The materials reviewed for Reveal Math Grade 1 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 3-4, Represent 2-Digit Numbers, Explore & Develop, Work Together, “How can you use a place-value chart to show how many?” Students are shown 7 groups of 10 and 6 ones with connecting cubes and a blank place value chart. This exercise supports conceptual development of 1.NBT.2, understand that the two digits of a two-digit number represent amounts of tens and ones. 

• In Lesson 4-6, Choose Strategies to Add, Launch, Notice & Wonder, students are shown an image of two glass jars with five marbles in one jar and seven marbles in the other jar. “What do you notice? What do you wonder? Teaching tip: Consider having students think about different strategies they can use to add, such as counting on, using addition doubles facts, and making a 10. This allows students to understand that there is more than one way to add two addends.” This activity supports conceptual development of 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10. 

• In Lesson 11-1, Use Mental Math to Find 10 Less, Explore & Develop, Activity-Based Exploration, “Have one student in each student-group choose a number card and ask student-groups to find 10 less than that number.” Students write an equation to represent the problem. “What did you notice about the number you started with, the total, and the number you ended with, the difference? How can you use mental math to find 10 less?” This activity supports conceptual development of 1.NBT.5, given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

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

• In Lesson 3-6, Compare Numbers, Practice & Reflect, On My Own, Exercise 1, “How can you compare numbers to show which is greater?” Students circle “is greater than, less than, or is equal to”. Students are given a picture of base-ten blocks showing 73 and 37. This activity supports conceptual understanding of 1.NBT.3, compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, <. 

• In Lesson 4-7, Use Properties to Add, Practice & Reflect, Exercise 5, “Which has the same sum as 3 + 6? A. 2 + 6 B. 4 + 3 C. 6 + 3” [6+ 3] Students solve a given addition problem and select an addition problem with the same sum, considering the role of the commutative property to solve the problem. This activity supports conceptual understanding of the cluster 1.OA.B, understand and apply properties of operations and the relationship between addition and subtraction.

• In Unit 9, Addition within 100, Math Probe, Exercise 2, students examine a portion of a number chart that has numbers 67 and 98 written in two boxes and a ? in another box that is in between those two numbers. “Circle the number that belongs in the ? box.” Answer choices include, “68, 69, 71, 77, 89, 97, none of these numbers”. [89] In the column to the right of the problem, students “Tell or show why” to justify their answer. This activity supports conceptual understanding of 1.NBT.5, given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

The materials reviewed for Reveal Math Grade 1 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. The materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. 

The materials develop procedural skill 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 1 progress toward 1.OA.6, add and subtract within 20, demonstrating fluency with addition and subtraction within 10. 

• In 5-2, Count Back to Subtract, Launch, Notice & Wonder, students see a “Balloon Pop” game board with 9 balloons and 3 being popped. Pose Purposeful Questions, “How many balloons are popped? How many balloons are there now? What do you know about the number of balloons before the game was played and the number of balloons showing in the picture?” Establish Goals to Focus Learning, “Let’s think about how we can use subtraction equations to show situations.” This activity provides an opportunity for students to develop procedural skill and fluency of 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10. 

• In Lesson 5-3, Count On to Subtract, Practice & Reflect, On My Own, Reflect, “How is counting on to subtract like counting on to add?” Math is Mindset, “What did you already know that helped you with today’s work?” Students explain how to use a number line for subtraction, and compare it using a number line for addition. This activity supports the development of procedural skill and fluency 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10. 

• In Lesson 7-1, Represent and Solve Add To Problems, Explore & Develop, Learn, Work Together, “5 raindrops fall. 6 more raindrops fall. How many raindrops fall? Show your thinking.” Students build fluency from conceptual understanding of addition to support the development of procedural skill and fluency of 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10.

• In Lesson 8-4, Represent and Solve More Take Apart Problems, Explore & Develop, Work Together, “Layla has 10 beads. Some are purple. The rest are orange. How many purple and how many orange? Show your thinking. ___ purple beads and ___ orange beads.” This exercise demonstrates the development of the cluster 1.OA.C, add and subtract within 20, relating to the procedural skill and fluency of 1. OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10. 

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

• In Unit 4, Addition within 20: Facts and Strategies, Fluency Practice, Fluency Check, Exercises 8-15, “What is the sum or difference? Write the number.” Students answer addition and subtraction equations within 10. Exercise 8, “8 - 1 = ___”, Exercise 9, “4 + 1 = ___.” These exercises provide an opportunity for students to demonstrate procedural skill and fluency of 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10. 

• In Lesson 5-3, Count On to Subtract, Differentiate, Reinforce Understanding, Differentiation Resource Book, Exercises 3-6, “What is the difference? 3. 8 - 4 =___, 4. 9 - 7 =___, 5. 6 - 4 =___, 6. 10 - 3  =___.” These exercises provide an opportunity for students to independently demonstrate procedural skill and fluency of 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10. 

• In Unit 8, Meanings of Subtraction, Fluency Practice, Fluency Talk, “How can you show 10 - 6 = 4? Explain your work.” This exercise provides an opportunity for students to independently demonstrate procedural skill and fluency of 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10. 

• In Unit 11, Subtraction within 100, Fluency Practice, Fluency Flash, Exercise 2, “What is the sum? Use doubles to help you add.  4 + 5 = ____.”  There is a picture of 4 red cubes and 4 red cubes and 1 yellow cube. This activity provides an opportunity for students to independently demonstrate procedural skill and fluency of 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10.

The materials reviewed for Reveal Math Grade 1 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 8-1, Represent and Solve Take From Problems, Explore & Develop, Learn, “Juanita has 12 sheep. She gives 4 sheep away. How many sheep does Juanita have now?” Pose Purposeful Questions, “What do you know from the problem? What is the question asking you to find?” This exercise allows students to develop and apply mathematics of 1.OA.1, use addition and subtraction within 20 to solve word problems.

• In Lesson 10-1, Represent and Solve Compare Problems, Launch Notice & Wonder, Pose Purposeful Questions, “What do you notice about the containers? What do you notice about the flowers in each container? How can you compare the flowers on the left to the flowers on the right?” With the teacher guiding the discussion, students look at a picture that has two children and two containers of flowers. The container on the left has 13 flowers in a random pattern. The container on the right has 17 flowers in a random pattern. This exercise allows students to develop and apply mathematics of 1.OA.1, use addition and subtraction within 20 to solve word problems.

• In Lesson 10-4, Solve Compare Problems Using Addition and Subtraction, Practice & Reflect, On My Own, “How can you make an equation to show the problem? Use ? for the unknown. Then solve.” Exercise 2, “Jackson has 6 fewer berries than Tammy. Tammy has 16 berries. How many berries does Jackson have? ____ berries.” This exercise allows students to independently apply mathematics of 1.OA.1, use addition and subtraction within 20 to solve word problems.

• In Lesson 12-10, Solve Problems Involving Data, Differentiate, Reinforce Understanding, Differentiation Resource Book, “Use the picture graph to answer the questions.” Exercise 1, “Which fruit did the most students choose?” Exercise 2, “Which fruit did the fewest students choose?” Exercise 3, “How many fewer students chose apples than bananas?” These exercises allow students to independently apply mathematics of 1.MD.4, organize, represent, and interpret data with up to three categories.

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 8-5, Solve Problems Involving Subtraction, Differentiate, Extend Thinking, Differentiation Resource Book, “How can you write a subtraction word problem with an unknown part or difference to match the picture? Solve the problem.” Exercise 1 shows a number line where the dot is on the 5 and the jumps end at 11. This exercise allows students to independently apply mathematics of 1.OA.1, use addition and subtraction within 20 to solve word problems.

• In Lesson 9-8, Add 2-Digit Numbers, Explore & Develop, Activity-Based Exploration, “Have two students select a number between 1-50 with ones digits from 5-9. Ask student-groups to write an equation to show adding their numbers. Students should explore using the base-ten blocks to find the sum.” This exercise allows students to develop and apply mathematics of 1.NBT.4, add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10.

• In Lesson 10-3, Represent and Solve More Compare Problems, Launch, Numberless Word Problem, “What math do you see in this problem?” Be Curious, “Mia buys more bananas than Carter. Mia buys some bananas. How many bananas does Carter buy?” Pose Purposeful Questions, “What do you know about the numbers of bananas each person buys? Which person has fewer bananas? If you know the number of bananas Mia buys, how could you find the number Carter buys?” This exercise allows students to develop and apply mathematics of 1.OA.1, use addition and subtraction within 20 to solve word problems.

• In Lesson 10-4, Solve Compare Problems Using Addition and Subtraction, Practice & Reflect, On My Own, Exercise 5, Extend Your Thinking, “Make a word problem to match the equation ? + 6 = 14. Use the word more.” This exercise allows students to independently apply mathematics of 1.OA.1, use addition and subtraction within 20 to solve word problems.

The materials reviewed for Reveal Math Grade 1 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 simultaneouslyThe materials reviewed for Reveal Math Grade 1 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 Unit 3, Place Value, Fluency Practice, Fluency Strategy, Exercise 1, “How can you draw to show how to add 7 + 1?  Write the number.  7 + 1 = ____.”  Students are shown seven connecting cubes + one more cube. This exercise provides an opportunity for students to develop procedural skill and fluency of 1.OA.6, add and subtract within 20.

• In Lesson 7-6, Solve Addition Problems, Practice & Reflect, On My Own, Exercise 5, STEM Connection, “Jordan visits a school for guide dogs. There are 14 dogs. More dogs join. Now there are 20 dogs. How many more dogs join? ____ guide dogs.” This exercise provides an opportunity for students to apply the mathematics of 1.OA.1, use addition and subtraction within 20 to solve word problems.

• In Lesson 9-1, Use Mental Math to Find 10 More, Assess, Exit Ticket, Exercise 1, “What is the sum? 10 +12 = ____” Students use their understanding of place value and the relationship between tens and ones, along with the provided number chart, to find the sum. Exercise 3, “A restaurant has 54 bread rolls. The cook buys 10 more bread rolls. How many bread rolls are there now? ____ bread rolls”. These exercises provide an opportunity for students to demonstrate conceptual understanding of 1.NBT.5, given a two-digit number, mentally find 10 more or 10 less than the number, without having to count.

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 5-4, Make a 10 to Subtract, Practice & Reflect, On My Own, Exercise 5, “What is the difference? 16 - 7 = ____”  Exercise 7, Error Analysis, “Jorge uses ten-frames to subtract 12 - 5. How can you help him find the correct difference?” Students see two ten-frames showing 12, with four crossed out on the first and two crossed out on the second. These exercises give students opportunities to demonstrate procedural skill and fluency and conceptual understanding of 1.OA.6, add and subtract within 20.

• In Unit 9, Addition within 100, students are provided opportunities to build conceptual understanding and procedural skill and fluency of place value to add and subtract. In Lesson 9-1, Use Mental Math to Find 10 More, Explore & Develop, Activity-Based Exploration, students use number cubes to explore the pattern of adding 10, “Have student-groups roll the number cubes to create a 2-digit number, using one number cube for the tens digit and the other for the ones digit. Ask student-groups to write an equation adding 10 to that number. Repeat the activity five times so that students can explore and identify a pattern they can use to add 10 to any 2-digit number. Ask student-groups to explain their pattern and how it can help them add 10 mentally.” Practice & Reflect, On My Own, “Is the equation true? Circle Yes or No.” Exercise 2, “32 + 10 = 42.” On My Own, “What is the sum?” Exercise 3, “10 + 51 = ___.” Exercise 8, “44 + 10 = ___.” These exercises provide an opportunity for students to develop and demonstrate conceptual understanding and procedural skill and fluency of 1.NBT.5, given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

• In Unit 11, Subtraction Within 100, students are provided opportunities to build conceptual understanding and apply the mathematics of subtraction within 100. In Lesson 11-2, Represent Subtracting Tens, Explore & Develop, Activity-Based Exploration, “Instruct student-groups to choose two tens cards. Have them write a subtraction equation where the greater number is the total and the unknown is the difference. Ask students to represent the equation with base-ten blocks or drawings to help them solve the equation. Tell them to pay close attention to the tens digit in each number of the solved equation. Have groups repeat the process several times using different tens cards.” 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 Unit 3, Place Value, Fluency Practice, Fluency Strategy, Exercise 1, “How can you draw to show how to add 7 + 1?  Write the number.  7 + 1 = ____.”  Students are shown seven connecting cubes + one more cube. This exercise provides an opportunity for students to develop procedural skill and fluency of 1.OA.6, add and subtract within 20.

• In Lesson 7-6, Solve Addition Problems, Practice & Reflect, On My Own, Exercise 5, STEM Connection, “Jordan visits a school for guide dogs. There are 14 dogs. More dogs join. Now there are 20 dogs. How many more dogs join? ____ guide dogs.” This exercise provides an opportunity for students to apply the mathematics of 1.OA.1, use addition and subtraction within 20 to solve word problems.

• In Lesson 9-1, Use Mental Math to Find 10 More, Assess, Exit Ticket, Exercise 1, “What is the sum? 10 +12 = ____” Students use their understanding of place value and the relationship between tens and ones, along with the provided number chart, to find the sum. Exercise 3, “A restaurant has 54 bread rolls. The cook buys 10 more bread rolls. How many bread rolls are there now? ____ bread rolls”. These exercises provide an opportunity for students to demonstrate conceptual understanding of 1.NBT.5, given a two-digit number, mentally find 10 more or 10 less than the number, without having to count.

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 5-4, Make a 10 to Subtract, Practice & Reflect, On My Own, Exercise 5, “What is the difference? 16 - 7 = ____”  Exercise 7, Error Analysis, “Jorge uses ten-frames to subtract 12 - 5. How can you help him find the correct difference?” Students see two ten-frames showing 12, with four crossed out on the first and two crossed out on the second. These exercises give students opportunities to demonstrate procedural skill and fluency and conceptual understanding of 1.OA.6, add and subtract within 20.

• In Unit 9, Addition within 100, students are provided opportunities to build conceptual understanding and procedural skill and fluency of place value to add and subtract. In Lesson 9-1, Use Mental Math to Find 10 More, Explore & Develop, Activity-Based Exploration, students use number cubes to explore the pattern of adding 10, “Have student-groups roll the number cubes to create a 2-digit number, using one number cube for the tens digit and the other for the ones digit. Ask student-groups to write an equation adding 10 to that number. Repeat the activity five times so that students can explore and identify a pattern they can use to add 10 to any 2-digit number. Ask student-groups to explain their pattern and how it can help them add 10 mentally.” Practice & Reflect, On My Own, “Is the equation true? Circle Yes or No.” Exercise 2, “32 + 10 = 42.” On My Own, “What is the sum?” Exercise 3, “10 + 51 = ___.” Exercise 8, “44 + 10 = ___.” These exercises provide an opportunity for students to develop and demonstrate conceptual understanding and procedural skill and fluency of 1.NBT.5, given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

• In Unit 11, Subtraction Within 100, students are provided opportunities to build conceptual understanding and apply the mathematics of subtraction within 100. In Lesson 11-2, Represent Subtracting Tens, Explore & Develop, Activity-Based Exploration, “Instruct student-groups to choose two tens cards. Have them write a subtraction equation where the greater number is the total and the unknown is the difference. Ask students to represent the equation with base-ten blocks or drawings to help them solve the equation. Tell them to pay close attention to the tens digit in each number of the solved equation. Have groups repeat the process several times using different tens cards.”

The materials reviewed for Reveal Math Grade 1 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 MP2. Within each of the lesson components, 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 4-8, Add Three Numbers, Practice & Reflect, On My Own, Exercise 9, Extend Your Thinking, “What are two other ways to order the addends 3 + 7 + 3 to find their sum?” [3 + 3 + 7; 7 + 3 + 3] Students engage with the full intent of MP1 as they use a variety of strategies and properties of operations to add three whole numbers.

• In Lesson 6-3, Compose Shapes, Explore & Develop, Work Together, “How can you make a flower using these shapes? Draw your new shape.” Shown are a half circle, a trapezoid, a rectangle, a quarter circle, a hexagon and a triangle. Students engage with the full intent of MP1 as they analyze and make sense of problems and engage in problem solving to make a composite shape using 2-dimensional shapes.

• In Lesson 12-6, Tell Time to the Half Hour, Launch, Notice & Wonder, Pose Purposeful Questions, two analog clocks are shown on the page, one shows 8:00 and one shows 8:30. “What time does the clock on the left show? How far does the minute hand have to move to go from 12 to 6?  Do you think the clocks are showing the same time? Explain.” Students engage in the full intent of MP1 as they analyze and make sense of the problem and determine if their answer makes sense.

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

• In Lesson 5-5, Use Near Doubles to Subtract, Launch, Notice & Wonder, students see a picture of two groups of crayons. The number of crayons are identical with one exception; the number of crayons on the left differs by one. Pose Purposeful questions, “What do you notice about the crayons in the first box? How many crayons are in each box? How do the numbers of crayons in each box relate to one another? What do we call two numbers that are the same? What do we call two numbers that are almost the same?” Students engage in the full intent of MP2 as they attend to the meaning of quantities and understand the relationships between problem scenarios and mathematical representations as they use near doubles to subtract.

• In Lesson 12-3, Strategies to Measure Length, Explore & Develop, Activity-Based Exploration, “Introduce the terms: measure and unit. Instruct student-groups to select 2 objects to measure using connecting cubes or counters as the unit to measure length. Have student-groups work collaboratively to use the materials provided to measure the lengths of their chosen objects.  For the purpose of this lesson, guide students to use only one unit per object. Ask students to state each of their measures by completing the following sentence frame. The [object name] is [number] [units of measure] long.” Students engage in the full intent of MP2 as they attend to the meaning of quantities when they use equal-size units to measure the length of objects in the classroom.

• In Lesson 12-9, Interpret Data, Launch, Notice & Wonder, “Students examine distinctions between a tally chart and picture graph as well as notice the information is the same while the format and organization is different.” Shown is a picture graph with 6 butterflies, 3 bees, and 8 grasshoppers. Another chart with tallies and digits shows the same information. Students engage in the full intent of MP2 as they understand the relationships between problem scenarios and mathematical representations, and explain the meaning of numbers and symbols in a tally chart and picture graph.

The materials reviewed for Reveal Math Grade 1 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, 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 5-7, Use Fact Families to Subtract, Differentiate, Extend Thinking, Differentiation Resource Book, students construct viable arguments as they look at fact families, “Could the fact be part of the fact family on the fact triangle? Write Yes or No. If the answer is no, explain your thinking.” Exercise 2, students see a triangle with 14 and 5, “9 - 5 = 4 sample answer: No. The fact triangle does not include 4 and 9.” 

• In Unit 6, Shapes and Solids, Unit Overview, Math Practices and Processes, Construct Viable Arguments and Critique the Reasoning of Others, provides guidance for teachers in engaging students in MP3, “Since problems such as sorting and building new shapes can often be solved in more than one way, students will also find themselves explaining why they selected one strategy over another. Some suggestions for guiding students to become more proficient at explaining their reasoning about shapes include: guiding students to focus on attributes that are defining attributes when they explain their reasons for identifying and naming shapes; having students practice critiquing each other in cooperative settings, such as working together to explore whether a closed flat shape could have 3 sides and 4 vertices; encouraging students to keep notes using words and drawings to organize the different types of shapes they have seen and the defining attributes for those shapes.” 

• In Unit 9, Addition within 100, Performance Task, Part D, students construct viable arguments when they explain their strategy for finding the sum of two addends in a word problem, “The children at the fair buy 54 apple snacks. They buy 21 peanut snacks. How many snacks dld the children buy? Explain how you found the total.” 

• In Lesson 13-5, Describe Halves and Fourths of Shapes, Explore & Develop, Work Together, students construct viable arguments as they justify their thinking about equal shares in given shapes. “Which shape has larger equal shares? Circle the shape. Explain your thinking.” There are two rectangles shown that are the same size. One rectangle is split into 4 equal shares. The other rectangle is split into 2 equal shares.

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 4-10, Understand the Equal Sign, Practice & Reflect, Exercise 3, Error Analysis, students critique the reasoning of others as they are shown two sets of pencils and asked, “Milo says he has more pencils than Ree. How can you show Milo that he has the same number of pencils as Ree?” A picture is shown with Ree’s pencils, 9 yellow and 6 blue, and Milo’s pencils, 8 yellow and 7 blue.  

• In Lesson 5-7, Use Fact Families to Subtract, Launch, Notice and Wonder, students critique the reasoning of others as they share what they notice about the given fact families and compare that to a classmate’s ideas, “How are they the same? How are they different?” Students see a triangle with 6, 4, and 2; another triangle with 4, 2, and 6; another triangle with 2, 4 and 6. Teaching Tip: “During the share out, have students restate what another student shared and say how it is alike or different from their idea. For example, ‘Can you say what she said in your own words? How is that alike or different from what you are thinking?’ This will foster a classroom of active listeners.”

• In Lesson 5-8, Find an Unknown in a Subtraction Equation, Practice & Reflect, On My Own, Exercise 7, students critique the reasoning of others as they look at a provided number line and answer, “Tony uses a number line to solve 15 - ? = 11. He says the unknown number is 6. Do you agree? Explain.” 

• In Lesson 6-1, Understand Defining Attributes of Shapes, Practice & Reflect, Exercise 8, Extend Your Thinking, students critique the reasoning of others as they see a picture of a square and are asked, “Carole says this shape is not a rectangle because the sides are all the same length. Do you agree with Carol?”  

• In Lesson 9-7, Regroup to Add, Launch, Notice & Wonder, Math is Mindset, “What can you do to be a good listener?” This guidance for teachers helps guide students in critiquing the reasoning of others as they discuss what they notice about the number of toys in a picture of a toy dispenser. Relationship Skills: Effective Communication, “As students engage in collaborative discourse around the Notice & Wonder routine, encourage them to actively and respectfully listen to one another. Invite students to think about and share what active listening looks and sounds like. As students discuss what they noticed and wondered, encourage classmates to listen as well as provide thoughtful feedback. Capitalize on opportunities to also model these behaviors when students are speaking.”

The materials reviewed for Reveal Math Grade 1 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate 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, the mathematical practice is labeled MPP Model with mathematics, rather than MP4: Model with mathematics. Additionally, the math practices are not identified within the lesson sections, therefore leaving the location of 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 Unit 3, Place Value, Performance Task, Number Cube Game, Part A, students model numbers by drawing base-ten blocks. “Caleb rolls the numbers 3 and 6. Draw base-ten blocks to show each of the numbers Caleb can make. Write the number the base-ten blocks show below each group of blocks.” 

• In Lesson 5-1, Relate Counting to Subtraction, Differentiate, Build Proficiency, Digital Station: Penguin Chill (Add and Subtract within 10), students use digital materials to model and  represent an equation and answer questions to practice addition and subtraction problems. “Three penguins are on the iceberg. Choose any group to join.” The choices are 1, 2, or 3. “Two penguins join. Get a hat for every penguin.” The student chooses five hats. “Five penguins are on the iceberg. Choose one of these groups.” The choices are 2, 2, or 2. “Two penguins swim away. Get a hat for every penguin.” The student chooses 3 hats. 

• In Lesson 6-4, Build New Shapes, Explore & Develop, Develop the Math, Compare and Connect, students use their knowledge of 2-dimensional shapes to create and describe a composite shape. “Pair students and provide them with two sets of the same shapes, i.e. triangles. Students work on their own to create a new shape, then compare. Then each partner describes how they made their new shape.” 

• In Lesson 13-3, Partition Shapes into Fourths, Explore & Develop, Pose the Problem, Learn, students describe how to partition a shape into equal shares and check to see if their representations make sense. “4 friends share a sandwich. How can the friend make equal shares?” Pose Purposeful Questions, “How can shapes be cut into equal shares? How can you determine if the parts are equal?” 

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 4-2, Count On to Add, Explore & Develop, Digital Guided Exploration: Count On to Add, Develop the Math, presentation slide 2.3, students choose an appropriate technological tool to help them add. Students are shown a picture of two children blowing bubbles; one child has 3 bubbles, the other child has 9 bubbles. “What tool can you use to count on?” Math is… Choosing Tools, “What other tools can you use to find a sum?” 

• In Lesson 5-8, Find an Unknown Number in a Subtraction Equation, Differentiate, Build Proficiency, Digital Additional Practice Book: Find an Unknown in a Subtraction Equation, students choose a strategy to find an unknown number in a subtraction problem, “How can you complete the equation? Tell or show how you solved, Exercise 4, “11 - ___ = 5.” There is space for students to draw the tool or show the strategy used for solving the problem. 

• In Unit 9, Addition Within 100, Math Practices and Processes, Use Appropriate Tools Strategically, “In this unit, students will be adding using various tools and strategies. After guiding students through each tool and how to use it, students will inherently use the one that they like best, even if it takes more time. Reinforcing their right to choose will give students confidence in what they are learning. Over time, math facts and adding/subtracting practices will become easier, and students will often abandon these tools and do more mental math. Base-ten blocks: Using blocks to break apart numbers helps students visualize the addition problem at hand. Base-ten blocks allow for easier counting by 10s using the largest number as the starting point and then adding ten rods and ones blocks representing the second number. Number Line: Using a number line requires students to have a good understanding of number relationships, especially if the number line is an open number line. Students are encouraged to start with the largest number in the addition or subtraction problem, and then use jumps involving 2s, 5s, or 10s to navigate to the answer in either direction depending on the type of problem.” 

• In Unit 9, Addition within 100, Unit Review, Performance Task, Reflect, students use knowledge of the tools and strategies presented in this unit to explain how to add multi-digit numbers, “What are some ways to add 2-digit numbers?”

The materials reviewed for Reveal Math Grade 1 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. Lastly, MP6 is identified in seven out of thirteen units. However, upon review, it was found that the materials provide additional opportunities for students to engage in the full intent of MP6 that were not identified for teachers.

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 Lesson 5-9, True Subtraction Equations, students attend to precision as they use various methods to determine if equations are equal. In Explore & Develop, Work Together, students work with a partner to discuss what the equal sign means and discuss how they can prove a subtraction equation is true, “Is this equation true or false? Explain. 9 - 4 = 11 - 6 (There is a question mark over the equal sign)”. In Practice & Reflect, On My Own, Exercise 4, “Is the equation true or false? Circle True or False. 16 - 7 = 18 - 9”. In Differentiate, Reinforce Understanding, Find Balance, Small Group, “Work with students in pairs. Provide a set of number cards 11-19. One student chooses a card and puts the number of connecting cubes in one of the balance scale baskets. The other student rolls the number cube and removes that amount from the basket. Students repeat for the other side of the balance scale. Students write an equation if the scales are balanced or adjust the cubes if the scales are not balanced.”

• In Lesson 6-3, Compose Shapes, Practice & Reflect, On My Own, Exercise 3, students attend to precision as they use pattern block shape outlines to compose new shapes from provided shapes (square, hexagon and quarter circle), “How can you use the shapes to make a new shape? Draw to show how.” 

• In Unit 12, Measurement and Data, Unit Overview, Math Practices and Processes, Attend to Precision, “When interpreting data, precision is extremely important. Data must be displayed in a way that is clear and easy to understand. By creating precise data displays, data can be analyzed and used to answer questions. Before becoming proficient in using these displays, students must understand the importance of precision. Titles, labels, and the key to picture graph all offer opportunities to attend to the precision. Within this unit students will need to attend to precision when indirectly measuring common objects and telling time. In this grade, students will be measuring using non-standard units. Students compare lengths, using the terms long/longer and short/shorter. They read and write time to the nearest hour and half hour. Some suggestions for helping attend to precision include having students: 

  • Review the important features of a data display including the title, labels, categories, and key when necessary. 

  • Try to analyze data displays without the necessary features to highlight the importance of them.  

  • Use various unit lengths to indirectly measure objects and state their measures. 

  • Interpret picture graphs they created and check for accuracy using a different method than students used to create it. 

  • Discuss the importance of precise measurement when telling time to the hour and half hour.”  

• In Lesson 12-7, Organize Data, Differentiate, Reinforce Understanding, Differentiation Resource Book, Exercise 1, students attend to precision as they determine what shape the real world object represents and write the name of the object under the correct column, “How can you sort these objects by shape? Write the object names in the chart.” The pictures that have corresponding words are: orange, puzzle, juice, block, blueberry, peanut butter. The chart has 3 shape categories listed horizontally across the top: sphere, cube, cylinder. 

Specialized language stands alone with vocabulary presentations in each lesson. When MP6 is identified for a lesson, MP6 specifically refers to precision with mathematics. Examples where the instructional materials attend to the specialized language of mathematics, including guidance for teachers to engage students in MP6 include:

• In Lesson 3-6, Compare Numbers, Explore & Develop, Activity-Based Exploration, students attend to the specialized language of mathematics as they work in small groups to compare pairs of 2-digit numbers and act out vocabulary. “Directions: Students pair up and compare their numbers to find which number is greater, using the base-ten blocks or drawings to show each number. Students record the numbers and circle the greater number. Students pair up with a new partner and repeat the activity. Have students consider their reasoning as they compare numbers. Math is... Precision: How do you know when you need to compare the ones? Did anyone find that neither number was greater? Explain.” In Bring it Together, Language of Math, “Add the vocabulary cards: compare, equal to, greater than, and less than to the math word wall. Have students act out the words equal to, greater than, and less than. Repeat as necessary and discuss.” 

• In Lesson 6-3, Compose Shapes, Explore & Develop, Activity-Based Exploration, students attend to the specialized language of mathematics as they compose and name new shapes, “Instruct students to use squares to make a rectangle. Then have students use triangles to make a rectangle.” Math is...Precision, “How can you name the new shapes you make?”  

• In Lesson 10-3, Represent and Solve More Compare Problems, Explore & Develop, Activity- Based Exploration, students attend to the specialized language of math as they understand and use appropriate math vocabulary when engaged in a discussion about a word problem. “Mia buys 2 more bananas than Carter. Mia buys 7 bananas. How many bananas does Carter buy?” Math is...Connections, “In a fact family, how is an unknown addend in an addition equation related to the difference in a subtraction equation?” Students then compare a similar problem, “Have students model, draw, write an equation, and solve the second problem. Carter buys 2 fewer bananas than Mia. Mia buys 7 bananas. How many bananas does Carter buy? Have students compare the two different wordings of these problems. They should conclude these are two different ways to word the same situation.”

• In Lesson 12-7, Organize Data, Explore & Develop, Language of Math, students attend to the specialized language of mathematics as they add the vocabulary word data to the math word wall, “Add the vocabulary card data to the math word wall.  Then have students complete the sentence: I can organize data into categories such as _____, _____, and _____.”

The materials reviewed for Reveal Math Grade 1 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 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. It should be noted there are ten specific identifications of MP8 out of 13 total Units. However, upon review, it was found that the materials provide additional opportunities for students to engage in the full intent of MP8 that were not identified for teachers. 

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 3-1, Numbers 11 to 19, Explore & Develop, Activity-Based Exploration, students look for and make use of structure as they represent teen numbers using a ten and some ones, “Students explore the structure of teen numbers. Student-groups choose one number card, then work together to show the number in the ten-frames.”  Math is...Patterns, “What patterns do you notice in the numbers you modeled?” 

• In Lesson 5-8, Find an Unknown Number in a Subtraction Equation, Explore & Develop, Guided Exploration, students discover the structure of the problem on a number line as well as how counting back lends structure to subtraction as they count back from 14 to 8, “Students use number lines to count on and count back to find the unknown in a subtraction equation.” Digital Guided Exploration, Presentation slide 2.4, “Let’s start counting back from 14. What number do you count back to? How many jumps?” Math is … Structure, “How can you check to make sure the unknown you found is correct?” Students are shown a number line labeled 0-20. 

• In Lesson 11-3, Subtract Tens, Explore & Develop, Activity-Based Exploration, students use number lines and a number chart to look for structure and make generalizations as they subtract tens from larger multiples of ten, “How can you use number lines to subtract tens? How can you use number charts to subtract tens?” 

• In Lesson 12-5, Tell Time to the Hour, Differentiate, Extend Thinking, Differentiation Resource Book, Exercise 1, students look for and use patterns as they draw hands on an analog clock with the awareness that telling time to the hour always involves the hour hand pointing to the hour number, and the minute hand pointing to the twelve, “Draw or write the time on the clocks for the event. Practice at 5:00.”

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 Lesson 4-3, Doubles, Assess, Exit Ticket, Item 3, students use repeated reasoning to use doubles facts to solve a word problem, “A box has 8 crackers. How many are in 2 boxes? ___ + ___ = ___ crackers.” 

• In Lesson 4-7, Use Properties to Add, Launch, Notice & Wonder, students look for and express regularity in repeated reasoning as they add the same numbers in different orders resulting in the same sum, and write equation pairs for 10. Students see 2 groups of 7 dinosaurs. Each group has 2 brown dinosaurs and 5 green dinosaurs. In the first group the 2 brown dinosaurs are at the bottom, and in the second group the 2 brown dinosaurs are at the top. “How are they the same? How are they different?” While sharing, the teacher also prompts students with purposeful questions such as, “How many toy dinosaurs are there? If we add to find the total would it matter which type of dinosaur we add first? Explain your thinking.” Differentiate, Extend Thinking, Differentiation Resource Book, Use Properties to Add, Exercise 3, “Write two different equations with the same addends and a sum of 10. Then write as many more different equation pairs with a sum of 10 as you can.” 

• In Lesson 7-3, Represent and Solve Put Together Problems, Practice and Reflect, On My Own, Exercise 6, Extend Your Thinking, students create a word problem and use regularity in repeated reasoning to solve, “Make a word problem to match the part-part-whole mat.” Students are shown a part-part-whole mat with 7 in each of the Part labels and ? in the Whole. 

• In Lesson 12-1, Compare and Order Lengths, Explore & Develop, Choose Your Option, Activity-Based Exploration, students use regularity in repeated reasoning to compare and order objects, “Students determine the comparable lengths of three objects. Instruct students to brainstorm different ways they can compare their objects. Some identifiable attributes might be use or purpose, color, shape, composition, weight, or size (length).  

  • The ______ is longer than the ______. 

  • The ______ is shorter than the ______.

  • The ________ is the longest object. 

  • The ________ is the shortest object.”

  Math is….Generalizations, “How can you use what you know to order the objects from longest to shortest.”

The materials reviewed for Reveal Math Grade 1 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 2, Number Patterns, Effective Teaching Practices, Use and Connect Mathematical Representations, “Making connections between different mathematical representations deepens a student’s understanding of the concept as well as the tools for problem solving.  Numerous representations are used to introduce and develop a student’s foundational knowledge of numbers through 120. Students make the connection between visual models and the numeral form of numbers. They use number charts, counters, connecting cubes, number lines, and drawings to help them visualize patterns in counting sequences. Students explore the number chart, identifying patterns in rows and columns of numbers. They notice how the digits change as they move across rows and down columns, counting on by 1s and 10s. Working with a variety of visual representations helps students build a conceptual understanding of place value and the sequence of written numerals. As you introduce each tool, model, or method, spend time questioning students to further their understanding.  

  • When introducing a tool, focus questions on the characteristics and patterns students see within the tool by allowing exploratory time.  

  • Pose questions that allow students to make connections between the different representation and the numerical form of the numbers

  • Provide opportunities for students to ask and answer their own questions based on what is still unclear about patterns in numbers through 120.” 

• In Unit 2, Number Patterns, Unit Overview, “Focus: Number Patterns: In this unit, students explore patterns in numbers to 120. Students will draw on their understanding of counting numbers to 100 and extend it to 120. They will notice that numbers greater than 100 follow the same pattern as numbers less than 100. The ones increase by 1 from 0 to 9 then repeat from 0. The tens stay the same until the ones restart at 0. Then the tens go up by 1 to 9. After 100, number patterns continue.” 

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 as well as content and mathematical practices. Examples include:

• Lesson 4-3, Doubles, Launch, Notice & Wonder, Pose Purposeful Questions, “The questions that follow may be asked in any order. They are meant to help advance students’ noticing and wondering about the two columns of buttons on the majorette’s jacket and are based on possible comments and questions that students may make during the share out. What do you notice about the number of buttons on each column? How many buttons are in each column? How can we find how many buttons there are in all?” 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 + 13).

• Lesson 6-4, Build New Shapes, Explore & Develop, Language of Math, “Have students practice the words: circle, hexagon, rectangle, square, and triangle by drawing and labeling a picture of each. Lead students to exchange drawings with a partner. Have partners describe one of the shapes to their partner until the partner can name the drawn shape.” 1.G.2, compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

• Lesson 13-1, Understand Equal Shares, Explore & Develop, Work Together, Common Misconceptions, “Students may think that because the partitioned parts are all the same shape that all parts are equal. Remind students equal shares means that the parts are the same size as well. Equal shares may or may not be the same shape as the whole.” 1.G.3, partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

The materials reviewed for Reveal Math Grade 1 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 2-3, Patterns on a Number Line, the materials identify standard 1.NBT.1, count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral. The lesson also identifies MP5, use appropriate tools strategically. 

• In Lesson 4-6, Choose Strategies to Add, the materials identify standard 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 + 13). The lesson also identifies MP5, use appropriate tools strategically, and MP3, construct viable arguments and critique the reasoning of others. 

The teacher materials contain explanations of the role of the specific grade-level mathematics, including prior and future content connections. Examples include:

• The Unit Overview includes the section, Coherence, identifying What Students Have Learned, What Students Are Learning, What Students Will Learn. In Unit 3, Place Value, Unit Overview, Coherence, “What Students Have Learned, Students composed and decomposed numbers up to 20 and explored representing 2-digit numbers up to 20. (Grade K) Students compared numbers 1 to 5. (Grade K) Students recognized patterns when reading and writing numbers. (Unit 2) What Students Are Learning, Students represent teen numbers with a ten and ones. Students group ones into tens and ones to make it easier to count and name the number. Students decompose 2-digit numbers in different ways. Students compare 2-digit numbers and then represent comparisons using the symbols >, <, and =. Students analyze the characteristics of a number line. Then compare two numbers on a number line.  What Students Will Learn, Students analyze other math symbols including the equal sign. (Unit 4) Students use number lines to subtract. (Unit 5) Students use number charts to add. (Unit 9)  Students represent 3-digit numbers. (Grade 2) Students add tens and ones. (Grade 2) Students compare 3-digit numbers (Grade 2).”

• 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. Each lesson overview contains a Coherence section that provides connections to prior and future work. In Lesson 11-1, Use Mental Math to Find 10 Less, Coherence, Previous, “Students subtracted single-digit numbers (Grade K). Students represented and solved various compare problems (Unit 10).” Now, “Students use mental math to find 10 less than a number. Students explain the patterns when finding 10 less.” Next, “Students use place value to subtract multiples of 10 from larger multiples of 10 (Unit 11). Students subtract within 100 (Grade 2).”

The materials reviewed for Reveal Math Grade 1 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 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, 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.”

• 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 researched 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 1 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 1, specifies the comprehensive materials list for the grade. The document specifies classroom materials (e.g., tangrams, index cards, building blocks, etc.), materials from a manipulative kit (e.g., pattern blocks, balance scales, student clocks, etc.), non-consumable teaching resources (e.g., place value chart, double ten frames, part-part-whole mat, etc.), and consumable teaching resources (tens cards, clocks, picture graph, etc).

In the Teacher Edition, each Unit Planner page lists materials needed for each lesson in the unit, for example, Unit 2, Number Patterns, Materials to Gather, each Lesson’s materials are given:

• Lesson 2-1 - Number Cards 1-120 Teaching Resource

• Lesson 2-2 - counters, Number Cards 1-120 Teaching Resource

• Lesson 2-3 - Number Cards 1-120 Teaching Resource, Number Chart 1-120 Teaching Resource, string or yarn, tape or clips

• Lesson 2-4 - blank number cubes (prepared with sides labeled 1, 1, 2, 3, 4, 4), Blank Number Lines 2 Teaching Resource, Number Cards 1-116 Teaching Resource

• Lesson 6-5 - counters, pennies, connecting cubes, or other small counting objects.

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

• Lesson 6-4, Build New Shapes, Materials, “The materials may be for any part of the lesson, Pattern Blocks 3 Teaching Resource, tangrams.” In Explore & Develop, Activity-Based Exploration, “Materials: Pattern Blocks 3 Teaching Resource (4 triangles, 1 semi-circle, 2 quarter circles, 1 square, 1 trapezoid, and 1 rectangle per student-group)”

• Lesson 9-2, Represent Adding Tens, Materials, “The materials may be for any part of the lesson, base-ten blocks, number cubes.” In Explore & Develop, Activity-Based Exploration, “Materials: base-ten blocks (9 tens rods and 9 ones units per student-group).”

The materials reviewed for Reveal Math Grade 1 meet expectations for including an assessment system that provides multiple opportunities throughout the grade 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 4, Addition within 20: Facts and Strategies, Unit Assessment (Form A), Item 1 lists “Use Doubles to Add (Sums to 20) “ 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 4, Addition within 20: Facts and Strategies, Performance Task, Ladybugs, “Students draw on their understanding of addition strategies to solve problems. Use the rubric shown to evaluate students’ work.”  Rubric, Part A, 2 Points: Student’s work shows proficiency with counting on to add using a number line. Student models and gives the correct sum. 1 Point:  Student’s work shows developing proficiency with counting on to add using a number line. Either a correct model or a correct sum is given. 0 Points: Student’s work reflects a poor understanding of counting on to add using a number line. Student fails to model the sum and gives an incorrect sum.”  

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 to use to remediate. 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 2, Number Patterns, Math Probe, Analyze The Probe, “Students circle the number that comes next in a pattern when counting by 1s. They justify their answers with words and/or symbols. Provide a number chart to 120 to help students as needed.” Students use the pattern of a number chart to identify what number comes next. Guidance is provided in an “If incorrect...Then” chart as to common misconceptions students have leading to an incorrect answer. Exercise 1 shows the counting sequence, “56, 57, 58, 59, 60, ___”.  “IF incorrect (student answers 59) THEN the student likely identifies the number 1 before (1 less) than the last number provided.” Sample Misconceptions, “Teacher: The students are counting by 1s from 55. They count 56, 57, 58, 59, 60,...Look at these numbers. Which number comes after 60? Student: 59. Teacher: Can you tell me or show how you know? Student: 56, 57, 58, 59, 60, 59, 60...I don’t know what comes next. [Student points to the number before 60 on the number chart.]” Take Action, “Revisit the use of a number chart and/or number line with counting activities in Lessons 2-1--2-3. Engage students in kinesthetic experiences. Create a number line on the floor to help students explore and  identify patterns.” Revisit the Probe, “Are there any questions that you still have about any of the items on this probe? Are there any answers you would like to change? Explain why you might want to change them.” Reflect on Your Learning provides students with a “thumbs up, thumbs sideways, thumbs down” to circle to show their understanding.

Exit Tickets are provided at the end of each lesson and evaluates students’ understanding of the lesson concepts and provides 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 5-8, Exit Ticket, “If students score 4 out of 4, Then have students do Additional Practice or any of the B or E activities.” The Build Proficiency (B) activities include Practice It! Game Station, Unknown Number Subtraction Race, Own It! Digital Station games, and Interactive Additional Practice. The Extend Thinking (E) activities include Use It! Application Station, Carnival Spinner Game, and Websketch Exploration.

The materials reviewed for Reveal Math Grade 1 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 6, Unit Assessment, Form A, supports the full development of 1.G.1 (Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size)) and MP7 (Look for and make use of structure) as students identify closed and open shapes as they draw lines to match provided shapes to the correct label. Item 1 “Is the shape closed? Match each shape to closed shape or not closed shape.” 

• In Unit 9, Addition within 100, Performance Task, is aligned to DOK 2 and supports the full development of 1.NBT.4 (Add within 100) and MP4 (Model with mathematics) as students demonstrate their understanding of addition with 2-digit numbers. School Fair, “Erik, Shaina, and Devon are working at the school fair. Part A. Devon works at the Bounce House. 25 children jump in the house during the first hour. 30 children jump in the house during the second hour. How many children jump in the house during the two hours? Make a drawing to represent the problem.”

• In Lesson 10-2, Represent and Solve Compare Problems Using Addition, Assess, Exit Ticket, supports the full development of 1.OA.1 (Use addition and subtraction within 20 to solve word problems) and MP2 (Reason abstractly and quantitatively) as students represent and solve compare word problems. Exercise 2, “Tran has 8 postcards. Simon has 3 more postcards than Tran. How many postcards does Simon have? Draw to show your thinking. ___ postcards”

• Unit 13, Equal Shares, Unit Assessment, Form A, supports the full development of 1.G.3 (Partition circles and rectangles into two and four equal shares) and MP3 (Construct viable arguments and critique the reasoning of others) as students analyze a circle partitioned into uneven shares and critique the reasoning of others. Item 10, “Chandra says this shape shows fourths. How can you help Chandra fix her mistake?”

The materials reviewed for Reveal Math Grade 1 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, and therefore are 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 1 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 1 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 1 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.