Kindergarten - Gateway 2

The instructional materials for Achievement First Mathematics Kindergarten meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Materials include problems and questions that develop conceptual understanding throughout the grade-level. Examples include: 

• Unit 2, Lesson 5, Introduction, students engage with K.G.6, compose simple shapes to form larger shapes, as they complete puzzles using geometric shapes. “Step 1 says I’m going to pick a puzzle. Step 2 says I need to Decide what shape might fit. T & T: How can I make sure that happens? Strategy 1: Keep trying shapes until one fits the space. Strategy 2: Look at the space you are trying to fill. What shape might fit because of its attributes? Then find the shape that has the same attributes. Remember! You can flip and turn the pattern blocks. What shape do you think would fit? Why did you pick that shape?”

• Unit 3, Lesson 2, Introduction and Workshop, students engage with K.CC.4, demonstrate understanding of the relationship between numbers and quantities, as they play “Counting Bags/Jars.” Students count the number of pattern blocks in the bag and then show the same amount using cubes. During the Workshop the teacher asks students, “How do you know this is the same amount / how are you showing the same amount?”

• Unit 5, Lesson 6, Introduction, students engage with K.CC.6, identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, as they play a game called Compare. During the Introduction, two cards are drawn (example, 7 and 9) and students are asked to pictorially show which is more or less by drawing circles on their whiteboards. The teacher asks, “How do you know from the picture?” A sample student response might be, “I know because in the picture you can see that there are extra circles in the row of 9 and the row of 7 is missing some.” 

• Unit 6, Lesson 3, Introduction, students engage with K.OA.1, represent addition and subtraction with objects, fingers, mental images, drawings, sound, acting out situations, verbal or equations, as the students complete a dice game while the teacher checks for understanding with questions leading the students to describe their thinking. Workshop, “Step 1: Roll 2 cubes; record their amounts. (“roll” a 4 and a 2) Show first cube: How many? 4 (record) Show second cube: How many? 4 (reccord) Show second cube: How many? (give time to count as needed) 2 (record) Lap 2: Conceptual: Which strategies are kids using? What misconceptions are arising? Check for Understanding: How did you solve? Why does that work? How does your equation match what you did?”

• Unit 8, Lesson 2, Workshop, students engage in K.NBT.1, compose and decompose numbers 11 to 19 into ten ones and some further ones, as students bundle objects into a group of ten and count on to determine the number of objects in a bag. The teacher is given suggestions for guiding the students to develop the concept of teen numbers. “What did you notice about the group of ten ones, loose ones and the way we write the numbers?”

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include: 

• Unit 6, Lesson 10, Exit Slip, students engage with K.OA.1, represent addition and subtraction with objects, fingers, mental images, drawings, sounds, acting out situation, verbal explanations, expression, or equations, as they find the difference between two numbers using manipulatives and represent with an equation. “Use your counters and tens frames to find the difference. Fill in the equation to show what you did.” Students are provided with the digits, 8 and 5, and given a blank equation to fill in. 

• Unit 7, Lesson 6, Exit Slip, students engage with K.OA.3, decompose numbers less than or equal to 10 into pairs in more than one way, as they look at a picture of a rekenrek and find another equation that is equal/the same. Problem 1, “Look at the picture and equation in box 1. Use your teddies to write another equation that is equal/the same.” In box 1, there is a picture of a rekenrek with 4 on the top and 1 on the bottom and the matching equation, 4 + 1 = 5. 

• In Unit 8, Practice Workbook G, students engage with K.NBT.1, compose and decompose numbers from 11 to 19 into ten ones and some further ones by using objects or drawings, as they independently draw pictures to show the decomposition of the number 18 into ten ones and 8 more ones. Problem 6, “Draw a picture to show 18 as ten ones and some more ones. Write a number sentence to match.”

• In Unit 9, Practice Workbook F, students engage with K.OA.4, by finding the number that makes 10 for any number 1 to 9 by using objects or drawings and recording the answer. Problem 3, “Draw circles and write a number to show how many more are needed to make 10.” Students are given 2, 4, 7, 6, 3, 1, 8, 9, and 5.

The materials for Achievement First Mathematics Kindergarten meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. These skills are delivered throughout the materials in the use of games, workshop, practice workbook pages and independent practice, such as exit tickets. 

The materials develop procedural skill and fluency throughout the grade-level. Examples include but are not limited to: 

• Unit 3, Lesson 12, Workshop Worksheet, students engage with K.CC.5, count to answer “How many?” questions about 20 things arranged in a line, a rectangular array, or a circle, as they count 18 pieces of silverware arranged in a rectangular array. “Mr. Lohela was having a dinner party. He set out the silverware. How many pieces of silverware did he set out?”

• Unit 6, Lesson 19, Introduction, students engage with K.OA.5, fluently add and subtract within 5, as they represent a story problem. “Step 1: Visualize. Make a mind movie while I read. There were 7 carrot sticks on Hubina’s plate. She ate 3 of them. How many carrot sticks are on her plate? Step 2: Represent and Retell. Now you need to show the story. You can use your cubes or your whiteboard and marker; it’s up to you. Remember to include what we know and what we need to figure out. When you are done, put your whiteboard and cubes flat and be ready to explain how you represented and how it matches the story.”

• Unit 7, Practice Workbook E, Making 3, 4, and 5 finger Combinations, students engage with K.OA.5, fluently add and subtract within 5, as they play a game to develop fluency within 5. “The teacher uses different finger flashes and students determine how many fingers are needed to make a target sum.” Once students understand the game, they play with a partner. 

• Unit 8, Practice Workbook E, students engage with K.OA.5, fluently adding and subtracting within 5, as they use fingers to calculate the missing addend. “Activity: Making 3, 4, and 5 Finger Combinations. T: I’ll show you some fingers. I want to make 3. Show me what is needed to make 3. (Show 2 fingers.) S: (Show 1 finger.)”

The materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include but are not limited to: 

• Unit 3, Practice Workbook C, students engage with K.CC.3, write numbers from 2 to 20, as they independently complete a number sequence filling in missing numbers from 10 -15. Problem 5, “Fill in the missing numbers, “10, 11, ____, ____, ____, ____.”

• Unit 6, Lesson 22, Assessment, students engage with K.OA.5, fluently adding and subtracting within 5, as they complete equations. Problem 5, “Solve. 2 + 3 = ___.”

• Unit 7, Lesson 8, Exit Slip, students engage with K.OA.3, decompose numbers less than or equal to 10 into pairs in more than one way, as they independently create equations with a sum of 10. “Show all of the ways you could make 10. (You may not need to fill in every equation.)” Blank equations equalling 10 follow the directions. 

• Unit 8, Practice Workbook E, students engage with K.OA.5, fluently add and subtract within 5, as they independently solve a series of addition problems with a sum of 2-5. Problem 1, “$$3 + 2 =$$____”

The materials reviewed for Achievement First Mathematics Kindergarten meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Students are given multiple opportunities to engage in real-world applications, especially during Math Stories, which include both guided questioning and independent work time, and Exit Tickets to independently show their understanding.

Materials include multiple routine and non-routine applications of the mathematics throughout the grade level. Examples include:

• Unit 2, Guide to Implementing AF Math, Math Stories, November, students engage with K.OA.2, solve addition and subtraction word problems, as they solve routine put-together/take apart-total unknown word problems. Sample Problem 1, “5 red crayons and 5 green crayons were in the basket. How many crayons were in the basket?” 

• Unit 3, Guide to Implementing AF Math, Math Stories, January, students engage with K.G.5, drawing shapes, in a non-routine problem. Sample Problem 13, “Ms. Chen draws a square and a triangle on the board. How many sides did she draw?”

• Unit 3, Lesson 25, Understand: Introduce the Problem, students engage with K.CC.5, count to tell how many, in a non-routine word problem, “4 friends line up in a row. Every friend is wearing sneakers. How many sneakers are there lined up in a row? Show and tell how you know.”

• Unit 7, Lesson 9, Understand: Introduce the Problem, students engage with K.OA.3, decompose numbers less than or equal to 10 into pairs in more than one way, by following the story problem protocol and using an efficient strategy to find all of the solutions to a routine word problem. “The grocer got another size box! He now has a box that holds exactly 9 apples. He has red and green apples that he needs to put into the box. What are all of the ways he could put red and green apples into his box?” 

Materials provide opportunities for students to independently demonstrate routine and non-routine applications of the mathematics throughout the grade level. Examples include: 

• Unit 3, Lesson 6, Exit Ticket, students engage with K.CC.2, count forward beginning from a given number within the known sequence, as they use a routine counting on strategy to add numbers on two dice. Problem 1, “Use a strategy to find the total. Write the total on the line.” Students are shown dice with six and three dots, respectively.

• Unit 7, Lesson 5, Exit Ticket, students engage with K.OA.2, solve non-routine addition and subtraction word problems within 10, as students calculate take apart problems with both addends unknown. “There are 8 kids on the bunk bed. Show as many ways they can be arranged on the top and bottom as you can.” Nine blank equations are provided for students, “___ + ___ = 8.”

• Unit 8, Lesson 6, Exit Ticket, students engage with K.NBT.1, compose and decompose numbers 11-19, in a routine story problem. “Marco picked a card that looks like this (image of two ten frames-1 with 10 dots and 1 with 6 dots). How many? Write the number on the line.”

The materials reviewed for Achievement First Mathematics Kindergarten 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 program materials. Examples include:

Conceptual understanding

• Unit 3, Lesson 16, Exit Slip, students engage with K.CC.5, count to answer “how many” questions about as many as 20 things, as they represent a quantity 10 -20 pictorially by using a strategy to keep track of the count. The Exit Slip shows the number 16 with two blank ten frames. Students are expected to draw circles on the ten frame to represent 16.

• Unit 6, Lesson 11, Exit Ticket, students engage with K.OA.1, represent addition and subtraction with objects, fingers, mental images, drawings, sound, acting out situations, verbal or equations, as they represent and solve subtraction problems while using counters and tens frames. “Use your counters and tens frames to solve. 7 - 3 =____ and 9 - 5 = _____” 

• Unit 9, Practice Workbook F, students engage with K.OA.3, decomposing numbers less than or equal to 10 into pairs in more than one way, as they draw pictures to show more than one way to make each number. Problem 4, “Draw a picture to show 2 ways to make each number. 6; 4; 7; 6; 3; 1; 8; 9; 5.”

Procedural skills (K-8) and fluency (K-6)

• Unit 6, Lesson 5, Exit Slip, students engage with K.OA.5, fluently add and subtract within 5, as they are given an image with two numbers to add them together, and a spot for an equation. “Cube 1 (6) Cube 2 (3) Equation _____ + _____= _____.” 

• Unit 7, Practice Workbook E, Shake and Spill, students engage with K.OA.5, fluently add and subtract within 5, as they spill five two-sided counters in a cup to find combinations of 5. “The students determine how many of each color is showing and record the sum using drawings or equations. The students should ‘shake and spill’ several times to show different pairs of numbers that sum to 5.” 

Unit 8, Practice Workbook E, students engage with K.OA.5, fluently adding and subtracting within 5, as they solve put together and take apart problems with the result unknown within 5. Problem 8, “$$2 + 2 =$$ ___.”

The materials reviewed for Achievement First Mathematics Kindergarten 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. The Standards for Mathematical Practice are identified and incorporated within mathematics content throughout the grade level. The Mathematical Practices are listed in the Unit Overviews as well at the beginning of each lesson. 

There is intentional development of MP1 to meet its full intent in connection to grade-level content. Examples include:

• Unit 3, Lesson 25, Pose the Problem, students make sense of a real world situation. “4 friends line up in a row. Every friend is wearing sneakers. How many sneakers are there lined up in a row? Show and tell how you know.”

• Unit 5, Lesson 9, Understand: Introduce the Problem, students make sense of a comparison in quantity. “I’m going to read you a problem. As I read, I want you to make a mind movie just like we do in Math Stories to visualize what is happening and what we need to figure out. Toy Trucks: Hector has 6 toy trucks. Eric has 5 toy trucks. Hector says he has a greater amount of toy trucks than Eric. Is Hector correct? Show and tell how you know.”

• Unit 8, Lesson 8, Assessment and Criteria for Success delineates expectations that students make sense of a problem. “Students should represent the story by either drawing a picture or writing numbers to represent the beads. They can compare the quantities of beads using concrete/pictorial methods (one-to-one matching) or more abstract methods (number line, counting sequence) or they should be able to explain their representation and solution and connect it back to the problem. For example, ‘I wrote 10+5 because Tanya has ten beads and 5 beads. Then I wrote 10+3 because Marie has 10 beads and 3 beads. I know that 10+5 is 15 and 10 + 3 is 13. I just know that 15 is bigger than 13. So YES, Tanya is correct.’”

• Unit 9, Lesson 1, Narrative, “Students engage with MP1 as they work to interpret the problem, plan solution pathways, and monitor progress. Teachers give students ample time to initiate and execute a plan before intervening; when/ if they do intervene, they ask probing questions that support students in problem solving broadly.”

There is intentional development of MP2 to meet its full intent in connection to grade-level content. Examples include:

• Unit 6, Lesson 1, Narrative, “Teachers help students engage with MP2 when they ask students to reason about the relationships between the quantities as they add two parts to make a whole. Students recognize that the addition symbol means put together and use that to model that when they put two quantities that are parts together they make a quantity that is the sum of those parts, the whole.”

• Unit 7, Lesson 6, Narrative, “Students reason abstractly and quantitatively (MP 2) when they describe the relationship between ways to decompose the same total as equivalent: ‘I can decompose 8 into 5 + 3 and 4 + 4, so 5 + 3 is the same as 4 + 4 because both are ways to make 8.’”

• Unit 8, Lesson 3, Narrative, “By having students draw a picture to show the value of quantity along with writing an equation, teachers help them transition from quantitative to more abstract reasoning.”

• Unit 8, Lesson 13, Narrative, “Students engage with MP2 today as they reason about the value of the digits of two-digit numbers; by requiring students to represent the tens and ones with concrete objects and/or pictures, teachers help students to understand the meaning of quantities and to shift from quantitative to abstract reasoning about numbers and their values.”

The materials reviewed for Achievement First Mathematics Kindergarten 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.

There is intentional development of MP3 to meet its full intent in connection to grade-level content. Examples include:

• Unit 3, Lesson 18, Introduction, teachers are provided guidance in helping students to construct viable arguments demonstrating how to find the total of three numbers rolled on dot cubes. “Step 2 is for us to find out the total. When we did this before, we only had to figure out the total for two dot cubes. How would we do it for three dot cubes? SMS (student might say): It’s the same! It’s just more dots, so I can count them all. If a student says this, have a student come up and demonstrate touching each dot and counting and a student showing the amounts on fingers and counting all. SMS (student might say): I can just see (subitize) and say the number on one dot and count on from there. Have a student demonstrate.”

• Unit 4, Lesson 4, Share/Discussion, during Workshop, students are picking two objects and determining which is heavier or lighter using either a balance or hefting. “Facilitate a discussion around a major misconception (i.e. an object that is longer/taller doesn’t always have to be heavier). Show non-example and related example: Which is correct? Why doesn’t ___’s work? OR, 2-3 students share their work/strategies: How did ___ compare their objects? How did ___ compare their objects? What is the same about these strategies? What is different? Why do both work?”

• Unit 5, Lesson 3, Mid-Workshop Interruption, students determine which number is more and which is less by building towers or matching one to one. “If >$$\frac{2}{3}$$ of students are successful, ask students to describe the relationship between 2 towers (green 8 and blue 3) in a turn and talk. Hunt for a student who says one tower is more and another who says the other tower is less. Share their answers and ask who is right; students should see that both students are right- the green tower is more and the blue tower is less. Discuss how this is true; students should articulate that they are opposites and that if one tower is more the other will always be less and vice versa. Challenge students to circle the amount that is more as well one the recording sheets moving forward. If >\frac{2}{3} of students are successful, call students back together to clarify expectations through a misconception protocol or role play.”

• Unit 7, Lesson 1, Introduction, students name and record (with equations) various ways to decompose the totals four and five. During a demonstration, the teacher tosses three red chips and one yellow chip. “Step 2: What do you see? The purpose of this question is to get students to generate the numbers they will use in their number sentences; feel free to use the questions below to help: If students say, ‘I see 4 counters/chips,’ ask, ‘What colors do you see?’ If students say, ‘I see red and yellow,’ ask, ‘How many red do you see? How many yellow?’ If students say, ‘I see red and yellow,’ but don’t notice the total, ask, ‘How many do you see altogether/How many does that make altogether?’”

• Unit 7, Lesson 11, Introduction, Play Again and Check for Understanding, teachers are instructed to pose a fictional problem for the students to analyze. “Rather than playing a full game, pose this problem: Mr. Lynch was playing the game, and he drew a 3, so he recorded like this: 3 + ____ = 10 (show). Then when he went to show that many on his tens frame, he realized that he didn’t have any! They had all been cut in half for an art project. So he used just the top half, like this (Show the top row of the tens frame with 3 counters on it.). Then, he said, ‘How many to ten?’ and he counted the empty squares. 2! He wrote 3 + 2 = 10. Does this work? Why not? SMS (student might say) That is how many to 5, not ten. The tens frame works because it has 10 squares in all, so if we show how many we have we can count the empty squares to figure out how many to ten. There are not 10 squares in all.”

The materials reviewed for Achievement First Mathematics Kindergarten meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

There is intentional development of MP4 to meet its full intent in connection to grade-level content. Examples include:

• Math Stories Guide, Promoting Reasoning through the Standards for Mathematical Practice, MP4, “Math Stories help elementary students develop the tools that will be essential to modeling with mathematics. In early elementary, students become familiar with how representations like equations, manipulatives, and drawings can represent real-life situations.” Within the K-4 Math Stories Representations and Solutions Agenda, students are given time to represent, retell, and solve the problem on their own. 

• Unit 6, Lesson 7, Introduction, Step 1, “There were 3 horses in the field. 4 more horses came out of the barn and into the field. How many horses are in the field now?” Step 2, “Now you need to show the story. You can use your cubes or your whiteboard and marker; it’s up to you. Remember to include what we know and what we need to figure out. When you are done, put your whiteboard and cubes flat and be ready to explain how you represented and how it matches the story.”

• Unit 9, Lesson 4, Task, “Beads in a Bowl: There are some beads in a bowl. The first friend takes 3 beads out of the bowl. The next friend takes 3 beads out of the bowl. The last friend takes 4 beads out of the bowl. There are no more beads in the bowl. How many beads were in the bowl? Show and tell how you know?” Narrative, “Students engage with MP4 as they use mathematical models to decontextualize the story. Students may represent pictorially or with equations, number bonds, or tape diagrams. Teachers also support the development of MP4 when they help students connect the models in the debrief and ask students which model is most useful for solving and why.”

There is intentional development of MP5 to meet its full intent in connection to grade-level content. Examples include:

• Unit 1, Lesson 15, Narrative, “Students engage with MP5 today as they are strategic about when they subitize and when they count to find the total. Teachers encourage students to be strategic and to consider the limitations of a strategy when he/she asks students why they are able to subitize a small amount but not a larger amount. In the workshop, teachers circulate and ask students which strategy they are using and why/why they are not using another strategy.”

• Unit 6, Lesson 12, Narrative, “Students engage with MP5 today when they choose from several strategies to represent and solve subtraction problems. Students can choose from any of the many strategies listed in the key points and have access to manipulatives, ten frames, whiteboards, and markers to use as tools. While there are target strategies for teachers to highlight, the class discusses multiple strategies and no strategy is preferred. The class relates different strategies to one another, noting that they ALL work for subtraction. By ensuring that students are choosing strategies themselves without teachers encouraging the use of a ‘preferred’ strategy, teachers help students to choose their own strategies and tools and explore their benefits and limitations.” 

• Unit 7, Lesson 11, Narrative, “Teachers help students to develop MP5 today by making a variety of tools available to students. They can use counters or other manipulatives, a tens frame, or fingers as tools. They can also draw pictures or use known facts or counting strategies to solve.” Exit Ticket, “How Many to Ten? Use your counters/cubes and tens frames to help you. (1. 7 + __ = 10; 2. 4 + __ = 10).” 

At times, the materials are inconsistent. The Unit and Lesson Overview narratives describe explicit connections between the MPs and content, but lessons do not always align to the stated purpose. 

The materials do not provide students with opportunities or guidance to identify and use relevant external mathematical tools and resources, such as digital content located on a website.

The materials reviewed for Achievement First Mathematics Kindergarten 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. 

There is intentional development of MP6 to meet its full intent in connection to grade-level content. Many problems present students with the opportunity to attend to precision within the mathematics and the reasoning of the answer. Examples include: 

• Unit 4, Lesson 7, Introduction, students are prompted to use sentence frames for precision to compare the capacity of two containers. “Which one held more rice? The _______ held more rice than the _______. I know because it held _____ scoops of rice and the _____ held ______ scoops of rice. Be sure to prompt for accurate comparative language.”

• Unit 1, Lesson 12, Introduction, Narrative, students play a game with a partner where they have to count cubes that have been placed on an image of a donut. “Students engage with MP6 as they use a strategy that ensure that they count each object exactly once, which is particularly challenging today as the objects are arranged in a circle. By helping students keep track of the count and by emphasizing the importance of doing so, teachers help students to attend to precision and to value that practice.”

• Unit 5, Lesson 12, Assessment and Criteria for Success, students use precision to count to 100. “Students should put the Fruit Loops down on their hundreds chart and count orally to 100. Students should put each Fruit Loop onto a string (that is taped to the desk) and count orally as they put each Fruit Loop on the string. Once students finish their necklace, they should double check and count each Fruit Loop to make sure they have exactly 100 Fruit Loops on their necklace.” 

The instructional materials attend to the specialized language of mathematics. The materials use precise and accurate mathematical terminology. Examples include:

• Unit 2, Lesson 9, Narrative, “Students engage with MP6 throughout the lesson as they use precise vocabulary to describe 2d and 3d shapes and their attributes. During the Introduction, students hunt around the room for 2D and 3D shapes. 'T&T: How could I know when I’ve found a shape?’ Students might say, ‘Look for the shapes that have the square/circle faces that you are looking for! You can also look for pointy vertices and/or the rounded or straight edges.’”

• Unit 4, Lesson 1, Introduction, Introduce the Math, “Today we’ll figure out how long (kinesthetic: make arms wide horizontally) or tall (kinesthetic: make arms wide vertically) things are by comparing two objects. That’s called length...When we are talking about how long or how tall things are, they can be LONGER (longer-choral response and motion: start with hands together and move apart) or SHORTER (shorter-choral response and motion: start with hands apart and move together).”

• Unit 5, Lesson 1, Assessment and Criteria for Success, students use the terms more, greater, the same, and equal to describe sets of objects. Questions are provided for teachers to support students in the use of these terms. “Teachers should circulate during workshop to gather data on student mastery. All students should be able to use the words, ‘more,’ ’greater,’ ‘the same,’ and ‘equal’ to describe their sets. Teachers should ask: 1. Which is more/has a greater amount of cubes? How do you know? 2. How can you describe this tower? (pointing to a tower that is more). 3. How can you describe these towers? (showing two towers that are the same).”

The materials reviewed for Achievement First Mathematics Kindergarten 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 grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP7 to meet its full intent in connection to grade-level content. Examples Include:

• Unit 2, Lesson 5, Narrative, “Students engage with MP 7 when they look at a composite shape and decompose it into smaller shapes using the structure of the shapes to help them.” 

• Unit 5, Lesson 10, Introduction, students engage with MP7 as they fill in a hundreds chart. “Fill in your hundreds chart. You will each get a blank chart like this one. You will count to yourself in a whisper voice and fill in the numbers. Today, we are starting with zero so I will write zero here (model writing zero off to the side before the first box). What comes next? I notice on your hundreds chart the numbers go across like this, so I will write 1 here. Now I’ll keep counting and writing the numbers (model up to 10).”

• Unit 8, Lesson 2, Introduction, Step 4, “What do you notice about the group of ten ones and loose ones and how we write the number? SMS (student might say): I notice that there is 1 group of ten ones and so there’s a one right there. Then there’s 4 loose ones so there’s a 4 right here.” The teacher replies, “Yes, this is called the tens place. There is the digit 1 here to show 1 group of ten ones. This is called the ones place. There’s the digit 4 here to show 4 loose ones.”

There is intentional development of MP8 to meet its full intent in connection to grade-level content. Examples Include:

• Unit 3, Lesson 23, Narrative, “Students look for and express regularity in repeated reasoning (MP 8) when they articulate that each number we say is one more than the one before and when they generalize that understanding in order to find one more than numbers beyond 20. Extension, “Play one more within a greater magnitude with cards from 20-50 (We know how to count to 50 now, so let’s try playing this game with some really big numbers!!!) Why does this work with all numbers?”

• Unit 5, Lesson 1, Narrative, “Teachers help students to use repeated reasoning (MP 8) to generalize how they can compare a set by prompting them to build towers and compare their lengths.”

• Unit 7, Lesson 8, Introduction, “(Show representation on recording sheet) How does this representation match the story? It shows that there are 10 apples in each box and that some are red and some are green. It shows that we need to figure out all of the ways we could fill the boxes with some red and some green. (Make sure students understand that each ‘Row’ or ‘rectangle’ represents a box.)” Mid-Workshop Interruption, “Which starting combination helped us find more solutions? 1 + 9. Why does that help us find all of the solutions? It is the smallest possible amount of [red or green] apples and the largest possible amount of [opposite color] apples. Then we add one [red or green] apple at a time and take away one [opposite color] apple at a time until we have the largest possible amount of [red or green apples] and the smallest possible amount of [opposite color] apples, so we know we have found all of the solutions.”