1st Grade - Gateway 2

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.”