6th to 8th Grade - Gateway 3

The materials reviewed for Math & YOU Grade 6 through Grade 8 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 provide 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 6, Chapter 6, Lesson 2, Teacher Edition, Data Talk, offers instructional guidance through teacher notes and suggested questions. The materials state, “Allow 1 minute for students to read the problem and study the data display independently. Then engage in a whole-class discussion. ‘What information is being shared by the data display? What mathematical statements can you make about the data? What do you find surprising?’ ‘If 1 cube = $3 billion, what do you think 1 million would look like in this data display?’"

• Grade 7, Chapter 4, Lesson 2, Teacher Edition, Support for All Learners, Supporting Student Learning, offers guidance for instruction through teacher notes and suggested questions. The materials state, “Use Quick Check exercises to assess understanding of key concepts of the lesson, and support emerging learners with additional resources. Remember to check students’ Self-Assessment to help inform your instructional decisions. Reinforce (TIER 1) Identify where students need support with the key learning of this lesson. Exercise 3: Apply the Multiplication and Division Properties of Equality to produce equivalent equations. Exercise 5: Solve equations using multiplication or division. Exercise 7: Apply equations involving multiplication or division to solve real-life problems.” 

• Grade 8, Chapter 5, Teacher Edition, Mathematics of the Chapter, Nick’s Notes, provides teachers with connections between the mathematics students are currently learning in the chapter and how those concepts will be applied in future grades and chapters. The materials state, “Why we’re doing it…, In higher mathematics courses, students will extend their understanding of solving systems of linear equations and learn to graph systems of linear inequalities. This foundational work will also support students when they solve systems of nonlinear equations graphically and algebraically. Essential Background, In the previous grade, students used one- and two-step equations to solve mathematical and real-life problems. In Chapter 1, students solved linear equations in one variable with one solution, no solution, and infinitely many solutions.”

The materials reviewed for Math & YOU Grades 6 through Grade 8 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 each chapter level 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 6, under Coherence Through the Grades, there is a chapter chart that shows the learning progression within the current chapter and how it connects to previous and future learning, both within and across grade levels. An example from the Digital Teaching Experience illustrates this connection includes:

  • Chapter 5, COHERENCE Through the Grades states, “Future Learning, Grade 6: Sections 6.1–6.3: Write and solve equations using addition, subtraction, multiplication, or division. Sections 6.1–6.3: Use substitution to determine whether a given value makes an equation true. Grade 7: Chapter 3: Apply properties of operations to simplify, add, subtract, and factor linear expressions with rational coefficients. Algebra 1: Sections 7.1 and 7.3: Interpret coefficients, constants, and factors of polynomial expressions.”

• Grade 7, Chapter 5, Mathematics of the Chapter, Nick’s Notes states, the teaching materials provide key points to the student’s learning of the chapter by explaining “What We’re Doing”, “Why We’re Doing It” and “Essential Background”, “Why we’re doing it… This study of proportional relationships is foundational for the study of functions, which is introduced in Grade 8 and continues into high school.”

• Grade 8, Chapter 7, Lesson 1, Practice, teacher materials provide some sidebar explanations of misconceptions when referencing certain problems. The materials state, “Students may incorrectly write the ordered pairs as (output, input) instead of (input, output). Encourage students to use each arrow as a guide. The arrow points from the first number in an ordered pair (the input) to the second number in the ordered pair (the output).”

The materials reviewed for Math & YOU Grades 6 through Grade 8 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 6 through Grade 8 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 6-8, 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 6 through Grade 8 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 what is 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 6, Chapter 2, Lesson 3, Nick’s Notes, Formative Assessment Tip, teacher materials offer guidance on how to make use of stickers listed in the materials list. “As students leave your room, give each student a sticker and tell them to place the sticker on their level of understanding. At the end of the last day of instruction, display a new copy of the Self-Assessment scale. Have students place stickers on the scale to rate their current understanding of the material. Compare the two rating scales to check for growth.”

• Grade 7, Chapter 10, Lesson 2, Nick’s Notes, Investigate, teacher materials offer guidance on items students may need to complete an activity. “Provide each pair of students with scrap paper, tape, scissors, and a can. It is more interesting when the cans are different sizes.” However, these materials are not listed in the Materials List PDF included in the Course Essentials.

• Grade 8, Chapter 1, Lesson 2, Nick’s Notes, Investigate, teacher materials provide the goal utilizing the materials listed in the pdf and the purpose. “Goal: Students will use algebra tiles to write and solve multi-step equations.”

The materials reviewed for Math & YOU Grades 6 through Grade 8 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 6, Chapter 4, Lesson 4, Nick’s Notes, Differentiating Instruction states, “In the Investigate, students used number sense, ratio reasoning, and percent models to find percents of numbers and missing percents. Emerging: Students may still be developing confidence in converting between fractions, decimals, and percents. Visual representations of percent problems will support their understanding of the problems. Provide guided instruction for each example and help students make sense of the percent models. Proficient: Students may be confident in using a percent model to represent a percent problem. Can they use an equation to represent the problem? After engaging with the Key Concepts, have students use the In-Class Practice exercises to assess their understanding.”

• Grade 7, Chapter 4, Lesson 4, Nick’s Notes, Differentiating Instruction states, “In the Investigation, students were introduced to the second and third success criteria. Now they will formalize their understanding of inequalities. Emerging: Students may struggle with the idea that there is more than one solution to a problem or with using the inequality symbols correctly. Have students work with peers to check solutions and emphasize the use of precise vocabulary during whole-class instruction.” 

• Grade 8, Chapter 7, Lesson 2, Nick’s Notes: Formative Assessment Tip, Four Square states, “A Four Square is often used as a study reference; however, it can be used to assess students’ understanding of a concept. One way to use a Four Square is to write a problem in the oval and label each of the four squares surrounding the oval as a related category. Related categories may include answer, meaning, application, algebra, numbers, model, graph, or equation. You can also ask students to illustrate each of the four depths of knowledge levels in the squares surrounding the problem. Consider hanging students Four Squares around the classroom so that they see a variety of ways to display information about a particular problem.”

The materials reviewed for Math and YOU Grades 6 through Grade 8 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 6, Chapter 4, Lesson 1, Problem 21, Practice, Dig Deeper, Exercise 21 states, “Complete the statement represented by each model. Use the results to write \frac{1}{3} and \frac{1}{7} as percents. Then explain how to use a model to write \frac{1}{n} as a percent.” Students are given two fractions greater than one, along with a model to help them solve the problem.

• Grade 7, Chapter 7, Lesson 2, Tier 1 Support, Dig Deeper PDF state, “Many card games involve making choices based on how likely it is to choose a certain card. By performing an experiment with a regular deck of cards, you will be finding the probability of certain outcomes when cards are chosen at random. A deck of cards has 4 suits. Each suit has 13 cards: a Jack, Queen, King, Ace, and the numbers 2 through 10. For this experiment, the Ace is worth 1, the Jack is worth 11, the Queen is worth 12, and the King is worth 13. Experiment Directions: Put all of the cards face down and spread them out. Choose two cards at random. Find the absolute value of the difference between the values of the cards. Replace the two cards. Mix the cards and repeat. Answer Exercises 1 and 2 before performing the experiment. 1. Make a list of all the possible outcomes and design a frequency table to record your results. 2. Make some predictions. Will all the outcomes be equally likely? If not, what outcomes will be most likely? least likely? Explain your reasoning. 3. Perform the experiment at least 60 times. Record the results in your frequency table from Exercise 1. 4. Make a bar graph of your results. Compare your results with your classmates. Were they similar? Explain. 5. Describe any patterns you notice. Did they fit your predictions? What outcomes are most likely? least likely? Explain. 6. Explain why it would be difficult to find theoretical probability for this situation. 7. What is the advantage to doing a large number of trials? Explain why doing more trials is especially important for this experiment. 8. You want to change the experiment. Instead of taking the absolute value of the difference, this time you will take the value of the first card minus the value of the second card. How would this change your results? Explain. 9. A friend asks you to play the following game. Two cards are chosen at random. If the absolute value of the difference is between 1 and 6, Player 1 gets a point. If the absolute value of the difference is between 7 and 12, Player 2 gets a point. If the difference is zero, both players get a point. Replace the cards, shuffle, and repeat. The first person to get 10 points wins. Explain why this game is not fair. Rewrite the rules to make the game more fair.”

• Grade 8, Chapter 10, STEAM Performance Task, Packaging Salsa states, “How can unused space be minimized when packaging an item? A company produces jars of salsa with the dimensions shown. The company plans to ship the jars in boxes. 1. Find the volume of each jar of salsa. Round your answer to the nearest whole number. 2. If a jar of salsa is placed in a box that is 10 centimeters long, 10 centimeters wide, and 10 centimeters tall, how much unused space will there be inside the box? Explain your answer. 3. The company plans to use the box shown to package and ship the jars of salsa. a. What is the maximum number of jars that will fit inside the box? Explain your answer. b. How much unused space will there be inside the box? Explain your answer. c. How much unused space will there be in a shipment of 12 boxes? Explain your answer. 4. The company was producing more jars than needed for each box. It was determined that there was a mistake made in the calculation for the maximum number of jars that can fit inside a box. Describe a possible error. 5. The salsa company does not want its jars to break in the shipping process. To ensure the jars are not damaged, the company has decided to fill the unused space with packing peanuts. The peanuts come in cylindrical containers in three different sizes. If the company wants to purchase one cylinder of packing peanuts for each shipment of 12 boxes, and the company wants the least amount of packing peanuts left over, which cylinder should the company choose? Explain your answer.” A jar with the height of 10 cm and the radius of the lid as 5cm is provided. A picture of a box is provided with the dimensions of 50 cm\times20 cm\times30 cm. Also, provided are dimensions of Cylinder A with radius of 25 cm and height of 45 cm. Cylinder B is provided with radius of 35 cm and height of 25 cm and Cylinder C with a radius of 15 cm and height of 80 cm.

The materials reviewed for Math and YOU Grades 6 through Grade 8 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, such as number cubes, algebra tiles, counters, grid paper, dice, and transparent paper. Within the Teaching Experience, point-of-use guidance consistently highlights opportunities for incorporating manipulatives.

Examples include: 

• Grade 6, Chapter 2, Lesson 5, students use number lines and place value charts to model addition and subtraction of decimals, discuss the increments shown, and determine sums and differences. Nick’s Notes Investigate states,“Students will use number lines and a place value chart to add and subtract decimals. Exercise 1 requires students to read and interpret decimals plotted on number lines. Have partners discuss the increments used for each number line before finding the sums and differences.”

• Grade 7, Chapter 7, Lesson 2, students conduct repeated trials using cubes in a mystery bag to explore experimental probability, record outcomes, and use the results to make and revise predictions. Nick’s Notes Motivate states, “Students will explore probability that is based on repeated trials of an experiment (experimental probability). Play Mystery Bag. Before students arrive, place 10 cubes of the same shape and size in a paper bag, 5 of one color and 5 of a second color. ‘There are 10 cubes in my bag. Can you guess what color(s) they are?’ not likely. Select one student to remove a cube from the bag. ‘Can you guess the color(s) of my cubes now?’ not likely. Replace the cube. Let the student pick again and see the color. Repeat your question. Do this 5-8 times until the student is ready to guess.”

• Grade 8, Chapter 3, Lesson 1, students use a pencil, paper, and a protractor to draw parallel lines and explore the angles formed when a line intersects them, identifying relationships among the angle measures. Nick’s Notes Investigate states, “Students will explore measures of angles created when a line intersects two parallel lines. Make sure students draw the lines so that they are parallel. This investigation can also be completed using a pencil, paper, and a protractor."

A variety of digital tools are found online in “Math Tools & Graphic Organizers” and include the following options: Algebra Tiles, Balance Scale, Desmos Geometry Tool, Desmos Graphing Calculator, Fraction Model, Number Line, Place Value, Probability Tools, Simulation, Scientific Calculator, and Four Function Calculator.