3rd-5th Grade - Gateway 2

The materials reviewed for Takeoff by IXL Grades 3 through 5 meet expectations for supporting the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

The materials reviewed for Takeoff by IXL Grades 3 through 5 include teacher-guided tasks during instruction and guided practice, as well as tasks in which students independently complete and respond to mathematical problems. During independent practice, students work in IXL Math. When students solve a question incorrectly, the platform provides a step-by-step explanation and allows students to watch a video tutorial or review a worked example before attempting the problem. 

Examples include:

• Grade 3, Unit 4: Area, Lesson 4.1, Independent Practice, students demonstrate conceptual understanding as they determine area as the number of unit squares that cover a two-dimensional figure without gaps or overlaps. Students find the area by tiling figures with unit squares. At the beginning of the lesson, students use unit tiles to cover rectangles and discuss how to measure how much space an object takes up. Throughout the activity, the teacher highlights similarities between length and area (e.g., units must be equal in size; there should be no gaps or overlaps). In #5–13, students use square tiles to represent and find the area of irregular figures. In #14, students explain the error in a given example. Find the area of figures made of unit squares (FLQ), “The shape is made of unit squares. What is the area of the shape?” (3.MD.5)

• Grade 4, Unit 8: Equivalent fractions and comparing, Lesson 8.1, Instruction: Learn to use models to find equivalent fractions. Students develop conceptual understanding as they represent and identify equivalent fractions. Teacher notes, “For #2, introduce the term equivalent. Explain that another way to say fractions are equal is to say they are equivalent. Highlight that the words equal and equivalent both originate from the Latin root equ, meaning even, equal, or level. (Consider pointing out that the word equation also has the same root and that both sides of an equation are balanced: they are equal in value.) Give each student a sheet of paper they can use to model equivalent fractions. Have them fold it down the middle into two equal parts. (Throughout the activity, students can fold the paper either lengthwise or widthwise.) Then have them unfold the paper, trace the crease with a pencil, and shade one of the equal parts. Discuss the fraction the shaded area represents. Students should note that they shaded \frac{1}{2} of the paper. Next, have students fold the paper into fourths and then unfold it, again tracing the creases they made. Discuss how the \frac{1}{2} that was shaded is now two of the four total parts, or \frac{2}{4} of the page. Make sure students understand that \frac{1}{2} and \frac{2}{4} are equivalent because they represent the same shaded area. Have students repeat the activity to find how many eighths \frac{1}{2} is equivalent to. (It's equivalent to \frac{4}{8}.)” (4.NF.1)

• Grade 5, Unit 2: Whole number division, Lesson 2.4, Instruction, Learning to divide using area models, students develop conceptual understanding as they use an area model to divide by a two-digit divisor. Teacher notes, “For #8, guide students to solve the problem with an area model. Have students start by choosing how to break up the rectangle. Highlight that they can use as many parts as they need. In particular, they don't have to use only two parts, like in #7. Students may choose parts of 240, 240, and 192. Or, if they can divide 480 by 24 in their heads, they may choose parts of 480 and 192. Explain that both models are correct, and discuss the pros and cons of each. It might be easier to divide 240 by 24, but it's faster to draw a model with fewer parts. When students have an area of 192 left, encourage them to divide by estimating and adjusting to find the length of that part. Point out that while you could keep taking away parts that have an area of 24, this would take a long time. For #9, point out that this problem is different from #7 and #8 because you are given the product and need to find one of the factors. Discuss how you can still use an area model to solve the problem. The number of chairs, 32, can represent the width of the area model, and the total cost, $3,488, can represent its area. You need to find the length of the model, which is the cost of each chair. Discuss ways you might try to find the length of the model. One way that works well is to divide the model into parts, the first of which can be a multiple of 100.” (5.NBT.6)

The materials reviewed for Takeoff by IXL Grades 3 through 5 meet expectations for providing intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.

The materials reviewed for Takeoff by IXL Grades 3 through 5 provide opportunities for students to develop procedural skills and fluency through teacher-guided instruction and guided practice. During independent practice, students work in IXL Math, where they solve procedural problems and receive immediate feedback. When students answer incorrectly, the platform provides step-by-step explanations and access to video tutorials or worked examples, allowing students to attempt the problem again. 

Examples include:

• Grade 3, Unit 3: Multiplication fluency, Lesson 3.3, Independent practice, students demonstrate procedural fluency with multiplication. Multiply 3 by numbers up to 10 (38K), “Multiply. 3\times5=___.” Students solve additional problems of the same type. (3.OA.7)

• Grade 4, Unit 2: Add and subtract whole numbers, Lesson 2.2, Instruction, Learn to add using the standard algorithm, students develop procedural skills and fluency as they use the standard algorithm to add whole numbers. Teacher notes, “For #4, discuss how Leo's work relates to Clare's work in #3. Both Leo and Clare add one place at a time. Clare writes each partial sum on a separate line, then finds the final sum at the end. Leo records all of his work on one line, so he adds the partial sums as he goes. Point out how each person writes the partial sum of 13 hundreds. Clare writes it below the line as 1 thousand and 3 hundreds. Leo only has space to write the 3 hundreds below the line. So he writes a small 1 at the top of the thousands column to keep track of the regrouped thousand. Explain that Leo's way is called the standard algorithm. An algorithm is a set of steps you can follow in order to solve a problem. For example, you used an expanded form algorithm to add in #2. The standard algorithm is one of the quickest ways to show your work in writing, especially when you're working with large numbers. Help students use the standard algorithm to find 3,495+2,381. Explain that you start from the right in case you have to regroup from a smaller place value unit to a larger place value unit. Here, the sum of 9 tens and 8 tens is 17 tens, so you record the 7 in the sum below and write a small 1 above the hundreds.” (4.NBT.4) 

• Grade 5, Unit 1: Whole number place value and multiplication, Lesson 1.5, Instruction: Learn to multiply using the standard algorithm. Students develop procedural skills and fluency as they use the standard algorithm to solve problems. Teachers notes, “For #7, guide students through Lucia's work to help them compare it to Jayden's work. Here are a few things to highlight: They both multiplied 2 ones by 8 to get 16 ones. Jayden wrote the 16 below the line. Lucia wrote 6 ones below the line, but she regrouped 10 ones into 1 ten and recorded it in the tens column at the top. That's because she knew she was going to add it to the tens from the next step. Then they both multiplied 3 tens by 8 to get 24 tens, or 240. Jayden wrote the 240 below the line. Lucia combined the 24 tens with the 1 ten she regrouped in the last step to get 25 tens. Lucia wrote 5 tens below the line, but she regrouped the 20 tens into 2 hundreds and recorded a 2 in the hundreds column at the top. Share with students that Lucia's method is the standard algorithm for multiplication. Students should be familiar with that term from addition and subtraction in earlier grades. Remind them that the standard algorithm is just one way to multiply, but it tends to be the fastest and most reliable, especially for large numbers.” Guided practice, Multiply using the standard algorithm, Teacher notes, “For #14–15, have students set up the multiplication vertically. Emphasize that students should line up the ones digits of the factors, just like they would when adding and subtracting.” Question 14, “An office company sells highlighters in packs of 5. How many highlighters would the company need to fill 281 packs?” Question 15, “The same office supply company sells glue sticks in packs of 4. In January, the company sold 572 packs of glue sticks. In August, the company sold 8 times as many packs of glue sticks as they did in January. How many glue sticks did the company sell in August?” (5.NBT.5)

The materials reviewed for Takeoff by IXL Grades 3 through 5 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade as reflected by the standards.

Multiple aspects of rigor are engaged simultaneously across the materials to develop students' mathematical understanding of individual lessons. Each lesson within the curriculum supports a variety of instructional approaches that incorporate conceptual understanding, procedural fluency, and application in a balanced way. According to Additional resources, Instructional design, A balanced approach to rigor, “Takeoff by IXL elevates students’ math learning using a balanced approach to mathematical rigor. With lessons thoughtfully crafted to develop students’ conceptual understanding, procedural fluency, and ability to tackle real-world applications, Takeoff not only provides in-depth exploration into each element of rigor but also intertwines the elements for a richer, more well-rounded understanding of math concepts.”

For example:

• Grade 3, Unit 8: Perimeter, Lesson 8.4, students demonstrate procedural skill and conceptual understanding by solving a problem with multiple solutions involving the area and perimeter of a pond and fence. Student book, Question 11, “Tiana is a landscape designer. She is designing a rectangular fish pond for a local park, and she must follow these rules: The pond must cover exactly 12 square feet. The pond must be surrounded by a rectangular stone patio. The patio must be surrounded by a rectangular grassy lawn. The lawn must be surrounded by 26 feet of fencing. Draw one way to design the pond, patio, and lawn. Be sure to label the length and width of each part.” (3.OA.7)

• Grade 4, Unit 3: Multiply by 1-digit numbers, Lesson 3.6, students demonstrate procedural skill and conceptual understanding by using an area model to multiply multi-digit numbers. Question 4, “Katy also wants to tile her front walkway. She plans to use 8 rows with 56 tiles each. How many tiles will she use for her walkway in all? Use an area model to solve.” (4.NBT.5)

• Grade 5, Unit 4: Volume, Lesson 4.2, Guided practice, Find the volume of rectangular prisms, students develop conceptual understanding and application as they interpret a real-world situation, reason about volume as layers of unit cubes, and determine the total number of robots in the crate. Teachers notes, “For #14, remind students to read the problem carefully when they write units for their answers. Since this problem asks for the number of robots that fit in the crate, not the volume of the crate, the answer is not 18 cubic feet.” Question 14, “Best Bots Toys packs its toy robots in cube-shaped boxes. Each box has a volume of 1 cubic foot. On Monday, Best Bots shipped a small crate filled with robots to one of its stores. The bottom crate was covered with a layer of 6 boxes. If the crate was 3 feet high, how many robots were in the crate?” (5.MD.C)

The materials reviewed for Takeoff by IXL Grades 3 through 5 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them, in connection to the grade-level content standards, as expected by the mathematical practice standards.

In Grades 3-5, MP1 is identified consistently across the year. Teachers receive guidance in the Teacher’s Guide, and each unit includes a Unit Overview with Math Practices guidance that highlights examples of the Mathematical Practices addressed within that unit. In addition, the Mathematical Practices are identified in the overview of each lesson.

Across the grades, students engage in open-ended tasks that support key components of MP1, including sense-making, strategy development, and perseverance. These tasks prompt them to make sense of mathematical situations, develop and revise strategies, and persist through challenges. They are encouraged to analyze problems by engaging with the information and questions presented, using strategies that make sense to them, monitoring and evaluating their progress, determining whether their answers are reasonable, reflecting on and revising their approaches, and increasingly devising strategies independently. 

Examples include: 

• Grade 3, Unit 4: Area, Lesson 4.5, Instruction: Find the areas of compound figures, students solve real-world problems involving area and composite shapes. Students persevere in solving complex problems. Teacher notes state, “For #12, students may solve in a variety of ways. Make sure students see that they need to find the difference between the area of the playground (57 square yards) and the area of the picnic section (27 square yards). If students ask, explain that "yd." is an abbreviation for yards.” Student book states, “11. Sam is putting new carpeting in his bedroom. Use the floor plan below to find how much carpet he will need.” Students use a floor plan of a composite shape made of two rectangles to solve the problem. Unit Outline, Math practices state, “Area is a brand new concept in third grade, and students will need substantial support in learning to make sense of problems that ask about area. Throughout the unit, draw attention to language that indicates students need to think about area and help them come up with strategies to persevere through some of the harder problems. In Lessons 4.5 and 4.6, students solve area problems involving compound figures. Encourage them to consider whether multiple approaches could work for the same problem. For example, they can think about when a figure can be split in more than one way. Throughout, help students monitor and evaluate their progress. Have them reread problems to make sure their work is leading them toward the missing value(s). If it's not, they should think about whether they need to make small changes or restart.”

• Grade 4, Unit 5: Factors, multiples, and patterns, Lesson 5.7, Instruction: Learn to extend and analyze shape patterns, students make sense of problems involving shape patterns. Teacher notes state, “For #3, provide students with toothpicks. Consider having students work in pairs or small groups. Each student or group of students needs at least 15 toothpicks. Have students build the next step in the pattern using toothpicks. Then discuss how students built the next step before having them draw it and complete the table. Be sure to surface these things: Every step in the pattern is a triangle. The number of toothpicks used to make each side of the triangle matches the number of the step. Because the number of toothpicks per side matches the step number, the number of toothpicks on each side grows by one with each step in the pattern. Explain that this is an example of a growing shape pattern. For #4, discuss how many toothpicks will be in the 5th, 6th, and 100th steps. Encourage students to think about how the pattern grows and how the number of toothpicks relates to the step number. For example, students may observe that the 100th step will be a triangle with 100 toothpicks per side. Or they may observe that the number of toothpicks is 3 times the step number. For #5, guide students to identify the number of toothpicks that could be used to make triangles in this pattern. Point out that those numbers are the multiples of 3 because each triangle has three equal sides. Encourage students to use the divisibility rule for 3 to determine which numbers are multiples of 3.” Student book states, “3. Jake made a pattern with toothpicks. Draw the next step in the pattern. Then complete the table.” Students see three toothpick triangles made with 3, 6, and 9 toothpicks. They also see a table labeled Pattern Step and Number of Toothpicks. The table shows the Pattern Step already filled in, and students complete the Number of Toothpicks row. “4. How many toothpicks will be in Step 5? Step 6? Step 100? 5. Which of these could be the number of toothpicks in a step of Jake’s pattern? 18, 24, 25, 40, 63, 81.” Unit Outline, Math practices state, “Students solve problems involving number and shape patterns throughout Unit 5. Support critical thinking skills by prompting them to explain how different representations are connected. For example, students focus on problems involving shape patterns in Lesson 5.7. Help them make sense of these problems by reasoning about how each shape in a pattern changes from one step to the next. It can be good to model your thinking aloud for students to see what this sort of reasoning looks like. Then they can identify entry points to solve the problem, such as developing a rule for the pattern. In #3, for example, students determine a rule for the number of toothpicks needed to build the triangles. Make sure students can justify why their rule works. In this case, they might describe how many toothpicks are needed to form one side of a triangle in the pattern. Because a triangle has three sides, the total number of toothpicks is three times that amount.)”

• Grade 5, Unit 3: Expressions, Lesson 3.4, Instruction: Learn to use expressions to solve word problems. Students make sense of multi-step word problems while incorporating their understanding of order of operations rules. Teacher notes state, “Explain that one way to solve a word problem with multiple steps is to write an expression that represents the answer and then evaluate it. This is good preparation for modeling real-world situations in middle school and high school. For #3, point out that though the stories use the same numbers, you need different operations to find the answers. So they're represented by different expressions. In the first story, there are 6 basketball courts and 4 basketballs at each one. So there are 6\times4 basketballs for the courts. There are also 10 extra basketballs. So there are 6\times10+4 basketballs in all. In the second story, there are 10 girls playing and 4 girls waiting for a turn, or 4+10 girls, at each court. Since this is the number of girls who are at each of the 6 courts, there are 6\times(4+10) girls altogether. Throughout, encourage students to identify the information that they know, what they need to find, and any hidden questions they need to answer along the way. This will help direct students' next steps while keeping the end goal in mind.” Student book states, “3. Everyone in Clara’s neighborhood is excited about their new community center. Read the problems about the community center. Then pick an expression and evaluate to solve each one. 6+4\times10, 6\times(4+10), 6\times4+10, (6+4)\times10. The center’s gym has six basketball courts. There are 4 basketballs for each court and 10 extra basketballs. How many basketballs are there in all? During a tournament, each of the 6 courts has 10 girls playing and 4 girls waiting a turn. How many girls are there altogether?” Unit Overview, Math practices state, “In this unit, students apply their understanding of the order of operations as they write and interpret expressions to solve multi-step word problems. Encourage them to analyze each situation, break it into smaller steps, and write expressions that represent those steps. Have students compare the problems in Lesson 3.4 #3, and emphasize that even when the numbers and operations are the same, the order may be different. Prompt students to explain how they decided which step to do first. This helps them look beyond the numbers to understand the full situation, which is an important part of making sense of problems and persevering in solving them.”

The materials reviewed for Takeoff by IXL for Grades 3 through 5 meet expectations for supporting the intentional development of MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

In Grades 3-5, MP2 is identified across the year. Teachers receive guidance in the Teacher’s Guide, and each unit includes a Unit Overview with Math Practices guidance that highlights examples of the Mathematical Practices addressed within that unit. In addition, the Mathematical Practices are identified in the overview of each lesson.

Across the grades, students participate in tasks that support key components of MP2, including reasoning with quantities, representing situations symbolically, and interpreting the meaning of numbers and symbols in context. These tasks encourage students to consider the units involved in a problem, analyze the relationships between quantities, and connect real-world scenarios to mathematical representations. Teachers are guided to support this development by modeling the use of mathematical notation, asking clarifying and probing questions, and facilitating conversations that help students make connections between multiple representations. 

For example: 

• Grade 3, Unit 5: Understand division, Lesson 5.1, Instruction, Introduction to division, students reason abstractly and apply division strategies to contextual situations, writing equations to represent those situations. Teacher notes state, “Provide students with sets of counters. Consider having students work in pairs or small groups. Each student or group of students will need at least 20 counters. For #2, guide students to model the problem using counters. Have them gather 18 counters and place them one at a time into 6 groups. They should find that there are 3 counters (or lollipops) in each group. Then have students complete the equation 18\div6=3. Explain that you can read the equation as ‘18 divided by 6 is 3,’ and discuss the meaning of division and the division symbol in this scenario. Be sure to highlight these things about the division equation: The first number is the total. It's called the dividend. It is the number being divided into equal groups. The second number is the number you are dividing by. It's called the divisor. In this scenario, it is the number of equal groups. The last number is the result of the division. It's called the quotient. In this scenario, it is the number in each group. For #5, start by having students describe what they notice about the balloons. They should notice that there are 3 groups of 9 balloons. Then have them discuss whether Anita's or Jason's story matches the balloons. In Anita's story, the number of balloons in each group is known. She used 9 as the divisor. In Jason's story, the number of groups is known. He used 3 as the divisor. Conclude that both word problems correctly describe the balloons.” Student book states, “2. Cassie is making treat bags for a party. She has 18 lollipops, and she wants to split them equally into 6 bags. How many lollipops should Cassie put into each bag? Model the problem with counters. Then complete the sentences. Cassie has 18 lollipops. She can make six groups of ____ lollipops. Complete the equation. 5. Anita and Jason each made up a division problem about these balloons. Whose problem matches the balloons? How do you know? Anita: There were 27 balloons, and I tied them into groups of 9. That’s 27\div9. Jason: I was thinking of a different story. There were 27 balloons, and I split them into 3 equal groups. That’s 27\div3.” Unit outline, Math practices state, “Beginning in Unit 5, third graders devote significant time to developing a conceptual understanding of division and exploring its use in real-world contexts. Highlight the two types of division situations—finding the number of groups and finding the size of each group. Lessons 5.1 and 5.6 include both types, giving students opportunities to reason abstractly and quantitatively as they connect division equations to real-world scenarios. In Lesson 5.4, shifting flexibly between equations, visual models, and word problems can support their understanding of why division by 0 is undefined. For example, in #10, students who struggle to draw a model for 3\div0 can come to understand the concept more clearly through the context: you can't split 3 muffins among 0 plates if there are no plates.”

• Grade 4, Unit 6: Divide by 1-digit numbers, Lesson 6.8, Instruction, Learn to interpret remainders, students reason abstractly as they solve division problems and determine how the remainder applies in a real-world situation, using the context to guide their interpretation. Teacher notes state, “For #2, have students use division to solve. Then have them compare the problems in #1–2. In #1, they divided by the number in each group ($4), so the quotient was the number of groups (14 groups of 4). In #2, they divided by the number of groups (5 tiers), so the quotient was the number in each group (9 cupcakes in each tier). Discuss whether the quotient (9) or the remainder (3) is the answer to the question. In this case, the problem asks for the number of cupcakes on each tier. The cupcakes on each tier are the equal groups, so the answer is the quotient. For #3, after students divide, discuss how you know that the answer can't be the quotient or the remainder. The quotient tells you there will be 10 full tables. The remainder tells you that if Amaya's aunt only borrows 10 tables, 5 people won't have a seat. Since she needs one more table to account for those 5 people, she needs to borrow a total of 11 tables. That's the number the question asked for.” Student book states, “Solve each problem. Then complete each sentence. Circle the part of the sentence that answers the question asked. 2. Amaya’s dad made 48 cupcakes and arranged them on a stand with 5 tiers. He put the same number of cupcakes on each tier, and he used as many cupcakes as he could. How many cupcakes are on each tier? There are ____ cupcakes on each tier, with ____ cupcakes left over. 3. Amaya sent out flyers to her family, and 65 people said they were coming to the reunion. Her aunt is borrowing tables that fit 6 people each. How many tables does her aunt need to borrow for everyone to have a seat? Amaya’s family will fill ____ tables, with ____ people at one last table. So her aunt needs to borrow ____ tables.” Guided practice, Interpret remainders, Teacher notes state, “For #6, consider discussing how you can use number sense to check whether your answer is reasonable. Since the answer is the amount left, it has to be less than 7 inches. Otherwise, Grandma Rose could have cut off another piece of string. In other words, the remainder must be less than the divisor. For #7–9, consider having students compare and contrast the questions. In #7, the answer is the quotient. In #8, the answer is one more than the quotient. In #9, the answer is the remainder. For #9, look out for students who try to divide before they add. Students can still get the right answer with this approach, but they'll need to be careful with their remainders. When you divide 52 by 9, you get 5 R7. When you divide 6 by 9, you get 0 R6. If you put the remainders together, you get 5 R13. But since 13 is enough for another group of 9, there are only 13-9=4 chairs left over.” Student book states, “6. Grandma Rose used 3,153 inches of string to make friendship bracelets for all of her grandchildren. She cut off 7-inch pieces of string and braided them together to make bracelets. How much string did she have left after cutting as many pieces as she could? 7. Amaya wants to fit in as much fun as she can, so she writes down how long she thinks every activity will take. Each round of her favorite game, charades, takes about 8 minutes. If they played charades for the entire game time, how many rounds would they finish? 8. She expects all 44 kids at the reunion to play hide-and-seek, so she creates teams ahead of time. Each team can have up to 5 people. What is the smallest number she needs for everyone to play? 9. So many people want to play musical chairs that Amaya plans to set up several games. She will use 52 chairs from the tables and 6 from her house. If she makes as many groups of 9 chairs as she can, how many chairs will she have left over?” Unit outline, Math practices state, “Throughout Unit 6, students solve division problems involving equal groups and multiplicative comparison. In Lesson 6.3, they begin reasoning about division situations with remainders. As they build fluency with calculations, encourage them to first focus on numbers without context to apply division strategies, then shift back to interpreting the result in the context at hand. Seeing these as separate steps helps them recognize how division strategies apply across different contexts. At the end of the unit, students apply conceptual understanding and procedural fluency to a range of real-world division problems. In Lesson 6.8, encourage them to use what they know about remainders to reason about word problems with numbers that don't divide evenly. In Lesson 6.9, remind students that strip models can help them visualize quantities and relationships in multiplicative comparison problems, making it easier to translate them into division equations. Lesson 6.10 features multi-step word problems that require students to synthesize everything they've learned in this unit. As they work through these problems, prompt them to refer back to the original context to ensure their steps make sense.”

• Grade 5, Unit 11: Divide decimals, Lesson 11.8, Instruction: Solve multi-step word problems involving decimal numbers. Students use strategies to solve division problems with decimal numbers and reason abstractly and quantitatively as they explain the symbols and quantities in the expressions. Teacher notes state, “For #2–3, discuss how to write an expression to solve each problem. Encourage students to think about the problems in parts to help write an expression, as they've done in previous lessons focusing on multi-step word problems. For example, you can break #2 into two steps. First, write an expression that represents how much money Kiera used to buy arcade tokens ($55.50-$30). Then, divide by 0.75 to find the number of tokens Kiera bought. Remind students to use parentheses to indicate which operations to perform first. For example, in #3, you have to figure out how many hours are left after the roller coasters before dividing by 2 to answer the question, so you should write the first part of the expression in parentheses.” Student book states, “Write and evaluate an expression to solve each problem. 2. Kiera had 55.50 in spending money. She saved 30 for food and rides, then used the rest to buy arcade tokens. If arcade tokens cost 0.75, how many tokens did Kiera buy? 3. From 1 p.m. to 4 p.m., Kiera ate lunch, played at the arcade, and rode 2 roller coasters. It took about 0.25 hours to wait in line and ride each roller coaster. If Kiera split the rest of the time between the arcade and lunch, how long did she spend at the arcade?” Guided practice, Use expressions to solve word problems. Teacher notes state, “For #7–8, consider discussing how the problems are similar and different. Surface that while both problems use multiplicative comparison, you use different operations to solve them. For #7, you multiply the area of Shore Souvenirs by 0.4 to find the combined area of Pelican Pizza Company and Coral Reef Creamery since the souvenir shop takes up more space than the two food shops. For #8, you divide to find the area of Balloons or Bust since it takes up less space than Coral Reef Creamery. For #9, consider discussing the meaning of each individual division expression. The cost of an adult season pass is $92, so the expression 92\div5 represents the cost of each visit for Jada's dad. The cost of a child season pass is $84, so the expression 84\div5 represents the cost of each visit for Jada.” Student book states, “7. Pelican Pizza Company and Coral Reef Creamery have the same area. Together, they take up 0.4 times as much space as Shore Souvenirs, which is 252 square meters. How big is the creamery? 8. Coral Reef Creamery takes up 1.5 times as much space as Balloons or Bust. What is the total area of all four stores? 9. Jada went to the Ocean View Aquarium. She liked it so much that she wants to go again with her dad. She wrote an expression to find the cost of each visit if they both got season passes and went 5 times. 92\div5+84\div5 Explain what each part of Jada’s expression represents.” Unit outline, Math practices state, “In Unit 11, students explore new methods for dividing decimals and apply these techniques to real-world problems. Encourage students to balance the concrete thinking required to understand the story with the abstract reasoning needed to compute quotients. Highlight problems, such as #7–9 in Lesson 11.5, that require students to consider the story before recording their answer (in this case, to decide whether the answer can be a decimal). Also emphasize problems like #9 in Lesson 11.8, which ask students to explain what each part of an expression represents. Such exercises help them connect real-world contexts with abstract mathematical representations.”

The materials reviewed for Takeoff by IXL for Grades 3 through 5 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

In Grades 3-5, MP3 is identified across the year. Teachers receive guidance in the Teacher’s Guide, and each unit includes a Unit Overview with Math Practices guidance that highlights examples of the Mathematical Practices addressed within that unit. In addition, the Mathematical Practices are identified in the overview of each lesson.

Across the grades, students participate in tasks that support key components of MP3. These include constructing mathematical arguments, analyzing errors in sample student work, and explaining or justifying their thinking orally or in writing using concrete models, drawings, numbers, or actions. Students are encouraged to listen to or read the arguments of others, evaluate their reasoning, and ask clarifying questions to strengthen or improve the argument. They also have opportunities to make and test conjectures as they solve problems. Teachers are guided to support this development by providing opportunities for students to engage in mathematical discourse, set clear expectations for explanation and justification, and compare different strategies or solutions. Teachers are prompted to ask clarifying and probing questions, support students in presenting their solutions as arguments, and facilitate discussions that help students reflect on and refine their reasoning.

Examples include:

• Grade 3, Unit 2: Understand multiplication, Lesson 2.3, Instruction, Represent equal groups with one or zero, students justify their reasoning, construct arguments to support their conclusions, and critique others’ work when multiplying by one and zero. Teacher notes state, “For #4, have students share their thoughts and then surface that Mateo is correct. Here are a few things to highlight during the discussion: If each group has one item, the total number of items will be the same as the total number of groups. If there is only one group, the total number of items will be the same as the number of items in the group.” Student book states, “4. Mateo sees a pattern. He explains it to Shelby. Mateo: I think 1 times a number is still that number. Shelby: Are you sure? What about really big numbers? Or, special numbers like 100? Do you agree with Mateo’s pattern? Why or why not?” Later in Lesson 2.3, students critique a student’s work as they make generalizations about multiplying by zero. Guided Practice, Multiply with one or zero, Teacher notes state, “For #21, have students explain whether or not they agree with Jamal.” Student book states, “21. Jamal is trying to model 0\times4. He says, ‘I can just draw nothing, since there are 0 groups.’ Do you think Jamal is correct? Why or why not?” Unit Outline, Math practices state, “As students develop a conceptual understanding of multiplication in Unit 2, they have many opportunities to critique others' reasoning. Help students make sense of new ideas by assessing claims made by others, such as in #4 in Lesson 2.3. In this example, Mateo claims that he has identified a pattern when multiplying by 1. Use this exercise to show students how to evaluate Mateo's claim and ultimately arrive at the identity property of multiplication. Students also have opportunities to practice evaluating others' work for errors. In #11 in Lesson 2.7, for example, students are asked to identify and explain the error Austin made in solving a multiplication equation with a number line (he started the jumps at 2 instead of 0). By correcting this error, students practice critiquing others' work and solidify their understanding of how to use number lines to represent multiplication.”

• Grade 4, Unit 3: Multiply by 1-digit numbers, Lesson 3.5, Instruction, Learn to use arrays to multiply, students formulate mathematical arguments and critique the work of others when working with multiplication of whole numbers. Teacher notes state, “For #2, have students compare the models. During the discussion, highlight that both models show 3\times15. They both have 3 rows with 15 in each row. Max showed 15 using 15 individual ones blocks. Emma showed 15 as a ten and 5 ones. Students should note that it is easier to find the total using Emma's model since you can find 3\times10 and 3\times5 separately and then add the two products. For #3, discuss how Emma's blocks were arranged in 3 rows, with 1 ten and 5 ones in each row. So, to show 3\times15 by shading an array, you can shade 3 rows, shading the full width of one grid and 5 additional boxes in the second. Then guide students to show their work with equations and the distributive property. Both Emma's model and the array show that to find 3\times15, you can think about 15 as 10 and 5, multiply each part by 3, then add the products.” Student book states, “2. Max and Emma used place value blocks to show 3\times15. How does each model show the product? Which model would you rather use to multiply? 3. Emma wants to represent her blocks with a drawing. Shade an array to show 3\times15. Use your array and the distributive property to find 3\times15. Think: 15 is ____ and ____. 3\times15=(____ \times____) + (____ \times____)=(____+____)=____” Students create the 3-by-15 array in the space provided. Later in Lesson 3.5, students have an opportunity to critique another student’s mistake and construct arguments about how the problem should be solved. Guided Practice: Use arrays and the distributive property to multiply. Teacher notes state, “For #12, have students discuss Chase's mistake. Have them compare their work to Chase's work. They should notice that when Chase tried to use the distributive property, he should have only split one of the factors. Use this problem as an opportunity to solidify the connection between splitting just one factor when you use the distributive property to multiply and splitting just one side of the array that represents the product.” Student book states, “12. Chase tried to use the distributive property to find 8\times19. Dustin spotted his mistake. Help Chase fix his mistake. Use the distributive property to find 8\times19.” Unit Outline, Math practices states, “Use Compare and talk activities to help students make connections between different models and strategies. For example, in #2 from Lesson 3.5, have students explore two different approaches for modeling multiplication with place value blocks. By comparing the approaches, students can see how using place value is more efficient. In #5 from Lesson 3.5, students discuss the claim that the distributive property can be used instead of an array. Have students decide whether Dustin's claim is logical to support their understanding of his new method. The examples of student work and mathematical thinking in this unit also provide opportunities for students to practice critiquing others' reasoning. Use #12 from Lesson 3.5, for example, to have students identify and explain an error in the use of the distributive property to solve a multi-digit multiplication problem.”

Grade 5, Unit 10: Multiply decimals, Lesson 10.1, Instruction: Use patterns and place value to multiply whole numbers by decimal numbers. Students construct mathematical arguments when multiplying a decimal number by 100 and identify patterns that support efficient work. Teacher notes state, “For #9, start by making sure that students see how Javon got his three answers. In particular, point out that you can find 0.3\times0.1 by finding \frac{1}{10} of 0.3 since 0.1 and \frac{1}{10}are equal. Then, discuss patterns that students see to help them finish Javon's equations. Surface that the digit 3 moves one place to the left as you go down the list of equations (or one place to the right as you go up), and guide students to use this pattern to complete the remaining equations. For example, you can find that 0.3\times0.001 by moving the 3 in 0.03 one place to the right. Finally, discuss why this idea works. Highlight that each equation has a factor of 0.3, and the other factor increases by a factor of 10 as you go down the list of equations. For example, 100 is 10 times as much as 10, so 0.3\times100 is 10 times as much as 0.3\times10=3. That means 0.3\times100=30. Similarly, 0.1 is 10 times as much as 0.01, so 0.3\times0.1 is 10 times as much as 0.3\times0.01.” Student book states, “9. Javon found 10 times and \frac{1}{10}of 0.3 by using place value. After writing his equations together, he wonders if there is a pattern. Help him finish the rest of the equations. Javon: My answers all have a 3, but it’s in different places. How can I tell where to put it in the other products?” Students see a list of problems ranging from 0.3\times0.01 through 0.3\times1000 in sequential order. Later in Lesson 10.1, students critique another person’s work. Guided Practice: Use patterns and place value to multiply whole numbers by decimal numbers. Teacher notes state, “For #31, make sure students connect Lara's idea back to the idea of repeated multiplication. Since the exponent describes how many times to multiply by 10, it can also describe how many times to move the decimal point. Students may also write 10^3in standard form and then think about the number of zeros to explain Lara's idea. As in #10, make sure they can explain why that idea works.” Student book states, “31. Lara has an idea for multiplying numbers by powers of 10 written with exponents. Lara: To multiply 10^3, I can move the decimal point to the right 3 times because the exponent is 3. Explain why Lara’s idea works.”Unit Outline, Math practices state, “Students get plenty of practice backing up their ideas in Unit 10, whether they're discussing with others or writing down their thoughts. Look for moments when students can analyze calculations or reasoning. In Lesson 10.1, students analyze and discuss patterns in decimal multiplication, make connections between related strategies, and write about why certain methods work. In Lesson 10.2, they use estimation to evaluate the reasonableness of answers and critique others' reasoning when estimates lead to flawed conclusions. Throughout the unit, encourage rich mathematical conversations and provide time for students to articulate their thinking in writing. These experiences help students build confidence in their ability to defend their ideas and to respond thoughtfully to the reasoning of others—essential skills for effective mathematical thinkers.”

The materials reviewed for Takeoff by IXL for Grades 3 through 5 meet expectations for supporting the intentional development of MP4: Model with mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

In Grades 3-5, MP4 is identified across the year. Teachers receive guidance in the Teacher’s Guide, and each unit includes a Unit Overview with Math Practices guidance that highlights examples of the Mathematical Practices addressed within that unit. In addition, the Mathematical Practices are identified in the overview of each lesson.

Across the grades, students participate in tasks that support the intentional development of MP4, including putting problems or situations in their own words and identifying important information, using the math they know to solve problems and everyday situations, modeling the situation with appropriate representations and strategies, describing how their model relates to the problem situation, and checking whether their answer makes sense, revising the model when necessary.

Examples include: 

• Grade 3, Unit 8: Perimeter, Lesson 8.4, Instruction: Solve problems involving area and perimeter, students model using area and perimeter, applying their knowledge of both to solve real-world problems. Teacher notes state, “For #2, discuss whether Mr. Lee has enough butcher paper and border trim to decorate his board. Make sure that students understand that to determine whether he has enough butcher paper, they'll need to find the bulletin board's area, and to determine whether he has enough trim, they'll need to find the bulletin board's perimeter. The area is 3\times5=15 square feet and the perimeter is 5 +3 +5 +3= 16 feet. That means Mr. Lee has enough butcher paper (barely!), but he doesn't have enough trim.” Student book states, “2. Mr. Lee plans to cover his bulletin board with green butcher paper and put yellow trim around the edge. He has 15 square feet of butcher paper and 15 feet of trim. Does Mr. Lee have enough paper and trim?” Students see a bulletin board with dimensions 5 feet by 3 feet. Teacher notes state, “For #4–5, have students draw a figure showing the information they know and what they need to find. For #4, make sure students recognize that the space is rectangular, one pair of opposite sides is 6 feet long, and the perimeter of the space is 30 feet. To find the length of the unknown sides, students can first subtract the two known side lengths from the perimeter (30-6-6=18) to find the remaining amount for both unknown sides. Then they can find half the remaining amount (18\div2=9) to find how long one unknown side is. For #5, make sure students recognize that the pit is rectangular, one pair of opposite sides is 4 feet long, and the area of the pit is 8 square feet. Since area = length \times width, students can use 4\times b=8 or the related equation 8\div4=b to find how long the unknown side is.” Student book states, “4. A paleontologist uses 30 feet of rope to mark off a rectangular space before a dig. The space is 6 feet wide. How long is the space? 5. The paleontologist plans to dig a smaller rectangular pit inside the roped-off space. The pit will cover an area of 8 square feet. If she makes the pit 4 feet long, how wide will it be?” For both problems, space is provided for students to draw pictures to model and solve the problems. Unit outline, Math practices state, “Throughout this unit, students use diagrams and equations to solve real-world problems about area and perimeter, which is also the primary focus of Lesson 8.4. For problems that don't provide a diagram, encourage students to draw and label their own. Creating a visual model not only helps students make sense of problems (MP1), but it can also help them visualize key information and decide how to solve the problem. Students also learn to model perimeter using addition, as well as multiplication for regular polygons. Seeing that both operations can be used to solve perimeter problems gives students extra flexibility in the way they choose to model a given scenario.” 

• Grade 4, Unit 7: Perimeter and area, Lesson 7.4, Instruction: Learn to solve multi-step word problems involving area and perimeter. Students model the math used to find perimeter and area in real-world problems. Teacher notes state, “Throughout this lesson, students will solve multi-step word problems involving area and perimeter. Encourage them to annotate the diagrams in the problems by labeling the information they know and using letters to represent unknown values. Have students record their work with formulas and equations. These problems can often be solved with a variety of approaches, so encourage students to look for efficient strategies that make sense to them. As in previous lessons, they may choose to represent their work using individual steps or with a single multi-operation expression. Both are fine. For #2, help students recognize that this problem requires two steps: 1) finding the missing side length and 2) finding the area.” Student book states, “2. Rosewood Park has a perimeter of 248 meters and a width of 54 meters. What is the area of the park? What do you need to find first? Find the area of the park.” Students see a picture of the park that they use to model the work needed to solve the problems. Later in Lesson 7.4, students find the area of two dog beds to determine which has the greater area, even though both have the same perimeter. Guided practice: Solve multi-step word problems involving area and perimeter. Teacher notes state, “For #9, Julia wants Fergus to have as much room as possible, so she should buy the dog bed with the greatest area.” Student book states, “9. Julia is buying a new bed for her dog, Fergus. She wants Fergus to have as much room as possible, but she isn’t sure whether to buy the square red bed or the rectangular blue bed. They each have a perimeter of 240 centimeters. Which one should she buy? How do you know?” Students see pictures of both beds. The shorter side of the blue rectangular bed is labeled 45 cm. Unit outline, Math practices state, “In Unit 7, students build on their previous experience with area and perimeter models. In early lessons, they transition from using diagrams and equations to writing area and perimeter formulas. Help students see that formulas are an efficient modeling tool and clearly communicate to others what missing value you are trying to find. As students learn to solve complex problems, encourage them to think about formulas flexibly to find different unknown values (e.g., missing side lengths or areas of compound figures). By using formulas in a variety of ways, students gain a deeper understanding of the relationship between area and perimeter, which prepares them to navigate multi-step problems in Lessons 7.3 and 7.4.”

• Grade 5, Unit 4: Volume, Lesson 4.4, Guided practice, Use volume formulas for rectangular prisms. Students use rectangular prisms to model real-world situations that involve finding volume using the formula. Teacher notes state, “For #16, explain that the container is nearly a rectangular prism, so you can pretend it actually is one to get a good estimate of the container's volume. You do not need to change the numbers to estimate the volume. When estimating the volume, have students write the volume formula they use and show their work. Have students explain whether the container's actual volume is greater than or less than the estimated volume.” Student book states, “16. Bridget has a new container for homemade ice cream. Estimate the container’s volume by modeling it as a rectangular prism. Do you think the container’s actual volume is greater than or less than the volume you estimated? Explain.” Students see a container with dimensions of 5, 10, and 2 inches. The container has rounded corners, which makes the volume less than it would be with straight corners. Unit outline, Math practices state, “Throughout Unit 4, students use rectangular prisms to model real-world problems that involve finding volume. Lesson 4.6, in particular, provides many opportunities for students to visualize and represent problem contexts. Recording information in a visual model can help students focus on the key quantities they need to solve the problems. For instance, in #5, students can label a rectangular prism with a height of 2 units, a width of 3 units, and a length of 2 units. By visualizing the prism, they can see that they need to multiply these quantities to find the volume of the box.”

The materials reviewed for Takeoff by IXL for Grades 3 through 5 meet expectations for supporting the intentional development of MP5: Use appropriate tools strategically, in connection with grade-level content standards, as expected by the mathematical practice standards.

In Grades 3-5, MP5 is identified across the year. Teachers receive guidance in the Teacher’s Guide, and each unit includes a Unit Overview with Math Practices guidance that highlights examples of the Mathematical Practices addressed within that unit. In addition, the Mathematical Practices are identified in the overview of each lesson.

Across the grades, students participate in tasks that support key components of MP5. These include selecting and using appropriate tools and strategies to explore mathematical ideas, solve problems, and communicate their thinking. Students are encouraged to consider the advantages and limitations of various tools such as manipulatives, drawings, measuring devices, and digital technologies, and to choose tools that best support their understanding and reasoning. They have opportunities to use tools flexibly for investigation, calculation, representation, and sense-making. Teachers are guided to support this development by making a variety of tools available, modeling their effective use, and encouraging students to make strategic decisions about when and how to use tools. Teacher materials prompt opportunities for student choice in tool selection, promote discussion around tool effectiveness, and support the comparison of multiple tools or representations. Teachers are also encouraged to highlight how different tools can yield different insights and to help students reflect on their tool choices as part of the problem-solving process.

Examples include: 

• Grade 3, Unit 11: Compare fractions, Lesson 11.4, Instruction: Learn to write whole numbers as fractions. Students work with number lines and models to write whole numbers as fractions. Teacher notes state, “For #6, have students start by splitting the distance between each pair of whole numbers into thirds. Then have them skip count by thirds (1 third, 2 thirds, 3 thirds, …), labeling \frac{1}{3}, \frac{3}{3}, and \frac{6}{3} as they go. Emphasize that \frac{3}{3} and \frac{6}{3} are equivalent to whole numbers by having students circle those fractions. Discuss how the number line makes it even easier to see that when you skip count by a fraction's denominator, you get numerators that result in fractions equivalent to whole numbers. For #7, students can count on from 6 thirds to find a fraction that is equivalent to 3. Some students may also recognize that they can find the numerator by multiplying 3\times3. Consider revisiting the idea from #5 that you can write any whole number as a fraction with a denominator of 3 by multiplying the whole number by 3 to find the numerator. Although students won't learn to interpret fractions as division until fifth grade, still validate students who notice that you can divide the numerator by the denominator to get the equivalent whole number.” Student book states, “6. Write and label \frac{1}{3}, \frac{3}{3}, and \frac{6}{3} on the number line. Circle the fractions that are equivalent to whole numbers. 7. Write a fraction that is equivalent to 3. Use the number line to help.” In the Guided practice, Write whole numbers as fractions, Teacher notes state, “For #11, have students identify the fractions that are equivalent to 2. Allow students to draw models as needed, but also encourage them to begin reasoning more abstractly. Highlight that \frac{12}{6} is equivalent to 2 because the numerator is equal to the total number of parts in 2 wholes that are cut into sixths. Students can use the same reasoning to explain why \frac{2}{1} is equivalent to 2, or they can apply the generalization from #8. For #12, students need to determine the number of halves in 6 wholes. They can draw a model with 6 wholes, cut each whole in half, and then count the equal parts. Alternatively, they can reason that 6 wholes each cut into 2 equal parts will have 12 parts in all. For #13–16, encourage students to find the missing numbers without drawing models. In #13, for example, each whole has 3 thirds, so 5 wholes will have 15 thirds (3\times5). For #15–16, consider highlighting that all fractions with a denominator of 1 are equal to whole numbers and that the whole number is the same as the numerator. Refer back to #8 as needed.” Student book states, “11. Circle all the fractions that are equivalent to 2. 12. Jack had 6 grapefruits. He cut each grapefruit in half. How many grapefruit halves did he make? Complete the equivalent fractions.” On #13–16, students complete equivalent fractions for whole numbers. Unit outline, Math practices state, “In Unit 11, students use a variety of tools—including number lines, fraction strips, and shape models—to explore fraction equivalence and comparison. In Lessons 11.4–11.7, highlight how each tool can support different kinds of reasoning. For example, in Lesson 11.4 #6, ask students to explain why a number line might be more helpful than a shape model for identifying fractions that are equivalent to whole numbers. Throughout the unit, give students time to reflect on why a particular model is useful, and encourage them to choose tools purposefully. Asking questions like ‘What did this model help you see?’ or ‘Would a different tool be more helpful here?’ can lead to more strategic and effective problem-solving.”

• Grade 4, Unit 14: Angles, Lesson 14.3, Instruction: Learn to measure and draw angles with a protractor. Students learn to use a protractor and determine why the tool is more useful for measuring angles. Teacher notes state, “For #3, discuss some similarities between the protractor and the ruler. Surface that both tools have number scales that increase by a constant amount. Also, the ruler and the protractor have unlabeled marks representing smaller measurements between the larger labeled marks. Next, discuss what makes a protractor a better tool than a ruler for measuring angles. Students may say that a ruler only measures length, and finding the distance between the two rays of an angle doesn't help you find its measure in degrees. Plus, the result will vary depending on how close to the vertex you place the ruler. Point out that tracing along the protractor's curved top would be like turning through a circle, which is how students measured angles in the previous lesson. Students should also notice that the scales on the protractor go up to 180, which corresponds to the number of degrees in an angle that turns through \frac{1}{2} of a circle. Walk students through the parts of the protractor. Tell them that the center point aligns with the angle's vertex. Highlight that the outer scale starts at 0\degree on the left and increases as you move to the right while the inner scale starts at 0\degree on the right and increases as you move to the left. Explain that having both scales makes it easier to measure angles that open in different directions. For #4, give each student a protractor and guide them to measure \angle JKL. Have students place the protractor's center point on the vertex, K, and align one of the rays with 0\degree. Students may align \overrightarrow{KL} with 0\degree on the outer scale or \overrightarrow{JK} with the inner scale. Then they can find which mark the other ray passes through and read the angle measure from the appropriate scale. Explain that students can write the angle measure inside the angle. Make sure they write the degree symbol here and throughout the rest of the lesson. Use this opportunity to emphasize the importance of reading the correct scale. In this case, the incorrect scale gives an angle measure that is much too small. Remind students that they can use their knowledge of angle types to check their measurements: since \angle JKL is obtuse, its measure must be greater than 90\degree.” Student book states, “3. Look at the protractor. How is it like a ruler? What makes it better for measuring angles? 4. Use a protector to measure \angle JKL.” Unit outline, Math practices state, “In Lesson 14.3, students learn how to use a protractor. Use #3 to point out the limitations of rulers and discuss why protractors are better for measuring angles. As students become more comfortable with a protractor, they can experiment with different ways to use one, such as measuring an angle without aligning either side to the 0\degree mark, as demonstrated in #8. Encourage students to reflect on all the available methods and apply them strategically as they complete the Practice using a protractor activity in Lesson 14.3.”

• Grade 5, Unit 11: Divide decimals, Lesson 11.3, Instruction: Learn to divide decimals by whole numbers using models. Students explore different models to solve division problems. Teacher notes state, “For #5, have students discuss Jon's and Luca's work. Surface the following: Jon divided using a place value sketch similar to the place value models students made in the previous problem. He used a total of 3 ones and 6 tenths to represent 3.6 (the dividend). He arranged the sketch in 3 equal groups (the divisor). The value of each group is 1.2 (the quotient). Luca used an area model. The total area is 3.6, the width is 3, and the length is 1.2. Luca used his area model to break 3.6 into smaller parts that are easy to divide, just like students did to divide whole numbers with area models in #1. While both models can help you divide, the area model is a more efficient way to break up the dividend. Drawing a place value sketch could take a long time when working with larger numbers.” Student book states, “5. Jon and Luca showed 3.6\div3=1.2 two different ways. How does each model show the dividend, divisor and quotient? Which model would you rather use to divide?” Students see that Jon used a sketch and Luca used an area model. Later in the lesson, students continue to decide which model to use. Guided practice, Use models to divide decimals by whole numbers, Teacher notes state, “For #13, students can find 715.2\div6 to solve the problem. Consider having some students share how they divided, and continue to highlight that area models with larger parts are more efficient. For #14, make sure students understand that this problem has two steps. First, subtract to find the cost of the hockey gear after the coupon. Then, divide that amount by 3 to find how much Henry will pay each month.” Student book states, “13. A group of 6 friends is taking a road trip to Grand Canyon National Park. Their planned route is 715.2 miles long. If they split the drive evenly, how many miles will each person drive? 14. Henry needs new hockey gear. He wants to buy gloves, pads, and a helmet. The set he likes costs $79.99, and he has a coupon to save $20. His mom agrees to help him buy the gear if Henry pays her back within 3 months. How much will Henry pay each month if he makes 3 equal payments? Unit outline, Math practices state, “As students learn to divide decimals in Unit 11, they also develop strategies for estimating decimal quotients. In Lesson 11.2, students explore when and how to estimate decimal quotients. Discuss the use of compatible numbers (numbers that are close to the original values and easy to divide). Look for opportunities, such as problem #13, for students to evaluate potentially compatible numbers and choose the most useful option. This prepares them to use estimation as a tool for checking the reasonableness of their answers, a strategy reinforced throughout Lesson 11.4. This unit also introduces multiple methods for dividing decimals, which students compare and evaluate in different contexts. This helps students develop the critical thinking skills needed to select the best tool for any given problem. For example, in Lesson 11.3, students break apart decimals using place value sketches and area models. Use problems like #5 to compare these methods and prompt students to explain their preferences. Discuss how place value sketches provide more concrete visual support, while area models are often quicker to draw. By thoughtfully selecting strategies, students develop their intuition and reasoning about when a given tool will be effective.”

The materials reviewed for Takeoff by IXL for Grades 3 through 5 meet expectations for supporting the intentional development of MP6: Attend to precision, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

In Grades 3-5, MP6 is identified across the year. Teachers receive guidance in the Teacher’s Guide, and each unit includes a Unit Overview with Math Practices guidance that highlights examples of the Mathematical Practices addressed within that unit. In addition, the Mathematical Practices are identified in the overview of each lesson.

Across the grades, students engage in tasks that support key components of MP6. These include formulating clear explanations, using grade-level appropriate vocabulary and conventions, applying definitions and symbols accurately, calculating efficiently, and specifying units of measure. Students are also expected to label tables, graphs, and other representations appropriately, and to use precise language and notation when presenting mathematical ideas. Teachers support this development by modeling accurate mathematical language, ensuring students understand and apply precise definitions, and providing feedback on usage. Lessons include opportunities for students to clarify the meaning of symbols, evaluate the accuracy of their own and others’ work, and refine their communication. Teachers are encouraged to facilitate discussions and structure tasks that promote attention to detail in both reasoning and representation.

Examples include: 

• Grade 3, Unit 13: Quadrilaterals, Lesson 13.2, Guided practice, Identify quadrilaterals, students use precision as they apply correct definitions and terminology to determine whether statements about quadrilaterals are true. Teacher notes state, “For #9, have students mark each statement as true or false. They can use the defining characteristics of quadrilaterals and of rectangles to determine that the first and third statements are true. If students aren't sure whether the second and fourth statements are true or false, encourage them to look at the shapes in #8 in order to identify a counterexample of each statement. For #10, consider having some students share their responses. In fourth grade, students will learn that this shape is a trapezoid. For #11, make sure students understand that Ted couldn't have drawn a rectangle or square since those shapes have right angles. For #12–14, have students use the set of polygons to answer the questions. Make sure students understand the attributes that define a rectangle or a parallelogram. A polygon must have 4 sides and exactly 4 right angles to be a rectangle. A polygon must have 4 sides and 2 pairs of parallel sides to be a parallelogram.” Student book states, “9. Mark each statement as true or false. All quadrilaterals have 4 sides. All rhombuses have 4 right angles. All rectangles have 4 right angles. All rectangles have 4 equal sides. 10. Heather says this quadrilateral is a parallelogram because it has parallel sides. Explain Heather’s mistake. 11. Ted drew a quadrilateral with 4 equal sides and no right angles. What shape did he draw? Rectangle, square, rhombus. Use the set of polygons to answer the questions. 12. Which polygons have right angles but are not rectangles? 13. Which polygons have parallel sides but are not parallelograms? 14. What is a characteristic that polygons B,C,D, and E have in common?” Unit outline, Math practices state, “Students attend to precision throughout Unit 13 as they use new vocabulary related to polygons. In Lessons 13.1–13.4, highlight opportunities for students to distinguish between shape categories. Discuss how different names reflect different levels of specificity (e.g., a quadrilateral is any four-sided figure, but a square is a special type of quadrilateral). Also, prompt students to name shapes in more than one way when possible. Be sure to model precise word choice when discussing shapes and their attributes, and expect students to do the same when communicating verbally and in writing.”

• Grade 4, Unit 1: Whole number place value, Lesson 1.3, Instruction, Learn forms of numbers, students attend to precision as they write numbers in word, standard, and expanded form, using proper terminology and conventions like hyphens instead of the word and in whole numbers. Teacher notes state, “For #2, provide each student or small group with a place value mat and a set of place value disks from 1 to 100,000. Include 10 of each disk. They will need either two printed copies of the place value mat or one copy in a protective sleeve with dry-erase markers. Guide students to fill out one place value mat with the number 125,348. Have them write the number in the standard form row on the mat, use disks to model the number, and then write the number in expanded form and in word form. When writing the number in word form, have the students read the number in standard form out loud. Discuss rules for writing numbers in word form, including conventions for using hyphens and not using the word and.” Students complete a table showing the number in standard, expanded, and word form. They continue to work with these forms later in Lesson 1.3. Guided practice, Write numbers in different forms, Teacher notes state, “For #5–7, have students write each number in expanded form and word form. For #13, students could choose to circle the entire number in expanded form instead of just one term, as the number in expanded form isn't equivalent to 609,336.” Student book states, “Write each number in expanded form and word form. 5. 59,216, 6. 84,023, 7. 608,033, 13. Lena was trying to write the number 609,336 in expanded form and word form, but she made a mistake. Circle her mistake. Expanded form: 600,000+90,000+300+30+6 Word form: six hundred nine thousand three hundred thirty-six. Explain how Lena could fix her work.” Unit outline, Math practices state, “Students must attend to precision, particularly with respect to place value, as they learn to write larger numbers in different forms. The impact of precision on communication becomes apparent in Lesson 10.3. For example, students need to recognize that to write 2,000+30+4 in standard form, they need to include a 0 in the hundreds place. Students should also attend to the specific rules for writing numbers in word form. When they read numbers aloud, make sure they properly say the numbers one period at a time. For example, they should read 5,362 as ‘five thousand three hundred sixty-two’ instead of ‘fifty-three hundred sixty-two’ or ‘five-thousand three hundred and sixty-two.’”

• Grade 5, Unit 1: Whole number place value and multiplication, Lesson 1.3, Instruction, Use patterns to understand powers of 10, students are introduced to exponents and use precise terminology while attending to the proper way to read numbers with exponents. Teacher notes state, “For #4, introduce powers of 10 as another way to express the numbers used in the warm-up. Use the example of 10^3 to define the terms base and exponent. Explain that 10^3 can be read as ‘10 to the third power’ or ‘the third power of 10.’ For #5, discuss the patterns in the table: The exponent is equal to the number of zeros in the whole number. Each time we multiply by 10, the whole number has one more zero and the exponent increases by 1. Each power of 10 is worth 10 times as much as the one that precedes it.” Student book states, “4. Complete the sentences. The ____ is used as the repeated factor. The ____ tells you how many times the base is used as a factor. 5. Complete the table to show powers of 10 in different forms. How does each power of 10 relate to the next largest power of 10?” Students see a table labeled Repeated Multiplication and Exponential Form. The table includes 10, 100, 1,000, and 10,000. Unit outline, Math practices state, “Students must attend to precision as they represent numbers in different forms in Unit 1. They revisit the standard, expanded, and word forms for numbers (Lesson 1.2) and learn how to represent powers of 10 using exponents (Lesson 1.3). Model how to use notation properly, as well as how to read numbers in different forms. Look for opportunities to highlight why it's important to read numbers correctly, like #26 from Lesson 1.3. (Reading 4\times10^4 kilometers as 4,000 kilometers instead of 40,000 kilometers makes a big difference!) Students must also be precise when estimating products. Prompt them to think carefully about how they round each factor and how that affects the estimate. In real-world problems, they should also consider whether rounding leads to an overestimate or an underestimate. For example, in Lesson 1.4, rounding up in #7 means Quinn will have more than enough carpet for one project, while rounding down in #8 means there won't be enough carpet for another.” 

The materials reviewed for Takeoff by IXL for Grades 3 through 5 meet expectations for supporting the intentional development of MP7: Look for and make use of structure, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

In Grades 3-5, MP7 is identified across the year. Teachers receive guidance in the Teacher’s Guide, and each unit includes a Unit Overview with Math Practices guidance that highlights examples of the Mathematical Practices addressed within that unit. In addition, the Mathematical Practices are identified in the overview of each lesson.

Across the grades, students engage in tasks that support key components of MP7. These include looking for and describing patterns, identifying structure within mathematical representations, and decomposing complex problems into simpler, more manageable parts. Students are encouraged to analyze problems for underlying structure and to consider multiple solution strategies. They make generalizations based on repeated reasoning and use those generalizations to solve problems efficiently. Teachers support this development by selecting tasks that highlight mathematical structure and prompting students to attend to and describe patterns they notice. Lessons provide opportunities for students to compare approaches, justify their reasoning, and reflect on how structure helps deepen their understanding. Teachers are encouraged to facilitate discussions and structure lessons in ways that promote the recognition and use of patterns and structure in problem-solving.

Examples include:

• Grade 3, Unit 3: Multiplication fluency, Lesson 3.6, Instruction: Learn strategies to multiply by 7. Students look for patterns and use structure to develop strategies for multiplying by seven. Teacher notes state, “For #4, have students draw a line that breaks the 7 columns into a set of 5 columns and a set of 2 columns. Each column has 4 dots, so students can find 4\times5 and 4\times2 and add the two smaller products to find 4\times7. For #5, share with students that the distributive property states that a product can be broken into the sum of two other products by splitting one of the factors. For example, you can break 7\times3 into (5\times3)+(2\times3), because 5+2=7. For #6, surface multiple ways to break apart the 7 rows, such as breaking it into 5 rows and 2 rows or into 4 rows and 3 rows. For this problem, consider using the latter so that students see a different way to break apart 7 than in #4.” Student book states, “4. Find 4\times7 by splitting the array apart into two smaller arrays. Step 1: Draw a line to show where to split the array. Step 2: Find the product each smaller array shows. Step 3: Add the products. 5. Complete the sentence. The distributive property says that a multiplication fact can be broken into the ____ of two other multiplication facts. 6. Find 7\times8 using the distributive property. First, show how you would break 7\times8 into easier products. Then, fill in the missing numbers. 7\times8=(x)+(x), 7\times8=__ + __ .” Unit outline, Math practices state, “In the first half of the unit, students use familiar structures to make sense of multiplication: skip counting and adding doubles to multiply by 2 (Lesson 3.1), using place value patterns to multiply by 5 and 10 (Lesson 3.2), and doubling twice to multiply by 4 (Lesson 3.4). Use these moments to draw out connections to prior knowledge. In Lesson 3.1 #7, for example, ask students to compare different ways of finding 2\times4—skip counting by 2, adding 4 twice, or using an array—and discuss how each method reflects multiplication. Highlight problems, like #7 in Lesson 3.2, which help students shift from familiar strategies to more abstract reasoning. In Lessons 3.5–3.8, students begin to decompose factors using the distributive property. Connect this strategy to the structure of arrays, showing how splitting arrays models breaking apart larger factors. In Lesson 3.7 #2, for example, students can represent 9\times8 as (9\times5)+(9\times3). In Lesson 3.9, introduce the associative property as a way to group factors when multiplying three numbers. Encourage students to explore different groupings and observe how the product stays the same.”

• Grade 4, Unit 4: Multiply by 2-digit numbers, Lesson 4.2, Instruction: Learn to use area models to multiply. Students use the structure of place value to set up two-digit multiplication problems with an area model, ensuring that the values in the tens and ones places are written correctly. Students also use the structure of area models previously used for two-digit-by-one-digit numbers. Teacher notes state, “For #3, discuss how to find the total number of squares. Students should recognize that they need to find 14\times18. Then have students count the squares in the array running across the top and down the side. They should recognize that, like the quilt, the array has 14 rows with 18 squares each. Discuss ways to quickly find the total number of squares. Highlight that since both dimensions are two-digit numbers, one good strategy is to break both numbers apart using place value. After breaking the numbers apart, have students divide the array into corresponding parts, find the number of squares in each smaller array, and add those products. For #4, surface these points in the discussion: Like the array in #3, an area model for 14\times18 that breaks up both numbers using place value would have four parts because each factor has two digits. Area models are much faster to draw, especially for large numbers.” Student book states, “3. Brad’s grandmother is helping him make a quilt for his bed. The finished quilt will have 14 rows with 18 squares in each row. How many squares will they use in all? 4. What would an area model for 14\times18 look like? Would you rather draw an array or an area model to multiply? Why?” Unit outline, Math practices state, “In this unit, students draw on the distributive property and the structure of the place value system to multiply by 2-digit numbers. These ideas become prominent in Lesson 4.2 as students work with area models. Be sure to discuss how breaking apart numbers by place value can help students multiply by 2-digit numbers. Highlight the role that distributive property plays too: Once you break up both numbers by place value, the distributive property helps you multiply the resulting parts. As students progress to other methods (partial products and the standard algorithm), continue to return to these foundational concepts of place value and the distributive property to help them make sense of their work.” 

• Grade 5, Unit 3: Expressions, Lesson 3.2, Instruction: Learn to write and interpret expressions. Students look for structure in the problems they solve and identify patterns that support their solutions. Teacher notes state, “For #8, highlight that each expression shares the quantity 48\div8. Have students discuss what happens to that quantity in each expression. Point out that when you subtract or divide by a whole number, the value decreases. When you add or multiply by a whole number, the value increases.” Student book states, “8. Without evaluating, compare the expressions to 48\div8. Which expressions are greater? Which expressions are less? (48\div8)-3, (48\div8)+3, (48\div8)\times3, (48\div8)\div3.” Unit overview, Math practices states, “In Unit 3, students make use of structure as they work with numerical expressions and the order of operations. Help students see how the order of operations provides a consistent structure for evaluating and writing expressions. Have them explore how small changes, such as moving parentheses or changing operations, can affect an expression's value (Lesson 3.1). Students can then leverage this structure to reason about expressions without evaluating them. For example, in #8, students compare the value of 48\div8 to other expressions with similar terms. Before they try to evaluate each one, prompt students to focus on how different operations change the original value. This type of reasoning strengthens students' ability to see and use structure, supports flexible thinking with numbers, and builds confidence in algebraic thinking.”

The materials reviewed for Takeoff by IXL for Grades 3 through 5 meet expectations for supporting the intentional development of 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. 

In Grades 3-5, MP8 is identified across the year. Teachers receive guidance in the Teacher’s Guide, and each unit includes a Unit Overview with Math Practices guidance that highlights examples of the Mathematical Practices addressed within that unit. In addition, the Mathematical Practices are identified in the overview of each lesson.

Across the grades, students engage in tasks that support key components of MP8. These include noticing and using repeated reasoning to make sense of problems, recognizing patterns, and developing efficient, mathematically sound strategies. Students are encouraged to create, describe, and explain general methods, formulas, or processes based on patterns they identify. Lessons provide opportunities for students to evaluate the reasonableness of their answers and refine their approaches through discussion and reflection. Teachers support this development by structuring tasks that highlight repeated reasoning, prompting students to make generalizations, and guiding them to build conceptual understanding, distinct from relying on memorized tricks. Instructional guidance also encourages teachers to model and elicit strategies that build toward formal algorithms or representations through consistent reasoning.

Examples include: 

• Grade 3, Unit 7: Applications of multiplication and division, Lesson 7.3, Instruction, Learn to multiply by multiples of 10, students look for patterns and use repeated reasoning to multiply by multiples of 10, applying their knowledge of multiplication facts and tens to solve problems efficiently. Teacher notes state, “Provide students with place value blocks. Consider having students work in pairs or small groups. Each student or group of students will need at least 10 rods (tens). For #2, ask students how they can show 4\times20 using place value blocks. Since they can think about 4\times20 as 4 groups of 20, they can make 4 groups of 2 tens each. (They can also make 4 groups of 20 ones, but that is less efficient.) Then have them use the model to complete the sentence and equation. Since 4 groups of 2 tens is 8 tens, 4\times20=80. For #3, have students draw a place value sketch to show 6\times50. Since there are 30 tens in all, 6\times50=300. For #5–6, guide students to multiply without drawing a model. Encourage them to pay attention to any patterns they see emerging. (For example, students might notice that they start by finding the product of the one-digit number and the number in the tens place of the two-digit number.) Use these patterns to transition to #7. For #7, discuss whether Molly's way will work for the product of any one-digit number and multiple of 10. (It will!) Like in earlier problems, you can think about the two-digit number as a group of tens and then state the product in terms of tens. That's like finding the basic multiplication fact and then putting a 0 on the end to represent that the product is a number of tens.” Student book states, “2. Use place value blocks to find 4\times20. Complete the sentence and the equation to describe your place value blocks. ____ groups of ____ tens is ____ tens. 4\times20=, 3. Draw a place value sketch to find 6\times50. Complete the sentence and the equation to describe your sketch. ____ groups of ____ tens is ____ tens. 6\times50=, 5. 3\times70=\times tens, 3\times70=__ tens, 4\times70=. 7. Molly has an idea for another way to find 9\times20. Molly: I can just find 9\times2 and then put a 0 on the end. Will Molly’s way work to multiply any 1-digit number by any number of tens? How do you know?” Unit outline, Math practices state, In Lesson 7.1, students use tables to identify multiplication patterns. Use questions that prompt them to describe how values change across a row or column and what operations explain the pattern. Show them how patterns they see can be used to make conjectures, which they can then test and use to make predictions. In #9–10, for example, students discover that multiplying any number of servings by 6 gives the number of strawberries, so they can extend this regularity to predict values that don't appear in the table. In Lesson 7.3, students begin multiplying with multiples of 10 by reasoning about place value and basic facts. Soon, they notice that a more efficient method is to multiply the basic fact and add a 0. Provide several examples to help students test and generalize this pattern.”

• Grade 4, Unit 3: Multiply by 1-digit numbers, Lesson 3.3, Instruction, Learn to multiply by multiples of 10, 100, and 1000, students notice and use repeated reasoning to find products of numbers multiplied by 10, 100, and 1,000. Teacher notes state, “For #3, discuss how to model 3\times4. Since there are 3 groups, the disks in each group should have a total value of 4. The digit 4 is in the ones place, so you can show 4 with 4 ones disks. For #5, encourage students to pay attention to any patterns they notice emerging. They will discuss these further in #6. For #6, have students identify the patterns they see in their models and equations. Surface these patterns in the discussion: Every model has 12 disks arranged in 3 groups of 4. The models show different products because the disks have different values. You can state the product in terms of the same place value unit as the multi-digit factor. For example, when you multiply by 4 hundreds, your answer is also in hundreds. Students may also notice that the number of zeros in the multi-digit factor is the same as the number of zeros in the product. Then have students use their patterns to find 3\times4,000. Encourage them to share their strategies. For example, students could imagine drawing 3 groups with 4 thousands disks each to get a product of 12 thousands. Or they could say that the product is 12,000 because 3\times4=12 and 4,000 has 3 zeros.” Student book states, “3. Draw equal groups of place value disks to show 3\times4. Complete the sentence and equation to describe your model. 3 groups of 4 ones is ____ ones. 3\times4=, 4. Try it again! Draw place value disks to show 3\times40. Complete the sentence and equation to describe the model you made this time. 3 groups of 4 ____ is ____. 3\times40=. 5. Do you see a pattern yet? Draw place value disks to show 3\times4,000. 6. Think and talk: What patterns do you notice in your models and equations? What do you think 3\times4,000 would be?” Unit outline, Math practices state, “In Lesson 3.3, students multiply by 10s, 100s, and 1000s using place value patterns. Guide them to recognize these patterns and come up with an efficient method for these problems. (See if they can generalize a rule about moving digits to the left or adding zeros on their own.) Throughout the rest of the unit, students progress through several methods of multiplying multi-digit numbers, including using place value arrays, area models, and partial products, before arriving at the standard algorithm. In Lessons 3.7 and 3.8, help students notice that these methods all share a conceptual basis and similar patterns of repeated calculations. As students build on each method, they will become more efficient and develop a conceptual understanding of multi-digit multiplication.”

• Grade 5, Unit 2: Whole number division, Lesson 2.7, Instruction: Learn to divide using the standard algorithm. Students look for and use repeated reasoning as they apply the steps for dividing a four-digit number by a two-digit number using methods introduced in previous lessons. Teacher notes state, “For #3, have students use the standard algorithm to solve. Highlight that while the dividend is a four-digit number, the algorithm works the same way. Since you can't divide the 2 thousands in 2,784 into groups of 12, consider the thousands and hundreds digits together. They represent 2,700, or 27 hundreds. Then decide how many groups of 12 you can make from 27 hundreds. After completing the first step, students should notice that the algorithm is the same as it was for three-digit dividends. For #4–5, have students divide using the standard algorithm. Throughout, encourage students to evaluate the reasonableness of their partial quotients in each step. For #5, in the second step, highlight that you need to record a 0 above the tens place of 2,153 since you can't divide 5 tens into groups of 7. Recording this 0 indicates that you made 0 groups of 7 from the tens. Draw similarities to recording zeros with the standard algorithm for addition or subtraction. For example, when you find 27-17, you write a 0 in the ones place to represent the 0 ones you have left.” Student book states, “3. Jackson High School is putting on a play. The cost of a ticket is $12, and the students have made $2,784 in ticket sales so far. How many tickets have they sold? 4. 3,465\div63=, 5. 2,153\div7=” Unit outline, Math practices, “Throughout Unit 2, students refine division strategies by identifying patterns and using repeated reasoning to work more efficiently. In Lesson 2.1, highlight what happens when zeros are added to or removed from the dividend or divisor. (For example, adding a zero to the dividend adds a zero to the quotient, while adding a zero to both numbers leaves the quotient unchanged.) In Lesson 2.4, as students divide with area models, encourage them to move from using small parts to larger, strategically chosen ones. In repeating this process, students build toward more efficient methods in Lessons 2.5 and 2.6. In Lesson 2.5, help students connect area models to partial quotients, and show how shifting from drawing and labeling sections to recording subtractions can streamline their work. As they begin using the standard algorithm in Lessons 2.6 and 2.7, prompt them to connect each step to strategies they've already used. This repetition supports students in building fluency and confidence with division.”