4th Grade - Gateway 2

The materials reviewed for Snappet Math Grade 4 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 4 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 1.2, Instruction & Guided Practice, Exercise 1j, students develop conceptual understanding as they learn to use place value understanding to read and write numbers. “1,111. Drag blocks to the chart to represent the number. Write the number in expanded form. The value of each digit has 1 more zero than the digit to its right.” Teacher tip, “Say: Starting with the tile in the ones column, look at the values on the number tiles. What changes as you go from right to left? [There is one more zero in each one.]” 4.NBT.2 (Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form..)

• Unit 3: Fractions, Lesson 3.1, Instruction & Guided Practice, Exercise 1t, students develop conceptual understanding of equivalent fractions by drawing models. “Are these two fractions equivalent? “ \frac{2}{3} ___ \frac{5}{6}.” Teacher tip, “Have students draw a diagram showing two-thirds. Ask: How can you generate an equivalent fraction? [Divide each part in half.] Is the fraction you generated five-sixths? [No, it was four-sixths.]” 4.NF.1 (Explain why a fraction \frac{a}{b} is equivalent to a fraction (n\times a)(n\times b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.)

• Unit 4: Operations with Fractions, Lesson 4.1, Instruction & Guided Practice, Exercise 1c, students develop conceptual understanding as they add fractions with unlike denominators. “Play this game in teams of two. On paper, divide a circle into 12 equal parts. Each player throws one number cube. Color as many pieces of the circle as the count on each cube. Each uses a different color for their parts. What part of the circle did you color? What part of the circle did your partner color? How much did you color together?” 4.NF.3c (Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction and/or by using the properties of operations and the relationship between addition and subtraction.)

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: Operations with Whole Numbers, Lesson 2.6, Independent Practice, Exercise 2i, students use place value to multiply numbers. “A school buys 8 large new blackboards for $397 each. How much do they cost in total? $$397\times8=$$ $___.” The teacher can support struggling students with teacher direction: “How do you know the product of a one-digit factor multiplied by a three-digit factor is reasonable? [Sample answer: 100 times 1 = 100 and 1,000 times 10 = 10,000, so the product must have three or four digits.]” 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.)

• Unit 4: Operations with Fractions, Lesson 4.8, Independent Practice, Exercise 2a, students subtract mixed numbers with like denominators. “Click the colored sections to subtract. 3\frac{6}{10}-1\frac{3}{10}=.” The teacher can support struggling students with teacher direction: “Elicit from students that the model represents the minuend. Ask: How do you know how many to take away? [Sample answer: I take away the subtrahend.]” 4.NF.3c (Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction and/or by using the properties of operations and the relationship between addition and subtraction.)

• Unit 4: Operations with Fractions, Lesson 4.10, Independent Practice, Exercise 2f, students write a fraction as a multiple of a unit fraction. “Write the fraction as a multiple of a unit fraction. \frac{5}{12}=5\times\frac{\square}{\square}.” The teacher can support struggling students with teacher direction: “Ask: How do you determine the unit fraction that is a factor? [The numerator is 1, and the denominator is the same as in the product.]” 4.NF.4a (Understand a fraction \frac{a}{b} as a multiple of \frac{1}{b}.)

The materials reviewed for Snappet Math Grade 4 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 2.1, Instruction & Guided practice, Exercise 1h, students develop fluency as they add multi-digit numbers. “$$2,123,698+46,000=$$___. 2,123,698+40,000=___. ___ +6,000=___.” Teacher tip, “Encourage students to write the problems vertically on their own paper. Ask: Why is it helpful to write the problems vertically? [Sample answer: Addition can be accomplished by place value.] Ask: When do you regroup? [Sample answer: If the sum of a place is greater than 9, regroup.]” 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm.)

• Unit 2: Operations with Whole Numbers, Lesson 2.2, Instruction & Guided practice, Exercise 1m, students develop fluency in multi-digit subtraction as they solve using place value. “$$300,000-197,528$$.” Teacher tip, “Ask: To be able to subtract eight 1s, how do you regroup? [Sample answer: Subtract 1 from the 100,000s and add 10 to the 10,000s. Then repeat for each place until the 1s.]” 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm.)

• Unit 4: Operations with Fractions, Lesson 4.13, Instruction & Guided practice, Exercise 1g, students develop procedural skill and fluency as they add fractions with unlike denominators of 10 and 10. “$$\frac{7}{10}+\frac{17}{100}=\frac{\square}{100}+\frac{17}{100}=\frac{\square}{100}$$” Teacher tip, “Explain that a dime is 1/10 of a dollar and a penny is 1/100 of a dollar. Ask: To get an equivalent fraction, why do you multiply the numerator and denominator of 4/10 by 10? [Multiplying the numerator and denominator by the same number is the same as multiplying by 1.] Why 10? [Sample answer: 100 is 10 times as many as 10.]” 4.NF.5 (Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.)

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 2: Operations with Whole Numbers, Lesson 2.1, Independent Practice, Exercise 2d, students develop fluency as they add multi-digit numbers. “$$5,934,672+631,489=$$___.” 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm.)

• Unit 2: Operations with Whole Numbers, Lesson 2.2, Independent Practice, Exercise 2j, students demonstrate fluency in multi-digit subtraction. “$$482,739-138,234$$.” 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm.) 

• Unit 6: Measurement and Data Lesson 6.4, Independent Practice, Exercise 2k, students demonstrate their fluency as they convert measurements using multiplication. “There are 10 mm in 1 cm. How many mm are there in 2 cm? 2 cm=___ \times10 mm= ___ mm.” 4.MD.1 (Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.)

The materials reviewed for Snappet Math Grade 4 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 1: Numbers, Lesson 1.7, Instruction & Guided Practice, Exercise 1c, students find factors and multiples of given values in a non-routine application. “A Matter of Factor. Explanation: A group of children stand together. Then the children stand in rows with an equal number in each row. The rest of the children say a multiplication equation to represent the equal rows. Write the equation on the board. The same children form equal rows in another way. Write the equation on the board. Continue this until there are no more possible equations. How can you make sure you have all the possible equations?” Teacher tip, “Have students work in small groups and arrange themselves in rows with an equal number of students in each row. Give each group an opportunity to present and share the factors, the two multiplication equations that can be written using the two factors, and the answer to the equation. Write the equations on the board. Challenge students to write a word problem describing a situation for the two multiplication equations using their two factors. This is to remind them that 2\times6 and 6\times2 have the same product but describe different situations.” 4.OA.4 (Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors.)

• Unit 2: Operations with Whole Numbers, Lesson 2.13, Independent Practice, Exercise 2m, students solve a multi-step word problem in a routine application. “There are 6 children. Each child receives the same number of grapes. There are 40 red grapes and 32 white grapes.  Each child receives ___ grapes.” Teacher tip, “Ask: What question do you need to answer before you can find how many grapes each child receives? [I need to know the total number of grapes.]” 4.OA.3 (Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.)

• Unit 5:  Solve Word Problems, Lesson 5.11, Independent Practice, Exercise 2e, students solve a real-world problem by multiplying a fraction by a whole number in a routine application. “Kristen’s balloon is floating away. It rises \frac{7}{8} of a meter every second. How far will Kristen’s balloon float in 8 seconds?” Teacher tip, “Ask: Why is there not enough blanks for a mixed number? [No fraction is needed because \frac{56}{8}=7]” 4.NF.4c (Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem.)

• Unit 6: Measurement and Data, Performance task, Exercise 1b, students solve a word problem by converting gallons to cups in a non-routine application. Problem 4, “A group of neighbors plans a barbecue. They expect 32 adults and 20 children to attend the barbecue. The neighbors fill some 5-gallon water coolers. They want to have enough water for each guest at the barbecue to drink 4 cups of water. How many coolers will they need to fill? Show your work.” Teacher tip, “Observe students as they work. Ask: How can you convert 5 gallons to cups? How many cups of water will the group need in all?” 4.MD.2 (Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.)

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 1.15, Independent Practice, Exercise 2a, students analyze a pattern of numbers to determine what the number will be later in the list using what they know about the pattern in a non-routine application. “2, 7, 12, 17, 22, Which of the following is true about the eighth term of the pattern? The second digit is 7. It is even. It is a multiple of 5.” 4.OA.5 (Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.) 

• Unit 5: Solve Word Problems, Lesson 5.2, Independent Practice, Exercise 2g, students solve multiplicative word problems in a routine application. “Sam has 5 times as many blue hats as red hats. He has 25 blue hats. How many red hats does he have? Complete the equations to solve. Let r represent the number of red hats.” 4.OA.2 (Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.)

• Unit 5: Word Problems, Lesson 5.4, Independent Practice, Exercise 2h, students divide and interpret the remainder to solve a routine division problem. “There are 42 cans of soup that need to be packed into boxes. Each box can hold 5 cans. How many boxes will be filled? How many cans will remain? 42\div5= ___ R ___” 4.NBT.6 (Find whole-number quotients and remainders with up to four-digit dividends and one-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 6: Measurement and Data, Lesson 6.10, Independent Practice, Exercise 2g, students solve word problems involving time in a routine application. “‘We leave at 3:43 PM. We have to cycle 1 hour 43 minutes.’” What time are we finished cycling? Set the clock to the right time.” 4.MD.2 (Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.)

• Unit 7: Geometry, Lesson 7.4, Independent Practice, Exercise 2i, students use parallel and perpendicular lines in a non-routine application. Students see a horizontal line. They then need to determine how many parallel and perpendicular lines they need to add to make a capital E. “How many extra lines do you have to draw to turn this line into a capital letter E? ___ parallel line(s), ___ perpendicular line(s)” 4.G.1 (Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.)

The materials reviewed for Snappet Math Grade 4 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 1: Numbers, Lesson 1.2, Instruction & Guided Practice, Exercise 1f, students extend their conceptual understanding as they use blocks and expanded form to understand place value. “2,222. Drag blocks to the chart to represent the number. Write the number in expanded form. The value of each digit has 1 more zero than the digit to its right.” 4.NBT.2 (Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form.)

• Unit 2: Operations with Whole Numbers, Lesson 2.2, Independent Practice, Exercise 2q, students develop procedural skill and fluency as they subtract multi-digit numbers. “The odometer shows 143,993 miles. How many more miles must Lindsey drive to reach 200,000 miles on the odometer?” Teacher tip, “Ask: When you finish regrouping, how many of each place should be in 200,000? [One 100,000, nine 10,000s, nine 1,000s, nine 100s, nine 10s, and ten 1s.]” 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm.)

• Unit 5: Solve Word Problems, Lesson 5.2, Independent Practice, Exercise 2d, students apply their understanding of division and multiplication to solve multiplicative comparison problems. “Mark has 6 times as many marbles as Layla. Mark has 36 marbles. How many marbles does Layla have? Complete the equations to solve. Let L represent the number of marbles Layla has. ____ \div ____$$= L$$, L=____ Layla has ____ marbles.” 4.OA.2 (Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.)

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, fluency, and teacher-guided conceptual understanding. Examples include:

• Unit 1: Numbers, Lesson 1.8, Independent Practice, Exercise 2d, students use procedural fluency with multiples and apply their understanding to find factor pairs. “Move the cars in different equal rows to help you complete the equations. 1 and ___ is a factor pair of 20. 2 and ___ is another factor pair of 20. 5 and ___ is another factor pair of 20.” 4.OA.1 (Interpret a multiplication equation as a comparison.) 4.OA.4 (Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors.)

• Unit 2: Operations with Whole Numbers, Lesson 2.1, Independent Practice, Exercise 2g, students use procedural fluency and apply their understanding as they add multi-digit numbers using the standard algorithm. “In Minnesota, 64,374 people had a leg injury. In Texas, 537,826 people had a leg injury. How many people had a leg injury in both states?” 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm.)

• Unit 4: Operations with Fractions, Lesson 4.2, Instruction & Guided Practice, Exercise 1e, students develop conceptual understanding alongside procedural skill and fluency as they subtract fractions. “$$\frac{8}{10}-\frac{1}{10}=$$___. You are taking away from the numerator. The denominator stays the same.” 4.NF.3.c (Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using the properties of operations and the relationship between addition and subtraction.)

The materials reviewed for Snappet Math Grade 4 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 1: Numbers, Lesson 1.3, Math practices, Exercise 4a, “In Exercise 4, students practice MP 1 (Make sense of problems and persevere in solving them). This mathematical practice specifically requires students to see relationships between various representations. Students could use the value of each digit in the number to help them write the number using words. If students struggle with place value, encourage them to write the number in a place-value chart or in expanded form. Call on a student sto share their answer for the question. [The relationship between the number and the words is based on the value of each digit in the number.]” The exercise states, “The building is 1,135 feet tall, Write the number using words. What is the relationship between the number and the words?”

• Unit 4: Operations with Fractions, Lesson 4.15, Math practices, Exercise 4b, “Exercise 4 gives students practice with MP 1 (Make sense of problems and persevere in solving them). This mathematical practice requires students to relate situations to concepts or skills previously learned.” “Here, students compare decimals less than a given decimal with comparing whole numbers less than a given whole number and counting back with whole numbers. The students can drag one X at a time to the grid to cross off one square. This decreases the decimal by 1 hundredth each time. Ask a volunteer to share their answer to the question. [Sample answer: Finding the 3 numbers less than a given number is like counting back. To count back from the whole number 68, subtract 1. To count back from the decimal 0.68, subtract 1 hundredth.]” The exercise states, “List the 3 greatest numbers less than 0.68 that can be represented on the grid. 0.__  0.__  0.__ How did you use what you know about whole numbers to help you solve this problem?”

• Unit 8: Geometric Measurement, Lesson 8.5, Math practices, Exercise 4a, “Exercise 4 has students practice MP 1 (Make sense of problems and persevere in solving them). Students make meaning of the problem and look for starting points. Students must assume that \angle ABD is a straight angle in order to solve this problem. Have students work with a partner to solve this problem. Call on a student-pair to share their calculations. [$$180-141=139$$]” The exercise states, Albert plans to hike from point A to point B to point C. If \angle DBC is 141\degree, what angle will he turn when he gets to point B? Albert must turn __$$\degree$$. Show your calculation.”

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 1.14, Math practices, Exercise 4d, “The intent of Exercise 4 is to allow students to practice MP 2 (Reason abstractly and quantitatively). In this example students will make sense of the relationship between the numbers in a pattern. Then they will use this relationship to generate the next numbers in the pattern.” “Give students time to think of their own pattern. Have the first partner enter the first three numbers of their pattern in the boxes. Then have the second partner explain the relationship between the numbers. Finally, have the second partner find the next numbers by entering them in the boxes. Have partners switch roles and repeat the activity.” The exercise shows five boxes for students to fill in with the pattern, and four boxes above for students to show the relationship. “Think of your own pattern. Write the first three numbers in your pattern. Have your partner explain the relationship among the numbers. Then have your partner find the next numbers.”

• Unit 6: Measurement and Data, Lesson 6.12, Math practices, Exercise 4a, “The intent of Exercise 4 is to allow students to practice MP 2 (Reason abstractly and quantitatively). They will create a logical representation of the problem using bills and coins to facilitate solving word problems. Have students use play money or draw the representation on paper. Then have students discuss their reasoning with a partner. [Sample answer: First I would draw 1 $20 bill, 2 $5 bills and 3 $1 bills each in their own column. Then I would draw the 3 $10 bills between the $20 bill and $5 bills and add another $5 bill to the $5 column. Then I would add up all the columns to get the total.]" The exercise states, "Patty has one $20 bill, two $5 bills, and three $1 bills. She gets three $10 bills and one $5 bill for her birthday. How much money does Patty have? $__. Explain how to use bills to represent the problem.”

• Unit 8: Geometric Measurement, Lesson 8.4, Exercise 4b, “In Exercise 4 students have an opportunity to apply MP 2 (Reason abstractly and quantitatively). This mathematical practice requires students to make sense of quantities and their relationships when finding the measurement of angles.” “In this problem, students make sense of the relationship of three angles that together form a straight angle. Students can drag a copy of the trapezoid to see that two of the angles are 55\degree. Call on a student to share how they found the measure of the angle. [Sample answer: I moved a copy of the red shape over to the other side of the angle, and saw that the measure of the angle on the right is also 55\degreee. Since the shapes together form a straight angle, I know that the angle in the center is [$$180\degree-55\degree=70\degree$$.]” The exercise states, “What is the measure of the blue angle? The angle is __$$\degree$$. Explain how you got your answer.”

The materials reviewed for Snappet Math Grade 4 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 2: Operations with Whole Numbers, Lesson 2.14, Math practices, Exercise 4c, “The purpose of Exercise 4 is to allow students to practice MP 3 (Construct viable arguments and critique the reasoning of others). They will listen to arguments and ask useful questions to help solve division problems using area models.” “Ask: What is different in this problem? [Sample answer: The area model is not given.] Encourage students to think about useful questions they could ask about creating the area model to help them solve the division problem. Then have groups share their questions with the class. [Sample answers: How many boxes should be in the area model?; What are some multiples of 5 that might be helpful?]” The exercise states, “Draw an area model on paper. Solve. 625\div5=__ What are two questions you could ask to help your group solve this problem?”

• Unit 5: Solve Word Problems, Lesson 5.7, Math practices, Exercise 4a, “The intent of Exercise 4 is to give students practice with MP 3 (Construct viable arguments and critique the reasoning of others). Students will ask clarifying questions or suggest ideas to improve an argument. Here, students analyze the work of another student and must compose a question that would help the other student correct their work. Have a volunteer share their question. [Sample answer: My friend forgot to add the 24 and 18 together first. Ask, ‘How many flowers did she have altogether?]” The exercise states, “Tia has 24 roses and 18 tulips. She plans to put an equal number of flowers into 6 vases. How many flowers will go into each vase? Your friend’s answer is 4 flowers in each vase. Think about what mistake your friend likely made. What question could you ask that would help your friend find the right answer?”

• Unit 7: Geometry, Lesson 7.3, Math practices, Exercise 4c, “The goal of Exercise 4 is for students to practice MP 3 (Construct viable arguments and critique the reasoning of others). Students will listen to the arguments of others and ask useful questions to determine if an argument makes sense.” “Here, students are considering another student’s assertion and indicating flaws that must have been part of that assertion. Have a volunteer share their answer. [No. Sample answer: It is possible for an obtuse triangle to be isosceles as long as the other two angles (which will be acute) are equal in measure.” The exercise states, “Libby said that it is impossible for an obtuse triangle to be isosceles. Is Libby correct? Why or why not?”

The materials reviewed for Snappet Math Grade 4 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 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 1.5, Math practices, Exercise 4a, students “practice of MP 6 (Attend to precision). Students will understand meanings of symbols used in mathematics.” “Pair students with a partner. Have them explain the meaning of the symbols >, <, and = to each other. Call on a volunteer from a pair of partners to explain the meaning of the symbol <. Then call on two different volunteers to explain the meaning of the symbols > and =. [< means less than. > means greater than. = means equal.]” The exercise states, “Compare the numbers. 2588 points and 2579 points; 2579 __ 2588” Students drag the symbol, >, <, or = to complete the equation. “Explain the meaning of the symbols >, <, and =.”

• Unit 4: Operations with Fractions, Lesson 4.16, Math practices, Exercise 4b, students “practice MP 6 (Attend to precision). This mathematical practice requires students to understand the meaning of symbols and label quantities appropriately.” “Here, students attend to the details of the symbols and the quantities. Students may struggle with this problem since 4.7 has one less decimal place. Ask students to use place value vocabulary in their answer. [Sample answer: The number must have 4 ones to be between 4.68 and 4.7. 69 hundredths is between 68 and 70 hundredths.]” The exercise states, “Fill in the correct number. 4.68<__$$<4.7$$ Explain how you determined your answer.”

• Unit 6: Measurement and Data, Lesson 6.14, Math practices, Exercise 4c, students “practice MP 6 (Attend to precision). They will understand the meaning of each X to construct and interpret a line plot.” “Have student volunteers share their response to see if there are different interpretations. [Sample answer: The phrase “greater than 6” means the Xs above 6\frac{1}{2} (1), the number of Xs above 7 (3), and the number of Xs above 7\frac{1}{2} (2). Then I added 1+3+1=5, to get the final answer.]” The exercise shows a line plot titled “Zucchini Plant Heights,” from 4 to 7\frac{1}{2}. “How many more plants are greater than 6 feet tall? __ Explain how you interpreted the line plot to answer the question.”

• Unit 8: Geometric Measurement, Lesson 8.3, Instruction & guided practice, Exercise 1d,  students “practice MP 6 (Attend to precision). This mathematical practice requires students to express numerical answers with a degree of precision appropriate for the problem context.” “Let’s sketch an angle. Step 1: Use the bottom of your protector to draw a ray. Mark the endpoint with a letter.  This will be the vertex of your angle.”

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 the 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 1.6, Math practices, Exercise 4c, “Exercise 4 allows students to practice MP 7 (Look for and make use of structure). Students will apply general mathematical rules as they round multi-digit numbers to the nearest thousand or hundred.” “Pair students with a partner. Have them work together to explain how the rules for rounding to the nearest thousand are similar to the rules for rounding to the nearest hundred. Be sure students understand that the rules are the same, except for putting the dash after the thousands or hundreds place in Step 1 and identifying the next digit (the digit to the right of the thousands or hundreds place) in Step 2.” The exercise states, “Explain how the rules for rounding to the nearest thousand are similar to the rules for rounding to the nearest hundred.”

• Unit 6: Measurement and Data, Lesson 6.7, Math practices, Exercise 4a, “The intent of Exercise 4 is to allow students to practice MP 7 (Look for and make use of structure). They will apply rules to convert customary measures of capacity and time. Encourage students to write out the relationships between units of capacity to help them remember the rules. Then ask a volunteer to share their explanation. [Sample answer: There are 4 quarts in 1 gallon. So, you multiply the number of gallons by 4 to get the number of quarts.]” The exercise states, “How many quarts (qt) are there in 6 gallons (g)? __ quarts Explain how you used a rule to solve the problem.”

• Unit 7: Geometry, Lesson 7.4, Math practices, Exercise 4c, “The intent of Exercise 4 is to give students practice with MP 7 (Look for and make use of structure). Students will look for the overall structure and patterns in mathematics.” “Now, students are asked to look for a pattern in the regular polygons that are shown. Have students record the number of sides and the number of pairs of parallel lines as necessary. Then have a student share their answer. [Sample answer: The number of sides is always double the number of pairs of parallel lines.] As an extension, ask students to consider regular polygons with an odd number of sides.” Three polygons are shown and the exercise states, “Examine each shape and count the number of sides and the number of pairs of parallel lines. What pattern do you see that relates the number of sides to the number of pairs of parallel lines?”

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: Operations with Whole Numbers, Lesson 2.20, Math practices, Exercise 4d, “Exercise 4 gives students practice with MP 8 (Look for an express regularity in repeated reasoning). They will focus on the overall process of dividing with remainders while still keeping track of the details of finding multiples, remainders, and checking solutions.” “Have students share their response. [Sample answer: You can multiply the quotient by the divisor and add the remainder to the product. If the sum of the product and the remainder are equal to the dividend, it is correct.]” The exercise states, 39\div7=__ remainder __. How can you check your quotient and remainder are correct?”

• Unit 4: Operations with Fractions, Lesson 4.7, Math practices, Exercise 4a, “In Exercise 4, students practice MP 8 (Look for an express regularity in repeated reasoning). Students will look for generalizations and shortcuts by using the commutative and associative properties to add.” “Here, students are asked to expand their understanding of properties as shortcuts to adding 3 fractions when there are different denominators. Invite a student to share their response. [sample answer: One of the fractions has a different denominator. The sum can still be found since \frac{1}{6} and \frac{5}{6} combine to make 1. Then, add  \frac{3}{8}. So, Garry spend 1\frac{3}{8} hours on homework.]” The exercise states, “Gary worked on his homework 3 different times. He spent \frac{1}{6} hour, \frac{3}{8} hour, and \frac{5}{6} hour on homework. How much time did he spend on homework? What did you notice that is different about this problem? Can you still use the commutative and associative properties to find the sum? Explain?”

• Unit 8: Geometric Measurement, Lesson 8.1, Math practices, Exercise 4a, “Exercise 4 has students practice MP 8 (Look for an express regularity in repeated reasoning). This mathematical practice requires students to find shortcuts for repeated calculations. Here, students can use more than one strategy that involves shortcuts. Call on students to share their answers and solution methods. [less than one whole pizza; Sample answer: I multiplied 7 by 45 and got 315. A full circle is 360 degrees, so James has less than a full circle, or less than one whole pizza.]” The exercise states, “After a party, James has 7 pieces of pizza left. Each of those pieces has an angle of 45\degree degrees. Does James have more than one whole pizza or less than one whole pizza left? Explain.”