K-2nd Grade - Gateway 2

The materials reviewed for Bridges in Mathematics 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 levels in Warm-Ups or Problems & Investigations. Students have opportunities to engage with these problems both independently and with teacher support. Teachers are provided with guidance through Unit Introductions and Math Teaching Practices. Examples Include: 

• In Kindergarten, Unit 7, Measurement and Teen Numbers, Module 2, Session 2, Problems & Investigations, students play Capture the Number to demonstrate conceptual understanding of teen numbers by building numbers on a double 10-frame, using the unit of 10 as a whole with some additional value. As students engage in the game, they follow the directions on the Work Place Instructions 7B Capture the Number print, which introduces a new element: two cards of each number, requiring players to avoid using a card that has already been drawn. On each turn, students are invited to build the quantity shown on the card using their number racks. Throughout the game, the strategy of counting "10 and some more" is emphasized to help students understand how to figure out how many dots are on the card and to find the matching numeral on the number path. Periodically, students are asked questions such as, “What number comes before this one? What number comes after? This number is between which two numbers?” These questions encourage students to think about the sequential relationships between numbers. The challenge question, “How many more to get to 20?” further supports students' conceptual understanding of teen numbers and their relationship to 20, helping students strengthen their understanding of number composition and progression. (K.NBT.1)

• In Grade 1, Unit 3, Adding, Subtracting, Counting & Comparing, Module 2, Session 1, Warm-Up, students develop conceptual understanding by using an addition problem to help solve subtraction problems. “As a quick warm-up, play I Have, You Need with the class with 10 as the target number. Tell the class that they’ll use finger formations. Then quickly review the game: You’ll say a number to let the class know how many you have, and they’ll say or show you on their fingers how many more you need to get to 10. 2. Start with 6, and then repeat with 9, 3, 5, 8, and 10 in quick succession. Teacher, ‘I have 6.’” Students, ‘You need 4 (showing 4 using finger formations)’” (1.OA.4)

• In Grade 2, Unit 1, Figure the Facts, Module 3, Session 2, Problems & Investigations, students demonstrate conceptual understanding of even and odd numbers by representing even numbers with equations. The lesson begins with a discussion in which students share what they know about even and odd numbers, with the teacher reminding them that even numbers can be shown as the sum of two equal addends, but odd numbers cannot. Students then show an even number on their number racks and think about how to write an addition equation to represent it. For example, one student explains, “I did 10. I used all the beads on the top row because there are 5 red and 5 white beads. That’s 5 + 5 = 10.” Another shares, “I did 12. See, there are 6 beads on top and 6 on the bottom, so that’s 6 + 6 = 12. Any time the top row and bottom row have the same number of beads, you know it has to be even.” The teacher then works with students to use doubles to help solve less familiar combinations, such as 5 + 6. Students model the problem on their number racks and share strategies, such as, “I can see it’s 10 with the red beads because 5 and 5, and then 1 more white one makes 11,” and, “Six and 6 is 12, and 1 less than that is 11.” The same process is repeated with additional combinations (e.g., 7 + 8, 6 + 7, 9 + 10, 4 + 3, 9 + 8), with students verbalizing strategies that reference doubles (e.g., “7+8 is 15 because 7+7 is 14 and 1 more is 15”). Students then read each recorded equation and determine whether the sum is odd or even, explaining their reasoning. Later in the module, in Session 5, Home Connections, students further apply their strategies to solve addition and subtraction problems such as 10 − 5, 8 + 4, and 14 − 10. (2.OA.3)

The materials reviewed for Bridges in Mathematics 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. 

Multiple procedural skill and fluency problems are embedded throughout the grade levels in Warm-Ups, Home Connections, Student Book, Number Corner, or Problems & Investigations. Students have opportunities to practice these skills both independently and with teacher support. Teachers are provided with guidance through Unit Introductions and Math Teaching Practices. Examples include:

• In Kindergarten, Number Corner, March, Teacher Guide, Day 10, Calendar Collector, Examining the Data, students develop procedural skills and fluency in adding sums to five. The materials state, “Complete the Calendar Collector update to fill the current row, then engage the class in discussing the data collected so far. What do students notice? What do they see? Give students a minute to think-pair-share observations. Here are some prompts and questions to pose during the discussion. How many lamb days did we have this week? How many lion days? Did we have more lamb days or more lion days this week? How many more? How do you know? Do you think we’ll get the same results next week? Why or why not? Is there any way to find out? Display the Lamb or Lion Weather? page and have students find the page in their Number Corner Student Books. Review the instructions with the class. Remind the class of the colors they chose to show the lamb days and the lion days. Explain that they will use the third and fourth rows later this month. Give students time to color in the 5-frame for Week 2, and have them write an equation to represent the week’s data. Ask students to compare the 5-frame and equation for Week 2 to those for Week 1. How are they the same? How are they different?” (K.OA.5)

• In Grade 1, Unit 2, Developing Foundational Facts, Module 2, Session 4, Problems & Investigation, students demonstrate procedural skills and fluency through a Work Place that develops strategies for solving addition and subtraction problems.  The materials state, “Explain that you have a new domino game to share with the class today that will help them practice addition strategies and facts. First you’ll play the game against the class, and then it will become a new Work Place. Show students the Sort the Sum game board, and summarize the game. Players take turns picking a domino from the draw pile and finding the sum of the two sides. If the sum is any number from 6 to 10, they place the domino in the matching column on the game board. If the sum is less than 6 or greater than 10, their turn ends. Play continues until all the white spaces on the game board are full. Have a couple of students turn the dominoes face-down in the middle of the discussion area, and thoroughly mix them. Play the game according to Work Place Instructions 2C Sort the Sum. Each time you take a turn and pick a domino from the draw pile, read the number of dots on both sides to the class and share your strategy for adding them. Teacher: ‘There are 4 dots on one side and 5 on the other. Let’s see … 4 + 5 is almost like 4 + 4, but 1 more. Four and 4 is 8, plus 1 more is 9. I got 9 on my domino, and 9 is greater than 6. Is there still room for a domino with 9 dots on our game board?’ Choose a different student to play for the class. Have that student read out the numbers on both sides of the domino as you record them on the board in the form of an addition expression. Have students share the answer, first in pairs and then as a class. Ask them to explain their strategies for finding the sum.” (1.OA.6)

• In Grade 2, Number Corner, March, Student Book, Fact Worksheet, students demonstrate procedural skills and fluency as they independently solve addition and subtraction problems up to 20 using strategies. After completing the problems, students choose one to show the strategy they used.  The materials state, “1. Complete the addition and subtraction facts. Use green for the facts you think are easy. Use orange for the facts you think are medium. Draw a red circle around the facts you think are challenging.” Some problems students solve include:  8 + 2, 8 - 4, 5 + 4, 14 - 7, and 7 + 8. 2. Write one of the facts that you circled in red. Show how to solve it using numbers, sketches, or words.” (2.OA.2)

The materials reviewed for Bridges in Mathematics 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.

Routine and non-routine applications of mathematics are embedded throughout the grade levels in Number Corner, Problems & Investigations, and Home Connections. These include both single- and multi-step problems that require students to apply concepts in new contexts. Students engage with these applications independently and with teacher support, particularly in the final lessons of each unit, which emphasize real-world scenarios. Across the curriculum, students have repeated opportunities to demonstrate their ability to apply mathematical concepts and skills. Examples include:

• In Kindergarten, Unit 6, Three-Dimensional Shapes & Numbers Beyond 10, Module 4, Session 1, Problems & Investigations, Investigations, students solve word problems with teacher support and generate an equation to represent their thinking. The teacher introduces the task by displaying the picture and asking, “What do you notice? What do you wonder?” Students share ideas and then refine the problem, identifying that, “I think we need to know about both buckets. Yes – then we can add them.” When quantities are provided, students determine, “A lot of friends wondered how many snowballs there were in all. Let’s answer that question in pairs.” Students use pictures, numbers, words, and math tools to solve, then share strategies before the class, with the lesson concluding in generating an equation to match the situation. (K.OA.2)

•  In Grade 1, Unit 2, Developing Foundational Facts, Module 3, Session 2, Home Connections, Dots & Dominoes, students independently apply addition in a non‑routine context by deciding how to configure blank dominoes to match a given situation. The task states, “3. Luis and Camila are playing dominoes. Luis turns over a domino with 3 dots on it, and Camila turns over a domino with 6 dots. How many dots are on both dominoes? Show your thinking. There are ____ dots in all.” Students read the problem, select dot configurations, and record their reasoning to determine the total. (1.OA.1)

• In Grade 2, Number Corner, January, Day 4, Calendar Grid, students apply their understanding of interpreting data and addition to solve real-world problems. During the activity, students examine a graph, identify the type of graph, and generate possible survey questions. They then respond to prompts such as, “How many students participated in this survey? How many more students chose dogs than cats?” Students write and solve equations to represent their reasoning, sharing and comparing strategies with peers. Multiple solution paths are recorded, including both addition and subtraction equations, to show how the data can be represented mathematically. (2.OA.1, 2.MD.10)

The materials reviewed for Bridges in Mathematics 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 sessions, modules, or units. Each unit within the curriculum supports a variety of instructional approaches that incorporate conceptual understanding, procedural fluency, and application in a balanced way.

Examples include:

• In Kindergarten, Unit 1, Numbers to 5 & 10, Module 3, Session 1, Problems & Investigations, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they work with 5- and 10-frames. In Terrific 10s, the teacher places 7 to 9 Unifix cubes on a 5-frame without indicating where to put the extra cubes, and students suggest “where to place the cubes.” Students then discuss, “Do you think the 5-frame is a good tool for counting more than five items? Why or why not? What might be a better tool for counting this many cubes?” When shown the 10-frame, students compare, “How are these two mats the same? How are they different?” and confirm that the number of cubes “stayed the same” when transferred from one frame to the other. Students count the spaces and respond to, “How many spaces are empty on the 10-frame? How many more do we need to make 10? How do you know?” They also interpret dot cards, answering “How many dots do you see?” and showing the number on their fingers, either by subitizing or counting. (K.CC.4, K.CC.5)

• In Grade 1, Unit 2, Developing Foundational Skills, Module 3, Session 3, Problems & Investigations, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they play Spin & Add. The teacher introduces the game by explaining that it will help students develop the foundational facts of +1 and +2 and practice counting on. As the class plays together, the teacher models, “I got a 6 this time. You spun a 2. We can count on to find the total, 6…7, 8,” and records the sum on the sheet. Students then play in pairs, spinning, adding, and recording their results. Before beginning, each student circles the numeral they believe will fill first and explains their choice, such as, “I picked 9 because that’s the number that just won,” and “Thirteen is the most. I think it will win.” At the end of the game, students compare results by posting winning numbers on a class display and answering questions like, “Which number came up as the winner most often? Were there some numbers that didn’t come up at all?” Students extend the activity by transferring data to a graph and considering, “Why are some numbers more likely to win than others?” (1.MD.4)

• In Grade 2, Unit 3, Addition & Subtraction Within 100, Module 3, Session 1, Problems & Investigations, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they solve routine real-world problems in The Seed-Finding Contest. The teacher reads the problem and asks students to think-pair-share what they know and wonder, recording their ideas on a chart. Students then work individually or in pairs to solve a selected question, using manipulatives, whiteboards, or paper to represent their thinking. When sharing strategies, students explain their reasoning with tools and equations. For example, Saul states, “First I made 42 and 24 with the pieces. Then I put the 20 from 24 together with the 42. That was 62. Then I added the last 4 and it was 66.” The teacher records the expression 42 + 24 and sketches the base ten pieces to show how quantities were combined. Students then connect models and symbols, such as noting that the loop around 42 and 20 shows the pieces were combined first. Another student, Akiko, explains, “I started like Saul, but I put all the tens together and then all the ones. Forty and 20 is 60, then 2 and 4 is 6. If you put those together, you get 66.” (2.OA.2)

The materials reviewed for Bridges in Mathematics 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.

Students across the K–2 grade band engage with MP1 throughout the year. It is explicitly identified for teachers in the Teacher Guide, including in the Concepts, Skills & Practices chart, and intentionally developed through the Warm-Up, Problems & Investigations, Home Connections, and Work Places sections of specific sessions. MP1 is also addressed in the Number Corner Teacher Guide.

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, use strategies that make sense to them, monitor and evaluate their progress, determine whether their answers are reasonable, reflect on and revise their approaches, and increasingly devise strategies independently. For example:

• In Kindergarten, Unit 5, Two-Dimensional Geometry, Module 3, Session 3, Problems & Investigations, students work together to fill hexagon spaces on a caterpillar-shaped board using pattern blocks. As they take turns spinning to select shapes, they make sense of the goal of filling the spaces exactly and begin to notice which shapes are more efficient. Students share observations, make predictions, and adjust their strategies based on the available shapes. They revisit their thinking, justify choices, and continue working even when a shape doesn't fit. Math Practices in Action states, “Games give students a wonderful opportunity to make sense of problems and persevere in solving them. In this case, the game itself is the problem, and students make sense of it by clarifying the objective and developing strategies for winning. They have many chances to play the game, giving them the opportunity to persevere in their efforts. This helps students develop a deeper understanding of the relationships among the pattern block shapes.”

• In Grade 1, Unit 4, Leapfrogs on the Number Line, Module 2, Session 3, Problems & Investigations, students use a 0 to 100 floor number line and frog stories to make sense of addition and subtraction through jumps of 10. They compare the number line to a cube train, explore movement using a character named Jumper, and act out stories to model mathematical situations. Students determine endpoints, write equations, and revise their work based on peer feedback. They use physical models, drawings, and shared reasoning to reflect on their strategies. Math Practices in Action states, “Listening to the problem more than once supports students in making sense of the problem. They first think about the context and then identify mathematical details. Having a strategy for tackling problems helps students persevere.”

• In Grade 2, Unit 8, Measurement, Data & Multidigit Computation with Marble Rolls, Module 2, Session 1, Problems & Investigations, students are introduced to an open-ended challenge using familiar materials to design a system that gets a marble to move on its own. After sharing ideas and predictions, they work in pairs to build, test, and adjust marble runs using tubes, tape, blocks, and other supplies. Students troubleshoot challenges, refine their approaches, and may use measuring tools to track how far the marble rolls. They reflect on outcomes through discussion and drawing. Math Teaching Practices state, “As students explore different combinations of ramps and props to see how far they can get a marble to travel, they use reasoning and evidence to tweak their setup. They apply problem-solving techniques and use what they’ve learned to make a marble go as far as possible.”

The materials reviewed for Bridges in Mathematics Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP2: Reason abstractly and quantitatively, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the K–2 grade band engage with MP2 throughout the year. It is explicitly identified for teachers in the Teacher Guide, including in the Concepts, Skills & Practices chart, and intentionally developed through the Warm-Up, Problems & Investigations, and Work Places sections of specific sessions. MP2 is also addressed in the Number Corner Teacher’s Guide.

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:

• In Kindergarten, Unit 3, Bikes & Bugs: Double, Add & Subtract, Module 2, Session 1, Problems & Investigations, students are introduced to Butterfly Race, a game that helps them practice recognizing dot patterns, using doubles combinations, and matching quantities to numerals. The game board features five trees representing ordinal positions. Students take turns drawing and briefly viewing ("flashing") 10-frame pairwise display cards. They identify the number of dots either by subitizing or counting and determine whether the corresponding numeral appears on the next tree. If it does, they move their butterfly (game marker) forward; if not, their turn ends. As the game progresses, students explain how they determined the total (for example, “4 plus 4 equals 8, and one more makes 9”) and share strategies with peers using the think-pair-share routine. The activity is later introduced as a partner Work Place game, reinforcing connections between quantities and numerals in a playful and structured context. Math Practices in Action states, “This game invites students to connect quantities, number names, and numerals. Making connections among these representations is essential in reasoning abstractly and quantitatively.”

• In Grade 1, Unit 2, Developing Foundational Facts, Module 1, Session 3, Problems & Investigations, students learn and play Domino Add & Compare, a game designed to support addition fluency and quantitative reasoning. Players take turns drawing a domino, identifying the quantity on each side, and writing an equation that shows the total. They also record an equation for their partner’s domino and compare the two sums. The equation with the greater total is circled, and the player with the higher total wins the round. In the case of a tie, both players keep their domino. The teacher models the game by playing against the class, guiding students to write equations, compare sums, and justify their thinking. Students use whiteboards to practice writing addition equations based on the dot patterns. After the demonstration, students play in pairs using a record sheet and a set of dominoes, independently writing and comparing equations. A challenge variation encourages advanced learners to combine two dominoes on each turn to find a total, further supporting reasoning about quantities and symbolic representations. Math Practices in Action states, “Students use symbolic notation to express mathematical ideas when they write equations to represent the dots on the domino. Connecting numerals to dot arrangements helps them begin to reason more abstractly.”

• In Grade 2, Unit 4, Measurement, Module 3, Session 4, Problems & Investigations, students revisit a problem involving Jessie’s height and the Giant’s total height, using a visual titled Some More Information to support recall. Students first share their answers without judgment, then participate in a structured strategy-sharing routine. They present and explain their thinking using tools, drawings, and equations, while peers ask questions, restate strategies, and identify similarities between different approaches. Students use place value and repeated addition to represent and justify how Jessie’s height (e.g., 48 inches) is scaled to determine the Giant’s height (e.g., 192 inches). In a follow-up task using What If Jessie Were a Different Height?, students apply and extend their reasoning to a new scenario, where Jessie is now 52 inches tall. Working with partners, they solve the problem using strategies different from their original approach, drawing on insights from class discussions. The activity emphasizes flexibility with numerical representation, reasoning about quantities, and making connections across problem scenarios and mathematical expressions. Equity-Based Practice states, “Students are asked to explain their solution strategies in a way that makes sense to others, as well as analyze and respond to the strategies of their peers. The intentional selection and sequencing of strategies by the teacher helps students to make connections between concrete and abstract representations of the problem.”

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

Students across the K–2 grade band engage with MP3 throughout the year. It is explicitly identified for teachers in the Teacher Guide, including in the Concepts, Skills & Practices chart, and intentionally developed through the Warm-Up, Problems & Investigations, and Work Places sections of specific sessions. MP3 is also addressed in the Number Corner Teacher Guide.

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.

According to the Teacher Guide, “Bridges materials support students in several critical mathematical practices, including using appropriate tools strategically, attending to precision, and looking for and making use of structure. For example, the Word Resource Cards and MLC’s free Math Vocabulary app help students attend to precision with their mathematical language, which in turn enables them to construct viable arguments and critique the reasoning of others.” For example:

• In Kindergarten, Unit 6, Three-Dimensional Shapes & Numbers Beyond 10, Module 1, Session 2, Problems & Investigations, students sort a collection of objects into categories of circles and spheres. As they place objects on sorting mats, the teacher prompts them to explain their reasoning by asking, “Why did you choose that mat?” and “What did you notice that was the same or different from the first item we placed?” Students then justify their choices aloud, and the class is invited to give a thumbs-up or thumbs-down in response, followed by discussion. Later, in a think-pair-share routine, students build understanding of the differences between two-dimensional and three-dimensional shapes. For example, one student states, “You can hold a sphere, like a ball,” while another explains, “It’s not flat like a circle.” These interactions include opportunities for students to construct and explain their mathematical thinking and to critique the reasoning of others. Math Practices in Action states, “Students engage in a debate about where each item belongs and why. The desire to convince others supports them in sharing their observations and justifying their thinking. These skills, in turn, contribute to the ability to make reasoned arguments and listen to those proposed by their classmates.”

• In Grade 1, Unit 6, Geometry, Module 4, Session 1, Problems & Investigations, students engage in the game “Pick Two to Make 20”, where they select two numbers from a set of three to create a sum as close to 20 as possible without going over. They “share their pick and justify it,” using number racks or derived fact strategies to support their thinking. Students work independently, in pairs, and as a class to evaluate combinations, explain their reasoning, and critique the reasoning of others. The teacher models strategies and invites students to “convince you otherwise” if an inaccurate choice is made, prompting justification and peer critique. This problem provides opportunities for students to construct arguments, defend their thinking, and evaluate the validity of other strategies through discourse. Math Practices in Action states, “Students learn to construct viable arguments and critique the reasoning of others when they share ideas on how to get as close to 20 as possible. Playing as a team motivates them to listen to one another so they have the best chance of winning against the teacher.”

• In Grade 2, Unit 3, Addition & Subtraction Within 100, Module 2, Session 2, Problems & Investigations, students solve a real-world problem involving how many blocks Jessie and his dad have left to travel. The scenario is introduced by displaying a paragraph and a single problem from the Visiting the Relatives print original that states, “Jessie and his dad are taking the bus to the triplets' apartment. The triplets live 75 blocks away. Jessie and his dad have already gone 38 blocks. How many more blocks do they have to go?” The teacher facilitates a discussion by prompting students to identify what the problem is asking and how to approach solving it. Strategies such as counting on from 38 or counting back from 75 are explored. The teacher models the equation “38+ ___ =75” and supports students in identifying the unknown value. Students start to visualize their thinking by using open number lines, with single hops from 38 to 75. One student should comment, “That took forever!” which then prompts the class to group hops by tens and look for more efficient strategies. Students then develop and share alternate approaches using larger hops, such as jumps of 10, 5, or 2. Math Practices in Action states, “Students reflect upon the variety of strategies they have shared, which helps them learn to construct viable arguments and critique the reasoning of others. Such reflection presses students to consider the efficiency of various strategies. It also promotes fluency with multidigit computation.”

The materials reviewed for Bridges in Mathematics Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP4: Model with mathematics, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the K–2 grade band engage with MP4 throughout the year. It is explicitly identified for teachers in the Teacher Guide, including in the Concepts, Skills & Practices chart, and intentionally developed through the Warm-Up, Problems & Investigations, and Home Connection sections of specific sessions. MP4 is also addressed in the Number Corner Teacher Guide.

Across the grades, students participate in tasks that support key components of MP4, 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:

• In Kindergarten, Unit 8, Computing & Measuring with Frogs & Bugs, Module 4, Session 1 Problems & Investigations, students engage in modeling by interpreting and representing dot images with equations. The teacher begins by displaying a dot card for 6 and asks students to “circle the smaller groups of dots to show how they saw them,” just as modeled during the warm-up. Students then “write an equation to match the way they circled the dots.” Students revisit the same image to find another way to group the dots and “write a new equation, just like they did in the warm-up.” They then repeat the process with a dot card for 10 and are invited to “turn and talk to a partner about how they saw the dots and what equations they wrote.” Later, students create their own dot image (keeping the total within 10), share it with two classmates, and record how their peers see the image. Classmates are encouraged to “make a quick sketch of the dots” and “write a corresponding equation for the way they saw the dots.” The session ends with a reflective discussion: “Is it possible to write more than one equation from one set of dots? Why or why not?” These steps guide students to identify quantities in dot images, group them meaningfully, and represent those groups using equations.  Math Practices in Action states, “Students model with mathematics when they generate equations to describe the combinations they see in the dots.”

• In Grade 1, Unit 4, Leapfrogs on the Number Line, Module 1, Session 3, Problems & Investigations, students are introduced to the number line as a mathematical model through a contextual story about Nevaeh and a frog in a community garden. The teacher displays a number line and a toy frog to represent “Little Frog,” explaining that “the number line is like the cool muddy area next to the stream in the garden where the little frog likes to hop.” Students use this model to act out and solve addition and subtraction problems. To begin, the class determines how many hops it takes for the frog to move from 0 to a given number, emphasizing that “the interval, or hop, is what is being counted.” Students then listen to and help complete a story: “Little Frog made four hops... then two more hops.” Using the number line, they track the frog’s movements from 0 to 6 and write a matching equation (4 + 2 = 6). The lesson then shifts to subtraction. The teacher poses a related scenario: “If Little Frog was on the 6 and took 2 hops back the other way, where do you think Little Frog landed?” Students act out this movement and record a subtraction equation (6 – 2 = 4). Later in the session, students work with the “Number Lines” student book page, using two colors to represent hops and creating both addition and subtraction equations from their actions. They also discuss how “the moves on the number line are inverse movements that ‘undo’ each other.” The session ends with a visualization activity where students imagine the frog hopping, resting, and returning, reinforcing their understanding of movement, number relationships, and mathematical modeling. Math Practices in Action states, “Drawing connections between the frog jumps and the number line helps students see how math can be used to represent familiar situations. Over time, students develop flexibility using the number line to model a given situation. They also grasp how to use the number line to perform addition and subtraction operations.”

• In Grade 2, Unit 7, Measurement, Fractions & Multidigit Computation, Module 3, Session 3, Problems & Investigations, students model a contextual comparison problem involving two sets of beads. After rereading the vignette “Boxing Up the Beads,” the teacher asks students, “How many more orange beads than green beads do they have?” Students refer to the image of packages containing “316 orange beads” and “135 green beads” as they solve the problem using a variety of representations, including base ten pieces, open number lines, and equations. One student explains, “First, I made 316 and 135 with the base ten number pieces. Then I figured out which ones matched each other… so there are 181 more orange beads than green beads.” The class records this thinking with visual models and the equation “316 – 135 = 181.” Other students solve the problem as an unknown addend, represented by the equation “135 + 181 = 316.” The teacher encourages students to discuss their strategies, asking, “Why can this problem be solved using addition or subtraction?” and “Did some strategies seem more efficient than others?” Students then select and solve another real-world problem from the “Know & Wonder” chart, while sharing and comparing their modeling approaches with peers. Math Practices in Action states, “Students are likely to use visual models such as base ten number pieces or open number lines to model this problem. The teacher’s use of equations to represent students’ thinking can help them make connections to more abstract representations.”

The materials reviewed for Bridges in Mathematics Kindergarten through Grade 2 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.

Students across the K–2 grade band engage with MP5 throughout the year. It is explicitly identified for teachers in the Teacher Guide, including in the Concepts, Skills & Practices chart, and intentionally developed through the Warm-Up, Problems & Investigations, and Work Places sections of specific sessions. MP5 is also addressed in the Number Corner Teacher’s Guide.

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.

According to the Teacher Guide, “Bridges materials support students in several critical mathematical practices, including using appropriate tools strategically, attending to precision, and looking for and making use of structure. For example, the Word Resource Cards and MLC’s free Math Vocabulary app help students attend to precision with their mathematical language, which in turn enables them to construct viable arguments and critique the reasoning of others.” For example:

• In Kindergarten, Unit 7, Measurement & Teen Numbers, Module 1, Session 2, Problems & Investigations, students explore weight comparison using real-world context and hands-on tools. The teacher begins by rereading a page from “Springtime Adventures”, where Amara wonders, “Do you think the three small rocks Amara is holding are heavier or lighter than Min’s large rock?” Students are prompted to share their ideas by giving a thumbs-up or thumbs-down and then think-pair-share how they might figure out which rocks are heavier. The class is introduced to a bag of real rocks, similar to those in the story. Students are invited to lift the bag and search the classroom for an object they believe is about the same weight. One student notes, “The block is heavier than the rocks,” while another adds, “The pan balance is tipped down on the block’s side. That means it’s heavier.” These interactions are supported by the use of a pan balance, allowing students to test and observe their predictions. Students sort their objects on a class graph labeled with heavier, lighter, or the same weight as the bag of rocks. They then transfer this data into personal bar graphs in their student books and record numerical values. Students then draw one item that is heavier, one that is lighter, and one that is the same weight as the rocks. The teacher supports tool use by modeling, prompting students with questions like, “How does the object compare?” and ensuring all students engage with the balance scale. Students are encouraged to choose tools, interpret their results, and represent findings using visuals and numbers. Math Practices in Action states, “One of the first steps toward using appropriate tools strategically is learning to use a variety of tools correctly and carefully. First, students learn to use many tools, such as measuring instruments, manipulatives, and technological devices. This will enable them to select which tool, if any, is most appropriate for a given task.”

• In Grade 1, Unit 1, Numbers All Around Us, Module 2, Session 1, Problems & Investigations, students construct and explore their own number racks to deepen their understanding of quantity and structure. Rather than providing pre-assembled tools, students are asked to build their own number racks because “doing it themselves helps them understand and trust the structure of the number rack.” Each student receives materials to build their rack: two chenille stems, ten red beads, ten white beads, a number rack board, and tape. Students thread beads onto the stems, placing “five white beads followed by five red beads” on each. Once assembled, they secure the stems to the board with tape, ensuring “the red beads are on the left and the white beads are on the right.” The teacher then invites students to experiment with the tool and “come up with a discovery to share,” first in pairs, then as a class. Students practice sliding beads to represent numbers, starting from a “starting position when all the beads are pushed to the right.” Using a demonstration rack, the teacher shows five beads and asks students to explain how they know it’s five. Strategies vary: “I counted the beads like this: 1, 2, 3,” one student says, while another explains, “Three is 2 less than 5, so I pushed all the red beads but 2.” The game “Show Me” follows, where students hear a number and display it using beads. The teacher notes which students count one by one versus those who move groups efficiently. The class practices showing several numbers (5, 3, 10, 9, 6, 4, 7, 8), often related to landmark numbers like 5 or 10. Students are encouraged to use increasingly efficient strategies, such as showing “10 in one or two pushes” and transitioning between numbers “without returning the beads to the starting position.” Math Practices in Action states, “Constructing their own number racks gives students a strong sense of ownership. They also develop an understanding of the structure of this tool, which they will use frequently throughout the year.”

• In Grade 2, Unit 6, Geometry, Module 4, Session 4, Problems & Investigations, students revisit a problem where three characters each request their sandwich to be cut into fourths in different ways, such as into squares, rectangles, or triangles. As the students reflect on the print original, they are prompted to consider Kylie’s concern, “My fourths look smaller than Gillian’s!” and Gillian’s claim that, “My fourths are the biggest!” The teacher introduces identical paper squares and demonstrates the three different folding methods: folding “in half horizontally, then vertically” to create square fourths, folding twice in the same direction to make rectangular fourths, and folding diagonally twice to form triangular fourths. Students then “prove whether or not the three different fourths are equal in size.” Students work in groups of three using paper squares, grid paper, colored tiles, and other tools such as rulers, pattern blocks, or geoboards. They brainstorm strategies, including folding and overlaying shapes, tracing onto grid paper to “count the number of square units,” and cutting and rearranging pieces. One student suggests, “You could trace the fourths on the grid paper and count the squares and triangles to see whether it comes out the same.” After exploring, each group creates a poster with their conclusion and evidence, such as drawing, measuring, or tiling. One student explains, “You have to explain how you know it’s true. You write about what you did or draw a picture.” While many students quickly confirm that the square and rectangular fourths are the same size using colored tiles, comparing the triangular fourth is more complex. Students work to determine equivalence through cutting, rearranging, or tracing. For example, a group notes that “we tried covering each of the fourths with tiles. It takes 4 to cover the square and 4 to cover the rectangle… but we can only fit 2 tiles on the triangle.” Through hands-on exploration and collaborative reasoning, students engage in a meaningful investigation of equal shares and geometric reasoning using multiple tools and strategies. Math Practices in Action states, “In this investigation into fractions, the teacher gives students the option to use a variety of tools. To make an informed choice, students must compare the advantages and limitations of various tools and come to a conclusion about which tool or tools make the most sense for them to use for this specific problem.”

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

Students across the K–2 grade band engage with MP6 throughout the year. It is explicitly identified for teachers in the Teacher Guide, including in the Concepts, Skills & Practices chart, and intentionally developed through the Warm-Up, Problems & Investigations, Home Connections and Work Places sections of specific sessions. MP6 is also addressed in the Number Corner Teacher Guide.

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. Sessions 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.

According to the Teacher Guide, “Bridges materials support students in several critical mathematical practices, including using appropriate tools strategically, attending to precision, and looking for and making use of structure. For example, the Word Resource Cards and MLC’s free Math Vocabulary app help students attend to precision with their mathematical language, which in turn enables them to construct viable arguments and critique the reasoning of others.” For example:

• In Kindergarten, Unit 1, Numbers to 5 & 10, Module 4, Session 1, Problems & Investigations, students explore the concept of patterns by creating and identifying them using animal sounds and movements. The lesson begins with a class discussion prompted by the question, “What do you know about patterns?” Teachers revoice student comments to reinforce the idea that “patterns follow a rule.” After recalling examples from the “Calendar Grid” and classroom environment, students are introduced to the day’s objective: creating patterns using sound. Students generate a list of animals and develop corresponding sounds and motions, such as “a hiss and a wiggle for a snake” or “a bark and a tail wag for a dog.” They are divided into groups and take turns performing their assigned sounds and motions in a sequence directed by the teacher. Volunteers form a pattern line, and classmates are asked to identify what comes next. One student responds, “Snake, dog, snake, dog … snake! I knew it was a snake because snake comes after dog and the dog was the last one.” Students then discuss whether inserting a different animal, like a lion, would fit the pattern. A student explains, “It goes dog, snake, dog, snake, not dog, lion.” This leads to a class consensus on the pattern rule: “It always repeats dog, then snake.” The activity concludes with students reflecting on their work and identifying the rule behind their pattern, reinforcing that patterns repeat according to a consistent rule. Math Practices in Action states, “The class has an opportunity to attend to precision by extending a repeating pattern. Young students are very invested in keeping the pattern going, which they can only do through precision.”

• In Grade 1, Unit 8, Changes, Changes, Module 3, Session 1, Problems & Investigations, students explore how physical changes affect the motion of paper by designing and testing their own paper gliders. The activity begins with students brainstorming ways to change a piece of paper. Suggestions include, “You could fold it up and make it small,” or “You could color the whole thing red or blue.” Students then consider how to change its location, such as moving it to their desk or “put[ting] it in [their] backpack.” The teacher shares a story about students who saw a bird “gliding overhead without flapping its wings” and were inspired to create paper gliders to introduce the concept of gliding. Students express interest in making their own, and the teacher models how to fold one step by step, including creating a curve in the paper by sliding it over a table edge. After observing how the glider flies, students are encouraged to suggest improvements, such as adjusting the curve to help it fly straighter. As students test and revise their designs, the teacher reminds them, “It might take some practice to get the glider to fly the way they want it to,” and encourages collaboration and perseverance. The activity provides opportunities for students to use precise vocabulary, follow and model sequential steps, and refine their work based on observed results. Math Practices in Action states, “Students must attend to precision in folding their gliders. Doing so not only draws upon their understanding of halves, but produces gliders that travel farther.”

• In Grade 2, Unit 2, Setting Foundations for Place Value & Measurement, Module 1, Session 3, Problems & Investigations, students estimate, count, and compare beans to practice working with numbers within 100. The lesson begins with students scooping what they believe is 50 beans from a container and counting their scoop by placing 10 beans into each portion cup. Teachers prompt them to explain their reasoning by asking how many cups are needed to hold 50 beans. One student shares, “With 10 in each cup, that’s 60, and then there are some extra ones,” while another explains, “If you had exactly 5 cups and no extras, that would be 50 because 10, 20, 30, 40, 50.” Students then complete a structured student book page by counting their beans by 10s and 1s and determining how many more or fewer they have than 50. The teacher encourages different strategies, such as counting on from 50 or counting back from their total to help support their thinking. For example, a student says, “I started at 50 and kept on counting to 63. It’s 13.” Students then combine their counts with a partner, determine whether they have more or fewer than 100, and calculate the difference. Teachers prompt discussion with questions like, “If I scooped 48 beans, how many would my partner have had to scoop for us to get a total of exactly 100?” Students engage in precise counting, use benchmarks like 10s and 100s, and explain their strategies clearly in this problem. Math Practices in Action states, “Students are especially motivated to attend to precision when they are trying to match a target number of beans. When counting with accuracy and efficiency by creating groups of 10 and more, they are deepening their place value understanding and developing skills related to computation with greater numbers.”

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

Students across the K–2 grade band engage with MP7 throughout the year. It is explicitly identified for teachers in the Teacher Guide, including in the Concepts, Skills & Practices chart, and intentionally developed through the Warm-Up, Problems & Investigations, Home Connections and Work Places sections of specific sessions. MP7 is also addressed in the Number Corner Teacher Guide.

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 by prompting students to attend to and describe patterns they notice. Sessions 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.

According to the Teacher Guide, “Bridges materials support students in several critical mathematical practices, including using appropriate tools strategically, attending to precision, and looking for and making use of structure. For example, the Word Resource Cards and MLC’s free Math Vocabulary app help students attend to precision with their mathematical language, which in turn enables them to construct viable arguments and critique the reasoning of others.” For example:

• In Kindergarten, Unit 2, Numbers to 10, Module 4, Session 2, Problems & Investigations, students explore and describe patterns by constructing a class quilt from individual blocks. After reviewing Amara’s quilt and discussing its shape, students are asked, “Do you think we could use our quilt squares to make a square or rectangular quilt? How could we do that?” As blocks are added, students predict and extend the pattern, responding to prompts like, “What would come next? How do you know?” Students identify and describe the structure of the quilt, noting repeated elements and spatial relationships. For example, one student shares, “It goes squares, butterfly, squares, butterfly!” while another observes, “The next row has a butterfly first!” As more blocks are added, the class discusses the repeating pattern, or “pattern core,” and considers rules such as alternating rows that begin with different blocks. The lesson continues with a class discussion where students repeat and build on their peers’ ideas. Observations include, “the patchwork blocks go in diagonal lines” and “there are 49 blocks altogether, 7 across and 7 up and down.” These reflect students’ developing ability to recognize structure and use repeated reasoning to describe and extend patterns. Math Practices in Action states, “Many students’ observations are likely to reflect what they notice about the regularity and patterns in the quilt. Some students will describe a repeating pattern, while others will focus on quantities, for example by describing the ways in which the number of butterflies changes in each row, column, or diagonal.”

• In Grade 1, Unit 7, One Hundred & Beyond, Module 2, Session 1, Problems & Investigations, students explore the concept of structure by identifying and extending patterns in a nature path made of connected cubes. The activity begins with students examining an image of a sidewalk under construction and considering a classmate’s observation that “some of the sections are dark and some are not.” They are prompted to ask, “I wonder whether the dark and light sections make a pattern,” which allows students to notice repeated structure. As they analyze the image, students recognize that “it goes 9 light sections and then 1 dark section each time” and connect this to counting by tens, stating, “I think it’s counting by 10s” and “the dark sections are like the decade numbers.” Teachers guide students to use ordinal numbers to describe section positions and help reinforce the idea of structure within the sequence. Students then build their own paths using cubes, mark every tenth cube with a sticker, and explain how this structure helps them count more efficiently. One student shares, “The stickers help because you can count by 10s instead of having to count all the cubes,” and another confirms, “You can count the stickers by 10s instead of counting each cube.” Students use structure to solve problems, explain their reasoning, and compare quantities. Math Practices in Action states, “Marking every tenth cube creates a structure that supports counting greater numbers efficiently. It allows students to count the cubes by 10s and then 1s.”

• In Grade 2, Unit 5, Place Value to 1,000, Module 2, Session 1, Problems & Investigations, students explore coin values by identifying structure within visual representations of coins and decomposing amounts in multiple ways. The session begins with students viewing brief flashes of images, such as a 5-frame filled with pennies, and recalling what they saw. After identifying “5 pennies” or “5¢,” students are asked what single coin is equivalent. One student responds, “1 nickel,” and the teacher confirms the equivalence by placing a nickel next to the image. The session continues with a 10-penny image, where students note, “I saw two groups of 5 pennies. That’s the same thing because 5 and 5 is 10.” This leads to the generalization that “2 nickels are worth the same as 1 dime.” Students are then prompted to build different combinations using money value pieces to match target amounts (e.g., 10¢, 20¢, 25¢), reinforcing the idea that numbers can be decomposed and recomposed in multiple valid ways. When determining how to make 25¢, students share combinations such as “5 nickels,” “2 dimes and a nickel,” and “10 plus 10 is 20, and then 5 more is 25.” The teacher supports students in articulating the relationships and structures by encouraging them to describe how they saw the amounts and how they built their solutions. Math Practices in Action states, “The number frames used in this session help students look for and make use of structure in ways that not only deepen their understanding of money but also help them use groups of 5, 10, and 25 as landmarks. These skills will be useful as students continue to add and subtract multidigit numbers.”

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

Students across the K–2 grade band engage with MP8 throughout the year. It is explicitly identified for teachers in the Teacher Guide, including in the Concepts, Skills & Practices chart, and intentionally developed through the Warm-Up, Problems & Investigations, and Work Places sections of specific sessions. MP8 is also addressed in the Number Corner Teacher Guide.

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, recognize patterns, and develop 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.

According to the Teacher Guide, “Bridges materials support students in several critical mathematical practices, including using appropriate tools strategically, attending to precision, and looking for and making use of structure. For example, the Word Resource Cards and MLC’s free Math Vocabulary app help students attend to precision with their mathematical language, which in turn enables them to construct viable arguments and critique the reasoning of others.” For example:

• In Kindergarten, Unit 3, Bikes & Bugs: Double, Add & Subtract, Module 2, Session 3, Problems & Investigations, students engage with the book “Munch, Crunch, What a Lunch!” to build an understanding of repeated reasoning by tracking how the number of insects grows on each page. As the story unfolds, students use number racks to model the growing groups and recognize patterns in the addition of one insect at a time. When asked, “How many more insects are on this page than the last page?” students respond with observations like, “There’s 1 more,” and, “One and 1 more is 2. That’s a double!” The teacher records each step as an equation, and by the end of the book, students are prompted to look at the entire sequence and reflect: “What do you notice? What do you wonder?” One student notes, “It’s counting up. Each number is 1 more than the number before. There’s always a 1.” This recognition of repeated structure helps students generalize the idea of adding 1 and lays the groundwork for understanding number patterns. The session closes with a comparison between this book and Butterfly Countdown, reinforcing the inverse relationship between counting up and counting down. As one student explains, “The first book shows taking away 1 and the second book shows adding 1.” Students notice repeated calculations, make generalizations, and reason about the structure of addition and subtraction through these experiences. Math Practices in Action states, “By adding 1 repeatedly, students come to see that each time you add 1, the result is the next number in the counting sequence. This is an early form of looking for and expressing regularity in repeated reasoning.”

• In Grade 1, Unit 8, Changes, Changes, Module 2, Session 1, Warm-Up, students engage in the warm-up “Fewer & More with Cubes” to build conceptual understanding of comparing quantities and begin to generalize patterns through repeated reasoning. Students work with partners to build cube trains based on teacher prompts, such as “Build a train with more cubes than mine” or “Build a train with fewer cubes than mine.” As students create and compare different trains, they begin to develop a sense of how length and color structure relate to quantity. For example, when shown a train of 20 cubes made from two 10-trains of different colors, students must construct a train with “fewer cubes,” prompting them to reason about number relationships repeatedly across multiple examples. Teachers help students formalize this structure by asking, “What does a train with more cubes than mine look like?” and students respond, “It’s longer than yours.” When a train of 12 cubes is broken into loose pieces, the teacher asks, “What would a set with more cubes than this look like?” and students recognize that a “bigger pile” indicates a larger quantity. This repeated work with visual and physical models helps students make generalizations about more and fewer, reinforcing consistent structure in number comparison and laying a foundation for efficient reasoning. Math Practices in Action states, “When students think about more in the context of a longer cube train or a bigger pile and fewer in the context of a shorter train or a smaller pile, they begin to generalize an understanding of more as a greater quantity and fewer as a lesser quantity of items.”

• In Grade 2, Unit 5, Place Value to 1,000, Module 3, Session 5, Unit 5 Assessment & Jump-a-Hundred: Introducing Work Place 5D, Problems & Investigations, students use repeated reasoning to navigate a number line and analyze distance from a target number. Students choose a starting number and then roll a die and flip a coin to determine how many jumps of 100 to make and in which direction. As students repeatedly add or subtract 100s, they begin to recognize patterns in how the digits change and how far numbers are from 500. Teachers prompt this reasoning by asking, “Which digits change after you make a series of jumps? Which digits stay the same?” and “How can we determine a winner?” Students track their movements and they justify and compare distances, saying things like, “We landed on 310. It’s 90 up to 400, and then 100 more to 500. We’re only 190 away from 500, but you’re 250 away.” These repeated calculations help students develop efficient strategies, such as combining smaller jumps or visualizing larger ones on the number line. As they analyze their decisions and outcomes in relation to the benchmark of 500, students begin to generalize, evaluate the reasonableness of their answers, and reflect on patterns in the base-10 structure. Math Practices in Action states, “Each turn in Jump-a-Hundred requires the player to jump forward or backward by a multiple of 100. This deepens students’ understanding of place value by highlighting the idea that adding or subtracting 100s affects the digit in hundreds place but does not affect the digits in the tens place or ones place.”