K-2nd Grade - Gateway 3

The materials reviewed for Bridges in Mathematics Kindergarten through Grade 2 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 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 K-2, 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, “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 1, Unit 7, One Hundred & Beyond, Module 2 Overview, teachers receive detailed prompts, anticipated student responses, and strategies for reinforcing place value concepts while students construct a number path to 120. For example, the Overview explains that “the class recreates the nature path with cubes, marking each multiple of 10. They use decade numbers as landmarks to help locate, identify, and compare other numbers by magnitude.” Session 3 notes further support teachers by providing specific questions and sample student reasoning: “Ask students to solve the expression mentally… Casey: ‘I thought about 4 bundles and 2 sticks, then imagined taking away one of the bundles. That left 3 bundles and 2 sticks, which is 32.’ Dustin: ‘If it was four 10-trains and 2 loose cubes, you could take away one of the 10-trains. Then you’d have 32.’”

In Grade 2, Unit 6, Geometry, Module 1 Overview, teachers are supported through margin notes that anticipate common strategies and misconceptions as students explore measurement, data collection, and multi-digit computation. For example, when students play Target 700, teachers are guided to elicit different approaches: “Ask students to calculate how far each team’s score is from 700. Invite them to get out the base ten number pieces if these seem helpful… Teacher: ‘What should I write to represent the difference between 845 and 700?’ Student A: ‘You write 845 – 700, and I know the answer already.’ Teacher: ‘Great! Keep it under your hat for a moment.’”

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

• Kindergarten, Unit 4, Paths to Adding, Subtracting & Measuring, Unit 4 Overview, Mathematical Background, Counting & Cardinality states,“The work students do in kindergarten using number paths supports the work they’ll do with number lines throughout the elementary grades. In first grade, students use both a number path and a number line to work with numbers within 120. They add and subtract using a closed number line beginning with increments of 1 and later increments of 10, as they represent and solve addition and subtraction problems. This early introduction and repeated experiences over time afford students the time needed to develop interval counting, thereby laying the foundation for significant work with number lines in later grades. In grades 2 and 3, students use the open number line to represent different strategies and as a tool for adding and subtracting. The flexibility of the open number line invites them to add and subtract using friendly numbers, including multiples of 10 and 100. To solve  56+14, for example, students might decompose the 14 into 10 and 4. On the number line, they represent their thinking by showing jumps of 10 and 4 to get to 70.”

• Grade 1, Unit 8, Changes, Changes, Unit 8 Overview, Mathematical Background, Data states, “Up until this point, work with telling time has been done primarily in Number Corner. The November and December Calendar Collector workouts and the March Calendar Grid workout focused on telling time to the hour and half hour. The activities in Unit 8, Module 1 provide final experiences with time and duration prior to grade 2, when students will learn to tell time to the nearest 5 minutes.”

• Grade 2, Unit 4, Measurement, Unit 4 Overview, Mathematical Background, Measurement states, “In grade 3 and beyond, students extend their learning about length measurement to measuring to the nearest half, fourth, and eighth of an inch. They also engage in similar measurement experiences with different measurable attributes including area, volume, angle, weight, and mass. Throughout, students continue to encounter concepts related to unit size and equivalence as they work with conversions in both customary and metric measurement systems.”

The materials reviewed for Bridges in Mathematics Kindergarten through Grade 2 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 current and adjacent grade levels.”

The materials reviewed for Bridges in Mathematics Kindergarten through Grade 2 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 Kindergarten through Grade 2 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 Kindergarten, Unit 6, Three-Dimensional Shapes & Numbers Beyond 10, Module 3, Session 1, Materials, Warm-Up states, “Ten & More Numerals display cards; demonstration number rack.”

• In Grade 1, Unit 7, One Hundred & Beyond, Module 2, Materials Preparation, Special Items states, “Session 1: Prepare a bag of Unifix cubes for every 2-3 students according to the instructions in the Preparation section for the session.”

• In Grade 2, Unit 3, Addition & Subtraction Within 100, Module 3, Session 2, Materials, Problems & Investigations states, “large base ten number pieces (see Preparation); Base Ten Triple Spin Spinner.”

The materials reviewed for Bridges in Mathematics Kindergarten through Grade 2 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 Kindergarten, Unit 7, Measurement & Teen Numbers, Module 1, Session 4, Assessment, Combinations to 5 & Equations Checkpoint is administered to individual students during Work Places. Teachers observe how students solve problems (e.g., automatic recall, counting, or using objects), record observations on the checkpoint form, and use the scoring guide to identify students for small-group reengagement activities.

• In Grade 1, Unit 2, Developing Foundational Facts, Module 1, Session 1, Assessment, Unit 2 Screener — Interview is conducted with one student at a time while peers work independently. Teachers use the screener and scoring guide to note which students need additional support and receive recommendations for specific Work Places or other activities to reinforce learning.

• In Grade 2, Assessment Guide, Unit 3, Addition & Subtraction Within One Hundred, Assessment. The Assessment answer key provides sample student responses and acceptable solution methods (e.g., using a number line in different ways). This guidance helps teachers evaluate student work and determine whether further instructional support is needed.

The materials reviewed for Bridges in Mathematics Kindergarten through Grade 2 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 Kindergarten, Work Place Collection, Work Place Guides & Instructions, Work Place 1E Spill Five Beans, Assessment & Differentiation, Multilingual learners states, “Use gestures and repetition to tie the words spill, count, and beans to the action of spilling beans and counting them. Many students can access this game counting in their home language. Use Work Place Sentence Frames, located on the Bridges Educator Site, to support students in discussing their work with others.”

• In Grade 1, Work Place Collection, Work Place Guides & Instructions, Work Place 3C Doubles Plus or Minus 1, Assessment & Differentiation states, “If you see that…: A student needs support to add or subtract 1. Differentiate: Support - Give the student Unifix cubes, a number rack, or both to use for adding or subtracting 1. Example: Tell them to use the cubes or number rack to double 3 to get 6. Then have them add 1 more cube (or bead) or take 1 cube (or bead) away.”

• In Grade 2, Work Place Collection, Work Place Guides & Instructions, Work Place 2A The Subtraction Wheel, Assessment & Differentiation states, “If you see that…: A student could benefit from support to use more precise mathematical language related to equations. Differentiate: When playing the sample round with students, write the equation on the board each time a team completes a fact on the subtraction wheel. Have students practice saying the equations aloud. Example: Write 10 - 8 = 2 and have students say: Ten minus eight is equal to two.”

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 Kindergarten through Grade 2 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 Kindergarten, Concept Quests, Unit 3, 3D, Show Me 5, students create multiple representations of the number five using objects and pictures. An extension invites students to “pick a number between 5 and 10 and make different pictures,” providing an additional challenge for advanced learners.

• In Grade 1, Unit 2, Developing Foundational Facts, Module 1, Session 3, Problems & Investigations, Playing Domino Add & Compare with a Partner, Challenge, “students turn up two dominoes on each turn and find the total number of dots on both dominoes.” The challenge specifies that students need “an entire set of double 6 dominoes or a half set of double 9.”

• In Grade 2, Concept Quest, Unit 5, 5B, Nine Coins 1, students solve for different combinations of nine coins that total one dollar and are prompted to find additional solutions. The extension asks, “Can you make $2.00 with 9 coins? If you can, with what coins? If you can’t, what is the closest you can get and with what coins?” providing opportunities for advanced students to extend their reasoning.

The materials reviewed for Bridges in Mathematics Kindergarten through Grade 2 meet expectations for providing manipulatives that are accurate representations of mathematical objects and are connected to written methods when appropriate. Examples include:

• In Kindergarten, Unit 1, Numbers to 5 & 10, Module 2, Session 1, Math Practice in Action, students explore the 5-frame as a visual model that is used throughout the year to support fluency with combinations within 5.

• In Grade 1, Unit 1, Numbers All Around Us, Module 2, Session 5, Problems & Investigations, students draw Double 10-Frame Dot Cards, solve the corresponding addition fact, and record the sum on their sheet. This connects the visual representation to written notation.

• In Grade 2, Unit 2, Setting Foundations for Place Value & Measurement, Module 1, Session 3, Problems & Investigations, students scoop and count 50 beans, grouping them into portion cups of 10, which reinforces place value and connects the concrete representation to numerical recording.