2nd Grade - Gateway 2

The materials reviewed for Snappet Math Grade 2 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Materials develop conceptual understanding throughout the grade level. According to the Snappet Teacher Manual, 1. Deeper Learning with Snappet Math, conceptual understanding is a part of the design of the materials. Balancing Rigor states, “Each lesson embeds Conceptual Learning as the foundation and is designed to progress students along the learning path that begins with Student Discovery, transitions to Applying in Concrete pictorial representations, and then provides opportunities for Processing in Abstract representations.” According to the Grade 2 Teacher's Edition Volume 1, “Snappet’s Student Discovery Phase of the lesson design helps teachers present important math concepts using hands-on manipulatives, games, and classroom activities. Virtual manipulatives are also provided for guided practice, and adaptive practice. The lesson design includes Concrete Pictorial Representations that utilize models and visuals during the lesson instruction. This approach helps teachers deliver high-quality instruction and builds a deeper understanding of math concepts for students.” Examples include:

• Unit 1: Numbers, Lesson 1.3, Instruction & Guided Practice, Exercise 1b, students develop conceptual understanding as they complete a place value chart to represent a number. “I have 5 hundreds, 9 tens, and 3 ones. What number do I have?” Teacher tip, “If students struggle to write the number, encourage them to draw the base-10 blocks to represent the problem. Ask: How many flats would you draw? [5]. How many rods? [9]. How many ones? [3].” 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.)

• Unit 2: Addition and Subtraction, Lesson 2.8, Instruction and guided practice, Exercise 1d, students develop conceptual understanding as they use addition to find the total number of objects in arrays. “Now count from top to bottom. The sheet has ___ columns of ___ stickers. Count on. There are ___ + ___ + ___ = ___.”  Teacher tip, “Guide the students to count the number of stickers in each row. Say: You can find the total number of stickers by adding the number of stickers in each row. Ask: What do you notice about the groups? [They are equal.] What strategy can you use to count equal groups efficiently? [Sample answer: I can skip count.]” 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)

• Unit 6: Measurement, Lesson 6.12, Instruction & Guided Practice, Exercise 1f, students develop conceptual understanding as they use a centimeter ruler as a number line to find sums and differences. “Lucy’s new pencil was ___ cm long. Now, her pencil is ___ cm. The pencil is ___ cm smaller than when it was new.” Teacher tip, “Have students look at the ruler. Elicit that both pencils end at the 17-centimeter mark. Ask: How can you find the difference in the lengths? [Find where the shorter pencil starts on the ruler.]” 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.)

According to Snappet, “Student Discovery, Lessons begin with hands-on learning. Research supports that new concepts are best learned using manipulatives in real, informal situations. This learning serves as the basis for conceptual understanding. Apply in Concrete Actual situations are presented as a concrete representation using models and visuals. Students learn to establish the relationship between the actual situation and the concrete representation.” Guidance is given for the teacher to use with students who are struggling to complete the Independent Practice items. In the Snappet Teacher Manual, Section 3.2, states, “When the teacher has completed the instruction for the day, students are given the opportunity to practice independently on their new skills. Each lesson includes approximately ten practice problems that are scaffolded for difficulty and are common for the whole class. Students are then presented with ten adaptive exercises that are customized to their skill levels….While students are working on their practice problems, the teacher can monitor the progress of their class in real-time. If the teacher notices a student or groups of students struggling with their exercises, they can intervene and provide support targeted to the needs of the students. At the same time, students that are “getting it” can move directly into adaptive practice and receive more challenging practice problems customized to their skill levels.” Examples include:

• Unit 2: Addition and Subtraction, Lesson 2.8, Independent Practice, Exercise 2e, students find the total number of candles using equal groups. “___ + ___ + ___.” Students answer, “___ total candles.” The teacher can support struggling students with teacher direction: “Guide students on to recognize that even though the candles are in groups of 4, they can be broken down into groups of 2, which is easier. Say: All even numbers can be counted by 2s.” 2.OA.4 (Use addition to find the total number of objects arranged in rectangular arrays and write an equation to express the total as a sum of equal addends.)

• Unit 5: Add and Subtract Within 1,000, Lesson 5.3, Independent Practice, Exercise 2e, students use a number line to solve an addition equation. “$$536+120=$$___.”  The teacher can support struggling students with teacher direction: “Ask: When you are adding numbers, why should you put the first number on the left side of the number line? [So you have room to make jumps to the right to increase the number.]” 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. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.)

• Unit 5: Add and Subtract Within 1,000, Lesson 5.8, Independent Practice, Exercise 2d, students add using strategies based on place value. “$$284+144$$. Add the hundreds together first. Then the tens. Last, the ones. 284+144=___$$+$$ ___ + ___ . 284+144=___.” The teacher can support struggling students with teacher direction: “Again, compare the digits of the addends and the sum by place value. Ask: Why don’t the sums match for some of the digits? [because there are 12 10s, and the 10s digit in the sum has to be 0 through 9]” 2.NBT.9 (Explain why addition and subtraction strategies work, using place value and the properties of operations.) 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 reviewed for Snappet Math Grade 2 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

According to the Snappet Teacher Manual, “In Snappet, students will build understanding by problem-solving using Models, Number Sentences, and Word Problems to develop mathematical fluency.” Process in Abstract: “Concrete situations are replaced with abstract mathematical symbols such as dashes, squares, or circles. Different schemas, models and step-by-step plans are often used for this. Learning takes place at a higher, more abstract level, preparing students for practicing procedural skills, developing fluency, and applying concepts flexibly to different situations.” The Instruction & Guided Practice problems provide ongoing practice of procedural skills within lessons. Examples include: 

• Unit 2: Addition and Subtraction, Lesson 2.4, Instruction & Guided Practice, Exercise 1n, students develop procedural skill and fluency as they add to 20 using strategies. “$$8+8=16$$, 8+7=___.” Teacher tip, “Ask: How can memorizing your doubles facts help you solve problems that are not doubles? [If you know your doubles facts, then you can quickly add a lot of numbers, even if you have to add one or take one away from an addend. Otherwise, you should probably use a different strategy.].” 2.OA.2 (Fluently add and subtract within 20 using mental strategies.)

• Unit 3: Add and Subtract Within 100, Lesson 3.3, Instruction & Guided Practice, Exercise 1f, students develop fluency as they add and subtract using a number sentence. “Add 58+6 using a number sentence. 58+6=___. What step(s) did you use?” Teacher tip, “Ask: Why did you add 2 first? [Because adding 2 gets to 60, then I can add 4 more to get 64.]” 2.NBT.5 (Fluently add and subtract within 100 using strategies based on place value, properties, operations, and/or the relationship between addition and subtraction.); 2.NBT.9 (Explain why addition and subtraction strategies work, using place value and the properties of operations.)

• Unit 3: Add and Subtract Within 100, Lesson 3.12, Instruction & guided practice, Exercise 1i, students develop procedural skill and fluency as they add within 100. “$$54+26+12=$$___. Think of partial sums. 4+__$$+$$__$$=$$__ ; 50+__$$+$$__$$=$$__ ; $$+$$$$=$$__.” Teacher tip, “Compare and contrast the partial sums method with the vertical addition method in a class discussion. Elicit that the partial sums method requires more writing and that vertical addition is essentially a shorthand version of partial sums.” 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 the Snappet Teacher Manual, Lesson Structure, “Automating and memorizing, Automating and memorizing is embedded in the learning goals of the Snappet program where this skill is important. The moment that Snappet recognizes the student has mastered the arithmetic knowledge and skill of the learning goal, the system automatically switches to tasks aimed at automation and memorization. This is accomplished by using exercises that students must completed in a given amount of time. Using this method, identifies whether a student knows the answer by automation or memorization or if they are still working out the calculations. If the student does not provide the correct answer in the given amount of time, then the program will allot more time for that exercise on the next attempt. The Snappet program will recognize when a student has sufficiently automated and memorized a goal and will adapt accordingly.” Students have opportunities to independently demonstrate procedural skills and fluency throughout the grade. Examples include:

• Unit 1: Numbers, Lesson 1.7, Independent Practice, Exercise 2f, students demonstrate procedural skill and fluency as they count backward from 1,000 by 1s. “___, ___, ___, 792, 793.” 2.NBT.2 (Count within 1,000; skip-count by 5s, 10s, and 100s.)

• Unit 2: Addition and Subtraction, Lesson 2.6, Independent Practice, Exercise 2h, students develop procedural skills and fluency as they subtract within 20 using mental strategies. “$$4-2=$$___.” 2.OA.2 (Fluently add and subtract within 20 using mental strategies.)

• Unit 5: Add and Subtract Within 1,000, Lesson 5.2, Independent Practice, Exercise 2d, “What is 100 less than 837? 727, 737, 827, 937?” 2.NBT.8 (Mentally and 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900.)

The materials reviewed for Snappet Math Grade 2 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics. 

Students have opportunities to engage with multiple routine and non-routine application problems with teacher support and independently. Snappet Teacher Manual, Performance Tasks, “Each grade-level course includes Performance Task Lessons that are designed to be a cumulative lesson encompassing multiple mathematical concepts. These lessons are designed as group projects or whole class discussion opportunities.” 

Examples of teacher-supported routine and non-routine applications of mathematics include:

• Unit 4: Solve Word Problems, Lesson 4.2, Independent Practice, Exercise 2f, students use drawings or equations to solve word problems in a routine application. “Faith made 88 cupcakes for the bake sale. In the first hour, she sold 45 cupcakes. During the day, she sold 33 more cupcakes. How many cupcakes does Faith have left? Use a model to solve the problem. Hint: Think of using a bar diagram, an array, an equation, etc. to answer the question. Faith has ___ cupcakes left.” Teacher tip, “Ask: Will Faith have more than or fewer than 88 cupcakes left? [less than] Do you add or subtract to find how many left? [subtract] How can you subtract? [Sample answer: I can subtract 88 – 45 and then subtract 33 from the difference.]” 2.OA.1 (Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.)

• Unit 4: Solve Word Problems, Lesson 4.7, Instruction & Guided Practice, Exercise 1a, students use play money to solve addition problems in a non-routine application. “Each partner adds less than 10 dollars in play money to their wallet. Then the partners add their money together.” Teacher tip, “Divide the class into pairs. Give each pair 18 in 1 bills of play money. In slide 1a, students put up to $9 in their wallets. Pairs add the amount in each wallet. In slide 1b, students write an equation to represent their addition in slide 1a. Remind students that the second box/answer should be an operation.” 1b, “Add less than 20 dollars in play money to your wallet. Then, give some of that money to your partner.” 2.OA.1 (Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.) 

• Unit 6: Measurement, Lesson 6.10, Instruction & Guided Practice, Exercise 1c, students solve word problems in a routine application. “In pairs, students use 3” and 8” lengths of ribbon or yarn to model word problems. They solve by drawing or writing equations on dry erase boards, then use a ruler to measure and check their work. Sample problems: 1. You have a 3” ribbon and an 8” ribbon. How much ribbon do you have in all? 2. How much longer is your ribbon than your partner’s ribbon? 3. You have 8” of ribbon. You cut off 2”. How much ribbon do you have left?” 2.MD.5 (Use addition and subtraction with 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings(such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.)

• Unit 7: Time and Money, Lesson 7.9, Instruction & Guided Practice, Exercise 1c, students solve word problems using money in a non-routine application. “Students work in groups of three or four. One student selects a random amount of bills and coins. Together, the group determines the total. Another student selects four items to take away from the first student’s total and returns it to the original set of money. Together the group determines how much was taken away and how much is left. Students work to write an equation to represent the total and the change.” 2.MD.8 (Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.)

Materials provide opportunities for students to independently demonstrate multiple routine and non-routine applications of mathematics throughout the grade level. Examples of independent demonstration of routine and non-routine applications of mathematics include:

• Unit 1: Numbers, Lesson 1.3, Independent Practice, Exercise 2c, students determine the value of digits based on where it is located on a place value chart in a non-routine application. Students see a place value chart with hundreds, tens, and ones.  There is a 5 in the H column, a 2 in the T column and a 9 in the O column.  “What is the value of the 5? ___” 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.)

• Unit 4: Solve Word Problems, Lesson 4.5, Independent Practice, Exercise 2j, students solve comparison problems in a routine application. “A puzzle has 10 more middle pieces than border pieces. There are 12 border pieces. How many middle pieces does this puzzle have?” 2.OA.1 (Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all position, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.)

• Unit 7: Time and Money, Lesson 7.9, Independent Practice, Exercise 2h, students solve problems involving money in a routine application. “Cole has three quarters, five dimes, and two pennies. He gives four dimes to his niece. How much money does he have left? ___ ¢” 2.MD.8 (Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, and using $ and ¢ symbols appropriately.)

• Unit 8: Data, Lesson 8.3, Independent Practice, Exercise 2h, students use a bar graph to solve comparison problems in a routine application. “Which grade has the fewest teachers?” Students select from answers “first, second, third, fourth” 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.)

• Unit 9: Geometry, Lesson 9.5, Independent Practice, Exercise 2j, students draw a cube using two squares and attach the squares with diagonals to make the cube in a non-routine application. Students have a dot section to draw the cube in. “Draw a cube.” 2.G.1 (Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.)

The materials reviewed for Snappet Math Grade 2 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

The materials address the aspects of rigor, however, not all are addressed equally. Heavy emphasis is placed on conceptual understanding, procedural skills, and fluency. All three aspects of rigor are present independently throughout the materials. Examples include:

• Unit 2: Addition and Subtraction, Lesson 2.2, Independent Practice, Exercise 2h, students develop procedural skill and fluency as they use the structure of ten to add “$$7+8=$$___.” 2.OA.2 (Fluently add and subtract within 20 using mental strategies.)

• Unit 4: Solve Word Problems, Lesson 4.5, Independent Practice, Exercise 2c, students apply their understanding of subtraction to solve a word problem. “There are 15 chickens on the farm. There are 40 roosters on the farm. How many more roosters are there? Complete the equation to solve the problem. ___$$-15=$$___ There are ___ more roosters.” Teacher tip, “Review that a comparison is used to find how many more or how many less one group has than another. Elicit from students that subtraction is used in a comparison problem.” 2.OA.1 (Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.) 

• Unit 5: Add and Subtract Within 1,000, Lesson 5.6, Instruction and guided practice, Exercise 1e, students extend their conceptual understanding using an open number line to add. “$$297+68=$$___ . Add 297+68 in two jumps. First the tens, then the ones.” Teacher tip, “Ask: Why does the slide show the first jump as only 3? [Because the result of adding 3 was a round number, 300, which makes it easier to add the rest of the number to get the sum]” 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. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study. Heavy emphasis is placed on procedural skills and fluency and teacher-guided conceptual understanding. Examples include:

• Unit 2: Addition and Subtraction, Lesson 2.8, Instruction & Guided Practice, Exercise 1b, students develop conceptual understanding alongside procedural skill and fluency as they count equal groups using objects. “How many stickers are in each row? How many stickers does this sheet have? Count by jumps.” Teacher tip, “Guide the students to count the number of stickers in each row. Say: You can find the total number of stickers by adding the number of stickers in each row. Ask: What do you notice about the groups? [They are equal.] What strategy can you use to count equal groups efficiently? [Sample answer: I can skip count.]” 2.OA.4 (Use addition to find the total number of objects arranged in rectangular arrays and write an equation to express the total as a sum of equal addends.)

• Unit 3: Add and Subtract Within 100, Lesson 3.2, Instruction & Guided Practice, Exercise 1j, students extend conceptual understanding and procedural skills as they use an empty number line to solve problems. “Add 35 and 23 on the number line. Use as few jumps as possible. 35+23=___.” 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.)

• Unit 8: Data, Lesson 8.2, Independent Practice, Exercise 2d, students use all three aspects of rigor, conceptual understanding, procedural fluency, and application as they read and interpret a picture graph. “Create a bar graph from the table.” Teacher tip, “Remind students that the scale on the left side of the bar graph is there to help identify the number of pencils in each category. The bar should touch the line that represents the amount. Ask: How do we read a picture graph and a bar graph differently when using the scale? [Sample answer: For the picture graph, we use the line or number above the last picture. The bar graph touches the line, so we use the last line the bar touches or where the bar stops.]” 2.MD.10 (Draw a picture graph and 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 Snappet 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. 

The Mathematical Practice Standards are identified in the Course Overview/Unit Pacing Guide, Teacher Guide, Unit Overviews, and Lesson Overviews. Each lesson has a Math Practices tab that provides 3-5 structured exercises supporting the intentional development of each Math Practice throughout the year. 

MP 1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with the teacher's support and independently throughout the units. Per Snappet Learning phases math, "MP1: Make sense of problems and persevere in solving them. Found in almost every math problem across the board. Students must understand the problem, figure out how to solve it, and work until it is finished. Standards encourage students to work with their current knowledge bank and apply the skills they already have while evaluating themselves in problem-solving. This standard is easily tested using problems with a tougher skill level than already mastered. While students work through more difficult problems, they focus on solving them instead of just getting to the correct answer." Examples include:

• Unit 2: Addition and Subtraction, Lesson 2.1, Math practices, Exercise 4c, “The goal of Exercise 4 is for students to gain practice with MP 1 (Make sense of problems and persevere in solving them). They will relate current situations to concepts or skills previously learned by using basic addition and subtraction facts to write fact families.” “This is an open-ended problem to help students recap what they have learned. Have students work with a partner. Have the first partner write a subtraction equation with a missing difference. Have a second partner write the three remaining equations in the fact family. [Sample answer: Switch the number being subtracted and the difference to write another subtraction equation. Write the subtraction equation as an addition equation with an unknown addend. Switch the order of the addends to write another addition equation. Using 11-4=?, the three other equations are 11-?=4, 4+?=11, and ?+4=11.] Then have partner pairs discuss what they previously learned to complete the fact family. [Use addition and subtraction facts.]” The exercise states, “Write a subtraction equation with a missing difference. Have your partner write the three remaining equations in the fact family. What have you already learned that can help you complete the fact family?”

• Unit 4: Solve Word Problems, Lesson 4.3, Math practices, Exercise 4a, “Exercise 4 provides students the opportunity to apply MP 1 (Make sense of problems and persevere in solving them) as they demonstrate their understanding of the variety of approaches that can be used to solve word problems. Ask: How do you know what operation to use? [Sample answer: Use addition because the problem states ‘sell in all’.] Ask: What numbers go in the tape diagram? [the addends, 24 and 20] Call on a student to share their strategy. [Sample answer: I know 20 is a multiple of 10, so I started with 24 and skip counted by 10 two times, saying ‘24, 34, 44.’]” The exercise states, “A snack bar sold 24 orders of popcorn and 20 bags of chips. How many snacks did it sell in all?” Students complete a tape diagram with “24” provided. “The snack bar sold __ snacks. What strategy did you use to find the sum?”

• Unit 8: Data, Lesson 8.1, Math practices, Exercise 4c, “The intent of Exercise 4 is to practice MP 1 (Make sense of problems and persevere in solving them). Students will see relationships between various data representations. Students will see the relationship between showing data using tally marks and picture graphs.” “Give students time to think about the question and discuss the answer with a partner. [Answers will vary. Sample answer: Both tally marks and picture graphs use 1 object (tally mark or picture) to represent each piece of data.]” The exercise states, "What is the same about showing data using tally marks and picture graphs?”

MP 2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with the support of the teacher and independently throughout the units. Per Snappet Learning phases math, “MP2: Reason abstractly and quantitatively. When trying to problem solve, it is important that students understand there are multiple ways to break apart the problem in order to find the solution. Using symbols, pictures or other representations to describe the different sections of the problem will allow students to use context skills rather than standard algorithms.” Examples include:

• Unit 1: Numbers, Lesson 1.12, Math practices, Exercise 4a, “Students practice MP 2 (Reason abstractly and quantitatively) in Exercise 4. They will make sense of quantities and their relationships. Allow time for students to look at the number line. Then have them look at the questions. Call on a volunteer to answer the questions out loud. Help students understand that there is one tick mark between 400 and 600. The space between 400 and 600 is 200. Half of that is 100. So, the value between each tick mark is 100. Then have students complete the number line.” The exercise states, “What is the value between each tick mark? How do you know? Fill in the numbers on the number line. 400 __ 600 __ 800” 

• Unit 5: Add and Subtract Within 1,000, Lesson 5.4, Math practices, Exercise 4c, “The goal of the Exercise is to practice MP 2 (Reason abstractly and quantitatively). Students will create a logical representation of a problem. They will identify convenient numbers by breaking them into two different groups of 10s or 1s that they can use to solve addition and subtraction problems that would normally require regrouping.” “Give partners time to subtract. Then have partners take turns explaining how they represented the problem. Note that students could represent the problem using a number line or number sentences. [Sample answer: Subtract 20: 524-20=503. Subtract 40 more: 503-40=463.]” The exercise states, “Subtract. How did you represent this problem? 523-60=__”

• Unit 9: Geometry, Lesson 9.7, Math practices, Exercise 4c, “Here, students reason abstractly and quantitatively to determine how to divide a fruit pizza to get the largest piece possible. Give students time to think about the problem. Then ask a volunteer to explain how Brent can divide the fruit pizza. [Divide the fruit pizza into 2 equal parts. Sample answer: When a rectangle is divided into 2 equal parts, each part is larger than if the rectangle is divided into 3 or 4 equal parts.] Ask: What fraction of the fruit pizza will Brent get? [$$\frac{1}{2}$$]” The exercise states, “Brent wants to divide the fruit pizza in equal parts. How can Brent divide the fruit pizza so that he gets the largest piece possible? Explain your reasoning.”

The materials reviewed for Snappet 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. 

The Mathematical Practice Standards are identified in the Course Overview/Unit Pacing Guide, Teacher Guide, Unit Overviews, and Lesson Overviews. Each lesson has a Math Practices tab that provides 3-5 structured exercises supporting the intentional development of each Math Practice throughout the year. 

MP 3 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students construct viable arguments and critique the reasoning of others as they work with the support of the teacher and independently throughout the units. Per Snappet Learning phases math, “MP3: Construct viable arguments and critique the reasoning of others. This standard is aimed at creating a common mathematical language that can be used to discuss and explain math as well as support or object to others’ work. Math vocabulary is easily integrated into daily lesson plans in order for students to be able to communicate effectively. “Talk moves” are important in developing and building communication skills and can include such simple tasks as restating a fellow classmate’s reasoning or even supporting their own reason for agreeing or disagreeing. Prompting students to participate further in mathematical discussions will help build student communication skills. Examples include:

• Unit 2: Addition and Subtraction, Lesson 2.7, Math practices, Exercise 4a, “The goal of Exercise 4 is to give students an opportunity to practice MP 3 (Construct viable arguments and critique the reasoning of others). Students will analyze addition and subtraction about the strategy they will use to solve the problem. Pair students with a partner. Have partners discuss the question. Make sure partners take turns explaining their reasoning. [Strategies will vary. Students may use the doubles strategy because the two addends are the same.]” The exercise states, “How will you solve the problem? Explain your reasoning. Then solve. 8+8=__”

• Unit 5: Add and Subtract Within 1,000, Lesson 5.5, Math practices, Exercise 4c, “Here, students will practice MP 3 (Construct viable arguments and critique the reasoning of others) by comparing methods. Ask: How did Bria solve this problem? [Sample answer: Bria started at 236. She made a jump forward of 100 to 336. She made a jump forward of 10 to 336. She made a jump forward of 10 to 346. She made a jump forward of 4 for 350. She made a jump forward of 3 to 353.] Ask: How did John solve his problem? [Sample answer: John broke 117 into hundreds, tens, and ones. He started at 236. He made a jump forward of 100 to 336. He made a jump forward of 10 to 346. He made a jump forward of 7 to 353.] Ask a student to answer the question. [Bria and John solved the problem correctly. Both methods result in the same sum.]” The exercise shows number lines representing Bria and John’s thinking. “There are 236 people in a grocery store. Then 117 people come in. Who solved the problem correctly? Explain.”

• Unit 9: Geometry, Lesson 9.1, Math practices, Exercise 4a, Exercise 4 gives students practice with MP 3 (Construct viable arguments and critique the reasoning of others). Students ask clarifying questions to determine if a figure is a polygon. Give students time to think about the first question. Ask a volunteer to share their clarifying questions. [Sample answer: Is it a closed shape? Does it have straight sides?] Have a student answer the clarifying questions. Ask: Is it a closed shape? [No] Ask: Does it have straight sides? [Yes] Ask a volunteer to answer the second question. [No. Sample answer: A polygon is a closed shape with straight sides. This shape is not closed, so it is not a polygon.]” The exercise shows an unclosed arrow. “What questions can you ask to help you decide if the shape is a polygon? Is the shape a polygon? Explain.”

The materials reviewed for Snappet Math Grade 2 meet expectations for supporting the intentional development of MP6: Attend to precision and the specialized language of mathematics for students, in connection to the grade-level content standards, as expected by the Mathematical Practice Standards.

The Mathematical Practice Standards are identified in the Course Overview/Unit Pacing Guide, Teacher Guide, Unit Overviews, and Lesson Overviews. Each lesson has a Math Practices tab that provides 3-5 structured exercises supporting the intentional development of each Math Practice throughout the year. 

MP 6 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students attend to precision and the specialized language of mathematics as they work with the teacher's support and independently throughout the units. Per Snappet Learning phases math, “MP6: Attend to precision. Math, like other subjects, involves precision and exact answers. When speaking and problem-solving in math, exactness and attention to detail is important because a misstep or inaccurate answer in math can be translated to affect greater problem-solving in the real world.” Examples include:

• Unit 1: Numbers, Lesson 1.13, Math practices, Exercise 4a, students “practice MP 6 (Attend to precision). This mathematical practice requires students to understand meanings of symbols used in mathematics.” “In this exercise, students will understand the meaning of the symbols >, <, and = when comparing numbers. Have students work with a partner to discuss the question. Students should understand that the symbol > means greater than.” The exercise states, “What does the symbol > mean?”

• Unit 5: Add and Subtract Within 1,000, Lesson 5.8, Math practices, Exercise 4b, students “practice MP 6 (Attend to precision) as they decompose 3-digit numbers to add them efficiently and accurately.” “Here, students consider how adding using decomposition gives an accurate sum. Give students time to think about the question. Call on a volunteer to share their answer. [Sample answer: Breaking apart numbers into hundreds, tens, and ones to add helps me make sure that all the hundreds, tens, and ones are accounted for in the sum.]” The exercise states, “Why does adding two 3-digit numbers by breaking apart each number into hundreds, tens, and ones result in the correct sum?”

• Unit 6: Measurement, Lesson 6.6, Math practices, Exercise 4a, students “practice MP 6 (Attend to precision). Students express the length of objects with a degree of precision.” “Here, students measure the length of a match to the nearest centimeter. Point out that the length of the match is between two numbers. Then have a volunteer share their thoughts with the class. [No. Sample answer: The match is between 5 and 6 on the ruler. It is closer to 6, so the length of the match is 6 centimeters.]” The exercise states, “Is the match 5 centimeters long? How do you know?”

• Unit 9: Geometry, Lesson 9.2, Math practices, Exercise 4c, students “practice with MP 6 (Attend to precision) as they understand the meaning of sides and angles to name a shape appropriately.” “Have a volunteer answer the first question. [Sample answer: The number of sides and angles tells you the name of a shape. A triangle has 3 sides and 3 angles. A quadrilateral has 4 sides and 4 angles. A pentagon has 5 sides and 5 angles. A hexagon has 6 sides and 6 angles. Ask a volunteer to answer the second question. [quadrilateral; Sample answer: The shape has 4 sides and 4 angles, so it is a quadrilateral.]” The exercise states, “How can you use the number of sides and angles to name a shape? What is the name of the shape? Explain how you know.”

The materials reviewed for Snappet Math Kindergarten 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 the grade-level content standards, as expected by the Mathematical Practice Standards. 

The Mathematical Practice Standards are identified in the Course Overview/Unit Pacing Guide, Teacher Guide, Unit Overviews, and Lesson Overviews. Each lesson has a Math Practices tab that provides 3-5 structured exercises supporting the intentional development of each Math Practice throughout the year. 

MP 7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and use structure as they work with the teacher's support and independently throughout the units. Per Snappet Learning phases math, “MP7: Look for and use structure. When students can identify different strategies for problem-solving, they can use many different skills to determine the answer. Identifying similar patterns in mathematics can be used to solve problems that are out of their learning comfort zone. Repeated reasoning helps bring structure to more complex problems that might be able to be solved using multiple tools when the problem is broken apart into separate parts.” Examples include:

• Unit 1: Numbers, Lesson 1.4, Math practices, Exercise 4d, “In Exercise 4 students practice MP 7 (Look for and make use of structure). This mathematical practice requires students to look for the overall structure and patterns in mathematics. In this example, students will notice patterns and structure when writing numbers in different forms.” “Have the first partner write 3 digits in the place value chart to represent a three-digit number. Have the second partner write the number in the place value chart in three different ways: standard form, expanded form, and word form. Have partners switch roles and repeat the activity. Then have partners work together to determine a pattern they notice when writing the same number in different forms. For example, partners may realize that all three forms show the place value of each digit in the number.” Students complete the place value chart. The exercise states, “Write the number in standard, expanded, and word form. What pattern do you notice when writing the same number in different forms?”

• Unit 3: Add and Subtract Within 100, Lesson 3.10, Math practices, Exercise 4a, “In Exercise 4, students will practice MP 7 (Look for and make use of structure) by applying general rules to use the standard algorithm to subtract. Here, students will think about when they need to regroup. Allow time for students to think about the questions. Call on a volunteer to answer the question. [Sample answer: Yes; Regroup when the 1s digit of the second number is greater than the 1s digit of the first number.]” The exercise shows 92-35= (set up vertically). “Do you need to regroup to subtract? How do you know?”

• Unit 9: Geometry, Lesson 9.8, Math practices, Exercise 4b, “The intent of Exercise 4 is for students to practice MP 7 (Look for and make use of structure) as they use structure to write parts of a whole as fractions.” “Give students time to name the fraction of the circle that is blue. Say: Use the +/- buttons to adjust the fraction. Have students think about the second question. Then have a volunteer share their thoughts with the class. [Yes. Sample answer: \frac{2}{3} of the circle is blue. Any 2 parts can be shaded to show \frac{2}{3}. So, shade 2 different parts.] Say: Explain another way to name the same fraction of the circle that is blue by adjusting the model. [Sample answer: I could make the circle have 6 equal parts and shade 2 different parts of that circle.]” The exercise states, “What fraction of the circle is blue? \frac{?}{?} Is there another way to show that the same fraction of the circle is blue? Explain.”

MP 8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and express regularity in repeated reasoning as they work with the teacher's support and independently throughout the units. Per Snappet Learning phases math, “MP8: Look for and express regularity in repeated reasoning. In mathematics, it is easy to forget the big picture while working on the details of the problem. In order for students to understand how a problem can be applied to other problems, they should work on applying their mathematical reasoning to various situations and problems. If a student can solve one problem the way it was taught, it is important that they also can relay that problem-solving technique to other problems.” Examples include:

• Unit 2: Addition and Subtraction, Lesson 2.9, Math practices, Exercise 4a, “In this Exercise, students will practice MP 8 (Look for an express regularity in repeated reasoning). Students will use repeated calculations to find the sum of the objects in a picture. Have students work with a partner to discuss the question. [Sample answer: Add the number of marbles in each group, 3, four times because there are 4 groups of marbles. 3+3+3+3=12, so there are 12 marbles.” The exercise states, “How can you use the number of groups of marbles and the number of marbles in each group to find the total?”

• Unit 5: Add and Subtract Within 1,000, Lesson 5.1, Math practices, Exercise 4c, “In Exercise 4, students will practice MP 8 (Look for an express regularity in repeated reasoning). They will use patterns and the structure of 10 and 100 to add and subtract multiples of 10.” “As students continue to work with a partner, have them discuss the question and find the sum. [Sample answer: Add 580+50 by making a 100. 580+20=600 and 600+30=630.] If additional support is needed, encourage students to use pencil and paper to draw a number line to help them solve this problem.” The exercise states, “How can  you make a hundred to add these numbers? 580+50-__”

• Unit 6: Measurement, Lesson 6.8, Math practices, Exercise 4c, “The purpose of Exercise 4 is for students to practice MP 8 (Look for an express regularity in repeated reasoning). Students convert meters to centimeters and convert centimeters to meters and centimeters.” “Continue to have students work with a partner. Have them take turns explaining their process. [Sample answer: Each meter is 100 centimeters. Subtract 100 from the centimeters. Subtract 100 from the centimeters two times and then add 2 to the meters. So, the bench is 2 meters and 35 centimeters long.]” The exercise states, “The length of a bench is 235 centimeters. What is the length of the bench in meters and centimeters? Explain your process?”