3rd-5th Grade - Gateway 2

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

Multiple conceptual understanding problems are embedded throughout the grade levels in Math Forums 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 Grade 3, Unit 5, Multiplication, Division & Area, Module 2, Session 2, students develop conceptual understanding of finding whole number quotients by interpreting division problems and representing them visually. As students engage in the Math Forum discussion, they focus on how the interpretation of division varies based on the given information. The teacher prompts the class to explore how the visual model for representing and solving division problems changes depending on the context. Teacher: “Andre, could you share with us how you solved problem 2?” Andre: “I made a picture with 5 places to put the cards.” Kyra: “Why did you do that?” Andre: “The problem said that 5 cousins were playing the game, so I made a place for each cousin. Then I started drawing cards - like 1 for you, 1 for you, 1 for you - and I kept going like that until I got up to 35.” Teacher: “Where do you see your answer?” Andre: “The answer is 7. Each cousin got 7 cards.” Teacher: “Nora, could you share with us how you solved problem 3?” Nora: “Yeah, I drew a picture of each set of cards.” Alana: “What do you mean?” Nora: “I drew sets of cards that each had 5 cards until there were 35 cards. There ended up being 7 groups.” Brett: “How did you know to do that?” Nora: “The problem said that everyone got 5 cards. I knew how many cards each person got, but not how many people there were. That is 5 cards for 7 people.” Teacher: “Where do you see your answer?” Nora: “The answer is 7. It’s how many groups of 5 I had to make to get to 35.” Teacher: “Can anyone talk about how this problem was similar to and different from problem 2? Take a moment and reread it if you need to.” Anton: “It was similar because it was still 35 cards. It was different because in problem 2 we knew how many groups there were, but with problem 3 we knew how many were in each group.” Once students have shared their strategies, the teacher invites them to observe all the strategies again and make additional connections. Students are asked: “How many groups?” and “How many in each group?” to help them distinguish what they know and what they are trying to find. (3.OA.2)

• In Grade 4, Unit 2, Multidigit Multiplication & Early Division, Module 3, Session 1, Problems & Investigations, students demonstrate conceptual understanding of multi-digit multiplication problems using the ratio table strategy. The lesson begins with a discussion about Riley’s and Raymond’s ratio tables, helping students understand how the ratio table represents multiplication and division. For example, Ocie explains, “I knew that the alligator traveled 22 feet each second because I looked at 220 to 242 and that’s 22.” This shows how the difference between multiples represents the multiplier, a key concept in division and multiplication. After students work on problems 1b and 1c, they share their strategies for solving the problems. Ayisha states, “I started with 198 and subtracted 22 to get 176. That’s 8 seconds. I kept doing that until I got to 5 seconds, which is 110,” demonstrating her understanding of using subtraction for division. Rosalia adds, “I remembered that 5s facts are half 10 facts. Since 10 times 22 is 220, then 5 times 22 is half of that, 110.” This highlights how students connect multiplication facts to solve division problems efficiently. The session concludes with students completing problems in the Student Book, reinforcing their understanding of multiplication and division in real-world contexts. For example, one problem asks, “Maggie has 15 chickens. Each chicken lays 24 eggs a month. How many eggs in all do Maggie’s chickens lay in a month?” Students are encouraged to estimate and justify their strategies, reinforcing their conceptual understanding of multiplication and division. (4.NBT.5)

• In Grade 5, Unit 5, Multiplying & Dividing Fractions, Module 3, Session 2, students develop conceptual understanding of multiplying two fractions through a Math Forum. The session involves a discussion where students share their work and explanations for solving fraction multiplication problems using visual models. Teacher: “Invite the student you selected for the set context to share their work. Encourage other students to ask questions and make connections.” Misha shares: “I wanted to make an array that could show both fifths and halves, so I chose a 4-by-5 array. One-fifth of the shirts have unicorns, so I circled one column in blue to show that. Half of the unicorn shirts also had a rainbow. Half of the column I circled in blue is two of the squares in the grid, which I marked in red.” Teacher: “So, is 2 the answer?” Misha: “Well, it’s \frac{2}{20}. Two out of 20 of the t-shirts have both a unicorn and a rainbow.” Kiki: “I got \frac{1}{20} of the t-shirts as an answer, which is equivalent to \frac{2}{20}.” The discussion highlights the flexibility of multiplication: Misha explains, “It said that \frac{1}{2} of the unicorn shirts had rainbows, too. The problem said that \frac{1}{5} of the t-shirts that had unicorns, so it seemed like I was finding half of a fifth and not a fifth of a half, right?” Joelynn adds: “Since it’s multiplication, you could do it in any order.” Teacher: “What makes this a multiplication problem?” Students: “I think it’s about finding a part of something.” The class then moves on to an area context problem, with Raven explaining: “I knew right away which model I was going to pick. I used the grids with thirds and fourths, since the dimensions of the fabric were \frac{1}{3} yard and \frac{3}{4} yard. When I shaded in the area, it was 3 of the little rectangles. Each is \frac{1}{12}, so the answer is \frac{3}{12}.” Pete asks: “What does the \frac{3}{12} mean in your problem?” Raven: “It’s \frac{3}{12} of \frac{3}{12} square yards, or /frac{1}{4} square yards.” Teacher: “And what makes this a multiplication problem?” Raven: “Because it’s about area. Just like a rectangle that’s 2 feet by 3 feet is 6 square feet because you multiply the dimensions to get the area. It works with fractions, too.” Finally, students solve problems from the Student Book, modeling fraction multiplication with pictures. Teacher: “Circle the picture that best represents each problem and solve the problem.” Problem 1: “\frac{4}{7}\times\frac{3}{4}=” Problem 2: “\frac{2}{3}\times\frac{1}{4}=” Students demonstrate conceptual understanding as they apply multiplication strategies and choose the correct visual model for fraction multiplication. (5.NF.4)

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 meet expectations for providing intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters. Examples include:

• In Grade 3, Unit 2, Introduction to Multiplication, Module 3, Sessions 3 and 4, students develop procedural skills to solve multiplications within 100, and then they demonstrate those procedure skills and fluency independently on a page in their student book. In Session 3, Number String, students practice the doubling strategy to help solve multiplication problems. The string starts with 2\times6=12, and then progresses to 4\times6=24, and then 8\times6=48. The materials state,“The focus of this number string is using doubling with known multiplication facts to solve other problems. Open by gathering students in your discussion area. Tell students that they will participate in a number string and introduce the context: Some visitors are coming to Ms. Walker’s third grade class to see them perform a play they have learned. The students want to set out some chairs for the visitors. Share the first problem: Students set out 2 rows of 6 chairs. How many chairs did they set out? Record 2 rows of 6 chairs on the whiteboard or chart paper. Ask students to solve the problem mentally and give a thumbs-up when they have an answer. When you see most students with thumbs up, ask two or three volunteers to explain how they figured it out. Represent students’ thinking on the copies of the 100-frame, using arrays. As students share their strategies for the problems in the number string, ask them to reflect on the shared thinking when a strategy aligns with the goals of the number string. For example: Why does it make sense to ____? Does anyone have a question for ____? How did you use doubling to solve these problems? What did you decide to double, and why?  How did you use each of the factors to solve the problem? Where do you see those factors in this representation? What connections do you notice among the strategies shared? Deliver the rest of the problems shown in the following table. Problems and answers are provided in the left column of this table for your convenience. When you present the problems to the students, do not include the products.” Students demonstrate procedural skills and fluency independently solving multiplication problems on a student book page. Some of the problems from the page are shown below. Session 4, Student Book states, “1b. Solve the following problems: 3\times5=___, 5\times2=___, 7\times5=___” (3.OA.7)

• In Grade 4, Unit 4, Addition, Subtraction, & Measurement, Module 3, Session 1, Subtraction Checkpoint, students demonstrate procedural skill and fluency with multi-digit subtraction problems by completing a subtraction checkpoint. The materials state, “1. Subtract. Use the strategy that seems most efficient to you. Show your thinking. 1,282-1,117=___, 349-198=___, 2. Solve this problem using the standard algorithm. 847-568=___.” (4.NBT.4)

• In Grade 5, Unit 4, Multiplying & Dividing Whole Numbers & Decimals, Module 1, Session 2, Number String, students develop procedural skills as they apply a strategy called the “Half 10 Strategy” to solve multi-digit multiplication problems involving decimal numbers. The number string begins with two-digit whole numbers and progresses to multiplying a whole number by a decimal.  The materials state, “Open the session by letting students know that today they will partici- pate in another multiplication and division number string, and then have time to visit Work Places. Deliver the number string. Pose each problem one at a time by writing the problem on the board. Give students time to find the products using mental math. Solicit and record all answers to a given problem, and then invite one or two students to share how they solved the problem. When possible, model students’ strategies on ratio tables at the board or display. Emphasize the connections between 5 and 15 groups of a given number to 10 groups of that number. As explanations of strategies are shared, press students to consider whether they are multiplying by groups of 15 or groups of 18. Wrap up by asking students to use the think-pair-share routine to explain how \times 10 problems can be used to solve \times5 (or other) problems, including decimal numbers. Have students share their thoughts and together generate a class summary. Have students write today’s date in their math journals and title it Using the Half 10 Strategy Number String.” (5.NBT.5)

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 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 Math Forums, 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 Grade 3, Unit 6, Geometry, Module 1, Session 1, Student Book, students apply their understanding of multiplication and division to solve real-world word problems about arrays and groups. The task states, “3. On one page of the stamp book, Robin has arranged her stamps in a 7-by-6 array. How many stamps are on this page? 4. On one page of the stamp book, Cody has 54 stamps. The stamps are organized in 6 groups. How many stamps are in each group?” (3.OA.3)

• In Grade 4, Unit 6, Multiplication & Division, Data & Fractions, Module 3, Session 2, Home Connections, students connect division with area and perimeter to solve real-world measurement problems. The task states, “Use what you know about area and perimeter to solve each problem. Show your thinking. Label your answers with the correct units. 1. The area of a rectangle is 306 square centimeters. One dimension is 6 centimeters. What is the other dimension? 2. The area of a rectangle is 612 square centimeters. One dimension is 6 centimeters. What is the other dimension? 3. The perimeter of a rectangle is 1,270 centimeters. One dimension is 6 centimeters. What is the other dimension?” (4.OA.3)

• In Grade 5, Unit 6, Graphing, Geometry, & Volume, Module 4, Session 1, Problems & Investigations, students solve real-world problems involving the area of flags by applying multiplication of fractions and mixed numbers. The lesson begins with prompts such as, “Ask the class to think about problem 1 … Ask students what the unit of measure square feet tells them about what they are finding. Confirm that they are finding the area of the banner.” Students estimate and justify possible areas, then examine a labeled sketch of the flag. They compare the sketch to prior work with 2-digit by 2-digit multiplication and discuss how dividing the rectangle into regions supports problem solving. Finally, students write and solve a multiplication equation for each region, combine partial products, and determine the area of the flag. (5.NF.4, 5.NF.6)

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

Multiple aspects of rigor are engaged simultaneously across the materials to develop students' mathematical understanding of individual 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 Grade 3, Unit 5, Multiplication, Division & Area, Module 3, Session 3, Problems & Investigations, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they represent division within 100 while playing Array Builder. Players spin two spinners, multiply the numbers to generate a product, and then build as many arrays as possible for that product. The teacher models by stating, “I spun a 4 and a 6. What is the area of my first array?” and students respond, “24!” Students explain that the array “has to be a 4\times6 array, but you can make it face either way.” As the class plays, students build arrays with linear units and colored tiles, sketch them, record the dimensions, and calculate points by adding the areas or multiplying the area of one array by the number of arrays built. Students then play in pairs, summarizing directions with partners and applying strategies for calculating final scores. (3.OA.1, 3.OA.7)

• In Grade 4, Unit 6, Multiplication & Division, Data & Fractions, Module 4, Session 3, Student Book, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they interpret and compare situations involving equal groups with fractions and decimals. For example, students answer, “Ryan spins a dime and a quarter. Elizabeth spins \frac{1}{10} and \frac{1}{100} . Who has the greater sum? How do you know?” and “Michael’s sum for his spins is \frac{47}{100}. Jana’s sum is \frac{4}{10}. Who has the greater sum? How do you know?” Students also solve, “Robyn has completely shaded the first decimal unit frame on her record sheet. She has shaded \frac{3}{4} of her second decimal unit frame. What final spin would completely fill her second frame?” (4.NF.2, 4.NF.3)

• In Grade 5, Unit 2, Adding & Subtracting Fractions, Module 1, Session 4, Home Connections, students develop all three aspects of rigor, conceptual understanding, procedural skill and fluency, and application as they solve multi-step word problems involving fraction addition and subtraction. Students respond to prompts such as, “Abby and Mariceli are preparing for a dance performance. On Monday, they practiced for \frac{2}{3} of an hour. On Tuesday, they practiced for \frac{5}{6} of an hour. How long did they practice on Monday and Tuesday together?” and “On Wednesday, Abby and Mariceli could not practice together, so they practiced separately. Abby practiced for \frac{11}{12} of an hour and Mariceli practiced for \frac{2}{3} of an hour. How long did they practice on Wednesday?” Students are directed to “Show your thinking for each problem,” using representations and equations to justify solutions. (5.NF.2)

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

Students across the 3–5 grade band engage with MP1 throughout the year. It is explicitly identified for teachers in the Teacher’s Guide, including in the Concepts, Skills & Practices chart, and intentionally developed through the Problems & Investigations, Assessments, Number Strings, Daily Practice, Math Forums, 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 Grade 3, Unit 3, Multidigit Addition & Subtraction, Module 1, Session 5, Problems & Investigations, students analyze and apply a variety of strategies to solve two-digit addition problems with understanding and flexibility. The session begins with students examining an Addition Strategies chart that features methods like Lucy’s Way, Travis’s Way, and Emma’s Way, each showing different approaches for solving 34+17. Students observe strategies such as place value splitting, using friendly numbers, and compensation. After a moment of silent reflection, students turn to a partner to discuss their observations. They then engage in a class discussion to summarize and name the strategies. As students explain their reasoning, they begin to articulate mathematical ideas in precise ways such as, Andre says, “I think Travis’s way is called add a friendly number and compensate. He added too much then jumped back to make up for it,” while Kiara contributes a new strategy by suggesting, “You can subtract 3 from 34 and add that 3 to 17. That would make the new problem 31+20.” The teacher adds these strategies to the class chart, reinforcing the value of student-generated methods. Students then choose two out of three new problem situations to solve with a partner using the strategies discussed or ones they invent. They estimate using rounding to assess the reasonableness of their answers and use tools such as base ten blocks to model their thinking. Math Practices in Action states, “Students get to choose the strategies and other tools they use to solve these problem situations. This provides them with opportunities to make sense of and identify entry points to the problem’s structure and numbers. It is important not to over-scaffold. Instead, allow students to pursue their own sense-making and pathways.”

• In Grade 4, Unit 1, Multiplicative Thinking, Module 3, Session 1, Problems & Investigations, students explore factors and develop strategies through both direct instruction and gameplay. The session begins with students revisiting factor posters and discussing prior learning about prime and composite numbers. They focus on identifying all the factor pairs of 80, using what they know about arrays and multiplication. Students take time to reflect on whether they have identified all the factor pairs of a number, then discuss their reasoning with a partner to verify their thinking and identify any missing pairs. This leads to a class discussion where students explain their thinking and justify whether or not they believe the list is complete. Students are then introduced to the game Products Four in a Row, which allows them to apply and refine their multiplication strategies. They work in pairs to strategically choose numbers and compute products with the goal of getting four in a row. As they play, students record equations, consider multiple paths to win, and revise their thinking based on their opponent’s moves. The teacher supports strategy sharing and encourages reflection by prompting students to discuss which factors they chose and why. The Math Practices in Action section explains, “Games of strategy are, in some sense, extended problems. When students spend time developing strategies for winning these games, they are becoming accustomed to making sense of problems and persevering in solving them. In the case of Products Four in a Row, their efforts also contribute to multiplication and division fact fluency.”

• In Grade 5, Unit 3, Place Value & Decimals, Module 3, Session 3, Problems & Investigations, students apply their understanding of decimal operations and metric measurement to solve multi-step, context-rich problems. The session begins with students reading and interpreting real-world situations, such as comparing snowball throw distances, determining total snowfall amounts, and calculating remaining snow after melting. Students are encouraged to show all of their work, estimate for reasonableness, and use tools such as base ten models and the Equivalent Measures student book page to support their thinking. They begin by solving independently, using strategies they choose, including diagrams, equations, and conversion tools. The problems require students to convert between metric units and solve with decimals, such as: “Greta threw her snowball 23.15 meters. Her teacher threw his 6.51 meters. How much farther did Greta throw her snowball than her teacher?” and “How many centimeters did Greta throw her snowball?” Students make sense of multi-step problems and persevere to find and justify their solutions. Students are then asked to prepare for an upcoming Math Forum by refining their written work and ensuring their reasoning is clear and well-communicated. The teacher collects work to select strategies for sharing and discussion in the next session. The Math Practices in Action section explains, “Students often work with partners or groups. In anticipation of sharing during the Math Forum in the next session, students are given an opportunity to make sense of problems on their own first. Knowing they might share their work and will see others’ solutions can motivate students to persevere through challenges.”

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 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 3–5 grade band engage with MP2 throughout the year. It is explicitly identified for teachers in the Teacher’s Guide, including in the Concepts, Skills & Practices chart, and intentionally developed through the Number Strings, Math Forums, Problems & Investigations, Work Places, and Assessments 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 Grade 3, Unit 7, Extending Multiplication & Fractions, Module 4, Session 1, Problems & Investigations, students play Racing Fractions, a partner activity designed to build understanding of fractional values and number line placement. Players take turns drawing two number cards and forming a fraction using the lesser number as the numerator and the greater number as the denominator, ensuring the result is less than or equal to 1. They then move a game marker along corresponding fraction number lines (halves, thirds, fourths, sixths, eighths) to match the value. The teacher models how to generate fractions, interpret their placement on a number line, and discusses the reasoning behind game rules (e.g., avoiding improper fractions). Students explain their reasoning, such as identifying equivalent fractions like 3/3 = 1, and make connections to previously learned content. During gameplay, players collect markers that reach 1, and the winner is determined by the number of collected markers after two rounds through the deck or when time expires. Teachers observe and differentiate support as needed, while students engage in reasoning about symbolic representation and quantitative relationships. Math Practices in Action states, “When they arrange the cards, students reason about fractions abstractly, and when they use the number lines, they reason about fractions quantitatively. Having access to models like the number line, while also beginning to think in more abstract ways, helps students develop a strong understanding of fractions.”

• In Grade 4, Unit 4, Addition, Subtraction & Measurement, Module 1, Session 1, Problems & Investigations, students use the “Tiny Squares” page, a scaled-down version of a 10000 grid, to build understanding of place value, number relationships, and rounding. The lesson begins with students estimating the total number of squares, then identifying and labeling benchmark numbers such as 10, 100, and 1000 with teacher support. Students locate specific numbers such as 5328, 3582, 4166, and 6614 on the grid and explain their reasoning, making connections between digits and their place values. They compare numbers using comparison symbols such as less than, greater than, and equal to, and participate in a class discussion to round numbers to the nearest ten, hundred, and thousand. Visual models and peer explanations help students make sense of quantities and the meaning behind each digit, supporting reasoning about numbers. Math Practices in Action states, “Students reason abstractly and quantitatively when they connect the tiny squares to written numerals that represent particular quantities. Doing so helps them build a sense of numbers of this magnitude so they can estimate and compute with understanding.”

• In Grade 5, Unit 2, Adding & Subtracting Fractions, Module 1, Session 4, Problems & Investigations, students use “Clock Fractions” tools to explore fractional parts of an hour and build fluency with equivalent fractions. They begin by identifying fractions like \frac{1}{2}, \frac{1}{4}, and \frac{1}{3} of an hour and expressing those amounts in minutes. Students work individually or with partners to find and record equivalent fractions using visual models, including those greater than one whole. Students then solve problems involving the addition and subtraction of fractions using both the “Clock Fractions” tools and written equations. They justify their answers using equivalent fractions and visual reasoning, including interpreting sums like \frac{1}{3}+\frac{1}{4} as \frac{7}{12} or \frac{35}{60}. The teacher facilitates discussion around the relationships between different representations. Students are then introduced to a game called “Clock Fractions”, where they spin to generate two fractions, write an equation, and fill in a clock model. Players use visual and symbolic reasoning to decompose fractions and complete three full clocks. The session ends with student reflection and discussion of strategies, deepening their understanding of fraction equivalence and addition. Math Practices in Action states, “Playing Work Places with the whole class is a valuable time for teachers to formatively assess their students. Teachers should observe how students use the clock model during the game, then use that information to decide which pairs to work with during future Work Place times.”

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 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 3–5 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 Number Strings, Math Forums, Problems & Investigations, Work Places, Daily Practice, Home Connections, and Assessments 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 Grade 3, Unit 3, Multidigit Addition & Subtraction, Module 2, Session 4, Math Forum, students participate in a Math Forum to share and analyze their strategies for solving multi-digit subtraction problems from the Estimate & Compare print original. The teacher begins by revisiting two problems, 381–142 and 465–298, and prompts students to reflect on the estimates they made during the previous session. Students share how they approached estimation, with one noting, “We thought about rounding each number to the nearest hundred, but then we thought the nearest ten might get us closer to the actual answer, so we used 380-140.” The discussion moves into evaluating the reasonableness of answers based on estimates. When an estimate of 300 differed significantly from an actual result of 239, a student explained, “We thought about how far each of the numbers were from the estimates and decided that they were both pretty different so it probably made sense that our answer was not that close.” Students then present and critique strategies such as using open number lines, base-ten models, and the constant difference strategy. One student justifies her approach by stating, “I made 298 into 300, so I knew that I needed to make 465 into 467. They will have the same difference.” Throughout the activity, students are encouraged to ask questions, compare methods, and reflect on efficiency and accuracy, guided by prompts such as “Which strategies are the clearest or seem the most efficient to you?” and “What kinds of subtraction problems are best to solve with a constant difference strategy?” Math Practices in Action states, “By this point in the year, students have taken part in several math forums. There is value in reminding them to be attentive and respectful when speaking and listening. Students should formulate questions to understand their classmates’ strategies and solution paths. These behaviors reinforce and contribute to a productive learning environment.”

• In Grade 4, Unit 1, Multiplicative Thinking, Module 2, Session 2, Problems & Investigations, students engage in a gallery walk to analyze and compare posters representing factor pairs for different numbers. The teacher begins the activity by prompting students to examine a poster of rectangles made with 18 tiles, asking questions such as “What are the dimensions of the rectangles?” and “Where do you see factor pairs?” to support student reasoning about multiplication and factors. Students then circulate around the room, observing classmates’ posters, recording observations in their journals, and identifying patterns such as why “some posters have many rectangles while others have only a few.”After the gallery walk, students work in small groups to complete the “Thinking About Factors” student page, using their posters to support their thinking and engaging in collaborative discussion. The activity culminates in a whole-class conversation about prime and composite numbers. One group initially claims, “All of the composite numbers are even,” prompting a peer to question the classification of 15. Through discussion, students clarify their understanding: “It’s prime if it only has one factor pair,” and “It is composite if it has more than one factor pair.” When another student adds, “You can only make one rectangle for prime numbers because they only have two factors,” the group revises their original idea. The teacher affirms this learning process, stating, “You went from having one idea about prime and composite numbers to understanding more about these two types of numbers.” Throughout this activity, students construct viable arguments, respond to peer reasoning, and revise their thinking. Math Practices in Action states, “When you give students the responsibility of defending and amending their own assertions, while also considering the thinking of others, you help them construct viable arguments and critique the reasoning of others. These discussions dramatically deepen students’ understanding of the mathematical concepts in question.”

• In Grade 5, Unit 5, Multiplying & Dividing Fractions, Module 1, Session 4, Problems & Investigations, students engage in a mathematical discussion about strategies used while playing the Target 1 Fractions game, where they create and evaluate expressions involving whole numbers and fractions to generate products close to 1. During a class debrief, the teacher presents a set of numbers, 4, 3, 2, 3, and 6, and asks students how they would select and arrange cards to meet the goal. One student explains, “I’d choose 2 for my whole number, and then maybe 4 and 6 for my fraction,” forming the expression 2\times\frac{4}{6}. Another student responds with a different approach, saying, “I like to use the commutative property and put the fraction first... \frac{2}{6} times 3... \frac{1}{3} of 3 is 1.” Students explain their reasoning, consider alternative strategies, and discuss which combinations are most effective. The teacher facilitates the conversation by posing follow-up questions and encouraging generalizations. Math Practices in Action states, “Students construct viable arguments and critique the reasoning of others when they share and compare their strategies for creating expressions and finding their products. Doing so invites students to make generalizations about multiplication with whole numbers and fractions.”

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 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 3–5 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 Number Strings, Math Forums, Problems & Investigations, Work Places, and Assessments 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 Grade 3, Unit 5, Multiplication, Division & Area, Module 4, Session 2, Problems & Investigations, students apply their knowledge of area to model classroom objects using square units. The teacher begins by revisiting questions from the previous session, such as “What do you think the term square units means?” and “Why might you want to know the area of something?” Students share ideas with a partner and then with the class. The teacher introduces the day’s task by explaining that students will use paper squares “in the same way that they used the tiles in the previous session” to estimate and measure the area of classroom objects. During a guided example, students predict how many paper squares it would take to cover a piece of copy paper. After placing squares on the page, one student comments, “If you think about cutting the extra off and putting it on the bottom, it’s about 6,” and another adds, “Yeah, I’d say the paper is about 6 squares big.” The teacher concludes, “So we can say that the approximate area of this paper is 6 square units, correct?” Students then work in pairs using the “Finding Areas Large & Small” activity pages to estimate area, measure using paper squares, and record the difference between their estimate and the measured amount. They are encouraged to reflect on their methods, compare strategies, and discuss how they handled irregular coverage. The lesson closes with a discussion about the similarities and differences between this task and their previous experience measuring area. Math Practices in Action states, “The questions you pose elicit ideas about several mathematical concepts. They might push students to justify their answers with fractional reasoning depending on how much of the surface remains uncovered. Such questions might also press students to justify their estimates mathematically using what they know about area concepts.”

• In Grade 4, Unit 6, Multiplication & Division, Data & Fractions, Module 4, Session 1, Problems & Investigations, students explore how to represent and interpret real-world data using measurement and line plots. The lesson begins with a class discussion about reading habits, where students share “how much they have read in the past few weeks,” including the number of books, pages, or time spent reading. The teacher then introduces a visual, “Ben’s & Tristan’s Books,” which shows two stacks of books. Students are prompted to compare the stacks: “Tristan and Ben both read the same number of books. Yet it looks like Ben might have read more than Tristan.” Students discuss the significance of book thickness and how it might represent how much someone reads. The class then considers a situation in which all students “recorded the height of their stack of books” for one month and are asked to determine “how tall their stack of books would be if they continued their collection for the entire year.” Students identify known and unknown information, suggesting strategies like addition and multiplication. One student proposes: “I would multiply the height of his stack of books by 12, because there are 12 months in a year.” Students then work with a table of measurements and transfer the data to a line plot. They determine that “each X on the line plot represents a student,” and label the number line in fourths. When plotting, one student clarifies, “The 3 means that the stack of books Ben read is 3 inches tall.” Students analyze the line plot to draw conclusions: “Which book stack height is most common?” and “Is there one book stack height that is the least common?” They justify their responses using the visual data, supporting their mathematical reasoning with real-world context. Math Practices in Action states, “This session and the next involve students in a mathematical modeling task. Students represent an everyday situation with mathematics, allowing them to make mathematical connections to their world while thinking critically and working collaboratively.”

• In Grade 5, Unit 4, Multiplying & Dividing Whole Numbers & Decimals, Module 2, Session 2, Problems & Investigations, students use ratio tables to model the cost and revenue of making and selling bracelets for a school fundraiser. The context is introduced by noting, “Sahra’s friend Bryan wants to make bracelets to help raise money for the Community Service Club.” Students begin by locating the “Bryan’s Bracelets” page and reviewing directions. They are prompted to solve two problems using ratio tables: one where “Bryan's beaded bracelets cost $2.25 each to make,” and another where he sells them for $3.50 each. Students are guided to fill in values for different quantities of bracelets and identify mathematical relationships, such as “finding the cost for two bracelets by doubling the amount it costs to make one.” They use a think-pair-share routine to develop multiplication and division equations. For example, students might generate equations like “2.25\times4=9.00” or “11.24\div5=2.25.” Teachers are encouraged to annotate ratio tables using student input and label operations with arrows to show multiplicative reasoning. As students work, they are encouraged to look for “patterns in the ratio table” and explain the order they use to complete the table. The task allows for multiple representations and invites students to solve the problems using different strategies, supporting the goal of helping students “talk out relationships between the numbers.” Math Practices in Action states, “An important facet of the mathematical modeling process is the ability to use the model to make predictions about the situation. In contexts like this one, the ratio table is a powerful model to use for recording known information, and using it to predict future sales or fundraising goals.”

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

Students across the 3–5 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 Number Strings, Math Forums, Problems & Investigations, Home Connections, and Assessments sections of specific sessions. MP5 is also addressed in the Number Corner Teacher Guide.

Across the grades, students participate in tasks that support key components of MP5. 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 Grade 3, Unit 4, Measurement & Fractions, Module 1, Session 6, Problem & Investigation, students engage in a “Measurement Scavenger Hunt” where they select and use appropriate tools to measure mass, volume, or length. The lesson begins with a class discussion about scavenger hunts, followed by a review of measurement concepts. Students are introduced to two spinners: one for the type of measurement (mass, volume, length) and one for the quantity (100, 250, 500, or 750). They record their results on a Measurement Scavenger Hunt sheet. Students explore the term millimeter as they start the activity. The teacher explains, “There are 10 millimeters in every centimeter,” and prompts students to calculate that “there are 1,000 millimeters in a meter” to build the context for selecting appropriate tools. Students work in pairs using classroom tools such as rulers, measuring tapes, pan balances, or measuring cups. After spinning both spinners, each pair must “mold, pour, or search for an object that matches the number they spun,” using clay for mass, water for volume, and classroom objects for length. They then measure the actual amount and compare it to their target, recording whether it was greater than or less than the quantity they spun. The teacher models the process with volunteers and asks guiding questions like, “How did you decide how much water to pour?” and “How can you use referents to help you make decisions?” Students are prompted to explain their thinking and tool selection throughout the activity. Math Practices in Action states, “Based on the units given, students need to decide whether to measure liquid volume, mass, or length. They must then choose the correct tool — be that a ruler, a pan balance, or a measuring cup — to accurately measure the items.”

• In Grade 4, Unit 2, Multidigit Multiplication & Early Division, Module 3, Session 1, Problems & Investigations, students determine the area of two chicken coops by multiplying side lengths. The first coop measures 12 feet by 13 feet, and the second is 6 feet by 26 feet. The task begins with a real-world context: “Maggie wants to make sure the animals on her farm have enough space.” Students are prompted to estimate the area, explain their thinking, and show their work. Students use problem-solving strategies and are encouraged to choose from a variety of tools, such as base ten grid paper, base ten linear and number pieces, sketches, or equations. Teachers prompt students to use “any strategy that works best for them” and to record their thinking visually or numerically. As they work, students explore models such as arrays, ratio tables, and partial products. A goal is for students to “seek efficient strategies” and try new approaches shared by peers. During the whole-class discussion, selected students share their estimates and solutions using different tools and methods. The class reflects on strategies, makes connections across models, and records one they would like to try next. Math Practices in Action states, “Students can use a variety of tools to help them solve multiplication problems. Leaving the choice up to students helps them learn to make appropriate decisions — in choosing not only the tool but also how they use it. While many tools are appropriate, they might not be very helpful if students don’t use them strategically.”

• In Grade 5, Unit 6, Graphing, Geometry & Volume, Module 2, Session 1, Problems & Investigations, students use geoboards, triangle classification charts, and structured partner work to strategically select and use tools to describe, construct, and classify triangles. During the partner activity, one student creates a triangle on a geoboard and describes it so their partner can recreate the same triangle without seeing it. Students use mathematical language, such as ordered pairs, angle types, and side lengths, to communicate precisely. For example, one student suggests, “Like over 2 and up 3 is one vertex,” while others describe whether a triangle has right, acute, or obtuse angles, or if it is scalene. Students then use classification charts to verify the angle types and side lengths of their constructed triangles, applying definitions such as, “If a triangle has three different length sides, then it must be a scalene triangle.” When a student thinks a triangle has a right angle, the class tests it by fitting a square pattern block into the corner. Throughout the lesson, students are encouraged to reflect on the tools and strategies that best support their reasoning and communication. Math Practices in Action states, “Students use appropriate tools strategically when they create triangles on the geoboards. The pegs on a geoboard are all spaced evenly. Because of this, students can be certain about the angles and relationships among the sides of each triangle. This information is essential for accurately describing and classifying the triangles.”

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 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 3–5 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 Number Strings, Math Forums, Problems & Investigations, Work Places and Assessments 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 Grade 3, Unit 6, Geometry, Module 2, Session 5, Problem & Investigation, students write and test riddles about quadrilaterals using precise mathematical language and reasoning. The class begins by recalling how they previously used clues to eliminate shapes, such as one student noting, “Trapezoid A is the only shape in our cards that has exactly 2 right angles.” Students discuss which attributes would make good starting clues and which might give away the answer too quickly. One student suggests starting with “not symmetrical,” since it eliminates some options without making the riddle too obvious. The class constructs a four-clue riddle, ending with a definitive characteristic to ensure only one shape matches all clues. They write: “My quadrilateral is not symmetrical. My quadrilateral has no congruent sides. My quadrilateral is not concave. (It doesn’t have an angle that goes inward). My quadrilateral has 2 right angles.” Students then test the riddle by sorting their shape cards and revising clues if needed. They’re guided to use accurate vocabulary like “congruent,” “symmetrical,” and “concave,” and are reminded to “check that mathematical terms are used correctly.” Math Practices in Action states, “Students benefit by working with a partner during this activity because they must collaborate to organize their clues. It also helps them practice using geometry vocabulary to communicate accurately with each other.”

• In Grade 4, Unit 3, Fractions & Decimals, Module 4, Session 3, Problems & Investigations, students practice placing fractions and decimals on a number line, with a focus on precision and mathematical reasoning. Students begin by drawing a horizontal number line in their math journals, labeling it from 0 to 2. They then estimate and record the location of 1, considering the placement of other values like 0.50. Students use number tags to represent values such as fractions and decimals, and collaboratively place them on a class number line. As they work, they justify their placements using mathematical language. Daria explains, “I chose 0.50. I tried to put it exactly halfway between 0 and 1 because 0.50 is the same as \frac{1}{2}.” Throughout the lesson, students adjust placements based on peer input and are encouraged to “draw tick marks… to show a more precise estimated location.” Teachers observe for accuracy and guide discussions around corrections when necessary. Math Practices in Action states, “Students attend to precision when they place numbers on the number line. Both the number tags on the class number line and the cut pieces in students’ math journals are easy to move around and adjust. Attending to precision with a flexible model invites students to carefully consider the relationships among these fractions and decimals.”

• In Grade 5, Unit 8, Solar Design, Module 1, Session 2, Problems & Investigations, students investigate how color affects the absorption of solar energy by conducting an experiment with thermometers wrapped in black and white paper. They begin by discussing the terms absorption and reflection and considering whether dark or light colors absorb more energy. To make predictions, students reflect on real-world experiences, such as choosing what color shirt to wear on a sunny day. Working in pairs, students set up thermometers following illustrated directions, then gather materials and conduct the experiment either outside or indoors using a high-wattage light source. They align the thermometers carefully to avoid shadows and record temperatures at one-minute intervals, ensuring accurate and consistent data collection. After the experiment, students review their results and document at least three observations in their journals. Questions such as, “Which thermometer absorbed more energy? How do you know?” and “What was the temperature change for the black paper? The white paper?” support them in interpreting the data precisely. Students use the term slope informally to describe how temperature changes over time in their graphs. With teacher guidance, they label axes, select a scale, and plot data points, then discuss trends they observe. For example, they are asked, “How does the slope of the line for each thermometer change from one minute to the next?” and “What does that tell you?” Students then create their own graphs using colored pencils and answer reflection questions. Math Practices in Action states, “It’s possible, but unlikely, that all the thermometers will have the same beginning temperature. Factors like the time it took to bring the assembly outdoors, the temperature of their hands while they carried the assemblies and how long students take to get ready to record data will all have an impact; there is also likely to be some slight variation between thermometers. Encourage students to write down the exact temperature they read on each thermometer, even if it is not consistent from one thermometer to the next. Recording data correctly is an important part of scientific experimentation.”

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 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 3–5 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 Number Strings, Math Forums, Problems & Investigations, Daily Practice, Home Connections, Work Places, and Assessments 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 Grade 3, Unit 2, Introduction to Multiplication, Module 2, Session 2, Problem & Investigation, students build and analyze cube trains and multiples strips to explore the structure of multiplication. After reviewing prior work with multiples of 4, students are encouraged to connect the visual model to multiplication equations. Corey explains, “The cube trains help us to see the groups and then the multiples strip shows the multiplication written out as a group repeated so many times,” illustrating how students use concrete models to uncover multiplicative structure. Students construct cube trains and multiples strips for different factors in pairs. They notice how multiples align across strips as they start to compare and recognize that multiples of 2, 4, and 8 all line up at 8 and 16. Quinn states, “Two is half of 4 so when you have 10 groups of 2 and 10 groups of 4, it makes sense that the 2s strip would be half as long as the 4s strip.” These observations prompt students to generalize relationships between factors and products. Students also examine shared products across strips, such as 24, and connect expressions like 4\times6 and 6\times4, building their understanding of the commutative property. Teachers guide students to make sense of structure by posing questions such as, “How many groups of 4 are equal to 28? How do you know?” Math Practices in Action states, “This visual representation of the multiples of 1–10 provides students with an opportunity to notice equivalent expressions, common multiples, and properties of multiplication. They can see, for example, that ten 7s is equal to seven 10s, which illustrates the commutative property of multiplication.”

• In Grade 4, Unit 5, Geometry & Measurement, Module 1, Session 2, Problems & Investigations, students examine and compare sets of angles to uncover underlying structure and develop precise mathematical language. Students are asked to identify which angle doesn’t belong and justify their reasoning. Mario observes that one angle “didn’t belong because all the other angles look like they’re opening to the right or left, but it’s opening up,” while Maya states, “most of the angles opened really wide except one.” These initial observations prompt students to begin generalizing about angle size and orientation. The teacher introduces the formal terms acute, right, obtuse, and straight angles, and students are guided to refine their language and reasoning using accurate definitions. They record their thinking in journals, sketch examples, and use tools such as the corner of a paper or pattern blocks to test and verify angle types. In a subsequent task, students label interior angles in pattern blocks and justify their classifications with partners, using shared strategies and representations. Math Practices in Action states, “Students will make use of structure as they discern commonalities among most angles in each group. As students identify common characteristics in a group of angles, support their thinking by asking whether there’s an angle that doesn’t share that characteristic.”

• In Grade 5, Unit 1, Expressions, Equations & Volume, Module 2, Session 1, Number String, students engage in a number string routine to deepen their understanding of multiplication by recognizing and reasoning about mathematical structure. The teacher begins with 4\times3, prompting students to visualize the problem. One student describes a 4-by-3 array: “There are 4 rows and 3 columns... If you count the little squares, there are 12 of them.” This leads to a discussion of area as a product and sets the foundation for identifying patterns in subsequent problems. As the number string progresses, students use the structure of previous problems to solve new ones. When solving 4\times6, they connect it to the first problem: “I looked at the 4\times3 array, and thought that the 4\times6 array would be the same on the 4 side but twice as long on the 3 side.” Barry generalizes this relationship, saying, “Since it’s twice as wide, the area is twice as big,” and Luis adds, “We doubled one side, so now the bigger rectangle is twice as wide. So it has twice the area, and 2 times 12 is 24.” Students continue applying this reasoning to solve 8\times6 and 4\times12, while they look for and make use of structure. Throughout the routine, students are encouraged to “think like mathematicians” by noticing patterns, testing ideas with arrays, and justifying their reasoning. Students repeatedly identify and apply structure to deepen their multiplicative thinking. Math Practices in Action states, “Many of the number strings this year are designed to help students look for and express regularity in repeated reasoning. By repeatedly doubling and halving in this number string, students are able to make generalizations about the properties of multiplication and develop strategies that contribute to computational fluency.”

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 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 3–5 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 Number Strings, Problems & Investigations, Daily Practice, Home Connections, 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 Grade 3, Unit 2, Introduction to Multiplication, Module 3, Session 3, Number String, students participate in a number string activity where they build on known facts to develop efficient multiplication strategies. The sequence begins with 2\times6=12, which becomes a foundation for later problems. As the string progresses, students use repeated reasoning, doubling known products to find new ones, such as 4\times6=2\times(2\times6) and 8\times6=2\times2\times2\times6. Teachers prompt generalization by asking questions like, “How did you use doubling to solve these problems?” and “What connections do you notice among the strategies shared?” Students use relationships like 2\times7=14 to solve 3\times7 by adding one more group of 7, and to solve 6\times7 by doubling: 6\times7=2\times(3\times7). As they repeatedly apply these strategies, students begin to generalize multiplication patterns, justify their reasoning, and reflect on the structure of multiplication using both arrays and equations. Math Practices in Action states, “When students solve problems that involve repeating the same kind of reasoning, they recognize patterns that help them develop generalized methods for completing certain kinds of calculations. In this case, they begin to see how they can use their knowledge of multiplying by 2 to multiply by 4, 8, and 3.”

• In Grade 4, Unit 7, Reviewing & Extending Fractions, Decimals & Multidigit Multiplication, Module 1, Session 5, Problems & Investigations, students explore patterns in equivalent fractions by repeatedly multiplying the numerator and denominator of \frac{2}{3} by the same number. Using fraction bars, they begin by comparing visual models and reasoning about changes in the number of parts. As students move from \frac{2}{3} to \frac{4}{6}, \frac{6}{9}, and beyond, they recognize that while the number of parts increases, the shaded portion stays proportional. One student explains, “Fraction bar d should have 12 parts with 8 shaded,” reasoning that they’re multiplying both the numerator and denominator by 4. Students articulate a general rule: “\frac{2}{3} times \frac{4}{4}... equals \frac{8}{12},” and “all the fraction bars have the same portion shaded.” They then apply this reasoning to extend the pattern, discuss how “multiplying by \frac{2}{2} or \frac{3}{3} is like multiplying by 1,” and write equations to justify their thinking. Students come to understand how multiplying a fraction by versions of 1 (like \frac{2}{2} or \frac{4}{4}) preserves its value as they reflect on repeated reasoning. They use visual models and equations to analyze structure, describe relationships, and generalize the process for creating equivalent fractions. Math Practices in Action states, “It is through repetition, and the search for patterns and rules within that repetition, that students come to understand that any time they multiply the numerator and denominator of a given fraction by the same number, the result is an equivalent fraction.”

• In Grade 5, Unit 3, Place Value & Decimals, Module 1, Session 3, Problems & Investigations, students explore repeated reasoning with place value and operations to develop generalizations about multiplying and dividing by powers of ten. The session begins with a game called “I Have, You Need”, where students determine which number, when added to a given one, totals 100. For example, when the teacher says, “I have 65,” students respond, “You need 35,” explaining, “Five more is 70, then 30 more is 100, so 35.” This sets the stage for reasoning about number relationships and structure. Students solve problems from “The Corner Grocery”, using multiplication and division with 10 to model real-world scenarios. For example, to find the cost of 10 packages at $0.80 each, students say, “I doubled $0.80 to get $1.60, doubled again to get $3.20, and again to get $6.40... added to $1.60 for 2 and I got $8.00 for 10.” Another student thinks in cents: “80 cents times 10 is 800 cents, and 800 cents is $8.00.” These strategies allow students to notice and use structure to reason about repeated operations. Students then analyze place value patterns through predictions and calculator verification. When multiplying 184\times10, they reason that “ten times 184 means all of the digits... are now ten times greater, or one place value to the left.” Teachers prompt deeper understanding by asking, “Does the decimal point move or do the digits move?” One student responds, “It’s the digits that move because they change their place value position.” Students generalize place value rules and correct common misconceptions as they use repeated reasoning to think about how digits shift and values change when multiplied and divided by 10.  Math Practices in Action states, “Students look for and express regularity in repeated reasoning by looking for patterns when they multiply and divide whole and decimal numbers by 10. This draws upon their understanding of place value and promotes computational fluency with multidigit numbers.”