K-2nd Grade - Gateway 2

The materials reviewed for Math & YOU Grades Kindergarten through Grade 2 meet expectations for supporting the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

Multiple conceptual understanding problems are embedded throughout the grade level within the Investigate, Discussions, In-class examples and Big Ideas Tasks. Students have opportunities to engage with these problems both independently and with teacher support. 

According to the Implementation Handbook, Foundational Beliefs, “Each lesson begins with opportunities for students to engage in investigation resulting in observations, conjectures, and discovery of informal strategies. These opportunities support students in developing an understanding of the mathematical concepts grounded in meaningful experience and connected to prior learning, building a sturdy conceptual foundation. From here, the focus turns to formalizing these ideas through explicit instruction of new mathematical terminology, formal strategies, and key concepts, while connecting back to students’ experiences during their investigation.”

Examples include: 

• Grade Kindergarten, Chapter 6, Lesson 8, Practice, students demonstrate conceptual understanding as they are given a number and use pictures, drawings, and equations to make 10. Exercise 5 states, “You need 10 stickers to win a prize. You have 8 stickers. Draw more stickers to make 10. Then write an addition sentence to match your picture. ____ + ____ = _____” (K.OA.4)

• Grade 1, Chapter 9, Lesson 2, Investigate, students develop conceptual understanding by using concrete and pictorial models to build and add two numbers. Lesson 2 states, “Look Back. Model the numbers 32 and 7. Look Ahead. Use the models to find the sum. 32 + 7 = _____.” Students use base-ten blocks to build numbers and then draw a picture to help them solve them. (1.NBT.4)

• Grade 2, Chapter 1, Lesson 2, Practice, students demonstrate conceptual understanding as they use a ten-frame and an equation to represent a number as a sum of two addends. Then they determine whether the sum is even or odd. Exercise 3 states, “Write the number as a sum. Is the sum even or odd?” The number 9 is shown with an empty ten-frame underneath. Below the ten-frame, the equation “9 = ___ + ___” is displayed. The words “Even” and “Odd” appear under the equation for students to circle. (2.OA.3)

The materials reviewed for Math & YOU Grades Kindergarten through Grade 2 meet expectations for providing intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.

Multiple procedural skill and fluency problems are embedded throughout the grade levels within In-Class Practice, which extends their learning from the Key Concept. The formal assessments offered at the lesson, chapter, multi-chapter, and course levels provide opportunities for students to independently demonstrate their procedural skills and fluency. 

According to the Implementation Handbook, Foundational Beliefs, “Students have opportunities to develop procedural fluency through targeted practice supported by question prompts designed to encourage reflection on the accuracy and efficiency of their strategy. Practice opportunities support students in solving tasks that incorporate procedures with connections, requiring students to think meaningfully about which strategies they are using and how they apply in the problem context, and to reason about the meaning of the resulting solution. Students regularly apply their learning in new real-world or mathematical contexts, focusing on how strategies extend to these contexts and interpreting the meaning of the solution in light of the situational context.”

Examples include: 

• Grade Kindergarten, Chapter 7, Lesson 5, In-Class Practice, students develop procedural skill and fluency as they complete subtraction sentences to solve problems without using manipulatives. Exercise 2 states, “Complete the subtraction sentence. Tell how you found your answer. 5 - 4 = ____.” The Teacher Edition, Guiding Student Learning states, “Observe as students work on the three exercises. Are they using models? Their fingers? Drawing a picture? When you finish, tell your partner how you found each answer.” (K.OA.5)

• Grade 1, Chapter 2, Lesson 4, Practice, students demonstrate procedural skill and fluency as they solve doubles addition problems written horizontally. Exercises show doubles addition problems presented in both horizontal and vertical formats. Exercise 3 states, “2 + 2 = ___,” and Exercise 4 states, “4 + 4 = ___.” Vertically written problems include Exercise 5, “5 + 5 = ___,” and Exercise 6, “3 + 3 = ___.” (1.OA.6)

• Grade 2, Chapter 5, Lesson 2, Practice Workbook, students demonstrate procedural skill and fluency as they subtract numbers within 100. Exercise 3 states, “67 - 35 = ___”(equation is written vertically). Exercise 5, “54 - 13 = ___” (equation is written horizontally). (2.NBT.5)

The materials reviewed for Math & YOU Grades Kindergarten through Grade 2 meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters.

Multiple routine and non-routine applications of mathematics are included throughout the grade level, with single and multi-step application problems embedded within lessons, including Investigate and Connecting to Real Life. Students engage with these applications both with teacher support and independently through Examples and Practice. Materials are designed to provide opportunities for students to demonstrate their understanding of grade-level mathematics when appropriate. 

For Example:

• Grade K, Chapter 12, Lesson 6, In-Class Practice, students apply their understanding by placing objects in a picture using the names of shapes and position words. Exercise 4 states, “Use the City Scene Cards to place the objects on the picture of a city, with buildings, cars, and people. Place a dog in front of the girl crossing the street. Place a tree beside the building that looks like a cube. Place an object that looks like a sphere above the buildings. Place an object that looks like a cone below the traffic light. Place a streetlight next to the girl on the sidewalk.” Laurie's Notes, Connect to Real Life states, “‘What do you see on this page? Tell your partner about any of the objects that look familiar.’ Provide students with the City Scene Cards and have them place the cards in the picture as you read each direction line. Have students glue the cards to their picture. Talk About It: ‘Why do you think it is important to know what words such as beside, in front of, and above mean?’ Listen for an understanding that position words help us locate things. Students may also mention issues of safety, such as, ‘The fire extinguisher is behind the door.’ Have students use the position words to describe some of the images in the picture, such as ‘the bicycles are in front of the taxi.’” (K.G.1)

• Grade 1, Chapter 11, Lesson 2, Practice, students apply their understanding as they use a tally chart showing the numbers of three categories of stuffed animals and answer questions. Exercise 1 states, “How many more dogs are there than penguins?” Exercise 2 states, “How many bears and dogs do you have in all?” Exercise 3 states, “You get 3 more penguins. How many penguins do you have now?” (1.MD.4)

• Grade 2, Chapter 1, Lesson 3, In-Class Practice, students apply their understanding by using repeated addition to solve a routine two-step comparison problem. Exercise 11 states, “You have two horses. One drinks 5 gallons of water each day. The other drinks 6 gallons of water each day. How many more gallons of water did the second horse drink in a week?” Laurie’s Notes Connect to Real Life states, “‘What is a horse farrier?’ A farrier trims a horse’s hooves and can put horseshoes on a horse. ‘When you get new shoes, how many shoes do you get?’ 2. ‘When a horse gets new shoes, how many shoes does the horse get?’ 4. Read the problem aloud to students. Have a volunteer retell the problem using their own words, sharing what information is known and what they are trying to find.” (2.OA.1)

The materials reviewed for Math & YOU Grades Kindergarten through Grade 2 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 as reflected by the standards.

Multiple aspects of rigor are engaged simultaneously across the materials to develop students' mathematical understanding of individual lessons. Each lesson within the curriculum supports a variety of instructional approaches that incorporate conceptual understanding, procedural fluency, and application in a balanced way. 

For Example:

• Grade K, Chapter 8, Lesson 1, In-Class Practice, students develop conceptual understanding and procedural skill and fluency as they use models to build and find a group of ten and some ones. An image of 18 butterflies is shown. Exercise 1 states, “Circle 10 butterflies. Tell how many more butterflies there are. Then write the numbers.” Teacher Edition, Guiding Student Learning states, “The directions for Exercise 1 are the same as the Example. ‘Look at the butterflies. What do you notice?’ There are three rows of five and then three more. ‘Which butterflies did you circle? What was your strategy for circling ten?’ ‘How many more ones did you have?’ 8 Supporting Learners: Give students a group of between 11 and 19 counters and a ten frame. Have students fill the ten frame. ‘Complete this sentence: I have ten ones and more ones.’”

• Grade 1, Chapter 5, Lesson 3, Practice, students demonstrate all three aspects of rigor, conceptual understanding, procedural skill and fluency, and application as they make sense of a real-world situation, determine the operations needed to represent the changes, and accurately compute to find how many people remain in the contest. Exercise 10 states, “There are 16 people signed up for a sand building contest. 9 drop out and 5 more sign up. How many people are in the contest now? Show your work.”(1.OA.1, 1.OA.6)

• Grade 2, Chapter 2, Lesson 3, Practice, students demonstrate procedural skill and fluency and application as they use mental math strategies involving addition and subtraction to solve problems. Exercise 12 states, “18 horses are on a farm. 9 are Paints. The rest are Mustangs. How many horses are Mustangs?” (2.OA.1, 2.OA.2)

The materials reviewed for Math & YOU Grades Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students across the K-2 grade band engage with MP1 throughout the year. MP1 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

Across the grades, students engage in open-ended tasks that support key components of MP1, including sense-making, strategy development, and perseverance. These tasks prompt them to make sense of mathematical situations, develop and revise strategies, and persist through challenges. They are encouraged to analyze problems by engaging with the information and questions presented, use strategies that make sense to them, monitor and evaluate their progress, determine whether their answers are reasonable, reflect on and revise their approaches, and increasingly devise strategies independently. 

An example in Kindergarten includes:

• Chapter 7, Lesson 8, Student Edition, students draw a representation of the situation, identify what needs to be found, and write a subtraction sentence to solve. They interpret the context (sandwiches, moles), represent the quantities, and use subtraction to determine the unknown. They explain their thinking by telling what they need to find. In-Class Practice Exercise 2 states, “Draw to show what you know. Tell what you need to find. Write a subtraction sentence to solve. 2. You make 5 sandwiches for your family. 2 are tuna sandwiches. How many are not tuna sandwiches? 3. You see 7 moles. 5 of them burrow underground. How many moles do you see now?” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students apply their learning from across the chapter to solve subtraction word problems. Students will use SMP.1 throughout the lesson as they make sense of what a question is asking and write a subtraction sentence to solve the problem. Support SMP.1 by asking questions to support students in interpreting the problem and in reflecting on if their answer is reasonable. Throughout the learning activities of this lesson, look for ways to support students in the mathematical practices as they learn. For example: Use In-Class Practice Exercises 2 and 3 to help students think about what they know and what they want to find. Suggested prompts: What do you know in this problem? How can you draw to show what you already know? What are you trying to find in the problem? How can your picture help you to find what you need?  How did your picture help you find your answer? What can you do to check that it is correct?”

An example in Grade 1 includes:

• Chapter 4, Lesson 3, Student Edition, students use a number line to determine which city in the problem receives the most rain. Students are provided with a number line labeled from 1 through 20. Students are then prompted to think about how their strategy could be used to solve other problems. In-Class Practice Exercise 9 states, “Denver receives 2 more inches of rainfall than San Jose. Sacramento receives 4 more inches than Denver. Which city receives the greatest amount of rainfall?” Teacher Edition, Talk About It, states, “‘How can you use the number line to help you solve the problem? Today, you used a number line and the count on strategy for sums up to 20. What is important to remember?’” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide, states, “As students become more familiar with modeling the counting on strategy using a number line, extend their thinking using SMP.1. Students will solve a complex, multi-step problem, in the context of data. The number line will serve as an excellent tool for organizing their thinking, while using the counting on strategy. Students will reflect on how number lines can help solve future problems. Use the Connect to Data Exercise to give students an opportunity to apply what they learned about counting on to solve a complex multistep problem. Suggested prompts: What makes this problem tricky? What do you know? What are you trying to find?  What do you notice that might help you? What strategy may make the problem easier to solve? How will you use the number line to help you to solve this problem? Do you think that a number line could help you to solve other problems? What types of problems?”

An example in Grade 2 includes:

• Chapter 6, Lesson 10, Student Edition, students use their knowledge of one minute equals sixty seconds to find possible correct solutions to the problem. Students are encouraged to determine if their solution(s) make sense in the context of the problem. They will also consider if the problem could have multiple correct answers. In the previous problem, students were provided the following times, in seconds: 342, 335, 340, 329. In-Class Practice Exercise 21 states, “Your time is less than all the other times but greater than 5 minutes. What is a possible time for you?” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide, states, “Students have opportunities to independently solve word problems that involve comparing numbers. Encourage SMP.1 by asking students to persevere in making sense of the problem and considering what tools or models they could use to help them solve the problem. In-Class Practice Exercise 21 requires students to think about what the question is asking and consider possible solution paths. Suggested prompts: What information do you have? What do you need to find out? (Students need to realize the given information is provided in the previous exercise. They also need to determine how many seconds are in 5 minutes to come up with possible solutions.) (After solving) Does your answer make sense in the problem context? Explain.  Compare solutions with a partner. Do you both have the same answer or a different one? Can both of you be right?”

The materials reviewed for Math & YOU Grades Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP2: Reason abstractly and quantitatively, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the K-2 grade band engage with MP2 throughout the year. MP2 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

Across the grades, students participate in tasks that support key components of MP2, including reasoning with quantities, representing situations symbolically, and interpreting the meaning of numbers and symbols in context. These tasks encourage students to consider the units involved in a problem, analyze the relationships between quantities, and connect real-world scenarios to mathematical representations. Teachers are guided to support this development by modeling the use of mathematical notation, asking clarifying and probing questions, and facilitating conversations that help students make connections between multiple representations. 

An example in Kindergarten includes:

• Chapter 5, Lesson 5, Student Edition, students see eight different types of vehicles with space provided to create a number bond that represents how they grouped the vehicles. In Problem 7, students see the same set of vehicles two more times, with space to create number bonds to show additional ways the vehicles could be grouped. “6. Circle the vehicles to show 2 groups. Then complete the number bond to match your picture. 7. Show 2 other ways you can circle the vehicles to show 2 groups. Then complete the number bonds to match your pictures.” Teacher Edition, Connect to Real Life states, “Have students discuss what the number bond shows in Exercise 6. You want students to recognize they need to take apart the whole into two smaller groups. How they group the vehicles is their choice.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide, states, “In this lesson, students continue composing and decomposing, this time with the quantity 8. Students are familiar with taking quantities apart and putting them back together. Students use SMP.2 as they begin to interpret number bonds in the context of a real-world scenario and relate each number in the number bond to a meaning relative to the context. Before completing In-Class Practice Exercises 1 - 5, give students the opportunity to precisely summarize their understanding of composing and decomposing. Suggested prompts: What do you notice about the number bonds on these pages? What is the same? What is different? Use the words part and whole to explain. Use the terms take apart and put together to explain what is the same and what is different about these number bonds. Explain.”

An example in Grade 1 includes:

• Chapter 10, Lesson 4, Student Edition, students use non-standard units of measure to estimate the lengths of two objects and explain their reasoning for choosing the units they used. In-Class Practice Exercise 4 states, “Complete the sentences using paper clips or color tiles. A workbook is about 6 ____ long. A workbook is about 12 ____ long.” Digital Teaching Experience, Supporting Mathematical Practices: Facilitation Guide, states, “In this lesson, students will measure items using two different nonstandard units. SMP.2 comes into play as students assign the correct units to fill in the blank in a sentence describing the length of a given item. Students will notice throughout the lesson that more of the smaller unit will be required to measure a given item and less of the larger unit will be required to measure the same item. Throughout the learning activities in this lesson, look for ways to support students in the mathematical practices as they learn. For example: For the In-Class Practice Exercise 4, students must place the correct units in blanks to accurately record the length of the same item measured two different ways. Suggested prompts: Before solving: What have you noticed about measuring with tiles and paper clips throughout this lesson? Estimate how many tiles long a workbook would be. Why did you choose that estimate? Estimate how many paper clips long a workbook would be? Why did you choose that estimate? What do you notice about your estimates for each unit? After solving: How can you prove that you placed the units in the correct place? What math language or tool can you use to prove your answer?”

An example in Grade 2 includes:

• Chapter 3, Lesson 6, Student Edition, students use partial sums to solve two-digit addition problems and reflect on their reasoning for the partial sums they applied. Investigate states, “Model the addends. Draw your models. 32+27.” Digital Teaching Experience, Supporting Mathematical Practices: Facilitation Guide, states, “In this lesson, students will use partial sums to add. To begin the lesson, students will use base ten blocks to model adding two numbers. SMP.2 will come into play as you use the models to help the students think about what quantities that the digits in each place value represent and how they can add using partial sums. Ask students to think about how many tens and how many ones are represented by the numbers. Throughout the learning activities of this lesson, look for ways to support students in the mathematical practices as they learn. For example: In the Investigate, students will model, quick sketch, and solve addition equations. Encourage students to think about what the numbers in the equation represent and how that can help them efficiently find the sum. Suggested prompts: Where is the 3 in 37 represented in your model? Where is the 7 represented? In this problem how many tens do you have all together? (5) Can you use your model to prove that? In this problem, how many ones do you have all together? (9) How does knowing this help you find the sum?”

The materials reviewed for Math & YOU Grades Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the K-2 grade band engage with MP3 throughout the year. MP3 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

Across the grades, students participate in tasks that support key components of MP3. These include constructing mathematical arguments, analyzing errors in sample student work, and explaining or justifying their thinking orally or in writing using concrete models, drawings, numbers, or actions. Students are encouraged to listen to or read the arguments of others, evaluate their reasoning, and ask clarifying questions to strengthen or improve the argument. They also have opportunities to make and test conjectures as they solve problems. Teachers are guided to support this development by providing opportunities for students to engage in mathematical discourse, set clear expectations for explanation and justification, and compare different strategies or solutions. Teachers are prompted to ask clarifying and probing questions, support students in presenting their solutions as arguments, and facilitate discussions that help students reflect on and refine their reasoning.

An example in Kindergarten includes:

• Chapter 8, Lesson 1, Student Edition, students develop strategies to identify groups of ten and then support their reasoning while critiquing the thinking of others. In-Class Practice Exercise 1 states, “Directions: Circle 10 butterflies. Tell how many more butterflies there are. Then write the number.” Students see a picture of 18 butterflies. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students develop strategies for identifying 10 objects in a group. Support SMP.3 by asking students to share their thinking with their peers. Challenge students to convince their classmates that they have identified a group of ten and discuss how their approach yielded the result. As students solve In-Class Practice Exercises 1 - 3, help them attend to the structure of each set of objects. Suggested prompts: Were you able to find a group of 10 butterflies (fish, beads)? What strategy did you use to find the 10 objects? How many more butterflies (fish, beads) are there? Why? If you circled a different group of 10 butterflies (fish, beads), would it change how many more there are? Why or why not? Listen for students to say there will always be a group of 10 and 8 (or 4) more.”

An example in Grade 1 includes:

• Chapter 8, Lesson 2, Student Edition, students determine which number choices match the base-ten representation of the number 50. In-Class Practice Exercise 8 states, “Circle the choices that match the model.” Students see 5 ten rods along with the following number choices: 50, 20 + 30, 2 tens + 3 ones, and 1 ten + 4 tens. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “SMP.3 comes into play as students work with a partner to use their understanding of place value to explain their thinking about adding tens. Students should be developing the ability to articulate their understanding concerning place value and combining groups of ten, as well as listening and critiquing the reasoning of their peers. During In-Class Practice Exercise 8, ask students to use math language to justify their thinking. Suggested prompts: Do you and your partner agree about which answers match the model? Take turns explaining how you know each answer matches the model. What math language can you use to help you explain? What questions do you have about your partner’s explanation? Work with your partner to show how you know that any incorrect answers do not match the model.”

An example in Grade 2 includes:

• Chapter 6, Lesson 8, Student Edition, students compare three-digit numbers and explain their reasoning for the comparison. In-Class Practice Exercise 3 states, “100 less than 729 is ____.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Throughout the lesson, students should be encouraged to work in small groups or with a partner. SMP.3 will come into play as you ask students to explain their thinking to one another. Students should compare their results and defend their answers. In-Class Practice Exercise 3 asks students to find the number that is 100 less than 729. Encourage students to compare their answers and explain their reasoning to one another. Suggested prompts: Do you agree or disagree with your partner’s answer? Explain. Can you explain to your partner how you know that 629 is 100 less than 729? I see that you used a model of the number to help you. Can you share with your group how you used that model to help you?”

The materials reviewed for Math & YOU Grades Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP4: Model with Mathematics, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the K-2 grade band engage with MP4 throughout the year. MP4 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

Across the grades, students participate in tasks that support the intentional development of MP4, including putting problems or situations in their own words and identifying important information, using the math they know to solve problems and everyday situations, modeling the situation with appropriate representations and strategies, describing how their model relates to the problem situation, and checking whether their answer makes sense, revising the model when necessary.

An example in Kindergarten includes:

• Chapter 6, Lesson 10, Teacher Edition, students listen to a story, draw a picture, and write an addition sentence to model the situation. Laurie’s Notes-Closure states, “I am going to tell a story. I want you to draw a picture for this problem and then write an addition sentence that will help you answer the question. I have 6 rocks in my collection. Some are black. The same number are gray. How many black and how many gray rocks are there?” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “As students solve both adding to and putting together addition word problems, they will model word problems using pictures, manipulatives, and addition sentences. Help students attend to how their models represent the context and support them in making connections between the various models they use. The Closure activity engages students in listening to a story, drawing a picture, and then writing an addition sentence to model the story. Suggested prompts: What information from the story did you use to draw your picture? How does your picture show that information? How did you write an addition sentence based on your picture? What parts of the addition sentence did you know? What parts did you have to figure out? How does your addition sentence help you to answer the question of how many rocks are black and how many are gray? How do you know that your answer makes sense?”

An example in Grade 1 includes:

• Chapter 6, Lesson 4, Student Edition, students skip count by tens and create a model to determine how many buses will stop. In-Class Practice Exercise 8 states, “A bus stops every 10 minutes. You see a bus leave. How many more buses stop in 40 minutes? ____ tens ____ ones, ____ more buses.” Teacher Edition, Talk About It states, “Discuss with your partner how you can show 40 minutes. How will your sketch help you answer the question?” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students use cube towers and counters to investigate groups of 10, counting by 10 and modeling decade numbers as they continue to develop their understanding of place value. Support students in using SMP.4 as they make connections between groups of ten concrete tools, the corresponding numeral, and the real-life quantities they represent. During In-Class Practice Exercise 8, students create a model to solve a real-world problem. Suggested prompts: Turn and talk with a partner. Describe the situation in your own words. What are the quantities in the problem? Can you create a model to represent the quantities and context of the problem? How will your sketch or model help you answer the question?”

An example in Grade 2 includes: 

• Chapter 4, Lesson 8, Student Edition, students use models to subtract two-digit numbers and find the difference. In-Class Practice Exercise 2 states, “A pest control expert set termite traps around an apartment. The first day, there were 87 termites in the traps. The second day, there were 95. Use a model to find how many more termites were in the traps the second day.” Teacher Edition, “How will you model Exercise 2? Talk to your partner about your model. If you each chose a different model, which one do you think is easiest to use? Why?” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “As students are solving problems, you can encourage SMP.4 by asking students to model their thinking mathematically and be mindful of the important stages in modeling. In-Class Practice Exercise 2 asks students to use a model to solve a word problem. Suggested prompts: What are the relationships you are trying to model? What tool will you choose to model your thinking? Can you write an equation to represent your thinking? Show how you used your model to answer the question. How will you interpret the numeric answer in the problem context?”

The materials reviewed for Math & YOU Grades Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP5: Use appropriate tools strategically, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the K-2 grade band engage with MP5 throughout the year. MP5 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

Across the grades, students participate in tasks that support key components of MP5. These include selecting and using appropriate tools and strategies to explore mathematical ideas, solve problems, and communicate their thinking. Students are encouraged to consider the advantages and limitations of various tools such as manipulatives, drawings, measuring devices, and digital technologies, and to choose tools that best support their understanding and reasoning. They have opportunities to use tools flexibly for investigation, calculation, representation, and sense-making. Teachers are guided to support this development by making a variety of tools available, modeling their effective use, and encouraging students to make strategic decisions about when and how to use tools. Teacher materials prompt opportunities for student choice in tool selection, promote discussion around tool effectiveness, and support the comparison of multiple tools or representations. Teachers are also encouraged to highlight how different tools can yield different insights and to help students reflect on their tool choices as part of the problem-solving process.

An example in Kindergarten includes:

• Chapter 11, Lesson 7, Student Edition, students build and explore shapes when given a picture or description, and they determine which shapes could be used to represent a cityscape. In-Class Practice Exercise 4 states, “Use your materials to build a part of the picture. Draw your picture or attach it to the page. Circle the shapes you used to make your picture.” Students see a picture of city buildings and vehicles drawn with 2-dimensional shapes. The pictures include a triangle, square, circle, hexagon, and rectangle. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students use a variety of materials (e.g., coffee stirrers, pipe cleaners, craft sticks, toothpicks, clay, marshmallows, etc.) to build two-dimensional shapes when given a picture and a description. You can vary the materials you make available to students for building their two-dimensional shapes. Encourage students to use the tools available in appropriate and strategic ways (SMP.5). Foster creativity but also help students to recognize that some materials might make a better vertex (e.g., a marshmallow) and others a better side length (e.g., a toothpick). Students use materials to build part of a picture provided in In-Class Practice Exercise 4. Support students in thinking about the tools they used. ‘What did you see in the picture that helped you to identify what materials you wanted to use? If you had chosen a different part of the picture, would you have chosen other materials?’ ‘Was there a part of the picture that was hard for you to create? What made it challenging?’ ‘Is there a tool that you didn’t use? Why didn’t you use it?’”

An example in Grade 1 includes:

• Chapter 10, Lesson 2, Student Edition, students use objects to compare the lengths of other objects and determine which tool will best help them measure an object’s length. In-Class Practice Exercise 4 states, “Choose an object to compare the lengths of the scissors and the crayon. Circle the longer object. What object did you use to compare?” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students will compare the length of two items using a piece of string or other tool. SMP.5 comes into play as students choose a tool to use to compare length. Students will reflect on the tools used and decide which tools are most efficient for comparing length. Students will need to consider factors such as ease of use and precision. In the Look Ahead students will think about which tools are useful for comparing lengths. Suggested prompts: ‘What other tools could you use to compare the lengths of the two combs? What is the same and different about the tools? Which tools do you think will be best for comparing the length of the combs? Try at least two different tools.’ ‘Which tools worked the best? What made it a good tool for comparing length? Do you think it would be a good tool for comparing the length of all objects? Explain.’”

An example in Grade 2 includes:

• Chapter 1, Lesson 5, Student Edition, students choose a model to help them solve a problem. Investigate states, “Choose a tool to model the story. Draw your model. A classroom has 4 groups of students. There are 3 students in each group. How many students are there in all? Use an array and repeated addition to find the total number of students.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “This lesson provides opportunities for students to problem solve. Allow students to choose any method - manipulatives, drawings, arrays, and/or equations - to help them reason about the mathematics of the problem situations. Encourage SMP.5 by asking students to articulate their choices and why they chose the tools that they used. In the Investigate, students will choose a tool to model a story problem. Encourage students to think about what tool is best for solving the problem. Suggested prompts: ‘What tool could you use to model this problem? (counters, picture, equation, etc.)’ ‘Compare the tool you used to solve the problem with the tools that other people in your group used. Which tool do you prefer for this problem? Why?’ ‘Discuss some advantages and disadvantages of each tool used to solve this problem. Will those advantages and disadvantages always be true, no matter what problem you are solving?’ Listen for students to realize that a tool may be preferred for one type of problem but have limitations when used for a different problem.’”

The materials reviewed for Math & YOU Grades Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP6: Attend to precision, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the K-2 grade band engage with MP6 throughout the year. MP6 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

Across the grades, students engage in tasks that support key components of MP6. These include formulating clear explanations, using grade-level appropriate vocabulary and conventions, applying definitions and symbols accurately, calculating efficiently, and specifying units of measure. Students are also expected to label tables, graphs, and other representations appropriately, and to use precise language and notation when presenting mathematical ideas. Teachers support this development by modeling accurate mathematical language, ensuring students understand and apply precise definitions, and providing feedback on usage. Lessons include opportunities for students to clarify the meaning of symbols, evaluate the accuracy of their own and others’ work, and refine their communication. Teachers are encouraged to facilitate discussions and structure tasks that promote attention to detail in both reasoning and representation.

An example in Kindergarten includes: 

• Chapter 8, Lesson 5, Student Edition, students understand and use symbols and definitions correctly to solve problems by creating addition sentences and equations. Practice Exercise 2 states, “Circle 10 hats. Draw dots in the ten frame to show how many hats are circled. Draw dots in the five frame to show how many more hats there are. Use the frames to write an addition sentence.” Practice Example 2, Teacher Note states, “‘Which part of the addition sentence tells you what to write first? Why?’ The equal sign comes before the plus sign, so I write the whole (14) first.” Students see 14 hats with a line drawn around 10 of them, along with space provided to write an addition sentence. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students continue to build confidence using addition sentences to model quantities. Support SMP.6 by asking probing questions about what each part of the addition sentence represents and in helping students decide what can change and what must remain the same in their addition sentences. Support students in attending to precision and relating quantities represented in different ways as they solve Practice Exercise 2. Suggested prompts: ‘How did you decide what number to write first? What clues did the addition sentence give you? What does the number mean about the hats?’ ‘Why is an addition sign used? What does the addition symbol tell you about the hats?’ ‘Is there a different addition sentence that could represent the same picture? Why or why not?’ Listen for students to explain that they could write 14=4+10.”

An example in Grade 1 includes:

• Chapter 3, Lesson 6, Teacher Edition, students explore the meaning of equality by testing whether equations are true or false, using precise mathematical language to define symbols and justify their reasoning. Dig In states, “Students will discover what makes an equation true. ‘I will say a number. Then I will say an expression. Clap if the expression I tell you is equal to the number. Pause for student questions. ‘The number is 4. Is it equal to 2+2? 4+0? 3+4?’ Play several rounds using different combinations of numbers and expressions. ‘What does equation mean?” Listen for ideas of equality. Show students equations with sums or differences on the right, the left, and vertically. ‘Are all of these equations?’ ‘Is the 2+3=4+1 an equation?’ Listen for any misconceptions that need cleared up. ‘So, is 2+3=4+1 true or false? What is 2+3?’ Write 5 underneath. ‘What is 4 + 1?’ Write 5 underneath. ‘5 = 5. So, 2+3=4+1 is true.’ ‘For an equation to be true, both sides of the equals symbol must have the same value.’” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students begin to understand the meaning of the equal sign as they use concrete tools and models to represent equations with an expression on both sides of the equal sign. Students must use SMP.6 as they notice the difference in the variety of ways an equation can be recorded and work to define each of the symbols in an equation. Students continue to work with precision as they prove equations with an expression on both sides of the equal sign are either true or false. In the Dig In, after posting the equations for the class to view, ask them to reflect and begin to create a definition of a true equation. Suggested prompts: ‘What do you notice about these equations? What do each of the symbols mean?” Listen carefully for how students define the equal sign. Ask them to use examples to establish an accurate definition. ‘Are all these equations true? What does true mean? What does false mean?’ ‘Is 2+3=4+1 a true equation? Can you prove that it is true using the words equation, expression, and equal?’”

An example in Grade 2 includes:

• Chapter 1, Lesson 3, Student Edition, students use equal groups and repeated addition to determine the number of objects in a situation. In-Class Practice Exercise 3 states, “Circle groups of 3. Then find the total number of fish.” Students see pictures of 15 fish and are given space to write a repeated addition equation to determine the total number of fish. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “As the lesson progresses, students will learn and be expected to use new vocabulary words such as equal groups and repeated addition. Encourage SMP.6 by asking students to use precise math language in addition to expecting that students write equations and find sums accurately and efficiently. For In-Class Practice Exercise 3, ask students to share their thinking and encourage them to use precise math language. Suggested prompts: ‘How many equal groups of fish are there? Can you use your equation to explain where you see the number of equal groups?’ ‘Explain what the term equal groups means to you. What is repeated addition and why does circling equal groups help you with that strategy?’ ‘Can you explain how you know there are a total of 15 fish?’”

The materials reviewed for Math & YOU Grades Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP7: look for and make use of structure, in connection with grade-level content standards, as expected by the mathematical practice standards.

• Students across the K-2 grade band engage with MP7 throughout the year. MP7 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

Across the grades, students engage in tasks that support key components of MP7. These include looking for and describing patterns, identifying structure within mathematical representations, and decomposing complex problems into simpler, more manageable parts. Students are encouraged to analyze problems for underlying structure and to consider multiple solution strategies. They make generalizations based on repeated reasoning and use those generalizations to solve problems efficiently. Teachers support this development by selecting tasks that highlight mathematical structure and by prompting students to attend to and describe patterns they notice. Lessons provide opportunities for students to compare approaches, justify their reasoning, and reflect on how structure helps deepen their understanding. Teachers are encouraged to facilitate discussions and structure lessons in ways that promote the recognition and use of patterns and structure in problem-solving.

An example in Kindergarten includes:

• Chapter 7, Lesson 3, Teacher Edition, students use number bonds to represent addition problems, and they apply structure and reasoning with number bonds to represent subtraction situations. Laurie’s Notes Closure states, “Draw the number bond shown on the board for all students to see. ‘Tell your partner a story that matches this number bond. Write a subtraction sentence for your story.’ The Support for All Learners page will connect you with content support to nudge students toward mastery.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “This lesson introduces students to subtraction problems that involve taking apart a group of objects. Students use SMP.7 as they take relate taking apart a number to a number bond and partner numbers. Students begin to make sense of how two different subtraction sentences can be written for any given number bond (i.e., 6-4=2 and 6-2=4). Use the Closure to engage students in using structure and quantitative reasoning as they tell a story and write a subtraction sentence for the number bond between 6, 4, and 2. Suggested prompts: ‘What does the number tell you about taking 6 apart in two groups? How did you use that in your story? How did you use that in your subtraction sentence?’ ‘What was the same about the story you and your partner told? What was different? Did those differences change your subtraction sentence?’ ‘Pick a different number bond for 6 and show a related subtraction sentence.’”

An example in Grade 1 includes:

• Chapter 12, Lesson 5, Student Edition, students decompose complex shapes into simpler shapes, noticing which shapes make up the larger figure. In-Class Practice Exercise 8 states, “Complete the model. How did you use the model to solve?” Students see a triangle and a rhombus and are asked how many of each shape are needed to form a hexagon. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students will decompose complex shapes into simple shapes and identify the component shapes in complex shapes. This can be very challenging for some students and not all students have the spatial skills to see the smaller components of a shape until after a line has been drawn. Students will engage in SMP.7 as they learn to see shapes as composed of smaller shapes. During In-Class Practice Exercise 8, students first need to understand the problem and then use structure to solve it. Suggested prompts: ‘What is the part-part-whole mat trying to show? How can you figure out how to complete the model?’ ‘Can you draw a picture or use pattern blocks to help you break up the shape into smaller shapes?’ ‘How did you decide how many triangles are needed?'"

An example in Grade 2 includes: 

• Chapter 7, Lesson 4, Student Edition, students use a number line to solve three-digit addition problems. Investigate states, “Use the number line to find each sum. 425+220=, 536+320= . Look Ahead, Use the number line to find each sum. 425+222=, 536+324= .” A number line is provided for each problem. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students use a number line to add three-digit numbers, but neither of the addends will be a multiple of ten. SMP.7 comes into play as students decompose one of the addends into hundreds, tens, and ones so that they can be added separately in three steps. Have students explain the structure they are using as they work to add using jumps on a number line. In the Investigate, ask students to look at the structure of each number they are adding and use place value to explain the jumps on the number line models. Suggested prompts: ‘How does the structure of each number that is being added help you decide which jumps to make on the number line? Can you show me where the numbers in the equation are represented on your number line?’ ‘Compare the two number line models for 425+220 and 425+222. What is similar? What is different? Use place value in your explanations.’”

The materials reviewed for Math & YOU Grades Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP8: Look for and express regularity in repeated reasoning, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the K-2 grade band engage with MP8 throughout the year. MP8 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

Across the grades, students engage in tasks that support key components of MP8. These include notice and use repeated reasoning to make sense of problems, recognize patterns, and develop efficient, mathematically sound strategies. Students are encouraged to create, describe, and explain general methods, formulas, or processes based on patterns they identify. Lessons provide opportunities for students to evaluate the reasonableness of their answers and refine their approaches through discussion and reflection. Teachers support this development by structuring tasks that highlight repeated reasoning, prompting students to make generalizations, and guiding them to build conceptual understanding, distinct from relying on memorized tricks. Instructional guidance also encourages teachers to model and elicit strategies that build toward formal algorithms or representations through consistent reasoning.

An example in Kindergarten includes:

• Chapter 7, Lesson 4, Student Edition, students use repeated calculations to add numbers that are one more than doubles. Investigate states, “Directions: Color the boxes to show how many there are to start. Cross out colored boxes to show how many are taken away. Tell how many are left. Tell what you notice.” Students see three problems that involve subtracting zero and three problems that involve subtracting one. Teacher Edition, Talk About It states, “‘Have students talk to their partner about what they notice whenever they take away zero. Listen for the concept of the difference being the same as the starting number. ‘Will this pattern always be true? Explain.’” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students investigate patterns involving subtracting 0, 1, or all of the quantity. Encourage SMP.8 as students look across several examples to express a rule for these three special subtraction cases. The Investigate encourages students to explore patterns for subtracting 0 and subtracting 1. You can support students in connecting this to patterns they noticed in the Dig In. Suggested prompts: ‘Tell your partner what you noticed about subtracting 0 on the Investigate. How did you see that with the number path during the Dig In?’ (Listen for students to say that taking 0 steps meant they stayed in the same location.) ‘Tell your partner what you noticed about subtracting 1 on the Investigate. How did you see that with the number path during the Dig In?’ (Listen for students to say they landed on the previous number.) ‘Do you think the patterns you noticed would work for larger numbers? Tell your friend why or why no.’”

An example in Grade 1 includes:

• Chapter 6, Lesson 9, Student Edition, students use repeated calculations to identify and write numbers using representations of the tens and ones place. In-Class Practice Exercise 5 states, “Match the model to its number.” Students see a column of ten rods and cubes representing the numbers 39, 103, and 93. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students will engage in SMP.8 as they synthesize everything they have learned about place value in two-digit numbers in this chapter. Questions like ‘Does that always work?’ or ‘Why does that work?’ are helpful to encourage generalizations about place value understanding and concepts that will continue through all variations of numbers. Use In-Class Practice Exercise 5 to encourage students to generalize an approach for solving problems. Suggested prompts: ‘How did you match the model with the number? What general process did you use? Were you able to match the models with the numbers without counting?’ ‘Describe the strategy you used for matching the models and the numbers. What math language can help you explain your strategy?’ ‘Will your strategy always work? Can you think of a time that your strategy might not work?’”

An example in Grade 2 includes:

• Chapter 6, Lesson 6, Student Edition, students skip count by 2s and 5s up to 1,000, noticing patterns that help them develop rules for skip counting. Investigate states, “Look Back, Count by ones, tens, and hundreds to 1,000. Look Ahead, Count by ones, tens, and hundreds to 1,000.” Students see number lines starting at 993, 930, 300, 988, and 970, and count forward by the given number. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students will count by twos and fives within 1,000. To begin the lesson, students will use a number line to count by twos and fives. SMP.8 will come into play as you ask students to notice patterns that can help them generalize and discover rules for skip counting by twos and fives. In the Investigate, students will use a number line to count by twos and fives. Ask students to look for patterns that might help them develop rules for counting by twos and fives. Suggested prompts: ‘Look at the first number line in the Look ahead. What do you notice about the numbers?’ (all the numbers are even) ‘Look at the second number line in the Look Ahead. What do you notice about the numbers?’ (they all end with a 0 or a five) ‘Can you use what you see on either of the number lines to help you come up with a rule for counting by twos or fives?’”