High School - Gateway 3

The materials reviewed for Math & YOU: Concepts & Connections 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.

• Each lesson includes learning targets that establish clarity for teachers, and formative assessments tie student performance to the intended learning outcomes.

• Videos support teacher content knowledge and pedagogical expertise by solidifying understanding of key concepts across the grades.

• Supporting the Mathematical Practices resources guide teachers to engage students in developing the MPs.

• 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: 

• Algebra 1, Chapter 5, Lesson 5, Instructional Guide, Paul’s Notes offers instructional guidance through teacher notes and suggested questions. The materials state, “Display and discuss the Key Concept. ‘Think of each side of the equation as a function. The functions have the same value when their graphs intersect, meaning f(x)=g(x) and the x-value of a point of intersection is a solution.’ Example 1: Some students may want to solve the equation using properties of equality. Encourage them to use graphing so that they can develop an understanding of multiple solution methods. ‘How is the intersection point of the graphs of the linear functions related to the solution of the original equation?’ The intersection point represents when the values of the functions are equal, so the x-value of the intersection point is the solution of the original equation. Remind students that they can check a solution by substituting it into the original equation to make sure it makes the equation true.”

• Geometry, Chapter 12, Lesson 2, Instructional Guide, Paul’s Notes Investigate offers instructional guidance through teacher notes and suggested questions. The materials state, “Students may recall the formulas for the volume of a prism and the volume of a cylinder, but do not provide them. Allow partners to make conjectures based on their discussions and prior experiences. Circulate and listen to students’ strategies for finding the volume of the deck of cards and the volumes of the stacks of coins. ‘How do you find the volume of a right cylinder?’ Students’ explanations should include multiplying the area of the circular base by the height of the cylinder. ‘How does your process for finding the volume change when the stack of coins does not form a right cylinder.’” Differentiating Instruction states, “Emerging: Students will benefit from holding and manipulating physical models to examine attributes. Prisms that are not rectangular are particularly helpful. Make connections between the structures of prisms and cylinders by saying, ‘You are finding the area of the base and then multiplying by the height.’ Students should have a list of area formulas and a calculator. Students may not have experiences with the concepts related to similar solids. Use models of cubes when finding surface areas and volumes of similar solids. Proficient: Students can use their algebraic skills to successfully work with the formulas in the lesson. Students may not have experiences with the concepts related to similar solids. If necessary, use models of cubes when finding surface areas and volumes of similar solid."

• Algebra 2, Chapter 7, Lesson 3, Instructional Guide, Support for All Learners, offers instructional guidance through teacher notes and suggested questions. The materials state, “Use the Quick Check exercises to assess understanding of lesson concepts 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 this lesson. Exercise 1: Simplify rational expressions and identify any excluded values. Exercise 4: Multiply rational expressions. Exercise 6: Divide rational expressions.”

The materials reviewed for Math & YOU: Concepts & Connections 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:

• Each chapter contains “Paul’s Chapter Insight Video,” which explains the mathematics of the chapter to the teacher and provides strategies and support for instruction.

• Co-author notes support teacher understanding of the mathematics of the grade.

• The “Mathematics of the Chapter” frontloads the content the students will be learning and how it will connect to current course work as well as future course 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 Instructional Guide embeds “Paul’s Notes,” which offer mathematical insights, tips, and strategies.

• The program materials also provide “Concepts and Tools” videos.

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. Examples include:

• Algebra 1, Chapter 8, Paul’s Chapter Insight Video describes the mathematics of the chapter. The video provides insights into how the presenter begins teaching transformations of quadratic functions by exploring changes in the coefficients and constants.

• Geometry, Chapter 13, Coherence Through the Grades provides a table that includes prior learning, current learning, and future learning. The resource lists connected lessons from Grades 6-8 and Algebra 1. For example, the materials state prior learning from Grade 6, “Sections 10.2 and 10.3: Display numerical data in histograms.” From Grade 7, the materials state, “Sections 5.4 and 6.2–6.6: Use proportional relationships to solve ratio and percent problems. Section 7.1: Use probability and relative frequency to describe the likelihood of an event. Use relative frequency to make predictions. Section 7.2: Develop a probability model using experimental or theoretical probability. Find and compare experimental and theoretical probabilities. Use simulations to find experimental probabilities. Section 7.3: Find the sample space of two or more events. Identify the favorable outcomes in the sample space. Section 7.3: Find probabilities of compound events. Section 7.4: Design and use simulations to find probabilities of compound events.” From Grade 8, the materials state, “Section 6.3: Make and use a two-way table to describe relationships between data.” From Algebra 1, the materials state, “Section 11.3: Use histograms to represent data sets. Sections 11.4 and 11.5: Make and use two-way tables to recognize associations and trends in data.” The materials state future learning as “Find probabilities using the geometric distribution. Find probabilities using the Poisson distribution. Calculate probabilities using normal distributions.”

• Algebra 2, Chapter 5, Teaching the Chapter with Learning Targets and Success Criteria includes a learning target and success criteria for the overall chapter as well as for each lesson. For example, Lesson 5.6: Composition of Functions lists the learning target as “Evaluate and find compositions of functions.” The success criteria state “Evaluate composition of functions. Find a composition of functions. State the domain of a composition of functions.”

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for including a year-long scope and sequence with standards correlation information.

Each course 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 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 the courses. 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 digital Teacher Toolkit provides a Modeling Course Overview. This resource describes the entirety of the modeling process and guides teachers in leading students through the process. It explains where to find and how to identify elements within the program that attend to the modeling process.

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: Concepts & Connections meet expectations for explaining the program’s instructional approaches, including reference to research-based strategies, and explaining the role of the standards.

The materials provide teachers with information about the program’s instructional approaches, an overview of the program with guidance for implementing it as intended, and the research-based strategies (with citations) that informed its development. Teachers access this information digitally in the Teacher Tool Kit: Course Essentials: Implementation Handbook and the Teacher Edition front matter. The front matter includes a QR code linking to the Why Did We Build Math & YOU? webpage. Each resource describes the program’s philosophy, cites the research used in its development, and connects 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. The Why Did We Build Math & YOU? webpage explains “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.

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.

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. The materials support teachers in making the following research-based practices a consistent part of classroom instruction:

• 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)

• 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)

The materials incorporate lesson-level supports such as prompts for discussion and reflection (e.g., Math Talk, 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.

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).”

Materials include and reference the role of the standards in the program. The Implementation Handbook: The Instructional Design – Using the Program (pages 38-60) describes how standards guide lesson design and coherence. The materials (page 38) 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 series. 

Lesson and chapter overviews show connections to standards’ progressions 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 the series and within chapters, showing how lessons build conceptual understanding, procedural skill, and application.

Teachers find explanations of how the series connects to the CCSS standards at point-of-use throughout the chapter and lesson resources in the Teaching Experience. These explanations describe how content supports conceptual development and progression across standards. The first page of each lesson in the print Instructional Guide provides a Lesson Overview. This overview identifies the CCSS Mathematics Content Standards and the WIDA English Language Development Standards addressed in the lesson. It also describes how the lesson fits within the progression of student learning by identifying key prerequisite skills built upon in the lesson. Co-Author Insight Videos expand on the mathematics taught in each chapter. These quick reference videos help teachers situate the learning and focus on the big concepts. The Concepts & Connections digital Teaching Experience includes these videos under each chapter in the learning path.

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

In the Digital Teacher Experience, Teacher Tool Kit: Course Essentials includes a Materials List PDF that 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:

• Algebra 1, Teacher Toolkit: Course Essentials, Materials List: Algebra 1 includes a Materials List that provides a comprehensive list of materials needed for each chapter and lesson. Materials range from scrap paper; data sets; algebra tiles; rulers; equation cards; and computers to items such as a thermometer and a cup of hot water. The list shows that the program expects access to technology, including one-to-one devices in many lessons. Graphing calculators or regression software appear in lessons within the statistics and probability units. Some resources, such as graphing and regression tools, are available online through platforms like Desmos.

• Geometry, Chapter 3, Lesson 4, Launch, Paul’s Notes–Launch, directs teachers to display a map and provides guiding questions that support students as they explore how to prove when lines in a coordinate plane are perpendicular or parallel. In the Investigate: Constructing Perpendicular Lines activity, students work with a partner using a piece of paper, which is listed as a required material in the Materials List. 

• Algebra 2, Chapter 6, Lesson 2, Graphing Natural Base Functions, Paul’s Notes–Graphing Natural Base Functions, directs teachers to have students use technology with a slider option to explore how the value of r affects the graph of y=ae^{rx}. This guidance supports students in explaining the changes that occur when different variables are manipulated.

Materials reviewed for Math & YOU: Concepts & Connections meet expectations for providing consistent opportunities to determine student learning throughout the school year but do not provide sufficient teacher guidance for evaluating student performance and determining instructional next steps.

The assessment system provides opportunities to determine student learning throughout the school year. Assessments vary in formality, length, and format and include the Pre-Chapter Test, Mid-Chapter Test, Chapter Performance Task, Big Idea Task, Chapter Test, Connecting Big Ideas, Multi-Chapter Test, End-of-Course Test, and Standards-Based Practice. The digital platform provides reports that support instructional decisions. Formative assessments occur across the course at the multi-chapter, chapter, and lesson levels, allowing teachers to monitor and support student learning on an ongoing basis. These embedded assessments provide evidence of student progress toward clear learning goals and enable teachers to make in-the-moment instructional adjustments.

The assessment system provides limited teacher guidance for evaluating student performance and determining instructional next steps. The Quick Check exercises provide insight into student progress, and in the lesson Support for All Learners section of the Teacher Edition, the Tier 1 guidance aligns the Quick Check exercises to the lesson Success Criteria. Paul’s Notes in the Teacher Edition include question prompts that make student thinking visible as students work through In-Class Practice exercises and provide questions that can be used as independent checkpoints on key lesson skills. Digital exams allow teachers to add comments but do not include guidance for providing feedback to assess student learning. Assessments are aligned to state standards; however, the materials do not include suggested interventions, recommendations for next instructional steps, or targeted instruction resources. Answer keys are provided for Chapter Tests, but there is no evidence of sample student responses to support teachers in determining next steps for instruction. The DAP Assessment Summary report and Standards report rate student performance by topic or standard as emerging, proficient, or advanced, but they do not include scoring guidance linked to specific tasks. The Implementation Handbook provides general guidance for interpreting formative assessments and using them to inform instruction, though these guidelines are not specific to individual assessments and offer minimal instructional strategies as next steps, such as additional practice opportunities, small-group work, or targeted support. The materials include suggestions for anticipating and addressing misconceptions and monitoring formative assessment work, but these supports do not extend to summative assessments. While the materials outline how each assessment type can shape learning, they do not provide chapter- or lesson-level intervention guides. The materials do not include consistent, specific guidance or strategies to help teachers use assessment data to inform instructional adjustments, target interventions, or plan enrichment. Opportunities for providing constructive feedback or identifying areas for improvement are limited within the digital assessment system, and the assessments do not include features that address varied learning environments, student populations, or instructional approaches.

The materials reviewed for Math & YOU: Concepts & Connections 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 materials include strategies intended to support the social-emotional needs of students in special populations. The Digital Teaching Experience, Implementation Handbook: High Expectations for All: Supporting Student Learning, describes how the curriculum is designed to embrace diversity and support student connection to mathematical content. In the Digital Teaching Experience, Teacher Toolkit: Course Essentials, Math Practice Connections to Social and Emotional Learning, teachers reference connections between the mathematical practice standards and SEL competencies, including working collaboratively, addressing misconceptions, and trying multiple problem-solving strategies.

Examples include:

• Algebra 1, Chapter 2, Lesson 4, Digital Teaching Experience, Support for all Learners states,“Learning is individualized and students may move in and out of proficiency levels with each skill and concept. The Support for All Learners resources provide a wide range of layered support for the diverse needs in your classroom. Reinforce (TIER 1) Use the In-Class Practice Quick Check exercises to assess understanding of key concepts of the lesson. You can also use the Lesson Extra Practice, Lesson Dig Deeper, and Differentiating the Lesson to reinforce or extend student learning. Re-engage (TIER 2) Use a Skill Builder to support students with prior skills. Intervene (TIER 3) Use a Skill Foundations to support students who have unfinished learning.”

• Geometry, Chapter 6, Lesson 2, Instructional Guide, Support for All Learners, India’s Notes, Equity in Action states, “When students are close to completing an assignment, they may begin to drift away from finishing it. They need your assurance to help them ‘make it happen.’ Provide guidance and scaffolding to encourage your students to finish with proficiency.” 

• Algebra 2, Chapter 4, Lesson 1, Instructional Guide, Formative Assessment Tip, Agree-Disagree Statement states, “This technique has two parts. First, students are given a statement with which they may agree or disagree, or they can indicate that they need additional information to decide. Students are then asked to explain why they agree or disagree, or why they need additional information. The second part of the technique involves students describing how they can investigate, figure out, or test their thinking. Agree-Disagree Statement gives students the opportunity to think about their own understanding of a concept or process. It helps students practice the skill of actively investigating their own thinking. In listening to their own words, and in listening to the thinking of others, students can solidify or modify their beliefs.”

The materials reviewed for Math & YOU: Concepts & Connections meet expectations for regularly providing extensions and/or opportunities for advanced students to engage with grade-level mathematics at greater depth.

The teacher materials include an Instructional Guide for each lesson that incorporates Differentiating Instruction Notes and Dig Deeper exercises. The activities and tasks in these sections are presented as optional extensions to core exercises. The materials include Performance Tasks and STEM Video Performance Tasks that require complex problem solving and connections across Science, Technology, Engineering, and Mathematics. The STEM Video Performance Tasks present mathematics within a STEM context and include accompanying tasks that require students to apply reasoning and mathematical skills to related problems.

Advanced learners select from optional extension tasks to explore topics or demonstrate understanding. These extensions remain connected to the core curriculum rather than introducing additional or separate assignments. The materials do not assign additional required work for advanced students; instead, they offer optional extensions intended to deepen or extend learning.

Examples include:

• Algebra 1, Chapter 7, STEM Performance Task, asks students to determine where a bird should drop its food so that it opens without falling into another bird’s mouth. Students apply their understanding of parabolas and gravity to model and justify the drop location.

• Geometry, Chapter 8, Lesson 2, Practice, Dig Deeper, Exercise 11 states, “A portion of an amusement park ride is shown. Find EF. Justify your answer.” The materials include an image of a roller coaster track suspended on two cylindrical pillars, one 40 ft high and the other 30 ft high. Both pillars meet the ground at a right angle. The materials also include a diagram of trapezoid ABCD, where CD represents the ground and AB represents the roller coaster track. The diagram labels AC and BD as diagonals that intersect at point E, and it shows a segment drawn from E perpendicular to the ground, meeting the ground at point F.

• Algebra 2, Chapter 1, Lesson 1, Instructional Guide, Investigate states, “Proficient: Students may remember the names of the parent functions and some of the characteristics of their graphs. Can they compare and contrast the key characteristics? Can they describe the transformations of functions?” This guidance prompts students to analyze parent functions, their key features, and their transformations.

The materials reviewed for Math & YOU: Concepts & Connections 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, where students use them to explore, communicate, and make sense of mathematical ideas. The materials integrate physical and virtual manipulatives, which students use to develop conceptual understanding and connect representations to written mathematics. Across the Student Edition, activities prompt students to use common secondary mathematics manipulatives, including algebra tiles, counters, grid paper, dice, and transparent paper. In the Teaching Experience, point-of-use guidance identifies where teachers incorporate manipulatives into instruction. Teachers access paper manipulatives in the Digital Teaching Experience by selecting Math Tools & Graphic Organizers. Available paper manipulatives include function and coordinate graphs, geometric shapes and nets, number lines, and grid paper.

Digital tools are available in the digital student experience, where students access them to think through and solve problems. Students select tools from Math Tools & Graphic Organizers, which include Algebra Tiles, Balance Scale, Desmos Geometry Tool, Desmos Graphing Calculator, Fraction Models, Number Line, Place Value, Probability Tools, and a Scientific Calculator.

Examples include: 

• Algebra 1, Chapter 4, Lesson 1, Instructional Guide, Launch, students manipulate a piece of string or rope on a coordinate plane to represent lines that satisfy given criteria. During the Launch, students identify key information used to write the equation of each line.

• Geometry, Chapter 12, Lesson 7, Instructional Guide, Investigate, students model solids of revolution by creating physical models, including taping a pencil to an index card and taping the straight side of a protractor to a pencil. Students then identify objects to tape to a pencil that model a cone and a cube when the pencil rotates. Paul’s Notes state, “The physical models in this Investigate will help students develop understanding of the types of solids that can be formed by rotating two-dimensional figures.” Students later sketch solids produced by rotating given figures around a specified axis without constructing a manipulative, building on their earlier work with physical models.

• Algebra 2, Chapter 9, Lesson 2, Instructional Guide, Launch, students use paper plates and string to explore radian measure. Instructional Guide, Paul’s Notes state, “Provide an opportunity for students to physically explore radian measure. Before class, gather paper plates and cut pieces of string so that they are the same length as the radius of the plates. Distribute a paper plate and a piece of string to each student. Ask, ‘How many pieces of string of the given length would it take to go around the edge of the plate?’”