1st Grade - Gateway 2

The materials reviewed for Eureka Math² Grade 1 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. 

Materials provide opportunities for students to develop conceptual understanding throughout the grade level. These opportunities are most often found within the Launch and Learn portions of lessons. Examples include:

• Module 2, Topic C, Lesson 12: Represent and find an unknown subtrahend in equations, Launch, students develop conceptual understanding as they represent subtraction as an unknown addend problem. “Put 8 pennies in one hand. Present the following situation. ‘I have 8 pennies.’ (Open your hand and show the pennies.) ‘See? Close your eyes. No peeking!’ Put 6 pennies in a location that is out of students’ sight. ‘Open your eyes.’ (Show the 2 pennies in your hand.) ‘What happened?’ (You don’t have as many pennies. Maybe you gave some away.) Summarize the story and represent it with an equation. ‘I hid some pennies. Let’s figure out how many I hid. Remember, I had 8 pennies.’ (Write 8.) ‘I hid some.’ (Make a minus symbol followed by an empty box.) ‘I have 2 pennies left.’ (Write = 2.) Have students work independently to solve. Provide materials such as number paths, pennies, and whiteboards for student use. Encourage students to self-select their tools. Select two students to share their thinking in the next segment. If possible, include a student who uses counting back. If no one counts back, then prepare to model it directly as part of the upcoming discussion. When students are finished working, transition to the next segment by framing the work. ‘Today, let’s share our strategies for figuring out how many pennies I hid.’” (1.OA.4) 

• Module 3, Topic D, Lesson 16: Identify ten as a unit, Learn, students develop conceptual understanding of place value and reason about groups of ten ones. “Distribute Unifix Cubes. Hold up the Hide Zero cards for 24. ‘Use your cubes to make 24. Make as many groups of 10 as you can.’ Allow students a moment to work. Then hold up the Hide Zero cards for 24 again. ‘How many tens did you make?’ (2) ‘What number is the same as 2 tens?’ (20) Pull the cards apart. Confirm that 2 tens is 20 and hold out the 20. (Hold out the 4.) ‘How many ones?’ (4) ‘How much is 20 and 4?’ (24) Put the cards back together to show 24. Ask students to set out the Tens and Ones removable they used in Fluency. Display a copy and model how to represent 24 in three ways as students follow along. Use the Hide Zero cards to show 24 again. Write 24 as the total in the number bond. ‘We can show 24 as two parts: 20 and 4.’ Pull the Hide Zero cards apart to show 20 and 4. Write 20 and 4 as the parts in the number bond. Ask students to point to where they see 20 and then 4 in their cubes.” (1.NBT.2a)

• Module 5, Topic D, Lesson 18: Determine if number sentences involving addition and subtraction are true or false, Learn, students develop conceptual understanding as they reason about the equal sign, and determine if equations involving addition and subtraction are true or false. “Display the false number sentence. 20+30=50+10. ‘Let’s figure out if this number sentence is true or false. What is 20=30?’ (50) Write 50 below 20+30. ‘What is 50+10?’ (60) Write 60 below 50+10. ‘The total for each expression on either side of the equal sign is not the same. This number sentence is false.’ Draw an X on the number sentence. ‘A number sentence is true when the expressions on both sides of the equal sign represent the same amount.’”  (1.OA.7)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. The Learn (Problem Set) and Land (Exit Tickets) portions of lessons consistently include these opportunities. When appropriate, teachers may use other portions of lessons for independent demonstration of conceptual understanding. Examples include:

• Module 3, Topic D, Lesson 24: Decompose the subtrahend to count back, Launch, students develop conceptual understanding as they subtract within 20 using strategies such as counting back. Students see an image of ten turtles and four frogs in a row. “Display the picture and the first line of the word problem. Read it aloud. Problem: John has ___ stickers. He gave Max 4 frog stickers. He gave Baz 4 turtle stickers. How many stickers did John give away? How many stickers does he have now? ‘Look at the picture. How many stickers does Jon have? How do you know?’ (14. He has 10 turtle stickers and 4 frog stickers. 10+4=14.) Fill in the total on the blank line in the sentence. Display the rest of the word problem and read it aloud. Invite students to turn and talk about the story. Prompt students to turn to the 5-group drawing and number path in their student book. Reread the problem one line at a time as necessary. Provide a few minutes for students to solve the problem and represent the story on their 5-group and number path. Show a 5-group drawing and number path. Review and record the solution to the problem in a way that emphasizes subtracting 4 and then another 4. ‘How did you use the dots to show the frog stickers given to Max?’ (I crossed off the last 4 dots. Those were the frog stickers.) ‘Let’s label this set of dots. If you did not label it yet, label yours as I label mine. What should we label these?’ Teacher Note (We can write F for frog stickers.) ‘How many stickers does Jon have now?’ (He has 10.) ‘Did we complete the problem?’ (Not yet. Jon also gives Baz 4 turtle stickers.) ‘What can we draw?’ (We can cross off 4 more dots. We can write T for turtle stickers.) ‘How many stickers did Jon give away? How do you know? (He gives away 8. I know because 4+4=8. How many does he have now? How do you know?’ (Now he has 6. They are left from the ten.) Record the number sentence 14-8=6 as you say the numbers in the context. ‘Jon had 14 stickers. He gave away 8, and now he has 6 stickers.’ Model representing the problem on the number path as students follow along. ‘Jon had 14 stickers. (Circle 14.) He gave away 4. (Hop back 4 and label it – 4.) Now he has 10 stickers. He gave away 4 more. (Hop back 4 and label it – 4.) He has 6 stickers left. What is the same about our drawing and our number path?’ (They both show 14 as the total. They both take away 4 and 4. That is 8. They both show 6 is how many stickers he has left. We got to ten both times.) Encourage students to take a moment to add to or revise their work as needed. Transition to the next segment by framing the work. ‘Today, we will subtract by breaking up a part and counting back to ten.’” (1.OA.6)

• Module 4, Topic C, Lesson 11: Compare to find how much shorter, Learn, students demonstrate conceptual understanding as they express the length of an object as a whole number of length units, by laying multiple copies of a shorter object end to end. “Ask students to turn to the two spiders in their student books. Identify the first one as a camel spider and the second one as a wandering spider. Invite students to share what they notice about the spiders. Expect a variety of observations, but highlight ideas related to length. ‘I wonder how much longer the wandering spider is than the camel spider. I also wonder how much shorter the camel spider is than the wandering spider. Let’s measure them and find out.’ Invite students to measure both spiders by using 10-centimeter sticks and cubes. Direct them to measure spider C by placing their cubes below the picture and to measure spider W by placing the cubes above its picture. Make sure students start measuring at the left endpoints. Tell students to record the lengths and leave their cubes in place. Invite students to share the lengths, and bring the class to agreement that spider C measures 10 centimeters and spider W measures 13 centimeters. Then display the two measurements. Students may revise their work as needed. ‘Which spider is longer? How do you know?’ (The wandering spider is longer. 13 centimeters is longer than 10 centimeters.) ‘Which spider is shorter? How do you know?’ (The camel spider is shorter. 10 centimeters is shorter than 13 centimeters.) Invite students to think–pair–share about how much longer the wandering spider is than the camel spider. ‘How much longer is the wandering spider than the camel spider? How do you know?’ (The wandering spider is 3 centimeters longer. There are 3 extra cubes.) Invite students to think–pair–share about how much shorter the camel spider is than the wandering spider. ‘How much shorter is the camel spider than the wandering spider? How do you know?’ (The camel spider is 3 cubes shorter. It needs 3 more cubes to be the same length as the other spider.) ‘What do you notice about the measurements for how much longer the wandering spider is than the camel spider and for how much shorter the camel spider is than the wandering spider?’ (They are the same.) Ask students to write a number sentence that represents how they found the difference in length: 10+3=13 or 13-3=10. Encourage them to draw a box around the number that represents the difference in length. Invite a few students to share their number sentences and explain their thinking. Read aloud each statement at the bottom of the page and have students complete it.” (1.MD.2)

• Module 5, Topic D, Lesson 19: Add tens to a two-digit number, Problem Set, students independently demonstrate conceptual understanding of place value as they add values of ten. “Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex. Directions and word problems may be read aloud. Students may use the Double Place Value Chart removable as they complete the problems. (Problem set picture shows 3 boxes of crayons and 4 more.) Read, Kit has 3 boxes of crayons and 4 more. She gets 2 more boxes. How many crayons does she have now? Draw, Write, Kit has _____crayons.” (1.NBT.4)

The materials reviewed for Eureka Math² Grade 1 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The materials develop procedural skill and fluency throughout the grade level, within various portions of lessons, including Fluency, Launch, and Learn. There are also opportunities for students to independently demonstrate procedural skill and fluency. Examples include:

• Module 1, Topic B, Lesson 12: Count on from 10 to find an unknown total, Fluency, Ready, Set, Add, students develop procedural skill and fluency in adding within 10. “‘Let’s play Ready, Set, Add.’ Have students form pairs and stand facing each other. Model the action: Make a fist, and shake it on each word as you say, ‘Ready, set, add.’ At ‘add,’ open your fist, and hold up any number of fingers.Tell students that they will make the same motion. At ‘add’ they will show their partner any number of fingers. Consider doing a practice round with students. Clarify the following directions: To show zero, show a closed fist at ‘add.’ Try to use different numbers each time to surprise your partner. Each time partners show fingers, have them both say the total number of fingers. Then have each student say the addition sentence, starting with the number of fingers on their own hand. Circulate as students play the game to ensure they are trying a variety of numbers within 5.” (1.OA.6)

• Module 2, Topic D, Lesson 18: Use related addition facts to subtract, Learn, Related Facts, students develop procedural skill and fluency with subtraction as they subtract by thinking about a related addition fact. “Ensure that students have Think Addition inserted in a personal whiteboard. Show Think Addition, and model filling in the subtraction equation to show 6-2=\square as students follow along. Point to the 2. ‘What goes with 2 to make the total 6?’ (4) ‘Let’s write that as an addition sentence.’ Guide students to write the related addition sentence: 2+4=6. ‘How does knowing 2+4=6 help us quickly, or efficiently, find the answer to 6-2?’ (To figure out 6-2, we can think of the part that goes with 2 to make 6. It’s 4.) ‘Yes, we can use the related addition fact we know to help us subtract.’ Complete the subtraction sentence and number bond for these related facts.” (1.OA.6)

• Module 6, Topic A, Lesson 3: Draw two-dimensional shapes and identify defining attributes, Fluency, Happy Counting by Ones from 100–120, students develop procedural skill and fluency with counting by starting from any number less than 120. “‘Let’s count by ones. The first number you say is 107. Ready?’ Signal up or down accordingly for each count. Continue counting by ones to 120. Change directions occasionally, emphasizing crossing over 110 and where students hesitate or count inaccurately.” (1.NBT.1)

Materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level. These are often found within the Problem Set or within Topic Tickets, Learn and Land lesson sections respectively. When appropriate, teachers may use other portions of lessons for independent demonstration of procedural skill and fluency. Examples include:

• Module 2, Topic A, Lesson 2: Subtract all or subtract 0, Problem Set, students demonstrate procedural skill and fluency as they solve problems where they subtract all or subtract 0 and use generalizations to help them solve related problems more efficiently. “Directions may be read aloud. As needed, students may use mental math, an anchor chart, their fingers, or cubes to solve. 1. Subtract. 5-5=__, 9-9=__, 7-7=__, 12-12=__. Write 2 number sentences like these.” Four problems are also included for subtracting 0. (1.OA.6)

• Module 3, Topic C, Lesson 14: Count on to make the next ten within 100, Land, Topic Ticket, students demonstrate procedural skill and fluency by using properties of operations as strategies to solve addition and subtraction problems. Teacher directions state, “Provide up to 5 minutes for students to complete the Topic Ticket. It is possible to gather formative data even if some students do not complete every problem.” (Students have a number line they use to show their jumps.) “Hop to 10 first to add. Write a 3-part number sentence. 8+6=__”  (1.OA.3)

• Module 5, Topic A, Lesson 1: Tell time to the hour and half hour by using digital and analog clocks, Learn, Problem Set, Problem 2, students demonstrate procedural skill and fluency with telling time. Directions state, “Draw lines to match the times.” Students see five analog clocks showing times to the hour or half-hour and times written in words or digital time formats. (1.MD.3)

The materials reviewed for Eureka Math² Grade 1 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. 

There are multiple routine and non-routine application problems throughout the grade level, including opportunities for students to work with support of the teacher and independently. While single and multi-step application problems are included across various portions of lessons, independent application opportunities are most often found within Problem Sets or the Lesson Debrief, Learn and Land sections respectively.

Examples of routine applications of the math include:

• Module 2, Topic C, Lesson 11: Represent and solve take from with change unknown problems, Learn, Act It Out, students solve routine addition and subtraction problems with the teacher’s guidance. “Ensure each student has a personal whiteboard and a bag of craft sticks. Tell the students that the craft sticks represent pencils. Display the word problem and read it aloud. ‘I put 9 pencils on the desk. Some fell off. Now there are 3 pencils. How many pencils fell off the desk? Let’s use our craft sticks to show this problem. 9 pencils are on the desk at first.’ Ask students to organize their sticks in a 5-group row or array but not as tallies. ‘Some fall off. Now there are 3 pencils. How can we show that?’ (We can move 3 sticks to their own group. We can take sticks off until there are 3 left.) Give students a moment to model with their sticks. ‘How many pencils fell off the desk? How do you know?’ (6 pencils. I took off one at a time until 3 were left. I took off 6 sticks. 6. I put 3 in one group. There are 6 in the other group.) ‘Let’s write a number sentence to match the problem. What should we write first? Why?’ We should write (9 first because there are 9 pencils on the desk in the beginning.) Record as students follow along on their personal whiteboards. ‘What should we write next? Why?’ (Write minus 6. That is how many pencils fall off.) ‘What should we write last? Why?’ (Write equals 3. There are 3 pencils left on the desk.) ‘What were we trying to figure out?’ (How many pencils fell off the desk) Draw a box around 6, the unknown. ‘The part that fell off the desk, or gets subtracted, is the unknown.’ Point to each component of the number sentence to summarize the problem. ‘There were 9 total pencils. 6 fell off, and now there are 3 pencils.’ Have students clean up their craft sticks.” (1.OA.1) 

• Module 3, Topic D, Lesson 19: Solve take from with change unknown problems with totals in the teens, Topic Ticket, students solve a routine subtraction problem independently. “Provide up to 5 minutes for students to complete the Topic Ticket. It is possible to gather formative data even if some students do not complete every problem.” Problem 3, “There are 16 frogs. Some frogs hop away. Now there are 12 frogs. How many frogs hop away?” (1.OA.1)

• Module 4, Topic A, Lesson 3: Compare the lengths of two objects indirectly by using a third object, Launch, students solve routine application problems as they order three objects by length and compare the lengths of two objects indirectly by using a third object. “Gather students and display the giraffe standing next to a tree. ‘I went to the zoo and saw a giraffe. What do you notice about the giraffe?’ (The giraffe is tall. It is next to a rock. It is eating leaves off the top of the tree.) ‘I visited other animals and then came back. When I came back, I saw a giraffe eating leaves from the same tree.’ Display the second giraffe. ‘What do you notice about the giraffe now?’ (It is eating leaves from the side of the tree. Before, it was eating from the top.) ‘Show thumbs up if you think this is the same giraffe I saw the first time. Show thumbs down if you think it is a different giraffe.’ Invite students to think–pair–share about whether the pictures show the same giraffe or a different giraffe. (The pictures show different giraffes. The first one is eating from the top of the tree. The second one is eating from the side of the tree. I think the pictures show the same giraffe. It finished eating from the top of the tree and then started eating from a different spot.) The pictures show different giraffes. How could we use the tree to see that the giraffes’ heights are different?’ The first giraffe is taller than the tree. It could reach the leaves on top of the tree. The second giraffe is shorter than the top of the tree. That's why it has to eat from the side of the tree. The first giraffe is taller than the second one.) Display both giraffes. Point to the first giraffe. ‘What can we say to compare giraffe A to the tree?’ (Giraffe A is taller than the tree.) Point to the second giraffe. ‘What can we say to compare giraffe B to the tree?’ (Giraffe B is shorter than the tree.)’ The giraffes are not standing together, and we can’t move them. We can use what we know to compare giraffe A to giraffe B. Here is what we know: Giraffe A is taller than the tree, and giraffe B is shorter than the tree.’ Have students think–pair–share about whether giraffe A is taller or shorter than giraffe B. (Giraffe A is taller than the tree, so it must be taller than giraffe B too.) Transition to the next segment by framing the work. ‘Let's compare the lengths of two objects that are not right next to each other. We will use a third object, like the tree in these pictures, to help us.’” (1.MD.1) 

Examples of non-routine applications of the math include:

• Module 1, Topic A, Lesson 4: Find the total number of data points and compare categories in a picture graph, Launch, Students use a non-routine strategy, with teacher guidance, in order to use attributes of a figure to sort. “Display the butterfly garden, and give students a moment to look at the image. ‘Mathematicians notice things about the world around them and ask lots of questions. Let’s look at this butterfly garden as mathematicians would.’ Have students think-pair-share about what they notice and what they wonder about the butterflies. Accept all responses, and consider writing them in a Notice and Wonder chart. Revoice student observations centered around attributes such as color, size, shape, and design. ‘I heard you say that some of the butterflies are the same in some ways. I wonder whether we can sort the butterflies like we sorted our cubes in the last lesson.’ Partner students, and ensure that each pair has a set of cut butterfly cards. Provide an opportunity for exploration by inviting partners to sort in ways that make sense to them. As they finish, gather the class, and discuss various ways to sort. Reiterate that there are many ways to organize these butterflies into groups.” (1.MD.4)

• Module 3, Topic D, Lesson 15: Count and record a collection of objects, Learn, Organize, Count, and Record, students independently solve non-routine application problems by organizing and counting objects and recording their process. “Partner students and invite them to choose a collection, any organizing tools they desire, and space to work. Circulate and notice how students organize, count, and record. Use the following questions and prompts to assess and advance student thinking about their plan. ‘What is your plan? Show (or tell) me how you are counting. How are you keeping track of what you already counted and what you still need to count? How does what you drew or wrote show how you counted your collection?’ If students finish early, provide some options for using their time productively: ‘Raise their hand so you can check in with them. Try another way to count and record. Switch with another set of partners and count to check each other’s collections. Share their recording with another pair of students and explain their thinking. Clean up and get another collection.’” (1.NBT.1)

• Module 6, Topic B, Lesson 9: Relate the size of a shape to how many are needed to compose a new shape, Topic Ticket, students solve non-routine problems as they work independently to create composite shapes. “1. Trace a composed shape (A picture with rectangles, trapezoids, triangles, and a hexagon is shown) How many sides does the composed shape have? Circle the shapes that made the composed shape. 2. Draw a triangle to make a composed shape. (An isosceles triangle is shown) What is the composed shape?” (1.G.2)

The materials reviewed for Eureka Math² Grade 1 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. 

All three aspects of rigor are present independently throughout the grade. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Module 1, Topic B, Lesson 10: Count on from 5 within a set, Learn, Find Five and Count On, students develop conceptual understanding when they recognize a group of 5 within a set and use it to count on to find the total. “Display the 7-dot card. Have students think-pair-share about the following question, ‘What are some ways to count on to find the total?’ Invite a few students to share responses with the class. If possible, include a student who mentions counting on from 5. Circle the row of 5 dots on the image. ‘I circled 5. 5 is one part of the total. What is the other part?’ (The other part is 2.) ‘Where do you see the parts of 5 and 2 in the picture?’ (There are 5 dots in the circled part on top and 2 on the bottom.) Have the class chorally count on from 5 to find the total. Circle the 5 as the first part students counted on from. Then record the counting on sequence so students can see it as they voice it. ‘What is the total number of dots?’ (7) Ask students to share the parts and the total again. As they share, draw a number bond to represent the part-total relationship. ‘What number sentence could we write to show how we counted on?’ (5+2=7).” (1.OA.5)

• Module 3, Topic A, Lesson 1: Group to make 10 when there are three parts, Learn, Group to Make 10, students develop procedural skill and fluency as they work with the teacher to group two addends that are partners to 10 and solve a three-addend equation. “Show three cups of pencils. Invite a student to count the pencils in each cup. Write the totals on sticky notes and place the sticky notes on the cups. ‘What equation can we write to show the total number of pencils in the three cups? Write it on your whiteboard. ($$5+5+3=$$__) Write 5+5+3=__. The parts in an addition equation or expression are called addends. Addends are parts we are adding.’ (Point to the 5, 5, and 3.) ‘What can we call the parts 5, 5, and 3 in this addition equation?’ (Addends) ‘Why are there three addends in our equation?’ (There are three addends because there are three cups, or three parts.) Ask students to work in pairs to find the total number of pencils. Listen for a variety of strategies, but in particular, listen for adding 5 and 5 to make ten. Have one or two students share. If no one suggests making ten, model it. ‘13. I counted on. Fiiiive, 6, 7, 8, 9, 10. Tennn, 11, 12, 13. I know 5+5=10 and 3 more is 13.’ Ask a student to combine both groups of 5 pencils into a single cup and label it with the number 10. Show the strategy of making ten first by drawing a number bond. Ask students to follow along on whiteboards. ‘5 and 5 are partners to 10. We can add 5 and 5 first to make ten. 10 and 3 make 13. Is grouping 5 pencils and 5 pencils into one cup helpful? Why?’ Ask a student to combine both groups of 5 pencils into a single cup and label it with the number 10. (Grouping 5 and 5 is helpful because it makes ten. It is easier to add 10 and 3 than 5, 5, and 3.) Return the pencils and sticky notes to their original cups. Rearrange the order of the cups to show 5, 3, and 5. ‘What changed about our cups?’ (You put them back, but now they are in a different order.) ‘What three-addend equation can we write?’ (5+3+5=__) Write 5+3+5=__ as students do the same. Ask them to work in pairs to find the total. Invite a student who uses partners to 10 to share. (I added 5 and 5 first; that is 10. I know that 10+3=13) Show the strategy of making ten first by drawing a number bond as students follow along. ‘Why did we group 5 and 5?’ (To make ten) ‘What is 10+3=?’ (13) Ask students to tell a partner the strategy of making ten first in their own words.” (1.OA.2)

• Module 5, Topic E, Lesson 22: Decompose both addends and add like units, Launch, students solve non-routine application problems by using place value knowledge to add. “Present the problem and use the Math Chat routine to engage students in mathematical discourse. Display the hands holding coins. (Image of two hands each with 2 dimes and 4 pennies) ‘Two friends each have some coins. What coins do they each have?’ (2 dimes and 4 pennies) ‘They are wondering how much money they have together.’ Give students 1 or 2 minutes of think time to find the total. Students may self-select tools such as coins, drawings, number sentences, number bonds, or mental math to find the total. Have them show a silent signal when they are ready. Invite students to discuss their ideas with a partner. Listen for students who find the total by combining tens with tens (dimes) and ones with ones (pennies). Facilitate a discussion by inviting two or three students to share their thinking with the class. Have students refer to the Talking Tool as needed. Revoice or demonstrate combining tens with tens and ones with ones. Draw the coins as shown in the sample. ‘Many of you combined the dimes and pennies to find the total. (Circle the tens.) How many dimes, or tens, are there?’ (4 tens) ‘How many is 4 tens?’ (40) Label the tens 40. (Circle the ones.) ‘How many pennies, or ones, are there?’ (8 ones) Label the ones 8. (Draw arms.) ‘What is 40 and 8?’ (48) Write the total and then write the number sentence 24+24=48. ‘Each hand holds 24 cents. Together, they have 48 cents.’ Point to the digits in the tens place: 2, 2, and 4. ‘2 tens and 2 tens make 4 tens. Did we make a new ten when we combined the ones?’ (No.) Transition to the next segment by framing the work. ‘Today, we will add two-digit numbers together by adding tens to tens and ones to ones. We will see if we make a new ten!’” (1.NBT.4)

Multiple aspects of rigor are engaged simultaneously throughout the materials in order to develop students’ mathematical understanding of grade-level topics. Examples include:

• Module 2, Topic D, Lesson 17: Use related addition facts to subtract from 10. Learn, Subtract Efficiently, students develop conceptual understanding alongside application as they use related facts to solve subtraction problems. “Display 10-8=__ and point to the gray box that represents the unknown. ‘Is the answer to a subtraction problem a part or the total? How do you know?’ (A part. 10 is the total and 8 is one part we know.) ‘Let’s think about addition and ask ourselves, What goes with this part to make the total, 10?’ Point to the 8. Have students write the answer on their whiteboards. Use the Whiteboard Exchange routine to review work and provide feedback. Tell students to turn their whiteboards over so the red side is up when they are ready. Say, ‘Red when ready!’ When most students are ready, tell them to hold up their whiteboard to show you their work. Give quick individual feedback, such as ‘Yes!’ or ‘Check your total.’ For each correction, return to validate the corrected work. Repeat with 10-3 and 10-2. ‘How are we figuring out these subtraction problems? Turn and talk to a partner.’ (We are thinking about the other part that makes 10. We think about the partners to 10. We know the addition fact that uses the same number bond.)” (1.OA.4)Module 4, Topic B, Lesson 5: Measure and compare lengths, Problem Set, Problems 1, 2 and 3, students develop conceptual understanding alongside application as they compare the lengths of two objects indirectly by using a third object. Students see a picture of a pair of scissors, a calculator and a centimeter cube to measure the items. Then they are directed to circle either “longer than” or “shorter than” to finish a sentence. “The (picture of scissors) is ___ the (picture of calculator). Write < or > to compare (picture of scissors) to (picture of calculator.)” (1.MD.1)

• Module 5, Topic A, Lesson 6: Add 10 or take 10 from a two-digit number, Learn, Ko’s Coins, students develop procedural skill and fluency alongside application as they add and subtract within 10. “Distribute one bag of pennies and dimes to each pair of students. Then play part 1 of the video, which shows Ko putting coins in her pocket and then finding another dime. Ask partners to use their pennies and dimes to find how much money Ko has in her pocket now. Remind them, if needed, that she has 2 dimes and 7 pennies. ‘How much money does Ko have? How do you know?’ (2 dimes and 7 pennies. 20, 21, … , 27 cents. 1 dime and 7 pennies is 17 cents. If we add 1 dime, we have 27 cents.) Play part 2 of the video, which shows Ko throwing a dime into a fountain at the park. Ask partners to use their pennies and dimes to find how much money Ko has in her pocket now. ‘How many cents does Ko have now? How do you know 17 cents?’ (She threw the dime she found into the fountain. She just took a dime away. Now she has 1 dime and 7 pennies. 10 cents less than 27 cents is 17 cents. Dimes are the same as ten cents. We can use dimes to show adding 10 or taking 10.) ‘Let’s do that some more.’ Have partners show 54 cents with dimes and pennies. ‘What is 10 more than 54? How do you know?’ (64 You just add a dime, or 10, to 54.) Display the two place value charts. Confirm that 10 more than 54 is 64. ‘54 is 5 tens and 4 ones. 64 is 6 tens and 4 ones.’ Record 54 and 64 in the charts as shown. Draw an arrow from 5 tens to 6 tens, and label it +10. When we add a 10, the digit in the tens place is 1 more. Leave the place value charts displayed but erase the recording. Have students show 42 cents with dimes and pennies. ‘What is 10 less than 42? How do you know?’ (32 It is 1 ten less than 42. You just take away a dime.) Confirm that 10 less than 42 is 32. Record 32 and 42 in the charts as shown. Draw an arrow from 4 tens to 3 tens and label it –10. ‘When we take a 10, the digit in the tens place is 1 less.’ As time allows, use the following suggestions to repeat the process: 74 cents (show 10 less) 85 cents (show 10 more)” (1.OA.6)

The materials reviewed for Eureka Math² Grade 1 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with these practices across the year and they are identified for teachers within margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Grade 1-2 Implementation Guide, “‘Promoting the Standards for Mathematical Practice’ highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP. Often, the suggested questions for a particular MP repeat. This intentional repetition supports students in understanding the MPs in different contexts.”

MP1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 1, Topic A, Lesson 5: Organize and represent categorical data, Learn, Sort and Count a Set, students build experience with MP1 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “When students select a set of objects, make a plan about how to count, carry out the plan, and adjust the plan as needed, they make sense of problems and persevere in solving them (MP1). Ask the following questions to promote MP1: How could you explain your plan in your own words? Is your plan working? Is there something else you could try?” Teacher directions state, “Pair students and invite them to find a workspace. Consider having students count on a work mat or chart paper to help them keep their materials in their own work area, allow for the organized objects to be moved and shared, and expedite clean-up. Have whiteboards or sticky notes available for students to track their counts. Circulate and ask questions such as the following that encourage organization and accuracy: What is your plan? Why did you choose that plan? Show or tell me how you are counting. How are you keeping track of what you already counted and what you still need to count? How can you organize to make counting easier? As students work, notice how they organize and count after sorting. The following samples show possible ways to sort objects. Select two work samples that demonstrate an accurate sorting method and one or more of these counting strategies to share in the next segment: Uses a strategy to keep track of the count (e.g., move and count), Uses a tool to group or organize (e.g., cups or number paths), Uses 5-groups to organize.”

• Module 5, Topic E, Lesson 21: Use varied strategies to add 2 two-digit addends, Land, Debrief, students build experience with MP1 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “As students work with addition expressions in which both addends are two-digit numbers, they make sense of problems and persevere in solving them (MP1). As the numbers that students are expected to work with get larger, the most efficient way to do so increasingly depends on what students are comfortable with and their own developing number sense. Encourage students to make sense of the problem as they see fit. The use of cubes can help students persevere if their number sense is not yet strong enough to lean on when working with large addends.” Students are presented with two Make 50 cards that show 25 and 25. Teacher directions state, “‘Show thumbs-up if you think these cubes make 50. Show thumbs-down if you think they do not make 50.’ Invite students to think–pair–share about the ways the cubes could be combined. ‘What are some ways we could combine these cubes to add them?’ Listen for students who share one of the three ways highlighted in the lesson. Invite them to share and record their thinking. Share any of the three strategies that students do not mention. (20 and 20 is 40. 5 and 5 is 10. 40 and 10 is 50. 25 and 20 is 45. 5 more is 50. 25 and 5 is 30. 30 and 20 is 50.) Write 25+25=50. ‘We made 50 again! Turn and talk. Which way is easiest for you to add two sets of cubes that have tens and ones? We can add tens and ones. We can also add the tens or the ones first. These are all ways to add 2 two-digit numbers.’”

• Module 6, Topic B, Lesson 8: Combine identical composite shapes, Learn, Compose a Shape, students build experience with MP1 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “Using composed shapes to make new, larger shapes can be challenging and gives students an opportunity to make sense of problems and persevere in solving them (MP1). These tasks require students to consider the smaller composed shape as a new unit and to manipulate it as a whole rather than as two distinct pieces.” Students are given two triangles cut out from a triangle removable. Teacher directions state, “‘Put your triangles together to see how many shapes you can make.’ Allow time for exploration. Encourage students to change the position of the triangles to make new shapes. Advance and assess student thinking by asking the following questions: What does it mean to make composed shapes? What is the new shape? How do you know? Invite two or three students to share how they composed new shapes (see possible compositions). Then direct students to open the student book to the recording sheet. ‘When we turn, flip, or move the triangles we can make new composed shapes.’ Model using the triangles to compose the shapes in the book, and have students name the new shapes.”

MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 2, Topic D, Lesson 15: Relate counting on and counting back to find an unknown part, Land, Debrief, students build experience with MP2 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “When students connect counting on and counting back to addition and subtraction number sentences, respectively, they reason abstractly and quantitatively (MP2). Later, these connections will help students make sense of addition and subtraction equations outside of any real-world context. The questioning throughout the lesson is designed to promote MP2 by having students verbalize these connections.” Teacher directions state, “Gather students with their Problem Sets. Choose one student’s work for the second number bond in problem 1 to display and discuss. Use the Five Framing Questions routine to invite students to analyze counting on and counting back. Notice and Wonder, ‘What do you notice about this work?’ (They counted on first. Counting back had a lot of hops.) Organize, ‘What steps did this student take?’ (They started at 7 and hopped up to 9. They started at 9 and hopped back to 2. They wrote addition and subtraction sentences to match.) Reveal, ‘Why can we count on or count back to find an unknown part?’ (Because the parts and total stay the same. You just use them in different ways.) Distill, ‘What is the difference between counting on and counting back to solve this problem?’ (If you count on, you don’t have as many hops to make. You only have to go up 2. If you count back, there are a lot of hops to get from 9 back to 2.) Know, ‘Why is counting on a more efficient strategy for solving this problem?’ (Counting on is better because there are fewer hops on the number path, or not as many fingers to put up.)”

• Module 4, Topic B, Lesson 6: Measure and order lengths, Learn, Compare Heights, students build experience with MP2 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “When students use measurements to order objects, they reason abstractly and quantitatively (MP2) because they use numbers to order the objects instead of direct comparison. This lesson creates a need for measuring by presenting scenarios where it is more convenient to measure objects than to compare them directly.” Teacher directions state, “Invite students to discuss what they know about the buddies’ plants. Point to Sam's plant and Wes's plant. ‘Is Sam's plant taller or shorter than Wes's plant? How do we know?’ (Sam's plant is taller than Wes's plant. 12 is greater than 9.) Record students’ thinking as 12>9. ‘Is Wes's plant taller or shorter than Sam's plant? How do we know?’ (Wes's plant is shorter than Sam's plant. 9 is less than 12.) Record students’ thinking as 9<12. Direct students to look at the bottom part of the ordering page in their student book. Read each statement aloud and invite partners to use any two plants to complete the statement. Tell students to write a comparison number sentence using the corresponding lengths. Consider the following example. ‘Look at the first statement as I read it aloud: Blank's plant is shorter than blank's plant. Which plants make the statement true? How do you know?’ (Sam's plant is shorter than Baz's plant. Sam's plant is 12 centimeters tall and Baz's is 14 centimeters tall.) Write a number sentence to show your thinking. Read it aloud to your partner. (12<14)”

• Module 6, Topic B, Lesson 6: Create composite shapes and identify shapes within two- and three-dimensional composite shapes, Learn, Composite Squares, students build experience with MP2 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “Students reason abstractly and quantitatively (MP2) when they identify two- and three-dimensional composite shapes and the shapes they are composed of. Doing this requires students to recognize the abstract boundary of the shape, ignoring the other shapes inside or outside of it. They attend to the quantitative attributes of shapes, such as the number of sides or corners, to identify and name the shape. Promote MP2 by helping students see both the composite shape and the parts that make it up.” Teacher directions state, “Display the squares image and have students turn to it in their student book. Pair students. Have them work together to find as many squares as they can. Ask students to outline the squares they see with a crayon.” (Some possible student responses are shown to teachers in a chart.) “Circulate and listen in to students’ thinking. Lead a discussion about the squares students see, using questions such as the following: Where do you see squares? How many small squares can you find? The bigger squares are composed shapes. They are made of smaller squares. Does anyone see a composed square made of other shapes? Do you see composed shapes that are not squares? Record their ideas by outlining the shapes they describe on the image.”

The materials reviewed for Eureka Math² Grade 1 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP3 across the year and it is explicitly identified for teachers within materials in margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Grade 1-2 Implementation Guide, “‘Promoting the Standards for Mathematical Practice’ highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP. Often, the suggested questions for a particular MP repeat. This intentional repetition supports students in understanding the MPs in different contexts.”

Students construct viable arguments and critique the reasoning of others in connection to grade-level content as they work with support of the teacher and independently throughout the units. Examples include:

• Module 1, Topic C, Lesson 14: Count on to find the total of an addition expression, Land, Debrief, students build experience with MP3 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “As students communicate their strategies, respond to their partner’s thinking, and ask questions, they construct viable arguments and critique the reasoning of others (MP3). If students need support, prompt them with the following questions: Tell (or show) your partner about your strategy. Why did you choose this strategy? What questions can you ask your partner about this strategy?” Teacher directions state, “Gather students with their Problem Sets. Have partners discuss a selected problem. To support student-to-student conversation, pair students, and assign one student to be partner A and the other partner B. Prompt partner A to use the ‘I can share my thinking’ section of the Talking Tool, and prompt partner B to use another section, such as ‘I can ask questions,’ to respond. (Sakon: My strategy was to count on my fingers. Val: How did you get 7? Sakon: Fooouuur, 5, 6, 7. (Holds up fist and then fingers.) Val: I did it a different way. I used the number path. I started at 4, and I landed on 7 too.) Then use any combination of the following questions to help students summarize their counting on understandings. ‘How can we count on when we cannot see all the parts?’ (We can use our fist for the first part and count on with our fingers for the second part.) ‘Why is it helpful to track the second number with your fingers?’ (We will know when to stop and get the answer.)”

• Module 2, Topic D, Lesson 16: Compare the efficiency of counting on and counting back to subtract, Land, Debrief, students build experience with MP3 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “Students construct viable arguments and critique the reasoning of others (MP3) when they discuss and defend the efficiency of strategies.The following questions provided in the sample dialogue promote MP3: Why is counting on or counting back more efficient for this problem? Why is the other counting strategy not as efficient for this problem? What can you do to figure out if counting on or counting back will be more efficient?” Teacher directions state, “Gather students. ‘We are going to play a game. When you convince someone, you try to get them to agree with your idea. First, I am going to try to convince you. Think about whether you agree with my strategy.’ Display the image of 11-9. ‘I decided to count back to solve this problem. I chose counting back because the problem is subtraction.’ Ask students to show a thumbs-up if they agree that you shared the best strategy and reason or to show a thumbs-down if they disagree with some part of what you said. ‘Some of you are not convinced. You disagree. Why?’ (I disagree. You should count on from 9.) ‘Say more. Why is counting on more efficient for this problem? Convince me.’ (9 is close to 11, so it is better to count on. Counting on is fewer hops on the number path or fingers.) ‘You convinced me! We should count on when the part we know is close to the total. When should we count back to solve? (When the part you know is just a small part of the total, When the part is 1, 2, or maybe 3, and the total is big) ‘We can count on and count back to subtract, but mathematicians think carefully about what strategy is efficient before they solve a problem.’”

• Module 3, Topic B, Lesson 7: Make ten when the first addend is 8 or 9, Learn, Make 10 at the Fair, students build experience with MP3 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “The game gives students the opportunity to construct viable arguments and critique the reasoning of others (MP3). As partner A finds the total in a problem, they explain their thinking to partner B, who in turn has a chance to critique partner A’s thinking and answer for accuracy. As you observe students playing the game, use the following questions to promote MP3: Why does your strategy work for you? What don’t you understand about your partner’s thinking? What questions can you ask about your partner’s thinking?” Teacher directions state, “Pair students. Make sure partners have their Make Ten Drawings in whiteboards, one Make Ten at the Fair, a die, and 2 two-color counters. If partner A lands on a problem, they use their Make Ten Drawing to find the answer by making ten, and then they explain their thinking to their partner…Partner B takes a turn.”

• Module 4, Topic A, Lesson 2: Reason to order and compare heights, Learn, Comparison Statements, students build experience with MP3 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “As students use the scientists’ observations to order owls and use their ordering to make their own comparisons, they construct viable arguments and critique the reasoning of others (MP3). When ordering the owls, partners may disagree about where a particular owl should go. Help students critique one another’s reasoning and reach the correct answer by asking them to think of questions they can ask their partner about why the owls should go in that order. When students make their own comparison statements involving owls that are not next to each other, listen for them to reason by comparing the owls to the owl in between them.” Teacher directions state, “Hang the owl posters in order by height, shortest to tallest. Tell students that these photos are enlarged to show the actual height of each owl… Then engage them in a discussion to compare the owls. ‘Owl E is first in order. What does that tell us about owl E?’ (It is the shortest owl.) ‘What does that tell us about the other owls?’ (All of the other owls are taller than owl E.) ‘Owl G is last in order. What does that tell us about owl G?’ (It is the tallest owl.) ‘What does that tell us about the other owls?’ (All of the other owls are shorter than owl G.) Display the sentence frames and read them aloud. (Sentence frames show ___is shorter than ___., ___ is taller than ___.) Point to owl E and owl B. Invite students to think–pair–share about the heights of owl E and owl B. ‘Is owl E shorter or taller than owl B? How do you know?’ (Owl E is shorter than owl B. Owl E is shorter than owl W, so owl E must be shorter than owl B.) Point to each owl as you revoice students’ reasoning. ‘Owl E is shorter than owl W, so owl E must also be shorter than owl B.’ Refer to the sentence frames again and ask students to compare the same two owls in a different way. ‘Owl E is shorter than owl B. What other true comparison can we say about these two owls?’ (Owl B is taller than owl E.) ‘If we know which owl is shorter, then we also know which owl is taller.’”

The materials reviewed for Eureka Math² Grade 1 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP4 and MP5 across the year and they are explicitly identified for teachers within materials in margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Grade 1-2 Implementation Guide, “‘Promoting the Standards for Mathematical Practice’ highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP. Often, the suggested questions for a particular MP repeat. This intentional repetition supports students in understanding the MPs in different contexts.”

MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students model with mathematics as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 2, Topic B, Lesson 6: Represent and solve related addition and subtraction result unknown problems, Launch, Related Subtraction Word Problem, students build experience with MP4 as described in the Teacher Note, Promoting the Standards of Mathematical Practice. “When students use a drawing to represent the story, label their drawing, and use their drawing to solve the problem, they model with mathematics (MP4). Ask the following questions to promote (MP4): How did you show the bugs that were on the leaf at the start? How did you show the bugs that flew on or off the leaf? How did you show the bugs that are on the leaf now?” Teacher directions state, “Invite students to turn to the next problem in their books. As with the last problem, have them solve the problem by using the Read–Draw– Write process. Again, begin by having the class follow along as you read the problem aloud. 9 bugs are on a leaf. 3 bugs fly off the leaf. How many bugs are on the leaf now? Have students retell the story to a partner. Then use the same process and questions as before to guide problem solving. ‘What did you do to figure out how many bugs are on the leaf now?’ (I drew 9 circles and crossed off 3. I see 6 are left. I drew 9 circles. I counted back 3: Niiiine, 8, 7, 6.) ‘Why are there fewer than 9 bugs now?’ (Some flew away.) Continue the class discussion by selecting two or three students to share their work. Use any combination of the following questions to help the class relate the number sentences to the drawings and the problem: ‘What does the first number in the number sentence tell us about? Where is that in the drawing? What math symbol does this number sentence use to show what happened? How does the math symbol match the drawing? Which number in the number sentence answers the question or shows the unknown? Where is the unknown in the drawing?’ Help the class agree on the correct number sentence. Then have students look at their own work. ‘What did we need to figure out?’ (We had to figure out how many bugs there are now.) ‘Where does your number sentence show that?’ (It’s the 6.) ‘Draw a box around 6. The bugs left on the leaf are the part that was unknown.’”

• Module 3, Topic E, Lesson 26: Pose and solve varied word problems, Learn, Who Has Most?, students build experience with MP4 as described in the Teacher Note, Promoting the Standards of Mathematical Practice. “Students who effectively organize information by using drawings, number bonds, or other mathematical tools model with mathematics (MP4).” Teach directions state, “Students identify what information they need to solve a mathematical question and represent their solution. Invite students to turn to problem 1 in their books. ‘Let’s think about this question: Who collected the most gems? Do we have enough information to answer that question from watching the video? Why?’ (No, we could see their bags were full, but we do not know how many gems they each collected.) Display the data that each person recorded. ‘Here is what Dad, Zan, and Kit wrote about their gems.’ Have students think–pair–share  about what they notice. Expect a variety of responses. ‘Our question is, who collected the most gems? Have we ever solved a problem like this before?’ (Yes.) ‘Turn and talk to your partner: What strategies could we use to solve this problem? What tools could we use?’ Invite partners or small groups to work together to solve the problem and show their thinking. Encourage them to use labeled drawings and number sentences in their work. Provide tools for students to self-select, such as 10-frames, counters, and number paths.”

• Module 6, Topic C, Lesson 10: Reason about equal and not equal shares, Launch, students build experience with MP4 as described in the Teacher Note, Promoting the Standards for Mathematical Practice. “As students use sticky notes to reason about equal shares of real-world objects, they model with mathematics (MP4). Real-world objects like food do not have perfectly straight sides or corners, which makes them more difficult to partition equally. Thinking about which shape an object resembles can help students make sense of how to partition the object.” In this activity students cut paper to show equal shares after watching a video about children sharing a brownie and teacher directions state, “‘Let’s cut a new brownie for the children in the video. How would you cut it to make the pieces fair? Use your scissors and sticky notes to try different ways of cutting the square brownie. Share your ideas with your partner.’ Have a few students who cut their sticky notes into equal shares display their work. If needed, cut sticky notes and show students the different ways to make equal shares. For each sample you show, ask these questions. ‘Is this fair? Why?’ Listen for responses that mention that the pieces are the same size, or equal.”

MP5 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students choose tools strategically as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 1, Topic A, Lesson 6: Use tally marks to represent and compare data, Launch, students build experience with MP5 as described in the Teacher Note, Promoting the Standards of Mathematical Practice, “When students choose their own tools and decide how they will count the objects in the video, they use appropriate tools strategically (MP5). Ask the following questions to promote MP5: Which tools could you use to help you keep count? Why did you choose that tool? Did it work well?” Teacher directions state, “Gather students and prepare them to watch a video by setting the context. Briefly explain that in this video, two siblings take a long ride in the car. To help pass time, they play a game called I See. One child looks for and counts bridges, and the other child looks for and counts signs. Before playing the video, have students discuss different ways the characters could keep track of their counts. ‘How could the children remember how many bridges or signs they count?’ Invite the class to play the game with the children as they watch the video. Partner students and assign one partner to count bridges and the other partner to count signs. Have students share with their partner how they will remember, or keep track of, their count. Do not provide guidance about the methods of tracking. Have students ready any tools they need, and then play the video. Have partners turn and talk about how many bridges and signs they counted. Then ask them to reflect on the method they used to count. ‘Tell your partner: Would you remember, or keep track of, your count the same way next time? Why?’ (No. I tried counting in my head, but I forgot where I was. Yes. I used my fingers, and it worked well. No. I used lines, but I had to keep looking down so I may have missed one.) Then transition to the next segment. ‘Let’s make a chart to show and compare what the children saw on their road trip.’”

• Module 4, Topic C, Lesson 12: Find the unknown longer length, Learn, Find the Longer Length, students build experience with MP5 as described in the Teacher Note, Promoting the Standards of Mathematical Practice. “Students use appropriate tools strategically (MP5) when they use cubes to measure the given object and then model the object of unknown length.”  Teacher directions state, “Students represent and solve a word problem with an unknown longer length. Have students turn to the butterfly problem in their student books. Read the problem aloud and invite students to turn and talk about it. Reread the first line of the problem and have students represent butterfly A's length by using a 10-centimeter stick. Then reread the second line. ‘How can we use cubes and sticks to show butterfly B's length?’ (We can use a 10-centimeter stick to match the length of butterfly A. Then we can add 4 more centimeters.) Ask students to show the length of butterfly B below butterfly A's length, aligning the endpoints. Then invite students to think–pair–share about butterfly B's length. ‘How long is butterfly B? How do you know?’ (14 centimeters I see 10 and 4; that is 14. 10, 11, 12, 13, 14.) Model drawing the measurements, then ask students to draw and label a picture with the measurements.”

• Module 5, Topic D, Lesson 15: Count on and back by tens to add and subtract, Launch, students build experience with MP5 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “As students decide which strategy to use to solve the sticker problem, they use appropriate tools strategically (MP5). Throughout the year, students have built up their mathematical ‘toolbox’ of strategies, which they can now apply to this new problem type.” In this activity students independently choose and justify tools and strategies to solve a word problem and teacher directions state, “Display the picture of the partially unrolled stickers to build context for students. ‘How many stickers are unrolled so far? How do you know?’ (There are 30 stickers unrolled. You can count the groups by ten: 10, 20, 30.) Display the word problem and read it aloud. ‘Miss Lin had a roll of 90 stickers. She gave 30 stickers to students. How many stickers are left on the roll?’ Have students retell the story to a partner. Ask them to engage with the problem by using the Read–Draw–Write process. Invite students to self-select strategies and tools such as base 10 rods, fingers, personal whiteboards, or number paths. Encourage all students, even those who can solve by using mental math, to justify their solutions with a representation. Circulate and notice the variety of student work. Select two students to share their work in the next segment. ‘Let’s talk about the different ways you solved this problem.’”

The materials reviewed for Eureka Math² Grade 1 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP6 across the year and it is explicitly identified for teachers within materials in margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Grade 1-2 Implementation Guide, “‘Promoting the Standards for Mathematical Practice’ highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP. Often, the suggested questions for a particular MP repeat. This intentional repetition supports students in understanding the MPs in different contexts.”

Students attend to precision in mathematics, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Module 2, Topic E, Lesson 23: Compare categories in a graph to figure out how many more, Land, Debrief, students build experience with MP6 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students attend to precision (MP6) when they generate mathematical questions about the sneakers graph. Look for students who show that they understand which questions can or cannot be answered based on the information they have. Promote this thinking by asking the following questions: Can you use the graph to find out which shoes cost more money? What addition and subtraction strategies can you use to answer questions about this graph?” Teacher directions state, “Display the graph of sneakers. Guide students through a 3-2-1 summary of the lesson. Have them think-pair-share about the following question.‘What are three questions you can ask about this graph? When you and your partner are ready, put up three fingers.’ (What is the graph about? How many more low-top sneakers are there than high-top sneakers? How many total sneakers are there?) Share the title Shoes in Our Class. Then have students think-pair-share about the following question. ‘What are two ways to figure out how many more low-top sneakers there are than high-top sneakers? When you and your partner are ready, put up two fingers.’ (You can see the extras that do not match. You can count how many more squares would make the groups equal.) Prompt students to figure out how many more low-top sneakers there are than high-top sneakers. Invite students to share the solution. Bring the class to the consensus that there are 3 more low-top sneakers than high-top sneakers. ‘Now think of one way to combine the groups of sneakers to find the total.’ Have students give a silent signal when ready. Invite students to share their strategy and solution. Bring the class to the consensus that there are 23 total sneakers. ‘Class, what makes a graph helpful for comparing groups?’ (You can see all the groups lined up. The squares on the number path make it easy to line up groups and see matches.)”

• Module 3, Topic E, Lesson 24: Decompose the subtrahend to count back, Launch, students build experience with MP6 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students attend to precision (MP6) when they rely on their familiarity with the partner of a given number to subtract by counting back using ten. Thinking of partners to determine how to break apart the subtrahend (instead of counting) is a way to calculate accurately and efficiently.” Teacher directions state, “Students solve a subtraction problem by counting back. Display the picture and the first line of the word problem. Read it aloud. Jon has 14 stickers. He gives Max 4 frog stickers. He gives Baz 4 turtle stickers. How many stickers did Jon give away? How many does Jon have now? ‘Look at the picture. How many stickers does Jon have? How do you know?’ (14. He has 10 turtle stickers and 4 frog stickers. 10+14=14.) Fill in the total on the blank line in the sentence. Display the rest of the word problem and read it aloud. Invite students to turn and talk about the story. Prompt students to turn to the 5-group drawing and number path in their student book. Reread the problem one line at a time as necessary. Provide a few minutes for students to solve the problem and represent the story on their 5-group and number path. Show a 5-group drawing and number path. Review and record the solution to the problem in a way that emphasizes subtracting 4 and then another 4. ‘How did you use the dots to show the frog stickers given to Max?’ (I crossed off the last 4 dots. Those were the frog stickers.) ‘Let’s label this set of dots. If you did not label it yet, label yours as I label mine. What should we label these?’ (We can write F for frog stickers.) ‘How many stickers does Jon have now?’ (He has 10.) ‘Did we complete the problem?’ (Not yet. Jon also gives Baz 4 turtle stickers.) ‘What can we draw?’ (We can cross off 4 more dots. We can write T for turtle stickers.) ‘How many stickers did Jon give away? How do you know?’ (He gives away 8. I know because 4+4=8.) ‘How many does he have now? How do you know?’ (Now he has 6. They are left from the ten.) Record the number sentence 14-8=6 as you say the numbers in the context. ‘Jon had 14 stickers. He gave away 8, and now he has 6 stickers. Model representing the problem on the number path as students follow along. Jon had 14 stickers. (Circle 14.) He gave away 4. (Hop back 4 and label it – 4.) Now he has 10 stickers. He gave away 4 more. (Hop back 4 and label it – 4.) He has 6 stickers left. What is the same about our drawing and our number path?’ (They both show 14 as the total. They both take away 4 and 4. That is 8. They both show 6 is how many stickers he has left. We got to ten both times.) Encourage students to take a moment to add to or revise their work as needed.”

• Module 4, Topic B, Lesson 5: Measure and compare lengths, Learn, Greater Than, students build experience with MP6 as described in the Teacher Note. Promoting the Standards for Mathematical Practice, “Students reason abstractly and quantitatively (MP2) when they use numbers rather than direct comparison to compare the lengths of their hands. The quantitative reasoning allows students to attend to precision (MP6) because it requires them to use measurements to compare objects that are too close in length to ‘just know or see’ which is longer. When using measurements to compare lengths instead of comparing objects directly, the comparison can be recorded using the >, =, and < symbols. This allows students to refer to the comparison even if the objects are no longer available to compare.” Teacher directions state, “Be sure students have the Comparison removable inserted into a personal whiteboard. Invite the pair of students whose hands are different lengths to share their measurements. Ask the student with the longer hand, student A, to share first. Then have the student with the shorter hand, student B, share next. Ask student A this question. ‘How long is your hand?’ (14 centimeters) Ask student B this question. ‘How long is your hand?’ (12 centimeters) Have students follow along as you record the lengths of the students’ hands on the first number sentence on the removable. Gesture to the numbers and ask these questions. ‘Which number is greater?’ (14) ‘Which number is less?’ (12) (Point to 14.) ‘Is this student's hand longer or shorter than their partner's hand?’ (Point to 12.) ‘The first hand is longer. ‘How do you know that this student's hand is longer?’ (14 is a bigger number than 12. 14 is greater than 12.) Instead of writing the words greater than, we can draw a symbol to represent them. ‘Draw with me. We make the open part of the symbol next to the larger number. We make the pointy part of the symbol next to the smaller number.’ Guide students to read the greater than number sentence chorally from left to right as they point to the numbers and to the symbol. Do not have students erase their boards. They will write the less than number sentence in the following segment.”

Students attend to the specialized language of mathematics, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Module 1, Topic B, Lesson 8: Count on from a known part and identify both parts in a total, Learn, Record Two Parts and a Total, students build experience with MP6 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “When students start with the next number after the part they count on from and are careful to count on the correct number of objects, they attend to precision (MP6). Ask the following questions to promote MP6: When counting on, what steps do you need to be extra careful with? What mistakes are easy to make when counting on?” Teacher directions state, “Have students follow along in their student books as you introduce the three problems, each using a different pair of dice. Use the following questions to guide students through each problem. ‘As you played, I noticed a few different ways that rolling the dice made the same total. This picture shows one roll. What are the two parts?’ (4 and 2) ‘Let’s count on to check the total.’ Have students chorally count on from 4, recording the count on the dice. Draw attention to how the counting sequence continues, rather than starts again, as you count on. ‘Why did we write 4, and then 5, 6, instead of writing 4, and then 1 and 2?’ (We already know we have 4, so we can keep counting instead of starting over.) Refer to the number bond as you activate prior knowledge about the model. ‘A number bond shows how parts come together to make a total. Let’s fill in this number bond to show the two parts and the total.’ Guide students to complete the number bond. ‘What is the total?’ Point to the total box. (6)  ‘What are the two parts?’ Point to the two part boxes above the arms. (4 and 2) ‘Notice that in this number bond, the parts are on the top and the total is on the bottom.’ Repeat the process with the next two problems. Release responsibility to the students as appropriate.”

• Module 4, Topic A, Lesson 1: Compare and order objects by length, Learn, Order by Length, students build experience with MP6 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “As students order objects by height, they attend to precision (MP6). They do this by using appropriate words like longer, taller, and shorter, and in practice as they align the endpoints of objects before comparing their lengths. Support student-to-student dialogue by inviting peers to agree or disagree, ask a question, give a compliment, make a suggestion, or restate an idea in their own words. The suggested questions in this segment are designed to promote MP6 and help students notice the ways they are attending to precision.” Teacher directions state, “Show a marker and point to each end. ‘Length is how long something is from one end to the other end.’ Set out two objects from the collection that are close in length. Do not align the endpoints. ‘How should I place these items to compare them?’ (They have to be lined up.) ‘Yes, to compare length, the endpoints, or the places where the objects start and end, must be lined up.’ Align the objects on one end and ask students which object is longer. Partner students and give each pair a classroom objects collection to compare. Have each pair of students select and align two objects, then tell which object is longer. ‘Why is it important to line up endpoints to compare two objects?’ (It's like the seesaw. If the things start in different places, then you can’t really tell which one is longer.) Make sure each student has an Ordering Mat. Tell partners to work together to order the five objects in their classroom objects collection from shortest to longest. Have them use the black line on the mat to line up the endpoints. They should not label or complete the comparative statements yet. As students work, look for seriation (ordering) strategies, such as: Placing the shortest and longest objects first, then working toward the middle. Comparing two objects, then combining pairs. Approximating, then moving one object at a time based on direct comparison of pairs of objects. Ask some of the following assessing questions: ‘How did you order your objects? How can you check to make sure the order is correct? What can you do if you see an object is out of order? Which object is shortest? Which is longest? How do you know? Why is it important to line up the endpoints?’ When students finish, show them an accurate work sample. Use the suggested assessing questions to guide discussion about how students ordered the objects. Ask students to revise their work if needed. Guide students to label each ordered item with a letter.”

• Module 6, Topic D, Lesson 16: Count and record totals for collections greater than 100, Launch, students build experience with MP6 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students attend to precision (MP6) when they write and say three-digit numbers. Students formally extend their understanding of place value to include the unit of hundreds in grade 2. However, they can still use place value concepts to precisely describe how to read and write three-digit numbers. For example, they can see that writing one hundred three as 130 or 1003 does not make sense because those numbers show 13 and 100 tens, respectively. Neither 13 tens nor 100 tens matches the collection. In contrast, writing the number as 103 matches the collection because it shows 10 tens.” Teacher directions state, “Display the quilt. Invite students to notice and wonder. Students may notice the quilt is composed of cubes with square faces. Explain that a 17-year-old named Adeline Harris collected signatures of famous people and used them to make this quilt. ‘How might you count this collection of cubes efficiently?’ (We can group them by tens and then count by tens.) ‘Let’s look at some ways to count larger collections.’ Display the four collections. Invite students to think–pair–share about how the different collections are grouped. (Three of them are in groups of tens and extra ones. The first collection is different because it is not grouped at all.) ‘Would you rather count a collection that is grouped or ungrouped? Why?’ (It’s better to count a collection that is grouped. Then you can count by tens and ones. That’s faster, and you don’t make so many mistakes.) Focus student attention on the pencil collection. Show the recording sheet. Write the title of the collection on the recording sheet and demonstrate estimating how many pencils are in the collection. Display the collection. ‘They grouped 10 pencils in each cup. Let’s count by tens and then by ones to find the total.’ Gesture to the cups and then to the pencils, respectively. (10, … , 100, 101, 102, 103) Show the recording sheet and write the total. Help students read the total aloud as one hundred three. Invite students to share how they could draw to represent the collection. Demonstrate making a math drawing to record the collection. ‘How many groups of ten did we count and draw?’ (10 tens) ‘What is 10 tens?’ (100) If students are unsure, then have them count by tens the math way to 100. Show the Hide Zero card for 100. ‘How many ones did we count and draw?’ (3 ones) Show the Hide Zero card for 3 next to the card for 100. ‘100 plus 3 equals 103. (Push the cards together.) ‘What happened when we pushed the cards together?’ (A zero got covered up with a 3. Now, the 100 looks like 10.) ‘The 0 in the ones place is hidden by the 3 ones. But we can see that there are 10 tens, or 100, and 3 ones.’ Slide apart the Hide Zero cards to show 100 and 3. ‘When we write 103, we write 10 tens and 3 ones. 10 tens is the same as 100. 3 more is 103.’ Begin an anchor chart that shows numbers written in standard, expanded, and unit form. Use 103 to interactively complete the first row of the chart.”

The materials reviewed for Eureka Math² Grade 1 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP7 and MP8 across the year and they are explicitly identified for teachers within materials in margin notes called, “Promoting the Standards for Mathematical Practice." According to the Grade 1-2 Implementation Guide, “‘Promoting the Standards for Mathematical Practice’ highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP. Often, the suggested questions for a particular MP repeat. This intentional repetition supports students in understanding the MPs in different contexts.”

MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and make use of structure as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 2, Topic B, Lesson 7: Count on or count back to solve related addition and subtraction problems, Learn, Create Penny Problems, students build experience with MP7 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students look for and make use of structure (MP7) when they notice that they start and end with the same number of pennies and when they use that idea to help them solve the problem. In particular, students should see that the equations are related because they include the same numbers but in a different order and with different operations. Ask the following questions to promote MP7: Why is it helpful to notice that you find and lose the same number of pennies? How do the equations show that you found and lost the same number of pennies?” Teacher directions state, “Make sure each student has a Penny Problems removable inside a personal whiteboard. Display Penny Problems and point to the first problem. One at a time, read the sentence frames aloud. Pause after each frame and invite students to choose and write in a number from 0 to 10. Then read the question aloud. Ask students to self-select tools and strategies, solve the problem, record their thinking, and share with a partner. Have students write a number sentence to represent their work. Invite partners to validate the accuracy of each other's work. Point to the start of the second problem. ‘You had ___ pennies.’ Guide students to write in the total from the first problem. ‘You lost ___ pennies.’ Guide students to write in the number of pennies that they found in the first problem. ‘How many pennies do you have now?’ Again, have students solve, record their thinking, share with a partner, write a number sentence to represent their work, and confirm their partner's accuracy. Some students may recognize that they can use the first problem to solve the second problem. Have partners work together to represent each student's pair of problems by counting up and back on their number paths. Then have students use the red side of their whiteboard to draw a single number bond that represents both of their problems. Students may use their number paths and their recordings as support. Share and discuss students’ work. The following sample dialogue provides a possible discussion. ‘Let's look at this work. What do you all notice?’ (They started with 7 pennies and ended with 7 pennies. They added 3 pennies and then took away 3 pennies. The same numbers are in both problems.) ‘They started with 7 pennies. How did they get back to 7 pennies?’ (They added 3 and then subtracted 3, so he got back to 7. They just took away the same number he added.) ‘Would they have gotten back to 7 if he lost 2 pennies? Why?’ (No, because he didn't find and then lose the same number of pennies. No, because 10-2=8, not 7.) ‘How does this number bond represent your addition word problem?’ (We added the parts 3 and 7 to get the total, 10.) ‘How does this number bond represent the subtraction word problem?’ (We started with the total, 10. We took away a part, 3, and we have a part left, 7.) ‘Why is there one number bond for these problems?’ (They have the same parts and total.)”

• Module 3, Topic A, Lesson 1: Group to make ten when there are three parts, Learn, Partners to Ten, students build experience with MP7 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students look for and make use of structure (MP7) when they look for ways to make ten and use structure to solve problems more efficiently.” Teacher directions state, “Play a game to review partners to 10. Give the following directions for the game: Students stand and face a partner. Partner A puts up some fingers, showing any number other than 0 or 10. Partner B looks at partner A’s fingers and thinks about partners to 10. Partner B shows the part that makes ten. Partners give each other a high ten because they made ten. Then they switch roles. Have pairs play for about a minute. Then tell students to ready their whiteboards for the Whiteboard Exchange routine. Display the picture of shells. ‘What three addends do you see?’ (7, 9, 1) ‘Which two addends are partners to 10?’ (9 and 1) Write the number sentence with three addends. Show how you made ten. Use the Whiteboard Exchange routine to provide immediate feedback: Tell students to turn their whiteboards over so the red side is up when they are ready. Say, ‘Red when ready!’ When most are ready, tell students to hold up their whiteboards to show you their work. Give quick individual feedback, such as ‘Yes!’ or ‘Check your total.’ For each correction, return to validate the corrected work. Choose a student or two to share their work, highlighting how they made ten by grouping two addends. Students may or may not use number bonds to show how they made ten. Some may circle the addends that make ten instead.”

• Module 5, Topic D, Lesson 17: Use tens to find an unknown part, Learn, Find an Unknown Addend, students build experience with MP7 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students look for and make use of structure (MP7) when they think of a problem in unit form (e.g., 3 tens+__=8 tens) and use a basic fact (e.g., 4+__=8) to find the unknown part, because they notice the similarity in structure between the two number sentences.” In this activity students connect counting on by tens to using a related fact to solve. Teacher directions state, “On the Add or Subtract Tens removable, write 40+__=80 at the top of the addition portion. Have students follow along. Tell students to draw quick tens to represent 40. ‘Let’s draw more tens to count on to 80.’ Draw tens one at a time as students follow along. Have them chorally count by tens from 40 to 80. ‘Circle the tens that we added. How many tens did we count on?’ (4 tens) ‘How many is 4 tens?’ (40) Guide students to record each known number of tens (4 and 8) in the correct spaces, leaving the unknown number of tens space blank. Point to the blank space. ‘What is the unknown number of tens? How do you know?’ (4. We counted on 4 tens.) Have students write 4 in the blank. Point to the blank space for the unknown in the original equation. ‘What is this unknown part? How do you know?’ (40. 4 tens is the same as 40.) Ask students to write 40 in the blank in the original equation. ‘Turn and talk. Which way is more helpful to you: finding an unknown part by counting on by tens, or thinking about the number of tens as an easier, related fact?’”

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and express regularity in repeated reasoning as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 1, Topic D, Lesson 21: Find all two-part expressions equal to 7 and 8, Launch, students build experience with MP8 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “When students predict how many two-addend expressions there will be with totals of 7 and 8, they look for and express regularity in repeated reasoning (MP8). Students make use of their previous work with finding all the expressions with totals of 0–5 to make this prediction. They notice the repeated pattern that each total has 1 more expression than the previous total and express regularity to predict that the pattern will continue.” Teacher directions state, “Gather students and display the Addition Totals chart. Point to the 0 column and activate prior knowledge by asking the following question. ‘How many expressions are there for 0?’ (1 expression) Point to totals 1–6 and repeat the same question for each total. (2, 3, 4, 5, 6, 7 (expressions)) ‘Think about the number of expressions for each total. What patterns do you notice?’ (Every time the total is 1 more, there is 1 more expression. There is 1 more expression than the total.) ‘How many expressions do you think we can make for 7? Why?’ (8 expressions. There are 7 expressions for 6, so there are probably 8 expressions for 7. 8 expressions. The total is 7, and there is usually 1 more expression than the total.) ‘How many expressions do you think we can make for 8? Why?’ Record students’ predictions to validate later in the lesson. Transition to the next segment by framing the work.”

• Module 5, Topic C, Lesson 11: Add the ones to make the next ten, Learn, How Many To Make Ten?, students build experience with MP8 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students look for and express regularity in repeated reasoning (MP8) when they find the unknown addend in a sequence of problems where the first addends all have the same digit in the ones place. Students come to understand that they can use partners to 10 to figure out which addend is needed to get to the next ten. This is the first step toward extending the make ten strategy to larger numbers.” In this activity students work in pairs using a Number Path to 120. Teacher directions state, “Once the number paths are ready, ask students to turn to the string of related problems in their student book. Direct their attention to the first problem. Consider guiding students with the digital interactive number path. ‘What is 7’s partner to 10?’ (3) Have students write the unknown addend in their books. Guide them to show 7+3=10 by using their fingers to hop on the number path. Direct them to the next problem. ‘Find 17 on the number path. What is the next ten?’ (20) ‘Hop to 20. How many times did you hop?’ (3) Have students write the unknown addend in their books. Use the same procedure for 27+3=30 and 37+3=40 Have students complete both the total (the next ten) and the unknown addend. Invite students to complete the last two problems on their own (47+3=50 and 57+3=60). Some students may continue to use the number path while others make use of the pattern. Display the list of equations with the 7s highlighted. ‘What do you notice?’ (I can count the totals by tens. We added 3 every time. The first addend always has 7 in the ones place.) ‘Why did we add 3 to make the next ten every time?’ (7 and 3 are partners to 10. They make ten.) Invite students to think–pair–share about what the next number sentence on the list would be. ‘What would be the next number sentence on our list?’ (67+3=70) Display the list of three equations with the tens place highlighted. ‘What happens to the number of tens when we make the next ten?’ (There is 1 more ten.) ‘Why does the number of tens grow by 1?’ (It happens because we made another ten with the ones.) ‘We used the ones from both addends to compose a new ten.’”

• Module 6, Topic C, Lesson 13: Relate the number of equal shares to the size of the shares, Learn, Equal Shares, students build experience with MP8 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students use the story of Azeez and his pizza to look for and express regularity in repeated reasoning (MP8). Through repeated experience, they notice and explain that the more people Azeez shares the pizza with, the smaller his share gets. This lesson shows the converse of the idea presented in lesson 9. In that lesson, students saw that the smaller the shape, the more of that shape it takes to compose a larger shape. Here students notice that the more pieces a shape is partitioned into, the smaller the pieces become.” Teacher directions state, “Display the picture of Azeez’s family. ‘Azeez’s children smell the pizza and come running into the kitchen. Partition the pizza to show how Azeez could share the pizza equally with his wife and children. After students finish partitioning the pizza, display the picture of the pizza cut into quarters. ‘This is how Azeez thinks he can share the pizza now. What is his idea?’ (He could cut the pizza into fourths to get 4 equal pieces.) Ask students to show thumbs-up if they partitioned their pizza into fourths. Invite students to revise their work if needed. ‘If Azeez cuts the pizza into fourths, his share would look like what is outlined in blue. What would his share of the whole pizza be?’ (1 fourth or 1 quarter) ‘If he shares the pizza with his wife, his share is 1 half of the pizza. If he partitions the pizza into fourths to share it with his children too, will his share get bigger or smaller? Why?’ (It would get smaller because there are more people sharing the pizza.) Display the picture of Azeez’s extended family. ‘Four more people come over. Now there are eight people sharing the pizza. If eight people share the pizza, what will happen to Azeez’s share? Why?’ (It will get smaller because there are a lot more people getting a share of the pizza.) Guide students to partition their pizza into 8 equal pieces. Then display the final pizza. ‘If Azeez cuts the pizza into 8 pieces, his share would look like what is outlined in blue. What would happen if he shares the pizza with more people?’ (His share will get smaller.) Display the three pizzas partitioned differently. ‘Which pizza has the largest parts? Why?’ (Pizza A, with 2 pieces, has the largest parts. It is cut in half for only two people.) ‘Which pizza has the smallest parts? Why?’ (Pizza C, with 8 pieces, has the smallest parts. It is cut to share with a lot of people.) ‘Which pizza would you choose? Why?’ (I want the pizza with 2 pieces so I can have a really big piece. I want the pizza with 8 slices because I like to share with my friends. I want the pizza with 4 slices because I have four people in my family.) ‘What happens when we partition a shape into more and more parts?’ (The pieces get smaller and smaller.)”