6th Grade - Gateway 3

The materials reviewed for Desmos Math 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.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Each unit contains a Unit Overview with a summary of the unit, vocabulary list, materials needed, and Common Core State Standards taught throughout the unit. Each Unit Overview page, also includes paper resources such as the Unit Facilitation Guide, Overview Video Guided Notes, and Guidance for Remote Learning to assist teachers in presenting. An example is included below:

• Unit 8, Unit Overview, the materials state, “Section 1: Visualizing Data (Lessons 1-6) Create dot plots and histograms to visualize data. Informally describe and compare data sets. Section 2: Mean and MAD (Lessons 7–10 + Practice Day + Quiz) Calculate the mean and mean absolute deviation (MAD) of a data set. Use mean and MAD to describe and compare data sets. Section 3: Median and IQR (Lessons 11–16 + Practice Day) Compare and contrast the mean and median as measures of center. Calculate the quartiles, interquartile range (IQR), and range of a data set. Create box plots to visualize data. Use median and IQR to describe and compare data sets.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. The Curriculum Guide, Lesson Guides and Teacher Tips, describe support for facilitation throughout the program. The materials state, “Each lesson includes support for facilitation, which can be found in different places on the lesson page. The Summary is an overview of the lesson and includes the length and the goals of each activity. The Teacher Guide is a downloadable PDF that accompanies every digital lesson. It includes screenshots of each screen as well as teacher tips, sample responses, and student supports. The Lesson Guide is a downloadable PDF that accompanies every paper lesson. It includes preparation details and materials for the lesson, as well as tips for purposeful facilitation of each activity. Teacher tips are suggestions for facilitation to support great classroom conversations. These include:

• Teacher Moves: Suggestions for pacing, facilitation moves, discussion questions, examples of early student thinking, and ideas for early finishers, as well as opportunities to build and develop the math community in your classroom.

• Sample Responses: One or more examples of a possible student response to the problem.

• Student Supports: Facilitation suggestions to support students with disabilities and multilingual students.”

Examples include:

• Unit 2, Lesson 4, Summary, the materials state, “About This Lesson The purpose of this lesson is for students to explore how to generate equivalent ratios in the context of balancing fruit on scales. This builds on what students learned in Lesson 3 about what equivalent ratios are. By the end of this lesson, students should be able to explain that multiplying each amount by the same number yields an equivalent ratio.” Lesson Summary, “Warm-Up (5 minutes) 

The purpose of the warm-up is to introduce the context of balancing fruits and for students to begin to generate equivalent ratios. Students adjust the numbers of apples and oranges on a scale to create several ways to balance the scale. Activity 1: Comparing Apples to Oranges (5 minutes) The purpose of this activity is for students to analyze a set of equivalent ratios and generate equivalent ratios for one relationship. This activity prepares students to explore several different ratios of fruits in Activity 2. Activity 2: Fruit Lab (25 minutes) 

The purpose of this activity is for students to explore strategies for generating equivalent ratios and determining whether two ratios are equivalent or not. Students first explore in the Fruit Lab, then analyze several different fictional students’ strategies for creating equivalent ratios. Students should leave this activity recognizing which operations do and do not create equivalent ratios. Lesson Synthesis (5 minutes) The purpose of the synthesis is for students to describe how to determine equivalent ratios that balance the scale when they know a ratio that does. Cool-Down (5 minutes)”

• Unit 4, Lesson 5, Lesson Guide, Warm-Up, the materials state “Overview: Students make connections between expressions and tape diagrams that represent ‘how many groups?’ Launch Invite students to work in pairs. Display the Teacher Projection Sheet. Facilitation Give students one minute to discuss Prompt 1 with a partner, then another 1–2 minutes to think individually about Prompts 2 and 3. Monitor for students who make connections to earlier lessons or to personal experience, particularly the scoops of flour from Lesson 3. Invite several students to share their thinking for each question. Consider focusing most of the discussion on how students used the tape diagram to represent their thinking, rather than on the answer to the question. Discussion Questions How did you decide how many groups there were? How can we show ________’s thinking on the tape diagram? How is this situation similar and different to ones we have seen so far in this unit? Readiness Check (Problem 3). If most students struggled, consider reviewing this problem. Invite students to share how they decided if each choice did or did not have the same value as the original.”

• Unit 6, Lesson 8, Screen 4, Not Equivalent, the materials state, “How would you convince someone that 2(3x+4) is not equivalent to 6x+4?” Teacher Moves,“Facilitation Invite students to consider why someone might think these two expressions are equivalent before focusing on why they are not (MP3). Discussion Questions What could you change about 6x+4to make it equivalent to 2(3x+4)? Math Community Consider inviting students to share what they think we can learn from looking at both correct and incorrect thinking.”

The materials reviewed for Desmos Math 6 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject. 

The Unit Overview Video is “A short video explaining the key ideas of each unit, including how the unit fits in students’ progression of learning.” The video is intended for teachers, and has adult-level explanations and examples of the more complex grade-level concepts via the “Big Ideas'' portion of the video. The examples that the presenters explain during the “Big Idea” portion of the overview video comes directly from lessons in the unit. 

All Unit Overview Videos end with an explanation and example from later grades. The presenters will show an example problem from beyond the course and explain how the problem on the screen connects to the “Big Ideas” of the current unit. For example:

• Unit 5, Unit Overview, Unit Overview Video, the presenter talks about the work that students are doing now will lay the foundation for the work in “Later Grades”, when students solve problems with positive and negative numbers. On the screen is a challenge problem from beyond the current course, the presenter explains the problem is about solving rational number problems on a number line. Additionally, the presenter explains that the goal is getting students to work with negative numbers and explains how it is similar to the work of extending students' understanding from whole numbers and fractions to decimals.

The Unit Facilitation Guide contains a section called “Connections to Future Learning,” which includes adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current course. For example:

• Unit 6, Unit Facilitation Guide, Connections to Future Learning, “Proportional Relationships (7.RP.A.3) In this unit, students work with multiple representations of real world situations. In Math 7, Unit 2, they will explore proportional relationships in multiple representations. For example, the cost of carpet is 1. 5 times the number of square feet. We can represent this relationship with the equation on the right.” The equation to the right states, “y=1.5x, x represents the number of square feet of carpet bought. y represents the cost of the carpet, in dollars.”

The materials reviewed for Desmos Math 6 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

The Math 6 Overview, contains the Math Grade 6 Lessons and Standards document which includes the following:

• Standards Addressed by Lesson - This is organized by unit and lesson. It lists the standards and Mathematical Practices (MPs) addressed in each lesson. 

• Lessons by Standard - This is organized by Common Core State Standards for Mathematics grouped by domains and indicates which lesson(s) addresses the standard. It also lists each MP and indicates which lessons attend to that MP.

The Curriculum Guide, Units, Unit Resources, states: “Each unit contains a Unit Overview page that includes resources to support different stakeholders. On each Unit Overview Page, you will find the following:”

• Unit Facilitation Guide: “A guide to support teachers as they plan and implement a unit. It includes information about how the unit builds on prior learning and informs future learning, as well as big ideas, lessons by standard, and key math practice standards. There is a brief summary of the purpose of each lesson along with other information that may be helpful for planning.”

• Unit Overview Video: “A short video explaining the key ideas of each unit, including how the unit fits in students’ progression of learning.” However, standards are not explicitly identified in the video. 

Examples from the Unit Facilitation Guide includes:

• Unit 3, Unit Facilitation Guide, Connections to Prior Learning, states, “The following concepts from previous grades and units may support students in meeting grade-level standards in this unit: Measuring and estimating lengths, volumes, and masses/weights in standard units. (2.MD.A, 3.MD.A) Multiplication of whole numbers by fractions and fractions by fractions. (4.NF.B.4, 5.NF.B.4) Understanding the concept of a ratio and using ratio reasoning to solve problems. (6.RP.A.1)”

• Unit 6, Unit Facilitation Guide, Connections to Prior Learning, states, “The following concepts from previous grades or earlier in Grade 6 may support students in meeting grade-level standards in this unit: Adding, subtracting, multiplying, and dividing decimals and fractions. (6.NS.A.1 , 6.NS.B.3) Using whole number exponents to represent powers of 10. (5.NBT.A.2) Evaluating expressions with addition, subtraction, multiplication, division, and parentheses or brackets. (5.OA.A.1) Graphing points in the first quadrant of the coordinate plane. (5.G.A.2)”

The Curriculum Guide, Lessons, Standards in Desmos Lessons, “A standard often takes weeks, months, or years to achieve, in many cases building on work in prior grade levels.

• Standards marked as “building on” are those being used as a bridge to the idea students are currently exploring, including both standards from prior grade levels or earlier in the same grade.

• Standards marked as “addressing” are focused on mastering grade-level work. The same standard may be marked as “addressing” for several lessons and units as students deepen their conceptual understanding and procedural fluency.

• Standards marked as “building towards” are those from future lessons or grade levels that this lesson is building the foundation for. Students are not expected to meet the expectations of these standards at that moment.”

For example:

• Unit 6, Lesson 7, Lesson Overview Page, Learning Goals, “Explain what it means for two expressions to be equivalent. Justify whether two expressions are equivalent.“ Common Core State Standards: Building On: 6.EE.A.2 Addressing: 6.EE.A.3, 6.EE.A.4, MP.3, MP.7 Building Towards: 7.EE.A.2

The materials reviewed for Desmos Math 6 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

Instructional approaches of the program and identification of the research-based strategies can be found in the Curriculum Guide, Courses, Our Philosophy. The materials state the following, “Every student is brilliant, but not every student feels brilliant in math class, particularly students from historically excluded communities. Research shows that students who believe they have brilliant ideas to add to the math classroom learn more.¹ Our aim (which links to Desmos Equity Principles) is for students to see themselves and their classmates as having powerful mathematical ideas. In the words of the NRC report Adding It Up, we want students to develop a ‘productive disposition-[the] habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.’² Our curriculum is designed with students’ ideas at its center. We pose problems that invite a variety of approaches before formalizing them. This is based on the idea that ‘students learn mathematics as a result of solving problems. Mathematical ideas are the outcomes of the problem-solving experience rather than the elements that must be taught before problem solving.’³ Students take an active role (individually, in pairs, and in groups) in developing their own ideas first and then synthesize as a class. The curriculum utilizes both the dynamic and interactive nature of computers and the flexible and creative nature of paper to invite, celebrate, and develop students’ ideas. Digital lessons incorporate interpretive feedback to show students the meaning of their own thinking⁴ and offer opportunities for students to learn from each other’s responses⁵. Paper lessons often include movement around the classroom or other social features to support students in seeing each other’s brilliant ideas. This problem-based approach invites teachers to take a critical role. As facilitators, teachers anticipate strategies students may use, monitor those strategies, select and sequence students’ ideas, and orchestrate productive discussions to help students make connections between their ideas and others’ ideas.⁶  This approach to teaching and learning is supported by the teacher dashboard and conversation toolkit (both are linked).”

Works Cited include:

• ¹ Uttal, D. H. (1997). Beliefs about genetic influences on mathematics achievement: A cross-cultural comparison. Genetica, 99(2–3), 165–172. https://doi.org/10.1007/bf02259520

• ² National Research Council. (2001). Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Academy Press. doi.org/10.17226/9822

• ³ Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray, H., Olivier, A., & Wearne, D. (1996). Problem solving as a basis for reform in curriculum and instruction: The case of mathematics. Educational Researcher, 25(4), 12–21. https://doi.org/10.3102/0013189x025004012

• ⁴ Okita, S. Y., & Schwartz, D. L. (2013). Learning by teaching human pupils and teachable agents: The importance of recursive feedback. Journal of the Learning Sciences, 22(3), 375–412. https://doi.org/10.1080/10508406.2013.807263

• ⁵ Chase, C., Chin, D.B., Oppezzo, M., Schwartz, D.L. (2009). Teachable agents and the protégé effect: Increasing the effort towards learning. Journal of Science Education and Technology 18, 334–352. https://doi.org/10.1007/s10956-009-9180-4.

• ⁶ Smith, M.S., & Stein, M.K. (2018). 5 practices for orchestrating productive mathematics discussions (2nd ed.). SAGE Publications.

Research is also referenced under the Curriculum Guide, Instructional Routines, when the materials says, “Some of the instructional routines, known as Mathematical Language Routines (MLR), were developed by the Stanford University UL/SCALE team.” There is a link embedded to read the research.

The materials reviewed for Desmos Math 6 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Math 6 Overview, Math 6 Year-At-A-Glance document, includes a list of frequently used materials throughout the year as well as lesson-specific materials. Each unit contains a Unit Overview which provides a list of materials that will be used for that particular unit. Additionally, materials that are needed for a lesson will be listed on the lesson page directly under the learning goals. Examples include:

• In Math 6 Year-At-A-Glance, Frequently Used Materials include: Blank paper, Graph paper, Four-function or scientific calculators*, Geometry toolkits**, Measuring tools (rulers, yardsticks, meter sticks, and/or tape measures), Scissors, Tools for creating a visual display, *Students can use handheld calculators or access free calculators on their devices at desmos.com, **Math 6 Geometry toolkits consist of tracing paper, graph paper, scissors, a ruler, a protractor (optional), colored pencils (optional), and an index card to use as a straightedge or to mark right angles.  

• \In Math 6 Year-At-A-Glance, Lesson-Specific Materials include: 6.1.13: Card stock (optional), 6.2.08: Stopwatch or other timer, 6.3.10: Tape or glue (for attaching cards to the Student Worksheet), 6.4.03: 2-cup, \frac{1}{2}-cup and \frac{1}{3}–cup measures (optional), 6.4.13: Unit cubes (optional), 6.5.05: Index cards or slips of colored paper.

The materials reviewed for Desmos Math 6 meet expectations for having assessment information included in the materials to indicate which standards are assessed. The Curriculum Guide, Assessments, Types of Assessments, states the following: “Formal Assessment The Desmos curriculum includes two types of formal assessments: quizzes and end assessments. Quizzes are typically five problems and assess what students know and can do in part of a unit. End assessments are summative assessments that are typically seven or eight problems and include concepts and skills from the entire unit. These include multiple-choice, select all, short answer, and extended response prompts to give students differing opportunities to show what they know and to mirror the types of questions on many current standardized tests.” Assessments within the program consistently and accurately reference grade-level content standards on the Assessment Summary. Examples include:

• Unit 1, Quiz, Screen 2, Problem 1, “Which shape has an area of 8 square centimeters?” Answer choices are the following: “A, B, C, D” A graph is provided with four shapes on it and a scale of 1 cm.  The Quiz Summary denotes the standard assesses as 6.G.1 and MP7.

• Unit 4, Quiz, Screen 10, Problem 5.2, “Determine the value of 5\div\frac{3}{4} .” The Quiz Summary denotes the standard assesses as 6.NS.1 and MP2.

• Unit 8, End Assessment: Form A, Screen 15, Problem 7, “Create a dot plot with: At least five points. A median of 6. A mean that is less than the median.” The Assessment Summary and Rubric denotes the standard assesses as 6.SP.4 ,6.SP.5c, and MP6.

The materials reviewed for Desmos Math 6 meet expectations for including an assessment system that provides multiple opportunities throughout the grade to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

All Quizzes and end assessments include a digital and paper option answer key, for correcting students’ work.  Each Quiz includes a “Quiz Summary” identifying the standards assessed, what is being assessed and which lesson(s) most align to each problem. Each end assessment includes an “Assessment Summary and Rubric,” which includes all components of the “Quiz Summary” and a rubric for interpreting student performance. Both the “Quiz Summary” and “Assessment Summary and Rubric” contains a section called, “Suggested Next Steps:” for following-up with students that struggle on a particular problem. Examples include:

• Unit 1, End Assessment: Form A, Screen 8, Problem 5.2, “What is the surface area of this prism? Explain or show your reasoning.” The Assessment Summary and Rubric, provide the following scoring guidance: “Problem 5.2, Standard 6.EE.A.2.C, 6.G.A.4, Meeting/Exceeding 4 Work is complete and correct. 52 square centimeters. E.g., In this prism, h=2, I=4, and w=3. There are two faces whose area are h\times l=8 square centimeters, two faces whose areas are h\times w=6 square centimeters, and two faces whose areas are l\times w=12 square centimeters. So the total surface area is 8+8+6+6+12+12=52 square cm. Approaching 3 Correct answer with minor flaws in explanation. Incorrect answer with logical and complete explanation. Developing 2 Correct answer with incomplete explanation. Incorrect answer with explanation that communicates partial understanding of area. E.g., Students who write 26 square centimeters may have calculated the sum of the areas of the visible surfaces only. Beginning 1 Incorrect answer with incorrect explanation or without an explanation. 0 Did not attempt.“ The Suggested Next Steps: If students struggle are, “Math Language Development Consider using the mathematical language routine Critique, Correct, Clarify to help students understand and communicate Sol’s mistake and how it could be corrected. Consider revisiting Lesson 9, Activity 1.”

• Unit 5, End Assessment: Form A, Screen 9, Problem 5, “Select the expression that has the greater value. Explain your reasoning.” Choices are, “2\cdot0.003, 0.2\cdot0.03, They have the same value.” The Assessment Summary and Rubric, provides the following scoring guidance: “Problem 5, Standard 6.NS.B.3, MP3, Meeting/Exceeding 4 Work is complete and correct. They have the same value. Both expressions are equivalent to 0.006. Approaching 3 Correct answer with minor flaws in explanation. Incorrect answer with logical and complete explanation. Students who choose either expression may have correctly calculated that one of them is equivalent to 0.006. Developing 2 Correct answer with incomplete explanation. Incorrect answer with explanation that communicates partial understanding of decimal multiplication. Students who say they have the same value but do not explain what the value is or how they know. Beginning 1 Incorrect answer with incorrect explanation or without an explanation. 0 Did not attempt.“ The Suggested Next Steps: If students struggle are, “Consider having students calculate the value of each expression, instead of estimating. Consider revisiting Lesson 5, Activity 1, Screen 7. Select only one representation to match the cards with, showing multiple representations.”

• Unit 8, Quiz, Screen 5, Problem 4.1, “These dot plots show the number of minutes it took Arnav and Kanna to walk to school last week. Whose data has a mean of 15 minutes? Show or explain your thinking.” Choices are, “Arnva, Kanna, Both, Neither”. The Quiz Summary, provides the following: “Problem 4 (Standards: 6.SP.A.3, 6.SP.B.5.C, MP3) This problem assesses students’ ability to reason about the mean of a data set and calculate the MAD of a data set from a dot plot. It corresponds most directly to the work students did in Lesson 8: Pop It! And Lesson 9: Hoops.” The Suggested Next Steps: If students struggle are, “On Problem 4.1, suggest that students find the mean for both sets of data. If they struggle on Problem 4.2 ask them what mean absolute deviation means mathematically. Consider revisiting Lesson 7, Activity 2, Screen 6 and Lesson 9, Activity 1, Screen 7.”

The materials reviewed for Desmos Math 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.

Formative assessments include Quizzes and End Assessments. All assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types 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. Examples Include:

• Unit 2, End Assessment Form A, Screen 4, Problem 3.1, develops the full intent of 6.RP.3 as students use ratio and rate reasoning to solve real-world problems. “Caleb’s favorite shade of green uses a ratio of 5 cups of blue paint to 3 cups of yellow paint. Caleb bought 12 cups of yellow paint. How much blue paint will he need to make his green? Use the sketch tool if it helps you with your thinking.” 

• Unit 5, End Assessment: Form B, Screen 10, Problem 6.1, develops the full intent of MP3 as students construct viable arguments and critique the reasoning of others. The problem states, “Here is the work Liam did to determine the least common multiple of 3 and 9. Explain why he is incorrect.”

• Unit 7, Quiz, Screen 3, Problem 2, develops the full intent of 6.NS.6a and 6.NS.6c as students identify and plot positive and negative numbers on a number line. “1. Drag each number to its approximate location on the number line. 2. Plot and label the opposite of each number on the number line.” Students are given a number line with 0 and 1 labeled, and given two numbers -4 and \frac{8}{3} to place on the number line.

The materials reviewed for Desmos Math 6 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

The Curriculum Guide, Support for Students with Disabilities, states the following about the materials: “The Desmos Math Curriculum is designed to support and maximize students’ strengths and abilities in the following ways:

• Each lesson is designed using the Universal Design for Learning (UDL) Guidelines…

• Each lesson includes strategies for accommodation and support based on the areas of cognitive functioning.

• Opportunities for extension and support are provided when appropriate.

• Most digital activities are screen reader friendly.

To support all students in accessing and participating in meaningful and challenging tasks, every lesson in the curriculum incorporates opportunities for engagement, representation, and action, and expression based on the Universal Design for Learning Guidelines.” The curriculum highlights the following six design choices that support access: “Consistent Lesson Structure, Student Choice, Variety of Output Methods, Concepts Build From Informal to Formal, Interpretive Feedback, and Opportunities for Self-Reflection.

The Desmos approach to modifying our curriculum is based on students' strengths and needs in the areas of cognitive functioning (Brodesky et al., 2002). Each lesson embeds suggestions for instructional moves to support students with disabilities. These are intended to provide teachers with strategies to increase access and eliminate barriers without reducing the mathematical demand of the task.” The materials use the following areas of cognitive functioning to guide their work: Conceptual processing, Visual-Spatial Processing, Organization, Memory and attention, Executive functioning, Fine-motor Skills, and Language.

These areas of cognitive functioning are embedded throughout the materials in the “Student Supports” within applicable digital lessons or listed under “Support for Students with Disabilities” in the Lesson Guide for some paper lessons. Examples include:

• Unit 2, Lesson 11, Lesson Guide, Activity 1: Sort‘em, students determine which questions from a variety of situations could be solved using equivalent ratios. “Support for Students with Disabilities Conceptual Processing: Processing Time Begin with a demonstration of the first problem to provide access to students who benefit from clear and explicit instructions. Check in with individual students, as needed, to assess for comprehension during each step of the activity.”

• Unit 5, Lesson 14, Screen 1, Warm-Up, “Abdel is grilling tofu dogs for his friends. His favorite tofu dogs come in packs of 8. His favorite buns come in packs of 6. What advice would you give to Abdel on how many packs to purchase?” Student Supports, “Students With Disabilities Conceptual Processing: Eliminate Barriers Use dogs and buns or objects like unit cubes and rods to demonstrate the situation described on this screen.”

• Unit 8, Lesson 4, Screen 5, Match-A-Plot, “Ebony made a dot plot and wrote this description. The center is at 7. There is a large spread. It looks like mountains. Create a dot plot that matches Ebony's description.”  Student Supports, “Students With Disabilities Conceptual Processing: Eliminate Barriers To assist students in recognizing the connections between new problems and prior work, consider asking them if any of the dot plots on the previous screen match Ebony’s description. Receptive Language: Processing Time Consider reading the prompt aloud and inviting one or more students to paraphrase it in their own words to support students who benefit from both reading and listening.”

The materials reviewed for Desmos Math 6 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.  

The Curriculum Guide, Lessons, provides an optional activity, “Are You Ready for More?” which is available in some lessons. “Are You Ready for More? offers students who finish an activity early an opportunity to continue exploring a concept more deeply. This is often beyond the scope of the lesson and is intentionally available to all students.”Additionally, some lessons’ screens provide ideas for early finishers in the Teacher Moves section. These ideas act as extensions to the activity that the student is currently working on, and allow them to engage with the activity at a higher level of complexity. Examples include:

• Unit 2, Lesson 3, Lesson Guide, Activity 2: Rice Around the World, students use equivalent ratios to adapt rice recipes from around the world. One example on the Student Worksheet, provides students with a recipe for Jollof Rice. The ingredients listed make one large bowl, and students must determine how much of each ingredient is needed to make two large bowls. Another example, provides students with a recipe for Arroz Con Leche. The ingredients listed serve four people and students must determine how much of each ingredient is needed for 12 people. The Lesson Guide, Activity 2: Rice Around the World states, “Early Finishers Encourage students to choose one of the recipes and determine the ingredients needed to make the dish for the whole class.”

• Unit 5, Lesson 14, Screen 4, LEAST Common Multiple, students learn how to determine the least common multiple (LCM) of two numbers by using different strategies. “The least common multiple (LCM) is the smallest number that is a common multiple of two numbers. What is the least common multiple of 6 and 15?” Students are provided a chart with the multiples of six placed in a circle and the multiples of fifteen placed in a square.  Teacher Moves, “Early Finishers Encourage students to determine as many pairs of numbers as they can that also have a least common multiple of 30.”

• Unit 8, Lesson 7, Screen 11, Are You Ready for More?, students create a data set in order to get a mean of seven. “Add at least four more points to create a dot plot that has a mean of 7. Click on the axis to add points. Then check your work. How many of these dot plots can you make? Note: You can also click on the points to remove them.” There is an interactive activity with a number line on the screen. It contains one dot above the number 3. Students can click on the axis to add more points. Teacher Moves, “Facilitation Invite students who finish Screens 5–10 early to explore this screen. Encourage students to share responses with each other in place of a whole-class discussion.”

The materials reviewed for Desmos Math 6 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics. 

The Curriculum Guide, Support for Multilingual Learners, states the following: “Desmos believes that there is a strong connection between learning content and learning language, both for students who are more familiar with formal English and for students who are less familiar. Therefore, language support is embedded into the curriculum in many different ways. In addition, the curriculum is built to highlight the strengths of each student and to surface the many assets students bring to the classroom. This resumption of competence is the foundation of all our work, and particularly of our support for multilingual students.” Curriculum Design That Supports Language Development, states “Every lesson in the curriculum incorporates opportunities for students to develop and use language as they grapple with new math ideas.” These opportunities are broken into the following four areas: 

• “Opportunities for Students to Read, Write, Speak and Listen

  • The Desmos Math Curriculum provides lots of opportunities for students to engage in all four language domains: speaking, listening, reading, and writing (e.g., text inputs, partner conversations, whole-class discussions). 

• Intentional Space for Informal Language

  • When students are learning a new idea, we invite them to use their own informal language to start, then make connections to more formal vocabulary or definitions.

• Math and Language in Context 

  • The Desmos Curriculum uses the digital medium to make mathematical concepts dynamic and delightful, helping students at all language proficiency levels make sense of problems and the mathematics.

• Embedded Mathematical Language Routines 

  • The Desmos 6-8 Math Curriculum is designed to be paired with Mathematical Language Routines, which support ‘students simultaneously learning mathematical practices, content, and language.’” 

Additionally, “Each lesson includes suggestions for instructional moves to support multilingual students. These are intended to provide teachers with strategies to increase access and eliminate barriers without reducing the mathematical demand of the task. These supports for multilingual students can be found in the purple Teacher Moves tab and in the Teacher Guide. These supports include: Explicit vocabulary instruction with visuals. Processing time prior to whole-class discussion. Sentence frames to support speaking opportunities. Instructions broken down step by step . Background knowledge or context explicitly addressed.”

Examples of these supports within the materials include the following:

• Unit 2, Lesson 11, Lesson Guide, Warm-Up, “Support for Multilingual Learners Receptive/Expressive Language: Eliminate Strategic Pairing Pair students to aid them in comprehension and expression of understanding.”

• Unit 5, Lesson 2, Screen 2, Show 0.45, Student Supports, “Multilingual Learners Receptive Language: Visual Aids Create or review an anchor chart that publicly displays tenths, hundredths, and thousandths in decimal and fraction form to aid in explanations and reasoning. Expressive Language: Eliminate Barriers Give students opportunities to practice saying the terms tenths, hundredths, and thousandths aloud.”

• Unit 8, Lesson 1, Screen 2, Warm-Up, Student Supports, “Multilingual Learners Receptive Language: Eliminate Barriers Consider reviewing the phrase ‘How much time do you spend _______’ to support students with comprehension throughout the lesson.”

The materials reviewed for Desmos Math 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. 

Virtual and physical manipulatives support student understanding throughout the materials. Examples include:

• Unit 1, Lesson 3, Student Worksheet, Activity 1: Area Strategies, students calculate the areas of parallelograms on a grid and reflect on their strategies. The activity states, “Use any strategy to determine the area of as many of these parallelograms as you can. Use the workspace below if it helps you with your thinking. Then record each area in the table.” Teachers provide students with a supplement sheet of all the parallelograms on graph paper, and scissors so that the students can cut out the parallelograms to help determine their areas.

• Unit 6, Lesson 1, Screen 1, Warm-up, students connect solving for an unknown with balancing a see-saw. The materials state, “Here are some weights on a see-saw. 1. Drag the movable point to adjust one of the weights. 2. Discuss what you notice and wonder.” Students are provided with a picture of a see-saw with two weights on one side and one weight on the other. The two weights are labeled “?” and “3 lb.” and the weight on the other side is labeled “7 lb.” Students can use a slider to manipulate the weight of the “?” and the see-saw moves based on the number the slider is on.