3rd-5th Grade - Gateway 3

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Throughout the Teacher’s Guide, margin notes and sidebar annotations provide guidance on equity-based practices, effective mathematics teaching practices, and the Standards for Mathematical Practice. These notes highlight routines, call out strategies for supporting discourse-rich classrooms, and offer reminders to connect lessons to larger instructional goals. Each module and session includes an Overview with specific suggestions for engaging students in the mathematical content. For example:

• In Grades 3-5, Teacher Guide, Productive Practices states, “Throughout these materials, you will find sidebar notes designed to engage you in professional learning about equity-based practices, effective mathematics teaching practices, and the standards for mathematical practice. These three sets of practices address the development of capable doers, knowers, and sensemakers in different — yet inextricably linked — ways. When looked at together, these reflect our commitment to equitable and effective teaching practices and authentic learning experiences for students.” Math Teaching Practice states, “These activities are designed to address the three effective mathematics teaching practices critical to providing engaging and challenging mathematical experiences to students: establish mathematics goals to focus learning, implement tasks that promote reasoning and problem solving, and build procedural fluency from conceptual understanding. Math teaching practice sidebar notes call out explicit connections to these practices.” In Equity-Based Practice,the materials state, “Critical to establishing a community of learners based on mutual respect are learning opportunities that invite mathematical exploration and that encourage collaboration enhanced by students’ strengths. Watch for sidebar notes that identify how particular activities go deep with mathematics or leverage multiple mathematical competencies.” Finally, in Instructional Routine, the materials state, “Think-pair-share is one of the instructional routines featured throughout Bridges that focuses on classroom discourse. Throughout your Bridges materials, sidebars like this highlight instructional routines and their role in nurturing a discourse-rich classroom community and offering opportunities for mathematical sensemaking.”

• In Grade 3, Unit 4, Measurement & Fractions, Overview, teachers are provided with structured overviews and module guidance that help them support student engagement in measurement and fraction concepts. The Overview notes that “Unit 4 opens with a series of hands-on activities in which students estimate and measure mass, liquid volume, and length in metric units. During Module 2, they solve measurement-related problem situations, tell time to the minute, and explore elapsed time problems. The third module introduces students to fractions, using several different models to build, compare, and investigate the relationships among fractions.” The unit culminates with opportunities to connect these skills as “students measure lengths to fractions of an inch and display measurement data on line plots.”

• In Grade 5, Unit 1, Expressions, Equations & Volume, Overview, teachers receive detailed guidance on how to introduce and extend concepts related to expressions, equations, and volume through a progression of modules. The Overview explains, “In Unit 1, students use volume measurement to review and extend a host of concepts and skills related to multiplication. In Module 1, students investigate different ways to arrange 24 cubes into a rectangular prism, prompting a deep look at the associative and commutative properties of multiplication. In Module 2, students deepen their understanding of volume concepts as they construct multiple rectangular prisms, including composite rectangular prisms to represent one volume.” This guidance continues into later modules where “students develop multidigit multiplication strategies to solve authentic and mathematical problems, and revisit multiplication and division through the lens of the area model.” and “explore dividing 3-digit numbers by 2-digit numbers in the final module as they revisit the relationship between multiplication and division.”

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 meet expectations for containing explanations and examples of grade-level concepts and/or standards and how the concepts and/or standards align to other grade levels so that teachers can improve their own knowledge of the subject.

The beginning of each unit, module, and session provides an overview of the concepts covered within the materials as well as background knowledge for the teacher. Examples include:

• In Grade 3, Unit 3, Multidigit Addition & Subtraction, Unit 3 Overview, Mathematical Background, Number & Operations states, “In Unit 1 and the first two months of Number Corner, third graders began important work of the grade by exploring strategies based on place value and properties of addition for adding and subtracting 2-digit numbers. Unit 3 extends these strategies to numbers within 1,000. Unit 3 introduces rounding multidigit numbers to the nearest 10 or the nearest 100, a skill for which the open number line is a powerful sensemaking tool. Students’ rounding skills are applied to estimating sums and differences, and assessing the reasonableness of their results. They investigate two new strategies, give and take for addition and constant difference for subtraction, both of which involve changing both numbers in a problem to make the computation easier. In Grade 3, proficiency with standard algorithms is not expected. While the standard algorithm for addition is introduced, students are encouraged to mindfully choose strategies based on the numbers involved in the problem.”

• In Grade 4, Unit 4, Addition, Subtraction & Measurement, Module 2, Overview states, “Students build on what they learned about addition in Module 1 as the focus shifts to subtraction. Number strings and problem situations help students deepen their understanding of subtraction strategies, including find the difference, take away, and constant difference. Students learn the standard algorithm for subtraction and compare it to other strategies they have explored. They learn one new Work Place in the module: Roll & Subtract 1,000. The teacher collects a work sample in Session 4 that provides information about students’ understanding of and proficiency with strategies for solving multidigit subtraction problems. Students will also show what they have learned in Module 2 with a brief checkpoint assessment at the beginning of the next module.”

• In Grade 5, Unit 6, Graphing, Geometry & Volume, Module 3, Session 1, Summary states, “This session opens with a checkpoint on the geometry skills and concepts covered in the previous module. Then students review what they have learned about volume so far this year, preparing them to extend and refine their understandings of this topic. At the end of the session, the teacher introduces and assigns the Measurement & Multiplication Review Home Connection. Module 3 Learning Goals, Students learn about the standard formulas for volume of rectangular prisms. Students write expressions to represent rectangular prism dimensions and connect to two volume formulas. Students measure rectangular prisms and calculate volume in standard units. Students use formulas to calculate volume to solve problem situations. Students develop an understanding that two volume formulas are equivalent and determine which to use in a given situation.”

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 meet expectations for including a year-long scope and sequence with standard correlation information.

The Bridges Educator Site includes a year-long scope and sequence document that details the entire school year, “A Scope & Sequence document lists the concepts, topics, and standards covered in each module of the Bridges Third Edition units and in each month of Number Corner Third Edition. (The scope refers to the content and skills within the curriculum, and the sequence refers to the order in which those topics are covered.)”

At each grade level, the Teacher Guide contains a section titled Developing Concepts & Skills Over Time, which explains how content is sequenced to build understanding, “Bridges in Mathematics recognizes that students need extended experiences with concepts over time to develop deep mathematical understandings. The continual development of content throughout Bridges units and Number Corner workouts follows a careful and intentional progression of instruction and is a hallmark of the program.”

Additional supports include:

• Scope & Sequence charts, which show the broad content development across Bridges units and Number Corner months.

• Curriculum Concept Maps, which “provide a broad view of how the major work of the unit is addressed and developed over the course of the year” and highlight connections across units and Number Corner workouts.

• Skills & Concepts Across the Grade Levels charts, which “shows the skills and concepts addressed in each unit, indicates the level at which each is addressed in the unit, and shows where and how the skill is addressed elsewhere in the current and adjacent grade levels.”

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 meet expectations for explaining the program’s instructional approaches, identify research-based strategies, and explain the role of the standards. At each grade level, examples include:

• The materials explain the program’s instructional approaches. The Teacher Guide describes Bridges in Mathematics Third Edition as “a comprehensive curriculum…that equips teachers to fully address national and state standards in a rigorous, engaging, and accessible manner.” The materials emphasize developing deep conceptual understanding, procedural fluency, and problem-solving skills, with the goal of positioning students as “mathematical doers, knowers, and sensemakers.” In addition, the program highlights the importance of building a community of learners, noting that when “all classroom members respect one another, believe their ideas count, and are supported in exploring these ideas, instruction and learning are transformed.” Teachers are guided to model respect, create an environment that supports risk-taking, and use routines such as think-pair-share to promote active engagement, collaboration, and mathematical discourse.

• The materials include and reference research-based strategies. The program draws directly on equity-based practices from Aguirre et al. (2013), which include going deep with mathematics and affirming learners’ identities, as well as on effective teaching practices outlined in Principles to Actions (NCTM, 2014) and Catalyzing Change in Early Childhood and Elementary Mathematics (NCTM, 2020). The Teacher Guide explains that Bridges “draws on research and leading mathematics education frameworks to support the development of high-quality curriculum,” grounding the instructional design in practices shown to support student understanding, achievement, and identity.

• The materials include and reference the role of the standards. Bridges is designed to “fully address national and state standards” and consistently references the Standards for Mathematical Practice (NGA & CCSSO, 2010) as central to how students engage with mathematical content.

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

Each grade level includes a Materials List outlining the manipulatives, print resources, and quantities needed for the year. In addition, each module contains a Materials Preparation section, and every session provides a table listing the specific items required. Examples include:

• In Grade 3, Unit 7, Extending Multiplication & Fractions, Module 3, Session 2, Materials, Problems & Investigations states, “colored tiles (12); adding machine tape (see Preparation); measuring tape.”

• In Grade 4, Unit 3, Fractions & Decimals, Module 3, Session 4, Materials, Problems & Investigations states, “spinner overlays (half-class set); more or less (half-class set); base ten number pieces (class set).”

• In Grade 5, Unit 7, Division & Decimals, Module 3, Materials Preparation, Special Items states, “Consider making extra copies of the Decimal Unit Frame print original. Keep them on hand for student use and for display.”

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 meet expectations for providing consistent opportunities to determine student learning throughout the school year. The assessment system provides sufficient teacher guidance for evaluating student performance and determining instructional next steps.

In the Assessment Guides for each grade level, the Formal Assessments section describes assessments administered throughout the year, including Beginning-of-Year Screeners, Checkpoints, and Unit Assessments. These assessments provide information about students’ prerequisite skills for upcoming instruction and their understanding of current grade-level standards. Scoring guides accompany the assessments and include guidance for evaluating student work and determining instructional next steps. The guides also reference Activities for Individual or Small-Group Reengagement that teachers may use to support student learning. Examples include:

• In Grade 3, Assessment Guide, Unit 7, Extending Multiplication & Fractions, Assessment. The Assessment answer key provides sample student responses for shading and labeling fraction strips. It also notes possible variations in student work (e.g., showing only fractional marks rather than shading each piece), giving teachers guidance for interpreting responses.

• In Grade 4, Unit 5, Geometry & Measurement, Module 3, Session 2, Assessment, Geometry Checkpoint includes directions for administering the assessment, observing students as they work, and recording results. Teachers are advised to use Checkpoint results to form small groups, adjust partnerships, and select targeted Work Places for reengagement.

• In Grade 5, Unit 4, Multiplying & Dividing Whole Numbers & Decimals, Module 1, Session 1, Unit 4 Screener includes modeled strategies students may use during the assessment, such as previewing the test, marking easier or more challenging items, and monitoring time. These directions provide guidance for assessment administration.

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 meet expectations for providing strategies and support for students in special populations to work with grade-level content and meet or exceed grade-level standards, which support their regular and active participation in learning. 

The Teacher Guide explains that Work Places are designed to provide opportunities for assessment and differentiation, stating, “Each Work Place is accompanied by a guide that identifies the skills and concepts involved as well as suggestions for assessment and differentiation.” These guides provide strategies and supports intended to promote student engagement in mathematical tasks. Examples include:

• In Grade 3, Work Place Collection, Work Place Guides & Instructions, Work Place 4B Tic-Tac-Tock, Assessment & Differentiation, Multilingual learners states, “Play a round of the game with students, modeling what to do with the spinners and clock. Write down a few of the sums and multiplication facts and label them hours and minutes, respectively, so students have a visual reminder of what to do. Model setting the time and filling in the clocks. Provide sentence frames for students to use to explain their thinking. Sentence frames can be found on the Bridges Educator Site. Suggest partnerships that allow students to play in the language they are most comfortable speaking or that provide support in mathematical discourse, interaction, language development, or understanding the game's rules.”

• In Grade 4, Work Place Collection, Work Place Guides & Instructions, Work Place 3C Decimal Four Spins to Win, Assessment & Differentiation states, “If you see that…: Students are adding the two fractions they spin without writing the tenths fraction as hundredths, resulting in incorrect sums. Differentiate: Support - Have students use base ten number pieces to represent the results of each spin. By noting that each base ten strip representing a tenth is divided into 10 units, students can see that \frac{2}{10}=\frac{20}{100}. Example: Teacher: Can you use the base ten number pieces to show \frac{2}{10} and \frac{35}{100}? Student: I can use the strips, so 1, 2 for the \frac{2}{10}, and 10, 20, 30, and then 5 of the little squares. But I am not sure what to do. Now I have 5 tenths and 5 hundredths. Teacher: How many hundredths are there in all? Student:  It's 10, 20, 30, 40, 50, 55 — 55 hundredths in all. OK, I can add the fractions if both fractions are hundredths.”

• In Grade 5, Work Place Collection, Work Place Guides & Instructions, Work Place 1D Quotients Win, Assessment & Differentiation, Multilingual Learners states, “Provide sentence frames for students to use to explain their thinking. Sentence frames can be found on the Bridges Educator Site. Suggest partnerships that allow students to play in the language they are most comfortable speaking or provide support in understanding the game's rules, mathematical discourse, interaction, and language development.”

An additional program, Bridges Intervention, provides support and intervention through problem solving, visual models, and fluency development. The program is organized into nine volumes addressing major number concepts and skills across K–5 and includes small-group activities, practice pages, and partner games. All volumes are available to Bridges’ educators so they can provide targeted support as needed. Math specialists, interventionists, resource teachers, and support staff can also use these resources to give students the time and support necessary to make sense of mathematics.

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 meet expectations for regularly providing extensions and/or opportunities for advanced students to engage with grade-level mathematics at greater depth.

The Teacher Guide at each grade level introduces Concept Quests as “a supplemental resource designed for kindergarten through grade 5” that provides “opportunities to extend and apply their learning.” Tasks are designed with “multiple entry points so that all students, regardless of backgrounds or ability, have equal access.” Kindergarten includes Excursion tasks, and grades 1–5 include both Excursions and Adventures. Excursions ask students to apply current mathematics in novel or authentic contexts, while Adventures are “more challenging” and extend understanding by connecting to other grade-level content or contexts. Each grade level includes seven sets of these tasks, found in Units 2–8 on the Bridges’ Educator Site.

In addition to Concept Quests, challenge problems and activities are embedded within sessions and assessments, offering further opportunities for advanced students to work with mathematics at greater depth. Examples include:

• In Grade 3, Concept Quests, Unit 6, E6E, School Garden, students solve a problem involving two rectangular garden beds with a total perimeter of 80 meters. The task asks them to find two possible arrangements, calculate the total area for each, and “use pictures and equations to explain your thinking.”

• In Grade 4, Concept Quests, Unit 7, A7B, The Deca Tree, students solve a multi-step problem involving exponential structure. The task describes a tree with “10 trunks,” each with “10 branches,” each branch with “10 twigs,” and each twig with “10 leaves.” After parts of the tree are removed, students determine how many leaves remain.

• In Grade 5, Unit 3, Place Value & Decimals, Module 1, Session 5, Student Book, Question 4, Challenge asks students to “use the digits 2, 4, and 6 to create six different decimal numbers” and then order the numbers from least to greatest.

The materials reviewed for Bridges in Mathematics Grade 3 through Grade 5 meet expectations for providing manipulatives that are accurate representations of mathematical objects and are connected to written methods when appropriate. Examples include:

• In Grade 3, Work Place Collection, Work Place Guides & Instructions, Work Place 6A Tangram Polygons, Work Place Instructions 6A Tangram Polygons, students build polygons with tangram pieces and record the number of pieces used on a sheet, connecting the physical model to written representation.

• In Grade 4, Unit 2, Multidigit Multiplication & Early Division, Module 1, Session 3, Problems & Investigations, students use base ten pieces to represent centimeters and decimeters, then apply these pieces to measure a sheet of paper, reinforcing the connection between physical measurement and standard units.

• In Grade 5, Work Place Collection, Work Place Guides & Instructions, Work Place 1D Quotients Win, students roll dice to generate division problems, sketch and label the problem on a record sheet, build a model with base ten pieces, and check their partner’s written calculations, linking manipulatives directly to multiple forms of written methods.