K-2nd Grade - Gateway 2

The materials reviewed for ClearMath Elementary 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. 

Multiple conceptual understanding problems are embedded throughout the grade level within the Activate, Explore, and Reflect sections, providing students opportunities to engage both independently and with teacher support. The Teacher's Implementation Guide, Student Tools for Learning, Making Math Concrete and Visible, notes that “Manipulatives bridge the gap between concrete experiences and abstract reasoning. They help students build number sense, visualize ideas, and internalize concepts through hands-on exploration.” The materials, therefore, ensure students can access and demonstrate conceptual understanding, including opportunities to do so independently when appropriate. Examples include:

• Grade K, Module 5, Topic 10, Lesson 9, Explore 2, students develop conceptual understanding as they work in pairs to match each teen ten-frame card with two corresponding equation cards. For example, a double ten-frame card displaying 13 is paired with the equations 13=10+3 and 10+3=13. In the Grade K, Teacher's Implementation Guide, Task Notes, teachers are prompted to ask, “Why does each ten-frame card match two equation cards?” and prompted to “Observe students as they match ten-frame cards with equations. Do students understand that teen numbers consist of 10 ones and some more ones? Do students understand that one equation decomposes the teen number and the other composes the teen number? Can students represent teen numbers in multiple ways?” (K.NBT.1)

• Grade 1, Module 1, Topic 2, Lesson 9, Student Practice Book, Questions 4–5, students demonstrate conceptual understanding as they use a picture to create addition story problems and explain their reasoning. They are prompted to write an addition story problem about the kittens and solve it using pictures, numbers, or words to explain their thinking. A picture with seven kittens is provided, showing four kittens in a basket and three kittens on a rug (1.OA.3).

• Grade 2, Module 2, Topic 5, Lesson 6, Student Practice Book, Question 1, students demonstrate conceptual understanding as they estimate a sum and then use an open number line to determine the actual sum. The problem is 82+14, with a number line provided showing points labeled 82, 92, and 96. Students extend their understanding of tens and hundreds to operate on the number line (2.NBT.7).

The materials reviewed for ClearMath Elementary 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 Grade-Specific Fluency Expectations map outlines fluency goals within and across grade levels for ClearMath Elementary. The Module and Topic Fluency Progressions in the Mathematical Progressions and Connections digital book describe how fluency develops across the courses. The Mathematical Progressions and Connections documents for each grade level state, “Fluency Progressions, Fluency encompasses number sense, precision, and efficiency. Number sense is understanding the relationships between quantities and having the ability to flexibly compose and decompose quantities. Precision entails recognizing that the solution is accurate and makes sense. Efficiency is developing strategies that promote using connections between concepts to solve problems in a streamlined way. These three pieces come together when students demonstrate fluency. These components are complex in and of themselves and therefore take time to develop and synthesize.” The fluency map highlights the grade-specific expectations within ClearMath Elementary. Additional fluency practice is available through MATHia Adventure, where students play games that support fluency development. Examples include: 

• Grade K, Module 4, Topic 9, Lesson 4, Explore 1, students develop procedural skills and fluency as they write equations to represent number path jumps. Directions state, “Look at each number path. Then, write an equation to match.” Teachers prompt students to identify the starting number, direction, and size of the jump, and explain whether the equation should use addition or subtraction: “What is the starting number of the jump? Explain your thinking. What is the direction of the jump? Explain your thinking. When a jump is backwards, do you use an addition sign or a subtraction sign to write the equation representing the jump? Explain your thinking. What is the jump size? Explain your thinking. What equation represents the jump?” (K.OA.5)

• Grade 1, Module 3, Topic 6, Lesson 10, Student Practice Book, students demonstrate procedural skills and fluency as they determine whether equations are true or false. They are asked to “Draw a line to show whether each equation is true or false,” with examples including 8 = 5 – 3, 7 + 2 = 5 + 3 + 1, and 10 – 5 – 2 = 8 – 4 – 1. (1.OA.6)

• Grade 2, Module 1, Topic 3, Lesson 4, Explore 1, Question 4, students develop procedural skills and fluency as they complete a table by finding half of given numbers (6, 8, 10, 18, 24, 36, 70). This supports fluency with addition and subtraction strategies within 100. (2.NBT.5)

The materials reviewed for ClearMath Elementary Kindergarten through Grade 2 meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters. 

Multiple routine and non-routine applications of mathematics are included throughout the grade level, with single- and multi-step application problems embedded within lessons, including Activate and Explore. Students engage with these applications both with teacher support and independently through tasks such as games, puzzles, and investigations. Materials are designed to provide opportunities for students to independently demonstrate their understanding of grade-level mathematics when appropriate. For example:

Examples of application of the math with teacher support include:

• Grade K, Module 3, Topic 5, Lesson 1, Explore 1, students apply their understanding and use manipulatives to act out joining problems. Directions state, “Tell students they will use the farm animal counters and the Farm Story Mat resource to act out joining stories. Have students model the following story in pairs as you read it aloud. The farmer saw two farm animals by the barn and one animal by the tree. How can you model this story using the counters and story map? The farmer put down the food for the animals, and they all came to eat. How can you model this action? How many farm animals came to eat? How can you determine the total number of farm animals?” The activity closes with a discussion: “What was the same about all of the joining stories you acted out? What was different about all of the joining stories you acted out?” (K.OA.2)

• Grade 1, Module 5, Topic 12, Lesson 5, Explore 1, students work independently on a non-routine problem as they create a draft design for the final marble run. The task states, “Draw the design of your marble run. Show 2 different routes. Use a different colored crayon for each route.” Students then build their marble run with connecting cubes and tape, measure the length of their ramp, and make predictions about the outcome. (1.MD.4)

• Grade 2, Module 5, Topic 12, Lesson 6, Reflect, Question 3, students apply their understanding of money to solve multi-step word problems. One problem states, “A pencil costs $1 and 50 cents, and a pen costs 2 and 36 cents. Malik would like to buy 2 pencils and 1 pen. He has $5. Does he have enough money? Explain your thinking.” In the Explore section, teachers display another prompt: “Colin buys a pen for 16\text{¢}. He pays with 2 dimes. How much change does he get back? How can you determine how much change Colin gets back?” (2.MD.8)

The materials reviewed for ClearMath Elementary 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 module within the curriculum supports a variety of instructional approaches that incorporate conceptual understanding, procedural fluency, and application in a balanced way. Examples include:

• Grade K, Module 5, Topic 12, Performance Task: Sports Shopping, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they reason about part–part–whole with number bonds, practice addition and subtraction within 10, and apply math in a real-world shopping scenario. “You have $10 to spend at a sports shop. There are many different items for sale at the sports shop.” (10 items pictured with prices written under them, e.g., soccer ball $4). “Directions: Create 2 different combinations of 2 items that you can buy from the sports shop to spend a total of $10. Complete the number bond model and write an equation to match each combination.” Outlines of number bonds and equations provided. “Your friend has $5 and buys 1 item at the shop. Directions: Write 1 item that your friend could buy for $5 or less. Then, write an equation to show how much money they have left after they buy it.” An outline of equation is provided. (K.OA.2, K.OA.3)

• Grade 1, Module 5, Topic 11, Lesson 1, Student Practice Book, Questions 5-10, students engage in procedural skill and fluency and conceptual understanding as they use colored rods to compose 10 with multiple addends, extend that reasoning to make 100, and complete equations such as “23+ ___ =100 and 85+ ___ =100.” (1.NBT.4)

• Grade 2, Module 4, Topic 10, Lesson 5, Reflect, Question 1, students engage with procedural skill and fluency and application as they subtract within 100 to solve a real-world problem: “Maria has 28 sunflower stickers in her collection. She gives 17 of them to her friend. How many sunflower stickers does Maria have left?” (2.NBT.5, 2.OA.1)

The materials reviewed for ClearMath Elementary Kindergarten through Grade 2 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.

The Teacher's Implementation Guide explains the curriculum’s approach to MP, Embedding Mathematical Habits of Mind states, “The first Habit of Mind is evident every day in every lesson as students engage in activities, solve problems, and analyze their solutions.” 

Opportunities for students to engage with MP1 appear throughout the year. The Teacher's Implementation Guide identifies lessons and sections in which to observe MP1 through Habits of Mind call-out boxes, topic overviews, and lesson facilitation notes. Each Topic Overview provides “student look-for’s” that describe how students will develop proficiency with the practice. Facilitation notes supply teacher prompts that connect MP1 to the grade-level content standards. The Assessment Guide also aligns MPs to Re-Engagement Lessons, where students reflect on their problem-solving.

The Teacher's Implementation Guide, Embedding Mathematical Habits of Mind states, “The Habits of Mind emphasize problem-solving, reasoning, constructing arguments, and modeling with mathematics to promote critical thinking and real-world application. ClearMath Elementary engages students in these practices by encouraging them to make sense of mathematics and demonstrate their reasoning through problem-solving, writing, discussion, and modeling.” ClearMath Elementary further states that “every lesson engages students in mathematical problem solving, guiding them to explore, discover, and reason. The interleaving of mathematical content and practices ensures students develop these habits of mind across all domains. The focus on discourse requires students to share ideas, evaluate reasoning, and refine understanding while using precise language and persevering in problem-solving.”

Examples include:

• Grade K, Module 1, Topic 6, Performance Task: Grapes with a Friend provides students with opportunities to engage in MP1 by interpreting a story problem, representing the situation with drawings, and writing equations. Students show different ways grapes could be added and then removed, determine if their answers make sense, and devise strategies independently.

• Grade 1, Module 3, Topic 6, Math in Action Center: Free Time Data provides students with opportunities to engage in MP1 by analyzing a graph showing how children spend free time and answering multi-part questions. Students make sense of the data, compare quantities, and interpret results to respond to real-world questions.

• Grade 2, Module 4, Topic 11, Lesson 4, Explore 1 provides students with opportunities to engage in MP1 by designing a town on grid paper. Students determine how to place required buildings, monitor and adjust their designs, calculate the area of buildings, and reflect on whether their final town layout makes sense.

The materials reviewed for ClearMath Elementary Kindergarten through Grade 2 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.

Opportunities for students to engage with MP2 appear throughout the year. The Teacher's Implementation Guide identifies lessons and sections in which to observe MP2 through Habits of Mind call-out boxes, topic overviews, and lesson facilitation notes. Each Topic Overview provides “student look-for’s” that describe how students will develop proficiency with the practice. Facilitation notes supply teacher prompts that connect MP2 to the grade-level content standards. The Assessment Guide also aligns MPs to Re-Engagement Lessons, where students reflect on their problem-solving.

The Teacher's Implementation Guide, Embedding Mathematical Habits of Mind states, “The Habits of Mind emphasize problem-solving, reasoning, constructing arguments, and modeling with mathematics to promote critical thinking and real-world application. ClearMath Elementary engages students in these practices by encouraging them to make sense of mathematics and demonstrate their reasoning through problem-solving, writing, discussion, and modeling.” ClearMath Elementary further states that “every lesson engages students in mathematical problem solving, guiding them to explore, discover, and reason. The interleaving of mathematical content and practices ensures students develop these habits of mind across all domains. The focus on discourse requires students to share ideas, evaluate reasoning, and refine understanding while using precise language and persevering in problem-solving.”

Example includes:

• Grade K, Module 1, Topic 2, Lesson 10, Explore 2, provides students with opportunities to engage in MP2 by attending to the meaning of quantities and representing situations symbolically. Students identify unknown numbers on a number path to 10 by drawing dots in blank quantity cards and writing the numbers in the blank spaces.

• Grade 1, Module 1, Topic 1, Lesson 5, Explore 1 and 2,  provides students with opportunities to engage in MP2 by representing quantities symbolically and understanding the relationships between problem scenarios and mathematical representations. Students first sort pictures of animals into categories, then create a picture graph of the categories, labeling the title and categories of the graph.

• Grade 2, Module 4, Topic 10, Lesson 4, Explore 3,  provides students with opportunities to engage in MP2 by considering units involved in a problem, attending to the meaning of quantities, and explaining what the numbers or symbols in an expression represent. Students use reasoning to compare quantities represented as counters, then extend their reasoning to picture equations with unknowns (e.g., smiley face + 7 < smiley face + 9 or smiley face – 3 > smiley face – 5), and discuss how their reasoning applies when comparing two expressions containing sums or differences of two-digit numbers.

The materials reviewed for ClearMath Elementary 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.

Opportunities for students to engage with MP3 appear throughout the year. The Teacher's Implementation Guide identifies lessons and sections in which to observe MP3 through Habits of Mind call-out boxes, topic overviews, and lesson facilitation notes. Each Topic Overview provides “student look-for’s” that describe how students will develop proficiency with the practice. Facilitation notes supply teacher prompts that connect MP3 to the grade-level content standards. The Assessment Guide also aligns MPs to Re-Engagement Lessons, where students reflect on their problem-solving.

The Teacher's Implementation Guide, Embedding Mathematical Habits of Mind states, “The Habits of Mind emphasize problem-solving, reasoning, constructing arguments, and modeling with mathematics to promote critical thinking and real-world application. ClearMath Elementary engages students in these practices by encouraging them to make sense of mathematics and demonstrate their reasoning through problem-solving, writing, discussion, and modeling.” ClearMath Elementary further states that “every lesson engages students in mathematical problem solving, guiding them to explore, discover, and reason. The interleaving of mathematical content and practices ensures students develop these habits of mind across all domains. The focus on discourse requires students to share ideas, evaluate reasoning, and refine understanding while using precise language and persevering in problem-solving.”

Examples include:

• Grade K, Module 4, Topic 9, Lesson 9, Activate and Explore: Center 2, provides students with opportunities to engage in MP3 by constructing viable arguments and critiquing the reasoning of others. Students work in pairs to determine an unknown number on the number trail based on finger clues given by their partner. Students justify their reasoning as they identify the mystery number and listen to their partner’s explanation to decide if it makes sense.

• Grade 1, Module 2, Topic 4, Lesson 3, Explore 1 and 2, provides students with opportunities to engage in MP3 by constructing viable arguments and critiquing the reasoning of others. Students compare dimes and pennies to connecting cubes representing numbers between 11-49, then work together to represent the number in dimes and pennies. Students explain their reasoning orally using concrete models.

• Grade 2, Module 3, Topic 7, Lesson 9, Explore 3, provides students with opportunities to engage in MP3 by constructing viable arguments and critiquing the reasoning of others. Students analyze sketches of faces of 3D shapes and justify their reasoning as they eliminate possibilities to identify a secret shape. Students also perform error analysis to determine whether eliminations were correct or incorrect.

The materials reviewed for ClearMath Elementary 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.

Opportunities for students to engage with MP4 appear throughout the year. The Teacher's Implementation Guide identifies lessons and sections in which to observe MP4 through Habits of Mind call-out boxes, topic overviews, and lesson facilitation notes. Each Topic Overview provides “student look-for’s” that describe how students will develop proficiency with the practice. Facilitation notes supply teacher prompts that connect MP4 to the grade-level content standards. The Assessment Guide also aligns MPs to Re-Engagement Lessons, where students reflect on their problem-solving.

The Teacher's Implementation Guide, Embedding Mathematical Habits of Mind states, “The Habits of Mind emphasize problem-solving, reasoning, constructing arguments, and modeling with mathematics to promote critical thinking and real-world application. ClearMath Elementary engages students in these practices by encouraging them to make sense of mathematics and demonstrate their reasoning through problem-solving, writing, discussion, and modeling.” ClearMath Elementary further states that “every lesson engages students in mathematical problem solving, guiding them to explore, discover, and reason. The interleaving of mathematical content and practices ensures students develop these habits of mind across all domains. The focus on discourse requires students to share ideas, evaluate reasoning, and refine understanding while using precise language and persevering in problem-solving.”

Examples include:

• Grade K, Module 3, Topic 5, Lesson 1, Explore 2, provides students with opportunities to engage in MP4 by modeling with mathematics. Students tell joining stories using picture cards of ducks, solve their partner’s joining story, and represent the problem using the picture cards. Students put the problem in their own words and check whether their answer makes sense and matches the original story.

• Grade 1, Module 4, Topic 9, Lesson 5, Explore: Center 3, provides students with opportunities to engage in MP4 by modeling with mathematics. Students play a center game by rolling number cubes to add or subtract from 10, record their starting number and equation, and model the situation using a 0–20 number line. Students discuss the problem in their own words and use the math they know to solve problems.

• Grade 2, Module 1, Topic 3, Lesson 8, Explore 2, provides students with opportunities to engage in MP4 by modeling with mathematics. Students use base-10 blocks and base-10 shorthand to add multiples of 10, then record their answers in equations. Teachers facilitate discussion about appropriate representations as students describe the model they used and how it relates to addition.

The materials reviewed for ClearMath Elementary Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP5: Use appropriate tools strategically for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Opportunities for students to engage with MP5 appear throughout the year. The Teacher's Implementation Guide identifies lessons and sections in which to observe MP5 through Habits of Mind call-out boxes, topic overviews, and lesson facilitation notes. Each Topic Overview provides “student look-for’s” that describe how students will develop proficiency with the practice. Facilitation notes supply teacher prompts that connect MP5 to the grade-level content standards. The Assessment Guide also aligns MPs to Re-Engagement Lessons, where students reflect on their problem-solving.

The Teacher's Implementation Guide, Embedding Mathematical Habits of Mind states, “The Habits of Mind emphasize problem-solving, reasoning, constructing arguments, and modeling with mathematics to promote critical thinking and real-world application. ClearMath Elementary engages students in these practices by encouraging them to make sense of mathematics and demonstrate their reasoning through problem-solving, writing, discussion, and modeling.” ClearMath Elementary further states that “every lesson engages students in mathematical problem solving, guiding them to explore, discover, and reason. The interleaving of mathematical content and practices ensures students develop these habits of mind across all domains. The focus on discourse requires students to share ideas, evaluate reasoning, and refine understanding while using precise language and persevering in problem-solving.”

Examples include:

• Grade K, Module 5, Topic 11, Lesson 13, Explore: Center 1 and 3, provides students with opportunities to engage in MP5 by using appropriate tools strategically. Students build three-dimensional structures using geometric solids and reflect on when it is or is not possible to position one solid above or below another. Students also build three-dimensional shapes using clay and toothpicks and discuss why spheres, cones, and cylinders cannot be built with toothpicks, recognizing the insight and limitations of different tools.

• Grade 1, Module 4, Topic 10, Lesson 12, Explore 2, provides students with opportunities to engage in MP5 by using appropriate tools strategically. Students solve addition problems using base-10 blocks and the Number Line Hotel, alternating between the two tools and reflecting on which tool works best depending on the problem. Teachers facilitate discussion as partners explain how they checked their work and how the choice of tools supported solving equations.

• Grade 2, Module 2, Topic 4, Lesson 3, Explore: Centers 1,2, and 3, provides students with opportunities to engage in MP5 by using appropriate tools strategically. Students measure classroom objects using inches, feet, and yards, then determine which unit and tool is most effective for each object. Teachers facilitate discussion on efficiency and accuracy as students justify their tool choices and recognize both the insights and limitations of the tools.

The materials reviewed for ClearMath Elementary Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP6: Attend to precision in connection to the grade-level content standards, as expected by the mathematical practice standards.

Opportunities for students to engage with MP6 appear throughout the year. The Teacher's Implementation Guide identifies lessons and sections in which to observe MP6 through Habits of Mind call-out boxes, topic overviews, and lesson facilitation notes. Each Topic Overview provides “student look-for’s” that describe how students will develop proficiency with the practice. Facilitation notes supply teacher prompts that connect MP6 to the grade-level content standards. The Assessment Guide also aligns MPs to Re-Engagement Lessons, where students reflect on their problem-solving.

The Teacher's Implementation Guide, Embedding Mathematical Habits of Mind states, “The Habits of Mind emphasize problem-solving, reasoning, constructing arguments, and modeling with mathematics to promote critical thinking and real-world application. ClearMath Elementary engages students in these practices by encouraging them to make sense of mathematics and demonstrate their reasoning through problem-solving, writing, discussion, and modeling.” ClearMath Elementary further states that “every lesson engages students in mathematical problem solving, guiding them to explore, discover, and reason. The interleaving of mathematical content and practices ensures students develop these habits of mind across all domains. The focus on discourse requires students to share ideas, evaluate reasoning, and refine understanding while using precise language and persevering in problem-solving.”

Examples include:

• Grade K, Module 5, Topic 11, Lesson 6, Explore 1 and 2, provides students with opportunities to engage in MP6 by attending to precision. Teachers show and accurately name cones, cylinders, spheres, and cubes as students say, read, and discuss the correct names and attributes of each shape. Students continue to use accurate names and attributes as they determine whether shapes can stack, roll, or slide on a flat or inclined surface.

• Grade 1, Module 4, Topic 7, Lesson 14, Explore: Center 1, provides students with opportunities to engage in MP6 by attending to precision. Students roll a number cube to build connecting cube trains, then compare the lengths using terms such as shorter and longer. Students record their comparisons on a game board, accurately representing their trains and using precise language to describe their observations.

• Grade 2, Module 3, Topic 8, Lesson 4, Explore: Center 1, provides students with opportunities to engage in MP6 by attending to precision. Students use pictures of analog clocks to match activities to the correct time to the nearest five minutes and determine whether the activity would take place in the A.M. or P.M. Students label times accurately and use precise language to explain their reasoning in a teacher-facilitated discussion.

The materials reviewed for ClearMath Elementary Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP7: Look for and make use of structure in connection to the grade-level content standards, as expected by the mathematical practice standards.

Opportunities for students to engage with MP7 appear throughout the year. The Teacher's Implementation Guide identifies lessons and sections in which to observe MP7 through Habits of Mind call-out boxes, topic overviews, and lesson facilitation notes. Each Topic Overview provides “student look-for’s” that describe how students will develop proficiency with the practice. Facilitation notes supply teacher prompts that connect MP7 to the grade-level content standards. The Assessment Guide also aligns MPs to Re-Engagement Lessons, where students reflect on their problem-solving.

The Teacher's Implementation Guide, Embedding Mathematical Habits of Mind states, “The Habits of Mind emphasize problem-solving, reasoning, constructing arguments, and modeling with mathematics to promote critical thinking and real-world application. ClearMath Elementary engages students in these practices by encouraging them to make sense of mathematics and demonstrate their reasoning through problem-solving, writing, discussion, and modeling.” ClearMath Elementary further states that “every lesson engages students in mathematical problem solving, guiding them to explore, discover, and reason. The interleaving of mathematical content and practices ensures students develop these habits of mind across all domains. The focus on discourse requires students to share ideas, evaluate reasoning, and refine understanding while using precise language and persevering in problem-solving.”

Examples include:

• Grade K, Module 2, Topic 4, Lesson 8, provides students with opportunities to engage in MP7 by looking for and making use of structure. Students sort objects into groups, count the objects, and work with patterns and units of repeat by identifying, creating, and extending patterns. Students explain why a shape comes next, identify the unit of repeat, and create their own repeating units, decomposing more complicated patterns into simpler elements.

• Grade 1, Module 5, Topic 11, Lesson 1, provides students with opportunities to engage in MP7 by looking for and making use of structure. Students use colored rods to compose tens and hundreds, building trains of rods that equal 100. Students connect adding ones to make a ten and adding tens to make a hundred, then write equations that represent the rod trains. Students generalize structures such as 1:10 and 10:100 and decompose more complex representations into simpler parts.

• Grade 2, Module 1, Topic 2, Lesson 1, provides students with opportunities to engage in MP7 by looking for and making use of structure. Students estimate large collections of objects using base-10 blocks, discuss how grouping supports both estimation and counting, and compare the usefulness of different group sizes. Students then estimate and count three bags of 115-155 objects and reflect on when estimating is valuable, when counting is valuable, and which strategy is most effective. Students explain the structures they use, make generalizations, and decompose larger problems into simpler steps.

The materials reviewed for ClearMath Elementary Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP8: Look for and express regularity in repeated reasoning in connection to the grade-level content standards, as expected by the mathematical practice standards.

Opportunities for students to engage with MP8 appear throughout the year. The Teacher's Implementation Guide identifies lessons and sections in which to observe MP8 through Habits of Mind call-out boxes, topic overviews, and lesson facilitation notes. Each Topic Overview provides “student look-for’s” that describe how students will develop proficiency with the practice. Facilitation notes supply teacher prompts that connect MP8 to the grade-level content standards. The Assessment Guide also aligns MPs to Re-Engagement Lessons, where students reflect on their problem-solving.

The Teacher's Implementation Guide, Embedding Mathematical Habits of Mind states, “The Habits of Mind emphasize problem-solving, reasoning, constructing arguments, and modeling with mathematics to promote critical thinking and real-world application. ClearMath Elementary engages students in these practices by encouraging them to make sense of mathematics and demonstrate their reasoning through problem-solving, writing, discussion, and modeling.” ClearMath Elementary further states that “every lesson engages students in mathematical problem solving, guiding them to explore, discover, and reason. The interleaving of mathematical content and practices ensures students develop these habits of mind across all domains. The focus on discourse requires students to share ideas, evaluate reasoning, and refine understanding while using precise language and persevering in problem-solving.”

Examples include:

• Grade K, Module 5, Topic 12, Lesson 10, Explore 1, provides students with opportunities to engage in MP8 by looking for and expressing regularity in repeated reasoning. Students estimate how many objects are in a collection of 101–120 items, then use repeated counting and skip counting to group and count the objects. Students evaluate the reasonableness of their estimates and adjust their strategies when estimating the next group of objects.

• Grade 1, Module 3, Topic 8, Lesson 7, Explore: Center 1, provides students with opportunities to engage in MP8 by looking for and expressing regularity in repeated reasoning. Students partition circles, squares, and rectangles into halves and fourths, noticing that repeated reasoning can be applied regardless of the shape. Teachers facilitate discussion as students explain the reasonableness of their answers and describe what they notice about the partitions.

• Grade 2, Module 1, Topic 1, Lesson 5, Explore: Center 1, provides students with opportunities to engage in MP8 by looking for and expressing regularity in repeated reasoning. Students roll a ten-sided number cube to create fact families, write related addition and subtraction equations, and generate word problems. Working in pairs, students repeat the process multiple times and evaluate the reasonableness of their partner’s story.