K-2nd Grade - Gateway 2

The materials reviewed for Takeoff by IXL Kindergarten through Grade 2 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 Kindergarten through Grade 2 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:

• Kindergarten, Unit 4: Introduction to addition, Lesson 4.2, Guided practice, Model and solve put together stories. Students develop conceptual understanding as they model and solve put-together stories. Teacher notes, “For #3, have students use two-color counters to model the following story: Marcus put 1 carrot on a plate and 2 carrots on another plate. How many carrots is that in all? Then have students count all the counters and write the numbers as they did in #2. For #4, have students use their own fingers to model the following story: Kenny holds up 5 fingers on one hand and 3 fingers on the other hand. How many fingers is that in all? Students should hold up their fingers to match those shown on the page. Then have students model the story with two-color counters and write the numbers.” (K.OA.1)

• Grade 1, Unit 4: Subtraction within 20, Lesson 4.1, Independent practice, students demonstrate conceptual understanding as they model counting back to subtract within 20 using number lines and lists, and use addition to check their work. Subtract by counting back - up to 20 (M2G), “Count back to subtract. The number line shows how. 20-3= ___” (1.OA.6) 

• Grade 2, Unit 8: Place value within 1,000, Lessons 8.1 and 8.3, students develop conceptual understanding as they represent and interpret place value to include hundreds. In Lesson 8.1, Instruction, Investigate the relationship between hundreds, tens, and ones, students use place value blocks to represent 100 as a group of ten tens. Teacher notes, “For #2, give students place value blocks and point out the hundred flats. Then have them discuss what they know about the number 100. They probably know a lot already! Here are some things they might say: ‘It comes after 99.’ ‘It's big.’ ‘It has two zeros in it.’ Once students have surfaced a wide range of things they know, transition to #3. Ask students how many ones are in 1 hundred (100). Point out that you can see all of the ones in the hundred flat; they've just been grouped together to make a hundred. Then ask how many tens are in 1 hundred (10). Point out that you can also see all of the tens in the hundred flat by looking at the columns or rows—they look like 10 ten rods laid side by side. For #4, have students compare Zoe's and Bruce's models. Surface that they both made 200, but Zoe used 2 hundreds and Bruce used 20 tens. Highlight that for each hundred that Zoe used, Bruce needed to use 10 tens.” In Lesson 8.2, students write and model three-digit numbers using place value models, sketches, and charts. In Lesson 8.3, students determine the value of a digit based on its position within a number. For example, in #10–13, students identify that a 3 in the ones place has a value of 3, while a 3 in the tens place has a value of 30, and that an 8 in the hundreds place has a value of 800, while an 8 in the tens place has a value of 80. (2.NBT.1)

The materials reviewed for Takeoff by IXL Kindergarten through Grade 2 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 Kindergarten through Grade 2 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 before students attempt the problem again. 

Examples include:

• Kindergarten, Unit 6: Addition and subtraction, students develop procedural skill and fluency as they add and subtract within 5. Lesson 6.10, Instruction, Learn about related addition and subtraction facts, Teacher notes, “For #3, have students look at the top picture as you read the first part of the story: There are 3 horses standing together. Then 1 more horse joins. How many horses are there now? Have students write an addition sentence that matches the story. Then have them look at the bottom picture as you can continue the story: There are 4 horses together. Then 1 horse runs away. How many horses stay? Discuss how the first and second parts of the story are related. Surface these points: Both parts include a total of 4 horses. In the first part, 1 horse joins. In the second part, that same horse leaves. Have students write a subtraction sentence that matches the second part of the story, and discuss how the two number sentences are related. Emphasize that they use the same three numbers and that adding 1 and taking 1 away are opposites. Explain that this is another example of related facts.” Guided practice, Add and subtract within 5, Teacher notes, “For #5-6, have students add or subtract and circle each pair of related facts with a different crayon. Some students might identify the related facts first and use the relationships to find some of their answers. Students will learn more about using related facts as a strategy to add and subtract in first grade. Here, it’s okay for students to solve first and then identify related facts by looking for number sentences that use the same three numbers.” Question 5, “Add or subtract. Circle the related facts,  1+2= ____,  1+3=____, 3+1=____, 3-2=____.”  Question 6, “Add or subtract. Circle the related facts. 2+1=____, 4+1=____, 3-1=____, 5-1=____.” Lesson 6.11, Guided practice, Add and subtract within 5, Teacher notes, “For #3, have students find the sum and differences using any strategy. Encourage them to choose strategies that they can do quickly and correctly.” Question 3, “Add or subtract. 1+2=____. 3-2=____, 4-1=___, 3+1=____.” Teacher notes, “For #6, explain that bees live in hives and students must draw a line from each bee to the hive where it belongs. Add or subtract to find the number of each bee’s hive.” Question 6, “Help each bee find its home. 1+4, 5-3, 2+1, 4-2, 4-1(2,5,3).” (K.OA.5)

• Grade 1, Unit 2: Addition and subtraction within 10, Lesson 2.9, Guided practice, Add and subtract within 10, students develop procedural skill and fluency as they add and sub within 20. Teacher notes, “For #8, have students add or subtract using any method, as in #3-6. Consider having students circle which problems, if any, they found difficult. Help students choose a strategy for those problems, or have students help each other. Consider putting problems the class finds more difficult into extra practice or a game to increase fluency.” Question 8, “Add or subtract. 7+2=____    2-0=____ 8-4=____   4+3=____  3+7=____   6-4=____” Teacher notes, “For #9, have students match the problems in the top row to problems in the bottom row with the same value. Question 9, “Match the problems with the same value. 10-3, 4+2, 3+5, 7-2; 9-3, 2+3, 1+6, 8-0.” Teacher notes, “For #10, have students add or subtract using any method. When they’re done, have them use the letters from the table to spell out the answer to the riddle.” Question 10, “Add or subtract. 5+5=____F, 8-2=,____ M, 6-1=____U, 4+0=____L, 6+2=____E, 7-4=____R, 10-8____A, 3+4=____S, 6+3=____T. Use the letters to solve the riddle. What is the best season for adding?” (1.OA.6) 

• Grade 2, Unit 1: Addition and subtraction within 20, Lesson 1.3, Independent practice, students demonstrate procedural fluency by adding and subtracting within 20 using a range of mental strategies. In prior lessons, students practice adding doubles and near doubles, counting on, and making ten. Make ten to add (YJE), “Which shows a way to find 6+7. Now use your choice to add 6+7= ___” (2.OA.2)

The materials reviewed for Takeoff by IXL Kindergarten through Grade 2 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:

• Kindergarten, Unit 6: Addition and subtraction, Lesson 6.1, Instruction, Add numbers using pictures, students develop procedural skill and conceptual understanding by using pictures to add two numbers with sums up to 10. Teacher notes, “For #2, read the addition sentence aloud as a question: What is 3 plus 2? Guide students to count the number of birds in each group to see that the picture matches the addition sentence. Consider having them circle each group. Then have students find the total by counting all the birds. Explain that the total in an addition sentence is called the sum. After writing the sum, lead students in reading the completed addition sentence aloud: ‘3 plus 2 equals 5.’ Ask students how pictures can help them find sums. Students should recognize that they can count all the objects to find the sum. Repeat the process for #3. Begin by reading the addition sentence as a question (‘What is 4 plus 1?’). Then have students count the birds in each group, find the sum, and say the completed addition sentence aloud (‘4 plus 1 equals 5’).” Question 2, Add. Count the birds to help. 3+2=____. Question 3, Add. Count the birds to help. 4+1=____.” (K.CC.4)

• Grade 1, Unit 2: Addition and subtraction within 10, Lesson 2.7, Guided practice, Add and subtract with 0, students develop procedural skill and conceptual understanding by solving a word problem that involves adding zero. Teacher notes, “For #10-11, have students fill in each number sentence to match the story and complete the sentence to answer the question.” Question 10 states, “Tess has no coins. Then her brother gives her 9 coins. How many coins does Tess have now? Write a number sentence. ____\bigcirc____\bigcirc____ Tess has ____ coins now.” Question 11 states, “There are 5 crabs on the beach. All 5 crabs leave. How many crabs are still on the beach? Write a number sentence. ____\bigcirc____\bigcirc____ There are ____ crabs still on the beach.” Teacher notes, “For #12, have students solve using any method.” Question 12. states, “There are 6 kids at the playground. It starts to rain, but 0 kids leave. How many kids are at the playground in the rain? ____ kids.” (1.OA.6)

• Grade 2, Unit 9: Addition and subtraction within 1,000, Lesson 9.4, students demonstrate procedural skill and conceptual understanding by using place value strategies to add three-digit numbers, breaking numbers apart and adding hundreds with hundreds, tens with tens, and ones with ones. Student book, Question 12 states, “On Thursday, the space museum sold 251 child tickets and 345 adult tickets. In all, how many tickets did the museum sell on Thursday?” Students use expanded form to solve. (2.NBT.7)

The materials reviewed for Takeoff by IXL for Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

In Grades K-2, 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: 

• Kindergarten, Unit 2: Numbers to 10, Lesson 2.7, Instruction, Correct mistakes: Up to 10, students critique the counting in the lesson and construct viable arguments explaining why the counts are incorrect. Teaching notes state, “Show students a page from Correct mistakes in counting: Up to 10. Introduce the fictional characters Dan and Emma, two kids who are learning to count (just like Beth and Max, who they met in Unit 1). Explain that Dan and Emma sometimes make mistakes. The class is going to help them fix the counting mistakes. Remind the class that it's okay to make mistakes, because fixing them helps us learn. For each group of objects, demonstrate how one of the characters counts, sometimes counting correctly, sometimes making a mistake. Point to the objects and count aloud. Include the following errors as you count: For #1, have Dan count the top bumper car twice, once at the start and once at the end. Keep a finger on the top car to make the mistake more apparent. For #2, have Emma count the apples correctly. For #3, have Emma miss counting the center tent. For #4, have Dan miss a number as he counts (e.g. 1, 2, 3, 4, 5, 7, 8, 9). For #5, have Dan count the ice cream correctly. For #6, have Emma repeat a number or two as she counts (e.g. 1, 2, 3, 3, 4, 5, 6, 6, 7, 8). After counting a group, ask students what they noticed. Have them say whether the counting was correct. If the counting was incorrect, have students explain what went wrong and how to fix it. Help the class celebrate when Dan or Emma counts correctly, and help the class encourage Dan or Emma to keep trying when they make mistakes.” Unit Overview, Math practices state, “Collaborative conversations in this unit give students a chance to reflect on their thinking and develop a deeper understanding of counting concepts. For example, in Lesson 2.7, students look for errors in Dan's and Emma's counting. Use this exercise to highlight common mistakes, such as skipping an object or repeating a number. Encourage students to explain why these errors might happen and how to correct them. As students discuss, prompt them to use what they know about the number sequence to support and communicate their ideas.” Students and teachers also have access to a worksheet, "Correct mistakes in counting: Up to 10," where the above mistakes are highlighted. 

• Grade 1, Unit 4: Subtraction within 20, Lesson 4.1, Instruction: Learn to subtract within 20 by counting back. Students examine two subtraction solutions, identify which one is correct, and critique the incorrect work. Teacher notes state, “For #6, students should recognize that Kelly is correct. Emphasize that when you subtract by counting back, you count back from the starting number—you shouldn't include it in the amount you count back. Steve wrote 3 numbers, but he only counted back by 2, which is incorrect. Ask students about other ways they can determine who is correct. Surface that they can subtract by counting back using another method (fingers or a number line) or use addition to check Kelly's and Steve's work.” Student book states, “6. Steve and Kelly both counted back to find 17-3. Who is correct? How do you know?” Students see both students’ work. Steve counts back using 17 as one of the three numbers, which makes his solution incorrect. Students have another opportunity to formulate a viable argument about which problem is easier to solve using the counting-back strategy. Guided Practice, Subtract within 20 by counting back states, “For #15, students should count back to find each difference. Have them circle the easier one and explain their reasoning.” Student book states, “15. Subtract. Which one was easier to find by counting back? 14-8=, 14-3=, Why do you think that was?” Unit Outline, Math practices state, “As students explore subtraction strategies within 20, they have multiple opportunities to construct arguments. For example, in #15 from Lesson 4.1, students explain which problem is easier to solve by counting back, reinforcing that it is more efficient to count back by smaller numbers. Similarly, in #9 from Lesson 4.7, students are asked to identify and explain which problem cannot be solved by making 10, promoting a deeper understanding of subtraction strategies. These activities provide students with practice in constructing written arguments while strengthening their conceptual understanding of subtraction. Students also critique others' reasoning during error analysis problems. In #13 from Lesson 4.4, students identify and correct Grace's mistake using a number line for subtraction. Encourage students to articulate Grace's error to help them strengthen their critical thinking skills.” 

• Grade 2, Unit 9: Addition and subtraction within 1,000, Lesson 9.4, Instruction, Learning to add by breaking apart both addends, students form viable arguments about which strategy works best for them when solving a multi-digit addition problem, and conduct error analysis of others’ work. Teacher notes state, “For #5, have students discuss whose method they think will work and why. Surface that all three of the approaches will result in the correct sum, though starting with the hundreds or ones is generally more organized. Then have students find the sum. Consider splitting the class into three groups and having each group use a different approach, starting with hundreds, tens, or ones. Then compare answers to show that no matter which place you start with, the sum is the same.” Student book states, “5. Anna, Leo, and Owen want to find 154+532. They all have different ideas for how to start. Anna: I will start by adding the hundreds, since they’re the biggest. Leo: I will start by adding the ones. Owen: I will start by adding the tens. I want to see if it works. Whose method will work? What is the sum?” Students see the problem in standard algorithm form and in expanded form. Later in the lesson, students conduct error analysis of another student’s work using the expanded form strategy. Guided Practice, Add using expanded form, Teacher notes state, “For #14, have students describe Jade's mistake. Then have them show how to fix the mistake by finding the correct sum.” Student book states, “14. Jade tried to find 571+416 using expanded form. What mistake did Jade make? Find the correct sum.” Unit Outline, Math practices state, “As students critique the reasoning of others in Unit 9, they learn to recognize and correct common mistakes made when dealing with larger numbers. For example, in #14 from Lesson 9.3, students examine a common place value error—Aaron confused the tens and ones in 304. Encourage students to explain the mistake and to suggest ways to avoid such mistakes in the future. Students also build their ability to construct arguments by comparing strategies for adding and subtracting within 1,000. Highlight opportunities to discuss different approaches, such as when breaking apart numbers to add (Lesson 9.4) and using number lines to subtract (Lesson 9.9). Ask students to justify the use of a particular strategy and why it might be more efficient than another. Encourage them to listen to each other's reasoning, ask clarifying questions, and respond with their own ideas.” 

The materials reviewed for Takeoff by IXL for Kindergarten through Grade 2 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 K-2, 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 at 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: 

• Kindergarten, Unit 10: Three-dimensional shapes, Lesson 10.10, Instruction: Describe objects in a scene, students model real-world objects with three-dimensional shapes and describe the objects using the shapes that compose them. Teacher notes state, “For #2, put students in small groups and have them take turns giving clues about an object for the rest of the group to identify. Like #1, students could simply point to the object that matches the clues if they don't know its name. The student giving clues should use the shape name and at least one position word. For example, here are some things students could say: This is shaped like a cube. It is behind the truck. (The house.) This is shaped like a sphere. It is below the moon. (The streetlight.) This is shaped like a cylinder. It is next to something shaped like a cone. (The water tank on the truck or the trashcan.) Find something that's shaped like a cone and is in front of the house. (The traffic cone or the light from the streetlight.)” Students see a city scene with a rectangular house and identify the shapes that compose the different real-life items. Unit Outline, Math practices state, “Students begin to see how geometric solids can represent objects in the real world in this unit. In Lesson 10.7, they match familiar objects to the solids they most closely resemble, such as a can to a cylinder or a box to a rectangular prism. Emphasize that real-world objects aren't perfect geometric shapes, but they can still be described and understood using shape names. In Lesson 10.10, students apply this reasoning to describe scenes, using solid names and spatial language to explain where and how objects appear in relation to one another. Through this work, students lay the foundation for using geometry to interpret and represent the world around them.”

• Grade 1, Unit 11: Addition and subtraction within 100, Lesson 11.10, Instruction: Represent and solve addition and subtraction stories within 100, students solve real-world problems using strategies such as the strip model to connect the mathematics to the situation. Teacher notes state, “For #2, have students read the story and identify what they know and what they need to find. Guide students to think about the strip model in the same way that they did for smaller numbers: it represents a whole (which you write on the line above) and can be split into parts. They know the two parts (the 21 kids in class and the 7 kids who join). They need to find the whole (the total number of kids in Ms. Joy's class). Have students discuss whether they need to add or subtract to solve the problem. They should understand that they need to put together two parts to find a whole, so they should add. Then have them write an equation using a ? for the unknown total. Students may write any related equation, but discuss how 21+7=? models the story well and makes it clear what operation to use to solve the problem. After students have written the equation, have them add using any strategy. After students find the sum, have them revisit the story and check whether their answer makes sense. Because more kids join Ms. Joy's art class, the sum should be greater than the starting number.” Student book states, “2. There are 21 kids in Ms. Joy’s art class. Then, 7 more kids join. How many kids are in Ms. Joy’s class now? Show the story. Write an equation.” Students use a strip model to solve the problem. Later in Lesson 11.10, students continue to model strategies to solve real-world problems. Teacher notes state, “In #4, students are given the whole (80 glue sticks) and one of the parts (30 glue sticks). Students should use a single-strip model and subtract to find the missing part. Students will likely use a place value sketch or subtract tens from tens. In #5, students are given two parts (35 boxes of crayons and 27 boxes of markers) and are asked to find the whole. Again, they will use a single-strip model, but this time they will add the parts to find the whole. Have students add using any strategy and record their work. Some good choices for this problem include using a place value sketch or adding tens to tens and ones to ones. After solving each problem, have students reread the problem and make sure their answer is reasonable. For example, in #4, the number of glue sticks left should be less than the number of glue sticks Ms. Joy started with because some were thrown away.” Student book states, “4. Ms. Joy has 80 glue sticks. She throws 30 of them away because they are dried out. How many glue sticks are left? Write an equation. 5. Ms. Joy has 35 boxes of crayons and 27 boxes of markers. How many boxes of crayons and markers does she have in all? Write an equation.” Unit outline, Math practices state, “Students model with mathematics in Lesson 11.10 as they solve story problems involving either addition or subtraction. As in Unit 5, encourage students to begin by using strip models to represent the relationships in each story. Visual models support students in identifying what's happening in the story and what they need to find. Reinforce the connection between models, equations, and stories by prompting students to explain how each part of a model relates to the numbers in an equation and the context of a story.”

• Grade 2, Unit 2: Equal groups, Lesson 2.3, Instruction: Learn to represent equal groups with repeated addition, students draw pictures to represent real-world problems involving equal groups. Teacher notes state, “For #2, guide students to draw an equal groups model. Students could draw four popsicle sticks three times and then circle each group. They could also draw a circle to represent each frame and then draw four popsicle sticks in each circle. Finally, guide students in completing the sentences. Have students count to find the total number of popsicle sticks. They may notice that they could also add three 4s or skip-count by 4s to find the total. If so, use these strategies to transition to the next problem. For #3, have students draw three circles to represent the three frames. Then have them draw five dots in each circle to represent the stickers. Point out that you can count every dot, skip-count by 5s, or add three 5s to find the total number of stickers. The repeated addition sentence relates to the strategy of adding three 5s. Explain how each addend represents the number of stickers in one group, and the sum represents the total number of stickers. Remind students that in order to add three numbers, they can find the sum of the first two addends, then add on the third addend.” Student book states, “2. Levi is making 3 picture frames. He needs 4 popsicle sticks for each frame. Draw to show how many popsicle sticks Levi needs in all. Complete the sentences. There will be ____ frames. Levi needs ____ popsicle sticks for each frame. Levi needs ____ popsicle sticks in all. 3. Next Levi needs 5 stickers for each of his 3 frames. Draw to show how many stickers Levi needs in all. Complete the sentences. There are ____ frames. Levi needs ____ stickers for each frame. Levi needs ____ stickers in all. Complete the repeated addition sentence. ____+ ____+ ____=_____” Later in Lesson 2.3, students complete a similar problem on their own. Guided Practice, Represent equal groups with repeated addition, Teacher notes state, “For #10, have students draw an equal groups model and write a repeated addition sentence, as in #3.” Student book states, “10. Eric is making 4 bookmarks. He needs 3 star stickers for each bookmark. Draw to show how many star stickers Eric needs in all. Write a repeated addition sentence.” Space is provided in all problems for students to draw pictures and equations as needed. Unit outline, Math practices state, “Throughout Unit 2, students model equal groups situations using drawings, arrays, and repeated addition equations. In Lessons 2.3, 2.5, and 2.6, make sure students use problem contexts to create equal groups of the correct size and write repeated addition equations that match their models. Highlight how different models can represent the same situation, and discuss which ones are most helpful in specific contexts. For example, in #3 of Lesson 2.5, Luca arranges muffins in rows and columns, so an array is a natural way to represent the situation. Encourage students to choose models strategically and explain how their models reflect the structure of the problem.”

The materials reviewed for Takeoff by IXL for Kindergarten through Grade 2 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 K-2, 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: 

• Kindergarten, Unit 3: Two-dimensional shapes, Lesson 3.5, Instruction: Describe squares. Students use precise mathematical terminology to identify two-dimensional shapes throughout the unit, including squares in this lesson. Teacher notes state, “For #4, have students look at the squares and discuss what they have in common. Highlight the following attributes of a square: Four straight, equal sides, Four square corners, No curved parts, Being flat (two-dimensional), Being closed, Review that size, color, and fill don't determine whether a shape is a square. Also, turning a shape doesn't make it a different shape. For #5, discuss how both shapes are rectangles because they have four straight sides and four square corners. The first shape is also a square because it has four equal sides. Explain that all squares are special rectangles. Have students circle the square.” In problem 4, students see several squares in different colors, locations, and orientations. In problem 5, students see a square and a rectangle. Later in Lesson 3.7, students use what they have learned about shapes to draw several shapes with the required attributes. Lesson 3.7, Instruction, Build and draw shapes, Teachers notes state, “For #4, have students build a shape with three straight sides. Ask them what shape they made (a triangle). Then have students draw their triangles on the dot grid. Have students share their different triangles and their strategies for building and drawing them. Some students may trace the triangle they built if it fits in the space. Others may draw freehand or with a ruler, or they may use the dots as vertices or to help draw straight lines. For #5, have students describe the attributes of a square. They should recall that squares have four equal sides and four square corners. Have students build a square and then draw it on the dot grid. Discuss how dot grids can help them draw equal sides. (Equal sides go through the same number of dots.) Then discuss how dot grids can be used to draw square corners. (A line that goes side to side through the dots and a line that goes up and down through the dots will form a square corner.) For #6, have students build a shape with 6 vertices and then draw it. Have students share a variety of correct shapes and strategies for building and drawing them.” Unit outline, Math practices state, “Throughout Unit 3, students practice using mathematical language to name and describe two-dimensional shapes. In Lessons 3.1–3.6, they learn to identify circles, triangles, rectangles, squares, and hexagons by attending to defining attributes, such as whether a shape has square corners, how many sides it has, and whether the sides are straight or curved. Encourage students to explain how they know a shape fits a category, and to use terms like sides, vertices, and square corners when supporting their reasoning. Make sure they also understand that attributes like size and color aren't useful for identifying shapes. Lessons 3.7 and 3.8 offer more opportunities to attend to precision as students build, draw, and compare shapes with specific attributes.”

• Grade 1, Unit 7: Measurement, Lesson 7.3, Instruction, Learn to measure length, students learn precise mathematical terms such as measure and unit and attend to precision when using smaller objects to measure the length of longer objects. Teacher notes state, “For #2, discuss what it means to measure length (to find how long something is). Explain that you can measure an item's length by lining up smaller same-sized items from one end to the other with no gaps or overlaps. The item you use to measure is called a unit. Let students know that today they'll use paper clips as units, but they can use other items as units as long as they're all the same length. Discuss the correct way to measure the length of the paint. Ask students to identify the error in each of the incorrect answer choices. Emphasize that to measure correctly, the paper clips must be lined up from one end of the tube to the other with no gaps or overlaps. Highlight that the paint is about the same length as 3 paper clips, so the paint is about 3 paper clips long. Explain that every measurement should include a number (3) and a unit (paper clips).” Students see four paint tubes, only one of which is measured correctly with 3 paper clips placed end to end beneath the tube. Later in Lesson 7.3, students review another student’s work to identify the mistake the student made while measuring. Guided practice, Measure length, Teacher notes state, “For #7, make sure students understand that John's measurement is incorrect because the keys aren't all turned the same way. They should see that this is problematic because the key isn't the same length on all sides. Relate this to measuring with paper clips, and highlight that all the paper clips need to be turned the same way. Next, give each student a copy of Measure with paper clips. Have students measure the length or height of each item, as they did in #3–6. As students work, make sure they are lining up the paper clips without gaps or overlaps.” Students see a picture of a pair of scissors measured with four keys, but the keys are not placed end to end beneath the scissors. Unit outline, Math practices state, “Throughout Unit 7, students develop precision in their language, tools, and representations while working with measurement, time, and money. When introducing new vocabulary—for example, comparison words such as shorter, longer, shortest, and longest (Lesson 7.1) or time expressions like o'clock (Lesson 7.5) and half past (Lesson 7.7)—model the terms clearly and encourage students to use them in conversation. Students learn that measuring length accurately requires them to line up units carefully, without gaps or overlaps. Use hands-on measurement activities to help them practice this skill, and discuss common measuring mistakes, like the ones shown in #7 from Lesson 7.3 and #4 from Lesson 7.4. The unit concludes with students identifying coins and representing their values using the cent symbol (Lesson 7.8). Emphasize how to correctly read and write money amounts to support clear, precise communication.”

Grade 2, Unit 6: Measuring length, Lesson 6.1, Instruction, Warm up, students measure with precision to find accurate solutions to real-world problems. Teacher notes state, “For #1, have students identify the error and discuss how to correctly measure the fork. (David should have lined up his paper clips end to end with no gaps or overlaps in order to get the most accurate measurement.)” Student book states, “1. David said this fork is 4 paper clips long. What mistake did he make? How could he fix it?” The paper clips are not placed end-to-end in the picture. Introduction to inch tiles, Teacher notes state, “For #4–5, have students measure the objects using their inch tiles. Make sure students line the tiles up with one end of the objects and make sure there are no gaps or overlaps between the tiles.” Student book states, “How long is each object? Use inch tiles to measure.” Students see a chopstick in problem 4 that is six inches long, and a piece of pasta in problem 5 that is two inches long. Unit outline, Math practices state, “Students' precision with measuring tools in Unit 6 directly affects their accuracy. In Lesson 6.1, when using inch tiles, emphasize the importance of placing them end to end without gaps or overlaps. As students transition to using rulers (Lessons 6.2, 6.3, and 6.8), model how to align the zero mark with the end of an object and how to read its length to the nearest whole unit. Regardless of the tool, prompt students to be mindful of how the measurement marks align with the object to ensure accurate measurements. Throughout the unit, reinforce the importance of recording measurements with correct units. Remind students that omitting units makes a measurement's meaning unclear and that using incorrect units can lead others to misinterpret an object's size. Being precise during both measurement and communication helps ensure that their results are understood by others.”

The materials reviewed for Takeoff by IXL for Kindergarten through Grade 2 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 K-2, 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: 

• Kindergarten, Unit 6: Addition and subtraction, Lesson 6.10, Instruction: Learn about related addition facts. Students look for patterns and structure as they use counting to solve addition problems. They also notice that the order of the addends does not change the sum. Teacher notes state, “For #1, guide students to make a linked-cube model that matches the top picture and have them write the corresponding addition sentence: 2+1=3. Then have students flip their model to match the bottom picture and write the reverse addition fact: 1+2=3. Point out that flipping the model didn't change its size. Similarly, changing the order of 1 and 2 didn't change the total. Discuss how the addition sentences are similar (they use the same numbers and 3 is the sum of both) and how they are different (the numbers are added in a different order). Explain that these are called related addition facts.” Students see three connecting cubes and space to write two equations for the same number of cubes. Instruction, Learn about related addition and subtraction facts. Teacher notes state, “For #3, have students look at the top picture as you read the first part of the story: There are 3 horses standing together. Then 1 more horse joins. How many horses are there now? Have students write an addition sentence that matches the story. Then have them look at the bottom picture as you continue the story: There are 4 horses together. Then 1 horse runs away. How many horses stay? Discuss how the first and second parts of the story are related. Surface these points: Both parts include a total of 4 horses. In the first part, 1 horse joins. In the second part, that same horse leaves. Have students write a subtraction sentence that matches the second part of the story, and discuss how the two number sentences are related. Emphasize that they use the same three numbers and that adding 1 and taking 1 away are opposites. Explain that this is another example of related facts.” Students see a picture of three horses in a field with one running toward them and another picture of three horses in the field with one running away. Unit outline, Math practices state, “Students rely on the structure of our number system to add and subtract efficiently in Unit 6. Show them how the counting sequence supports quicker addition and subtraction strategies in Lessons 6.3 and 6.9—for example, counting on from the larger number instead of counting the total when one addend is small. Students can apply a similar principle as they decompose numbers in Lessons 6.4–6.6. Point out that once they break apart numbers in one way, they can adjust each part by one to find new decompositions. Throughout the unit, highlight how recognizing the structure of numbers leads to efficient problem solving. Students also begin to see the structure in the relationship between addition and subtraction. In Lesson 6.10, emphasize that related facts use the same three numbers. While students aren't expected to use related facts to add and subtract yet, understanding fact family relationships lays a foundation for fluency in first grade.”

• Grade 1, Unit 2: Addition and subtraction within 10, Lesson 2.5, Instruction: Relate addition and subtraction. Students look for patterns and structure in fact families to solve addition and subtraction problems within 10. Teacher notes state, “For #2, explain that a set of four related addition and subtraction sentences makes a fact family, just like related people make a family. Have students look again at the number sentences in #1 and ask them what they notice about them. Make sure that students understand what it means to notice something. Highlight the following ways the number sentences are related: All four number sentences use the same three numbers. The addition sentences show that you can add in any order. (This is a review from Lesson 2.1.) The sum of the addition sentences is the same as the first number in the subtraction sentences. It is also the largest number. Point out that students could have used the same cube train to show all four sentences of the fact family. Consider having some students show one of their cube trains from #1 and demonstrate how it shows the fact family. For #4, have students complete the model by finding the sum. Remind students that they can think of 4 and 5 as the two parts of the whole, 9. Then have students use the strip model to write the four number sentences in the fact family. Highlight the following points: The addition sentences describe what happens when you put the parts together. The subtraction sentences describe what happens when you take one of the parts away from the whole. Explain that when you know one fact, you can find all its related facts because they will use the same three numbers. For #6–7, have students complete the related number sentence using the given fact. Encourage them to use the part/whole relationships between the numbers in the given fact to determine the missing number in the related fact. Explain that you can solve some addition or subtraction problems more quickly by starting with a related fact that you know or can easily find.” Student book states, “2. The number sentences above form a fact family. What do you notice about them? 4. Complete the model.” Students see a representation of 4 plus 5 equaling 9 and space to complete the four problems in the fact family. “Complete each related fact. 10-4=6, 6+4=____. 7. 4+4=8, 8-4=____” Unit outline, Math practices state, “Students make use of structure as they develop efficient strategies for adding and subtracting within 10. Reinforce the importance of the number sequence in counting on (Lesson 2.2) and counting back (Lesson 2.4). Use tools such as number lines or number lists to make this structure visible—model how one addend is the starting point, the second addend is the number of jumps (or numbers listed), and the sum is the landing point (or final number). Use fact families (Lesson 2.5) to explicitly draw students' attention to the inverse relationship between addition and subtraction. Highlight how knowing one fact can help uncover others within the same family. Support students in applying this understanding to find missing addends (Lesson 2.3) and to subtract by counting on (Lesson 2.6). Encourage them to consider the numbers in a problem when choosing between counting on or counting back to subtract. In Lesson 2.8, introduce and model number pairs that make 10. Emphasize 10 as a natural benchmark for simplifying calculations. Because 10 is central to our place value system and the structure of teen numbers, the making 10 strategy will become especially useful in students' continued work with addition and subtraction.”

Grade 2, Unit 4: Subtraction within 100, Lesson 4.7, Instruction: Learning to model subtraction with regrouping. Students use the structure of place value to solve subtraction problems within 100. Teacher notes state, “For #2, have students model the problem using place value blocks. Then demonstrate how to draw a place value sketch to show the work you did with the place value blocks. Consider using two different colors in your sketch and providing each student with two different colored pencils to use in their own sketches. Help students see that 2 tens and 6 ones remain in the sketch. Students may be tempted to say that there are 3 tens remaining since only 2 tens were crossed out initially. Be sure to point out that another ten was used when we regrouped it into 10 ones. For #3, have students model the problem using place value blocks, then draw a place value sketch to show their work, as they did in #2. Again, help students to see that there are 3 tens and 4 ones remaining, so the answer is 34. For #5, discuss Ann's method. She first determined if she needed to regroup by looking at the ones. Since there are more ones in 27 than 62, Ann regrouped one of the tens from 62 into 10 ones before she started subtracting. Then she subtracted tens from tens and ones from ones. Guide students to use Ann's way to find 76-4949. Since there are more ones in 49 than there are in 76, you will need to regroup. Throughout, encourage students to evaluate the reasonableness of their results in each step. For example, it makes sense that 7 tens and 6 ones can be regrouped as 6 tens and 16 ones because 60+16=76.” Student book states, “2. Find 51-25. Use place value blocks to solve. Draw a place value sketch to show your work. There are ____ tens and ____ ones left over. 51/25=. 3. Find 52-18. Use place value blocks to solve. Draw a place value sketch to show your work. 52-12=. 5. Ann is trying to find 62-47. She has an idea for a faster way to record her work. Ann: First, I can check if I need to regroup. Then I can regroup before I start subtracting. Use Ann’s way to find 76-49.” Students see that Ann has set up a place-value problem that shows taking one ten and adding ten ones to the ones place. Unit outline, Math practices state, “Throughout Unit 4, students rely on the structure of the base-ten number system to understand subtraction. They break numbers into tens and ones (Lessons 4.2 and 4.6), subtract with place value blocks (Lessons 4.7 and 4.8), and organize work in place value charts (Lesson 4.8). Highlight how these methods are supported by the structure of the base-ten system. They all rely on subtracting like units—tens from tens and ones from ones—whether with models, written work, or mental math. Students also learn strategies for subtraction that leverage the relationship between addition and subtraction. This includes compensation in Lesson 4.3 and counting on to subtract in Lesson 4.4. Encourage students to use addition to gain confidence in their answers. For example, even if counting on to subtract isn't a student's preferred strategy, they can use it to check the answer they found with another strategy.”