Kindergarten - Gateway 1

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

The curriculum is divided into six modules and each includes an Observational Assessment and a Module Assessment. In Kindergarten, guidance for the Module Assessment, in the Teacher Edition states, “Administer this assessment only to students whose observational assessments show inconsistent proficiency throughout the module. Use the suggested language, or support students in their native language to better ascertain the student’s understanding of the math content. If a student is unable to answer the first few questions, end the assessment and retry after more instruction.” Examples of grade-level items from Module Assessments include:

• Module 1, Module Assessment, Item 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)

• Module 2, Module Assessment, Item 3, “Clear the work mat and remove all the shapes. On the work mat, construct a square oriented like a diamond from equal-length straws. ‘What is the name of this shape?’ Teacher note: Square and rectangle are acceptable answers; diamond is not. Provide students with 4 more straws of equal length and 4 straws of half the length to construct the square if needed. ‘Make a rectangle.’ Teacher note: A highly proficient student might not reconstruct the square but simply say that it is already a rectangle.” (K.G.2, K.G.5)

• Module 4, Module Assessment, Item 1, picture shows five birds in a tree and one bird flying away. “Place cubes, marker, and number bond in front of the student. Show the bird scene. ‘Look at the birds. What parts do you see? Fill in the number bond to match.’ Teacher note: Students may use cubes, pictures, or numbers to complete the number bond. Point to a part in the number bond that the student has filled in. ‘What does this tell us about?’ (Point.) Teacher note: Listen for students to describe the reasoning behind their sort. “Birds” is not descriptive enough. Elicit the attributes of the part.” (K.OA.1)

• Module 6, Module Assessment, Item 3, a picture shows eight birds randomly placed on the top portion of the page and ten pigeons, in two rows of five, pictured below. “Place the bird picture in front of the student. Then tell this story: ‘There were 8 blue birds flying and 10 pigeons walking on the ground. How many birds are there?’ Write a number sentence that tells about all the birds. Prompt students to write a number sentence to show their thinking and explain it. Point to different parts in their number sentence and use the following questions to check for understanding. Which birds does this number tell about? Where is the total number of birds in your number sentence? Where are the parts?” (K.OA.2, K.NBT.1)

The materials reviewed for Eureka Math² Kindergarten meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards. 

According to the Kindergarten Implementation Guide, “Each lesson is structured in four sections: Fluency, Launch, Learn, and Land. Lessons are designed for one 50-minute instructional period. Fluency provides distributed practice with previously learned material. It is designed to prepare students for new learning by activating prior knowledge and bridging small learning gaps. Launch creates an accessible entry point to the day’s learning through activities that build context and often create productive struggle that leads to a need for the learning that follows. Every Launch ends with a transition statement that sets the goal for the day’s learning. Learn presents new learning related to the lesson objective, usually through a series of instructional segments. This lesson component takes most of the instructional time. Suggested facilitation styles vary and may include direct instruction, guided instruction, group work, partner activities, interactive video, and digital elements. The Problem Set, an opportunity for independent practice, is included in Learn. Land helps you facilitate a brief discussion to close the lesson. Suggested questions, including key questions related to the objective, help students synthesize the day’s learning.” 

Instructional materials engage all students in extensive work with grade-level problems through the consistent lesson structure. Examples include:

• Module 1, Lessons 11, 12, 25, and 27 and Module 6, Lesson 3 engage students in extensive work with K.CC.3 (Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 [with 0 representing a count of no objects]). Module 1, Lesson 27: Write numerals 9 and 10, Launch, students practicing writing the numerals 0-8. “Display the Baseball Bears digital interactive. ‘The blue and red teddy bears are having a home run competition. They get 1 point for every home run they hit. Our job is to keep score. What can we do to show how many points each bear scores?’ Listen to student ideas. If students do not suggest writing the numerals, show a copy of the Scoreboard, and describe how it is used to keep track of runs in baseball. Distribute a Scoreboard to each student. ‘The bears don’t have any points yet. What’s the math word for none?’ (Zero) Prompt students to write 0 for each bear on the Scoreboard. Demonstrate the numeral formation while saying the number rhyme if needed. The blue team will go first. In the digital interactive, swing to let a bear take its turn at bat. If the bear hits a home run, prompt students to change their Scoreboards. If the bear does not hit a home run, it gets an out. ‘The blue bear scored a point! How should we change our Scoreboards?’ (We should change the blue bears’ 0 to a 1.) Consider having students hold up their personal whiteboards so that you can quickly validate the accuracy of their work. Swing for the blue bear until it has 3 outs. A red bear will automatically come to bat. Repeat the process, demonstrating numeral formation as needed, until students have to write 9. Transition to the next segment by framing the work. ‘We need to change the score to 9, but we haven’t practiced writing 9 or 10. Let’s learn those numbers now.’” Module 6, Lesson 3: Write numerals 11-20, Land, students use a dry erase marker and board to write the numerals 11-20. “Review Problem Set answers as a class, asking students to say each number the regular way and the Say Ten way. Pause at the leaf problem. ‘What does the 1 in 12 represent, or tell us about?’ (It tells us about the 10 leaves on the stem.) Continue reviewing the answers to the other problems. Display Puppet’s work on the bee problem. ‘This is how Puppet wrote 17. Did Puppet write it correctly?’ (No, Puppet wrote one hundred seven. Puppet didn’t hide the 0 with the 7.) ‘Turn and ask your partner: What would you tell Puppet so Puppet can write 17 correctly?’ (I would tell Puppet to use the Hide Zero cards. Puppet needs to hide the 0 with the 7.)”

• Module 2, Lessons 2, 3, and 4 engage students in extensive work with K.G.2 (Correctly name shapes regardless of their orientations or overall size). Lesson 2: Classify shapes as triangles or non triangles, Fluency, Show Me Attributes, students act out a variety of shape terms. “Review the body movements for corners, straight sides, curved sides, closed, and open. ‘I’ll say a word. You use your body to show the word. Ready? Show me closed. Show me open. Show me straight. Show me curved. Show me corners.’ Continue having students show attributes in any order.” Lesson 4: Classify shapes as rectangles or non rectangles, with square rectangles as a special case, Learn, students learn and discuss the attributes of a rectangle. “Show the Shapes chart from previous lessons. Hold up a rectangle so that the longest sides are horizontal. ‘This is a rectangle. Is it open or closed?’ (It’s a closed shape.) ‘Show me with your body: Are the sides of the rectangle curved or straight?’ (Shows the movement for straight) Touch and count the sides as a class. Repeat with the corners. Invite students to track the count by using their fingers. ‘What if I turn the shape? (Turn so the shortest sides are horizontal.) Is it still a rectangle?’ (Yes.) ‘How do you know it is still a rectangle? It looks different now.’ (It’s still the same shape. You just turned it. It still has 4 sides and 4 corners.) Add rectangle to the Shapes chart. Display the three shapes. Ask students to think–pair–share about the following question. Tell them to explain their reasoning. ‘Look at these shapes. Do you see any rectangles?’ Lead a discussion by calling on pairs to share. Reasoning may include the following: These two look like rectangles. (Points to the shapes in the middle and on the right.) But that one doesn’t look right. (Points to the shape on the left.) They all have 4 straight sides and 4 corners. They are all rectangles. The sides are leaning on that one. (Points to the first shape) ‘Let’s all look at the sides of the first shape.’ Place the rectangle from the 2D shape set on the floor. ‘If we sit any side of a rectangle on the floor, there should be 2 sides going straight up and down. If the sides are not straight up and down, then the shape is not a rectangle.’ Revisit the three displayed shapes and bring the class to consensus that the first shape is not a rectangle, but the last two are.”

• Module 6, Lessons 2, 7, 9, and 21 engage students in extensive work with K.NBT.1 (Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation [such as 18=10+8]; understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones). Lesson 2: Find 10 ones in a teen number, Launch, students compare the efficiency of counting objects in different configurations. “Display the flower picture. ‘How many petals are on this flower? What strategy can help us count them?’ (We can touch and count each petal. You can start at the top and count all the way around. We should put an x on the petal we count first so we don’t forget.) ‘Let’s count by ones and touch each petal as we count.’ Mark the start and then touch and count the 14 petals. Then display the pigeon picture. ‘How many pigeons are there? What strategy could help us count them?’ (Let’s find 10 and count the other birds.) Identify 10 and count on with the class. ’There are 14 pigeons. Say it the Say Ten way.’ (Ten 4)  Display the balloon picture. ‘How many balloons are there? (14) Wow, how did you know that so fast? (I counted on. I saw a group of 10 and then counted the rest. I saw 10 and 4. That makes 14.) Which picture was the easiest to count? Why?’ (The birds because they were all in a line and we counted 10. The balloons because I could see 10 and 4 and I knew that was 14.)” Lesson 7: Decompose numbers 10-20 with 10 as a part, Learn, students show decomposing a teen number by using a number bond. “Make sure students have a marker and a personal whiteboard with a Number Bond removable inside. ‘I have some tools that might make it easier for us to see how many cubes there are.’ Present a 10-frame carton. ‘Count the slots as I touch them. Ready?’ (1, 2, 3, … , 10) Place 1 blue Unifix Cube in each slot until the carton is full. ‘How many blue cubes?’ (10) Place the 3 yellow cubes into another carton without counting. ‘Count with me. We will start with 10. Ready?  Tennnn … (Gesture over the full carton.) 11, 12, 13 (Point to each yellow cube crisply.)’  Make 13 by using Hide Zero cards and place the cards where the total goes in the number bond. Invite students to write the total in their number bond. Silently pull the cartons apart as shown. Then pull apart the Hide Zero cards and place them in the parts of the number bond. Write the total with a marker. ‘Make your number bond match mine.’ Hold up the card with 10. ‘What does this number tell about?’ (The blue cubes. One of the parts.) Hold up the card with 3. ‘What does this number tell about?’ (The yellow cubes. The other part.) Hold up both cards to make 13. ‘What does this number tell about?’ (All of the cubes. The total).” Lesson 21: Count and compare sets with more than 10 objects, Fluency, Whiteboard Exchange: Decompose Teen Numbers, students represent teen numbers as 10 ones and some more ones as they build fluency with decomposition. “Make sure students have a personal whiteboard with a Double 10-Frame inside. Display the blank Double 10-Frame. ‘Show me 16.’ Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the answer. Repeat the process with the following sequence: 17, 15, 13, 9, 11, 10, 12, 20. As students work, notice who can adjust by simply erasing or drawing more dots, and which students must erase the board each time.”

The instructional materials provide opportunities for all students to engage with the full intent of all Kindergarten standards through a consistent lesson structure. Examples include:

• Module 2, Lessons 10 and 13 engage students with the full intent of K.G.5. (Model shapes in the world by building shapes from components [e.g., sticks and clay balls] and drawing shapes). Lesson 10: Construct a circle, Launch, students make observations about different types of wheels. “Display the picture of wheels and ask students to name the three real-world objects. Point to each object as it is named. (A Ferris wheel, a bicycle wheel, a ship steering wheel) ‘What is the same about all of them?’ (They are all round. They are all wheels. They have circles on them. They have lines going from the middle to the outside.) ‘Why do you think all of these wheels are shaped like a circle?’ (My bike tire is a circle so it can spin. They have to go around and around.) Have students point out the center point of each wheel. If students did not mention the spokes of the wheels, point those out as well. Transition to the next segment by framing the work. ‘Today, let’s use the parts of a wheel to help us build circles.’” In the Learn section, students use equal-length straws to create the outer points of a circle. Lesson 13: Draw flat shapes, Learn, students analyze the Navajo blanket and compare what they see with what they know about shapes. “Display the full picture of the Navajo blanket. ‘This is the whole blanket. It was woven by a Native American from the Navajo tribe over one hundred years ago. What do you notice?’ (There are more rectangles. I notice small and big rectangles. They are not all the same size.) Display the picture of the loom and blanket. ‘The weaver used a special tool, called a loom, to make the blanket. The loom was probably made of tree branches, like the one you see here.’ Invite students to think–pair–share about the following question. ‘Take a close look at the sides and corners of the shapes. Did the person who made this blanket make perfect rectangles?’ (Some look more like squares and some look like rectangles. The sides are wavy. They aren’t straight. I don’t think they are rectangles.) ‘Artists like weavers sometimes use curved lines and rounded corners when they make shapes. When mathematicians draw shapes, they use a special tool called a straightedge. Using a straightedge helps mathematicians be sure they make straight lines and pointed corners. Students use a straightedge to trace polygons. Let’s draw shapes by using a straightedge. Display Dot Paper and hold up a straightedge for students to see. Watch as I line up my straightedge with the top and bottom of the dotted line. I don’t cover up the dots. I need to see them so that I can connect the dots. I hold the straightedge in place with one hand while I draw a line with the other hand.’ Distribute Dot Paper and a straightedge to each student. Ask students to trace the six shapes with their straight edge tools. Observe as they work. Encourage students to use their straightedge on both the dotted and solid lines to trace each entire shape. Students draw and name polygons.”

• Module 3, Lessons 2 and 7 engage students with the full intent 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). Lesson 2: Compare lengths of simple straight objects by using longer than, shorter than, and about the same length as, Learn, students work with the teacher to compare two objects’ lengths. “Display two colored pencils laying horizontally, so that the shorter pencil appears to be longer than the longer pencil. Hide the bottom parts of the pencils with a piece of paper. ‘Look at these pencils. Which is longer? (We don’t know. You have to move the paper.) Probe students who say they don’t know or want to see under the paper for their reasoning. Highlight responses that involve seeing or aligning endpoints. Remove the paper to reveal the bottom parts of the pencils. Align the pencils and have students make longer and shorter comparison statements. ‘Which pencil is longer?’ (The blue one.) ‘We can say that the blue pencil is longer than the orange pencil. Can you say which is shorter?’ (The orange pencil is shorter than the blue pencil.)” Lesson 7: Compare weights by using heavier than, lighter than, and about the same weight as, Land, Debrief, students work to determine which items in a pair are heavier. Students see pairs of pictures: piece of paper or a notebook, easel and clipboard, and crayons and paint. “‘Pretend that you are going for a long walk. On this walk, you will carry a backpack with three objects inside. I’ll give you some objects to choose from. You will want your backpack to feel light, so keep that in mind as you choose each object.’ Display the picture of the note pad and piece of paper. Invite students to think–pair–share after each image. As they share, prompt them to tell which object is heavier or lighter.”

• Module 5, Lessons 7 and 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). Lesson 7: Find the total in an addition sentence, Fluency, students practice 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?’” Lesson 14: Find the difference in a subtraction sentence, Fluency, Show Me the Math Way: Subtract, students develop subtraction fluency with 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.”

The materials reviewed for Eureka Math² Kindergarten 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 65 percent of instructional time to the major clusters of the grade. 

• The number of modules devoted to major work of the grade (including supporting work connected to the major work) is 5 out of 6, approximately 83%.

• The number of lessons devoted to major work of the grade (including supporting work connected to the major work) is 109 out of 131, approximately 83%. 

• The number of days devoted to major work of the grade (including supporting work connected to the major work) Is 109 out of 131, approximately 83%. The number of lessons and the number of days is identical in Grade K as assessments are not included in the total number of days. Teachers have the option of using module assessments for students who have not demonstrated mastery across lessons.

A lesson-level analysis is most representative of the instructional materials as the lessons include major work and supporting work connected to major work. As a result, approximately 83% of the instructional materials focus on major work of the grade.

The materials reviewed for Eureka Math² Kindergarten meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. 

Materials are designed so supporting standards are connected to the major work standards and teachers can locate these connections on a tab called, “Achievement Descriptors and Standards” within lessons. Examples include:

• Module 1, Topic A, Lesson 3: Classify objects into two categories and count, Learn, Count Each Group, 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.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). Students sort items into eating utensils and drawing tools, and answer “how many” questions. “Use the eating utensils and drawing tools to model how to count each group by using the number path. By carefully placing each item on the number path, students practice one-to-one correspondence. ‘We count to find how many are in each group. The number path is a tool that can help us count. Let’s count the things we use to draw as we put them on the number path, like this.’ Demonstrate saying only one number as you move each item onto the number path. ‘I moved each object to make sure I only counted it once. We said one number for each object. When we do this, let’s call it move and count. Let’s move and count again, but this time let’s whisper and shout like we did earlier. Who remembers when we shout?’ (On the last number) Clear the number path. Move each drawing tool to the number path as students count. ‘Turn and tell your partner how many drawing tools there are. Say a complete sentence like this: There are ___ drawing tools in all.’ (Wave hand over drawing tools.) (There are 3 drawing tools in all.) Repeat with the group of eating utensils, stopping after students count to confirm students’ understanding. ‘You counted 1, 2, 3, 4. Which number tells how many?’ (The number 4, The last number you said) ‘The last number I said, 4, tells me how many. What do we have 4 of?’ (4 things we use to eat) ‘Yes. 4 tells us about all the things we use to eat, the whole group. Why do you think it might be important to know how many things are in a group? Why would you want to count them?’ (To see if there is enough for everyone, So you know if you lost some).”

• Module 2, Topic A, Lesson 5: Communicate the position of flat shapes by using position words, Fluency, Make 4 with Rectangles and Beans, connects the supporting work of K.G.2 (Correctly name shapes regardless of their orientations or overall size) to the major work of K.CC.4 (Understand the relationship between numbers and quantities; connect counting to cardinality) and K.OA.5 (Fluently add and subtract within 5). In the activity, students use their knowledge of rectangles and the number of sides to count 4. “Invite students to fold their Rectangles removable on the black lines, so that only one rectangle is facing up. ‘Touch and count the corners of the rectangle. (Point to a corner.)’ (1, 2, 3, 4) ‘Touch and count your beans.’ (1, 2, 3, 4) ‘Put 3 of your beans on the corners of the rectangle. Keep the other bean in your hand. How many beans are on your rectangle?’ (3) ‘How many beans are in your hand?’ (1) ‘Raise your hand when you can say the sentence to make 4. Start with 3. (Gesture to the 3 beans on the rectangle.)’ Wait until most students raise their hands, and then signal for students to respond. (3 and 1 make 4.) Continue with the following sequence.” Four additional exercises to practice fluency within 4 are included.

• Module 3, Topic D, Lesson 21: Describe and compare several measurable attributes of objects and sets, 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. 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 number relates 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.’”

• Module 4, Topic D, Lesson 25: Extend Growing Patterns, Learn: How many triangles?, connects the supporting work of K.G.6 (Compose simple shapes to form larger shapes) to the major work of K.CC.4 (Understand the relationship between numbers and quantities; connect counting to cardinality) and the major work 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). Students use pattern block triangles to discover a growth pattern by counting and comparing. “Partner students and distribute 10 green pattern block triangles to each pair. Have pairs work together for 2—3 minutes to recreate each tower. Circulate and support students as needed. ‘How many triangles are in the first tower?’ (1) ‘How many are in the second tower?’ (3) ‘How many are in the third tower?’ (6) ‘At first we had 0 triangles. We added 1 triangle to make our first tower.’ Write 0 next to the first tower. From 0 to the first tower, draw an arrow labeled +1. Invite students to think—pair—share about how they can find how many more triangles are in the second tower than in the first. (I can see that the second tower has 2 more than the first tower. Both the first and second towers have 1. I can count the extra triangles in the second tower to see how many more it has. The first tower has 1 and the second tower has 3. I know that 1+2=3, so there are 2 more triangles in the second tower.)...’ Let’s say how many we added each time.’ Gesture to each tower as students count +1, +2, +3. ‘How many more will be in the next tower? How do you know?’ (I think there will be 4 more. The pattern is like counting, 1, 2, 3, so next is 4. I think there will be 4 more. First it got 1 bigger, then 2 bigger, and then 3 bigger.) ‘There will be 4 more. Without looking, what are some ways to find how many triangles are in the next tower?’ (We can add 4 more triangles to the 6 triangles that are in the third tower 6+4=10. I can start at 6 and use my fingers to count 4 more. 6, 7, 8, 9, 10) Build the fourth tower and have students confirm that there are 10 triangles.”

The instructional materials reviewed for Eureka Math² Kindergarten meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. 

Kindergarten lessons are coherent and consistent with the Standards and teachers can locate standard connections on a tab called, “Achievement Descriptors and Standards” within lessons. Examples include:

• Module 1, Topic E, Lesson 22: Count sets in scattered configurations and match to a numeral, Launch, connects the major work of K.CC.A (Know number names and the count sequence) to the major work of K.CC.B (Count to tell the number of objects). Students use different counting strategies to count a group of concrete objects and a group in a picture. “Display 7 bear counters in a scattered configuration. Consider using two colors to highlight a 5-group. ‘I wonder how many bears there are. What can we do to find out?’ (We can count them.) ‘How should we count the bears so we don’t make a mistake?’ (We can touch and count. We can move them as we count. We can move them into a line.) ‘Good ideas. I will move them into a line as we count them together.’ Move the bears into a line as the class counts chorally to 7. ‘How many bears?’ (7 bears) ‘That’s right. Moving and counting the bears makes it easy to count without missing any bears or counting a bear twice.’ Set aside the bear counters for Land. Display the picture of bears in the forest. ‘Look at the picture of bears. I wonder how many bears are in this picture. Can we move and count them as we did before?’ (No. They are stuck in the picture.) ‘How should we count the bears in the picture so we don’t make a mistake?’ (We can mark the first bear we count, keep going, and stop when we get back to the first one we marked. We can cross each bear off as we count it.) ‘Good ideas. Let’s cross off the bears. Count as I cross off each bear. How many bears are there?’ (7 bears) Transition to the next segment by framing the work. ‘Today, let’s look at pictures and use our counting strategies to find how many.’”

• Module 2, Topic C, Lesson 11: Construct and classify polygons, Launch, connects the supporting work of K.G.A (Identify and describe shapes) to the supporting work of K.G.B (Analyze, compare, create, and compose shapes). Students explore 4-sided polygons and then construct them. “Set up 4 or 5 stations by clearing all materials from workspaces, except for a generous supply of full and half-length coffee stirrers. Gather students around the first station with Puppet. ‘Let’s build flat shapes with straws. What flat shapes could we build?’ Students will likely name a square, a rectangle, and a triangle. Use one of their suggestions and have Puppet construct a rectangle or square. Ask students to name the shape and tell how they know it is a rectangle. Have Puppet construct a 4-sided shape that is not a square or a rectangle. The polygon should not be easy for students to name. ‘Puppet built an interesting shape. What is the same about this shape and the rectangle?’ (They both have 4 straight sides.) Place the 4 card on the table and let students know that this table is for shapes with 4 sides and 4 corners. ‘Someone said we could build a triangle with the straws. Should we build a triangle at the table for shapes with 4 sides and 4 corners?’ (No) ‘What number belongs on the triangle table?’ (3) Place the 3 card on another table. Elicit possible numbers of sides and corners for the other tables and label them. ‘Could we build a circle with the straws?’ (No, circles don’t have straight sides.) Transition to the next segment by framing the work. ‘Today, we will make shapes with straight sides and corners.’”

• Module 3, Topic B, Lesson 8: Use a balance scale to compare two objects, Learn, Scavenger Hunt, connects the supporting work of K.MD.A (Describe and compare measurable attributes) to the supporting work of K.MD.B (Classify objects and count the number of objects in each category). Students find objects around the classroom to weigh and determine which is heavier. Students then complete a table with a list of heavier and lighter objects. “Group students. Give each group a scale and a set of Weight Comparison cards. Briefly review norms for using and caring for the scale, such as not pressing down on the buckets and not adjusting the calibration mechanisms. ‘When the music starts, move around the room and find two objects. Your objects must be small enough to fit inside the baskets on the balance scale.’ Play a familiar song to create clear starting and stopping points for the scavenger hunt. When the song ends, students should be near their scale with their objects. Direct groups to put their objects in a central location. ‘When it is your turn, you can pick two objects from your group’s collection. Place them on the balance scale gently. Use your cards to show which is heavier and which is lighter, and then say which it is in a complete sentence. Everyone in the group gets to draw your objects in their books.’ Guide students if needed. Help groups choose who goes first. Circulate and observe. Use the following assessing and advancing questions, as needed: Which object is heavier and which is lighter? How do you know? How did you show which is heavier and lighter on your recording sheet? What do you think could be heavier than ___? Why do you think that? How can you test your idea?”

• Module 6, Topic B, Lesson 11: Represent teen number decompositions as 10 ones and some ones and find a hidden part, Learn, How Many Are Hiding?, connects the major work of K.NBT.A (Work with numbers 11-19 to gain foundations for place value) with 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 hide part of the total, tell a story, and represent the situation. “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 inserted. 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.”

The materials reviewed for Eureka Math² Kindergarten 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.

Content from future grades is identified within materials and related to grade-level work. These references are consistently included within Topic and Module Overviews and less commonly found within teacher notes at the lesson level. Examples include:

• Module 1, Topic G: Analyze the Count Sequence, Topic Overview, connects K.CC.A (Know number names and the count sequence) and K.CC.B (Count to tell the number of objects) to work in future grades. “Topic G celebrates kindergarten students’ growth with counting concepts while opening a door to more sophisticated ways of using the number system. Their work to uncover the pattern of 1 more and the pattern of 1 less in the count sequence is the first conceptual step on a path leading to counting on strategies in grade 1.” (1.OA.C)

• Module 3: Comparison, Module Overview, After This Module, connects K.MD.A (Describe and compare measurable attributes) to work in Grade 1, Module 4. “Students explore indirect comparison by using the length of one object to compare two other objects. They begin to measure length by using the standard units of centimeter cubes.” (1.MD.A)

• Module 6, Topic A, Lesson 2: Find 10 ones in a teen number, Debrief, Teacher Note, 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 vocabulary that will be solidified in Grade 1. “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.” (1.NBT)

Materials relate grade-level concepts from Grade K explicitly to prior knowledge. These references can be found consistently within Topic and Module Overviews and less commonly within teacher notes at the lesson level. In Grade K, prior connections are often made to content from previous modules within the grade. Examples include:

• Module 2, Topic A, Lesson 2: Classify shapes as triangles or nontriangles, Launch, Teacher Note, connects K.G.A (Identify and describe shapes) to prior knowledge students may have before entering Kindergarten. “Depending on prior math experience, kindergarten students may name all, some, or none of the triangles in the pictures. They are more likely to recognize exemplars, or ‘typical’ triangles, in familiar orientations. They are less likely to recognize exemplars or variants in unusual orientations.”

• Module 5: Addition and Subtraction, Topic C Overview, connects work with K.CC.2 (Count forward beginning from a given number within the known sequence) to Fluency work they have already been doing in Kindergarten. “The topic also presents opportunities to practice and apply key kindergarten standards. Counting from a number other than 1 (K.CC.2) 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 (K.OA.4).”

• Module 6: Place Value, Module Overview, Before This Module, connects K.NBT.1 (Compose and decompose numbers from 11 to 19 into ten ones and some further ones) to previous work from Modules 1 and 5. “In Kindergarten Module 1, students learn to integrate all four elements of the number core as they count and create sets. Their tools encourage them to think of numbers relative to 10. In Kindergarten Module 5, students use addition and subtraction sentences to represent composition and decomposition of numbers to 10. They look for and make use of patterns. They continue to develop fluency with counting within 100. Through counting collections, students discover that grouping objects makes it easier to both track and count.”