3rd-5th Grade - Gateway 2

The materials reviewed for Math & YOU Grades 3 through Grade 5 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 3, Chapter 4, Practice Workbook, Operations and Algebraic Thinking: Relate Multiplication and Division, students demonstrate conceptual understanding as they find the missing factor in multiplication equations. Exercise 4 states, “Which equation has the same unknown value as 63\div7=___?” Answer choices include, “A. 63=7\times___, B. 63=___\times7, C. ___ \div7=63, D. 7= ___ \div63.” (3.OA.6)

• Grade 4, Chapter 1, Lesson 1, In-Class Practice, students develop conceptual understanding as they generalize place value understanding for multi-digit whole numbers and establish a foundational understanding that supports strategies for computing products and quotients of multi-digit whole numbers throughout the grade. In Exercises 2-13, students identify the value of an underlined digit in a multi-digit number. For example the materials state, “Exercise 2. 93,517,” with the 5 underlined. In-Class Practice 14, students compare the values of two identical digits in a multi-digit number. A Supporting Learners note provides teachers with guidance to support students by using an anchor chart that includes the standard, word, and expanded forms of a six-digit number to help students think about numbers in various ways. The note also suggests that students continue using base-ten blocks, quick sketches, and Place Value Mats. For Exercise 14, the Supporting Learners note recommends that students write the value of each digit to make the pattern visible. (4.NBT.A)

• Grade 5, Chapter 13, Lesson 1, Practice Workbook, students demonstrate conceptual understanding as they explain how shapes with different dimensions can have the same volume. Exercise 11 states, “Your friend says the two figures have the same volume. Is your friend correct? Explain.” The first figure is a rectangular prism made of six cubes. It measures 3 inches long, 2 inches high, and 1 inch wide. The second figure is a rectangular prism made of six cubes arranged in a single row. It measures 6 centimeters long, 1 centimeter high, and 1 centimeter wide. (5.MD.C)

The materials reviewed for Math & YOU Grades 3 through Grade 5 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 3, Chapter 7, Lesson 7, Practice Workbook, students demonstrate procedural skills and fluency as they add three numbers within 100. Exercise 10 states, “Which problem can you solve without regrouping?” Four problems are written vertically for students to solve, “282 + 274 + 260 = ___, 141 + 155 + 399 = ___, 195 + 300 + 315 = ___, 261 + 215 + 323 = ___” (3.NBT.2)

• Grade 4, Chapter 2, Lesson 6, Practice, students demonstrate procedural skills and fluency as they solve subtraction problems using the standard algorithm and check for accuracy using addition. Exercise 1 states, “Find the difference. Use addition to check your answer. 3,473 - 2,760.”(4.NBT.4)

• Grade 5, Chapter 4, Lesson 5, In-Class Practice, students develop procedural skill and fluency as they solve in-class practice problems. In the Teacher Guide, Guiding Student Learning notes support teachers in helping students compare and reflect on strategies, building fluency throughout the lesson. The materials state, “Exercises 1–3: Have students solve using an area model and then solve using partial products with regrouping. ‘Which strategy is more efficient? Which strategy makes more sense?’ Exercises 4–6: Monitor as students practice independently. Which students are using the area model efficiently? Which students can regroup and calculate accurately?” Student Edition, Exercise 1 states, “402\times221” is written vertically.” (5.NBT.5)

The materials reviewed for Math & YOU Grades 3 through Grade 5 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. 

Examples include: 

• Grade 3, Chapter 4, Performance Task: Career - Protected by Landscaping, students apply their understanding by using multiplication and division to solve a multi-step, real-world problem involving water use. Exercise 2 states, “On Monday, a 100-gallon rain barrel was half full when it started raining. The barrel collected 16 more gallons of rain water. A landscaper uses 8 gallons of water every 2 days to water the shrubs and other plants. How much water will be left in the rain barrel after 2 weeks without any rain? Show how you know.” Nick’s Notes Performance Task, Group Engagement states, “Ask students to imagine that they are a landscape architect. ‘What important information does a landscape architect need to consider when designing a landscape?’ Exercise 2: Students may be overwhelmed by the amount of information given in this problem. Suggest that they first find the number of gallons of water in the rain barrel after the rain. Ask guiding questions, ‘How many gallons can the rain barrel hold?’ 100 ‘How can you find half of that amount?’ Sample answer: 50 + 50 = 100, so half of 100 is 50. ‘How many gallons of water are in the rain barrel after it collects 16 more gallons or rain water?’ 66.” (3.OA.3)

• Grade 4, Chapter 3, Big Ideas Task, students apply their understanding as they use the four operations with whole numbers to solve problems. Students use multiplicative comparisons and the four operations to find and compare the amount of food eaten relative to body weight for various animals, including an animal of their choice. Exercise 1 states, "A hummingbird weighs 18 grams and eats 36 grams of nectar every day. A pygmy shrew weighs grams and eats 12 grams of food every day. a. Your friend says that a hummingbird eats 3 times as much as its body weight every day. Use a model to tell whether your friend is correct. Explain your choice of model. b. Which animal eats more food relative to its body weight? Use a model to explain. c. Choose an animal of your choice and research its average weight. If the animal ate 3 times as much as its body weight, how much food would this be? d. Research how much food the animal in part (c) actually eats every day. Does it eat more than or less than 3 times as much as its body weight every day? Explain how you know." (4.OA.1)

• Grade 5, Chapter 8, Lesson 5, Practice, students apply their understanding as they add and compare fractions. Exercise 10 states, "Compare the sum of the weights of two rock sparrows and the sum of the weights of two yellow-throated sparrows. Show your work." Students are provided with data showing the fractional weights of various sparrows in a table. (5.NF.2)

The materials reviewed for Math & YOU Grades 3 through Grade 5 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 3, Chapter 13, Practice Workbook, Lesson Extra Practice, 13.2, students develop conceptual understanding and application as they solve area and perimeter word problems. Exercise 11 states, “You put painter’s tape around two rectangular windows. The windows are each 47 inches long and 32 inches wide. How much painter’s tape do you need?” (3.MD.8)

• Grade 4, Chapter 3, Lesson 8, In-Class Practice, students demonstrate procedural skill and fluency and application as they apply a problem-solving plan to understand and solve a problem about spending a budget on equipment for a science lab. Exercise 2 states, “You are restocking the science lab at your school. You have 4,500 to spend on microscopes and balances. You bought 3 microscopes that each cost 889 and 6 balances that each cost $236. How much do you have left to spend?” Nick’s Notes, In-Class Practice states, “As different contexts are presented, can students successfully implement the problem-solving plan? Can they identify the known information? Can students identify other quantities they need to find before they can answer the actual question? Are students applying multiplication strategies efficiently? Can they determine whether an answer is reasonable?” (4.OA.3)

• Grade 5, Chapter 9, Lesson 5, In-Class Practice, students develop all three aspects of rigor, conceptual understanding, procedural skill and fluency, and application as they solve a problem about winning dogs at a dog show. Exercise 24 states, “One-half of the sporting and working dogs in the dog show won awards. What fraction of the dogs at the event were winning sporting or working dogs?” Students work to understand the context and the information provided in the table. Students develop conceptual understanding of fraction multiplication as they identify the meaning of the fractions in the table, representing the fraction of dogs at the event that were a certain type of dog, and then multiply the sporting and working fractions by \frac{1}{2} to find the fraction of dogs at the event that were winning sporting or working dogs. Procedural skill and fluency are demonstrated as students use a rule to multiply the fractions. Teacher Edition, Talk About It states, "Exercise 24: 'What information is shown in the data table?’(fractions of different types of dogs at a dog show) ‘What mathematical questions might you ask about the data?' Allow students time to share their initial thoughts." (5.NF.4)

The materials reviewed for Math & YOU Grades 3 through Grade 5 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 3-5 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 Grade 3 includes:

• Chapter 4, Lesson 8, Student Edition, students interpret relative distances between towns, represent the situation to determine how far Benbow is from the halfway point, and explain their reasoning to justify the solution. Students must make sense of the information and decide on a strategy to solve it. Practice Exercise 6 states, “You are halfway between Cooks Valley and Phillipsville. How far are you away from Benbow? Explain.” Teacher Edition states, “How can you make sense of this problem? What can you do to represent the given information?” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide, states, “In this lesson, students have additional opportunity to apply division facts from the chapter to solve real-life problems. To begin the lesson, students review the problem-solving plan. Then as students independently practice problem solving, students engage in SMP.1 by persevering to make sense of problems on their own and determine what they already know and what they need to find out.”

An example in Grade 4 includes:

• Chapter 9, Lesson 6, Student Edition, students use a feeding table to determine how many days two cats can be fed from a given bag of food. Students are provided with a table that shows various weights of a cat and a fractional amount of food they each will eat daily. Students will need to select a strategy that makes sense to use in solving the problem as well as checking their solution to be sure it makes sense in the context of the problem. In-Class Practice Exercise 5 states, “You have a 6-pound cat and a 9-pound cat. You have a 34-cup bag of cat food. The cats eat the recommended amounts of food each day. What is the greatest number of days you can feed the cats from the bag? Show your work.” Teacher Edition, Talk About It, “What do you know? What do you need to find? What are some strategies you could try to find the solution? Have students share and discuss with the whole class. Which strategy made the most sense to you?” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students are given real-life scenarios that require them to solve multi-step word problems involving multiplying a fraction by a whole number. There are opportunities for students to engage in SMP.1 as they think about what the questions are asking them to do and persevere to find tools and strategies to solve them.”

An example in Grade 5 includes:

• Chapter 4, Lesson 1, Student Edition, Students apply multiplication strategies to understand and solve two problems about orange orchards, deciding how to mathematically model the scenario and decide on a strategy that makes sense for solving. Practice Exercise 6 states, “You have an orange orchard that has 4 rows with 7 trees in each row. Each tree produces 250 oranges. If you sold each orange for $2, how much money would you make? 7. You decide to sell the oranges in bags of 8 oranges each. You sell 875 bags at $8 each. How much more money would you have made if you sold all the oranges individually” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide, states, “The lesson provides ample problem-solving opportunities for students to apply appropriate multiplication strategies to solve real-world problems. Enable SMP.1 by providing time for understanding of problems and persevering in finding a solution pathway. After students have solved Practice Exercises 6 and 7, have them discuss their problem-solving approaches. Suggested prompts: Describe in your own words what the problems are asking. How does the solution to Exercise 6 help you with Exercise 7? How can a model help you to solve each problem? What does each part of your model represent in the context of the problem? Is your answer reasonable mathematically? Explain why it makes sense in the context of the situation. How can you check for reasonableness?”

The materials reviewed for Math & YOU Grades 3 through Grade 5 meet expectations for supporting the intentional development of MP2: Reason abstractly and quantitatively, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students across the 3-5 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 Grade 3 includes:

• Chapter 10, Lesson 8, Student Edition, students reason abstractly to find the mass of rolls of nickels and quarters. To solve the problem, they must distinguish between the dollar value of the rolls and the number of coins in each roll. In-Class Practice Exercise 12 states, “A nickel has a mass of 5 grams. A quarter has a mass of about 6 grams. Find the mass of a roll of nickels and a roll of quarters. Roll of nickels: ____, Rolls of quarters: ____” Students are shown an image of a roll of nickels valued at $2.00 and a roll of quarters valued at $10.00. Teaching Edition, Talk About It states, “SMP.2 How many nickels are in $2? How many quarters are in $10? How does this help you find which roll has the greater mass?” Students engage in MP2 by reasoning quantitatively about the relationship between the value and mass of coins, using the given values of $2.00 for a roll of nickels and $10.00 for a roll of quarters to determine how many of each coin are in a roll and compare their total masses.

An example in Grade 4 includes:

• Chapter 9, Lesson 2, Student Edition, students use models to explain their reasoning when solving problems involving the multiplication of whole numbers and fractions. In-Class Practice Exercise 20 states, “You spend \frac{5}{4} hours each day for 7 days feeding the birds. How many hours do you spend feeding the birds in all? Use a model to check your answer.” Teaching Edition, Talk About It states, “‘Tell your partner what the problem is about.’ After students share with their partner, discuss the context with the whole class. ‘On your whiteboard, draw a model of \frac{5}{4}.’ Monitor students as they work. ‘Show your partner your model and explain to them how you know your model is correct.’ Check that students use two wholes to model \frac{5}{4}.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide, states, “Students will engage in SMP.2 as they use the models to explain how each of the quantities in the equations are represented, and when problem solving, to make meaning of the quantities in the mathematical and real-world contexts. For In-Class Practice Exercise 20, allow students to turn and talk to articulate the quantitative relationships in the problem and contextualize their answer in the real-life context. Suggested prompts: Tell your partner what the problem is about. Can you draw a model to represent the problem? What does the shaded portion of your model represent in the real-life context? Once you compute to find an answer, what are the units for that quantity? Use a complete sentence to answer the problem.”

An example in Grade 5 includes:

• Chapter 7, Lesson 6, Student Edition, students determine how much each person pays when six friends equally share the total cost of a water taxi ride that includes a per-mile charge and a tip. In-Class Practice Exercise 1 states, “1. You and 5 friends equally share the cost of taking a water taxi to an island. The charge is $10.35 per mile. You travel 8.4 miles and tip the captain $15. How much do each of you pay?” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students will solve word problems involving decimals. Students will reason abstractly and quantitatively (SMP.2), making sense of the quantities and their relationship in problem situations, taking quantities out of the problems to work with (decontextualize), and then contextualize the quantities to interpret their answer in the situation. Encourage the flexible use of properties and solution strategies within the class when solving problems. Use In-Class Practice Exercises 1-5 to remind students they are reasoning abstractly and quantitatively. Suggested prompts: As you read each problem, what does each quantity represent in the context? How are the quantities related? What operations will you perform? How will your answer help you to solve the problem? What are the units? How does your answer make sense in the problem context?”

The materials reviewed for Math & YOU Grades 3 through Grade 5 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 3-5 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.

An example in Grade 3 includes:

• Chapter 3, Lesson 1, Student Edition, students solve problems involving multiplication by 3 and then reflect on and critique the strategies their peers used to find solutions. In-Class Practice Exercise 2 states, “Find the product. 3\times6= .” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Throughout the lesson students should share their ideas and strategies with their classmates. Encourage SMP.3 by asking them to compare their strategies with their classmates and consider which strategy they prefer and why. In Class Practice Exercise 2 asks students to find the product 3\times6. Ask students to share their thinking and listen to the ideas of others. Suggested prompts: How did you find 3\times6? Explain to the person sitting next to you. Did your partner solve 3\times6 the same way or a different way than you? What questions do you have about your partner’s explanation? Talk with your partner about which strategy you prefer, and why.”

An example in Grade 4 includes:

• Chapter 7, Lesson 6, Student Edition, students explain and show their reasoning when comparing two fractions, and they critique their partner’s thinking. In-Class Practice Exercises 1, 2, and 4, state, “Compare using <, >, or =. 1. \frac{7}{8}__\frac{3}{4} 2. \frac{6}{10}__\frac{4}{5} 4. \frac{8}{10}__\frac{4}{5}.” Digital Teaching Experience, Supporting The Mathematical Practices: Facilitation Guide states, “As students are learning to compare fractions throughout the lesson, take opportunities to encourage SMP. 3 by asking students to explain and justify their thinking and actively listen to the reasoning of their peers. In-Class Practice Exercises 1-6 ask students to compare two fractions. Ask students to justify their reasoning to a partner. Suggested prompts: How do you know that six-twelfths is greater than one-third? Explain or show your thinking to your partner. Listening partners: Does your partner’s explanation make sense? Why or why not? How did you compare three-eighths and two-fifths? Did you compare it the same way as your partner or a different way? Do you agree with your partner? Why or why not?”

An example in Grade 5 includes:

• Chapter 9, Lesson 6, Student Edition, students critique the work of a peer who multiplies two mixed numbers using an area model. The model shown has \frac{5}{3} shaded from top to bottom and \frac{5}{2} shaded from left to right. In-Class Practice Exercise 9 states, “Your friend uses the model to find 2\frac{1}{2}\times1\frac{2}{3}. Is your friend correct? Explain.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students should be able to construct viable arguments and critique the reasoning of others (SMP.3). The You Be the Teacher problem can be an excellent springboard for classroom discourse; have students work in pairs to critique the mathematical thinking represented by the student work. In the In-Class Practice Exercise 9, place students in groups to construct viable arguments and critique the reasoning of others. Suggested prompts: Is the student’s work correct? What part(s) of their work represent correct mathematical thinking? What part(s) represent erroneous thinking? Explain. What advice can you give to this student when using a model to multiply mixed numbers?’”

The materials reviewed for Math & YOU Grades 3 through Grade 5 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 3-5 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 Grade 3 includes:

• Chapter 6, Lesson 7, Student Edition, students use multiplication to solve real-world word problems and show the models they used to find the product. Practice Exercise 7 states, “You and your friend want to buy 2 amusement park tickets that cost 30 each. You save 2 each week. Your friend saves $4 each week. If you combine your money, how long will it take you and your friend to save enough money to buy the tickets?” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Throughout the lesson, students have several opportunities to solve two-step story problems involving multiplication and division that can be represented in many ways. Encourage SMP.4 by asking students to create a model (can include arrays, drawings, and/or equations) to represent the problem scenario and use the model to solve the problem. Practice Exercise 7 involves a complex scenario; deciding how to model the problem is essential in solving it. Suggested prompts: How will you create a model to represent each person’s savings? Allow students to be creative in how they model the mathematics in this problem and allow them to choose independently. How will you use your model to find the answer? How will you know when both people have saved enough? Can you solve another way to check your answer?”

An example in Grade 4 includes:

• Chapter 3, Lesson 7, Teacher Edition, students consider different models for multiplication to help them determine the solution. Nick’s Notes-Dig In states, “Goal: Students will compare multiplication expressions and identify different ways to find products. Distribute a whiteboard and marker to each student. Write the equations 4\times17=68 and 2\times2\times17= on the board. ‘Find the missing product with your partner.’ Some students may begin to multiply on their whiteboards, while others may recognize that the equations have the same product. ‘Did anyone find the answer quickly? Explain.’ ‘If the equation was written as 2\times17\times2 , would the product still be 68? Explain.’ Yes, the order of the factors does not affect the product. Students may be able to apply the Commutative Property of Multiplication but not recall the name. Write the expression 2\times(50\times4) on the board. ‘Which factors should you multiply first?’ Write three more expressions on the board: (2\times50)\times4, (2\times4)\times50, 4\times(2\times50). ‘Compare the four expressions. How are they alike?’ All the expressions have the same three factors and the same product. ‘How are they different?’ The order of the factors and the placement of the parentheses are different. Write the following expressions on the board: 3\times54\times2, 4\times2\times98, and 2\times125\times4. Turn and Talk: ‘How would you find each product? Would you keep the order the same? Would you rearrange the factors?’ Talk About It, ‘Properties of operations allow you to work flexibly with multiplication. Today you will apply different properties to find products. How can properties of multiplication help you when using mental mat.’” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “At different times in the lesson, students may struggle to make sense of the properties. Have students engage in SMP.4 by asking them to use an area model or base ten blocks to model the problem and help them identify which property they can use to multiply each expression. In the Dig In, students will compare multiplication expressions and identify different ways to find products. Students can use models to help them understand the connections between the expressions. Suggested prompts: Can you use an area model to help you find the product? How does a model help you understand why 4\times17=2\times2\times17? Turn and talk with your partner. Could you use an area model to check your work?”

An example in Grade 5 includes:

• Chapter 3, Lesson 6, Student Edition, students use multiplication models with decimals to solve a real-world word problem involving money. Practice Exercise 5 states, “You buy one of each item. You pay with five 5 bills and receive 0.77 in change. What is the price of the glowing liquid kit?” Students are shown a table with the following information: Beaker – 7.99, Ultraviolet Light – 13.29, Glowing Liquid Kit – ? Teacher Edition states, “How can you use the information in the table? Write an equation to find the price of the glowing liquid kit. How can you represent the amount paid and the unknown price of the glowing liquid kit in your equation?” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students will model with mathematics (SMP.4). Students will develop a model for solving real-life problems by using expressions and equations. These models will help students attend to the relationships between quantities. It is important that students use their model to interpret the results and check for reasonableness. Use real-life contexts to help students represent multi-step word problems. When solving Practice Exercise 5, encourage students to think about the stages of modeling with mathematics. Suggested prompts: How are the quantities from the table related? How can you create a model to represent the quantities and the problem? How can you use your model to draw conclusions about the situation? What does your numeric answer mean in the context? How can you check the reasonableness of your answer?”

The materials reviewed for Math & YOU Grades 3 through Grade 5 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 3-5 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 Grade 3 includes: 

• Chapter 10, Lesson 2, Student Edition, students find elapsed time and determine whether a number line is a useful tool for solving the problem. In-Class Practice Exercise 4 states, “Find the elapsed time. Start: 5:15 P.M. End: 5:41 P.M.” Students are provided with a number line to use in solving the problem. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “As the lesson progresses, students will also learn to use number lines to find elapsed time. Encourage SMP.5 by having students think about the different tools and models they can use to find elapsed time. In-Class Practice Exercise 4 asks students to find the elapsed time. Ask students to think about how different tools can help them find the elapsed time. Suggested prompts: ‘How can a number line be a useful tool for finding elapsed time?’ ‘What other tools might help you find the elapsed time?’ ‘What would modeling the times on a clock show you that a number line would not? Which tool do you prefer for finding elapsed time? Why?’”

An example in Grade 4 includes:

• Chapter 9, Lesson 3, Student Edition, students find the missing number to solve multiplication problems with whole numbers and fractions. They determine which tools and strategies will best support their solutions. In-Class Practice Exercise 7-9 states, “Find the missing number. Use a model to help. 7. \square\times\frac{3}{8}=\frac{18}{8}. 8. 5\times\frac{?}{5}=\frac{15}{5}.”9. 7\times\frac{8}{?}=\frac{56}{3}.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Throughout the lesson, you can encourage SMP.5 by asking students to consider which tools or strategies they can use to help them solve the problems and to strategically choose the tools that will be most useful. In-Class Practice Exercises 7 through 9 ask students to use a model to find the missing number in an equation. Encourage students to consider the different tools that could help them. Suggested prompts: ‘What tools could you use to model these equations?’ (number line, area model, fraction tiles) ‘How did using the tool help you to solve the problem?’ ‘Find a partner who used a different tool to solve the problem. Compare their model to yours. How are they similar and different?’”

An example in Grade 5 includes:

• Chapter 14, Lesson 1, Student Edition, students decide which tools to use in order to solve geometry problems. In-Class Practice Exercise 10 and 12 states, “10. Draw one triangle for each category. Acute Triangle, Obtuse Triangle, Right Triangle. 12. Find the angle measures of each triangle. What do you notice about the sum of the angle measures of a triangle?” Teaching Edition, Talk About It states, “Exercise 10: “What tool can help you draw the triangles? Explain.’ Sample answer: I can use a protractor to draw 3 acute angles, 1 obtuse angle, or 1 right angle” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students will classify triangles by their angles and their sides and draw triangles to meet certain classification categories. As always, continue to affirm that students should make independent strategic choices about the tools they use (SMP.5) as they identify and draw triangles. Tools in this lesson might include rulers, protractors, and geometric drawing software. Students should be able to explain the benefits and limitations of the math tool they use. In the In-Class Practice Exercises 10 and 12 students will use appropriate tools strategically. Suggested prompts: ‘In Exercise 10, what tool can help you to draw each triangle? What aspect of the tool will help you to draw the characteristics for each category?’ ‘In Exercise 12, can you name different tools that could help you find the angle measures?’ (protractor or pattern blocks) ‘Explain how you will use the tool for each triangle. How many angles do you need to measure?’”

The materials reviewed for Math & YOU Grades 3 through Grade 5 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 3-5 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 Grade 3 includes:

• Chapter 4, Lesson 2, Student Edition, students determine the correct operation (multiplication or division) to complete a problem and precisely justify their choice. In-Class Practice Exercise 11 and 12 state, “Complete each equation using \times or \div . 11. 6\bigcirc2=3, 12. 6\bigcirc2=12.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “The lesson focuses on students using their knowledge of fact families to help them complete multiplication and division equations. Encourage SMP.6 by asking students to compare their work with others and explain mathematically how they know their answer is accurate. In-Class Practice Exercises 11 and 12 ask students to complete the equation using \times or \div . Encourage students to compare their answers and explain how they know they are accurate. Suggested prompts: ‘Compare your answers to your partner’s answers. Do you agree or disagree?’ ‘Can you explain using math language how you knew what symbol to put in the equation?’ ‘Can you prove your equation is accurate with a picture or a model?’”

An example in Grade 4 includes:

• Chapter 6, Lesson 3, Student Edition, students compare the factors and multiples of a number to explore the relationship between them. In-Class Practice Exercise 11 and 12 states, “Complete the Venn diagram.” Students see two Venn diagrams. In Exercise 11, the circles are labeled Factors of 40 and First Six Multiples of 5. In Exercise 12, the circles are labeled Factors of 32 and First Six Multiples of 4. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students build an understanding of the relationship between factors and multiples. Throughout the lesson, encourage students to engage in SMP.6 by using precise math language. As students build understanding of factors and multiples, they should use those terms while communicating with one another. Students also engage in precision when considering how they know they have found all the factors of a number. In-Class Practice Exercise 11 asks students to complete a Venn diagram comparing factors of 40 and multiples of 5. Encourage students to communicate precisely using clear math language with one another and calculate accurately and efficiently. Suggested prompts: ‘Turn and talk to your partner about the Venn diagram. What is a factor and what is a multiple?’ ‘How will you know if you have found all of the factors of 40? Is there anything you know about the number 40 that can help you find the factors?’”

An example in Grade 5 includes:

• Chapter 3, Lesson 1, Student Edition, students solve addition and subtraction problems with decimals in order to complete a puzzle. In-Class Practice Exercise 13 states, “Estimate the sums and differences to answer the question. What did 0 say to 8?” Students see four addition and four subtraction problems with decimals. Examples include: 78.98 - 27.19 and 257.9+106.17. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students will attend to precision (SMP.6) throughout this section. They need to be able to articulate and apply strategies to estimate accurately. Encourage the use of accurate and precise mathematical language recognizing the words roughly, approximately, or about as clues that an estimate is acceptable over an exact answer. Use In-Class Practice Exercise 13 to discuss precision. Suggested prompts: ‘What do you notice about the expressions in Exercise 13? How are they similar? How are they different?’ (they are all decimals; they involve addition and subtraction) ‘What do you notice about the letter values (in purple)?’ (they are whole numbers) ‘Explain how this will impact how you choose to estimate the sum or difference of the decimals.’”

The materials reviewed for Math & YOU Grades 3 through Grade 5 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 3-5 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 Grade 3 includes:

Chapter 7, Lesson 7, Student Edition, students look for friendly numbers and apply strategies to solve addition problems with a max of four numbers. Investigate states, “Reorder the addends so the addition is easier. 8+5+2=, 48+85+52= . Reorder the addends so the addition is easier. 348+285+152= .” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students will add three or more multi-digit numbers by adding like place values. To begin, students review basic addition facts and think about how they can reorder addends to efficiently add three numbers by making a ten. SMP.7 will come into play here and throughout the lesson as students recognize what numbers they can use to make a ten and then apply that to multi-digit numbers as well. In the Investigate, students will reorder addends to efficiently add three numbers. Suggested prompts: ‘Look at the equation in the Look Back. Which addends did you add first to find the sum? Why?’ ‘How is the equation in the Look Back related to the equation in the Look Ahead? Which addends did you add first in the Look Ahead?’ ‘Why can you change the order of the addends and still get the same sum? Which addition property are you using?’”

An example in Grade 4 includes:

• Chapter 5, Lesson 3, Student Edition, students solve division problems by using place value to decompose more complex numbers. In-Class Practice Exercise 4 states, “Find the quotient. 72\div4= .” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Throughout the lesson, students will use the models to see the way they can break numbers into smaller parts so that they can divide. SMP.7 will come into play when using place value blocks and quick sketches helps them see the structure of the numbers and how they can be decomposed to find the quotient. During In-Class Practice Exercise 4, students are asked to find the quotient. Encourage students to use their understanding of place value to decompose the numbers to divide. Suggested prompts: ‘Make a quick sketch of 72. How many tens are there? How many ones?’ ‘How many tens can you put in each group? What could you do with the remaining tens?’ ‘Why was it necessary to trade some tens in for ones? Can you think of another problem that would require you to do this?’”

An example in Grade 5 includes:

• Chapter 9, Lesson 6, Student Edition, students use the structure of the area model to solve multiplication problems involving mixed numbers and fractions. Investigation states, “Look Back, Shade and label the model to represent \frac{3}{2}\times\frac{5}{2}. Then find the product. Look Ahead, Shade and label the model to represent 1\frac{1}{2}\times2\frac{1}{2}. Then find the product.” For both problems, students use an area model to support solving the multiplication problem. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students continue using the structure of area models (SMP.7) to understand aspects of fraction multiplication, this time with mixed numbers. Allow the meaning of the area models to sink in for students to feel confident in the methods for computing the products. Provide sense-making experiences for students to find the structure and patterns as they rewrite mixed numbers as improper fractions to multiply. Use the Investigate to help students use the structure of an area model to grasp the concept of multiplying mixed numbers. Suggested prompts: ‘What do the products in the Look Back and the Look Ahead have in common? What is the same about the area models? What is different?’ ‘How do the models show how multiplying mixed numbers is related to multiplying improper fractions?’”

The materials reviewed for Math & YOU Grades 3 through Grade 5 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 3-5 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 Grade 3 includes:

• Chapter 13, Lesson 5, Student Edition, students compare and calculate the area and perimeter of a rectangle. Investigate states, “Look Ahead, Find the perimeter of each rectangle. Are the perimeters the same? Perimeter of Rectangle 1: ____units, Perimeter of Rectangle 2: ____units. Can you draw a third different rectangle with an area of 18 square units? If so, what is the perimeter?” Teacher Edition, Nick’s Notes-Investigate states, “‘What do you notice about the perimeters of the rectangles?’ they are different ‘As the perimeter increases and the area stays the same, what do you notice about the shape of the rectangle?’ The rectangle becomes longer or narrower, stretched out, and less like a square” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students focus on rectangles with the same area and compare their perimeters. To begin the lesson, students will use toothpicks and square tiles to make rectangles that have the same perimeter. SMP.8 will come into play as you ask students to make observations about the rectangles and use those observations to make predictions and generalizations. In the Investigate, encourage students make predictions and generalizations based on the observations they make. Suggested prompts: ‘What do you notice about the perimeters of the rectangles?’ ‘What generalizations can you make about what happens as the perimeter of a rectangle increases, but the area stays the same?’ (After students draw their third rectangle with an area of 18) ‘Before calculating the perimeter of the rectangle, do you predict that it will be more or less than the perimeter of the first two rectangles? Explain.’”

An example in Grade 4 includes:

• Chapter 6, Lesson 2, Student Edition, students identify multiples of 3 and 9 and analyze patterns to develop divisibility rules. Investigate states, “Look Back, A multiple is the product of a number and any other counting number. Shade all the multiples of 3. Shade all the multiples of 9. Look Ahead, What do you notice about the sum of the digits in each multiple of 3? What do you notice about the sum of the digits in each multiple of 9?” Students see two multiplication tables up to 50 that they will use for shading. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students will use division to find factor pairs. To begin the lesson, SMP.8 comes into play as students notice patterns to help them discover the divisibility rules for 3 and 9. As the lesson progresses, students will be introduced to more divisibility rules. In the Investigate, students will shade multiples of 3 and 9 to develop understanding of the divisibility rules for 3 and 9. Ask students to look for patterns to help them discover the rules. Suggested prompts: ‘Look at the numbers that you shaded in the table. What do you notice? What do you wonder?’ ‘Why do you think there are more multiples of 3 shaded than 9?’ ‘How do you know if a number is a multiple of 3? Can you write a rule for it?’”

An Example in Grade 5 includes:

• Chapter 1, Lesson 2, Student Edition, students use repeated reasoning to write multi-digit numbers in different forms and compare the value of the digits. Investigation states, “Look Back, Use base ten blocks to model 4,442. Draw your model. Then write the values. Look Ahead, Write the value of each digit. Use the pattern to help you write the value of the digits.” In the Look Back, students draw base-ten blocks to represent the number. In the Look Ahead, students use place value to represent the number. Teacher Edition, Nick’s Notes-Investigate states, “In the Look Ahead, how might you describe the 4s digits in the hundreds and tens place by using the relationship between place value positions? Describe the relationship between the 4s in the thousands and tens place positions. Will this strategy work to compare the 4s in the thousands and hundreds place positions? Discuss your thinking with a partner.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students will look for and express regularity in repeated reasoning (SMP.8) as they work to develop generalizations with place value positions to understand the relationship between each place. Allow students to do the work of finding the patterns between the place value positions. Provide ample sense making experiences for students to explore numbers in the place value position. In the Investigate, students will use repeated reasoning. Suggested prompts: ‘How might you describe the 4’s digit in the hundreds and tens place by using the relationship between place value positions? What about the 4’s in the thousands and tens place positions?’ ‘Will this strategy work to compare the 4’s in the thousands and hundreds place position? Discuss.’ ‘How can you prove your strategy? Will it always work?’”