6th Grade - Gateway 3

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 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 Teacher Portal provides comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Program Information provides information “About the Program” including the Program Overview, Unit Summaries, Program Components, and Planning Tips. There is also a document titled “Features to Engage Students” as well as a section dedicated to explaining the Activity Routines. Examples include:

• Program Information, Planning Tips, “Planning a Unit: Begin with an overview of the entire unit. Leaf through the TE, paying close attention to the answer key. This preview helps to understand the nature of the content and the work students will do. Preview the Slide Decks (or Slide Deck Alternatives) and Lesson Notes that appear after the corresponding student page. Look for teacher-guided opportunities to engage the students in discussions and collaboration. Assess formally or informally if students need extensive review of prerequisite content or “just-in-time” support. Look at the Getting Ready exercises at the beginning of each lesson, Essential Skills in Other Resources, and even Skill Boosters as sources of refreshment or intervention. Look at the planning suggestions, especially the estimated class hours for each lesson. Identify pages and activities that seem reasonable for students to finish in any given class period. Think about which pages (or portions of pages) might be appropriate for independent work or homework. Students need not do every problem on every page. Teachers make these instructional decisions based upon student needs.”

• Features to Engage Students includes examples throughout the course of Kinesthetic Activities, Technology Activities, Activity Routines, Puzzles/Games/and Card Sorts, and Real-life Problems. 

The Teacher Edition includes sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. The Unit Planning for each unit provides teachers with pacing, resources, and assessment options, materials, digital slide decks, reproducibles, and preparations to make ahead of time. There are also tips about math background, standards correlations, mathematical practices, strategies for different learners, connections among the big ideas, and explanations of rigor. Examples include:

• Unit 1, “Unit Pacing: Up to 14 class hours, 1.0 Opening Problem: Beach Cleanup (< 1 hour); 1.1 Introduction to Data Analysis (3 hours); 1.2 More Measures of Center and Spread (3 hours); 1.3 Data Displays (3 hours); Review (3 hours); Assessment (1 hour).”

• Unit 3, Strategies to Support Different Learners, MathLinks Examples, Know Your Learners, “Here are some specific ways that MathLinks supports special populations. [3.4] For struggling learners, distribute the first group of R3-2 Measurement Cards (customary measurement) for the sorting activity, and then assign Practice 7. Repeat later with the second group of cards (metric measurement), followed by Practice 8.”

• Unit 10, Big Ideas and Connections explain how the grade level material is organized around seven big ideas. The ideas build on past work and prepare students for the future work. For example, “MathLinks: Grade 6 is organized around seven big ideas. This graphic provides a snapshot of the ideas in Unit 10 and their connections to each other. Extend the number system to include negatives. Explore relationships between inputs and outputs.”

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

Correlation information is present for the mathematics standards addressed throughout the grade level/series. This information can be found in General Resources online as well as both teacher and student editions. Examples include:

• Teacher Portal, Program Information, Program Overview-Each unit is linked to the content standards covered. 

• Teacher Portal, Program Information, Unit Summaries-Each unit is connected to the content standard. This is the same information as the Program Overview except for there is a narrative summary of the unit's content.

• Teacher Portal, Correlations, Lessons, Quizzes, and Tests-Each standard is listed and connected to the lessons that incorporate it as well as quiz and test questions that address it. 

• Teacher Portal, Correlations, Tasks and Projects-Each task and project are aligned to Domains, Big Ideas, and Mathematical Practices.  

• Teacher Edition, Unit Planning-The standards developed within each unit, including each sub section, are listed as well as identifying specific uses of mathematical practices. Standards and mathematical practices are also identified in each lesson.

• Student Packet-The back of each packet lists the standards worked on within the unit and the short version of the mathematical practices. Each lesson identifies the standards and practices as well. 

Explanations of the role of the specific grade-level/course-level mathematics are present in the context of the series. Each unit planning section in the Teacher Edition includes a section titled “Big Ideas and Connections” which provides a snapshot of the ideas in the grade level and the connections within the specific unit. It also details how the ideas connect to both Prior Work and What’s Ahead. For example: 

• Unit 1, Teacher Edition, Teaching Tips, Big Ideas and Connections, of the seven Big Ideas for grade 6, emphasis in unit 1 is on “Use statistical measures and displays to describe center and spread.’ and ‘Gain computational fluency with positive rational numbers.” “These ideas build on past work and prepare students for the future. Some of these include: 

  • Prior Work: Represent and interpret data, especially line plots (Grades 4, 5); Add, subtract, multiply and divide whole numbers and decimals to hundredths (Grade 5)

  • What’s Ahead: Use random sampling to draw inferences about population(s) (Grade 7); Investigate bivariate data (Grade 8); Interpret categorical and quantitative data (HS); Make inferences and justify conclusions about statistical experiments, surveys, and observational studies (HS); Use statistics as a tool when mathematical modeling (HS).”

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

Materials explain the instructional approaches of the program. These are easily found online under Program Information in Design Principles. These documents explain how MathLinks integrates two foundational philosophies of Focus/Coherence/Rigor and Universal Design for Learning into the development of their program. For example: 

• Universal Design for Learning and MathLinks includes a complete description of how MathLinks has incorporated Universal Design for Learning principles and framework. “Universal Design for Learning (UDL) is a framework to improve and optimize teaching and learning for ALL students (CAST, 2018). Its three main principles are to (1) provide multiple means of engagement, (2) provide multiple means of representation, and (3) provide multiple means of action and expression. Here are some examples of how these principles are built into the MathLinks design.” A chart provides connections between UDL Principles with examples from MathLinks.

• The UDL document also provides a chart for Strategies to Support Special Populations. This connects UDL instructional strategies and MathLinks features. For example the UDL Strategy “Know Your Learner” includes: “Understand student attributes that support or interfere with learning. Determine preferred learning and interaction styles. Assess student knowledge of prerequisite mathematics content. Check for understanding continuously. Provide differentiation opportunities for intervention or enrichment to reach more learners. Encourage students to write about their attitudes and feelings towards math. Use contexts that link to students’ cultures.” The Corresponding MathLinks Features include: “Use Getting Started and Spiral Review to assess prerequisite skills and retention of new knowledge, and Monitor Your Progress, The MathLinks Rubric, and Unit Reflection as student self- assessment tools. (SP) Journal suggestions are referenced. (TE) Look at Essential Skills and Nonroutine Problems for intervention and enrichment. Many Projects offer opportunities for student choice based upon student needs, interests, or cultures. Quizzes provide traditional methods for assessment, and the Extra Problems sets provide extra practice when needed, or may be used to create customized assessments. (UR) For those who need it, Skill Boosters are designed to fill gaps in knowledge without losing pace on grade-level work. (GR).” 

• The UDL document also addresses the concept of “unfinished learning” and the adapted work from Fossum (2017). A table breaks information into two categories: Common Instructional Misstep and MathLinks Alternative. An example includes, “Common Instructional Misstep: Trying to address every gap a student has; MathLinks Alternative–Pre-assessments and Essential Skills prioritize most essential prerequisite skills and concepts for upcoming content.”

• Focus, Coherence, and Rigor provides a thorough description of each aspect including references to specific problems throughout the series (details found in Gateway 1). In addition, there is a description of how the three aspects connect, including integration from the CA Mathematical Framework (2023), with narrative examples of major domains for the grade.

Materials include and reference research-based strategies. In Program Information, all of the explanatory documents include references to specific sources. In addition, there is a References and Index section under General Resources. For example:

• Focus, Coherence, and Rigor cites a Rutgers University study led by Dr. Roberta Shore (Young, 2009); Achieve the Core (2014, 2015); CA Math Framework (2023); NCTM position paper (2014); EdReports Evidence Guide (2021); and CCSS-M.

• Resources and References contains a bibliography that lists references to strategies and other approaches used in the curriculum. Some of the resources cited include: 

  • “Achieve the Core. (2014) “Mathematics: Focus by Grade Level: 6.” Retrieved from: https://achievethecore.org/content/upload/SAP_Focus_Math_6.pdf

  • Boaler, J., Munson, J., Williams, C. (2016) Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching

  • SFUSD. (2019) "UDL Guidelines in Math." Retrieved from: https://www.sfusdmath.org/universal-design-for-learning.html

  • Zwiers, J. Diekmann, J. (2017) "Principles for the Design of Mathematics Curricula: Promoting Language and Content Development." Retrieved from: https://ul.stanford.edu/sites/default/files/resource/2021-11/Principles%20for%20the%20Design%20of%20Mathematics%20Curricula_1.pdf”

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

There are numerous places to find the list of materials needed, both for the year and for each unit. 

The Teacher Portal, Program Information, Materials and Copy Lists, provides lists of generic materials needed for the year as well as specific materials for each unit, including what to make copies of. For example:

• Teacher Portal, Program Information, Materials and Copy Lists, provides a list of generic materials needed for the year as well as specific materials for each unit, including what to make copies of. For example: Unit 2: Materials include “Small square tiles (2.1), 24 per pair; Two-color counters (2.2), 20 per pair; paperclip or game piece (2.2), 1 per student.” Copies include “Factor Game (2.1), 1 per pair;  Product Game (2.2), 1 per pair; Match and Compare Sort Cards (Review), 1 per pair.”

• Teacher Edition, Unit Planning provides a list of materials that will be used for that particular unit. For example: Unit 4, Unit Planning, Materials: Post-its [4.0] (1 per student, optional); 3 × 5 cards [Review] (1 per student); Play coins [4.2] (nickels, dimes, quarters, optional); General supplies (e.g., colored pencils, markers, rulers, tape, scissors, graph paper, calculators, chart paper)

• There is a recommended student supply list for parents in the Parent Support section.

• Teacher Portal, Program Information-print version, Table of Contents (p.17), Shopping Lists gives an extensive list of materials. “These shopping lists are compiled for the convenience of users who wish to order and assemble MathLinks supplies for Grade 6. The following materials are typically obtained from a mathematics manipulative catalog.” (Linking cubes or small objects, 2-color counters, Square tiles, Play coins, Large Number Line (positive and negative), Small laminated number lines)

• “The following materials are typically available in the school warehouse, office supply stores, or other local stores. The list estimates quantities for a full year.” (11x17 paper, Chart paper (or butcher paper), 3x5 cards (for class activities), Sticky notes (medium size), 5x7 cards, Envelopes or small zip lock bags, Paper clips (small), Meter stick, Play-Doh, paint, or food coloring (primary colors), Measuring tools (length, liquid, weight in customary and metric))

• “The following materials are recommended as part of each student’s supply list. If students do not purchase supplies, then we recommend that teachers add these to Class Supplies.” (3-ring binder (including storage of student packets), Spiral notebook (for journal and non-packet work), Graph paper, Lined paper, Colored paper (various colors), 3x5 cards (individual vocabulary cards), Markers, Highlighter markers, Colored pencils, Pencils, Erasers, Ruler (inches and centimeters), Scissors, Tape, Glue stick, Calculator)

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

Assessments include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series. There are a variety of item types in the Formative and Summative Assessments such as: multiple choice, select all, short answer/fill in the blank, extended response prompts, graphing, mistake analysis, matching, constructed response and technology-enhanced items. These allow opportunities for “multiple means of expression (e.g. talking, writing, drawing typing responses).” Examples Include:

• Pre-Assessment, “intended to give a snapshot of some of the main essential skills, or prerequisites, for units. They are not intended to be comprehensive. Each pre-assessment is in a multiple-choice format.”

• Quizzes, “ntended to be used at the end of each unit. There are two forms of quizzes per unit. They assess work done with procedural problems and constructed responses similar to those found in Student Packets.”

• Tests/Cumulative Tests,“Multiple choice, selected response, and short answer problems that assess grade-level content standards are organized into 10 unit-specific test files. They are intended to be combined to make cumulative assessments of some desired length.”

• Tasks, “engage students in solving multi-part problems around a theme or context, and the MathLinks Rubric is often an appropriate tool for evaluating this work.”

• Projects, “are authentic, multi-day experiences involving skills, concepts, and problem solving across one or more domains. They may incorporate research, require the collection and display of data, or the creation of a product. The instructions for the project typically also serve as the scoring guide.”

• Extra Problems, “are organized by lesson, are appropriate for extra practice, or may be used to create additional assessments.”

• Journals, “may be used as a journal prompt or ‘exit ticket’. These problems generally require students to explain their thinking related to an important concept in the lesson.”

• Monitor Your Progress, “appears at the end of every lesson. This prompts teachers to ask students to assess their learning on the front cover of the Student Packet. There, a 3-2-1-0 scale gives students and teachers a quick measure of student confidence for each lesson goal.”

• Unit Reflection, “Students make connections related to the concepts learned and reflect on their own learning.”

• Rubric-Worthy Problems with The MathLinks Rubric, “Problems with multiple parts that are built around a theme or context typically include some of the following: (1) doing procedures, (2) creating of using representations (e.g. pictures, numbers, symbols, graphs), (3) interpreting data, (4) constructing responses, (5) explaining reasoning. Many rubric-worthy problems appear in Student Packets. Use the MathLinks Rubric Activity Routine to promote class discussions and shift some responsibility for peer feedback and self-reflection to students for some problems.”

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 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.

Both physical and virtual manipulatives are used, though not extensively. Typically, manipulatives are used, or suggested, during development of conceptual understanding. Within lessons, there is significantly more use of manipulatives as representational strategies such as drawings, graphs, charts, tallies, diagrams. Use of manipulatives is consistently connected to written methods, and students are often expected to explain their grasp of content as they gain understanding and work through problems. For example:

• Teacher Edition, Unit Planning, Materials, every unit includes a list of manipulatives and reproducibles needed for the unit. For example, Unit 6 indicates square or cubes (optional), card sorts, matching cards, square puzzle, and general supplies–colored pencils, markers, rulers, tape, scissors, graph paper, calculators, chart paper.

• Unit 2, Lesson 1, Lesson Notes S2.1a: Building Rectangles, “Students explore the connection between factors of several numbers and the possible dimensions of a rectangle with a given area. Small square tiles are recommended. For this work, we will use natural numbers only….For (2), allow time for students to work in pairs to draw all of the rectangles with the given areas and record their dimensions. Encourage students to build rectangles with tiles first if it is helpful.”

• Digital Tools such as are available in Technology Reproducibles on the Teacher Portal. For example, Unit 4, Technology Reproducibles, Exploring Fraction Division, Open Source, “Use virtual fraction bars to estimate the number of times one fraction “goes into” another. Use before 4.3.” A link is provided.

• Unit 6, Lesson 1, Lesson Notes S6.1b: Exponential Notation, “As we introduce exponents here, we use the words ‘square numbers’ and ‘cubic numbers’ numerically and geometrically. Students may benefit from building figures with square tiles or cubes before drawing them.”