8th Grade - Gateway 3

The materials reviewed for Desmos Math 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. 

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. Examples include:

• Unit 2, Unit Overview, “Section 1: Dilations (Lessons 1–4 + Quiz), Describe dilations precisely in terms of their center of dilation and scale factor. Apply dilations to figures on and off of a coordinate grid. Section 2: Similarity (Lessons 5–8 + Quiz), Identify similar figures and properties of similar figures using transformations. Section 3: Slope (Lessons 9–10 + Practice Day), Explain slope in terms of similar triangles on the same line and determine the slopes of lines.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of specific learning objectives. The Curriculum Guide, Lesson Guides and Teacher Tips, describe support for facilitation throughout the program. “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 3, Lesson 5, Summary, “About This Lesson The previous lesson looked in depth at an example of a linear relationship that was not proportional and then examined an interpretation of the slope as the rate of change for a linear relationship. In this lesson, slope remains important. In addition, students learn the new term vertical intercept or y-intercept for the point where the graph of the linear relationship touches the y-axis.” “Lesson Summary: Warm-Up (5 minutes) The purpose of the warm-up is to introduce students to the general context in this lesson (the relationship between flag height and time) and to connect visual and graphical representations of one specific flag. Activity 1: Flags, Part 1 (5 minutes) The purpose of this activity is for students to relate the starting height and speed of a flag to a graph showing the flag’s height over time (MP2). Activity 2: Flags, Part 2 (15 minutes) The purpose of this activity is for students to make connections between various representations (including graphs, tables, and expressions) of two flags’ height and time (MP4). Students will use repeated reasoning of a flags height at specific times to develop an equation modeling this relationship (MP8). Activity 3: Flags, Part 3 (10 minutes) The purpose of this activity is for students to strengthen their understanding of how the parameters in a linear equation affect the positive vertical intercept and slope of a graph. Lesson Synthesis (5 minutes) The purpose of the synthesis is for students to discuss how to use a graph or an equation to identify the vertical intercept and slope of a given scenario and make sense of them in context. Cool-Down (5 minutes)”

• Unit 5, Lesson 7, Lesson Guide, Activity 1: Awards, “Launch This activity has two parts: answer the questions, then create a visual display. Tell students to continue working in their groups of 2–3. Throughout Activity 1, students will need to work together to answer the questions, as each representation holds a “piece of the puzzle. Teacher Moves Circulate through the room as students work, offering help as needed. Routine (optional): Consider using the routine Compare and Connect to support students in making sense of multiple strategies and connecting those strategies to their own.”

• Unit 7, Lesson 8, Screen 3, Challenge #1, “What number is represented by the point on the number line?” Teacher Moves, “Activity Launch Tell students that their task in this activity is to look at a zoomed-in number line and determine what number is represented by the point. Teacher Moves Consider using the student view in the dashboard to show students the type of feedback they’ll receive when they submit an answer. Challenge students to get the correct answer in as few tries as possible by reflecting carefully on the feedback they receive after each attempt. Facilitation Consider using pacing to restrict students to Screens 3–8.”

The materials reviewed for Desmos Math 8 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 contains 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. For example:

• Unit 5, Unit Overview, Unit Overview Video, the presenter talks about thethree “Big Ideas” of the Unit (Introduction to Functions, Representing and Interpreting Functions, and Volume), and the goal(s) of each “Big Idea”. Examples are provided from lessons, while the presenter talks about key vocabulary, and how the “Big Ideas” are connected to previous and future units within the grade. 

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 1, Unit Facilitation Guide, Connections to Future Learning, “Function Transformations (HSF.BF.B.3) In this unit, students perform transformations on figures. In high school, students will perform transformations on functions. For example, the equation of the solid parabola is f(x)=x^2. By translating the parabola 2 units right and 1 unit up, the equation of the dashed parabola is g(x)=(x-2)^2+1.” A graph with both parabolas sketched is provided.

The materials reviewed for Desmos Math 8 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

The Math 8 Overview contains the Math Grade 8 Lessons and Standards document which includes:

• 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, “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 videos. 

Examples from the Unit Facilitation Guide include:

• Unit 3, Unit Facilitation Guide, Connections to Prior Learning, “The following concepts from previous grades may support students in meeting grade-level standards in this unit: Deciding whether or not quantities are in a proportional relationship. (7.RP.A.2.a) Using proportional relationships to solve problems. (7.RP.A.3) Writing equations to describe proportional relationships. ( 7.RP.A.2.c) Solving problems with positive and negative numbers. (7.EE.B.3) Applying transformations to lines. (8.G.A.1.a and 8.G.A.1.c)”

• Unit 8, Unit Facilitation Guide, Connections to Prior Learning, “The following concepts from previous grades or units may support students in meeting grade-level standards in this unit: Writing and evaluating numerical expressions involving whole-number exponents. (6.EE.A.1) Reading, writing, and comparing decimals. (5.NBT.A.3) Calculating the areas of right triangles and other polygons. (6.G.A.1) Determining distances in the coordinate plane. (6.G.A.3)”

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 8, Lesson 1, Lesson Overview Page, Learning Goals, “Recall how to calculate the area of a triangle. Calculate the area of a square with vertices at the intersection of grid lines using strategies like ‘decompose and rearrange’ and ‘surround and subtract’.” Common Core State Standards: Building On: 6.EE.A.1, 6.G.A.1 Addressing: 8.NS.A.2 Building Towards: 8.NS.A.2, 8.EE.A.2, 8.G.B, 8.G.B.6

The materials reviewed for Desmos Math 8 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. “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, “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 8 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Math 8 Overview, Math 8 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 t needed for a lesson are listed on the lesson page. Examples include:

• In Math 8 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, and Tools for creating a visual display. *Students can use handheld calculators or access free calculators on their devices at desmos.com. **Geometry toolkits consist of tracing paper, graph paper, colored pencils, scissors, a ruler, a protractor, and an index card to use as a straightedge or to mark right angles.   

• In Math 8 Year-At-A-Glance, Lesson-Specific Materials include: 8.1.13: Masking tape or blue painter’s tape, thick markers, 8.5.10–15: Models of cylinders, cones, and spheres (optional), 8.6.02: Rulers, meter sticks, or tape measures marked in centimeters.

The materials reviewed for Desmos Math 8 meet expectations for having assessment information included in the materials to indicate which standards are assessed. The Curriculum Guide, Assessments, Types of Assessments, “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 2, End Assessment: Form A, Screen 4, Problem 3, “Here is Triangle 1. Triangle 2 also has a 30\degree angle. Explain or show why Triangle 1 and Triangle 2 might not be similar to each other.” The Assessment Summary and Rubric denotes the standard assesses as 8.G.5 and MP3.

• Unit 5, Quiz 1, Screen 5, Problem 3.2, “Jaleel wrote a book. He wants to print some copies for his friends and family. The printing company charges a one-time fee of $200, plus $2 for each printed book. Is the number of books he prints a function of total cost? Explain your thinking.” The Quiz Summary denotes the standard assesses as 8.F.1, MP2 and MP6.

• Unit 8, Quiz, Screen 5, Problem 4, “Drag the movable points to the correct position on the number line.”  Student are given the points \sqrt[3]{9}, \sqrt{10}, \sqrt{16}, \sqrt[3]{27}.  The Quiz Summary denotes the standard assesses as 8.NS.2 and MP7.

The materials reviewed for Desmos Math 8 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 6, Problem 5.1, “Is shape A congruent to shape B? Use the digital tracing paper if it helps with your thinking. Explain your reasoning using translations, rotations, and/or reflections.” The Assessment Summary and Rubric, provide the following scoring guidance: “Problem 5.1, Standard 8.G.A.2, MP1, Meeting/Exceeding 4, Student successfully answers the question and includes a logical and complete explanation. Yes. I can reflect shape A, rotate it, and then translate it onto shape B. 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. Beginning 1 Incorrect answer with or without incorrect explanation. 0 Did not attempt.“ The Suggested Next Steps: If students struggle are, “Consider having students use the digital tracing paper to try out many different transformations. Help students get started by suggesting they try aligning only one part of the figures as a first step. Consider revisiting Lesson 9, Activity 1.”

• Unit 4, Quiz, Screen 5, Problem 3.2, “Liam, Anika, and Sai are each solving the same equation for x. Original equation: 12x+4=20x-12 The result of Anika’s first step was 3x+1=5x-3. Describe the first step Anika made for the equation.” The Quiz Summary, provides the following: “Problem 3 (Standards: 8.EE.C.7, MP3) In this problem, students describe the reasoning of others in solving a linear equation with one variable. This problem corresponds most directly to the work students did in Lesson 4: More Balanced Moves.” The Suggested Next Steps: If students struggle are, “Consider reminding students of valid balancing moves, then ask them which ones were used by Liam, Anika, and Sai. Consider revisiting Lesson 4, Activity 1.”

• Unit 6, End Assessment: Form A, Screen 5, Problem 3.2, “Use the sketch tool to draw a scatter plot that includes: At least six points. A negative, nonlinear association.”The Assessment Summary and Rubric, provides the following scoring guidance:“Problem 3.2, Standard 8.SP.A.1, MP1, Meeting/Exceeding 4 Work is complete and correct. Plot shows at least six points that are not on the same line, with a generally negative trend. Approaching 3, Work shows conceptual understanding and mastery, with minor errors. Students who plot at least five points that are not on the same line but with a positive trend may not understand positive and negative associations. Developing 2,Work shows a developing but incomplete conceptual understanding, with significant errors. Students who plot a linear negative association may need additional support understanding the difference between linear and nonlinear associations. Beginning 1,Weak evidence of understanding. Student plot a positive linear association. 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 the terms positive, negative, linear, and nonlinear as they relate to correlation in a data set. Consider revisiting the Cool-Down in Lesson 7.”

The materials reviewed for Desmos Math 8 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 examine 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 8 and 9, Problem 6.1 and 6.2, assesses MP1 as students make sense of problems and persevere in solving them. “All of the labeled points in the graph are on the same line. Determine the slope of the line.” Students are given a line with four sets of ordered pairs marked. Two of the ordered pairs have variables instead of numbers, so students will need to solve for slope. Problem 6.2 has the same graph as problem 6.1, and a table which tasks students to “Determine the values of a and b.”

• Unit 4, Quiz, Screen 7 and 8, Problem 4.1 and 4.2, assesses 8.EE.7 as students solve real-world problems in which two conditions are equal. Problem 4.1, “Imani and Esteban each have different audiobook club memberships. After listening to 4 audiobooks, whose book club costs more?” Students choose from: Imani, Esteban, or “They cost the same amount”. Problem 4.2, “After how many audiobooks with both book clubs cost the same total amount?”

• Unit 6, End Assessment: Form B, Screen 9, Problem 6.2, assesses 8.SP.4 as students use a two-way table to generate a relative frequency table. “This two-way table shows the number of adults and children who prefer pizza or hot dogs. Complete the relative frequency table by row. Round to the nearest percent. Use paper and a calculator to help you with your thinking.” A table is provided showing the preference of hot dogs and pizza for both adults and children. Students use this data to complete the relative frequency table.

The materials reviewed for Desmos Math 8 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, 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 1, Lesson 3, Screen 2, Challenge #1, “Use a sequence of transformations to transform the pre-image (shaded) onto the image.” Student Supports, “Students With Disabilities, Visual-Spatial Processing: Visual Aids, Provide printed copies of the representations for students to draw on or highlight.”

• Unit 4, Lesson 4, Lesson Guide, Warm-Up, students determine whether the move described in each statement maintains the equality of an equation. Student Supports, “Students With Disabilities, Memory: Processing Time, Provide sticky notes or mini whiteboards to aid students with working memory challenges.”

• Unit 7, Lesson 3, Lesson Guide, Activity: Power Pairs, students play a card game where they match up equivalent expressions. “Support for Students With Disabilities, Conceptual Processing: Eliminate Barriers Demonstrate the steps for the activity or game by having a group of students and staff play an example round while the rest of the class observes. Memory: Processing Time Provide sticky notes or mini whiteboards to aid students with working memory challenges.”

Paper lessons in Unit 5 do not have this section.

The materials reviewed for Desmos Math 8 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.” Examples include:

• Unit 2, Lesson 7, Screen 8, Are You Ready for More?, students examine angle measurements in triangles to determine whether or not two triangles are similar. “Here are two similar triangles. Santino says that in similar triangles, if you match up two pairs of sides at a vertex, then the third sides are always parallel. Is Santino correct? Explain your thinking.” The screen contains an interactive activity with two similar triangles. The students can move one around to do the investigation. There is a button, “Try New Triangles” that allows students to generate new sets of similar triangles. Teacher Moves, “This screen is designed to help differentiate the lesson by giving an extra challenge to students who finish Screen 7 ahead of time before the class discussion on Screen 9. Because only a subset of your class will complete this screen, we recommend you don't discuss it with the entire class.”  

• Unit 4, Lesson 11, Screen 10, Are You Ready for More?, students use the context of balanced hangers to determine the solution to a system of equations. “Find values for x and y so that both hangers balance. Press ‘Try It’ to see if the hangers balance.” The screen contains a table for students to fill in the value for x and y and a “Try It” button. The screen also contains a diagram with a hanger. On the left side, there are three triangles labeled x and three squares labeled 3. On the right side is another hangar. On the left side of that hangar are two triangles labeled x. On the right side there are four circles labeled y. When students input values for x and y and press “Try It”, the diagram animates to show whether it is correct or not. Teacher Moves, “This screen is designed to help differentiate the lesson by giving an extra challenge to students who finish Screen 9 ahead of time before the class discussion on Screen 11. Because only a subset of your class will complete this screen, we recommend you don't discuss it with the entire class.”  

• Unit 5, Lesson 12, Screen 9, Are You Ready for More?, “students use functions to explore how changing a cylinder’s radius or height impacts its volume.” “Explore the relationship between radius and height when volume is fixed. On paper, write what you notice and wonder.” The screen contains a graph of “Cylinders With 180\pi cu. cm Volume”. The x-axis is Radius (cm) and the y-axis is Height (cm). There is a point on the graph that corresponds to the cylinder on the left. Students can change the radius of the cylinder and the height changes as well to keep the volume constant. Teacher Moves, “This screen is designed to help differentiate the lesson by giving an extra challenge to students who finish Screens 5–8 ahead of time before the class discussion on Screen 10. Because only a subset of your class will complete this screen, we recommend you don't discuss it with the entire class.”

The materials reviewed for Desmos Math 8 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, “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 1, Lesson 4, Lesson Guide, Activity 2: Make My Transformation, “Support for Multilingual Learners Lighter Support: MLR 2 (Collect and Display) While students are working, circulate and collect examples of how students describe the transformations. Display these while students are working so that they can incorporate some into their discussions.”

• Unit 6, Lesson 8, Screen 10, Lesson Synthesis, Student Supports,“English Language Learners Lighter Support: MLR 8 (Discussion Supports) As students describe the line of fit, the individual points, or the associations, restate students' ideas as questions (i.e., using the discussion questions) in order to demonstrate mathematical language, clarify, and involve more students. Press for details by asking students to elaborate on an idea or to give an example from the image.”

• Unit 8, Lesson 3, Screen 2, Squaring Lines, Student Supports,“English Language Learners MLR 2 (Collect and Display) Circulate and listen to students talk during pair work or group work, and jot notes about common or important words and phrases, together with helpful sketches or diagrams. Record students’ words and sketches on a visual display to refer back to during whole-class discussions throughout the lesson.”

The materials reviewed for Desmos Math 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.

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

• Unit 1, Lesson 11, Student Worksheet, Activity 2: Tear It Up, students learn about the relationship of interior angles of triangles. The activity states, “1. On a blank sheet of paper, use a straightedge to draw two very different triangles. 2. Mark the vertices of each triangle and cut the triangles out. Then rip the three vertices off of the triangle. 3. Arrange the vertices of each triangle so that the three vertices meet with no overlap. 4. Compare your results with your classmates’ results. What do you notice about the sum of the angles in a triangle?”

• Unit 5, Lesson 10, Screen 4, Cone and Cylinder, students adjust the height of three-dimensional shapes to reason about volume. The materials state, “Adjust the height so the objects have the same volume. Then press ‘Try It.’” Students are provided a virtual workspace where they can manipulate the height of a cylinder, the cone height cannot be adjusted as it is filled with a purple liquid. Once students adjust the cylinder to the desired height and click the “Try It” button, the cone goes over the cylinder and begins to drain the liquid into the cylinder. If the cylinder is too big or too small the students can adjust the cylinder and try again.