.
MLR6 Three Reads: Students are guided to read the problem three separate times with three separate purposes with quick discussions between each read.
MLR7 Compare and Connect: Students identify, compare, and contrast their own understandings with other students’ mathematical approaches, representations, concepts, examples, and language.
MLR8 Discussion Supports: Teachers provide a variety of supports to foster inclusive whole-class discussions, such as:
Revoicing or rephrasing.
Pressing for details.
Providing sentence frames.
Providing multimodal instructional suggestions (e.g. reading, writing, speaking, listening, pointing, gesturing, acting out, etc).
Using choral responses.
Modeling a think-aloud.
Providing think time.
However, while the materials note that the language domain of writing is addressed through routines such as MLR1 Stronger and Clearer Each Time, writing is not as consistently emphasized as listening and speaking. Structured writing tasks are less consistently present across lessons compared to listening and speaking tasks, which may limit opportunities for balanced development across all four language domains (see the report for 2g.MLL).
For example, in Grade 2, Unit 7, Adding and Subtracting within 1,000, Lesson 4, Activity 2, students work in partners to use place value understanding to add and subtract by place. The activity begins with a structured whole-class discussion where students compare two worked examples of two students’ solution strategies showing adding by place value. The materials direct the teacher to give students 1 minute of quiet think-time, followed by 3 minutes of a partner discussion before facilitating the whole-class discussion comparing the two worked examples. This structured think-time provides MLLs with time to translate in their heads, if needed, and an opportunity to orally rehearse their responses with a partner before participating in the whole-class discussion. After the whole-class discussion, students work independently on four problems with access to base-ten blocks, and then compare their answers and solution strategies with their partners. A note titled Access for English Language Learners suggests the use of MLR7 Compare and Connect which directs the teacher to lead a discussion comparing, contrasting, and connecting the different approaches by asking, “Did anyone solve the problem the same way, but would explain it differently?” This piece of evidence demonstrates how the materials embed supports that enable MLLs to regularly engage with grade-level mathematics in a language-rich environment; the materials do not dilute mathematical rigor but rather equip students with linguistic scaffolds that allow for their full and complete participation. This example illustrates how the materials integrate language support alongside math content.
Beyond the lesson level, the Course Guide, 2. Problem-Based Teaching and Learning describes how the material’s problem-based instructional design fosters the full and complete participation of all students. This section outlines the material’s approach to mathematics teaching and learning, including narratives describing the following principles: All Students are Capable Learners of Mathematics, Learning Mathematics by Doing Mathematics, and Community Building. Specific actions to support the principle of Community Building and the development of a math learning community are outlined in the Course Guide, 7. Key Structures in This Course. Here, a chart displays vital student actions and teacher moves to build a positive mathematics community as put forth by Phil Daro and the Strategic Education Research Partnership (SERP) Institute. This is especially pertinent in English language development; language development research states that MLLs understanding of clear, explicit learning goals helps to facilitate their language development by setting an authentic purpose for using language.
In addition, the Course Guide states that in order to extend the invitation to do mathematics to all students, explicit development of the math learning community is required. This is suggested in the first six lessons, where every grade embeds norm-building and reflection questions for teachers and students to collectively identify what it looks like and sounds like to do math together. For example, the Lesson Preparation for Grade 2, Unit 1, Adding, Subtracting, and Working with Data, Lesson 1 directs the teacher to co-create a Math Community poster that features columns labeled Doing Math and Norms with spaces for the class to describe both teacher and student actions. The materials state, “In upcoming lessons, students will add to and revise these ideas, including drafting classroom goals and expectations for the ‘Norms’ column. Keep the poster displayed in the classroom.” Additionally, the lessons in this unit feature Teacher Reflection Questions such as:
“Today’s lesson provided an opportunity to learn from your students. How were you able to incorporate your students’ lived experiences into the lesson?”
“Think about who participated in math class today. What assumptions are you making about those who did not participate? How can you leverage each of your students’ ideas to support them in being seen and heard in tomorrow’s math class?”
“Think about a time you recently made a mistake during math class. How did you leverage your mistake to show that mistakes are just learning in process?”
“Who did math today in class and how do you know? Identify the norms or routines that allowed those students to engage in mathematics. How can you adjust these norms so that all students do math tomorrow?”
Criterion 1.2: Coherence
Each grade’s materials are coherent and consistent with the Standards.
The materials reviewed for Illustrative Mathematics® v.360, Kindergarten through Grade 2 meet expectations for coherence. They address the major work of the grade, connect supporting content to the major work, and make meaningful connections across clusters and domains. The materials also clearly highlight how grade-level content builds on knowledge from prior grades and lays the foundation for future learning.
Indicator 1e
When implemented as designed, the majority of the materials focus on the major clusters of each grade.
The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade. The instructional materials devote at least 75 percent of instructional time to the major clusters of the grade.
In Kindergarten:
The approximate number of units devoted to major work of the grade (including assessments and related supporting work) is 6 out of 8, approximately 75%.
The number of lessons devoted to major work of the grade (including assessments and related supporting work) is 124 out of 139, approximately 89%.
The number of days devoted to major work of the grade (including assessments and related supporting work) is 136 out of 155, approximately 88%.
The number of days devoted to major work of the grade (including assessments and supporting work, excluding optional lessons) is 119 out of 138, approximately 86%.
In Grade 1:
The approximate number of units devoted to major work of the grade (including assessments and related supporting work) is 7 out of 8, approximately 88%.
The number of lessons devoted to major work of the grade (including assessments and related supporting work) is 124 out of 148, approximately 84%.
The number of days devoted to major work of the grade (including assessments and related supporting work) is 138 out of 164, approximately 84%.
The number of days devoted to major work of the grade (including assessments and supporting work, excluding optional lessons) is 135 out of 156, approximately 87%.
In Grade 2:
The approximate number of units devoted to major work of the grade (including assessments and related supporting work) is 7 out of 9, approximately 78%.
The number of lessons devoted to major work of the grade (including assessments and related supporting work) is 130 out of 150, approximately 87%.
The number of days devoted to major work of the grade (including assessments and related supporting work) is 144 out of 168, approximately 86%.
The number of days devoted to major work of the grade (including assessments and supporting work, excluding optional lessons) is 112 out of 136, approximately 82%.
An instructional day analysis is most representative of the instructional materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each unit. As a result, in Kindergarten, approximately 88% of the instructional materials focus on the major work of the grade. In Grade 1, approximately 84% of the instructional materials focus on the major work of the grade. In Grade 2, approximately 86% of the instructional materials focus on the major work of the grade.
Indicator 1f
Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.
The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.
Materials are designed so that supporting standards/clusters are connected to the major standards/ clusters of the grade. These connections are listed for teachers within the Pacing Guide and Dependency Diagram document.
An example of a connection in Kindergarten includes:
Unit 3, Flat Shapes All Around Us, Lesson 5, Activity 1, connects the supporting work of K.MD.3 (Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.) to the major work of K.CC.6 (Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies). Students identify examples of circles and triangles. The activity states, “‘Choose 1 triangle that you colored in. Tell your partner 1 thing that you know about that shape.’ 30 seconds: quiet think time. 30 seconds: partner discussion. ‘Write a number to show how many triangles you colored.’ 1 minute: independent work time. ‘Choose 1 circle that you colored in. Tell your partner one thing that you know about that shape.’ 30 seconds: quiet think time. 30 seconds partner discussion. ‘Write a number to show how many circles you colored.’ 1 minute: independent work time. ‘Did you color more triangles or more circles? How do you know?’”
An example of a connection in Grade 1 includes:
Unit 2, Addition and Subtraction Story Problems, Lesson 13, Activity 1 connects the supporting work of 1.MD.4 (Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another) to the major work of 1.OA.1 (Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions). Students determine whether comparison statements about data are true or false and explain how they know. The activity states, “Read the task statement. ‘Priya and Han made some statements about their data. Your job is to decide whether you agree or disagree. Once you decide, circle it on your paper.’” A chart titled “Favorite Art Supply” is displayed. Student Task Statement, “A group of students is asked, ‘What is your favorite art supply?’ Their responses are shown in this chart. 1. More students voted for crayons than markers. 2. Fewer students voted for crayons than paint. 3. 1 more student voted for paint than crayons. Show your thinking using drawings, numbers, or words. 4. 1 fewer student voted for paint than markers. Show your thinking using drawings, numbers, or words. 5. 3 more students voted for markers than crayons. Show your thinking using drawings, numbers, or words. If you have time: Change the false statements to make them true.”
An example of a connection in Grade 2 includes:
Unit 2, Adding and Subtracting Within 100, Lesson 1, Activity 1 connects the supporting work of 2.MD.10 (Draw a picture graph and a bar graph with single-unit scale) to the major work of 2.OA.1 (Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem). Students use information from a bar graph to compare different methods for solving addition and subtraction problems within 100. Student Task Statement, “Use the bar graph to answer the questions. 1. How many students chose art or reading? Show your thinking using drawings, numbers, or words. 2. How many more students chose video games than art?” A bar graph shows art, reading, and video games with values between 16 and 32.
Indicator 1g
Materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.
The instructional materials for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for including problems and activities that connect two or more clusters in a domain, or two or more domains in a grade.
Connections between major works are present throughout the grade-level materials where appropriate. These connections are listed for teachers in the Course Guide in the Dependency Diagram, and may appear in one or more phases of a typical lesson: warm-up, instructional activities, lesson synthesis, or cool-down.
In Kindergarten, Cool-downs are not always included; instead, an Observation section provides 'Look Fors' to formatively assess learning throughout the lesson. An example of a connection in Kindergarten includes:
Unit 4, Understanding Addition and Subtraction, Lesson 4, Warm-up connects the major work of K.CC.B (Count to tell the number of objects) to the major work of K.OA.A (Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from). Students count two groups of numbers to find a total. The Activity states, “‘How many students would rather be a bird? How do you know?’ Share responses. Demonstrate or invite students to demonstrate counting. ‘How many students would rather be a fish? How do you know?’ Share responses. Demonstrate or invite students to demonstrate counting.”
In Grade 1, Cool-downs are not always included; instead, an Observation section provides 'Look Fors' to formatively assess learning throughout the lesson. An example of a connection in Grade 1 includes:
Unit 4, Numbers to 99, Lesson 8, Activity 2 connects the major work of 1.NBT.A (Extend the counting sequence) to the major work of 1.NBT.B (Understand place value). Students match cards that show different base-ten representations. The Launch states, “‘Today we are going to match cards that show the same 2-digit number. For example, look at these 3 cards. Which 2 representations show the same 2-digit number? Why doesn’t the other one show the same number?’ (The first 2 cards both show 4 tens and 1 one or 41. The last card isn’t the same because it only shows 1 ten. It has the same digits, but they mean something different.)” Activity, ‘“This set of cards include base-ten diagrams, words that show ___ tens and ___ones, expressions, and 2-digit numbers. Match the cards that show the same 2-digit number.’” Three representations are provided: An image of four 4 tens and one connecting cube, 40+1 (as an expression) and 1 ten and 4 ones (written in words).
In Grade 2, an example of a connection includes:
Unit 3, Measuring Lengths, Lesson 11, Cool-Down connects the major work of 2.MD.B (Relate addition and subtraction to length) to the major work of 2.NBT.B (Use place value understanding and properties of operations to add and subtract). Students solve subtraction problems within 100 with the unknown in all positions. Student Task Statement, “Priya had a piece of ribbon that was 74 inches long. She cut off 17 in. How long is Priya’s ribbon now? Show your thinking using drawings, numbers, or words. Use a diagram if it helps. Don’t forget the unit in your answer.”
Indicator 1h
Content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.
The instructional materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.
The Course Guides contain a Scope and Sequence explaining content standard connections. Prior and Future connections are identified within materials in the Course Guide, Section Dependency Diagram, which states, “an arrow indicates the prior section that contains content most directly designed to support or build toward the content in the current section.” Some Unit Overviews, Lesson Narratives, and Activity Syntheses describe the progression of standards for the concept being taught. Each Lesson contains Preparation identifying learning standards (Building on, Addressing, or Building toward).
An example of a connection to future grades in Kindergarten includes:
Unit 6, Numbers 0-20, Lesson 7, Preparation connects K.CC.5 (Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects) to work with relating counting to addition and subtraction in 1.OA.5. Lesson Narrative, “Students write a number to represent a quantity greater than 10 for the first time. Students use full 10-frames and some more to identify and create numbers 11–19. Students may count all of the dots or counters to determine the teen number, or they may count on from 10. Counting on to determine the total is not an expectation in kindergarten.”
An example of a connection to future grades in Grade 1 includes:
Unit 6, Length Measurements Within 120 Units, Lesson 9, Warm-up connects 1.NBT.1 (Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral) to work with understanding three-digit numbers in Grade 2. The Narrative, “The purpose of this Choral Count is to invite students to practice counting by 1 from 90 to 120 and notice patterns in the count. Keep the record of the count displayed for students to reference throughout the lesson. When students notice the patterns in the digits after counting beyond 99 and explain the patterns based on what they know about the structure of the base-ten system, they look for and express regularity in repeated reasoning (MP7, MP8). Students will develop an understanding of a hundred as a unit and three-digit numbers in grade 2.”
An example of a connection to future grades in Grade 2 includes:
Unit 6, Geometry, Time, and Money, Lesson 9, Preparation connects the work of 2.G.3 (Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape) to the work with fraction equivalence in Grade 3. Lesson Narrative, In previous lessons, students partitioned circles and rectangles into halves, thirds, and fourths, and identified an equal piece of different shapes as half of, a third of, or a fourth of the shape. “In this lesson, students continue to practice partitioning circles and describe halves, thirds, and quarters of circles, using the language ‘half of,’ ‘a third of,’ and ‘a quarter of’ to describe a piece of the shape. They also use this language to describe the whole shape as a number of equal pieces. Students recognize that a whole shape can be described as ‘2 halves,’ ‘3 thirds,’ or ‘4 fourths.’ This understanding is the foundation for students’ work with a whole and fraction equivalency in grade 3.”
An example of a connection to prior knowledge in Kindergarten includes:
Unit 5, Composing and Decomposing Numbers to 10, Lesson 2, Preparation connects K.OA.3 (Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from) to work composing shapes in Kindergarten Unit 3. Lesson Narrative, “In a previous unit, students made designs with pattern blocks and counted how many of each pattern block they used. In this lesson, students make and share a design with the same total number of pattern blocks but different numbers of individual pattern blocks.
An example of a connection to prior knowledge in Grade 1 includes:
Course Guide, Scope and Sequence, Unit 1, Adding, Subtracting, and Working with Data connect 1.OA.5 (Relate counting to addition and subtraction) and 1.OA.6 (Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten; decomposing a number leading to a ten; using the relationship between addition and subtraction; and creating equivalent but easier or known sums) to work sorting objects by attributes from Kindergarten. “Students also build on the work of kindergarten as they engage with data. Previously, students sorted objects into given categories, such as by size or shape. Here, students use drawings, symbols, tally marks, and numbers to represent categorical data. They go further by choosing their own categories, interpreting representations with up to three categories, and asking and answering questions about the data.”
An example of a connection to prior knowledge in Grade 2 includes:
Unit 5, Numbers to 1,000, Lesson 1, Preparation connects 2.NBT.1 (Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones) to work with place value concepts in Grade 1 and previous work composing and decomposing tens in Grade 2. Lesson Narrative, ”In grade 1, students were introduced to a ten as a unit made of 10 ones. They used that understanding to represent two-digit numbers and add within 100. Students used connecting cubes to make and break apart two-digit numbers. In previous units in grade 2, students used the words compose and decompose as they made and broke apart tens when they added and subtracted within 100. In this lesson, students are introduced to the unit of a hundred. Building on the understanding that they can use 10 ones to compose a ten, students learn they can compose a hundred using 10 tens.”