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.

The materials reviewed for Reveal Math Grade 4 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include:

• The Implementation Guide provides a program guide, which includes a program overview, the program components, unit features, instructional model, lesson walk-through, and a brief description of the different unit components, such as Math is…, focus, coherence, rigor, and language of math.  

• The Implementation Guide provides pacing for each unit; mapping out the lessons in each unit and how many days the unit will take.

• The Unit Planner contains an overview of the Lessons within the unit, Math Objective, Language Objective, Key Vocabulary, Materials to Gather, Rigor Focus, and Standard.

• The Unit Overview provides a description for teachers as to how the unit connects to Focus, Coherence, and Rigor. 

• Within each lesson, the Language of Math section, provides teachers with specific information about the vocabulary used in lessons and how to utilize vocabulary cards to enhance learning experiences. 

• In Unit 5, Numbers and Number Patterns, Unit Overview, Effective Teaching Practices, Facilitate Meaningful Math Discourse, “Student discourse is an interactive process of collaborative exploration, exchange of ideas, and building of shared understanding. The teacher is a facilitator and performs the following actions: 

  • Engages students as they explore and share ideas and strategies.

  • Observes what students are doing and saying and forms a plan for the whole-class discussion.

  • Creates an environment in which students take ownership.

  The student is leader, partner, problem-solver, and communicator, and performs the following actions:

  • Presents and explains ideas, strategies, and reasoning to peers.

  • Listens respectfully to peers and engages them in constructive argument.

  • Compares approaches and solutions with peers for shared understanding.”

• In Unit 7, Division Strategies with Multi-Digit Dividends and 1-Digit Divisors, Unit Overview, Math Practices and Processes, Reason Abstractly and Quantitatively, “During the course of a lesson, the teacher should encourage students to reason abstractly and quantitatively when approaching the mathematics. A goal in teaching is to enable students to make sense of the problems and mathematics by themselves. For example, students may reason abstractly when using an area model to find partial quotients.” 

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. The materials provide information about planning instruction, and give suggestions for presenting instructional strategies and content, as well as mathematical practices. Examples include:

• In Lesson 5-2, Understand Prime and Composite Numbers, Notice & Wonder, Teaching Tip, “Have students work in pairs to discuss objects in groups. Have them think of instances when they may have put real-life objects or manipulatives in equal groups, and what those representations may have looked like.”

• In Lesson 5-6, Analyze Features of a Pattern, Notice & Wonder, Teaching Tip, “Have student pairs take turns stating similarities and differences between the two sequences. Then compile a list of the similarities and differences from the class. This encourages participation on the part of all students as well as encouraging them to state an answer without consideration of whether it is the “correct answer.”

The materials reviewed for Reveal Math Grade 4 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

Correlation information is present for the mathematics standards addressed throughout the grade level. Examples of how individual units, lessons, or activities throughout the series are correlated to the CCSSM include:

• In the Digital Teacher Center, Program Overview: Learning & Support Resources, Implementation Guide, Correlations, identifies the standards included in each lesson. This guide also indicates whether the standards are considered major, supporting, or additional standards. 

• Each Unit Planner includes a pacing guide identifying the standards that will be addressed in each lesson.

• In Lesson 6-4, Multiply 2-Digit by 1-Digit Factors, the materials identify focus standard 4.NBT.5, multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers. The lesson also identifies MPs 1 and 4. 

• In Lesson 11-5, Solve Problems Involving Fractions and Mixed Numbers, the materials identify focus standard 4.NF.4c, solve word problems involving multiplication of a fraction by a whole number. The lesson also identifies MPs 1 and 8. 

Explanations of the role of the specific grade-level mathematics are present in the context of the series, and teacher materials provide information to allow for coherence across multiple course levels. This allows the teacher to make prior connections and teach for connections to future content. Examples include:

• The Unit Overview includes the section, Coherence, identifying What Students Have Learned, What Students Are Learning, What Students Will Learn. In Unit 2, Generalize Place-Value Structure, What Students Have Learned, “Students learned the place value of 3-digit numbers in different forms. (Grade 2), Students have had experiences comparing quantities and numbers that represent quantities since Kindergarten. Students compared two 3-digit numbers. (Grade 3), Students rounded whole numbers to the nearest ten and hundred. (Grade 3)” In What Students Are Learning, “Students generalize the base-ten place-value structure, and explain that a digit in one place represents ten times the value of the digit in the place to its right., Students represent multi-digit whole numbers using forms such as standard, expanded, and word forms., Students compare two multi-digit numbers using place value and round multi-digit numbers to an appropriate place for the given estimation need.” In What Students Will Learn, “Students expand their understanding of place value as they explore relationships between digits in decimal fractions. (Grade 5), Students represent decimals to thousandths in different forms. (Grade 5), Students apply their understanding of place value as they compare two decimals written in decimal notation, and round decimals to an appropriate place for the given estimation need. (Grade 5)”

• Each lesson begins by listing the standards covered within the lesson, indicates whether the standard is a major, supporting or additional standard and identifies the Standards for Mathematical Practice for the lesson. Each lesson overview contains a coherence section that provides connections to prior and future work. In Lesson 9-3, Add Fractions with Like Denominators, Coherence, Previous, “Students learned that fractions are numbers made up of equal parts of a whole (Grade 3)., Students composed and decomposed fractions (Unit 8).” Now, “Students model addition of fractions with common denominators, students add fractions with like denominators.” Next, “Students subtract fractions with like denominators (unit 9)., Students add and subtract fractions with unlike denominators (Grade 5).”

The materials reviewed for Reveal Math Grade 4 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

The materials explain the instructional approaches of the program. Examples include:

• Digital Teacher Center, Program Overview: Learning & Support Resources, Teacher Welcome Letter Template specifies “Reveal Math, a balanced elementary math program, develops the problem solvers of tomorrow by incorporating both inquiry-focused and teacher-guided instructional strategies within each lesson.” 

• Teacher Guide, Volume 1, Welcome to Reveal Math, the overall organization of the math curriculum has five goals:

  • “The lesson model offers two instructional options for each lesson: a guided exploration that is teacher-guided and an activity-based exploration that has students exploring concepts through small group activities and drawing generalizations and understanding from the activities.

  • The lesson model incorporates an initial sense-making activity that builds students’ proficiency with problem solving. By focusing systematically on sense-making, students develop and refine not just their observation and questioning skills, but the foundation for mathematical modeling.

  • Both instructional options focus on fostering mathematical language and rich mathematical discourse by including probing questions and prompts.

  • The Math is… unit builds student agency for mathematics. Students consider their strengths in mathematics, the thinking habits of proficient “doers of mathematics,” and the classroom norms that are important to a productive learning environment.

  • The scope and sequence reflects the learning progressions recommended by leading mathematicians and mathematics educators. It emphasizes developing deep understanding of the grade-level concepts and fluency with skills, while also providing rich opportunities to apply concepts to solve problems.”

The Implementation Guide, located in the Digital Teacher Center, further explains the instructional approaches of specific components of the program. Examples include:

• Unit Features, Unit Planner, “Provides at-a-glance information to help teachers prepare for the unit. Includes pacing: content, language, and SEL objectives; key vocabulary including math and academic terms; materials to gather; rigor focus; and standard (s).”

• Unit Features, Readiness Diagnostic, “Offers teachers a unit diagnostic that can be administered in print or in digital. The digital assessment is auto-scored. Assesses prerequisite skills that students need to be successful with unit content. Item analysis lists DOK level, skill focus, and standard of each item. Item analysis also lists intervention lessons that teachers can assign to students or use in small group instruction.”

• Unit Features, Spark Student Curiosity Through Ignite! Activities, “Each unit opens with an Ignite! Activity, an interesting problem or puzzle that: Sparks students’ interest and curiosity, Provides only enough information to open up students’ thinking, and Motivates them to persevere through challenges involved in problem solving.”

• Instructional Model, “Reveal Math’s lesson model keeps sense-making and exploration at the heart of learning. Every lesson provides two instructional options to develop the math content and tailor the lesson to the needs and structures of the classroom.” Each lesson follows the same structure of a “Launch, Explore & Develop, Practice & Reflect, Assess and Differentiate.” Each of these sections is further explained in the instruction manual.

• Number Routines, in each lesson there is a highlighted number routine for teachers to engage students with. These routines “are designed to build students’ proficiency with number and number sense. They promote an efficient and flexible application of strategies to solve unknown problems…”

The Implementation Guide, located in the Digital Teacher Center, discusses some of the research based features of the program. Examples include: 

• Implementation Guide, Effective Mathematical Teaching Practices, “Reveal Math’s instructional design integrates the Effective Mathematics Teaching Practices from the National Council of Teachers of Mathematics (NCTM). These research-based teaching practices were first presented and described in NCTM’s 2014 work Principles to Action: Ensuring Mathematical Success for All.”

• Implementation Guide, Social and Emotional Learning, “In addition to academic skills, schools are also a primary place for students to build social skills. When students learn to manage their emotions and behaviors and to interact productively with classmates, they are more likely to achieve academic success Research has shown that a focus on helping students develop social and emotional skills improves not just academic achievement, but students’ attitudes toward school and prosocial behaviors (Durlak et al., 2011)...”

• Implementation Guide, Support for English Learners, Lesson-level support, English Learner Scaffolds, each lesson has an “English Learner Scaffolds” section to support teachers with “scaffolded instruction to help students make meaning of math vocabulary, ideas, and concepts in context. The three levels of scaffolding within each lesson - Entering/Emerging, Developing/Expanding, and Bridging/Reaching are based on the 5 proficiency levels of the WIDA English Language Development Standards.”

• Implementation Guide, Math Language Routines, throughout the materials certain language routines are highlighted for teachers to encourage during a lesson, these routines were developed by a team of authors at Center for Assessment, Learning and Equity at Standard University and are “based on principles for the design of mathematics curricula that promote both content and language.” In the implementation guide, the material lists all eight Math Language routines and their purposes, “MLR1: Stronger and Clearer Each Time - Students revise and refine their ideas as well as their verbal or written outputs.”

• Implementation Guide, Math Probe - Formative Assessment, each unit contains a Math Probe written by Cheryl Tobey. Math Probes take time to discover what misconceptions might still exist for students. Designed to ACT, “The teacher support materials that accompany the Math Probes are designed around an ACT cycle - Analyze the Probe, Collect and Assess Student Work, and Take Action. The ACT cycle was originally developed during the creation of a set of math probes and teacher resources for a Mathematics and science Partnership Project.”

The materials reviewed for Reveal Math Grade 4 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

The Digital Teacher Center, Program Resources: Course Materials, Planning Resources, Materials List: Grade 4 specifies the comprehensive materials list. The document specifies classroom materials (e.g., markers, grid paper, dominoes, construction paper, etc.), materials from a manipulative kit (e.g., geoboards, base-ten blocks, transparent spinner, etc.), non-consumable teaching resources (e.g., number cards 1-10, place-value chart to millions, 10x10 grids, etc.), and consumable teaching resources (e.g., activity cards, problem solving tool, alphabet letters, etc.).

In the Teacher Edition, each Unit Planner page lists materials needed for each lesson in the unit, for example, Unit 10, Addition and Subtraction Strategies with Mixed Numbers, Materials to Gather:

• “Lesson 10-1 - fraction tiles, transparent spinner

• Lesson 10-2 - Blank Number Lines 2 Teaching Resource, fraction circles, fraction tiles, index cards, paper strips

• Lesson 10-3 - fraction circles, fraction tiles, number cubes, paper strips

• Lesson 10-4 - Blank Number Lines 2 Teaching Resource, fraction circles, fraction tiles, index cards, paper strips

• Lesson 10-5 - Blank Number Lines 2 Teaching Resource, fraction circles, fraction tiles, number cubes, paper strips

• Lesson 10-6 - index cards, Problem Solving Tool Teaching Resource.”

At the beginning of each lesson in the “Materials” section, a list of materials needed for each part of the lesson is provided: 

• In Lesson 2-4, Round Multi-Digit Numbers, Materials, “The materials may be for any part of the lesson: Activity Cards Teaching Resource, Blank Number Lines Teaching Resource, Place-Value Chart to Millions Teaching Resource.”

• In Lesson 4-3, Solve Comparison Problems Using Multiplication, Materials, “The materials may be for any part of the lesson: connecting cubes, counters, number cubes.”

• In Lesson 7-2, Estimate Quotients, Materials, “The materials may be for any part of the lesson: index cards, spinners labeled 2-9.”

The materials reviewed for Reveal Math Grade 4 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Each unit, beginning with Unit 2, offers a Readiness Diagnostic that assesses the content of the unit and gives teachers a snapshot of the prerequisite skills the students already possess. Each unit also includes a Unit Assessment that evaluates students’ understanding of and fluency with concepts and skills from the unit. In the Teacher Edition, an Item Analysis lists each item’s DOK level, skill focus, content standard, and a Guided Support Intervention Lesson that teachers can assign or use for small groups or remediation. For example:

• In Unit 7, Division Strategies with Multi-Digit Dividends and 1-Digit Divisors, Unit Assessment (Form B), Item 13 lists “Interpret Remainders in Word Problems” as the Guided Support Intervention Lesson. This resource can be located in the Digital Teacher Center in the Targeted Intervention section of the Unit.

Unit Performance Tasks include a scoring rubric that evaluates student work for each section on a 2, 1, or 0 point scale. No follow-up guidance is provided for the Performance Task. For example:

• In Unit 2, Generalize Place-Value Structure, Performance Task Part B, Rubric (2 points), “2 Points: Students’ work shows proficiency with writing numbers through the millions in expanded form. Solution and explanation are accurate. 1 Point: Students’ work shows developing proficiency with writing numbers through the millions in expanded form. Solution shows one inaccurate number or explanation. 0 Points: Students’ work shows weak proficiency with writing numbers through the millions in expanded form. Solution and explanation are inaccurate.”

Math Probes analyze students’ misconceptions and are provided at least one time per Unit, beginning with Unit 2. In the Teacher Edition, “Authentic Student Work” samples are provided with correct student work and explanations. An “IF incorrect…, THEN the student likely…Sample Misconceptions” chart is provided to help teachers analyze student responses. A Take Action section gives teachers suggestions and resources for follow up or remediation as needed. There is a “Revisit the Probe” with discussion questions for students to review their initial answers after they are provided additional instruction, along with a Metacognitive Check for students to reflect on their own learning. For example:

• In Unit 11, Multiply Fractions by Whole Numbers, Math Probe Which is Greater?, Analyze The Probe, Targeted Concept, “Determine the product of a whole number and a fraction by reasoning about the magnitude of numbers.” For example, “Problem 2, Circle a or b to show which is greater. a. 3 x $\frac{1}{2}$ OR b. 7 x $\frac{1}{4}$.” Targeted Misconceptions: “Some students have difficulty determining an estimated magnitude of a product when working with fractions. They may overgeneralize about the size of fractions in general or misjudge the magnitude of a particular fraction. Other students may multi[ly both the numerator and denominator when multiplying a fraction by a whole number.” Sample Student work is provided, along with “IF incorrect...THEN the student likely…” explanations of the sample misconception are provided.

Exit Tickets are provided at the end of each lesson and evaluate students’ understanding of the lesson concepts and provide data to inform differentiation. Each includes a Metacognitive Check allowing students to reflect on their understanding of lesson concepts on a scale of 1 to 3, with 3 being the highest confidence, and beginning in Unit 2, include an Exit Skill Tracker that lists each item’s DOK, skill, and standard. The Exit Ticket Recommendations chart provides information regarding which differentiation activity to assign based on the student’s score. For example, “If students score…Then have students do” which provides teachers information on what Differentiation activities to use such as Reinforce Understanding, Build Proficiency or Extend Thinking. For example:

• In Lesson 14-3, Draw and Measure Angles, Exit Ticket, Item 1, “What is the measure of the angle? Use a protractor to measure the angle.” Exit Ticket Recommendations: “If students score 2 of 2, Then have students do Additional Practice or any of the B (Build Proficiency) or E (Extend Thinking) activities. If students score 1 of 2, Then have students Take Another Look or any of the B activities. If students score 0 of 2, Then have students do Small Group Intervention or any of the R (Reinforce Understanding) activities.”

The materials reviewed for Reveal Math Grade 4 meet expectations that assessments include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series. 

Reveal Math offers a variety of opportunities for students to demonstrate the full intent of grade-level standards and mathematical practices. While content standards and DOK levels are consistently identified for teachers in the Teacher Edition, and content standards are labeled for students in digital assessments, the standards for mathematical practice are not identified for teachers or students. It was noted that although assessment items do not clearly label the MPs, students are provided opportunities to engage with the mathematical practices.

Unit Readiness Diagnostics are given at the beginning of each unit, beginning with Unit 2. Formative assessments include Work Together, Exit Tickets, and Math Probes. Summative assessments include Unit Assessment Forms A and B, and Unit Performance Tasks at the end of a unit. Benchmark Assessments are administered after multiple units, and an End of the Year (Summative) Assessment is given at the end of the school year. Examples include:

• In Unit 4, Multiplication as Comparison, Performance Task, Ticket Sales, supports the full intent of 4.OA.2 (Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number…), and MP4 (Model with mathematics) as students model a situation with an appropriate representation and use an appropriate strategy. “Part C, The theater sold a combined total of 36 tickets to its drama and historical performances. The theater sold 5 times as many drama tickets as spy show tickets. How many drama tickets were sold? How many tickets to the historical performance were sold? Show your work by writing equations with a symbol for the unknown.”

• In Unit 11, Multiply Fractions by Whole Numbers, Unit Assessment Form B, Item 11 supports the full intent of 4.NF.4b (Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number), and MP5 (Use appropriate tools strategically) as students choose an appropriate tool or strategy to solve the problem. “Michelle picks $2\frac{5}{6}$ quarts of blackberries on Monday. On Tuesday, she picks 4 times as many blackberries as on Monday. How many quarts of blackberries did she pick on Tuesday?

• In Lesson 13-1, Relate Metric Units, Assess, Exit Ticket, Item 5, supports the full intent 4.MD.1 (Know relative sizes of measurement units within one system of units...), and MP7 (Look for and make use of structure) as students look for patterns to make generalizations and solve problems. “Choose whether each measurement statement is True or False. 24 kilograms = 2,400 grams; 35 kilometers = 35,000 meters; 49 meters = 49,000 centimeters, 52 liters = 52,000 milliliters.”

• Summative Assessment, Item Analysis, Item 5 supports the full intent of 4.OA.3 (Solve multistep word problems posed with whole numbers having whole-number answers using the four operations), and MP2 (Reason abstractly and quantitatively) as students consider the units involved and represent the problem situation symbolically. “Carlos has 264 stickers in his collection divided equally among 6 books. He trades Zoe 2 full books of stickers in exchange for 61 of her stickers. How many total stickers does Carlos have now? A. 176 stickers, B. 237 stickers, C. 325 stickers, D. 413 stickers.”

The materials reviewed for Reveal Math Grade 4 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

There are multiple locations of supports for students in special populations at the unit and lesson level. These supports are specifically aligned to lessons and standards, making them engaging in a variety of ways. They also scaffold up to the learning instead of simplifying or lowering expectations.

The Implementation Guide, Support for English Learners, identifies three features at the Unit level:

• “The Math Language Development feature offers insights into one of the four areas of language competence - reading, writing, listening, and speaking - strategies to build students’ proficiency with language.”

• The English Language Learner feature provides an overview of the lesson-level support.”  

• The Math Language Routines feature consists of a listing of the Math Language Routines found in each lesson of the unit.” 

The Implementation Guide, Support for English Learners, also identifies three features at the Lesson level:

• Language Objectives: “In addition to a content objective, each lesson has a language objective that identifies a linguistic focus for the lesson for English Learners. The language objective also identifies the Math Language Routines for the Lesson.”

• English Learner Scaffolds: “English Learner Scaffolds provide teachers with scaffolded instruction to help students make meaning of math vocabulary, ideas, and concepts in context. The three levels of scaffolding within each lesson - Entering/Emerging, Developing/Expanding, and Bridging/Reaching are based on the 5 proficiency levels of the WIDA English Language Development Standards. With these three levels, teachers can scaffold instruction to the appropriate level of language proficiency for their students.”  

• Math Language Routines: “Each lesson has at least one Math Language Routine specifically designed to engage English Learners in math and language.”  

The Implementation Guide, Differentiation Resources, provides a variety of small group activities and resources to support differentiation to sufficiently engage students in grade level mathematics. Examples include: 

• Reinforce Understanding: “These teacher-facilitated small group activities are designed to revisit lesson concepts for students who may need additional instruction.”  

• Build Proficiency: “Students can work in pairs or small groups on the print-based Game Station activities, written by Dr. Nicki Newton, or they can opt to play a game in the Digital Station that helps build fluency.”  

• Extend Thinking: “The Application Station tasks offer non-routine problems for students to work on in pairs or small groups.”   

The Implementation Guide, Differentiation Resources, provides a variety of independent activities and resources to support differentiation to sufficiently engage students in grade level mathematics. Examples include:

• Reinforce Understanding: “Students in need of additional instruction on the lesson concepts can complete either the Take Another Look mini-lessons, which are digital activities, or the print-based Reinforce Understanding activity master.”

• Build Proficiency: “Additional Practice and Spiral Review assignments can be completed in either print or digital environment. The digital assignments include learning aids that students can access as they work through the assignment. The digital assignments are also auto-scored to give students immediate feedback on their work.”  

• Extend Thinking: “The STEM Adventures and Websketch activities powered by Geometer’s Sketchpad offer students opportunities to solve non-routine problems in a digital environment. The print-based Extend Thinking activity master offers an enrichment or extension activity.”  

The Teacher Edition and Implementation Guide provide overarching guidance for teachers on how to use the supports provided within the program. Examples include:

• Teacher Edition, Volume 1, Lesson Model: Differentiate, for every lesson, there are multiple options for teachers to choose to support student learning. Based on data from Exit Tickets, students can reinforce lesson skills with “Reinforce Understanding” opportunities, practice their learning with “Build Proficiency” opportunities, or extend and apply their learning with “Extend Thinking” opportunities. Within each of these opportunities, there are options of workstations, online activities and independent practice for teachers to elect to use. 

• Implementation Guide, Targeted Intervention, “Targeted intervention resources are available to assign students based on their performance on all Unit Readiness Diagnostics and Unit Assessments. The Item Analysis table lists the appropriate resources for the identified concept or skill gaps. Intervention resources can be found in the Teacher Center in both the Unit Overview and Unit Review and Assess sections.”  The Item Analysis can be found in the Teacher Edition. Intervention resources include Guided Support, “Guided Support provides a teacher-facilitated small group mini-lesson that uses concrete modeling and discussion to build conceptual understanding” and Skills Support, “Skills Support are skills-based practice sheets that offer targeted practice of previously taught items.”  Both of these can be located in the Digital Teacher Center.

The materials reviewed for Reveal Math Grade 4 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

Each unit opens with an “Ignite!” activity that poses an interesting problem or puzzle to activate prior knowledge and spark students’ curiosity around the mathematics for the unit. In the Digital Teacher Center, “What are Ignite! Activities?” video, contributing author Raj Shah, Ph.D., explains, “An Ignite! Activity is an opportunity to build the culture of your classroom around problem-solving, exploration, discovery and curiosity.” The activity gives teachers, “the opportunity to see what the students can do on their own, without having to pre-teach them anything.” This provides an opportunity for advanced students to bring prior knowledge and their own abilities to make insightful observations. 

The Teacher Edition, Unit Resources At-A-Glance page includes a Workstations table which, “offers rich and varied resources that teachers can use to differentiate and enrich students’ instructional experiences with the unit content. The table presents an overview of the resources available for the unit with recommendations for when to use.” This table includes Games Station, Digital Station, and Application Station. 

Within each lesson, there are opportunities for students to engage in extension activities and questions of a higher level of complexity. The Practice & Reflect, On My Own section of the lesson provides an Item Analysis table showing the aspect of rigor and DOK level of each item. The Exit Ticket at the end of each lesson provides differentiation that includes extension through a variety of activities.

Additionally, there are no instances of advanced students doing more assignments than their classmates.

The materials reviewed for Reveal Math Grade 4 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

The materials provide strategies for all students to foster their regular and active participation in learning mathematics, as well as, specific supports for English Learners. 

In the Implementation Guide, Support for English Learners, Unit-level support, “At the unit level are three features that provide support for teachers as they prepare to teach English Learners. The Math Language Development feature offers insights into one of the four areas of language competence - reading, writing, listening, and speaking - and strategies to build students’ proficiency with language. The English Language Learner feature provides an overview of lesson-level support. The Math Language Routines feature consists of a listing of the Math Language Routines found in each lesson of the unit.” The Unit Overview also includes a Language of Math section highlighting key vocabulary from the unit. These sections provide an overview of the strategies present within the unit ,and give guidance as to possible misconceptions or challenges that EL students may face with language demands. Included within the Unit Review is a Vocabulary Review that includes an Item Analysis for each item as well as what lesson/s the term was found in.  

At the lesson level, there are supports to engage ELs in grade-level content and develop knowledge of the subject matter. These involve oral language development and reading and writing activities. The Teacher Edition and Implementation Guide outline these features. Examples include:

• Language Objective, “In addition to a content objective, each lesson has a language objective that identifies a linguistic focus of the lesson for English Learners. The language objective also identifies the Math Language Routine of the lesson.”

• Math Language Routine, “Each lesson has at least one Math Language Routine specifically designed to engage English Learners in math and language.” Math Language Routines (MLR), listed and described in the Implementation Guide include: Stronger and Clearer Each Time, Collect and Display, Critique, Correct, and Clarify, Information Gap, Co-Craft Questions and Problems, Three Reads, Compare and Connect, Discussion Supports.

• English Learner Scaffolds, “English Learner Scaffolds provide teachers with scaffolded instruction to help students make meaning of math vocabulary, ideas, and concepts in context. The three levels of scaffolding within each lesson - Entering/Emerging, developing/Expanding, and Bridging/Reaching are based on the 5 proficiency levels of the WIDA English Language development Standards. With these three levels, teachers can scaffold instruction to the appropriate level of language proficiency of their students.”

• Language of Math, ”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.”

• Number Routines such as “Would You Rather?” or “Math Pictures” and Sense-Making Routines such as “Notice and Wonder” or “Which Doesn’t Belong?” provide opportunities to develop and strengthen number sense and problem solving through discussion or written responses.

Most materials are available in Spanish such as the Student Edition, Student Practice Book (print), Student eBook, Math Replay Videos, eGlossary, and Family Letter (digital).

The materials reviewed for Reveal Math Grade 4 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent, and when appropriate, are connected to written methods.

Physical manipulatives needed for each unit and lesson can be found in the Teacher Edition, Unit Planner, at the beginning of each unit under “Materials to Gather”. Each lesson also identifies needed materials in the “Materials” section on the first page of each lesson.

Virtual manipulatives can be found online under “e-Toolkit”. Manipulatives are used throughout the program to help students develop a concept or explain their thinking. They are used to develop conceptual understanding and connect concrete representations to a written method.