K-2nd Grade - Gateway 3

The materials reviewed for Math & YOU Grades 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.

The materials provide comprehensive teacher guidance for presenting both student and ancillary materials. Examples include:

• The Implementation Handbook includes an Overview, Foundational Beliefs, Program Resources, The Student Learning Journey, and The Instructional Design: Using the Program. The handbook outlines the intended implementation of the program, providing general pacing guidance and approaches for differentiating learning pathways.

• Embedded throughout the Teaching Experience are notes and supports providing specific content-related guidance to aid teachers in supporting students in accessing and learning the mathematical concepts.

• Expert co-author notes at point-of-use providing insights on the mathematical content throughout the learning experiences

• “Mathematics of the Chapter” in each lesson provides context for how strategies support students in the learning objectives.

• Videos offer support for teacher content knowledge and pedagogical expertise to help solidify their understanding of key concepts across the grades

• Math Practice Resources provides guidance for teachers to engage students in developing the SMPs

• Intervention Library provides teachers guidance on how to use Skill Builder and Skill Foundations support for the entire K-12 curriculum series. 

The materials provide sufficient annotations and suggestions connected to the specific learning objectives. Examples include: 

• Grade K, Chapter 3, Lesson 3, Teacher Edition, Investigate, Dig In: Laurie’s Notes provide precise language to use during lesson delivery with questions to ask (using ? icons), teacher actions, and student actions. The materials state, “Show the Five Frame Card for 5. ‘What would one more look like? Tell your partner.’ Have counters and linking cubes available for students to explain their thinking. Display the Ten Frame Card for 5. Explain that the model is 2 five frames. ‘How could you show one more than five on this model?’ Invite volunteers to use counters or linking cubes to show one more than five. Introduce the number 6 and show a die with the six dots facing up. ‘Describe the pattern to your partner.’ Give students time to share and then discuss as a class. Talk About It: Now that students have seen the quantity of six, chorally count to 6 several times. Students can lap, tap their knee, or bob their head for each count. The last time they count, have students hold up six fingers. Make sure students understand that it takes fingers on both hands to model the number 6.”

• Grade 1, Chapter 11, Lesson 2, Teacher Edition, In-Class Practice: Laurie’s Notes on Differentiating Instruction offer look-for’s and listen-for’s at different levels of student understanding to support teachers in adjusting instruction. The materials state,“Emerging students may need help understanding what the questions are asking or connecting the question to a mathematical operation. The questions do not need an equation and can be answered from counting on the chart. Have students think about the operation (addition or subtraction) that would be used to answer the question. Exercise 5: This is a two-step question. Guide students to find how many students have already answered and then how many more need to answer for a total of 9 students. Proficient students can read the tally charts and understand how to answer the questions. They may need support in thinking of the steps in a multi-step question. Exercise 5: Have students explain their solution process to a partner. Advanced students can read a tally chart and understand how to answer the questions. Exercises 3 and 4: Have students write the equation that they would use to solve each of the questions.”

• Grade 2, Chapter 6, Lesson 7, Teacher Edition, In-Class Practice, Talk About It, provides a green call-out box that explicitly connects Exercise 13 with SMP.4. The materials state, “What model can we use to help understand what the question is about? Tell your partner. If students don’t suggest a number line, draw one. Elicit information to help represent the information known. Students may need to use trial and error to solve for the jump size (number in each row) and that is okay. It is a helpful strategy.”

The materials reviewed for Math & YOU Grades Kindergarten through Grade 2 meet expectations for containing explanations and examples of grade-level/course-level concepts and/or standards and how the concepts and/or standards align to other grade/course levels so that teachers can improve their own knowledge of the subject.

The materials contain adult-level explanations and examples of the more complex grade/course-level concepts so that teachers can improve their own knowledge of the subject. Examples include:

• The program materials provide video resources, “Everyday Connections” and “Laurie’s Chapter Insights” at most chapter levels that explain the math of the chapter to the teacher and provide teacher’s strategies and support within the chapter. 

• The “Mathematics of the Chapter” front loads the content the students will be learning and how it will connect to current grade-level work as well as future grade-level work.

• Each lesson includes an overview with the Content Standards for Mathematics (examples and explanations of the standards), Coherence (explanations of how the lesson fits into students’ learning arc and how to connect new material to known concepts), and Rigor (student learning levels expected in the lesson).

The materials contain adult-level explanations and examples of concepts beyond the current course so that teachers can improve their own knowledge of the subject. For example:

• Grade K, Digital Teaching Experience, Concepts and Tools Videos, 5, 10, and 20 Frames show teachers how to use these tools to illustrate key concepts within Kindergarten and in Grade 1. Kindergarten videos include Model a Number on a Five Frame, Compare Numbers on Ten Frames, and Use a Ten Frame to Subtract 5. Grade 1 videos include Making 10 on a Ten Frame and Getting to 10 to Subtract on a Double Ten Frame. The videos identify when Five and Ten Frames are used, build teachers’ understanding of the tools’ designs and purposes, and discuss potential misunderstandings.

• Grade 1, Chapter 6, Mathematics of the Chapter, Laurie’s Notes state, “In kindergarten, students counted by ones and by tens up to 100… In this chapter, we build on students’ experience by counting and writing numbers to 120. Most importantly, this chapter moves from counting by ones and tens to understanding place value… The language involved in place value can be confusing for first grade students. The terms (digit, tens, tens place, ones, ones place) seem to overlap and are often misused or confused. This is especially true with digit as opposed to number. Tens and tens place have related meanings, but the tens place is a location in a number and implies a value of a digit whereas tens are specifically a quantity of tens that could be used to compose a number.”

• Grade 2, Chapter 8, COHERENCE Through the Grades, traces fluency in subtraction from Grade 1 (Prior Learning) to Grade 2 (Current Learning) to Grade 3 (Future Learning): “Grade 1, Chapter 5, Subtract within 20, demonstrating fluency for subtraction within 10 (1.OA.C.6). Grade 2, Chapter 8, Lesson 8.6, Fluently subtract within 100 using strategies (2.NBT.B.5). Grade 3, Lessons 7.8-7.10: Fluently subtract within 1000 using strategies and algorithms (3.NBT.A.2).”

The materials reviewed for Math & You Grades Kindergarten through Grade 2 meet expectations for including a year-long scope and sequence with standards correlation information.

Each grade includes a Pacing Guide that provides a recommended number of days for lessons and assessments. The guide outlines suggested pacing for teaching the entire course and includes the grade-level standards addressed throughout the school year. The pacing information appears digitally in the Teacher Toolkit: Course Essentials within the learning path.

Standards correlation information is provided through multiple resources that appear consistently across grade bands. The Standards Correlation (by Standard) resource identifies where each standard is a primary or secondary focus within lessons and highlights opportunities for students to engage with the Standards for Mathematical Practice. The Standards Correlation (by Course) resource identifies the content standards aligned to each lesson and indicates whether the standard is a primary or secondary focus. Both resources are available in the Digital Teaching Experience and the Teacher Toolkit: Course Essentials.

The Implementation Handbook, Enact a Chapter: Preparing, Align Content, outlines each chapter’s learning targets and suggested pacing. At the chapter level, the COHERENCE Through the Chapter chart in the Teacher Edition lists the relevant content standards and designates the lesson where each standard is addressed.

The materials reviewed for Math & YOU Grades Kindergarten through Grade 2 meet expectations for explaining the program’s instructional approaches, including reference to research-based strategies, and explaining the role of the standards.

Materials explain the instructional approaches of the program in the Implementation Handbook, Teacher Edition front matter, and the Why Did We Build Math & YOU? page, which outlines the rationale for the program’s design and connects the instructional approaches to research-based strategies. Each resource describes the program’s philosophy and cites research used in the program’s development while connecting that research to the materials. The webpage details the instructional approaches organized around the program’s four pillars: Conceptual Foundation, Engaging Content, Teaching Support, and Innovative Digital Experience.

Across lessons, instructional approaches follow a consistent sequence that supports student learning. Lessons begin with opportunities for investigation that prompt students to make observations, conjectures, and informal strategies connected to prior learning. Next, instruction formalizes ideas through explicit introduction of new terminology, strategies, and key concepts while connecting back to the initial exploration. Students then develop procedural fluency through targeted practice that emphasizes accuracy, efficiency, and reflection on strategy use. Finally, students apply learning in new real-world or mathematical contexts, interpreting solutions in light of the situation. This structure reflects a coherent and intentional design aligned to the program’s Foundational Beliefs: Building Mathematical Rigor, Fostering Productive Mathematical Thinkers, Empowering Teachers, and Supporting All Learners while Maintaining Expectations.

Materials include and reference research-based strategies. The program explains the reasoning behind its selected instructional strategies, illustrating how they enhance learning outcomes and support evidence-based teaching practices. The Implementation Handbook and Why Did We Build Math & YOU? webpage describe “What Research Informs the Pillar?” and “How Is the Pillar Visible within Math & YOU?” Drawing on research from the National Research Council (2001), NCTM (2014, 2023), and the Common Core State Standards (NGA & CCSSO, 2010), the program reflects the consensus that mathematical proficiency involves conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition.

By prioritizing conceptual understanding as the foundation for developing fluency and problem-solving skills, Math & YOU aligns with research-based best practices that promote coherent and connected learning. The Implementation Handbook section, Empowering Teachers, emphasizes the use of research-based teaching practices and identifies specific frameworks that inform daily instruction. Teachers are supported to include the following research-based practices a consistent part of classroom instruction:

• Provide Teacher Clarity (Hattie) and Establish and Communicate Mathematical Goals to Focus Learning (NCTM)

• Implement Tasks that Promote Reasoning and Problem Solving (NCTM)

• Use and Connect Mathematical Representations (NCTM)

• Incorporate Classroom Discussion (Hattie) and Facilitate Meaningful Mathematical Discourse (NCTM)

• Pose Purposeful Questions (NCTM)

• Build Procedural Fluency from Conceptual Understanding (NCTM)

• Support Productive Struggle in Learning Mathematics (NCTM)

• Provide Meaningful Feedback (Hattie) and Elicit and Use Evidence of Student Thinking (NCTM)

• Encourage Spaced Practice (Hattie)

In Grades K-2, materials also incorporate lesson-level supports such as prompts for discussion and reflection (e.g., Talk About It, Data Talk) that encourage students to articulate their reasoning and teachers to facilitate mathematical discourse. These supports, introduced at point-of-use, reflect the program’s alignment with research emphasizing communication, reasoning, and metacognition in mathematics learning.

Materials include and reference the role of the standards in the program. The Implementation Handbook: The Instructional Design – Using the Program describes how standards guide lesson design and coherence. The materials state, “For each lesson, the Standards Correlation (by Course) identifies the content standard(s) that align to the lesson, denoting whether the standard is addressed as a primary or secondary focus of the lesson.” The materials describe how the Standards for Mathematical Content and the Standards for Mathematical Practice work together to support rigor and coherence across the grade band.

Connections to standards’ progressions are visible in lesson and chapter overviews through features such as Coherence Through the Grades, Mathematics of the Chapter, and Learning Targets with Success Criteria. These features explain how the standards progress across grades and within chapters, showing how lessons build conceptual understanding, procedural skill, and application.

Appendix A, Research Foundation, further explains the program’s grounding in research on how students learn mathematics. It states, “The Math & YOU program was thoughtfully designed from a strong research-based foundation based on how students learn mathematics. A summary of key research results informing each of the research-based beliefs is provided in this section…. Mathematical rigor entails students’ development of and connections between procedural fluency, conceptual understanding, and application (NGA & CCSSO, 2010).”

The materials reviewed for Math & YOU Grades Kindergarten through Grade 2 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

In the online Teacher Tool Kit: Course Essentials, the Materials List PDF provides a comprehensive list of supplies for every chapter and lesson. Supplies range from consumable to reusable, and each item corresponds to items needed for the class, small groups, or individual students. In the teacher materials, notes indicate when additional items are required; however, some of these items may be difficult for teachers to obtain, and no suggested alternatives are provided.

Examples include:

• Grade Kindergarten, Chapter 11, Lesson 11, Laurie’s Notes, Investigate, teacher materials provide guidance for using the listed supplies. The materials state, “Provide partners with scissors and glue. Have students cut out the six Two-Dimensional Shape Cards and place the shapes on the correct side of the page: curves or no curves.”

• Grade 1, Chapter 1, Lesson 1, Laurie’s Notes, Dig In, provide guidance for using the listed supplies. The materials state, “Have students model two or three simple addition problems. While some students act out the scenario in the center of the circle, the remaining students can model the problem with their counters.” 

• Grade 2, Chapter 13, Lesson 2, Laurie’s Notes, Dig In, provides guidance on using materials to support hands-on learning. The materials state, “Use bendable straws to make a square and a rectangle. Make a slit in the top part of the straw so it fits inside another straw. To make a rectangle, shorten two of the four straws. If you do not have straws, you can use strips of file folder-weight paper with brass fasteners.”

The materials reviewed for Math and YOU Grades Kindergarten through 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. 

Teacher materials provide notes for differentiating instruction in every lesson to target various student levels. These notes include suggestions for emerging students and some contain suggestions for the proficient level. Targeted intervention resources are provided with three different levels at every lesson level. An intervention library is provided as a bank of Skill Builder and Skill Foundations prerequisite support and these can be assigned based on each student’s learning needs.

The Implementation Handbook, High Expectations for All: Supporting Student Learning, provides an overview of the ways in which the curriculum “is designed to embrace diversity and to help all students feel connected to the mathematics they are learning.” Equity in Actions videos offer additional insights for teachers to help establish practices that promote learning for all students. 

Examples include:

• Grade Kindergarten, Chapter 10, Lesson 3, In-Class Practice, Laurie’s Notes, Differentiating Instruction states, “Emerging students are still not confident in the counting sequence to 100. Counting on is challenging, and they often need to start at the beginning of the row to count on. Pair students with a stronger counter. Missing numbers in the ones column only have one visual clue. Have students point to the number that is one less than the hidden number. Proficient students are able to count to 100 following a hundred chart or number path and can regularly identify a number missing in the hundred chart. Have students explain to their partner their strategy for finding a missing number. Advanced students are able to count to 100 fluently and can regularly identify a missing number in the hundred chart. Extension: ‘Without looking at the chart, what number is above 68? What number is below 72? What number is after 34? What number is before 51?’”

• Grade 1, Chapter 5, Lesson 1, In-Class Practice, Guiding Student Learning states, “Supporting Learners: If students struggle with Exercise 1, gather a small group to complete Exercises 2 and 3. Provide number lines for Exercises 4 to 7 as needed. The number line model allows students to show the hops back without counting back. Some students will draw and count the hops and see which number they end on. This is effective for the number line, but not for the strategy itself. If students are doing this, have them count back out loud after drawing hops to practice. ‘Exercise 1 gives you the starting point. What do you notice about Exercises 2 and 3?’ Supporting Learners: Students may want to use ten frames and linking cubes to model counting back and then transfer what they did to the number line.”

• Grade 2, Chapter 3, Lesson 8, In-Class Practice, Laurie’s Notes, Differentiating Instruction states, “Emerging students are not secure with place value or may not recognize the need to regroup. They may add correctly but write the sum incorrectly. Exercise 5: Have students work with a partner or in a small group. Provide base ten blocks. Have students describe how the partial sum for the ones is different in each equation. Proficient students are able to recognize when regrouping is needed in an addition problem. They are beginning to make sense of how to record their thinking about regrouping. Exercise 6: Have students check their sums with a partner before ordering the results. Advanced students are able to describe how regrouping is shown in an addition problem and can explain why regrouping is needed. Exercise 8: Have students explain their strategy for finding the missing numbers. Extension: ‘Write an addition problem involving 2 two-digit numbers that does not involve regrouping. Explain why regrouping is not needed.’”

The materials reviewed for Math and YOU Grades 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.

Supports are located within lessons in sections titled Differentiating Instruction Notes and Dig Deeper exercises, as well as in Differentiating the Lesson PDFs in the Digital Resources. Activities and tasks in these sections are presented as optional extensions to other exercises. Performance Tasks provide opportunities for complex problem-solving. There is no evidence of advanced students being assigned additional work; instead, all extensions are offered as optional opportunities to deepen or extend learning.

Examples include:

• Grade Kindergarten, Chapter 4, Lesson 3, In-Class Practice, Laurie’s Notes, Differentiating Instruction states, “Advanced students are able to explain how they are comparing two numbers and state the relationship using correct comparative language. Provide students with 10 blue and 10 red linking cubes. Have students use the linking cubes to make their own comparison problems for their partners to solve. Have partners use whiteboards to write the numbers and circle the number that is greater, or cross out the number that is less.”

• Grade 1, Chapter 7, Lesson 5, In-Class Practice, Dig Deeper, Exercise 15 states, “Your friend is thinking of a number that is less than 83 and greater than 74. The number has 6 ones. What is your friend’s number? Can there be more than one answer? Explain.” Laurie’s Notes, Differentiating Instruction, “Proficient students can compare numbers and explain the comparison based on a number line. They may continue to confuse the less than and greater than symbols. Exercises 15: Challenge students to use a number line to defend their answer. Advanced students can compare numbers and explain the comparison based on a number line. They use the inequality symbols fluidly. Exercise 15: Have students prove how they know they found the classmate’s number. Students can also make mystery number puzzles for each other similar to this exercise.”

• Grade 2, Chapter 6, Performance Task: Career-Order in the Court! states, “Extension: Have students write the rows of the spreadsheet on separate strips of paper. Exercise 1b asks students to order the clients by case hours, so have students sort the strips. Extend this by having them sort alphabetically. By writing on strips of paper students are learning that it’s important to keep records together. Add another case so the total number of hours is 500. ‘What will the case ID be?’”

The materials reviewed for Math & YOU Grades Kindergarten through Grade 2 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Manipulatives are embedded across lessons, providing opportunities for students to explore, communicate, and make sense of mathematical ideas. The materials integrate both physical and virtual manipulatives to support conceptual understanding and strengthen connections to written mathematical representations. The program offers a Manipulative Kit; however, activities are also designed for use with commonly available classroom manipulatives. This flexibility allows teachers to implement lessons effectively with materials already available in most classrooms. Within the Teaching Experience, point-of-use guidance consistently highlights opportunities for incorporating manipulatives.

Examples include:

• Grade Kindergarten, Chapter 1, Lesson 3, students use linking cubes to model the numbers three and four, adding one more cube each time as they count aloud with a partner to represent the concept of one more. Laurie’s Notes Dig In states, “Students will model the numbers 3 and 4. Play ‘One More Please.’ One partner has three linking cubes. Partner A hands one cube to Partner B and counts, ‘1.’ Partner B says, ‘One more please.’ Partner A hands over a second cube and counts, ‘2.’ Partner B says, ‘One more please.’ Partner A hands a third cube to partner B and counts, ‘3.’ Have students reverse roles.”

• Grade 1, Chapter 8, Lesson 2, students use base-ten blocks to model adding ones in order to build understanding of adding decade numbers. Investigate states, “Look Back: Use base 10 blocks to model each sum. Then complete the equations. 7+1\square. 3+2=\square. Look Ahead: Model each sum. Then complete the equations: 70+10=\square. 30+20=\square. Laurie’s Notes Investigate states, “Direct students to use base ten blocks to model the pairs of equations. Then have them draw quick sketches in the given space for each equation.”

• Grade 2, Chapter 1, Lesson 4, students arrange counters in equal rows to create rectangular arrays, describe the rectangles to a partner, and explain how changing the arrangement affects the shape. Laurie’s Notes Dig In states, “Students will be introduced to arrays as they arrange counters in a rectangular shape. ‘Explain what a rectangle is to a partner.’ Display 8 counters in 3 rows. ‘I arranged 8 counters. Does this look like a rectangle?’ [no] ‘How can we arrange the 8 counters so it looks like a rectangle?’ [Move 2 counters so there are 2 or 4 equal rows. Arrange the 8 counters so it looks like a rectangle. How would you describe the rectangle?” The introduction to arrays in Grade 2 is supported by the Concepts and Tools Video in the Digital Teaching Experience, Identify the Number of Rows and Columns in an Array and Identify Equal Groups. The video explains arrays, related vocabulary, and possible misconceptions for teachers, using both drawings and common objects (i.e., toy cars) as models.

A variety of digital tools are found online in “Math Tools & Graphic Organizers” and include the following options: Balance Scale, Flash Cards, Four Function Calculator, Fraction Model, Geoboard, Linking Cubes, Money, Number Frames, Number Line, Pattern Blocks, Place Value, Rekenrek.