2nd Grade - Gateway 2

The materials reviewed for Reveal Math Grade 2 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 2-1, Understand Hundreds, Explore & Develop, Guided Exploration, “Students extend their understanding of place value to the hundreds place and build on their understanding that a group of 10 tens makes 1 hundred.” Math is Modeling, “Why is a tens rod a good way to show each student’s fingers?” Students work with a partner to skip count by 10s to find the value of 10 tens. Students record their thinking on paper using drawings or numbers. Facilitate Meaningful Discourse, “What do you notice about the number of ten rods and the value of the ten rods? What can you use to represent all 100 fingers? Why do you think we call the base-ten block for one hundred a flat? How do tens relate to a hundred?”  This activity supports conceptual development of 2.NBT.1, understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones.

• In Lesson 6-5, Use a Number Line to Subtract, Differentiate, Extend Thinking, Differentiation Resource Book, Exercise 1, “Malena sells jackets and gloves at a store. How can you use the information in the table and draw a number line to show the difference? Explain your answer.” Exercise 1, “How many more jackets are sold on Thursday than Tuesday?” This exercise supports conceptual development of 2.MD.6, represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.) 

• In Lesson 9-3, Represent Addition with 3-Digit Numbers with Regrouping, Explore & Develop, Bring it Together, “How can you use base-ten blocks to regroup 10 ones? How can you use base-ten blocks to regroup 10 tens?” During this lesson, students use base ten blocks to represent and solve 3-digit addition equations with regrouping. This provides students with the opportunity to develop conceptual understanding of 2.NBT.7, add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. 

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

• In Lesson 2-1, Understand Hundreds, Practice & Reflect, On My Own, Reflect, “Why is it helpful to group 10 tens as 100?” Students use their understanding of 100 as 10 groups of 10. Each student writes a reflection and students share their reflections with their classmates. This supports conceptual understanding of 2.NBT.1, understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones.)

• In Lesson 3-4, Understand Even and Odd Numbers, Practice & Reflect, On My Own, Exercise 5, “Cleo is washing strawberries. Is the number of strawberries even or odd? Explain how you know.” Students are shown five groups of two strawberries. This activity supports conceptual understanding of 2.OA.3, determine whether a group of objects (up to 20) has an odd or even number of members.

• In Lesson 5-6, Use a Number Line to Add, Practice & Reflect, On My Own, Exercise 1, “How can you use a number line to add? Fill in the numbers to complete the equation. 1. 33 + ___ = ___.” Students complete the equation by using bars shown on the number line. This supports conceptual understanding of 2.MD.6, represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

• In Lesson 9-2, Represent Addition with 3-Digit Numbers, Practice & Reflect, On My Own, Exercise 2, “What is the sum?  Use base-ten shorthand to show your work. 206 + 481 = ____.”  This activity supports conceptual understanding of 2.NBT.7, add and subtract within 1000.

The materials reviewed for Reveal Math Grade 2 meet expectations that the materials develop procedural skills and fluency throughout the grade level. The materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. 

The materials develop procedural 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 2 progress toward 2.OA.2, fluently add and subtract within 20 using mental strategies, and 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

• In Unit 4, Fluency Practice, Fluency Strategy, Exercise 1, “How can you make a 10 to find 7 + 6? Explain.”  Fluency Flash, “What is the sum?  Make a 10 to add.” Exercise 2, “9 + 4 = ___.”  These activities provide an opportunity to develop procedural skill and fluency of 2.OA.2, fluently add and subtract within 20 using mental strategies.

• In Lesson 5-1, Strategies to Add Fluently Within 20, Explore & Develop, Learn, “How can you find the total number of snack bars using mental math? You can use the strategies you know to find 8 + 5 using mental math. One way to find the sum is to count on. Another way to find the sum is to decompose one addend to make a 10.” This activity supports the development of 2.OA.2, fluently add and subtract within 20 using mental strategies. 

• In Lesson 6-3, Represent Subtraction with 2-Digit Numbers, Explore & Develop, Work Together, “There are 55 napkins in a stack. Some people use 31 of them. How can you use base-ten blocks to find how many napkins are left in the stack?” Teacher guidance encourages teachers to, “Point out different strategies students may use to demonstrate subtraction as they arise. Strategies might include removing or covering up the change number of base-ten blocks.” This activity builds procedural skill and fluency of 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction, from conceptual understanding of addition and subtraction.

• In Lesson 6-4, Represent 2-Digit Subtraction with Regrouping, Explore & Develop, Activity- Based Exploration, “Directions: Present this problem to students: 51 - 14 = ? Instruct students to represent and solve the equation using base-ten blocks.” Math is Thinking, “How can you take away 4 ones when there is only 1 ones unit?” This activity provides an opportunity for students to develop procedural skill and fluency of 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

• In Unit 6, Strategies to Fluently Subtract within 100, Math Probe, Directions, “Determine if the strategy shown is a correct approach to do this subtraction: 45 - 17. Do not actually perform the calculations.” Exercise 1, “45 - 10 - 7. Does this strategy work? Circle Yes or No. Explain why you chose Yes or No.” This exercise shows the development of the cluster 2.NBT.B, use place value understanding and properties of operations to add and subtract, relating to the procedural skill and fluency of 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

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

• In Lesson 5-3, Strategies to Fluently Add within 100, Practice & Reflect, On My Own, Exercise 7, “What is the sum? Show your thinking. 31 + 13 = ___.” This exercise provides students with an opportunity to independently demonstrate procedural skill and fluency of 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

• In Unit 6, Strategies to Fluently Subtract within 100, Unit Assessment, Form A, Item 14, “Armond eats 17 berries. He eats 9 blueberries. The rest are blackberries. How many blackberries does Armond eat? Show your work using addition to subtract.” Students have used this strategy throughout Unit 6 and this provides an opportunity for students to demonstrate procedural skill and fluency of 2.OA.2, fluently add and subtract within 20 using mental strategies.

• In Unit 6, Strategies to Fluently Subtract within 100, Unit Review, Performance Task, Reflect, “What are some different strategies for subtracting 2-digit numbers? Which strategy do you think is the most helpful?” This exercise allows students to independently demonstrate procedural skill and fluency of 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

• In Unit 7, Measure and Compare Lengths, Fluency Practice, Fluency Check, Exercise 3, “What is the sum or difference?  12 - 5 = ____.” This activity provides an opportunity for students to demonstrate procedural skill and fluency of 2.OA.2, fluently add and subtract within 20 using mental strategies.

The materials reviewed for Reveal Math Grade 2 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 4-2, Represent and Solve Take From Problems, Explore & Develop, Learn, “Jon brings some juice boxes to share with his class. His classmates drink 8 of the juice boxes. There are 11 juice boxes left. How many juice boxes did Jon bring?” Pose Purposeful Questions, “What words in the problem help you determine if you need to add or subtract? What is the unknown? How do the quantities relate to one another?” Students use a bar diagram to represent the problem. This exercise provides an opportunity for students to develop the mathematics of 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems. In Assess, Exit Ticket, Exercise 2, “Write an equation to represent the problem using ? for the unknown. Then solve. Kim has 11 lemons. She uses some lemons to make lemonade. There are 7 lemons left. How many lemons does Kim use to make lemonade? Equation: ______ Solve: _____.” This exercise allows students to independently apply mathematics of 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems.

• In Lesson 7-11, Solve More Problems Involving Length, Explore & Develop, Learn, “Diane draws a line 26 centimeters long. Oliver draws a line 15 centimeters long. How much longer is Diane’s line than Oliver’s line?” Pose Purposeful Questions, “What information is given in the problem? Have you solved similar problems to this before? Explain. What operation do you think you will use to solve this problem?” This exercise allows students to develop the mathematics of 2.MD.6, represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

• In Lesson 8-2, Solve Money Problems Involving Coins, Practice & Reflect, On My Own, Exercise 9, Extend Your Thinking, “Paris had some coins. Her mom gave her 2 dimes and 3 nickels. Now Paris has 49¢. How much money did Paris have to begin with? This exercise allows students to independently apply mathematics of 2.MD.8, solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

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

• In Lesson 4-5, Represent and Solve Take Apart Problems, Differentiate, Build Proficiency, Digital Additional Practice Book: Represent and Solve Take Apart Problems, Exercise 3, students independently engage in a non-routine application problem by writing a word problem about an everyday situation in which one addend is unknown, “a. Write a word problem that has an unknown addend. b. Use an equation to solve your word problem.” This exercise allows students to independently apply mathematics of 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems.

• In  Lesson 4-7, Represent and Solve Compare Problems, Practice & Reflect, On My Own, Exercise 6, Extend Your Thinking, “a. Write a word problem that compares two numbers using the word fewer. b. Use an equation to solve your word problem.” This exercise allows students to independently apply the mathematics of 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems.

• In Lesson 5-2, More Strategies to Add Fluently within 20, Explore & Develop, Activity-Based Exploration, “Distribute an even quantity of counters between 14 and 18 to each pair or small group of students. Give different quantities to each group to allow for a range of possible equations. Groups use the counters to represent a doubles fact with all their counters. Have students write the doubles fact shown by their counters and have them create and write down a near doubles fact that can be shown by their counters.” This exercise allows students to develop and apply mathematics of 2.OA.2, fluently add and subtract within 20 using mental strategies. 

• In Lesson 5-10, Solve One- and Two-Step Problems Using Addition, Launch, Numberless Word Problems, “What could the question be? Keisha has some flowers. Dale has some flowers. Bruce has some flowers.” Pose Purposeful Questions, “What operation can you use to solve this problem? How do you know? What information do you need to answer your question?” This exercise allows students to develop and apply the mathematics of 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems.

The materials reviewed for Reveal Math Grade 2 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, Use Arrays to Add, Unit Review, Exercise 13, “How can you skip count to find the number of cubes in the array?  Fill in the total.  ____ cubes.” Students are shown an array of 3 rows and 5 columns. This item provides an opportunity for students to demonstrate the conceptual understanding of 2.OA.4, use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

• In Lesson 4-8, Represent and Solve More Compare Problems, Math Probe, Exercise 3, “30 fish are in a big tank. Some are red and some are blue. There are 10 blue fish in the tank. How many red fish are in the tank? Solve the problem. Circle the correct equation. a. ? - 10 = 30, b. ? + 10 = 30, c. 30 + 10 = ?. Explain your choice.” Students show their work and explain their answer in the space provided. This exercise provides an opportunity for students to apply mathematics of 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems.

• In Unit 9, Strategies to Add 3-Digit Numbers, Readiness Diagnostic, Exercise 6, “What is the sum of 30 + 6? A. 9, B. 24, C. 36, D. 44.” Exercise 7, “What number is 10 more than 47? A. 37, B. 48, C. 57, D 147.” These exercises provide an opportunity for students to demonstrate the procedural skill and fluency of 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

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

• In Lesson 6-1, Strategies to Subtract Fluently within 20, Practice & Reflect, On My Own, Exercise 1, “How can you count on to subtract? Fill in the difference. 15 - 6 = ___.” Students are shown a number line with a dot at 6 and jumps ending at 15. Exercise 3, “How can you count back to subtract? Fill in the difference. 14 - 8 = ___.” Students are shown a number line with a dot at 14 and jumps ending at 6. Exercise 8, Extend Your Thinking, “Kylie has some erasers. She gives 4 of them to her sister. Now Kylie has 7 erasers. How many erasers does Kylie start with? Complete the subtraction equation and fill in the answer. Explain how you solved the problem.  ___ - 4 = 7.  Kylie started with ___ erasers.” Students explain how they solved the problem in the space under the word problem. These exercises give students opportunities to develop procedural skill and fluency and apply the mathematics and conceptual understanding of 2.OA.2, fluently add and subtract within 20 using mental strategies. 

• In Lesson 7-2, Measure Length with Feet and Yards, Differentiate, Build Proficiency, Digital Additional Practice Book: Measure Length with Feet and Yards, Exercise 1, students are provided with an image of a hammer and a ruler lined up underneath it, “What is the length of the hammer?” Students see the hammer as measuring 12 inches long and use their understanding that 12 inches equals one foot to write their answer. In Exercises 5 and 6, students apply what they know about measurement to explain their thinking. Exercise 5, “What unit would you use to measure the length of a doll? Explain.” Exercise 6, “What tools would you use to measure the length of a cafeteria table? Explain your thinking.” These exercises provide opportunities for students to demonstrate procedural skill and fluency as they measure length with a ruler and apply the mathematics of 2.MD.1, measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

• In Lesson 11-1, Understand Picture Graphs, Differentiate, Digital Additional Practice Book: Understand Picture Graphs, Exercise 6, “Molly read 5 books, Sal read 8 books, and Xin read 1 less book than Sal. How can you represent this data using a picture graph?”This exercise provides an opportunity for students to demonstrate procedural skill and fluency, and apply the math and conceptual understanding of 2.MD.10, draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

The materials reviewed for Reveal Math Grade 2 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 5-7, Decompose One Addend to Add, Explore & Develop, Activity-Based Exploration, “Have students work in groups of two. One partner will choose a 2-digit number less than 50. The other partner will choose a number less than 50 that ends in 6, 7, 8, or 9. Groups will write an addition expression with their numbers, and decompose both addends to solve. Then have students discuss if they would get the same sum decomposing only one addend. Instruct students to explore decomposing one addend and counting on using the teaching resource to show the addition.” Students engage with the full intent of MP1 as they analyze and make sense of the problem, and use a variety of strategies to decompose one addend to add.

• In Lesson 6-9, Solve One-Step Problems Using Subtraction, Practice & Reflect, On My Own, Exercise 6, Extend Your Thinking, “Write a one-step subtraction word problem with 2-digit numbers. Use any strategy to solve it.” Students engage with the full intent of MP1 as they use a variety of strategies to solve a one-step word problem.

• In Lesson 11-3, Solve Problems Using Bar Graphs, Launch, Pose Purposeful Questions, students are shown a bar graph that represents animals (duck, squirrel, rabbit, goose) and number of animals (8 duck, 6 squirrel, 2 rabbit, 3 goose). “What information do you have?  What part of the bar graph is missing? Why is that missing part important for analyzing the data?” Students engage in the full intent of MP1 as they analyze, discuss, and make sense of a bar graph with missing information.

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

• In Lesson 7-8, Relate Centimeters and Meters, Practice & Reflect, On My Own, Exercise 1, “What is the length of the whiteboard in meters?” Students measure. “Will the measurement of the whiteboard have more centimeters or more meters? Circle the answer. Students engage in the full intent of MP2 as they consider units involved in a problem, and attend to the meaning of quantities while measuring the classroom whiteboard.

• In Lesson 8-3, Solve Money Problems Involving Dollar Bills and Coins, Differentiate, Digital Additional Practice Book: Solve Money Problems Involving Dollar Bills and Coins, Exercise 1, “What is the value of the group of coins?” Students see a picture of 9 coins: 2 quarters, 3 dimes, 1 nickel, and 3 pennies. Students engage in the full intent of MP2 as they consider the units involved, find the total value of coins by skip counting, and then add the values.

• In Lesson 8-5, Be Precise When Telling Time, Explore & Develop, Work Together, “As students share their responses, ask them to explain how they decided whether the activity would normally happen during the a.m. or p.m. Evan completes the following activities. Does he complete each activity in the a.m or p.m.? Soccer practice 5:15 ___; math class 11:20 ___; homework 6:40 ___.” [p.m., a.m., p.m.] Students engage with the full intent of MP2 as they understand the relationships between problem scenarios and mathematical representations of a.m. and p.m..

The materials reviewed for Reveal Math Grade 2 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 2-3, Read and Write Numbers to 1,000, Explore & Develop, Activity-Based Exploration, students construct viable arguments as they use base-ten blocks, numerals, words, and decomposition to create 3-digit numbers and justify their values. Math is... Explaining, “How can you prove that each number form shows the same number? Students justify their thinking and broaden their understanding of 3-digit numbers.” 

• In Lesson 5-4, Use Properties to Add, Explore & Develop, Learn, students construct viable arguments as they learn that addends added in any order result in the same sum, “Leah and Rosh find 15 large paperclips and 21 small paperclips. How can they show how many paperclips they find altogether? Leah and Rosh use base-ten blocks and equations to show their thinking. Leah 15 + 21 = 36 (one ten and five ones and two tens and one one are shown in base-ten blocks). Rosh 21 + 15 = 36 (two tens and one one and one ten and five ones are shown in base-ten blocks).” Bring It Together, Elicit Evidence of Student Thinking, “Explain the differences and similarities between both representations. Does the order of the addends matter? How do you know? How can you show that you can add numbers in any order?”  

• In Lesson 9-4, Decompose Addends to Add 3-Digit Numbers, Launch, Which Doesn’t Belong?, students construct viable arguments as they discuss and are shown: 237 + 141, 2 hundreds 3 tens 7 ones and 1 hundred 4 tens and 1 one in base ten blocks, 2 hundreds 3 tens 7 ones and 1 hundred 4 tens and 1 one in base-ten shorthand, and 200 + 100 + 30 + 40 + 7 + 1. Teaching Tip, “Have students discuss similarities and differences they notice with a partner. Encourage them to determine and justify multiple options that don’t belong. This may help students to consider and understand different perspectives prior to engaging in discussion with the whole group.”  

• In Lesson 10-8, Explain Subtraction Strategies, Practice & Reflect, On My Own, Exercise 8, Extend Your Thinking, students construct viable arguments when they justify an efficient strategy to solve, “Juan wants to sell 364 tickets to a school play. He already sold 198 tickets. How many tickets does Juan have left to sell? Use two different subtraction strategies to solve and explain what strategy is more efficient for this problem.”

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

• In Lesson 5-4, Use Properties to Add, Practice & Reflect, Exercise 9, Error Analysis, students critique the reasoning of others as they use the Commutative Property, “Mae says the sum of 23 + 30 is 53. Dan says 30 + 23 has a different sum. How do you respond to Dan?”  

• In Lesson 9-4, Decompose Addends to Add 3-Digit Numbers, Practice & Reflect, Exercise 5, Error Analysis, students critique the reasoning of others. “Imani adds 125 + 38 by place value. She decomposes the addends as 100 + 20 + 5 and 300 + 80. Imani says the sum is 505. How do you respond to her?” 

• In Lesson 6-7, Adjust Numbers to Subtract, Practice & Reflect, On My Own, Exercise 7, students critique the reasoning of others as they explain their thinking, “Beth uses the adjusting strategy to solve 89 - 71. She writes 90 - 70. Is Beth adjusting the numbers correctly? Explain why or why not.” 

• In Lesson 7-3, Compare Lengths Using Customary Units, Explore & Develop, Learn, students critique the reason of others as they determine a way to compare the lengths of two bracelets to answer the question, “Serena thinks the two bracelets are the same length. Jamal thinks his bracelet is longer. How can Serena and Jamal find out who is correct?”

• In Lesson 10-8, Explain Subtraction Strategies, Launch, Notice & Wonder, Math is...Mindset, “How can you exchange ideas with someone who may think differently than you?” This guidance for teachers helps guide students in critiquing the reasoning of others as they explore different strategies to solve the same problem. Relationship Skills, Advocacy, “After students work through the Notice & Wonder routine independently, have them share their reasoning with a partner and advocate for their chosen strategy. If students have chosen different strategies or found different solutions, invite them to work together to understand one another’s reasoning. Remind students that strong learners are willing to learn from not only their teachers but also their peers.”

The materials reviewed for Reveal Math Grade 2 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 2, Place Value to 1,000, Unit Overview, Math Practices and Processes, Model with Mathematics, “This unit provides students with the opportunity to model mathematics in a number of different ways. Students use base-ten blocks and the place-value chart to understand the value of each digit in a 3-digit number and determine different ways to decompose. They are also able to connect their answer to a situation which helps them determine if their answers are reasonable and, if not, they are able to go back and adjust their process to come up with a more appropriate response. This may make some students uncomfortable, so some suggestions for building student’s confidence in applying place value understanding include: Relate models back to the problem situation to form connections. Discuss the similarities and differences between different representations so students can identify those they understand and why.”

• In Lesson 7-10, Solve Problems Involving Length, Explore & Develop, Work Together, students use strategies they know to model and represent a real world situation. “Adele has 33 yards of ribbon. She uses some ribbon. Now she has 16 yards of ribbon. How much ribbon does Adele use? Make a drawing and write an equation to help you solve the problem.”

• In Unit 10, Strategies to Subtract 3-Digit Numbers, Performance Task, Part B, students represent a real world situation and use their knowledge of subtraction with regrouping to solve, “There are 256 people that work at the aquarium. There are 137 paid workers. The rest of the workers are volunteers. How many volunteers work at the aquarium? Represent and solve the problem.” 

• In Lesson 11-6, Show Data on a Line Plot, Explore & Develop, Work Together, students see a table titled, “Length of Hair”, with columns for length (in inches) and number of students and make a representation of the data provided. “How can you represent the data using a line plot? Draw a line plot.” 

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 6-3, Represent Subtraction with 2-Digit Numbers, Explore & Develop, Activity- Based Exploration, students use a variety of tools to solve 2-digit subtraction problems, “Distribute materials to groups of 3 students. Write this problem: There are 45 sandwiches at the deli. Some people buy 21 of them. How many sandwiches are left? Groups will write an equation to represent the problem. Then have them represent and solve the equation using one of the tools they were given. Each student in the group should use a different tool to solve the problem.” Teachers provide base-ten blocks and Blank Open Number Lines Teaching Resource. 

• In Lesson 7-1, Measure Length with Inches, Practice & Reflect, Digital On My Own: Measure Length with Inches, students use technological tools to move the objects presented above a ruler to measure each length and explain their thinking. Exercise 1, students are shown a paintbrush, “Move the object to measure. What is the length of the paintbrush? Use the inch ruler to measure. Enter the measurement and the units.” Exercise 4, students are shown a glue stick, “Move the object to measure. What is the length of the glue stick? Use the inch ruler to measure. Show your work or explain your thinking. Will the glue stick fit in a box that has a length of 3 inches?” Students use the writing tool in the provided space to show their work or explain their thinking. 

• In Unit 9, Strategies to Add 3-Digit Numbers, Unit Overview, Math Practices and Processes, “At the elementary level, students are introduced to different tools that can be used to help them solve problems more efficiently. Students must understand how to use these tools, as well as why they are helpful. As tools are used, it is important to relate the representation shown in the tool back to the problem students are trying to solve. Students should be given the opportunity to practice with each tool until they are competent with it. Once students are comfortable with the tools, they can be given the choice of which tools to use to solve. Students should learn that some tools are better used with certain problems than others and some students may have preferences for certain tools. This helps students see that there can be multiple solving strategies that can be successful.” 

• In Lesson 9-5, Decompose One Addend to Add 3-Digit Numbers, Explore & Develop, Guided Exploration, students choose a tool to help them use decomposing an addend to add 3-digit numbers, “How can you determine which addend to decompose? Why might we decompose 328 instead of 625? Think About It: Which tool can you use to help you add hundreds, tens and ones?”

The materials reviewed for Reveal Math Grade 2 meet expectations that there is intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.  

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

The instructional materials address MP6 in the following components:

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

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

• In Lesson 3-2, Patterns When Skip Counting by 5s, Explore & Develop, Guided Exploration, students attend to precision as they skip count by 5s, “Students will skip count by 5s and identify and describe the place-value patterns using number charts and number lines.”  Math is...Precision, “Why would you choose to skip count by 5s instead of counting by 1s? Have students discuss if it is faster to count a large number by 1s or to skip count by 5s. This discussion will help students build understanding of efficient strategies for counting.” 

• In Lesson 8-5, Be Precise When Telling Time, Explore & Develop, Activity-Based Exploration, students attend to precision as they use a timeline to connect daily activities to a time and determine if it occurs in the a.m. or p.m., “How can you determine the order of events during the day? What activity might you do at midnight? What activity might you do at noon? Using the timeline, how would you define a.m. and p.m.?” 

• In Lesson 11-1, Understand Picture Graphs, Practice & Reflect, On My Own, Exercise 1, students attend to precision as they use data presented in a tally chart titled “Favorite Sport” to create a picture graph, “How can you represent the data using a picture graph? Use the tally chart to make a picture graph.” 

• In Lesson 11-4, Collect Measurement Data, Explore & Develop, Activity-Based Exploration, students attend to precision as they collect and represent hand measurements of classmates using an inch ruler and organizing data in a tally chart, Directions: Have students measure a classmate’s hand so every student’s hand gets measured in the group. Remind students to measure their classmate’s hand twice. Math is...Precision: Why should you measure a length more than once? Students understand the need to confirm measurements for accuracy and precision.” 

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

• In Lesson 2-4, Decompose 3-Digit Numbers, Explore & Develop, Language of Math, students attend to the specialized language of mathematics as they understand what the word decompose means, “Help students build understanding of the word decompose by considering the prefix de- and the word compose. The prefix de- can mean to undo an action. Give students separate index cards with the words compose, frost, and clutter and another with the prefix de-. Have student pairs work together to identify the meanings of each word with and without the prefix de-.” 

• In Lesson 5-7, Decompose One Addend to Add, Explore & Develop, Pose the Problem, students attend to the specialized language of mathematics as they discuss the decomposition of one addend in the addition expression 45 + 27, “What are the addends? What does decompose mean in your own words?”

• In Lesson 8-5, Be Precise When Telling Time, Explore & Develop, Language of Math, students attend to the specialized language of mathematics as they understand the meaning of a.m. and p.m., “Add the vocabulary cards: a.m. and p.m. to the math word wall. Have students work with a partner to each create a sentence using a.m. and p.m.”

• In Lesson 12-4, Understand Equal Shares, Differentiate, Reinforce Understanding, Differentiation Resource Book, students attend to the specialized language of mathematics as they use “halves” and “fourths” to describe the partitioning of shapes into equal shares, Exercise 1, students see a rectangle with no lines on it, a rectangle with a vertical line dividing it into two halves, and a rectangle that is divided into fourths by a horizontal and a vertical line, “How can you partition the shape into 4 equal shares? Draw a line to partition the rectangle into ____ . [halves] Then draw a line that creates ____. [fourths]”

The materials reviewed for Reveal Math Grade 2 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. 

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 Unit 3, Patterns within Numbers, Unit Overview, Math Practices and Processes, Look for and Make Use of Structure, “Identifying patterns in numbers is a trait of mathematically proficient students. Starting in Grade 1, students identified patterns when counting to 120. Students now build their skills to identify patterns within 1,000. Students first identify patterns when counting by 1s by completing a number chart. This then leads students to identify patterns when counting by 5s, 10s, and 100s. This understanding of patterns, and how they can be used, helps students develop number sense, which is the foundation for their future work in addition and multiplication. To help students identify and use these patterns, students need opportunities to interact with them. Some suggestions for identifying patterns includes: 

  • Encourage students to identify, share, and discuss all patterns they see when counting. 

  • Provide students with opportunities to use these patterns to find unknown numbers within a skip counting sequence. 

  • Students discuss how they can use the patterns they identified to add numbers.” 

           This guidance for teachers allows students to engage in the full intent of MP7, look for 

           and make use of structure.

• In Lesson 8-1, Understand the Value of Coins, Practice & Reflect, On My Own, Exercise 8, Extend Your Thinking, students look for and explain the structure of the problem as they skip count and then find the total value of a group of coins, “Alan has 2 dimes and 3 nickels. How can you find the total value of Alan’s coins? Explain your thinking.” 

• In Lesson 12-1, Recognize 2-Dimensional Shapes by Their Attributes, Explore & Develop, Activity-Based Exploration, students look for patterns and structures as they make generalizations to recognize that they can use sides, angles and vertices to identify polygons. Students are shown polygons on teaching resource 2-Dimensional Shapes, “How can you group the shapes by different attributes? What do some of the shapes have in common? Which shapes are triangles? Why? Which shapes are quadrilaterals? Why? Which shapes are pentagons? Why? Which shapes are hexagons? Why? What attributes can you use to identify different shapes?” 

•  In Lesson 12-6, Partition a Rectangle into Rows and Columns, Practice & Reflect, On My Own, Exercise 2, students use patterns to partition a rectangle into rows and columns and develop a repeated addition equation to correspond to it, “How many rows, columns, and squares is the rectangle partitioned into? Write an equation to find the total number of squares?” 

Examples of intentional development of students looking for and expressing regularity in repeated reasoning, including guidance for teachers to engage students in MP 8 include:

• In Lesson 3-7, Use Arrays to Add, Practice & Reflect, On My Own, Reflect, students notice repeated calculations to understand algorithms and make generalizations as they relate arrays and repeated addition, “How are arrays and repeated addition related?” 

• In Lesson 6-7, Adjust Numbers to Subtract, Assess, Exit Ticket, Item 1, students use repeated reasoning to evaluate the reasonableness of their answers as they subtract 2-digit numbers by adjusting numbers, “Which way shows how to adjust the numbers to subtract? Choose all correct answers. 58 - 23.” The answer choices are “a. 60 - 21, b. 60 - 25, c. 55 - 20, d. 61 - 20.” 

• In Lesson 7-9, Estimate Length Using Metric Units, Explore & Develop, Activity-Based Exploration, students look for and express regularity in repeated reasoning as they use everyday items in the classroom as tools to estimate length. Teachers prompt students to use some ones units to measure items in centimeters, and to use their own arm span to measure items in meters, “What item(s) did you use to measure in centimeters? In meters? Why was it helpful to have several 1-centimeter items? Can you still estimate the measurement with only one unit? Why might estimating length be useful?”

• In Unit 10, Strategies to Subtract 3-Digit Numbers, Unit Overview, Math Practices and Process, Look for and Express Regularity in Repeated Reasoning, “Many of the strategies and methods used in this unit show students shortcuts for solving efficiently. By recognizing place-value patterns, students are able to subtract tens or hundreds mentally. Strategies such as decomposing or adjusting numbers also allow for mental subtraction. Students should see that adjusting the problem into friendlier numbers allows for easier solving. By allowing students time to explore these different methods and compare them, they should see that these general methods which are used for 2-digit numbers can simply be adjusted for 3-digit numbers. Some suggestions for helping students see these patterns and shortcuts include: 

  • Having students use multiple methods to solve the same problem and comparing their solutions with one another.  

  • Providing multiple opportunities to use each strategy to identify solving patterns that are always true. 

  • Discussing how they used the strategies for 2-digit numbers and 3-digit numbers and whether they think these same strategies could be used for even greater numbers.”