4th Grade - Gateway 2

The materials reviewed for Reveal Math Grade 4 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The materials develop conceptual understanding throughout the grade level, with teacher guidance, through discussion questions and conceptual problems with low computational difficulty. Examples include:

• In Lesson 5-2, Understand Prime and Composite Numbers, Explore and Develop, Activity- Based Exploration, “Prepare index cards with the numbers 11, 14, 16,19, and 21. Have students work in pairs or small groups, and distribute a set of cards to each group. Have student-groups find and record all the factor pairs of each number using a method of their choice. ‘How did you know if you found all the factor pairs? What do you notice about the number of factor pairs each number has? Are some numbers easier to find all the factor pairs for than others? Why do you think that is?’” This activity supports conceptual understanding of 4.OA.4, find all factor pairs for a whole number in the range 1-100.

• In Lesson 8-2, Generate Equivalent Fractions using Models, Activity-Based Exploration, “students use fraction models to explore the relationship between the numerators and the denominators of equivalent fractions. Students work with a partner using manipulatives of their choice to represent and complete the equations.” This helps build conceptual understanding of the standard of 4.NF.A, extend understanding of fraction equivalence and ordering.

• In Lesson 10-5, Subtract Mixed Numbers, Pose the Problem, “What operation can you use to find a solution? What are some strategies and representations you have used before that could help solve the problem?“ These questions help develop conceptual understanding of standard 4.NF.3, understand a fraction a/b with a > 1 as a sum of fractions 1/b.

The materials provide opportunities for students to independently demonstrate conceptual understanding through concrete, semi-concrete, verbal, and written representations. Examples include:

• In Lesson 3-5, Strategies to Subtract Multi-Digit Numbers, Own My Own, Exercise 11, “What two different strategies can you use to solve the equation? How are the two strategies similar? How are they different? 15,736 - 10,302 = ?” This exercise provides an opportunity for students to independently demonstrate conceptual understanding of 4.NBT.4, fluently add and subtract multi-digit whole numbers using the standard algorithm. 

• In Lesson 6-7, Multiply Two 2-digit Factors, Differentiate, Build Proficiency, Student Practice Book, Item 1a, “Solve using an area model. 12 x 18 = ___.” This activity provides an opportunity for students to independently demonstrate conceptual understanding of 4.NBT.5, multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations.

• In Lesson 9-5, Subtract Fractions with Like Denominators, On My Own, students use a number line to solve, “Henry’s home is \frac{7}{8} mile from school. He stops at the library on his way home. The library is \frac{4}{8} mile from the school. How much farther does Henry need to travel to get home? Use the number line to find the difference.” This activity provides an opportunity for students to independently demonstrate conceptual understanding of 4.NF.3d, solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.

The materials reviewed for Reveal Math Grade 4 meet expectations that the materials develop procedural skills and fluency throughout the grade level. The materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. 

The materials develop procedural skills and fluency throughout the grade with teacher guidance, within standards and clusters that specifically relate to procedural skill and fluency, and build fluency from conceptual understanding. Examples include:

• Fluency Practice exercises are provided at the end of each unit. Each Fluency Practice includes Fluency Strategy, Fluency Flash, Fluency Check, and Fluency Talk. “Fluency practice helps students develop procedural fluency, that is, the ‘ability to apply procedures accurately, efficiently, and flexibly.’ Because there is no expectation of speed, students should not be timed when completing the practice activity.” Fluency Practice exercises in Grade 4 progress toward 4.OA, use the four operations with whole numbers to solve problems, and 4.NBT.4, fluently add and subtract multi-digit whole numbers using the standard algorithm.

• In Lesson 3-3, Understand an Addition Algorithm, Activity-Based Exploration, students work in pairs or small groups to solve, “the addition problem _,___ +  _,___ = 3,978 written vertically” develops procedural skill and  fluency of 4.NBT.4, fluently add multi-digit whole numbers using the standard algorithm.

• In Unit 7, Division Strategies with Multi-Digit Dividends and 1-Digit Divisors, Fluently Practice, Fluency Talk, “How would you add two numbers that have a different number of digits?” Students use place value strategies along with the addition algorithm to solve the problem. 4.NBT.4, fluently add and subtract multi-digit whole numbers using the standard algorithm.

• In Lesson 10-5, Subtract Mixed Numbers, Number Routine: Find the Missing Values “Students build number sense by using solved equations to find unknown values in related division equations. These prompts encourage students to talk about their reasoning: What patterns do you notice in the quotients? What do you notice about the number of zeros in the quotients? How does the number of zeros in the products compare with the number of zeros in the dividend.” Students build fluency by using equations to find unknown values, 4.NBT.6, find whole number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value...

The materials provide opportunities for students to independently demonstrate procedural skill and fluency. Examples include:

• In Lesson 3-4, Understand an Addition Algorithm Involving Regrouping, Own my Own, Exercise 6, (written vertically) “12,058 + 4,867 = ?” Students have an opportunity to independently demonstrate procedural skill and fluency of 4.NBT.4, fluently add and subtract multi-digit whole numbers using the standard algorithm.

• In Lesson 3-5, Strategies to Subtract Multi-Digit Numbers, Assess, Exit Ticket, students find the difference between two 4-digit numbers using the standard algorithm, “What is the difference? 17,392 - 5,261 = ___” Students have an opportunity to independently demonstrate procedural skill and fluency of  4.NBT.4, fluently add and subtract multi-digit whole numbers using the standard algorithm.

• In Lesson 6-6, Multiply Two Multiples of 10, Own my Own, Exercise 7, “20 x 90 = ?” Students have an opportunity to independently demonstrate procedural skill and fluency of  4.NBT.5, multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations.

• Units 7-14 all have a Fluency Practice section. For example, Unit 10, Fluency Practice, Exercise 12, “Find the sum or difference 354 - 287.” This provides students a chance to independently demonstrate 4.NBT.4, fluently add and subtract multi-digit whole numbers using the standard algorithm.

The materials reviewed for Reveal Math Grade 4 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Additionally, the materials provide students with the opportunity to independently demonstrate multiple routine and non-routine applications of the mathematics throughout the grade level. 

The materials provide specific opportunities within each unit for students to engage with both routine and non-routine application problems. In the Digital Teacher Center, Program Overview: Learning & Support Resources, Implementation Guide, Focus, Coherence, Rigor, Application, “Students encounter real-world problems throughout each lesson. The On My Own exercises include rich, application-based question types, such as ‘Find the Error’ and ‘Extend Thinking.’ Daily differentiation provides opportunities for application through the Application Station Cards, STEM Adventures, and WebSketch Explorations. The unit performance task found in the Student Edition offers another opportunity for students to solve non-routine application problems.” 

The materials develop application throughout the grade as students solve routine problems in a variety of contexts, and model the contexts mathematically within standards and clusters that specifically relate to application, both dependently and independently. Examples include:

• In Lesson 3-4, Solve Comparison Problems Using Division, Explore & Develop, Learn section, “An apple costs 35 cents. A banana costs 12 cents. How many times as much does an apple cost compared to a banana? Use a bar diagram and an equation to represent and solve the problem.” This exercise allows students to develop and apply mathematics of 4.OA.1, interpret a multiplication equation as a comparison.

• In Lesson 4-4, Solve Comparison Problems Using Division, Practice & Reflect, Problem 11, “Cory learned that the airport is 5 times farther from his home than the library. He knows the airport is 30 miles from home. What is the distance from Cory’s home to the library?” This exercise allows students to develop and apply mathematics of 4.OA.2, multiply or divide to solve word problems involving multiplicative comparison.

• In Lesson 6-8, Solve Multi-Step Problems Involving Multiplication, Differentiate, Build Proficiency, Student Practice Book, Item 3, “Omar has $150 to buy new netting for the soccer nets. He buys 9 yards of netting. Each yard of netting costs $14. How much money does he have left after buying the netting?” This exercise allows students to develop and apply mathematics of 4.OA.3, solve multistep word problems with whole numbers and having whole-number answers using the four operations....

The materials develop application throughout the grade as students solve non-routine problems in a variety of contexts, and model the contexts mathematically within standards and clusters that specifically relate to application, both dependently and independently. Examples include:

• In Lesson 4-3, Solve Comparison Problems Using Multiplication, Launch, Numberless Word Problem, “What math do you see in this problem? Penelope and Madison are pitching at a softball tournament. Penelope strikes out 3 times as many batters as Madison. How many batters could Penelope have struck out?” This exercise allows students to develop and apply mathematics of, 4.OA.2, multiply or divide to solve word problems using multiplicative comparison.

• In Lesson 6-8, Solve Multi-Step Problems Involving Multiplication, Extend Thinking, Exercise 1, “Given each problem, fill in two 2-digit numbers and then solve. Show you work. Students and school staff purchase supplies from the school bookstore. 1. Marcus buys packages of pencils and packages of pens. Each package of pencils contains ___ pencils. Each package of pens contains ___ pens. Marcus buys ____ packages of pencils and ____ packages of pens. How many total pencils and pens does Marucs buy?” This exercise allows students to develop and apply mathematics of 4.OA.3, solve multi-step word problems posed with whole numbers and having whole-number answers using the four operations 

• Online Resource, Lesson 13-45, Solve Problems That Involve Units of Measure, Extend Thinking, Application Station, A-Maze-ing Progress, “A computer programmer wrote the following code to get a character through a maze: You are going to draw the maze based on the code. The whole maze is contained within a rectangular grid. Each square block of the grid is one square foot. Once your maze is complete, answer the following questions: What is the perimeter of your maze in feet? What is the perimeter of your maze in yards? What is the area of your maze in square feet? What is the area of your maze in square yards? 1. How did you determine the perimeter in feet and in yards? 2. How did you determine the area in square feet and in square yards?” This exercise allows students to develop and apply mathematics of 4.MD.2, use the four operations to solve word problems involving distances, intervals of time.

The materials reviewed for Reveal Math Grade 4 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. Additionally, multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout each grade level. 

All three aspects of rigor (conceptual understanding, procedural skill & fluency, and application) are present independently throughout the grade level. Examples include:

• In Lesson 2-2, Read and Write Numbers to One Million, Activity-Based Exploration, “Students explore how place-value structure can help them read and write multi-digit numbers. Prepare sets of index cards with the following cards in each set: 1, 2, 3, 4, 5, 8, four, eight, fifteen, thirty-two, hundred, hundred, thousand, as well as two cards with commas. Distribute place-value charts and a set of index cards to each student-group. Student groups will use all index cards to represent a 6-digit number in standard form and word form. Have students rearrange the cards to create at least one more 6-digit number with the cards.” This exercise provides the opportunity for students to extend their conceptual understanding of place value as they read and write multi-digit wheel numbers.

• In Lesson 7-6, Understand Remainders, On My Own, Problems 3-7, students develop procedural skill and fluency with identifying quotients and remainders. For example, Problem 3, “929 ÷ 3 = ___.” Problem 5, “3,225 ÷ 8 = ___.” Problem 7, “8,437 ÷ 7 = ___.”

• In Lesson 13-4, Convert Units of Time, On My Own, Problem 7, students apply their understanding of converting larger units of time to smaller units of time to solve problems. “Salma volunteered for 4 hours last weekend. How many minutes did Salma volunteer?”

The materials provide a balance of the three aspects of rigor as multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the grade level. Examples include:

• In Lesson 6-6, Multiply Two Multiples of 10, Additional Practice, Exercise 5, students extend their conceptual understanding of place value and properties of operations to build procedural skill and fluency to find products of two multiples of 10. “A food bank collected 50 food items every day for 30 days. How many total food items did they collect? Show and explain two ways to solve the problem.” 

• In Lesson 9-6, Solve Problems Involving Fractions, On My Own, Problem 8, students extend their conceptual understanding of adding and subtracting fractions with like denominators to solve real-world problems. “Santosh walked \frac{9}{10} mile. He realized he dropped his scarf, so he walked back \frac{3}{10} mile. Then he walked another \frac{5}{10} mile. How far is Santosh from where he started?

• In Lesson 11-2, Understand Multiplying a Fraction by a Whole Number, On My Own, Problem 10, students develop conceptual understanding and fluency with multiplying a fraction by a whole number and apply this understanding to solve real world word problems. “Each leg of a relay race is \frac{3}{4} mile. There are 4 legs. How many miles is the relay race? Justify your answer.”

The materials reviewed for Reveal Math Grade 4 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, in both sections, the mathematical practice is labeled as MPP Reason abstractly and quantitatively, rather than MP1 or MP 2. Within each of the lesson components, the mathematical practices are not labeled or identified, leaving where they are specifically addressed up for interpretation and possible misidentification.

The materials provide intentional development of MP1: Make sense of problems and persevere in solving them, in connection to grade-level content. Examples include: 

• In Lesson 5-1, Understand Factors of a Number, Reinforce Understanding, Exercise 3, “Draw a model to find all the possible arrangements. A bakery has 36 pastries to arrange on its shelves. How can the pastries be arranged if the owner wishes to put an equal number of pastries on each shelf and arrange them on 2 to 6 shelves?” This exercise is an intentional development of MP1, make sense of problems and persevere in solving them, as students draw models to solve an open-ended problem. 

• In Teacher’s Guide, Lesson 6-4, Multiply 2-Digit by 1-Digit Factors, Guided Exploration, “Students use an area model to represent decomposing a 2-digit factor by place value and finding partial products to multiply.” Additionally, students answer: “How can estimation help you solve the problem? What multiplication equation can you write? How did you determine how to decompose the factor? Which decompositions are the most helpful for 2-digit factors?” Students engage with MP1 as they use a variety of strategies to make sense of the problem.

• In Lesson 7-8, Solve Multi-Step Problems using Division, Launch, Numberless Word Problem, Be Curious, “Kim is making bouquets. There will be some roses in each bouquet. She had some roses. She gave some roses to her mother.” Students engage with MP1 as they work to understand the information presented in a numberless problem, and use a variety of strategies to solve the problem.

The materials provide intentional development of MP2: Reason abstractly and quantitatively, in connection to grade-level content. Examples include:

• In Unit 3-3, Understand an Addition Algorithm, Practice and Reflect, On My Own, Exercise 10, “A band played two concerts with a total attendance of 9,698 people. The first concert had 4,467 people in attendance. How many people attended the second concert? Write an addition equation to solve.” Students engage with MP2 as they understand the relationship between problem scenarios and mathematical representations.

• In Unit 6, Multiplication Strategies with Multi-Digit Numbers, Math Probe, Estimate Products, Exercise 1, “Use reasoning to choose the closest estimate of the product. Circle the best estimate: 23 x 48 a. 80, b. 800, c. 100, d. 1,000, e. 20, f. 200 Explain your reasoning.” Students engage with MP2 as they consider units involved in a problem and attend to the meaning of quantities. 

• In Lesson 7-2, Estimate Quotients, Guided Exploration, “Students use their understanding of compatible numbers to estimate quotients and to estimate a range for quotients of 3- or 4- digit dividends and 1-digit divisors.” Additionally, students answer, “How do you know if an estimate will be greater or less than the exact quotient? How is finding a compatible number to use to estimate a quotient different from finding a compatible number to estimate a product?” Students engage in MP2 as they understand the relationship between compatible numbers and estimates.

The materials reviewed for Reveal Math Grade 4 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, in both of these sections, the mathematical practice is labeled MPP: Construct viable arguments and critique the reasoning of others, rather than MP3 Construct viable arguments and critique the reasoning of others. Additionally, the math practices are not identified within the lesson sections, therefore leaving where they are specifically addressed up for interpretation and possible misidentification.

Examples of intentional development of students constructing viable arguments in connection to grade-level content, including guidance for teachers to engage students in MP3 include:

• In Lesson 2-1, Understand the Structure of Multi-Digit Numbers, Explore & Develop, Develop the Math, Guided Exploration, Learn, Work Together, students construct viable arguments as they recognize the relationships in a multi-digit whole number. “How can you describe the relationship between the values of the digits 3 in this number? 3,830 Explain.”

• In Unit 4, Multiplication as Comparison, Math Probe, Exercise 3, students construct viable arguments as they multiply or divide to solve word problems involving multiplicative comparison. “The 4th-grade class is selling hats and shirts to raise money for a new fish tank. So far, they have sold 27 hats and 9 shirts. How many times as many hats as shirts have they sold? Choose all that apply. 27 x 9 = ?, 9 x ? = 27, 27 ÷ 9 = ?, 9 + ? = 27. Explain why you chose the equation or equations.”

• In Lesson 10-3, Add Mixed Numbers, Own My Own, Exercise 12, students construct a viable argument to justify their thinking. “Extend your Thinking, What are possible missing numbers? Justify your answer.  1\frac{൞}{5} + 2\frac{൞}{5}= ൞\frac{1}{5}”

Examples of intentional development of students critiquing the reasoning of others in connection to grade-level content, including guidance for teachers to engage students in MP3 include:

• In Lesson 5-2, Understand Prime and Composite Numbers, Own My Own, Exercise 13, students critique the reasoning of others. “Scott says he can arrange 71 marbles into equal groups in more than 2 ways. Do you agree with Scott? Explain your reasoning.”

• In Lesson 6-2, Estimate Products, Differentiation, Additional Practice, Exercise 6, students critique the reasoning of others. “A store sells 2,875 pounds of fruit each month. The store owner estimates that 8,000 pounds of fruit will sell in 4 months. Is the store owner’s estimate reasonable? Explain.”

In Lesson 9-2, Represent Adding Fractions, Explore & Develop, Develop the Math, Guided Exploration, Learn, Work Together, students solve word problems involving addition and subtraction of fractions to critique the reasoning of others. “Macie says if she combines the juice into one bottle, she will have a total of \frac{1}{2} bottle of juice. How can you respond to Macie? Explain.”

The materials reviewed for Reveal Math Grade 4 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, in both sections, the mathematical practice is MPP Model with mathematics, rather than MP4. Additionally, the math practices are not identified within the lesson sections, therefore leaving where they are specifically addressed up for interpretation and possible misidentification. 

Examples of intentional development of students modeling with mathematics in connection to grade-level content, including guidance for teachers to engage students in MP4 include:

• In Lesson 3-9, Solve Multi-Step Problems Involving Addition and Subtraction, Differentiate, Reinforce Understanding, Exercise 1, “Solve. Show your work. The carnival sells tickets each day. Based on the data in the table, how many more tickets did they sell on Saturday and Sunday than on Wednesday, Thursday, and Friday?” Students engage with MP4 as they use the math they know to solve problems and everyday situations.

• In Lesson 7-3, Equal Shares, Guided Exploration, the teacher asks, “How could you distribute the counters so there is the same number in each group? How does your representation connect to the division equation? How does it connect to the multiplication equation? What real-world experience can you relate this situation to? How do you know your answer makes sense?” Students engage with MP4 as they describe models and how they relate to the problem situations, and check to see if their answers make sense.

• In Lesson 10-2, Represent Adding Mixed Numbers, Guided Exploration, Math is...Modeling, students answer, “How do representations help you to understand how to add mixed numbers?” “Students use the representation to make sense of a strategy they can use to add mixed numbers.”

Examples of intentional development of students using appropriate tools strategically in connection to grade-level content, including guidance for teachers to engage students in MP5 include:

• In Lesson 4-4, Solve Comparison Problems Using Division, Activity-Based Exploration, “Write a division equation with a symbol for the unknown on each index card. Prepare another set of cards with word problems involving multiplicative comparison based on the equations you wrote. Provide sets of index cards to each group of students and ask them to match the word problem to a division equation. Then ask them to exchange one matched set word problem and division equation with another pair of students and ask them to solve it. Student pairs represent each word problem by using bar diagrams, counters and related multiplication equations to solve the problems.” Students engage in MP5 as they justify which tools/strategies they chose to represent their work. 

• In Lesson 5-6, Analyze Features of a Pattern, Guided Exploration, students answer “Why is a table a useful tool to determine patterns?” “Students explain that organizing numerical information in a table helps to analyze the relationships between the numbers and identify the features of the pattern.”

• In Unit 10, Addition and Subtraction Strategies with Mixed Numbers, Performance Task, Part B, “The clinic gives each of their large breed dogs 3\frac{1}{4} cups of food each day. They feed the dogs two times each day. The dogs are given less than 2 cups of food with each feeding. How much food does the clinic give to the dogs at each feeding? Use a representation to justify your answer.” Students engage in MP5 as they select a strategy/representation to solve a real-world problem.

The materials reviewed for Reveal Math Grade 4 meet expectations that there is intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics for students in connection to the grade-level content standards, as expected by the mathematical practice standards.  

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, the mathematical practice is labeled MPP Attend to precision, rather than MP6: Attend to precision. Additionally, the math practices are not identified within the lesson sections, therefore leaving where they are specifically addressed up for interpretation and possible misidentification.

The instructional materials address MP6 in the following components:

• In the Digital Teacher Center, Program Overview: Learning & Support Resources, Implementation Guide, Language of Math, Unit-level Features, “The Language of Math feature highlights math terms that students will use during the unit. New terms are highlighted in yellow. Terms that have a math meaning different from everyday means are also explained.” Math Language Development, “This feature targets one of four language skills - reading, writing, listening, speaking - and offers suggestions for helping students build proficiency with these skills in the math classroom.” Lesson Level Features, “The Language of Math feature promotes the development of key vocabulary terms that support how we talk about and think about math in the context of the lesson content.” Each Unit Review also includes a vocabulary review component which references specific lessons within the unit.

Examples of intentional development of MP6: attend to precision, in connection to the grade-level content standards, as expected by the mathematical practice standards, including guidance for teachers to engage students in MP6 include:

• In Unit 6, Multiplication Strategies with Multi-Digit Numbers, Unit Review, Review, Exercise 16, “Mandy receives 1,375 points for each level completed in an online math game. She completes 6 levels. How many points does she receive? Use partial products to solve.” Students attend to precision as they multiply whole numbers.

• In Lesson 8-3, Generate Equivalent Fractions using Number Lines, Explore & Develop, Bring it Together, Language of Math, “the word interval is a noun meaning a space between two things or a gap. Discuss with students the usage of the word with regard to a number line. Have students use the word in a sentence and then share.” Students attend to precision as they create a number line with accurate intervals. 

• In Lesson 13-11, Solve Problems Involving Data on a Line Plot, Launch, Notice & Wonder, Be Curious, “What do you see? How is the data on one line plot similar to the data on the other line plot? How is it different? Why do you think there are more Xs in the top line plot than the bottom line plot? What math operations could be performed using the data?” Students attend to precision as they express the similarities and differences about data seen on two line plots.

Examples of where the instructional materials attend to the specialized language of mathematics, including guidance for teachers to engage students in MP6 include:

• In Lesson 3-3, Understand an Addition Algorithm, Activity-Based Exploration, “How are the values of the digits in the addends related to the value of the digits in the sum? How can this relationship help you add multi-digit numbers efficiently?” Students attend to the specialized language of mathematics as they learn the difference between horizontal and vertical.

• In Lesson 4-3, Solve Comparison Problems Using Multiplication, Language of Math, “Students need multiple opportunities to practice the language of multiplicative comparison. Emphasize the words as many times as, time as much as, times as long as signaling a comparison.” In the Guided Exploration, “Have students consider other words, such as times less than that could be used to describe the relationship between the number of batters struck out by Penelope and Madison.”

• In Lesson 5-2, Understand Prime and Composite Numbers, Own My Own, Exercise 4, “Is the number prime or composite? Explain your reasoning. 31” Language of Math, “To help students remember the meanings of prime number and composite number, point out the word composite has more letters than prime, and there are more factor pairs in a composite number than a prime number. Have students discuss other possible ways to remember definitions.”

The materials reviewed for Reveal Math Grade 4 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, the mathematical practice is labeled MPP Look for and make use of structure, rather than MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning. Additionally, the math practices are not identified within the lesson sections, therefore leaving where they are specifically addressed up for interpretation and possible misidentification. 

Examples of intentional development of students looking for and making use of structure, to meet its full intent in connection to grade-level content, including guidance for teachers to engage students in MP7 include:

• In Lesson 2-1, Understand the Structure of Multi-Digit Numbers, Exercise 3, “How can you describe the relationship between the values of the underlined digits? 258 and 2,180.” Students engage with MP7 as they use structure to describe the relationship between digits and their place value position. 

• In Lesson 5-2, Understand Prime and Composite Numbers, Explore & Develop, Develop the Math, Guided Exploration, “Have students work in a small group to discuss how to arrange the 12 basketballs and then the 17 soccer balls. They can use counters to model the different arrangements. Have them share how they used the arrays to identify the factor pairs. Think about it: How does building rectangular arrays help you identify factor pairs? How do you know if you have created all possible arrays? What do you notice about the way the soccer balls can be arranged? Facilitate Meaningful Discourse: How can you use the term factor pair to explain why a number is prime or composite? How can arranging items in equal groups help you understand whether a number is prime or composite? Do you need to list all the factor pairs of a number to determine if it is prime or composite? Explain. Are all odd numbers prime? Explain why or why not? Have students work in small groups to make a list of prime numbers between 1 and 100. Students can make a list of prime numbers between 1 and 100. Students can make use of manipulatives, equations, and a list of other factor pairs? What patterns do you notice in prime numbers?” Students engage with MP7 as they look for and explain the structure within mathematical representations.

• In Lesson 13-1, Relate Metric Units, Explore & Develop, Learn, Work Together, “Mr. Decker needs 7 liters of paint for his classroom art project. How many milliliters of paint does he need?” Students engage with MP7 as they “explain how patterns in the structure of the metric system help them convert from a greater unit to a smaller unit.”

Examples of intentional development of students looking for and expressing regularity in repeated reasoning, including guidance for teachers to engage students in MP8 include:

• In Unit 3, Addition and Subtraction Strategies and Algorithms, Unit Review, Exercise 8, “An office building sells for $350,000. A house nearby sells for $245,000. How much more money does the office building sell for?” Students engage in MP8 as they move from repeated subtraction calculations of place value decomposition to the standard algorithm.

• In Lesson 13-7, Solving Problems Using a Perimeter Formula, Explore & Develop, Work Together, “Keira uses 48 yards of fencing to enclose a flower garden that has a length of 12 yards. What is the width of the flower garden? Write an equation to show your work.” Students engage with MP8 as they create, describe, explain a general formula, process, method, algorithm, model, etc. to solve the word problem.

• In Lesson 14-8, Classify Triangles, On My Own, Exercise 9, “A triangle has an angle equal to 140°. How can you classify this triangle? Why can you only classify it by its angle?” Students engage in MP8, by writing a generalization about classification of shapes.