The materials reviewed for Snappet Math Grade 3 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 3 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 2: Multiplication, Lesson 2.1, Instruction & Guided Practice, Exercise 1c, students develop conceptual understanding as they learn the meaning of multiplication. “How many times?! Explanation: Give each group of four students 20 counters. Ask students to evenly divide the counters between all group members. How many counters does each student have? How many times is the number of counters repeated for the group members? Repeat with multiples of 4.” 3.OA.1 (Interpret products of whole numbers, e.g., interpret $5\times7$ as the total number of objects in 5 groups of 7 objects each.)

• Unit 5: Fractions, Lesson 5.1, Instruction & Guided Practice, Exercise 1h, students develop conceptual understanding of fractions as they discuss sharing a pizza equally between friends. “Cut a pizza into 6 equal parts. Each part is $\frac{\square}{\square}$.” 3.NF.1 (Understand a fraction $\frac{1}{b}$ as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction $\frac{a}{b}$ as the quantity formed by a parts of size $\frac{1}{b}$.)

• Unit 8: Area, Perimeter, and Geometry, Lesson 8.3, Instruction & Guided Practice, Exercise 1e, students develop conceptual understanding of area by using square units. “Inches are larger than centimeters. How many square inches is the figure? 4 square inches. How many square centimeters is the figure? 25 square centimeters.” Teacher tip, “Have students use the square inches to cover the square first. Then have them use the square centimeters. Ask: Why do you need more square centimeters to cover the same region? [Answer: Centimeters are smaller than inches.]” 3.MD.6 (Measure areas by counting unit squares (square cm, square m, square in, square 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 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 3: Division, Lesson 3.5, Independent Practice, Exercise 2a, students relate division to equal groups and images. “Divide 12 muffins equally among 3 boxes. $12\div3=$___. There are ___ muffins in each box.” The teacher can support struggling students with teacher direction: “Tell students to move the counters into the boxes to answer the problem.” 3.OA.2 (Interpret whole-number quotients of a whole number; e.g., interpret $56\div8$ as the number of objects in each share when 56 objects are partitioned equally into 8 shares or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.)

• Unit 5: Fractions, Lesson 5.2, Independent Practice, Exercise 2f, students write fractions using images. “___ out of ___ boxes are colored. So, $\frac{\square}{\square}$ of the shape is colored.” The teacher can support struggling students with teacher direction: “Draw $\frac{3}{4}$ on the board using the diagram from 2e and the diagram from 2f with two additional parts shaded. Ask: Do these pictures represent the same fraction? Why or why not? [Yes, even though the pictures look different, they both show three equal parts shaded out of four.]” 3.NF.1 (Understand a fraction $\frac{1}{b}$ as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction $\frac{a}{b}$ as the quantity formed by a parts of size $\frac{1}{b}$.)

• Unit 5: Fractions, Lesson 5.4, Independent Practice, Exercise 2l, students represent fractions on a number line. “Drag the arrow to $\frac{2}{4}$.” The teacher can support struggling students with teacher direction: “Ask: Why is it easier to place the arrow in the correct location when the number line has ticks on it? [You can point the arrow at the tick mark that exactly marks the correct location.]” 3.NF.2b (Represent a fraction $\frac{a}{b}$ on a number line diagram by marking off a lengths $\frac{1}{b}$ from 0. Recognize that the resulting interval has size $\frac{a}{b}$ and that its end point locates the number $\frac{a}{b}$ on the number line.) and 3.NF.3c (Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.)

The materials reviewed for Snappet Math Grade 3 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 1: Addition, Subtraction, and Patterns, Lesson 1.4, Instruction & Guided practice, Exercise 1l, students develop fluency as they add using strategies based on place value. “$$547+176$$.” Teacher tip, “Have students explain each step, stating when they are working on 1s, 10s, or 100s in each step. This will help them retain a depth of understanding that could get lost over time. Challenge students to focus on the structure of the place values so they know how to accurately line up addends vertically, no matter how many digits there are.” 3.NBT.2 (Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.)

• Unit 3: Division, Lesson 3.17, Instruction & guided practice, Exercise 1j, students develop procedural skill and fluency as they fluently divide by 8. “___$$\times8=56$$ $56\div8=$___.” Teacher tip, “Ask: What division strategy does the thought bubble suggest? [Rewrite the division equation as a missing-factor multiplication equation.] What is the missing factor in the multiplication equation? [7]” 3.OA.7 (Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division.)

• Unit 8: Area, Perimeter, and Geometry, Performance task, Exercise 1a, Question 1, students develop procedural skills and fluency as they multiply to find the length of a garden. “The students in third grade are planting a rectangular vegetable garden at school. Here are the measurements of the garden: 18 feet length and 9 feet width. How large is the garden? Show how you found your answer. ___ square feet.” Teacher tip, “Ask students questions such as Can you answer this question by finding the area? Can you answer this question by finding the perimeter? Students will likely multiply the length and width of the rectangle.” 3.MD.7b (Multiply side lengths to find the areas of rectangles with whole-number side lengths in the context of solving real-world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning); 3.OA.7 (Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations.)

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: Addition, Subtraction, and Patterns, Lesson 1.3, Independent Practice, Exercise 2h, students demonstrate procedural skill and fluency as they add and regroup within 1000. “$$258+353=$$___.'' 3.NBT.2 (Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.)

• Unit 2: Multiplication, Lesson 2.9, Independent Practice, Exercise 2f, students demonstrate procedural skill and fluency as they use strategies to multiply. “$$5\times2=$$___, $6\times2=$___. half, double, $1\times$ more, $1\times$ less.” 3.OA.7 (Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations.)

• Unit 8: Area, Perimeter, and Geometry, Lesson 8.5, Exercise 2i, students demonstrate procedural skills and fluency as they find the length of a rectangle using the area formula. “Marcus painted a rectangle with an area of 36 square ft. The width of the rectangle is 6 ft. How long is the rectangle?” 3.MD.7b (Multiply side lengths to find the areas of rectangles with whole-number side lengths in the context of solving real-world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning); 3.OA.7 (Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations.)

The materials reviewed for Snappet Math Grade 3 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics. 

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

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

• Unit 4: Solve Word Problems, Lesson 4.4, Independent Practice, Exercise 2d, students solve word problems involving division in a routine application. “Rose gave 25 toys to children. Each child received 5 toys. How many children did Rose give toys? Rose gave toys to ___ children.” 3.OA.3 (Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities.)

• Unit 5: Fractions, Performance task, Exercise 1d, Problem 9, students solve a word problem by using multiplication and division to find the cost in a non-routine application. “Three friends buy a large pie. They equally share the cost and each takes the same number of pieces of the pie. How much will each friend pay? How many pieces will each friend take?” Teacher tip,  “Students will likely use a division fact or a related multiplication fact to find each answer. If students use a division fact, Ask: How did you choose which operation to use to solve the problem? If students use related multiplication facts, ask: How did you know which multiplication fact to use?” 3.OA.3 (Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities.)

• Unit 6: Measurement, Lesson 6.3, Instruction & Guided Practice, Exercise 1i, students solve word problems involving elapsed time in a routine application. “We leave at 10:35 am. We drive for 2 hours 25 minutes. What time do we arrive home? Set the clock to the right time.” Teacher tip, “Ask: Is 10:35 AM the starting time or the ending time? [Starting]” 3.MD.1 (Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.)

• Unit 8: Area, Perimeter, and Geometry, Lesson 8.7, Instruction & Guided Practice, Exercise 1c, students find the area of composite or irregular shapes in a non-routine application. “Can you break this figure into three rectangles? Color the unit squares to show how. What is the area of this figure? Could you find the area without counting the squares?” Teacher tip, “Say: This composite figure is made up of more than one two-dimensional shape. Draw the figure on graph paper and color each rectangle a different color. Share your work with the class. There is more than one way to divide the figure into three different rectangles. Ask: How can we use the area of each rectangle to find the area of the entire figure? [Sample answer: We can add the area of the three rectangles together.]” 3.MD.7d (Find the areas of composite figures by decomposing them into non-overlapping rectangles and adding the areas of the parts.)

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: Addition, Subtraction, and Patterns, Lesson 1.8, Exercise 2j, students use their knowledge of patterns to determine the next numbers in a list in a non-routine application.  Students will need to notice that the number does not increase by the same value, but that it increases by 3, 4, 5, etc., as the numbers move across the list. “6, 9, 13, ___, ___” 3.OA.9 (Identify arithmetic patterns (including patterns in the addition table or multiplication table) and explain them using properties of operations.)

• Unit 4: Solve Word Problems, Lesson 4.1, Independent Practice, Exercise 2c, students use multiplication and division to complete an equation in a routine application. “A flower shop has 40 flowers and 5 vases. An equal number of flowers will go in each vase. How many flowers will go in each vase? Complete the equation. __ __ 5 = __ flowers in the vase.” 3.OA.3 (Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.)

• Unit 6: Measurement, Lesson 6.12, Independent Practice, Exercise 2d, students solve a subtraction problem in a routine application. “Shelly packed a suitcase that weighs 45 kg.  She takes out 28 kg of luggage.  How much does the suitcase weigh now? Write a subtraction equation with a ? for the missing number. 45 - __. Answer: __ kg.” 3.MD.2 (Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using a drawing (such as a beaker with a measurement scale) to represent the problem.)

• Unit 7: Data, Performance task, Exercise 1b, students measure pencils to the nearest quarter inch and then create a line plot using the measurements in a non-routine application.  Students see 8 pencils to measure, and a number line with quarter inch marks on it to make the line plot. “The students collect some pencils. Measure each pencil to the nearest $\frac{1}{4}$ inch. Plot the lengths on the line plot.” 3.MD.4 (Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units-whole numbers, halves, or quarters.)

• Unit 8: Area, Perimeter, and Geometry, Lesson 8.10, Independent Practice, Exercise 2d, students find the perimeter of a garden in a routine application. “What is the perimeter of this garden? The perimeter is + + = feet.” 3.MD.8 (Solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.)

The materials reviewed for Snappet Math Grade 3 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: Addition, Subtraction, and Patterns, Lesson 1.5, Instruction & Guided Practice, Exercise 1j, students develop procedural skill and fluency as they use regrouping to subtract. “Find $485-127$ by regrouping. Step 1: Subtract the ones place. Not enough ones? Trade 1 ten for 10 ones.” Teacher tip, “Encourage students to recognize that they are only regrouping when there is not enough of one value on the top. Check in with students. Before moving to the next slide, Ask: After trading a 10 away from the 10s place, how would you subtract that place? [Since 1 is being taken away from the 8, 7 is left. Subtract 2 from 7 to get 5.]” 3.NBT.2 (Fluently add and subtract within 1,000, using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.)

• Unit 4: Solve Word Problems, Lesson 4.6, Independent Practice, Exercise 2g, students apply their understanding as they solve two-step word problems using operations and equations. “There are 5 hens in the coop. Each hen laid 6 eggs. The farmer collected 16 eggs. How many eggs are left in the coop?” 3.OA.8 (Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity.)

• Unit 8: Area, Perimeter, and Geometry, Lesson 8.4, Instruction & Guided Practice, Exercise 1k, students extend their conceptual understanding as they relate the methods of counting unit tiles, repeated addition, and multiplication to find the area of the same shape. “Each square has an area of 1 square meter. Add the rows to find the area. $6+$___$$+$$___$$=$$___ square meters. Multiply to find the area. ___ $\times6=$___ square meters.” Teacher tip, “Review that multiplication is a shortcut for repeated addition. Ask: When you add to find the area, what do you add? [Sample answer: Add the greater dimension the number of times equal to the lesser dimension.]” 3.MD.7 (Relate area to the operations of multiplication and addition.)

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 1: Addition, Subtraction, and Patterns, Lesson 1.3, Instruction & Guided Practice, Exercise 1m, students engage with conceptual understanding and procedural skills as they use strategies to solve addition problems. “$$349+166$$” Teacher tip, “Encourage students to add using the strategies they just learned and visualizing the number in expanded form and the place-value chart.” 3.NBT.2 (Fluently add within 1,000 using strategies that reinforce the structure of place value.)

• Unit 2: Multiplication, Lesson 2.8, Instruction & Guided Practice, Exercise 1c, students develop conceptual understanding alongside procedural skill and fluency as they multiply by 10. “Write a multiplication equation for each group of 10-dollar bills. Look at each factor that is not 10, and look at the product. What pattern do you notice in the numbers? Try to use the bills and pattern to multiply $5\times10$, $6\times10$, $7\times10$, $8\times10$, $9\times10$, $10\times10$.” Teacher tip, “Have students drag and drop the $10 bills to model the multiplication equations. Have them model 1 x 10, then 2 x 10, all the way to 10 x 10. Ask: What patterns do you notice? [Sample answers: The numbers increase by 10 each time.] Have students work in pairs, with one writing a multiplication equation with 10 and the other modeling the result with the money. Ask: How does this help us understand how to multiply by 10? [It shows us that it is similar to multiplying by 1, but we add a 0 at the end.]” 3.OA.7 (Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division, or properties of operations.)

• Unit 4: Solve Word Problems, Lesson 4.1, Independent Practice, Exercise 2f, students use all three aspects of rigor, conceptual understanding, procedural fluency, and application as they use equations in word problems. “A builder uses 54 stones to build a wall. The wall is 6 rows high. How many stones go in each row, if each row has the same number of stones? Complete the equation that solves the problem. ___ ___ ___ = ___ stones.” Teacher tip, “Ask: How do we know that this is a multiplication and division word problem, not a subtraction or addition word problem? [Possible answer: We have equal groups.] Can we count the bricks to solve the problem? [No; $6\times6$ is 36. Our total is 54.]” 3.OA.3 (Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities.)

The materials reviewed for Snappet Math Grade 3 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the Mathematical Practice standards. 

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

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

• Unit 2: Multiplication, Lesson 2.12, Math practices, Exercise 4a, “Exercise 4 has students work with MP 1 (Make sense of problems and persevere in solving them). Students relate multiplying by 4 to concepts or skills previously learned. Encourage students to recall what they learned about multiplying by 2 when there is 1 less group of 2. Ask a volunteer to share their answer. [Sample answer: When multiplying by 2, to find the product of 1 less group of 2, I found 2 less than the known product. Since I am multiplying by 4, I will find 4 less than the known product, or $20-4=16$. So, there are $4\times4=16$ candles.]” The exercise states, “There are $5\times4=20$ candles. How can you use what you already learned to quickly find the number of candles?”

• Unit 3: Division, Lesson 3.15, Math practices, Exercise 4c, “The intent of Exercise 4 is to provide students with practice applying MP 1 (Make sense of problems and persevere in solving them) as they relate dividing by 6 to previously learned skills.” “Have students work in pairs to draft a response to the question. Call on a student-pair to share their answers. [Students may say they could draw an array, draw boxes and tally marks, use counters, use multiplication facts, or rewrite the division equation as a multiplication equation with a missing factor.]” The exercise states, “$$36\div6=$$__ What are two different strategies you have learned that you could use to solve this problem?”

• Unit 8: Area, Perimeter, and Geometry, Lesson 8.10, Exercise 4c, students “practice MP 1 (Make sense of problems and persevere in solving them). Students will analyze a problem to see what information is given and determine a starting point.” The exercise states, “Both rectangles have a perimeter of 24 units. Which sides need to be labeled to determine if you have the same length and width? Explain.” 

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: Addition, Subtraction, and Patterns, Lesson 1.6, Math practices, Exercise 4a, “The intent of Exercise 4 is to allow students to practice MP 2 (Reason abstractly and quantitatively). Students will better understand the meaning of the quantities above the H, T, O in an HTO chart when regrouping to subtract across zeros. Have students work with a partner to answer this question.” The exercise states, “How do you subtract from a number that has a zero in the 10s and 1s places?”

• Unit 3: Division, Lesson 3.12, Math practices, Exercise 4c, “Exercise 4 gives students practice applying MP 2 (Reason abstractly and quantitatively) as they demonstrate their ability to decontextualize situations by representing them symbolically.” “This is an open-ended problem. Have students work in pairs to decide upon a division equation. Then have them develop a scenario for their word problem. Remind them they also need to draw a model of their choosing to represent their word problem. Call on several student-pairs to share and demonstrate their word problems and models with the class.” The exercise states, “Write a word problem and draw a model to represent it.”

• Unit 7: Data, Lesson 7.2, Math practices, Exercise 4a, “Exercise 4 allows students to apply MP 2 (Reason abstractly and quantitatively). This mathematical practice requires students to make sense of quantities and their relationships. Pair students with a partner. Give students time to think about the question. Then allow time for each partner to explain how they determined the scale of the bar graph. Call on a volunteer to explain their reasoning to the class. Be sure students understand that the numbers on the bar graph skip by 3s, so the scale is 3.” The exercise states, “What is the scale of the bar graph? Explain to a partner how you know.”

The materials reviewed for Snappet Math Grade 3 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: Multiplication, Lesson 2.16, Math practices, Exercise 4c, “In Exercise 4 students practice MP 3 (Construct viable arguments and critique the reasoning of others). Students will choose a useful strategy to determine the product and defend their reasoning when multiplying by 7.” “Give students time to find $5\times7$. Say: You can drag bags of 7 apples to help you. Then call on a volunteer to explain how they can justify their conclusion, or reasoning. [Sample answer: The factor 5 is half the factor 10. So, the product of $5\times7$ is half the product of $10\times7$.]” The exercise states, “You know $10\times7=70$. What is $5\times7$? $5\times7=$__ Justify your reasoning.”

• Unit 6: Measurement, Lesson 6.10, Math practices, Exercise 4a, “The intent of Exercise 4 is to allow students to practice MP 3 (Construct viable arguments and critique the reasoning of others). They will explain the questions they ask themselves to choose an appropriate unit of measure. Have students discuss the clarifying questions they would ask themselves in order to make a decision about which unit of measure, milligrams, grams, kilograms, or metric tons, to use. Call on a student to share their answer. [Sample answer: I would use kilograms because milligrams are for very tiny items, grams might be appropriate for 1 carrot, and tons are similar to a car which is much heavier than carrots. So, kilograms is the best option.]” The exercise states, “What unit of measure would you use to describe the mass of a bunch of carrots? Explain.”

• Unit 8: Area, Perimeter, and Geometry, Lesson 8.9, Math practices, Exercise 4a, “Exercise 4 has students practice MP 3 (Construct viable arguments and critique the reasoning of others). This mathematical practice requires students to justify conclusions with mathematical ideas. Some students may agree with the answer of 30 since walking around the outside of the garden would be a way to measure the perimeter. However, this is an incomplete answer if the units are not given. For example, 30 shoes could be the perimeter of the garden.” The exercise states, “Mr. Noschang walked around the outside of his garden making his shoes touch end to end while he walked. He counted 30 steps. He said that the perimeter of his garden is 30. Do you agree or disagree? Explain.”

The materials reviewed for Snappet Math Grade 3 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 2: Multiplication, Lesson 2.3, Math practices, Exercise 4b, students “practice with MP 6 (Attend to precision). Students understand the meaning of the multiplication symbol when using arrays and equal groups to model multiplication.” “Here, students use the meaning of the multiplication symbol to explain the meaning of an equation involving multiplication. Have students share their answer. [Sample answer: The equation means 6 times 3 is equal to 18.]” The exercise states, “What does the equation mean? $6\times3=18$”

• Unit 5: Fractions, Lesson 5.7, Math practices, Exercise 4a, students “practice MP 6 (Attend to precision). They will express numerical answers accurately and efficiently when renaming fractions.” “Pair students with a partner. Have them rename the fraction and then discuss how they know they renamed the fraction with precision. [Sample answer: The model shows that each circle is divided into 5 equal parts. All of the 15 equal parts are shared. There are 3 whole circles.]” The exercise states, “Rename the fraction. $\frac{15}{5}=$__ wholes. How can you be sure you renamed the fraction the right way?”

• Unit 6: Measurement, Lesson 6.11, Math practices, Exercise 4c, students “practice MP 6 (Attend to precision). They will communicate precisely and use clear mathematical language to determine mass by reading scales.” “Encourage students to use mathematical language they have learned in this unit to clearly and precisely explain their reasoning [Sample answer: The non-digital scales have lines to represent each kilogram or every 5 kilograms. This means you can only get a precise mass to either 1 or 5 kilograms. The digital scale measures weight to the tenth of a kilogram. Tenths is more precise than a whole number, so the digital scale gives a more precise mass.]” The exercise states, “How does a digital scale give a more precise mass for a person?”

• Unit 8: Area, Perimeter, and Geometry, Lesson 8.1, Instruction & guided practice, Exercise 1i, students “practice MP 6 (Attend to precision). This mathematical practice requires students to be precise when using mathematical language and discussing their reasoning.” Students see a rhombus on the screen. “Look at the sides of this shape. What shape is it? How do you know? This is a ____, because ____.”

The materials reviewed for Snappet Math Kindergarten meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to the grade-level content standards, as expected by the Mathematical Practice Standards. 

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

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

• Unit 2: Multiplication, Lesson 2.9, Math practices, Exercise 4b, “Exercise 4 has students practice MP 7 (Look for and make use of structure). Students apply rules when multiplying by 2 to find products.” “Ask: What do you notice about numbers being multiplied by 2? [Sample answer: 5 is 1 less than 6.] Ask a student to answer the question. [Sample answer: When a number being multiplied by 2 is 1 less than the other, then the product will be 2 less than the other. So, the product of $5\times2$ is 2 less than the product of $6\times2$, or $12-2=10$.] Test student’s knowledge of the rule by asking: What number times 2 has a product of 8? [4]” The exercise states, “What rule can you use to find $5\times2$? $6\times2=12$; $5\times2=$__”

• Unit 3: Division, Lesson 3.9, Math practices, Exercise 4a, “Exercise 4 allows students to practice MP 7 (Look for and make use of structure) as students learn to apply general mathematical rules to specific situations. Call on a student to share their response to the question. [A number divided by 1 is itself.] Take some time to review the rule for dividing with 1 as a divisor.” The exercise states, “Equally divide 6 marbles. $6\div1=$__; __ marble(s) in the bag. What is a rule for dividing with 1 as a divisor?”

• Unit 8: Area, Perimeter, and Geometry, Lesson 8.8, Math practices, Exercise 4c, “Exercise 4 has students practice MP 7 (Look for and make use of structure). This mathematical practice requires students to be flexible in their use of operations and understand quantities.” “Here, students are reasoning about the relationship between area and perimeter. By trying to find out if Tony is correct or incorrect, students will attempt to create two different quadrilaterals that have the same perimeter and area and should conclude that that it is impossible since the only other rectangle with an area of 4 units is 1 unit by 4 units, which does not have a perimeter of 8.” The exercise states, “Tony says that he can make more than one quadrilateral that has an area of 4 square units and a perimeter of 8 units. Do you agree or disagree with Tony? Explain your answer.”

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: Addition, Subtraction, and Patterns, Lesson 1.7, Math practices, Exercise 4b, “The intent of Exercise 4 is to allow students to practice MP 8 (Look for an express regularity in repeated reasoning) by understanding the broader application of patterns.” “Ask: Are the numbers all even? [No] Ask: Are the numbers all odd? [No] Ask: Are some of the numbers even and some odd? [Yes] Guide students in noticing that the numbers change between odd and even.” The exercise shows a multiplication chart with arrows showing skip-counting by 3s. “What do you notice about the pattern when skip-counting by 3?”

• Unit 3: Division, Lesson 3.10, Math practices, Exercise 4a, “Exercise 4 provides students with an opportunity to use MP 8 (Look for an express regularity in repeated reasoning) to find a pattern when they see different multiples of 10 being divided by 10. Call on a student to describe an organized way of dividing the apples into 10 groups. Ask: How can you use grouping to divide by 10? [I can make groups of ten and then count how many groups there are.] Call on another student to respond to the question on the screen. [The quotient is the dividend without the zero.]” The exercise states, “Divide 30 apples into 10 equal groups. __ apples in each group. $<strong>30\div10=</strong>$ What pattern do you notice?”

• Unit 7: Data, Lesson 7.1, 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 will use repeated calculations to interpret picture graphs.” “Have students work with a partner. Have each partner take turns explaining their thinking. Some students may choose to use repeated addition and others may choose to use multiplication. One way students could explain their thinking is to add the number of strawberries and lemons, $4+3=7$. Then multiply 7 by 8. Lead a class discussion to review the relationship between repeated addition and multiplication.” The exercise shows a picture graph for “Favorite fruit.” “Each picture $=8$ Explain to a partner how to find the number of friends that picked strawberry and lemon.”

The materials reviewed for Snappet Math Grade 3 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students to guide their mathematical development. The Documentation section of the materials provides comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include:

• Snappet Teacher Manual, 3.1 Teacher Guide, “A Teacher Guide is available for every instructional lesson in Snappet, both digitally and on paper. The Teacher Guide contains the lesson overview, math content standards alignments, materials needed, vocabulary, EL/SEL strategies, common errors, and step-by-step support for teaching the lesson. Consistent design: The Teacher Guide, like the lesson itself, always has the same structure and is therefore, easy and clear to follow. From the Teacher Guide, the teacher has access to the learning path for every learning objective with constant visibility into the progress of the class.  Full support: The learning phases explained in the teacher manual are also visible while teaching the lesson in the digital environment. This gives the teacher the support they need not only while planning their lessons, but also while teaching their lessons. Easy to print: The teacher manual is easy to print by course or by lesson. Each downloadable and printable Teacher Guide is customized with the most up-to-date information about the progress and skill development for each student.”

• Instructional videos include 1-2 minute videos showing how to use the software, 5-minute videos of the classroom condensed to show each segment of the lesson, and full lesson videos. 

• Grade 3-Pacing Guide provides the number of weeks to spend on each Unit and a Materials list for each Unit.

Materials include sufficient and useful annotations and suggestions that are presented within the context of specific learning objectives. Preparation and lesson narratives within the Unit/Lesson Overviews and Teacher Tips provide useful annotations. Examples include:

• Grade 3-Unit Overviews, Unit 8 Overview: Area, Perimeter, and Geometry, Understanding the Math, “Geometry is essential for helping children understand spatial relationships. It teaches them to determine how large a room is, how far away a desk is, or which way to move. Geometry allows students to connect mapping objects in the classroom to real-world contexts regarding direction and place. Measurement helps describe our world using numbers. We use numbers to describe simple things like length, weight, and temperature, but also complex things such as pressure, speed, and brightness.”

• Unit 3: Division, Lesson 3.9, Instruction & guided practice, Exercise 1c, Teacher Tip, “(SEL) Give each student ten counters. Have students attempt to divide ten counters into ten equal groups. Ask them to write a division equation to show 10 ÷ 10  and its quotient. Have them try different dividends that have the same divisor.”

• Unit 7: Data, Lesson 7.2, Lesson Overview, Common Error (CE), “Look for the (CE) label for ideas of where to apply this suggestion. If students misread the graph by looking at the number below the top of the scale instead of right at the top, then tell them their answer is too small. If students still struggle, have them start at 0 and see that the value of the graph increases by one multiple of the scale as it reaches each gridline, so the value of the gridline at the top is the correct value.”

The materials reviewed for Snappet Math Grade 3 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject. 

Snappet Math provides explanations for current grade-level concepts within the Understanding the Math and Learning Progressions components of the Unit Overviews. Prior, current, and future standards are connected within the Lesson Overview of each lesson. Additionally, each Lesson Overview includes Deepening Content Knowledge Beyond Grade Level, which provides explanations and examples of more complex grade-level concepts and concepts beyond the current course. Examples include:

• Unit Overviews, Unit 2 Overview: Multiplication, Understanding the Math, “Understanding that multiplication is repeated addition of the same number is essential for success with division. Interpreting products of whole numbers into multiple groups of the same size of objects helps solidify the meaning and use of multiplication. Multiplication fluency allows students to make connections to real-world contexts involving money, cooking, gardening, sports, and other topics where making a number larger through multiplication has an impact on the situation. A solid foundation in multiplication facts for single-digit numbers allows for success in solving a wide variety of math problems.” 

• Unit Overviews, Unit 4 Overview: Solve Word Problems, Learning Progression, “In prior grade levels, students solved addition and subtraction word problems that had missing or extra information, writing equations to represent different situations (2.OA.A.1). They compared two-digit numbers using a number line, place value, and symbols (1.NBT.B.3). They solved word problems involving money (2.MD.C.8). In this grade level, students will continue to multiply and divide within 100 fluently using the relationship of multiplication and division. They will multiply and divide to solve one- and two-step word problems, using equal groups, arrays, bar diagrams, and equations. They will also use their knowledge of multiplication and division to understand and solve comparison word problems by pulling information directly from the word problem and/or from visual representations. They will identify information in a word problem as either given or missing, and then they will decide if an equation or visual is best to use for solving. In future grade levels, students will solve more complex multi-step word problems (4.OA.A.3). They will estimate to solve addition/subtraction word problems (4.OA.A.3). They will solve word problems involving money (4.MD.A.2, 5.NBT.B.7). They will use multiplication and division to solve word problems involving decimals (5.NBT.B.7) and fractions (5.NF.B.3).”

• Unit 7: Data, Lesson 7.1, Lesson Overview, Deepening Content Knowledge Beyond Grade Level, “Statistics and Data Analysis: The skills learned in analyzing scaled picture graphs form the foundation for more advanced studies in statistics and data analysis. In higher grades, students will engage with more complex data sets, learning to create and interpret histograms, box plots, and scatter plots. They will also begin to understand concepts such as mean, median, mode, range, and standard deviation.”

• Unit 8: Area, Perimeter, and Geometry, Lesson 8.9, Lesson Overview, “In prior lessons, students have measured using an inch ruler (2.M.A.1); solved word problems involving length (2.MD.B.5). In this lesson, students will use a ruler, tape measure, and string to measure the outline of an object and make the connection that the outline of the object is the perimeter (3.MD.D.8); find the perimeter using unconventional forms of measurement (3.MD.D.8). In future lessons, students will choose an appropriate unit of measurement (4.MD.A.1); solve problems with perimeter and area of rectangles (4.MD.A.3).”

The materials reviewed for Snappet Math Grade 3 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Correlation information is present throughout the grade level and can be found in the Pacing Guide, Unit Overviews, and each Lesson Overview. Explanations of the role of the specific grade-level mathematics in the context of the series can be found in each Lesson Overview under The Specific Role of the Standard in the Overall Series. Examples include:

• The Pacing Guide provides a table separated by unit and includes columns identifying previous skills, grade-level skills, and future skills. The skills are grouped by standard and are linked to identify lesson(s) standard alignment.  

• Unit Overviews identify the standards addressed in each unit and a lesson standard alignment. The Unit Overviews also include a learning progression that links current standards to previous and future standards for each unit.

• Unit 6: Measurement, Lesson 6.4, Lesson Overview, The Specific Role of the Standard in the Overall Series, “Foundation for Time-Based Reasoning: The standard 3.MD.A.1 is crucial as it lays the foundational skills for time-based reasoning, an integral aspect of everyday life and scientific inquiry. By mastering elapsed time problems, students develop an essential skill set for understanding sequences of events, duration, and the concept of causality.”

• Unit 8: Area, Perimeter, and Geometry, Lesson 8.8, Lesson Overview, Mathematical Content Standards, “3.MD.D.8 Solve real world and mathematical problems related to finding the perimeter of a polygon, finding unknown lengths, and comparing figures with the same perimeter, and different areas, or same areas and different perimeters.”

The materials reviewed for Snappet Math Grade 3 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. Instructional approaches of the program are described within the Teaching using the Snappet Method document. The four lesson components—Instruction and Guided Practice, Extend Learning using Math Practices, Independent and Adaptive Practice, and Small Group Instruction are described. Examples include:

• Instruction and guided practice, “The lesson design for instruction follows the CRA approach to teaching: Concrete, Representational, Abstract. The exercises begin with Activate Prior Knowledge exercises which are designed to be used as real-time feedback opportunities during the introduction of the new lesson. This is followed by Student Discovery where manipulatives, games, or activities will be introduced to prepare students minds and bodies for new learning. These activities are followed by instruction slides that provide opportunities for students to think out loud, think pair share, co-craft questions, and talk about the new concept in a variety of ways. Instruction is followed by Guided Practice exercises where students can try it on their own while being supported by the teacher. The Guided Practice exercises also give the teacher the opportunity to identify if students are ready to begin practicing independently and to identify any common errors that might be occurring. Following Instruction and Guided Practice, teachers can go deeper into the mathematics by introducing the Math Practices exercises.”

• Extend lLearning using Math Practices, “Teachers will utilize the exercises available in Math Practices to go deeper in the complexity of student learning. These exercises are designed to be non-routine, open-ended, and an extension of the discussions that occurred during the lesson. Often, these exercises will extend beyond the Student Discovery activities. It is recommended to group students into groups of 2 (K-2) or 3 (3-5) to encourage students to discuss their thinking and give evidence for their reasoning.”

• Independent and Adaptive Practice “Students continue their learning of the concepts during independent practice. Independent Practice exercises are written at grade level and act as a “diagnostic assessment” to determine the appropriate level of Adaptive Practice. Adaptive practice offers 5 levels of difficulty that are defined by the quintile measures. Level 3 is considered grade-level proficient. Quality is the goal over quantity. It is recommended that only 1-3 sets (10-30 questions) of adaptive practice exercises be completed in any one practice session. Once students have reached their target goals and attained their desired level, they should either practice on a different concept or finish practice for the day.”

• Small Group Instruction, “Every lesson includes a Small Group Instruction intervention lesson for students that are struggling with the concept. This becomes evident when students are not able to progress during adaptive practice. Student initials will appear in yellow and will be identified as being “stuck” on their progress towards their target goals. It is recommended to provide reteaching to these students in a small group setting using the exercises in the small group instruction section. These exercises are scaffolded to provide support for struggling students. Once you have completed this lesson with students and they have demonstrated understanding using the guided practice exercises in the small group lesson, you can continue to monitor the students progress by having them continue to practice adaptively on the lesson.”

Research-based strategies within the program are cited and described in the Snappet Teacher Manual within Research-based Strategies in Snappet Math. Snappet Math states, “The Snappet Math curriculum integrates a series of rigorously research-based instructional approaches and strategies explicitly designed to facilitate effective K-5 mathematics education. Informed by eminent educational researchers and institutions, including the National Council of Teachers of Mathematics (NCTM) and the Institute of Education Sciences (IES), the key strategies are as follows:...” Examples include: 

• Concrete-Pictorial-Abstract (CPA) Approach, “This method involves the sequential use of concrete materials, pictorial representations, and abstract symbols to ensure thorough understanding (Bruner, 1966). Snappet's curriculum employs and explicitly references the CPA approach in the lesson phases ‘Apply in a concrete pictorial representation’ and ‘Apply in an abstract representation.’”

• Problem-Solving Instruction, “Snappet encourages students to engage with real-world problems, enhancing the relevance and application of mathematical concepts and procedures (Jonassen, 2000). Guidance is provided on various problem-solving strategies (Polya, 1945) in both instruction & guided practice and during independent practice.”

• Formative Assessment, Feedback, and Error Correction, “Regular assessments help to understand a student's learning progress, provide opportunities to give feedback, and adjust instruction (Black & Wiliam, 1998). Feedback is one of the most powerful influences on learning and achievement (Hattie, 2003), and correcting common errors has been identified as a factor that positively influences student achievement (Smith & Geller, 2004). Due to Snappet’s elaborate and immediate feedback system, every activity serves as a formative assessment. During instruction and guided practice, student responses appear on the Interactive Whiteboard in real-time for all students and the most common errors made by the students are summarized and highlighted. This feedback allows teachers to identify and correct common errors quickly, promoting student understanding and success. For every lesson and standard, both the teacher and students get continuous feedback on the current performance and progress. The immediate and actionable feedback, along with prompt error correction, is integral to promoting student achievement and progress in the Snappet Math curriculum.”

• Direct Instruction, “Direct instruction is a major factor in contributing to student achievement (Rosenshine, 2012). This involves clear, concise teaching where the teacher models what is to be learned and provides guided practice with immediate feedback. The Snappet Math curriculum incorporates this approach, with teachers provided with detailed lesson plans, strategies for explicitly teaching concepts, and resources for modeling mathematical thinking. The interactive nature of Snappet also allows for real-time guided practice and these exercises are explicitly referenced in every lesson with the guided practice icon ( ), aligning with the principles of direct instruction.”

The materials reviewed for Snappet Math Grade 3 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. The program provides a Material List, and specific lessons include a Materials heading needed to support instructional activities within the Lesson Overview. Examples include:

• Grade 3-Material List, “The list below includes materials used in the 3rd Grade Snappet Math course, excluding printed materials and templates. The quantities reflect the approximate amount of each material that is needed for one class. More detailed information about the materials needed for each lesson can be found in the Lesson Overview.” A table lists the Materials, Unit(s), and Approximate Quantity Needed, “Paper cups; 3, 6; 13 cups per pair of students.” 

• Unit 2: Multiplication, Lesson 2.5, Lesson Overview, Materials, “Per pair: 24 counters.”

• Unit 6: Measurement, Lesson 6.5, Lesson Overview, Materials, “Per group: a drinking glass, a small bottle, a milk container, a larger bottle, a measuring cup, empty cups, a mug, water.”