5th Grade - Gateway 2

The materials reviewed for Snappet Math Grade 5 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 5 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 1.7, Instruction & Guided Practice, Exercise 1j, students develop conceptual understanding as they learn the meanings of patterns in the number of zeroes when a decimal is divided by a power of 10. “$$12kg\div1=12kg$$. 12kg\div1,000=___ kg. “3 zeros so 3 spaces to the left.” Thus, 1 pencil weighs ___ kg.” Teacher tip, “Have a student volunteer write or say a word problem to describe the situation illustrated in slide 1j. [Sample response: If 1,000 pencils weigh 12 kilograms, how much does 1 pencil weigh?]” 5.NBT.2 (Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.)

• Unit 4: Fractions - Add & Subtract, Lesson 4.1, Instruction & Guided Practice, Exercise 1h, students build conceptual understanding of adding fractions with unlike denominators using a pictorial model. “$$\frac{1}{4}+\frac{3}{8}=?$$” Teacher tip, “Encourage students to use the models as they give the addends. Ask: How do you find an equivalent fraction to make the denominator greater? [Sample answer: Multiply the numerator and denominator by the same number.]” 5.NF.1 (Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.)

• Unit 7: Measurement and Geometry, Lesson 7.4, Instruction & Guided Practice, Exercise 1c, students develop conceptual understanding by finding the volume of rectangular prisms by counting unit cubes. “Measuring with Cubes. Directions: Perform this activity in small groups. Find a small box in the classroom. Use cubes to measure the amount of ‘cube space’ inside the box. Measure it twice. How did you pack the cubes inside the box? Did you get the same measurement each time? Why or why not? How many cube units does the space measure?” 5.MD.3 (Recognize volume as an attribute of solid figures, and understand concepts of volume measurement.) 5.MD.4 (Measure volumes by counting unit cubes, using cubic cm, cubic in., cubic ft, and improvised units.)

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 that are struggling to complete the Independent Practice items. 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: Operations with Whole Numbers, Lesson 2.7, Independent Practice, Exercise 2g, students divide two-digit numbers by one-digit numbers by breaking them apart into smaller parts. “You have 96 flowers. You make bouquets of 8 flowers. You can make more than 10 bouquets, because 80\div8=10. You can create 96\div8=___ bouquets.” The teacher can support struggling students with teacher direction: “Make sure students realize that they can count the flowers in the bouquets to verify they have the right number. Each bouquet has 8 flowers, and 12 bouquets will have 96 flowers.” 5.NBT.6 (Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.)

• Unit 4: Fractions - Add and Subtract, Lesson 4.2, Independent Practice, Exercise 2b, students subtract fractions with different denominators. “$$\frac{7}{8}-\frac{3}{4}=?$$” An arrow points from the denominator 4 to the second denominator 8 to show common denominators. “$$\frac{7}{8}-\frac{\square}{\square}=\frac{\square}{\square}$$.” The teacher can support struggling students with teacher direction: “Encourage students to draw the models, so they can be shaded. Ask: How can you determine by how many to multiply the numerator and denominator? [Sample answer: Divide the greater denominator by the lesser denominator to find the factor that was used. Then multiply the numerator of the fraction with the lesser denominator by that same factor.]” 5.NF.1 (Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.)

• Unit 5: Fractions - Multiply and Divide, Lesson 5.1, Independent Practice, Exercise 2b, students compare the product of a fraction or mixed number. “5 pieces of \frac{3}{4}cake. 5\times\frac{3}{4}=3\frac{3}{4}. Which statement is correct? The product is less than one factor. The product is less than both factors. The product is greater than both factors.” The teacher can support struggling students with teacher direction: “How do you compare a whole number with a mixed number? [The mixed number is greater unless the whole number is greater than the whole number part of the mixed number.]” 5.NF.5a (Interpret multiplication as scaling (resizing), by comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.) 5.NF.5b (Interpret multiplication as scaling (resizing), by explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence \frac{a}{b}=\frac{(n\times a)}{n\times b)} to the effect of multiplying \frac{a}{b} by 1.)

The materials reviewed for Snappet Math Grade 5 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: Operations with Whole Numbers, Lesson 2.2, Instruction & Guided practice, Exercise 1h, students develop fluency as they use the standard algorithm to find the product of two numbers. “$$7\times742$$.” Teacher tip, “Ask: How do you determine what digit to write when you multiply 7 with the numbers in the tens column? [I multiply 7 times 4 and then add 1 to the product to get 29. I write 9 in the product and 2 above the hundreds digit.]” 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm.)

• Unit 3: Operations with Decimals, Performance Task, Exercise 1a, Question 1, students develop procedural skills and fluency as they use the standard algorithm to multiply. “Jareem goes to a baseball game. He notices that there are 25 seats in his row and 13 rows in the section. How many seats are there in the section? Show your work.” Teacher tip, “Students should use the standard algorithm to multiply. Ask: How can you use estimation to check your answer?” 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm.)

• Unit 6: Expressions and Patterns, Lesson 6.6, Instruction & guided practice, Exercise 1k, students develop procedural skill and fluency as they use rules to generate a numeric pattern. “Use the rule to complete the pattern. add 4. 0, ___, ___, ___, ___, ___, ...” Teacher tip, “Ask: Where do you start? [start at 0 and add 4] What is next? [keep adding 4]” 5.OA.3 (Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.)

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 1.11, Independent Practice, Exercise 2q, students demonstrate procedural skill and fluency as they round decimals to any place. “Rounds to 12.4. Rounds to 12.5. Answers: 12.43, 12.54, 12.405, 12.45.” 5.NBT.4 (Use place value understanding to round decimals to any place.)

• Unit 2: Operations with Whole Numbers, Lesson 2.3, Independent Practice, Exercise 2f, students demonstrate procedural skills and fluency as they use the standard algorithm to find the product of two two-digit numbers. “$$26\times72=$$___.'' 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm.)

• Unit 6: Expressions and Patterns, Lesson 6.4, Instruction & Guided Practice, Exercise 2e, students demonstrate procedural skill and fluency as they write algebraic expressions. “Write an expression for the statement. “Triple the quotient of 16 and 12”. “$$3$$___(___ \div ___)” 5.OA.2 (Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.)

The materials reviewed for Snappet Math Grade 5 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 2: Operations with Whole Numbers, Lesson 2.6, Independent Practice, Exercise 2l, students solve a word problem involving division in a routine application. “There are 480 flowers planted in 8 rows. How many flowers is that per row? __ flowers per row.” Teacher tip, “Divide students into small groups and have each person write a division problem using tens like the ones in this lesson. Students who need help can use the same contexts as these last two slides. Then have students exchange problems and answer them.” 5.NBT.6 (Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.)

• Unit 4: Fractions - Add and Subtract, Lesson 4.7, Instruction & Guided Practice, Exercise 1f, students solve word problems by adding and subtracting fractions in a routine application. “David shares 3 pizzas. Sharon eats \frac{3}{4} pizza. Bobby eats \frac{1}{2} pizza. Ruby eats \frac{7}{8} pizza. How much pizza is left? Explain how you would solve the problem.” Teacher tip, “Ask: What is the question we are trying to answer? [How much pizza is left?] What are the steps to figure it out? [Add the fractions that show how much they ate. Subtract that total from 3.] Sample answer: Add how much Sharon, Bobby, and Ruby eat and then subtract from 3.” 5.NF.2 (Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.)

• Unit 7: Measurement and Geometry, Lesson 7.3, Instruction and guided practice, Exercise 1c, students solve problems involving length, weight, and capacity in a non-routine application. “David makes pancakes for breakfast. He uses 1000 mg of flour for every pancake. He makes 6 pancakes. How many g of flour does David use? Step 1: Look at the problem. What operation can you use to solve? To find the total of an equal number of groups, _____.” Teacher tip, “Remind students of the steps for solving word problems: identify the information given in the problem; identify what you are being asked to find; use the information to write and solve an equation. Ask: In this problem, what do you know? [milligrams of flour for one pancake and the total number of pancakes] What are you being asked to find? [total grams flour used] What operation will you use to solve the problem? [multiplication]” 5.MD.1 (Convert among different-sized standard measurement units within a given measurement system, and use these conversions in solving multi-step, real-world problems.)

• Unit 8: Line Plots and the Coordinate System, Performance task, Exercise 1b, students play a game plotting points to find the buried treasure in a non-routine application. Problem 4, “Javier and Liya play a game called “Buried Treasure.” These are the rules: Choose three locations to bury your treasure. Mark each treasure with four pegs. Take turns guessing the coordinates of the other player’s treasures. The first player who identifies the four corners of one of the other player’s treasures wins. Complete the chart. Who wins the game?” Teacher tip, “Students should plot Liya’s guesses on Javier’s board, and vice versa, in order to determine whether each guess is correct. Ask: Does either player locate a corner of a treasure box? Does either player find all four corners of a treasure box?” 5.G.2 (Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpreting coordinate values of points in the context of the situation.)

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 5: Fractions - Multiply and Divide, Lesson 5.7, Independent Practice, Exercise 2e, students multiply two fractions to solve a real-world problem in a routine application. “Blanche opened a pizza box. Inside, there was \frac{4}{5} of a pizza. Blanche ate \frac{2}{3} of the remaining pizza. How much of the pizza did Blanche eat?” 5.NF.6 (Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.)

• Unit 5: Fractions - Multiply and Divide, Lesson 5.10, Exercise 2b, students use fraction models and multiplication of the numerator to divide a whole number by a unit fraction in a routine application. “Divide both pizzas into fifths. How many pieces of pizza are there? 2\div\frac{1}{5}= ____ pieces of pizza.” 5.NF.7c (Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions.)

• Unit 6: Expressions and Patterns, Lesson 6.4, Independent Practice, Exercise 2g, students determine the mathematical equation given a series of operations in a non-routine application.  “Double the sum of 6 and 5, and multiply to 8 divided by 2. 2___(6___5)___(___÷___)” 5.OA.2 (Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.)

• Unit 7: Measurement and Geometry, Lesson 7.3, Independent Practice, Exercise 2k, students solve a problem involving weight and capacity in a non-routine application. “How can Timothy fill a large container with smaller containers of 200g, 300g, 500g each without exceeding a 9kg weight limit? Write down two solutions below:” 5.MD.1 (Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.)

• Unit 8: Line Plots and the Coordinate System, Lesson 8.4, Independent Practice, Exercise 2d, students interpret coordinate values of points in the context of the situation in a routine application. “How much farther is the park from the bank than the post office is from the bank? From the bank to the park: ___ miles From the bank to the post office: ___ miles The park is ___ - ___ = ___ mile(s) farther.” 5.G.2 (Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpreting coordinate values of points in the context of the situation.)

The materials reviewed for Snappet Math Grade 5 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 aspects 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: Operations with Whole Numbers, Lesson 2.2, Independent Practice, Exercise 2g, students develop procedural skill and fluency as they solve multi-digit multiplication problems. “$$9\times285$$.” 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm.)

• Unit 5: Fractions - Multiply and Divide, Lesson 5.9, Instruction & Guided Practice, Exercise 1m, students extend their conceptual understanding as they learn to divide a fraction by a whole number. “$$\frac{3}{4}$$ pizza divided between three children. How much pizza does each child get?” Teacher tip, “Remind students that this context matches the activity they completed during the Student Discovery exercise. The division equation here is the division equation that they could have used to represent the situation.]” 5.NF.7a (Interpret the division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (\frac{1}{3})\div4, and use a visual fraction model to show the quotient.)

• Unit 7: Measurement and Geometry, Lesson 7.6, Independent Practice, Exercise 2h, students apply the volume formula to find the volume of rectangular prisms. “A sandbox is 6 feet long, 4 feet wide, and 2 feet high. How much sand does it take to fill the sandbox? ____ ft^3 of sand.” Teacher tip, “Ask: Which formula will you use to solve this word problem? Explain. [$$l \times w \times h=V$$, because the problems gives all three dimensions.]” 5.MD.5b (Apply the formulas V=l \times w \times h and V=B \times h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real-world and mathematical problems.)

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: Operations with Whole Numbers, Lesson 2.4, Instruction & Guided Practice, Exercise 1c, students engage with conceptual understanding and procedural fluency as they use the standard algorithm to multiply. “In The Classroom. An event planner sells 432 passes for a conference. Each pass cost $56. What is the total amount? Model the Problem: Use place-value blocks to model 432. Discuss with your group how you can use the model to find 432 \times$56. Write your ideas on your whiteboard, and find the product together. Compare your answer with answers from other groups.” Teacher tip, “Ask: Why can’t you use equal groups to model the problem? [I would need to make 56 equal groups, which is too many to be practical.] Discuss as a class the different ideas that different groups had. As part of the discussion, ask students about how they might model ten thousands using place-value models.” 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm.)

• Unit 4: Fractions - Add and Subtract, Lesson 4.3, Independent Practice, Exercise 2o, students engage with conceptual understanding, procedural fluency, and application as they estimate sums and differences of fractions. “Paul cut \frac{3}{5} of a yard off a plank. The plank was 2\frac{1}{8} yards long. About how long is the remaining part?” Teacher tip, “Elicit from students that they add to join groups and subtract to take some away. Remind students to round to the nearest 0, \frac{1}{2}, or 1.” 5.NF.2 (Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.)

• Unit 5: Fractions - Multiply and Divide, Lesson 5.2, Instruction & Guided Practice, Exercise 1m, students engage with conceptual understanding and procedural skills by multiplying a whole number by a fraction using fraction models, arrays, and multiplication facts. “Calculate \frac{1}{3} of 9, or \frac{1}{3}\times9. Draw the model below on a piece of paper. Circle the rows that you need to solve the problem. \frac{1}{3}\times9= ___.” 5.NF.4a (Interpret the product (\frac{a}{b})\timesq as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a\times q \div b. For example, use a visual fraction model to show (\frac{2}{3})\times4=\frac{8}{3}, and create a story context for this equation…)

The materials reviewed for Snappet Math Grade 5 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. It means that 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 the problem instead of just getting to the correct answer." Examples include:

• Unit 1: Numbers, Lesson 1.15, Math practices, Exercise 4a, “The intent of Exercise 4 is to allow students to practice MP 1 (Make sense of problems and persevere in solving them). Students will plan a solution pathway for comparing and ordering decimals by determining which place to start comparing. Have students consider a time they made a mistake solving a problem and how having a solution strategy may have helped them prevent that mistake. Call on a student to share their response to the question. [Students may respond that having a plan prevents you from quickly picking what you think is the correct answer without thinking through all the details which may prevent silly mistakes.]” The exercise states, “Select the heaviest bag. 5.267 lb; 5.268 lb; 5.169 lb; 5.256 lb Think about how you solved this problem. Why is it important to have a plan for solving different types of problems?”

• Unit 3: Operations with Decimals, Lesson 3.7, Math practices, Exercise 4b, “Exercise 4 has students practice MP 1 (Make sense of problems and persevere in solving them). This mathematical practice has students checking to see if a strategy or approach makes sense.” “Here, students are asked to make sense of each step of a solution process. Have a student answer the first question. [Sample answer: It is too low since the dividend was rounded down.] Call on a student to answer the second question. [Sample answer: I divide by 4 because the amount that was rounded has to be “shared” by the divisor.] Have a student share their response to the third question. [Sample answer: I rounded the dividend down. I need to account for rounding down by adding the quotient.]” The exercise states, “Answer the question for each step. Original problem: $16.12 \div4 =? $16.00 \div 4 = $4.00. Is $4.00 too high or too low? $0.12 \div 4 = $0.03. Why do I divide by 4? $4.00 + $0.03 = $4.03 Why do I add instead of subtract?

• Unit 5: Fractions - Multiply and Divide, Lesson 5.8, Math practices, Exercise 4c, “In Exercise 4 students get practice with MP 1 (Make sense of problems and persevere in solving them). Acting out the problem will provide students with a way to explain the meaning of the problem to themselves.” “In this problem, students must think critically to determine the dividend based on the divisor and quotient given. Students can duplicate the pitchers of water and can drag the characters to parts of the pitcher as they work to find the answer.” The exercise states, “Some pitchers of water are shared equally by 4 friends. Each friend gets 1\frac{3}{4} of a pitcher of water. How many pitchers are there?”

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 teacher's support 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 1.4, Math practices, Exercise 4a, “The intent of Exercise 4 is to allow students to practice MP 2 (Reason abstractly and quantitatively). They will understand and describe the relationship between quantities. Have students think-pair-share about the question before writing an answer [a centimeter is one hundredth of a meter]” The exercise states, “Vicky is 1 meter and 2 centimeters tall. What is the relationship between meters and centimeters?”

• Unit 3: Operations with Decimals, Lesson 3.10, Math practices, 4b, “The goal of Exercise 4 is for students to practice with MP 2 (Reason abstractly and quantitatively). This mathematical practice has students attending to the meaning of quantities in addition to computing them.” “Here, students consider how the algorithm for multiplication works for both decimal and non-decimal numbers. Have a student answer the first question. [Sample answer: I use the same steps because the decimal point does not affect the steps.] Have a student answer the second question. [Sample answer: When one or both of the factors contains a decimal point, I have to think about where the decimal point belongs in the final product.]” The exercise states, “Explain how finding the two products is the same: 26.4\times9 and 264\times9. Explain how finding the two products is different: 26.4\times9 and 264\times9.”

• Unit 6: Expressions and Patterns, Lesson 6.5, Math practices, Exercise 4a, “The intent of Exercise 4 is to allow students to practice MP 2 (Reason abstractly and quantitatively) as they explore how to make meaning of the symbols within two different expressions to compare their values without performing the calculations. Pair students with a partner. Have partners discuss what is similar and what is different about the two expressions, Then have them describe how they know which expression is greater. [Students should recognize that the second factor is the same. Because Expression 1 has a greater whole-number first factor than Expression 2, its value will be greater.]” The exercise states, “Compare the two expressions below. Expression 1 40\times257; Expression 2 20\times257 Without simplifying the expression, explain how you know which expression is greater.”

The materials reviewed for Snappet Math Grade 5 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 class mathematical discussions will help build student communication skills. Examples include:

• Unit 1: Numbers, Lesson 1.8, Math practices, Exercise 4d, “The intent of Exercise 4 is to allow students to practice MP3 (Construct viable arguments and critique the reasoning of others). Students listen to the arguments of others and ask clarifying questions to determine if an argument makes sense.” “Have partners work independently to decide who is correct. Then have them share their decisions and justifications. Pairs must come to agreement using arguments before responding to the question. [Pam is correct. If I align the numbers on the right by decimal point and add vertically, I do not get the number on the left. This is because there is a 0 in the tens place, but in the decomposition 4 is written in the hundredths place when it should be written in the thousandths place.]” The exercise states, “Micah says the number below is decomposed correctly. Pam says it is incorrect. 6.304=6+0.3+0.04 Who is correct? How do you know?

• Unit 3: Operations with Decimals, Lesson 3.13, Math practices, Exercise 4d, “The goal of Exercise 4 is for students to practice MP 3 (Construct viable arguments and critique the reasoning of others). Students follow specific steps when dividing with decimals. They ask clarifying questions about presented methods for dividing with decimals and suggest ideas to improve/revise the method.” “Continue to have students work with a partner. Have partners solve the problem individually. Then, have each partner take turns sharing their solution method, asking each other questions about the solution method, and giving each other suggestions to revise their method, if needed.” The exercise states, “$$623\div0.89=$$___ What questions could you ask your partner about their solution method?  What suggestions can you give your partner to revise their work?”

• Unit 7: Measurement and Geometry, Lesson 7.8, Math practices, Exercise 4a, “Exercise 4 provides students with an opportunity to practice MP 3 (Construct viable arguments and critique the reasoning of others). Students will analyze problems and use definitions to classify polygons. Here, students need to recall the definitions of the terms regular, parallel, and pentagon and apply those definitions to determine the feasibility. Ask a volunteer to share their response. [No. Sample answer: Since a pentagon only has 5 sides, it is impossible for it to have 3 pairs of parallel sides.]” The exercise states, “Fionna asks you to create a regular pentagon that has 3 pairs of parallel sides. Is this possible? Why or why not?”

The materials reviewed for Snappet Math Grade 5 meet expectations for supporting the 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.

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 are 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 1.12, Math practices, Exercise 4a, students “practice MP 6 (Attend to precision). Students express numerical answers with a degree of precision appropriate to the situation.” “Lead a class discussion on why recording a person’s height to the hundredth place may be better than recording it to five or six decimal places. Elicit that a tape measure or other measuring tool cannot measure accurately to so many digits and hundredths is easier to communicate but still precise.” The exercise states, “When measuring an object, why might you round to the nearest hundredth?”

• Unit 4: Fractions - Add and Subtract, Lesson 4.1, Math practices, Exercise 4a, students “practice MP 6 (Attend to precision). Students will label quantities appropriately when creating equivalent fractions. Here, students must recognize that not all fractions can be rewritten to have the same denominator of one of the existing addends.” The exercise states, “Can you add these fractions by changing the first addend into a fraction that has the same denominator as the second addend and is equal to the first addend? Explain.”

• Unit 5: Fractions - Multiply and Divide, Lesson 5.6, Math practices, Exercise 4b, students “practice with MP 6 (Attend to precision). Students consider precision in the multiplication of mixed numbers in the context of the problems presented.” “Here, students are attending to the precision in regard to the fraction part of the product. Students may benefit from working with a partner on this problem. Have a volunteer share their response. [Sample answer: I can eliminate 1\frac{2}{5} since 5 is not a factor of 12, the denominator of the product.]” The exercise states, “Select the two mixed numbers whose product is 4\frac{1}{12}\cdot1\frac{2}{3}; 1\frac{2}{5}; 1\frac{3}{4}; 2\frac{1}{3} Was it possible to eliminate one of the mixed numbers as a possible factor? Explain.”

• Unit 7: Measurement and Geometry, Lesson 7.6, Math Practices, Exercise 4a, students “practice MP 6 (Attend to precision). Students will understand the meaning of symbols and label quantities accurately when using the volume formula.” “Here, students are asked to compare the use of the different formulas to demonstrate their understanding. Ask a student to share their answer. [No. Sample answer: There is no difference since you end up multiplying the 3 numbers either way.]” The exercise states, “Find the volume of the figure two ways, filling in the numbers in the formulas. Is there a difference in these methods? Explain.”

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 support of the teacher 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 1.3, Math practices, Exercise 4c, “The intent of Exercise 4 is to allow students to practice MP 7 (Look for and make use of structure). They will look for and describe the structure and patterns in mathematics.” “Have students work in small groups to come up with a list of possible strategies. Strategies may include the following: (1) multiplication by 10 adds a zero and division by 10 removes a 0. (2) Think about the related multiplication problem by asking yourself: What number times 10 gives me the dividend? (3) Express the division as a fraction and simplify. or (4) Use multiplication to check the answer to the division problem.” The exercise states, “$$25\times10=$$__; 250\div10=__ Imagine that your friend missed math class today. How could you describe at least two strategies your friend might use to solve division of a number by 10?”

• Unit 3: Operations with Decimals, Lesson 3.14, Math practices, Exercise 4a, “The goal of Exercise 4 is to give students practice with MP 7 (Look for and make use of structure). Students use structure and patterns when dividing with decimals. Allow time for students to look at the process Jasmine uses. Be sure students complete the problem. Have a student share their response. [Sample answer: Jasmine multiplied by 5.75 and 0.25 by 20. Multiplying 5.75 and 0.25 by the same number results in a division problem with the same quotient as the original problem.] Ask: What other numbers could Jasmine have multiplied 5.75 and 0.25 by to find the quotient? [Sample answer: 100, 4]” The exercise states, “Jasmine divides 5.75 by 0.25. 5.75\div0.25; $$\div$$$$=$$__ Describe Jasmine’s method. Why does her method result in the correct quotient?”

• Unit 5: Fractions - Multiply and Divide, Lesson 5.5, Math practices, Exercise 4c, “The goal of Exercise 4 is to provide students practice with MP 7 (Look for and make use of structure). Students will see fraction multiplication as an algorithm using the components of the factors.” “In this problem, students will work with the numerator and denominator of the product as separate components in order to write pairs of factors. [Sample pairs: \frac{3}{8} and \frac{5}{3}, \frac{3}{6} and \frac{5}{4}, \frac{15}{12} and \frac{1}{2}] Ask several students to share their process. [Sample explanation: I thought about the factors of 15 and the factors of 24. I used those factors to create fractions where the numerators were a factor pair for 15 and the denominators were a factor pair for 24.]” The exercise states, “Write at least 4 pairs of fractions that have a product of \frac{15}{24}. Explain your process.”

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 1: Numbers, Lesson 1.6, Math practices, Exercise 4c, “The intent of Exercise 4 is to allow students to practice MP 8 (Look for an express regularity in repeated reasoning). Students look for patterns in the products of the same decimal number multiplied by 10, 100, and 1,000 in order to draw conclusions about how to multiply more fluently.” “Students should recognize that even though they don’t have all three products to see the full pattern, they know that 100 has two zeros so they should move the decimal point 2 place values to the right. Ask: What would happen if there was only one digit to the right of the decimal place before you multiplied by 100? [You would need to insert a 0 at the end of the number before moving the decimal point.]” The exercise states, “Students may struggle to apply what they have learned about the structure of multiplying by 10, 100, 1,000 without numbers. Encourage students to pick different types of numbers as examples and multiply them to see what happens before answering the question. Note that the statements are sometimes true. They are true when the first factor is a whole number.” The exercise states, “A number multiplied by 10 has one 0 at the end. A number multiplied by 100 has two 0’s at  the end. A number multiplied by 1,000 has three 0’s at the end. Are these statements always, sometimes, or never true? Explain.”

• Unit 4: Fractions - Add and Subtract, Lesson 4.4, Math practices, Exercise 4b, “The goal of Exercise 4 is for students to practice MP 8 (Look for an express regularity in repeated reasoning). Students will look at both the overall process and details when adding and subtracting complex unlike fractions.” “Here, students consider one method of finding common denominators that will always work to add or subtract fractions with unlike denominators. Have students work on an answer and then discuss with a partner. [Yes. Sample answer: Since multiplying the two denominators always creates a denominator that is a multiple of both numbers.]” The exercise states, “Carlos wrote these directions for creating like denominators. 1. Multiply the numerator and denominator of the first fraction by the denominator of the second fraction. 2. Multiply the numerator and denominator of the second fraction by the denominator of the first fraction. Will this method always work? Explain.”

• Unit 7: Measurement and Geometry, Lesson 7.4, Math practices, Exercise 4c, “The intent of Exercise 4 is to give students an opportunity to practice MP 8 (Look for an express regularity in repeated reasoning). Students should note how repeated calculations can lead to generalizations and shortcuts when finding volume.” “This problem requires a level of abstraction as students see the repeated calculations of stacking the figure that is shown. Some students may provide a more efficient explanation than others. [Sample answer: It is not possible because the base has 12 square units. I could only make prisms that are 12, 24, 36, 48, 60,... square units.]” The exercise shows a two by six layer of blocks and states, “Could you build a rectangular prism that is 50 square units using this figure as your base? Each cube is 1 unit$$^3$$ .” Students mark, “yes” or “no.” “Explain how you determined your answer.”