3rd-5th Grade - Gateway 3

The materials reviewed for Math & YOU Grades 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.

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 3, Chapter 10, Lesson 1, Teacher Edition, Nick’s Notes Dig in, includes sufficient and useful annotations to tell time. The materials state, “Goal: Students will read and set clocks using five-minute intervals. Display a demonstration clock that reads 2:20. ‘What time does the clock show?’ 2:20 ‘How did you determine the time?’ Listen for the hour hand being between 2 and 3 and counting by fives from the 12 to the 4. Display 2:35 on the demonstration clock and have students tell the time and explain their answer. Distribute student clocks to pairs of students. ‘Work with your partner to set your clock to show 3:20.’ Pause. ‘Explain your strategy.’ Have students demonstrate and explain additional times that are multiples of 5, such as 1:15 and 4:25. ‘What do each of the tick marks on the clock represent?’ one minute. ‘How can you set the clock to show 3:22?’ Listen for setting the clock to show 3:20 and then moving the minute hand ahead two tick marks to show 3:22. ‘Work with your partner to set your clock to show 8:17.’ Pause. ‘Explain your strategy.’”

• Grade 4, Chapter 2, Lesson 3, Teacher Edition, Nick’s Notes Dig In, provides annotations and suggestions on how to aid students with adding numbers within 1,000. The materials state, “Goal: Students will add by ones, tens, and hundreds to get close to 1,000. Write ‘ ____ + 900 = 1,000’ on the board. ‘What is the missing number?’ 100 ‘How do you know?’ Allow think time before asking students to share their answers and reasoning. Repeat the questioning for the equations   ____  + 750 = 1,000 and 425 +  ____ = 1,000. ‘We are going to play a game called Target 1,000. Each player will have their own set of Number Cards (0–9). Shuffle the cards and draw six cards at random. Then arrange the cards you drew to make 2 three-digit numbers whose sum is as close to 1,000 (over or under) as possible. The player closest to 1,000 wins the round.’ Play one round as a whole class to ensure that students understand the game. Then have pairs of students play three rounds. Circulate as students are playing the game and ask questions to gauge their understanding. ‘What strategy are you using to get close to 1,000? Are there certain digits you need in each place value?’ Talk About It ‘Today you will add multi-digit numbers. Tell your partner what addition strategies you find helpful.’”

• Grade 5, Chapter 12, Lesson 2, Teacher Edition, Practice Notes, provides suggestions on how to understand points on a coordinate plane for Lesson Practice. The materials state, “Provide copies of the Coordinate Plane Instructional Resource as additional support, if needed. Remind students to look for any information about what each unit represents.”

The materials reviewed for Math & YOU Grades 3 through Grade 5 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 “Nick’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 3, Chapter 1, Mathematics of the Chapter states, “This chapter develops an understanding of multiplication and division through multiple representations: equal groups, equal-sized groups, number lines, skip counting, and writing equations. This understanding builds the foundation for future use with multi-digit whole numbers, integers, fractions, and decimals. Students begin to build connections between prior knowledge of repeated addition and multiplication as they see how prior knowledge can be applied to a new concept. This interconnectedness of mathematics is important for understanding the progression of numbers and operations, rather than compartmentalizing mathematics as a set of non-related topics.” 

• Grade 4, Chapter 1, Nick’s Notes-Overview states, “What we’re doing…Students will continue investigating the base ten number system and place value understanding for multi-digit whole numbers less than or equal to 1,000,000. Through exploration and discussion, students will go beyond just knowing the name of each place value to discover that the value of a digit in one place is ten times the value of the same digit in the place to its right. Students will read, write, and compare numbers from 100 to 1,000,000 and understand the role of the comma in naming each period (thousands, ones). Context-based estimation will support students as they round multi-digit numbers to any place. Why we’re doing it…This chapter provides students with opportunities to extend their understanding of numbers through 1,000,000. When students understand the structure of place value, they compose and decompose larger numbers more easily. By the end of Grade 4, students will use this knowledge to fluently add and subtract multi-digit whole numbers to 1,000,000. Rounding multi-digit numbers to any place allows students to check the reasonableness of an answer and utilize a number line as a tool. Essential Background In Grade 2, students learned to read and write whole numbers to 1,000. Then they used the symbols <, >, and = to compare two whole numbers. In Grade 3, they continued to work with whole numbers to 1,000 by exploring addition and subtraction with strategies and algorithms based on place value, properties of operations, and the relationship between addition and subtraction. They were also introduced to rounding whole numbers to the nearest 10 or nearest 100.”

• Grade 5, Chapter 3, Lesson 3, Coherence states, “In this lesson, students will use place value to add two decimals. Some of the problems will require regrouping. Just as with whole numbers, it is important that students understand the need to line up addends by place value. Students will use estimating strategies from Lesson 3.1 to assess the reasonableness of their answers.”

The materials reviewed for Math & You Grades 3-5 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 3 through Grades 5 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 make 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 3 through Grade 5 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 3, Chapter 9, Lesson 1, Nick’s Notes, Dig In, provides guidance on how teachers use the listed materials to support the lesson. The materials state, “Students will fold pieces of paper to relate halves and fourths. Distribute a strip of paper and a crayon to each student.”

• Grade 4, Chapter 13, Lesson 3, Nick’s Notes, Investigate, explains how teachers use rulers to guide students in classifying triangles by side length. The materials state, “Students will sort triangles into three groups using their side lengths. Provide each student with a centimeter ruler so that they can measure the sides of the triangles. Suggest that students label each side with its length to help them look for patterns.”

• Grade 5, Chapter 3, Lesson 2, Nick’s Notes, Dig In, describes how teachers use Place Value Mats and base-ten blocks to help students model and understand decimal values and regrouping. The materials state, “Use a Place Value Mat to display flats, rods, and units for the numbers 1.34, 0.48, and 2.3. For each ask, ‘What is the decimal?’ For the fourth example, use 3 rods and 11 units. ‘What is the value of this decimal?’ 0.41 ‘What do you notice about this model compared to the first three models?’ The 10 units can be regrouped or exchanged for 1 rod. Present an addition problem where regrouping is needed. ‘Use your blocks to model 0.26 + 0.37 on your mat and tell your partner your strategy for finding the sum.’”

The materials reviewed for Math & YOU Grades 3 through 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. 

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 3, Chapter 2, Lesson 2, In-Class Practice, Guiding Student Learning, “ Exercise 1 states, have students first describe what 6\times5 means and then finish drawing the 6 jumps of 5. Starting at 0, use the jumps to skip count aloud, referencing the same product in the table as you count. ‘There are 6 jumps of 5. You can use the model to skip count the six jumps 5, 10, …, 30 to find the product, or you can say 6\times5=30.’ Exercises 3 and 6: Remind students to apply the Commutative Property of Multiplication to use the table to find 5\times4 and 5\times6. Supporting Learners: Help students connect each problem to the completed table by skip counting to the product.”

• Grade 4, Chapter 4, Lesson 1, Tier 1 Support, Differentiating the Lesson PDF states, “Lesson Procedure, Write the problem 7\times3 on the board. Have students model 7\times3 using base ten blocks. Ask ‘How will your model change if we multiply 7\times30?’ [use rods instead of units] Have students model this using base ten blocks, keeping their model for 7\times3 on their desk. Circulate and monitor student understanding. Assist where needed. Ask, ‘How are these models similar?’ [they both have 7 groups of 3 blocks] ‘How are these models different?’ [the value of the blocks have changed] ‘What do you notice about the products?’ Sample answer: 21 is in both answers, but has a 0 at the end when multiplied by a multiple of ten. Repeat this process using other one-digit numbers, changing one factor to a multiple of ten, until students begin to notice the pattern of the 0 being added to the end of the beginning equation. Explain the game Times Tens to the students. Players flip one number card and roll one die. Students multiply the number on the die by 10, then multiply this by the number on their card. For example, if a 3 is rolled, 3\times10=30. Students would then multiply their card number by 30. Arrange students in partner groups. Distribute number cards 1–9 and dice. Allow students to continue using their base ten blocks as needed.”

• Grade 5, Chapter 12, Lesson 2, In-Class Practice, Nick’s Notes, Differentiating Instruction states, “Emerging students can find the distance between two points in a coordinate plane but may have difficulty if the ordered pairs have not been plotted. They may still reverse the x- and y-coordinates when plotting points. Exercise 8: Provide students with a coordinate plane to plot the two known points. ‘Plot two more points that are between the two points you just plotted. What are the coordinates?’ Proficient students can accurately plot ordered pairs and determine the distance between two points on a vertical or horizontal line. Exercise 8: Encourage students to complete this exercise without graphing. Advanced students can accurately plot ordered pairs. When given two ordered pairs, they can efficiently find the distance and determine whether the line is horizontal or vertical. Exercises 8 and 9: ‘Without graphing, which two points are on a vertical line? How do you know?” (4, 2) and (4, 8); They have the same x-coordinate. “Which two points are on a horizontal line? How do you know?” (0, 6) and (5, 6); They have the same y-coordinate.”

The materials reviewed for Math & YOU Grades 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.

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 and STEAM Video Performance Tasks provide opportunities for complex problem-solving and for making connections across Science, Technology, Engineering, Art, and Mathematics. The STEAM Video Performance Tasks highlight mathematics within a STEAM context and include accompanying tasks that prompt students to apply reasoning and skills to solve related problems. 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 3, Chapter 5, Performance Task: Career-Order in the Court! states, “Dogs grow just like humans. They start as puppies, become adults, and then enter old age. But dogs age faster than humans. In general, to find the age of a small dog in dog years, multiply its age in human years by 5. To find the age of a large dog in dog years, multiply its age in human years by 7. 1.1. A small dog and a large dog are both 9 years old in human years. How old is each dog in dog year? 2. The table shows the usual life spans of 6 dog breeds in dog years. Find each life span in human years. Organize the data in a bar graph. 3. Write and answer a question about your graph. 4. Cats also age faster than humans. The table shows the age of a cat for each human year. a. A cat is 6 years old in human years. If the pattern in the table continues, how old is the cat in cat years? b. A cat is 10 years old in human years. How is the cat in cat years? c. A small dog, a large dog, and a cat are each 6 years old (in human years). Order the animals by their ages (in animal years) from least to greatest. 5. Use Technology Use the Internet or another resource to answer each question. a. What is another dog breed that is not listed in the table? b. About how long, how tall, and how heavy is one of these dogs? c. What is one interesting fact about this dog breed? d. About how long would you expect this breed to live?”

• Grade 4, Chapter 11, Lesson 2, In-Class Practice, Nick’s Notes, Differentiating Instruction states, “Advanced students can recall the equivalent metric measures needed to write metric masses and capacities using smaller metric units, and they can compare measures in different units. Exercise 22: ‘Make a table of equivalent measures in liters and milliliters from 250 milliliters to 2,000 milliliters using increments of 250 milliliters. Compare your table with your partner’s table.’”

• Grade 5, Chapter 3, Lesson 4, In-Class Practice, Nick's Notes, Differentiating Instruction states, “Advancedstudents can subtract decimals given in both vertical and horizontal formats with regrouping. They correctly align place values even if the numbers have a different number of place values. Extension: ‘Write a subtraction problem that requires regrouping of two or more place values. Exchange problems with your partner.’”

The materials reviewed for Math & YOU Grades 3 through Grade 5 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 3, Chapter 8, Lesson 5, students use accurate fraction strip representations of fractional parts and after modeling with the strips, write the corresponding fractions on the board. Nick’s Notes Dig In states,“Students will combine Fraction Strips and use the unit fraction to count equal parts. Distribute two \frac{1}{2} Fractions Strips to each student. ‘Place your Fractions Strips end to end.’ Model counting by halves as you point to each part and have students repeat after you. ‘One-half, two halves, three halves, four halves.’ Then write the fractions \frac{1}{2}, \frac{2}{2}, \frac{3}{2}, and \frac{4}{2} on the board.”

• Grade 4, Chapter 5, Lesson 3, students use base-ten blocks and quick sketches as accurate representations of place value, connecting physical models to written methods. Nick’s Notes Investigate states, “Students will use base ten blocks and make quick sketches to explore dividing numbers into equal groups. ‘How many tens are in 39?’ 3 ‘How many ones are in 39?’ 9 ‘Use base ten blocks to model the number and then make a quick sketch.’ Circulate and check students' work. ‘How many hundreds are in 575?’ 5 ‘How many tens are in 575?’ 7 ‘How many ones are in 575?’ 5 ‘Use base ten blocks to model the number and then make a quick sketch.’ Circulate and check students' work.”

• Grade 5, Chapter 14, Lesson 2, students use rulers and protractors to create accurate visual representations of quadrilaterals, connecting these to written geometric descriptions. Nick’s Notes Investigate states, “Students will draw quadrilaterals that meet descriptions of their sides or their angles. Distribute rulers and protractors to students. Circulate and monitor as students work, asking them to explain how each quadrilateral meets the given description. Encourage students to use tick marks and right-angle marks, where appropriate.”

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.