3rd-5th Grade - Gateway 1

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

The Assessment Guide provides an assessment map that identifies when and where each grade-level standard is assessed throughout the school year. The materials are designed to assess skills incrementally, following a typical developmental progression.

The Assessment Guide provides a comprehensive overview of the types and frequency of assessments at each grade level. Grades 3 through 5 include a variety of formative and summative assessments, including observational assessments, screeners, Mid-Unit Checkpoints, and End-of-Unit Assessments. There are additional assessments available on the Bridges Educator Site; however, these supplemental items were not considered in the evaluation of grade-level alignment for this indicator. Examples include:

• In Grade 3, Unit 4, Measurement & Fractions, Module 4, Session 4, Unit 4 Assessment, Problem 12 states, “Write the correct symbol >, =, or < to compare the fractions. Draw a sketch to show your thinking.” “a.\frac{6}{6}__1” “b. \frac{2}{3} ___ \frac{1}{2}” “c. \frac{2}{3} ___ \frac{4}{6}” “d. \frac{1}{3} ___ \frac{1}{8}” (4.NF.2)

• In Grade 4, Unit 3, Fractions and Decimals, Module 3, Session 4, Assessment, Fraction & Decimal Checkpoint, Problem 7 states, ”Mr. Jackson made several batches of homemade glue for a class project. He needed 14 cup of flour for each batch. He made 10 batches. How much flour did he use in all? Write a multiplication equation to solve this problem.” (4.NF.4b)

• In Grade 5, Unit 5, Multiplying and Dividing Fractions, Module 4, Session 6, Unit 5 Assessment, Problem 5 states, “Rose lives 2\frac{1}{4} miles from her cousin. One day, she rode her bike to her cousin's house. When she got \frac{2}{3}of the way there, her bike broke, and she had to walk the rest of the way. How far did she ride her bike before it broke? Show your thinking. Write an equation that shows the answer.” (5.NF.6)

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

Formative and summative assessments across Grades 3-5 include screeners, Checkpoints, work samples, and End-of-Unit Assessments. These tools measure student progress toward key lesson-level concepts and broader unit goals. They are designed to assess the full intent of grade-level content and practice standards and utilize a variety of item types, such as written responses, performance tasks, and observational work samples. Unit 8, which integrates mathematics and science, does not include formal assessments, but the materials provide guidance for collecting student work samples to monitor learning and understanding.

In Grades 3-5, each unit features a variety of assessment opportunities, including work samples and formal paper-and-pencil tasks. These typically include a screener at the beginning of the unit, one or more Mid-Unit Checkpoints, and a Unit-End Assessment. In addition to the embedded unit assessments, optional assessments are available on the Bridges Educator Site to support teachers in monitoring student growth throughout the year.

 Examples include:

• In Grade 3, Unit 5, Multiplication, Division & Area, Module 4, Session 6, Unit 5 Assessment, develops the full intent of 3.OA.3 (Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem). Problem 5, “Solve each problem. Show your thinking and write the answer. Then write two equations to match the problem. a. The pet store just got 60 new dog bowls. Devon is putting the bowls into stacks of 10. How many stacks can he make if he uses all 60 bowls? Devon can make ____ stacks. Multiplication Equation: ____. Division Equation:____. b. The pet store has 7 stacks of Frisbees for dogs. If there are 5 Frisbees in each stack, how many Frisbees is that in all? The pet store has ____ Frisbees for dogs. Multiplication Equation: ____. Addition Equation: ____.”

• In Grade 4, Unit 4, Addition, Subtraction & Measurement, Module 1, Session 7, and the Unit 4 Assessment, develop the full intent of 4.NBT.2 (Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.). Place Value and Addition Checkpoint, Problem 3 states, “Fill in the circles with <, >, or = to make true comparison statements.” Students compare 54,385 to 54,853; 10,059 to 10,015; and 407,221 to 470,008. Unit 4 Assessment, Problem 2, “Fill in the table.” “Base Ten Numeral,” “Number Name,” and “Expanded Form.” There are two rows. The first row contains “679,527” under the “Number Name” column, and students must complete the other two columns. The second row includes “four hundred seven thousand, two hundred thirty-five” in the “Number Name” column, and students must again complete the remaining columns.

In Grade 5, Unit 2, Adding & Subtracting Fractions, Module 4, Session 4, Unit 2 Assessment, develops the full intent of MP.3 (Construct viable arguments and critique the reasoning of others). Problem 10 states, “Tonya ran \frac{2}{3} of a mile at the high school track and then walked \frac{2}{6} of a mile. a. How far did Tonya go in all? Show your thinking. Include units in your answer. b. When Danny solved this problem, he got \frac{4}{9} of a mile for his answer. Is this a reasonable answer? Why or why not?”

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials provide students with extensive work with grade-level problems through the daily structure of Bridges sessions. Each session includes one or more of the following components: Warm-Up, Problems & Investigations, Work Places, and Assessments. A 60-minute instructional block is allocated for these activities, with an additional 20 minutes dedicated to Number Corner routines. Throughout the year, students engage with grade-level standards through scaffolded lessons, repeated practice in Work Places, and fluency development in Number Corner. The Problems & Investigations component emphasizes conceptual understanding and application, while Number Corner reinforces foundational skills. Examples of extensive work with grade-level problems to meet the full intent includes:

• In Grade 3, Unit 3, Multidigit Addition & Subtraction, Module 1, Session 1, and Unit 5, Multiplication, Division & Area, Module 1, Sessions 4 and 5, engages students with the full intent of 3.OA.3 (Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem). In Unit 3, Module 1, Session 1, Home Connections, students solve multiplication and division word problems. “Problem 5. Mai is planting flowers in the school garden. There are 4 flower boxes. In each one, Mai is going to plant 4 marigolds and 3 cornflowers. a. How many flowers will Mai plant in all? Show your thinking. b. Write an equation to represent this problem. Use the letter f to stand for the number of flowers Mai plants in all. Problem 6. CHALLENGE. Will is helping to plant vegetables in the school garden. He can plant 7 seeds in each planting box. He has 45 seeds in all. a How many planting boxes can he fill? Show your thinking. b How many seeds will Will have left over?” In Unit 5, Module 1, Session 4, Student Book, students continue to solve multiplication and division word problems. “Problem 1. Lisa, Imani, and Kahlil were picking flowers for their aunt. If they each picked 8 flowers, how many flowers did they pick in all? Show your thinking. Problem 2. Frank collected 18 beautiful shells for his 3 cousins. If he gave each cousin the same number of shells, how many shells did each cousin get? Show your thinking. Problem 3. CHALLENGE. Four friends were making cards to sell at the holiday sale. Each friend made 9 cards. They put all their cards together and then bundled them in groups of 6 cards to sell. How many bundles of 6 cards did they have to sell? Show your thinking.” In Unit 5, Module 1, Session 5, Student Book, students are given word problems and then 4 choices for each operation. They must pick the choice that represents the problem given. The materials state, “Pick the equation you could use to solve each problem. Then solve the problem. Problem 1. Ray’s cat had 6 kittens. His neighbor adopted 2 of them. How many kittens does Ray have left? Problem 2. Marsha was given 6 toys for her new cat. The toys came in packages of 2. How many packages of toys did Marsha receive? Problem 3. Carlos’s oldest cat is 6 years old. His youngest is 2 years old. How much older is his older cat?”

• In Grade 4, Unit 4, Addition, Subtraction & Measurement, Module 1, Session 5 and Module 2, Session 3 engages students with the full intent of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm). In Module 1, Session 5 Student Book, students solve multidigit addition problems using the standard algorithm. The materials state, “Problem 5. Solve two or more of these problems using the standard algorithm for addition. Record your thinking in the space. 567+274, 345+289+57, 409+356, 6,578+795” In Module 2, Session 3, Student Book, students solve multidigit subtraction problems using the standard algorithm. The materials state, “Problem 4. Choose at least two of these problems and solve them using the standard algorithm for subtraction: 2025-1984, 1987-1864, 2071-1993, 2005-1967.”

In Grade 5, Unit 4, Multiplication & Division Strategies, Module 3, Session 5, Home Connections, engage students with the full intent of 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm). Students are provided problems that have been started using the standard algorithm and are asked to finish the problem and/or supply the missing numbers to complete the algorithm. The materials state, “Problem 1. Maria is practicing using the standard algorithm for multiplication to solve problems. She knows she needs to multiply the ones first, but then she needs help. Help Maria finish these problems she started.” Students see the problems 38\times28 and 84\times37 written vertically to be solved using the algorithm. In Problem 2, they are given the problems 23\times11 and 15\times12, with some numbers missing in the algorithm. “Fill in the spaces to complete the problems.”

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

According to the Teacher’s Guide, “a Bridges classroom features a combination of whole-group, small-group, and independent problem-centered activities.” Daily instruction “takes the form of a 60-minute Bridges session and a 20-minute Number Corner workout.” 

The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade as included in the following grade-level breakdown: 

In Grade 3: 

• The approximate number of units devoted to major work of the grade (including assessments and related supporting work) is 8 out of 8, approximately 100%.

• The number of modules devoted to major work of the grade (including assessments and related supporting work) is 31 out of 32, approximately 97%. 

• The number of sessions devoted to major work of the grade (including assessments and related supporting work) is 140 out of 160, approximately 88%.

• Bridges sessions require 60 minutes per day, with a total of 140 sessions focused on the major work of the grade. This accounts for 8,400 minutes out of 9,600 total Bridges session minutes. Number Corner, delivered alongside Bridges each day, requires 20 minutes per day and is implemented over 170 instructional days. Of those, 151 days focus on the major work of the grade, contributing 3,020 minutes. Altogether, 11,420 of the program’s 13,000 total instructional minutes, approximately 88%, are devoted to the major work of the grade.

In Grade 4: 

• The approximate number of units devoted to major work of the grade (including assessments and related supporting work) is 7 out of 8, approximately 88%.

• The number of modules devoted to major work of the grade (including assessments and related supporting work) is 29 out of 32, approximately 91%. 

• The number of sessions devoted to major work of the grade (including assessments and related supporting work) is 137 out of 160, approximately 86%.

• Bridges sessions require 60 minutes per day, with a total of 137 sessions focused on the major work of the grade. This accounts for 8,220 minutes out of 9,600 total Bridges session minutes. Number Corner, delivered alongside Bridges each day, requires 20 minutes per day and is implemented over 170 instructional days. Of those, 121 days focus on the major work of the grade, contributing 2,420 minutes. Altogether, 10,640 of the program’s 13,000 total instructional minutes, approximately 82%, are devoted to the major work of the grade.

In Grade 5: 

• The approximate number of units devoted to major work of the grade (including assessments and related supporting work) is 8 out of 8, approximately 100%.

• The number of modules devoted to major work of the grade (including assessments and related supporting work) is 30 out of 32, approximately 94%. 

• The number of sessions devoted to major work of the grade (including assessments and related supporting work) is 146 out of 160, approximately 91%.

• Bridges sessions require 60 minutes per day, with a total of 146 sessions focused on the major work of the grade. This accounts for 8,760 minutes out of 9,600 total Bridges session minutes. Number Corner, delivered alongside Bridges each day, requires 20 minutes per day and is implemented over 170 instructional days. Of those, 118 days focus on the major work of the grade, contributing 2,360 minutes. Altogether, 11,120 of the program’s 13,000 total instructional minutes, approximately 86%, are devoted to the major work of the grade.

An instructional minute analysis is the most representative measure of the materials, as both the Bridges sessions and the Number Corner include major work, supporting work connected to major work, and embedded assessments throughout each unit. As a result, in Grade 3, approximately 88% of the instructional materials focus on major work of the grade. In Grade 4, approximately 82% of the instructional materials focus on major work of the grade. In Grade 5, approximately 86% of the instructional materials focus on major work of the grade.

The instructional materials for Bridges in Mathematics Grade 3 through Grade 5 meet expectations for including problems and activities that connect two or more clusters in a domain, or two or more domains in a grade. 

Connections among the major work of the grade are present throughout the materials where appropriate. These connections are listed for teachers in the Teachers Guide within each Module Concepts, Skills & Practices chart, and may appear in one or more phases of a typical lesson: Problem & Investigation, Number String, Math Forum, Work Places, Assessments, or Home Connections. 

In Grade 3, an example of a connection includes:

• Unit 2, Introduction to Multiplication, Module 3, Session 3 connects the major work of 3.OA.A (Represent and solve problems involving multiplication and division) to the major work of 3.OA.D (Solve problems involving the four operations, and identify and explain patterns in arithmetic). Students engage in a Number String activity where they use known multiplication facts and doubling strategies to solve new problems. For example, they start with 2\times6=12 and build to 4\times6=24 and 8\times6=48 by successively doubling. They also explore problems like 3\times7 and 6\times7, discussing how arrays and factor relationships support efficient computation. Numbers Strings Sample Summary states, “You can use facts you know to figure out other facts by doubling. If you know 3\times4 is 12, you can figure out 6\times4 by doubling 12 to get 24. Arrays help you see or record the strategy.” In the Problem & Investigation section, students examine a comic strip titled Cat Food Catastrophe depicting a humorous scenario at Mateo’s Pet Store. Students observe and discuss the image, sharing noticings like “There are 6 cans of cat food in each box,” and wondering things like “How many cans of cat food were there to start with?” They record these observations and generate mathematical questions based on them, then determine how many cans were knocked over.

In Grade 4, an example of a connection includes:

• Unit 7, Reviewing & Extending Fractions, Decimals & Multidigit Multiplication, Module 2, Session 2 connects the major work of 4.NF.A (Extend understanding of fraction equivalence and ordering) to the major work of 4.NF.C (Understand decimal notation for fractions, and compare decimal fractions). Students play a game called Put It on the Strip: Decimals, where they select decimal fraction labels and work collaboratively in teams to determine and justify where each label belongs on a decimal strip. Students then analyze the placements using a scoring guide to determine point values based on the location of their fractions. As they compare values such as \frac{48}{100} and \frac{6}{10}, they engage in discussions about equivalency and use strategies like rewriting fractions with common denominators. In Problems & Investigations, Scoring Guide, “Maya: ‘You can see that \frac{48}{100} is between those two places, so we get 3 points. Yeah, we know that because we can multiply \frac{4}{10} by \frac{10}{10}. That’s \frac{40}{100}. ’” Students then play a partner game called Scoot the Marker, where they flick a marker along their decimal strip and record its distance to score points based on accuracy. They measure these distances in hundredths and compare them to ranges defined in the scoring guide, further deepening their understanding of decimal magnitude. As part of gameplay modeling, the teacher encourages students to “engage in reading and reporting the distances as fractions of a meter (for example, forty-three hundredths meter, six-tenths meter, and so on).”

In Grade 5, an example of a connection includes:

• Unit 8, Solar Design, Module 2, Session 3 connects the major work of 5.NBT.B (Perform operations with multi-digit whole numbers and with decimals to hundredths) to the major work of 5.NF.B (Apply and extend previous understandings of multiplication and division to multiply and divide fractions). Students analyze a line graph comparing the heat retention of different model houses with various window placements and materials. They interpret which configurations heat up or cool down the fastest and why. Problems & Investigations state, “Which house heated up the most? (House D, the one with the large window toward the sun with water storage inside.) The fastest? (House A, the one with the large window toward the sun but no water storage.) The least? (House B, the one with the small window toward the sun, but house C, the one with a large window shaded by an awning, is a close second.” Students then apply this understanding to design their own model house windows. They calculate the total surface area of their walls and determine that their windows must total \frac{1}{8} of that area, 42 square inches. Students use strategies to find combinations of window dimensions to meet this requirement, such as “4.5\times3=13.5… There is 28.5 sq. in. left for more windows.” Students then sketch their window plans, cut out the openings in cardboard walls, and calculate how to evenly divide a plastic shower curtain to form the window coverings.

The instructional materials reviewed for Bridges in Mathematics Grades 3 through 5 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. 

The Teacher’s Guide includes an introduction that explains the connections to content standards. Prior and future standard connections are identified throughout the Teacher’s Guide, Unit Overview, and the "Skills & Concepts Across the Grade Levels" section, which states that it “provides a quick snapshot of the expectations for students’ learning during this unit, as well as information about how these skills are addressed in Bridges.” Each Unit Overview outlines the progression of standards related to the concept being taught and shows how they are addressed through Warm-Ups, Problems & Investigations, Home Connections, and Assessments. These standards are identified at the session or workplace level within each unit and module. The Skills & Concepts Across the Grade Levels section also indicates whether a skill is introduced, developing, proficient, reviewed and extended, or supported.

An example of a connection to future grades in Grade 3 includes:

• Unit 2, Introduction to Multiplication, Overview, connects the work of 3.OA.1 (Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each) and 3.OA.3 (Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem) to the foundational development of multiplication concepts in Grade 3, where students begin to model and interpret multiplication using equal groups, arrays, and contextual problems. Mathematical Background, Make Sense of Operations, Properties & Patterns states, “Multiplication and division contextual problems fall into four groups, three of which (see the following chart) are foci of instruction in Grades 3 and 4. (A fourth group, the combination problem structure, is taught alongside probability in the middle grades.) Unit 2 introduces multiplication with equal groups and arrays (gray cells in the following chart); these concepts resurface in other units and throughout Number Corner. Division as it relates to equal groups and arrays, and area as a measurement, are addressed in Units 5 and 7. Multiplicative comparison is introduced in Grade 3 via doubling, but is not targeted for proficiency until Grade 4.”

An example of a connection to prior knowledge in Grade 3 includes:

• Unit 3, Multidigit Addition & Subtraction, Overview, connects 3.NBT.2 (Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction) to the progression of place value understanding and operational fluency across Grades 2 and 3, where students build on their ability to unitize tens and apply strategies for adding within 100 to develop fluency and flexibility with addition and subtraction within 1,000 by unitizing hundreds and using place value strategies. Mathematical Background, Making Sense of Operations, Properties & Patterns state, “Important work of third grade includes fluently and flexibly adding and subtracting within 1,000. This goal builds upon understandings of place value and properties of addition, as well as addition and subtraction strategies developed in previous grades. Throughout Grade 2, students develop place value understanding by unitizing tens and learn to apply this understanding to adding within 100 and subtracting multiples of 10 from multiples of 10 in the range of 10–90. Third graders expand their place value understanding to unitize hundreds and develop strategies for adding and subtracting within 1,000.”

An example of a connection to future grades in Grade 4 includes:

• Unit 6, Multiplication & Division, Data & Fractions, Overview, connects the work of 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models) and 4.NBT.6 (Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models) to the conceptual and procedural foundations built in Grades 3 and extended into Grade 5, where students build fluency with multi-digit operations and apply these strategies to increasingly complex problem types, including those involving measurement and volume. Mathematical Background, Geometry, Measurement & Data states, “The same principles of measurement that guided students’ development of ideas around area are expanded to volume in fifth grade as they progress through similar stages of conceptual development within the third dimension. They build rectangular prisms to find the volume, initially counting individual cubes. Then they build the base of a rectangular prism and add layers, repeatedly adding the area of the base for each layer that follows to find the volume. By the end of grade 5, students use rulers to measure the three dimensions of a rectangular prism and learn to use a formula to find the volume, including finding missing dimensions. In fourth grade, however, the focus is on becoming proficient with concepts about area, but the same ideas about measurement both precede and follow grade 4 work.”

An example of a connection to prior knowledge in Grade 4 includes:

• Unit 1, Multiplicative Thinking, Overview, connects 4.OA.1 (Interpret a multiplication equation as a comparison, e.g., interpret 35=5\times7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations) to Grade 3 concepts such as understanding multiplication and division as inverse operations, applying properties of multiplication, and developing fluency with multiplication and division facts within 100. Mathematical Background, Concepts state,  “In preparation for multidigit multiplication and division, the mathematical focus of Unit 1 is a review and extension of skills and concepts introduced in grade 3: multiplication and division as inverse operations, properties of multiplication, and multiplication and division facts within 100. In grade 4, students use these relationships and their understanding of operations to expand the meaning of multiplication to include multiplicative comparison (‘times as many’ or ‘times as much’). Their understanding of multiplication is also deepened through number theory as they explore prime and composite numbers and the relationship between factors and multiples.”

An example of a connection to future grades in Grade 5 includes:

• Unit 1, Expressions, Equations & Volume, Overview, connects the work of 5.NBT.6 (Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models) to future grade-level concepts in Grades 6 and 7, where students formalize proportional reasoning through work with ratios, rates, and proportional relationships across mathematical and real-world contexts. Mathematical Background, Place Value Operations & Making Sense of Operations states, “Fifth graders explore a wide variety of proportional situations that build their capacity to think multiplicatively. Covariation is an essential idea underlying proportional reasoning, and in grade 6, students will focus more attention on reasoning about ratios and rates both within a particular context and between different contexts. In grade 7, they will analyze and use proportional relationships across mathematical and authentic contexts. For now, flexibly using multiplicative relationships to find products and quotients sets students up for success in later grades.”

An example of a connection to prior knowledge in Grade 5 includes:

• Unit 5, Multiplying & Dividing Fractions, Overview, connects 5.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction) to foundational work with unit fractions and number lines in Grade 4 and early Grade 5, where students developed an understanding of fractions as quantities and built conceptual knowledge of multiplication involving fractions and whole numbers. Mathematical Background, Fraction Operations state, ​”In this unit, fraction multiplication and division builds on work that began in fourth grade — multiplying unit fractions by whole numbers — as well as on the partitioning work done with number lines earlier in fifth grade during Unit 2. Prior to grade 5, fractions were viewed from a part-whole — or measurement — perspective, which helped students recognize a fraction as a countable quantity.”