K-2nd Grade - Gateway 1

The materials reviewed for Eureka Math² Grades Kindergarten through Grade 2 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

According to the Grades K-2 Implementation Guide, “All Eureka Math² assessments are considered formative because they are intended to inform instruction. The assessments may also be considered summative when you choose to use the data to produce a grade or report that becomes part of a student, school, or district record. Typical Module Assessments consist of 3–5 interview-style items that assess proficiency with the major concepts, skills, and applications taught in the module. Module Assessments include the most important content, but they may not assess all the strategies and standards taught in the module. Many items allow students to show evidence of one or more mathematical practices. Give this assessment when a student shows inconsistent proficiency over the course of a module based on notes you make using the Observational Assessment Recording Sheet. Module Assessments provide suggested language for the interview-style items. As needed, and if possible, consider assessing students in their home language. When students are unable to answer or they respond incorrectly to the first few questions, end the assessment, and retry after more instruction. “ In Grade 1 and Grade 2, the Assessment System includes lesson-embedded Exit Tickets, Topic Quizzes, and Module Assessments. According to the Grades 1-2 Implementation Guide, Assessment,“Exit Tickets are short assessments that close lessons. These assessments use at least one problem or question to assess whether a student has learned the basic skills and concepts needed for success in upcoming lessons. Items reflect the minimum that students must demonstrate to meet the lesson objective. A Topic Ticket replaces the Exit Ticket in the final lesson of each topic. Topic Tickets are short sets of items that assess proficiency with the major concepts and skills from the topic. Many items allow students to show evidence of one or more of the mathematical practices. Students complete Topic Tickets independently, with directions or problems read aloud as necessary. Typical Module Assessments consist of 6–10 items that assess proficiency with the major concepts, skills, and applications taught in the module. Module Assessments represent the most important content, but they may not assess all the strategies and standards taught in the module. Many items allow students to show evidence of one or more of the mathematical practices.” 

Examples of grade-level items from Module Assessments include:

• Kindergarten, Module 1, Module Assessment, Problem 3, “Give the student the bag of 10 objects. Hold up the Hide Zero 7 card. ‘(Hold up the 7 card.) Count out this many. If you get 1 more, how many will there be? Point to the number that shows 1 more than 7.’” (K.CC.3, K.CC.4a, K.CC.4b, K.CC.4c)

• Grade 1, Module 3, Module Assessment, Problem 4, students see base ten blocks showing two tens and six ones and an empty number bond with ___ tens ___ ones underneath. “Write as tens and ones.” (1.NBT.2)

• Grade 2, Module 4, Module Assessment, Problem 13, “Tim finds 35+23+17+20 Look at Tim’s work. Show a different way on your paper.” (2.NBT.6)

The materials reviewed for Eureka Math² Kindergarten through Grade 2 meet expectations for having assessment information included in the materials to indicate which standards are assessed. Materials consistently identify the standards and practices assessed for formal assessments.

The publisher states, “Every formal Eureka Math² assessment item is tagged to at least one CCSS standard, at least one associated Achievement Descriptor, and at least one Mathematical Practice.” Under the overarching Assessments tab in the Great Minds Library, the Standards link for Topic Quizzes and Module Assessments identifies the standards addressed on each assessment. The digital version also identifies the aligned standard below each problem in the Topic Quizzes and Module Assessments; the PDF print versions do not include these standard alignments. The publisher also states, “Across each grade level, all eight SMPs were tagged to multiple items, allowing students to demonstrate the full intent of the grade-level practices.” For each Module Assessment, the digital platform provides a dot chart titled “Mathematical Practice Alignment to Module Assessment Items” that identifies the Mathematical Practices aligned to individual assessment items. Examples include:

• Kindergarten, Module 6, Module Assessment, Problem 1 states, “Write 15 on a whiteboard. ‘What number is this?’ Place a connected 10-stick and 16 loose cubes in front of the student. ‘Use cubes to show me this number.’ Teacher note: Note whether students use the connected 10-stick or count out 15 from the loose cubes. (Point to the 1.) ‘Show me the cubes this digit tells about.’ (Point to the 5.) ‘Show me the cubes this digit tells about. Write the next number.’” (K.CC.3, K.CC.4c, K.CC.5, K.NBT.1, MP2, MP7)

• Grade 1, Module 2, Module Assessment, Problem 1 states, “Max had 10 nuts. He got some more. Now he has 16 nuts. How many nuts did Max get?” (1.OA.1, MP1)

• Grade 2, Module 4, Module Assessment, Problem 8 states, ”___-297=613 (2.NBT.7, MP5, MP6, MP7)

The materials reviewed for Eureka Math² Grades 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.

According to the Kindergarten Implementation Guide, Assessment, “The assessment system in kindergarten helps you understand student learning by generating data from many perspectives. The system includes a recording sheet to guide your observations during lessons and Module Assessments. This sheet lists the module’s Achievement Descriptors, or ADs.” These formal assessments consistently list grade-level content standards for each item. Students have opportunities to demonstrate the full intent of the standards using a variety of modalities (e.g., oral responses, writing, modeling, etc.). The Grades 1-2 Implementation Guide explains that the assessment system uses multiple assessments and sources of data to develop a complete picture of student understanding. The materials state, “On its own, a single assessment does not show a complete picture of student progress” and direct teachers to “use a combination of observational and scored assessments to understand and report on overall student performance.” The materials further explain that some standards are developed and assessed across multiple modules and that students are not expected to demonstrate full proficiency until all related content has been taught. Across the series, Kindergarten students engage with Module Assessments (interview style), Topic Tickets, and Performance Tasks that include a variety of item types and modalities. Module Assessments and Topic Tickets items also identify associated grade-level content standards and Mathematical Practices. In Grade 1 and 2, students complete Exit Tickets, Topic Tickets, Module Assessments, Performance Assessments, and Benchmark Assessments. Collectively, these assessment opportunities allow students to demonstrate the full intent of grade-level standards and Mathematical Practices across the series.

Examples include: 

• Kindergarten, Module 5, Module Assessment, Problem 4 states, “Add and subtract to fill in the blanks.” Students solve 5+0=____, 2+2=____, 4-1=____, 3-3=____. The Teacher Note states, “Note whether students use fingers, objects, or drawings. These supports are acceptable and encouraged when students are challenged with new math concepts or higher numbers. For sums and differences within 5, kindergarten students should know the answer without supports most of the time.” The materials assess the full intent of K.OA.5 as students fluently add and subtract within 5.

• Grade 1, Module 3, Topic A, Topic Ticket, Problem 2 states, “Lan has 8 red apples. Meg has 2 yellow apples. Ned has 4 green apples. How many apples do they have? Draw. Write. They have ____ apples.” The materials assess the full intent of 1.OA.2 as students solve a word problem that calls for addition of three whole numbers whose sum is less than or equal to 20.

• Grade 2, Module 4, Topic D, Topic Ticket, Problem 2, supports the full intent of MP3 (Construct viable arguments and critique the reasoning of others) as students reason the work of another student. They are asked to explain what about the work is not efficient. “Ling wants to add 10 to both numbers to find 690-298 Lan says this is not an efficient strategy. Tell why Lan says it is not an efficient strategy.”

The materials reviewed for Eureka Math² Grades 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 students with consistent opportunities to engage in the full intent of all Grade K-2 standards. According to the Kindergarten and Grades 1–2 Implementation Guides, each lesson consists of four sections: Fluency, Launch, Learn, and Land. Fluency provides distributed practice with previously learned material and prepares students for new learning by activating prior knowledge and bridging small learning gaps. Launch activities create an accessible entry point to the day’s learning by building context and often creating productive struggle. Each Launch ends with a transition statement that establishes the learning goal. Learn presents new learning related to the lesson objective through a series of instructional segments and includes opportunities for direct instruction, guided instruction, group work, partner activities, interactive video, and digital elements. The Problem Set, included in Learn, provides students with an opportunity for independent practice. During Land, teachers facilitate a brief discussion using suggested questions, including key questions related to the objective, to help students synthesize the day’s learning. In Grades 1–2, Land also includes a Debrief and an Exit Ticket that provide information about student understanding to support instructional decisions.

Examples include:

• Kindergarten, Module 5, Topic A, Lesson 7 and Topic B, Lesson 14 engage students with the full intent of K.OA.1 (Represent addition and subtraction with objects, fingers, mental images, drawings, sounds [e.g., claps], acting out situations, verbal explanations, expressions, or equations). In Lesson 7, Fluency, students show a number the math way, pop up some more, then say an addition sentence to build addition fluency within 5. “Let’s play pop up! I’ll say a number. You’ll pop up that many fingers the math way.” In the second Fluency activity, students make 5 to build fluency with writing addition sentences. “Have students form pairs. Distribute a set of cards with numerals and objects 0–5 to each pair and have them play according to the following rules: Lay out six cards. Partners take turns matching cards that make 5. If no cards make 5, draw an additional card until a match is made. Write the corresponding addition sentence. Place the matched cards to the side and add two more cards from the deck. Continue taking turns until no more matches can be made.” In the Learn section, students choose tools and strategies to find the total of an expression. “Write 4+3=___ and ask students to find the total. Have students work independently to represent and solve the problem. Provide materials such as Unifix Cubes, 10-frames, number paths, and personal whiteboards. Encourage students to self-select their tools. They may also choose to draw or use their fingers. Circulate and observe student strategies. Select two or three students to share in the next segment. Look for work samples that help advance the lesson’s objective by using the count all and count on strategies to find a total. Gather the class to discuss the selected work samples. Show the samples side by side. If one of the selected samples involved fingers, allow the student to demonstrate the action. Invite students to think–pair–share about the following question, ‘What do you notice about this work?’” In Lesson 14, Fluency, Show Me the Math Way: Subtract, students show a number the math way, hide some, then say a subtraction sentence to develop subtraction fluency within 10. In the Learn section, students choose strategies and tools to find the difference. “Write 7-3=___. Ask students to complete the number sentence. Invite them to draw or select math tools, such as Unifix Cubes, 10-frame cartons, number paths, fingers, or 10-frames. Circulate and observe as students work. Select students to share in the next segment. If possible, select work samples that use different tools.” The student book includes examples of subtraction problems. “Invite students to self-select tools to complete the Problem Set. Space is provided for drawing, but students may or may not choose to draw. Before releasing the class to work independently, ask students to notice what is different about the last two number sentences on the back page.”

• Grade 1, Module 1, Topic A, Lesson 3 and Module 2, Topic E, Lesson 23 engage students in extensive work with 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). In Module 1, Lesson 3: Sort to represent and compare data with three categories, Learn, students generate data by sorting and counting, and then represent the data on a graph. “Partner students and distribute a bag of cubes to each pair. Direct students to sort by color and count each group. Depending on the available workspace, consider having students sort their cubes on a personal whiteboard so they can easily move their cubes as needed. When most pairs are finished, ask students to turn to the My ___ graph and get out crayons and a pencil. Display the incomplete graph to guide the class in graphing their own cube collections. First, invite students to complete the title frame with a word or phrase that tells what they are graphing. Then model graphing the first category. ‘How can we use these number paths to make a graph to show our color groups?’ (We can use one path for each color. We can use our crayons to color how many cubes there are in each group.) ‘How can we label this number path to show the red cubes?’ (We can write the word red or draw a red square next to it.) ‘How can we show how many red cubes we have?’ (We can color the same number of squares as there are red cubes. We can write the total in the box.) Have students use their cubes to complete their graphs independently. Observe and support as needed. When students finish, ask them to put away their cubes. Use the following prompt to have them turn and talk about the graph. ‘What does the graph tell you about the cubes?’ Listen for students to share the totals of each category and comparisons of categories. Some may even find the total of all their cubes.” In the Land section, students sort to represent and compare data with three categories. “Display the Apple Count graph. Facilitate a discussion about what information graphs provide and what we can learn from them.” Module 2, Lesson 23: Compare categories in a graph to figure out how many more. In Launch, students create a three-category vertical bar graph with sticky notes and ask questions as a class. “Scatter and display the farm animal counters. Ask students to turn and talk about what they notice. Then ask students to share ideas about how to sort the animals into groups by using attributes such as the following: color, animal type, number of legs, where the animal lives on a farm, actual size of the animal. ‘Let’s group them by different kinds of animals, or animal types.’ Sort the farm animal counters into piles of cows, hens, and pigs. ‘How can we organize our groups to compare them?’ (We could line them up: cows, hens, pigs.) ‘Let’s line them up like a graph. Let’s line up groups carefully so we can see matches. They can go up and down or side to side.’ Organize the groups into columns that follow this order: cows, hens, pigs. Have students count as you place each animal in a line. Use a sticky note to record the total number of animals in each group. Place the total above each column. ‘The lines of animals go up and down instead of side to side. Can we still compare the groups?’ (Yes.) Have students take out their books and turn to the first three-category graph. ‘We can graph our sort by using the number paths and labels on this page. What did we sort?’ (Farm animals) ‘Let’s write that as the title of the graph.’ Help students recall what they know about the parts of a graph from Module 1 by asking questions such as the following: How should we use the number paths to show our sort? Why are there pictures of a cow, a hen, and a pig on this graph? Notice the word totals. What belongs in each gray box? Allow time for students to complete the graph by using the class sort. Then transition students to the next segment by framing the work. ‘Today, we will ask and answer questions about the data on this graph.’” 

• Grade 2, Module 3, Topic D, Lessons 14 and 18 engage students with the full intent of 2.MD.7 (Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.). In Lesson 14, Learn, students determine the difference between a.m. and p.m. “Display the timeline. ‘Look at this timeline of a day. What do you notice?’ (I notice that a whole day is 24 hours. I notice that the day is split into 2 equal halves. I notice that a.m. takes up half of the day and p.m. takes up the other half of the day.) ‘The day is divided into two equal parts: a.m. and p.m. Both are 12 hours long. The morning hours, referred to as a.m., begin at midnight, when we are typically sleeping, and end at noon, the middle of the day, around lunch. The afternoon and evening, from noon to midnight, is referred to as p.m. We typically think of a.m. as the morning. However, a.m. starts at midnight when we are sleeping, so it is dark outside for part of the a.m. (Display the timeline with Dark outside labeled for early a.m.) In most places, it starts getting dark again in the evenings when we go to bed and fall asleep.” In the Land, Problem Set, students are shown a time along with a picture describing an event. Students must determine when this event would normally take place and whether it is a.m. or p.m. (In Problem 1, students see a person lying in bed. A window shows it is dark outside. In Problem 2, students are shown playing on a playground.) “1. 9:30, a.m. p.m., 2. 1:30 a.m., p.m.” In Lesson 18, Learn, students apply their ability to count by fives to tell time on a clock by five-minute intervals. The teacher uses a demonstration clock. “‘What unit does each tick mark on the clock represent?’ (Each tick mark represents 1 minute.) ‘How many minutes are there between each number on the clock? (Gesture to the numbers on the clock.)’ (5 minutes) ‘Count by 5 minutes with me.’ (0 minutes, 5 minutes, 10 minutes, 15 minutes, … 60 minutes) ‘How many minutes are in 1 hour?’ (60 minutes) ‘When the minute hand gets to the 12, 60 minutes have elapsed, or gone by. Then a new hour begins.’ Show 7:00 on the demonstration clock. ‘Watch what happens to the hour hand as I move the minute hand.’ Show 7:05 on the demonstration clock. ‘How many minutes have elapsed since seven o’clock?’ (5 minutes) ‘We say this time as “seven oh five,” and we write it like this. (Write 7:05.) What do you notice about the hour hand?’ (It moved away from the 7 a little bit.) Move the minute hand around the clock, asking students to write the time on their whiteboards at each 5-minute interval (7:10, 7:15, 7:20, …  7:55). Highlight the position of the hour hand relative to the minute hand. Stop at 7:30. ‘How many minutes until the next hour?’ (30 minutes) ‘What do you notice about the hour hand?’ (It’s halfway between the 7 and 8.) Stop at 7:55. ‘How many minutes until the next hour?’ (5 minutes) ‘What do you notice about the hour hand?’ (It’s really close to the 8. It’s almost touching it.) Show 8:00 on the demonstration clock. ‘What time is it now?’ (8:00) ‘Where is the hour hand?’ (It is directly on the 8.) Direct students to write the time they see. ’True or false: This time is written as 7:60.’ (False) Invite students to think–pair–share about why the time is not written as 7:60. (It’s eight o’clock now, so the hour should be an 8. No minutes in the eight o’clock hour have gone by, so it starts at zero again. As soon as the minute hand hits 60 minutes, the hour changes and the minutes restart at zero.) ‘When 60 minutes have passed, the minutes are renamed as 1 hour. The hour changes, and the minutes go back to zero to show that a new hour has started.’ Invite students to turn and talk about why we never see 60 minutes displayed on a digital clock.”

The materials reviewed for Eureka Math² Grades K 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 as included in the following grade-level breakdowns. 

Kindergarten:

• The approximate number of modules devoted to major work of the grade (including assessments and related supporting work) is 4 out of 6, approximately 67%.

• The approximate number of lessons devoted to major work of the grade (including assessments and related supporting work) is 106 out of 140, approximately 76%. 

• The approximate number of instructional days devoted to major work of the grade (including assessments and related supporting work) is 106 out of 140, approximately 76%. In Kindergarten, the number of lessons and instructional days is the same because the materials do not include module assessments in the total instructional days. The K Implementation Guide, Assessment, Module Assessment, states, “Give this assessment when a student shows inconsistent proficiency over the course of a module based on notes you make using the Observational Assessment Recording Sheet.” The Eureka Math² Pacing Guidance Kindergarten document further identifies the Module Assessments as optional and excludes them from the total instructional days: “Module 1 Assessment (Optional)*, *Not included in total days of instruction.”

Grade 1: 

• The approximate number of modules devoted to major work of the grade (including assessments and related supporting work) is 6 out of 6, approximately 100%.

• The approximate number of lessons devoted to major work of the grade (including assessments and related supporting work) is 120 out of 140, approximately 86%. 

• The approximate number of instructional days devoted to major work of the grade (including assessments and related supporting work) is 152 out of 175, approximately 87%.

Grade 2:

• The approximate number of modules devoted to major work of the grade (including assessments and related supporting work) is 4 out of 6, approximately 67%.

• The approximate number of lessons devoted to major work of the grade (including assessments and related supporting work) is 110 out of 140, approximately 79%. 

• The approximate number of instructional days devoted to major work of the grade (including assessments and related supporting work) is 138.5 out of 172, approximately 81%.

An instructional day analysis across Kindergarten through Grade 2 is most representative of the instructional materials, as the days include major work, supporting work connected to major work, and the assessments embedded within each chapter. Approximately 76% of the materials in Kindergarten, 87% of the materials in Grade 1, and 81% of the materials in Grade 2 focus on major work of the grade.

The materials reviewed for Eureka Math² Grades 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 are connected to the major standards of the grade. Teachers can locate these connections within the “Achievement Descriptors and Standards” tab in lessons.

Examples include:

• Kindergarten, Module 3, Topic D, Lesson 21, Teach Book, Learn, connects the supporting work of K.MD.2 (Directly compare two objects with a measurable attribute in common, to see which object has "more of"/"less of" the attribute, and describe the difference) 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). Students select measurable attributes to compare objects and sets. “Distribute a mystery bag to each table or group of students. Ensure the comparison tools are available and accessible to pairs. ‘You and your partner will each reach into the mystery bag and take out one thing. Your job is to compare the things you and your partner take out in as many ways as you can. Use our list to help you remember the different ways you can compare.’ Invite students to self-select math tools they need to make their comparisons, such as a balance scale or number path. Students may compare as many items as time allows. Share a procedure that is appropriate for your classroom for returning objects or sets and selecting new items. Observe and offer guidance as needed. As students work, use the following prompts to assess and advance student thinking: Point to the list to tell me about how you compared your things. What strategy did you use to compare length/weight/number? You labeled your groups with numbers. How do the numbers help you compare? If you could only use numbers to compare, what would you do? What is another way you can compare? After students have compared several items, ask them to record one of their comparisons on the recording sheet in their student books. Tell students to leave their objects and recording sheet out for the next segment of the lesson. Clean up other items.” In Gallery Walk, “Students compare objects by using numbers and make comparison statements about their work. Gather the class and remind them of the protocol for a gallery walk. Remind students to look but not touch, as they would in a museum or gallery. They can hold their hands behind their backs as a reminder. Ask students to whisper comparison statements for the work they see on the gallery walk. Remind them of the comparison chart if needed. Once the class completes the gallery walk, have students think–pair–share about the following question. ‘Could numbers be used in all of these comparisons? Why?’ (Yes. You could count how many are in each group. I’m not sure. I have a pencil and a glue stick.) Distribute a set of Hide Zero cards to each pair. ‘Use your cards to put numbers next to your objects.’ If the work compares the lengths of two items such as a pencil and a glue stick, let students reason about how numbers relate to the situation. (e.g., There is 1 pencil and 1 glue stick.) Once all students place the numbers, draw students’ attention to their completed work. ‘We can use numbers to tell about all our comparisons! If we hide our objects and just look at the numbers, can we still make a comparison? Let’s try it.’ Give each student a piece of paper. Have them use the paper to cover the objects in their comparison. The numbers should still be visible. ‘Let’s walk and look at each other’s work one more time. This time you will only be able to see the numbers. As you walk, compare the numbers by whispering a statement by using the words greater than, less than, or equal to.’”

• Grade 1, Module 2, Topic E, Lesson 20, Teach Book, Learn, 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) and 1.OA.7 (Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false). In the activity, students look at objects (crackers) organized into two groups and the mathematics that would need to be done to make the number in each group equal. “Have students turn to the picture of crackers in their student books. Display the picture of Zoey and Adrian. ‘Zoey and Adrian are packing a snack. What is different about their crackers?’ (Zoey has 6 crackers. Adrian has 9 crackers.) ‘What is the same about their crackers?’ (They both have the same kind of cracker. They have 6 crackers that match up. They are lined up.) ‘We can match Zoey’s 6 crackers to 6 of Adrian’s crackers. (Draw a line to match up each pair of crackers.) What could they do to make sure they have the same number of crackers? Turn and talk.’ Ask students to draw on their whiteboards to show two ways that Zoey and Adrian can have the same number of crackers. They do not need to write a number sentence yet. Look for students who use strategies similar to those modeled in Launch. Invite two students to explain their ideas. (I drew 3 more crackers for Zoey. Now they both have 9 crackers. I crossed off 3 of Adrian’s crackers. Now they both have 6 crackers.) Ask students to look at their work to compare it with the two sample strategies given. Some students may choose to revise or complete their work. ‘What number sentence can we write to show what the first student did?’ (6+3=9). Have students record the number sentence above Zoey’s crackers. Then use the following process to guide students to represent what the second student did by using the order that matches the picture. ‘Zoey has 6 crackers. (Write 6.) Adrian has 9 crackers. We take away 3. (Write = 9-3.) Look at the number sentence we just wrote: 6=9-3. 6 is the same total as 9 minus 3. Is that true? How do you know?’ (Yes, it is true. 9 minus 3 equals 6.)”

• Grade 2, Module 6, Topic B, Lesson 5, Teach Book, Learn, connects the supporting work of 2.OA.3 (Determine whether a group of objects [up to 20] has an odd or even number of members) to the major work of 2.NBT.2 (Count within 1000; skip-count by 5s, 10s, and 100s). Students use skip-counting to test their ideas and articulate why unequal groups cannot be organized into an array. “Present the following statement: ‘Unequal groups can be rearranged into rows or columns to make an array.’ Use the Always Sometimes Never routine to engage students in constructing meaning and discussing their ideas. Give students 1 minute of silent think time to evaluate whether the statement is always, sometimes, or never true. Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Then facilitate a class discussion. Invite students to share their thinking with the whole group. Encourage them to provide examples and nonexamples to support their claim.” Later in the Debrief portion of the lesson, students compose arrays with rows and columns and use a repeated count to find the total. “‘Is this an array? Why?’ (No, it is not an array because the rows are not equal groups. The columns aren’t either. There are some squares missing. It is not an array because the tiles are not organized in equal groups.) ‘Is it useful to organize objects into an array? How?’ (Yes, organizing objects into an array helps because you can skip-count to find the total. Yes, organizing objects into an array helps you see the number of groups and the number of objects in each group. Yes, organizing objects into an array helps you write a repeated addition equation to find the total.)”

The instructional materials for Eureka Math² 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 among the major work of the grade appear throughout the materials where appropriate. Each lesson includes Achievement Descriptors that describe what students should be able to do by the end of the lesson and identify the associated standards. The Teach books include a Module Achievement Descriptors by Lesson resource at the end of each module that identifies the Achievement Descriptors addressed in each lesson, allowing teachers to trace the standards addressed across lessons and modules. 

Examples include:

• Kindergarten, Module 6, Topic B, Lesson 11, Teach Book, Learn, connects the major work of K.NBT.A (Work with numbers 11-19 to gain foundations for place value) 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 represent teen numbers as 10 ones and some ones, decompose the numbers into a ten and additional ones, and use self-selected tools and mathematical representations to identify a hidden part and write a corresponding number sentence. In How Many Are Hiding?, “Partner students. Make sure each pair has 20 Unifix Cubes, two 10-frame cartons, and a whiteboard with a Number Bond removable inside. Give the following directions for the activity. Consider modeling a problem with a partner before letting pairs work at their own pace. Partner A uses 20 cubes and the 10-frame cartons. Partner B uses a whiteboard with the removable insert. Partner A models a teen number with cubes in the 10-frame cartons. Then partner B writes the total and parts in the number bond. Partner B closes their eyes and partner A hides one of the 10-frame cartons. Partner B opens their eyes and covers the part in their number bond they think is hiding. Partners confirm that their parts and total match and work together to write a number sentence. Partners switch roles and repeat. Circulate and support students as needed. Encourage students to talk about how the numbers in the number bonds match the cubes. Ask students to tell what the numbers in the bond and sentence refer to.”

• Grade 1, Module 5, Topic E, Lesson 24, Teach Book, Land, connects the major work of 1.OA.D (Work with addition and subtraction equations) to the major work of 1.NBT.C (Use place value understanding and properties of operations to add and subtract). Students decompose addends into tens and ones and use place value strategies, including decomposing an addend to make the next ten, to solve addition problems with two two-digit numbers. In the Debrief, “Display 18+12= solved three different ways. Show one way at a time. The first way is by adding like units. The second way is by adding tens first. The third way is to make the next ten. In all three ways, one or both addends were decomposed. Invite students to analyze each work sample by using a variation of the Five Framing Questions routine. The responses provided for the following questions show a possible answer for each work sample. Have students notice and organize. ‘How did this student find the total?’ Display all three samples at the same time. Help students reveal the strategies. ‘Let’s focus on breaking apart addends. Where do you see addends broken apart in the work?’ Help students to distill the information and know how these strategies help them to add. ‘How do you think breaking apart addends helped these students?’ Guide students to further know how these strategies help them. ‘What are some ways to make an easier problem?’”

• Grade 2, Module 1, Topic E, Lesson 22, Teach Book, Learn, connects the major work of 2.OA.A (Represent and solve problems involving addition and subtraction) to the major work of 2.NBT.A (Understand place value). Students apply place value understanding to solve add-to, change-unknown word problems by representing numbers with place value units and counting on by ones, tens, and hundreds to benchmark numbers. In Represent and Solve an Add To with Change Unknown Word Problem, “Direct students to the problem in their books. Chorally read the problem with the class: ‘Ming biked 64 miles. He wants to bike 100 miles. How many more miles should Ming bike?’ Direct the class to use the Read-Draw-Write process to solve the problem. Invite students to Think–Pair–Share about what they could draw to represent the problem. Circulate and observe as students work. Select a few students to share their work. Look for examples to highlight multiple solution strategies, including using place value units to count on. As students share their strategies, ask them to explain their rationale. The student work samples shown demonstrate several possible solution strategies. Consider asking these or similar questions to connect the representations: ‘Where do you see 64 in each representation? 36? The total? When did you change units? Why? Where do you see a benchmark number? How does your drawing match the situation?’”

The instructional materials reviewed for Eureka Math² 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. 

Each Teach Book begins with a Module Overview that includes a topic-by-topic summary of the development of learning throughout the module. The Module Overview also contains Before This Module and After This Module coherence sections that look backward and forward to show the module's place in the development of learning within and across grade levels. In addition, The Why Behind the Lesson Design explains the organization of the module's content and structure, including how the module connects to previous and future grade-level learning. Each Topic Overview summarizes the development of learning within the topic and typically describes how the content connects to previous or upcoming learning, including learning from prior or future grade levels. Some Teacher Notes within lessons also explain connections between current learning and prior and future mathematical concepts.

An example of a connection to future grades in Kindergarten includes:

• Module 6, Topic A, Lesson 2, Teach Book, Debrief connects K.CC.A (Know number names and the count sequence) and K.NBT.A (Work with numbers 11-19 to gain foundations for place value) to future work in Grade 1, where students use the term digit when working with place value (1.NBT). The Teacher Note states, “Young students often refer to two-digit numbers as having two numbers. If this happens, casually introduce the term digit. For example, ‘We can call the 1 and 4 you see in 14 digits. 14 is the number. 1 and 4 are the digits.’ Students will not be responsible for using digit as terminology until grade 1.”

An example of a connection to prior knowledge in Kindergarten includes:

• Module 5, Topic C, Topic Resources, Topic C Overview, connects K.CC.2 (Count forward beginning from a given number within the known sequence) to previous fluency work in Kindergarten, where students count forward beginning from a given number rather than beginning at 1. The Topic C Overview states, “The topic also presents opportunities to practice and apply key kindergarten standards. Counting from a number other than 1 is familiar because students have exercised this skill in Fluency throughout the year. Now, through practical application, they begin to explore its usefulness in finding a total. With growing confidence in their ability to count from a number other than 1, they may be inclined to count on when approaching add to with change unknown problems or finding partners to 10.”

An example of a connection to future grades in Grade 1 includes:

• Module 6, Module Resources, Module 6 Overview, Part 1 connects 1.G.A (Reason with shapes and their attributes) to future work in Grade 2, where students identify defining attributes of two- and three-dimensional shapes, partition shapes into equal shares, and use fraction language when telling time (2.G.A, 2.MD.7). The Coherence: After This Module states, “Students begin using the number of angles in a shape as a defining attribute of flat shapes. They use the number of faces, edges, and vertices as defining attributes of solid shapes. Fraction work expands from working with halves and fourths to include thirds. Students refine their understanding of equal shares as they see that equal shares are the same size, but not always the same shape. Grade 2 students relate fractions to telling time by using the language quarter past and quarter to.”

An example of a connection to prior knowledge in Grade 1 includes:

• Module 1, Topic D, Lesson 18, Teach Book, Learn, connects 1.OA.D (Work with addition and subtraction equations) to previous work in Kindergarten, where students decompose and compose numbers less than or equal to 10 (K.OA.A). The Teacher Note states, “Students decomposed and composed 5 in kindergarten. The two-part expressions that equal 5 are intentionally used in this lesson so that students can extend their thinking to equality between expressions. Using familiar combinations will allow them to attend to new, more complex mathematical concepts.”

An example of a connection to future grades in Grade 2 includes:

• Module 5, Topic B, Topic Resources, Topic B Overview, connects 2.MD.A (Measure and estimate lengths in standard units) and 2.MD.B (Relate addition and subtraction to length) to future work in Grade 3, where students partition the interval from 0 to 1 into equal parts (3.NF.A) and solve measurement problems involving intervals on number lines (3.MD.B). The Topic Overview states, “Now that students have a solid conceptual understanding of length, they use a number line to represent distances, for example, the distance a rocket travels. Students apply their knowledge of a ruler to a number line, where they refer to the space between each tick mark as an interval. By using the distance between points and their ability to skip-count by fives and tens, students identify unknown numbers on a number line with a given interval. They reason about how the size of each interval affects the number of intervals, discovering that the more intervals a length is divided into, the smaller each interval is. This lays the foundation for partitioning the interval from 0 to 1 into equal parts in grade 3.”

An example of a connection to prior knowledge in Grade 2 includes:

• Module 2, Module Resources, Module 2 Overview, connects 2.OA.A (Represent and solve problems involving addition and subtraction) to previous work in Grade 1, where students represent and solve addition and subtraction problems within 20 and develop strategies for adding within 100 (1.OA.A). The Module 2 Overview states, “Grade 1 students deepen their problem-solving skills as they use tape diagrams and drawings to represent and solve more complex problems within 20, which include start unknown problem types. Students build on their work in module 5 by extending addition strategies to larger numbers within 100. The focus is on making easier problems by decomposing one or both addends. Students may add like units, add tens then ones, or vice versa, or make the next ten. They use various tools and recording methods, such as number bonds, the number path, and the arrow way to support their strategy work.”