The materials reviewed for Desmos Math 7 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

In the Curriculum Guide, Courses, Conceptual Understanding, “Lessons develop students’ conceptual understanding by inviting them into familiar or accessible contexts and asking them for their own ideas before presenting more formal mathematics.” The materials include problems and questions that allow for students to develop conceptual understanding throughout the grade level. Examples include:

• Unit 2, Lesson 2, Screen 4, Two Strategies, students analyze proportional relationships in a real world setting to complete a table of values (7.RP.A). “Here are two different strategies for finding the number of balloons for the rubber duck. Discuss how Ariel and Emma would use their strategies to finish their tables.” Two tables are provided, one shows Ariel’s strategy and one shows Emma’s strategy. 

• Unit 5, Lesson 2, Screen 6, Settle a Dispute, students use a vertical number line to apply and extend previous understandings of addition and subtraction to add and subtract rational numbers (7.NS.1). “Marc and Naoki are trying to evaluate $3-(-2)$. Marc says, ‘This is like adding 2 anchors, so the submarine goes DOWN to 1.’ Naoki claims, ‘This is like removing 2 anchors, so the submarine goes UP to 5.’ Who is correct?” Choices include: Marc (1), Naoki (5), Both, or Neither. An image of a submarine in water with a vertical number line next to it is provided. The submarine is placed at three on the number line. There is a point at five on the number line labeled Naoki and a point at one labeled Marc.

• Unit 8, Lesson 3, Screen 3, Update Your Prediction, students analyze a simulation to predict the contents of a mystery bag (7.SP.6). “The graph shows the results from 200 picks. Use these results to decide how many of the 10 blocks you think are green.”

The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. Examples include:

• Unit 4, Quiz, Screen 4, Problem 3, students use proportional reasoning to solve multistep ratio and percent problems (7.RP.3). “To make a certain shade of paint, Anya mixed $\frac{2}{3}$ cups of white paint with $2\frac{2}{3}$ cups of blue paint. How many cups of blue paint should she mix with $\frac{3}{4}$ cups of white paint to make the same shade?” 

• Unit 6, Lesson 2, Screen 9, Which Restaurant?, students use a bar diagram to solve real-world  problems with equations (7.EE.B). On previous screens students selected tape diagrams to determine how much Raven and her three siblings can spend on a meal if they have $44. One restaurant charges a $3 service fee for each meal, and another charges a $6 service fee for their order. “Here are diagrams that represent the situations on the previous screens. Figure out the value of x and y in the diagrams. Enter your values in the table below. Explain to a neighbor what your values say about which restaurant the siblings should choose.” Two tape diagrams are provided. The tape diagram labeled Burrito Express has four equal boxes labeled x and a box labeled six. The entire tape diagram is labeled 44. The tape diagram labeled Salads-R-Us has four equal boxes labeled . The entire tape diagram is labeled 44.

• Unit 7, Lesson 5, Practice Problems, Screen 3, Problem 2, students construct triangles from three measures of angles or sides and determine the conditions of a unique triangle, more than one triangle, or no triangle, as they determine a possible third length of a triangle given two side lengths (7.G.2). “One side of a triangle is 5.5 inches long. Another is 10.5  inches long. Which of the following could be the length of the third side? Select all that apply.” Answer choices include: 3 inches, 5 inches, 7 inches, 10 inches, 12 inches, and 20 inches.

The materials reviewed for Desmos Math 7 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

In the Curriculum Guide, Courses, Procedural Fluency, ”In order to transfer skills, students should be able to solve problems with accuracy and flexibility. Several structures in the Desmos Curriculum support procedural fluency: repeated challenges, where students engage in a series of challenges on the same topic, challenge creators, where students challenge themselves and their classmates to a question they create, and paper practice days that use social structures to reinforce skills before assessments.”

Materials develop procedural skills and fluency throughout the grade level. Examples include:

• Unit 3, Lesson 3, Screen 6, From Radius to Circumference, students develop procedural skill and fluency using the formula for circumference of a circle to solve problems (7.G.4). “The radius of this circle is 7 centimeters. What is its approximate circumference?” A picture of a circle with radius of 7 cm labeled.

• Unit 5, Lesson 6, Screen 12, Cool-Down, students develop fluency in multiplying rational numbers (7.NS.2). “Determine the value of each expression.” Choices include: $(-5)\sdot4$, $6\sdot(-2)$, $(-3)\sdot(-7)$. 

• Unit 6, Lesson 13, Screen 9, Write an Inequality, students construct a simple inequality by reasoning about the quantities (7.EE.4). “Write an inequality that represents this graph.” Students are given a number line with an open circle on 19 with a line heading to the right.

The materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Examples include:

• Unit 1, Lesson 2, Practice Problems, Screen 3, Problem 1.2, students solve problems involving scale drawings of geometric figures (7.G.1). “For each scaled copy, write the scale factor from triangle T to that triangle. If the triangle is not a scaled copy, leave it blank.” The screen contains a triangle T with side lengths 3, 4, and 5. There are 6 other triangles labeled A through F with their side lengths labeled. There is a table with two columns, Triangle and Scale Factor, for the student to enter work. 

• Unit 4, Quiz, Screen 2,  Problem 1, students use proportional relationships to solve multi step percent problems (7.RP.3). “The value of a car decreases over time. This year, Faaria’s car is worth $22,000. If the value of Faaria’s car decreases by 8%, what will her car be worth next year?” Students are given choices ranging from $4,400 to $23,760.

• Unit 8, Lesson 7, Screen 5-6, 3-Day Vacay and Make a Simulation, students design and use a simulation to generate frequencies for compound events (7.SP.8). Students are given a display which shows the percent chance for rain over the period of three days. Screen 5, “Ivan is planning a 3-day vacation. Here is the forecast at his destination. What do you think is the probability that it will rain at least once during these 3 days? Drag the slider to show your guess.” The slider ranges from 0% to 100%. Screen 6, “Ivan wants to design a simulation to estimate the probability of rain during his vacation. Add blocks to each bag to match the probability of rain on each day.”  An interactive is provided with Fri 25%, Sat 25% and Sun 40%.  Under each forecast is a bag and blocks representing rain and sunshine.

The materials reviewed for Desmos Math 7 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics. 

In the Curriculum Guide, Courses, Application, students “have opportunities to apply what they have learned to new mathematical or real-world contexts. Concepts are often introduced in context and most units end by inviting students to apply their learning…” Materials include multiple routine and non-routine applications of mathematics throughout the grade level. Examples include:

• Unit 2, Lesson 9, Screen 6, Better Mileage, students explain what point (x, y) on the graph of a proportional relationship means in terms of the situation (7.RP.2d). “Kaya wants to buy a new vehicle that gets better gas mileage than her truck. Which vehicle should she pick?” A graph is provided with Maximum Distance (miles) on the y-axis and Gas (gallons) on the x-axis. There is a line labeled “Truck,”and a point labeled “Vehicle A”, and a point labeled “Vehicle B”, on the graph. Students click on either Vehicle A or B and explain their thinking on why they would buy that vehicle.

• Unit 5, Lesson 10, Screen 2, Greater than Zero, students use their knowledge of rational numbers to make the given inequality true (7.NS.1). “Make a true inequality by dragging the cards. Then press Check My Work.” Students are given the inequality [  ]([  ][  ] + [  ]) > 0 and the card choices are: -1, 2, 3, -4, -5, 6, 7, -8.  

• Unit 7, Lesson 13, Screen 8, Are You Ready for More, students solve real-world problems involving volume, and surface area of right prisms (7.G.6).  “On paper, complete the following tasks: 1. Sketch a container that can hold 120 cubic units of popcorn while using as little paper as possible. 2. Label your container's dimensions. Then calculate the amount of paper it uses.” A graphic is provided of a popcorn container in a right prism with height  (h) , width (w), and length (l).  Students have the option of unfolding the container. 

Materials provide opportunities for students to independently demonstrate multiple routine and non-routine applications of mathematics throughout the grade level. Examples include:

• Unit 4, Lesson 3, Screen 6, Sheets of Stickers, students compute unit rates with ratios of fractions (7.RP.1).  “Cho is considering buying stickers by the sheet. Four sheets cost 14. How much would sheets cost?” 

• Unit 5, Lesson 8, Practice Problems, Screen 5, Problem 3.2, students apply and extend previous understandings of multiplication and division of fractions to multiply and divide rational numbers (7.NS.2). “A machine that drills holes for wells drilled to a depth of -72 feet in one day (24 hours). If the machine drilled at a constant rate, what was the depth after 15 hours?”

• Unit 6, Lesson 3, Screen 11, Lesson Synthesis, students use variables to represent quantities in a real-world or mathematical problem (7.EE.4). “Explain how the number 9 is important in each representation. In the story… In the equation… In the tape diagram…” Students are given the following three cards:  Story - Jaylin buys 3 bags of bagels. The store gives her 5 bagels for free, making it 32 bagels total. Equation $-3x+5=32$. Tape Diagram - Students are given a tape diagram divided into four pieces, the three pieces labeled x are of equal size the last piece is labeled 5. All pieces together must equal 32. 

• Unit 8, Lesson 11, Screen 7, Write a Headline, students use the data from a sampling method to make generalizations about a population in the form of a headline (7.SP.1). “Pick a sampling method below and use the sample to write a headline about how much time Americans spend with their friends per day.” Students are given the following sampling methods to choose from: “Ask 10 students in one high school classroom”, “Ask 10 random people at a nursing home”, and “Dial 10 random phone numbers and ask whoever answers.” Clicking on each sampling method generates a different Mean and dot plot.

The materials reviewed for Desmos Math 7 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

All three aspects of rigor are present independently throughout Grade 7. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Unit 2, Lesson 5, Screen 9, Lesson Synthesis, students demonstrate conceptual understanding as they explain what they know about proportional relationships by creating their own situation given an equation (7.RP.2). “Make up a situation that could be represented by the equation $r=10p$.  Be sure to explain what r, 10, and p represent in your situation.” 

• Unit 5, Quiz 1, Screen 4, Problem 3.1, students develop procedural skill and fluency while understanding subtraction of rational numbers as the additive inverse (7.NS.1c).  “Determine the value of the variable that makes the equation true. $15-a=17$.”

• Unit 8, Lesson 5, Screen 8, 100 Rolls, students approximate the probability of a chance event be running the simulation and collecting data (7.SP.6). “Roll the number cube as many times as you want. What do you think is the probability that Player 1 wins?” Students are able to roll a number cube 100 times by pressing the “Roll 100 Times” button, the results are recorded in a chart. Player 1 wins if an even number is rolled, and Player 2 wins if an odd number is rolled.

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single unit of study throughout the materials. Examples include:

• Unit 3, Practice Day 2, Student Workspace Sheet, Problems 1-5, engage students in conceptual understanding, procedural skill and fluency, and application as they use the formulas for the area and circumference of a circle to solve real-world problems (7.G.4 & 7.G.6). Students solve a real-world problem (application) related to circumference and area of circles. Students must demonstrate conceptual understanding by determining which questions refer to circumference and area and then sort the questions from smallest measurement to largest. At the end of the activity students must calculate the circumference and area of a circle using their estimates. Problems 1-5 tasks students with the following: “1. Sort the cards into two groups based on whether you would use the circumference or the area of a circle to answer the question. Record your answers below. 2. Sort the cards in each group from smallest measurement to largest. Record your answers below. 3. Select one circumference card and one area card to examine more closely. What information do you need? Estimates for this information: 4. Use your estimates to calculate an answer to your circumference question. 5. Use your estimates to calculate the answer to your area question.” 

• Unit 4, Lesson 6, Student Worksheet, Activity 2: Sea Green Turtle, students build conceptual understanding and application as they use proportional relationships to solve multi step percent problems (7.RP.3). “Some beaches where green sea turtles come ashore to lay eggs have been made protected sanctuaries so the eggs will not be disturbed. This year, there were 234 nesting turtles at a sanctuary. That number is a 10% decrease compared to last year. Create each representation to show how many nesting turtles were at the sanctuary last year. How many nesting turtles were at the sanctuary last year?” Students are asked to create a double number line, a table, and an equation.

• Unit 5, Practice Day 1, Task Cards, Task 2: Cafeteria Food Debt, Problems 1 and 2, students engage in procedural skill and fluency alongside application as they solve a real-world problem involving addition and subtraction of rational numbers (7.NS.1d).  “At the beginning of the month, Emika had $24 in her school cafeteria account. The table below shows how her account balance changed over the course of three weeks. 1. Complete the table for weeks 2-3. 2. How much would Emika have to deposit into the account during week 4 so that the final balance is positive?” A table is included with the columns: Week, Beginning Balance ($), Final Balance ($), Expression for Difference Between Final and Beginning, and Change ($). The row labeled Week 1 is complete. In row 2, students are given the beginning and final balances and must write the expression and calculate the change. In row 3, students are given the beginning balance and the change, and must write the expression and final balance. 

• Unit 6, Lesson 17, Student Worksheet, Activity 1: Orange Juice and Donuts, Problems 1-3 engage students in conceptual understanding, procedural skill and fluency, and application as they solve word problems leading to inequalities of the form $px+q>r$ or $px+q<r$, where p, q, and r are specific rational numbers and interpret the solution in the context of the problem (7.EE.4b). “Kiandra wants to surprise some friends before school with orange juice and donuts. At the store, an orange juice costs $2.15 and a donut costs $0.75 . There is no sales tax. The store has a $10 purchase minimum for credit cards. Kiandra used her credit card to pay. How many friends might she have bought treats for? 1. Write an inequality that describes Kiandra’s situation. 2. Solve the inequality you wrote. 3. What does the solution to your inequality mean in this situation?”

The materials reviewed for Desmos Math 7 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

The MPs are explicitly identified for teachers in the Math 7 Lessons and Standards section found in the Math 7 Overview. MP1 is identified and connected to grade-level content, and intentionally developed to meet its full intent. Students make sense of problems and persevere in solving them as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Lesson 10, Student Worksheet, Activity 1: Measure Your Classroom, students plan a solution pathway to solve a problem involving scale drawings. “With your team, on blank paper: 1. Write a plan for how you will gather, record, and check your measurements. 2. Sketch the outline of your classroom and any permanent objects. Include all measurements.” Activity Facilitation, “Display Sheet 2 of the Teacher Projection Sheets to the class and review the image as an example of what they will produce as a team in Activity 1. Consider reading each question for teams to consider aloud and discussing any questions that might be unclear. Next, invite students to work together to make a plan for creating their rough sketch and gathering the measurements they need (MP1). Follow with a brief whole-class discussion to hear strategies groups have incorporated into their plans.” Students make sense of the problem as they work together to make a plan for creating the group rough sketches, and gathering measurements.

• Unit 5, Lesson 5, Screen 2, Puzzle #1, students use perseverance and creativity to solve a puzzle involving adding and subtracting signed numbers (MP1). “Make a true equation by dragging and flipping the cards. Try to use as few flips as possible. Then press ‘Check my Work.’” Students are provided an equation tool with the equation, an unknown plus an unknown is equal to an unknown subtracted from an unknown. The original card choices are: 1, 2, 3, 4, -5, -6, -7 and -8. All cards can be flipped from positive to negative and vice versa. 

• Unit 6, Lesson 12, Student Worksheet, Activity 1: Three Reads, students solve multi-step equations. “Kyrie is making ___ invitations to their school’s Community Day. They have already made ___ invitations, and they want to finish the rest of them within a week. Kyrie plans to spread out the remaining work so that they can make the same number of invitations each day.” Students answer: “1. With a partner, discuss what this situation is about. 2. Draw a tape or hanger diagram to represent this situation. 3. Given these values, adjust your diagram. Then use your diagram to figure out how many invitations Kyrie should make each day.” Students make sense of the verbal descriptions of situations.

• Unit 8, Lesson 10, Screen 7, See Some Samples, students use a slider to examine the relationship between the sample mean and population mean. “1. Drag the point to collect crabs and see the mean width for the sample. 2. Discuss the advantages and disadvantages of using a large sample.” Students make sense of the relationship between the sample mean and the population mean as they think about the advantages and disadvantages of using a large sample, as well as the role of randomness in sampling.

MP2 is identified and connected to grade-level content, and intentionally developed to meet its full intent. Students reason abstractly and quantitatively as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Lesson 7, Student Worksheet, Activity 1, Will it Fit?, students use the measurements of a scale drawing to calculate the actual dimensions of a figure. “Here is the scale drawing that Karima presented to her neighborhood park’s board of directors. 1. The scale for Karima’s drawing is 2 cm to 5 m. Explain what this means in your own words. 2. Will Karima’s court fit in the 20-by-20-meter square area the park directors designated for the court? Use your measuring tools and the table below to help you with your thinking. Round each measurement to the nearest tenth of a centimeter. Explain how you know whether or not the court will fit.” Students find the scale and actual measurements of the Length of the Court, the Width of the Court, the Hoop to 3-pt Line, and the 3-pt. Line to the Side Line. Students reason abstractly and quantitatively as they find the measurement of the scale drawing and convert to the dimensions of the actual court. 

• Unit 2, Lesson 5, Screen 4, Cake Calculations, students represent a proportional relationship symbolically with an equation. “A cake recipe says to use 3 cups of flour for every 2 cakes. Write a proportional equation to calculate the amount of flour needed, f, for any number of cakes, c.”

• Unit 5, Lesson 7, Screen 6, What It Means, students connect the concepts of rate, time, and position. Students are given a picture of a turtle on a number line and scaled by 10s with the interval -30 ft to 30 ft and -3.2 minutes in a white box. “One student wrote the following equation to determine Tam's position on the previous screen: $(-5)(-3.2)=16$. Explain what each number represents in the scenario. -5 represents…, -3.2 represents…, 16 represents…” Teacher Moves, “The purpose of this discussion is to connect rate, time, and position with numerical expressions and equations, and to explain why the product of two negative values is positive (MP2).”

• Unit 8, Lesson 5, Screen 4, Keeping Track of Heads, students read and interpret graphs of probabilities. Students are shown a video in which a coin is flipped, then the “Fraction of Heads (So Far)” is graphed along with the “Number of Flips.” Students answer: “1. What does the point $(10, \frac{1}{2})$ mean? 2. How might a graph help you determine if a coin is fair?”

The materials reviewed for Desmos Math 7 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

The MPs are explicitly identified for teachers in the Math 7 Lessons and Standards section found in the Math 7 Overview. Students engage with MP3 in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Lesson 2, Screen 6, Anushka's Robot, students explain the relationship between lengths in a figure and corresponding lengths in a scaled copy. “Anushka built a robot and made a copy that is not a scaled copy. Explain Anushka's strategy. What advice would you give Anushka to help her make her new robot a scaled copy?”  Students are given an image with two robots, one with dimensions labeled 3, 4, 1 and the other labeled with dimensions 9, 10, 1. Teacher Moves, “Consider asking students how they know the two figures are not scaled copies. If it does not come up naturally, consider mentioning that the angles of the two figures are different. Give students 2–3 minutes to inspect Anushka’s robot and record their responses. Encourage students to read others’ responses and/or discuss their response with a partner and decide if others' strategies were similar to or different from their own. If time permits, consider using the snapshots tool or dashboard's teacher view to display several student suggestions for different ways to make a scaled copy. Routine (optional): Consider using the routine Critique, Correct, Clarify to help students communicate about errors and ambiguities in math ideas and language.” Students engage with MP3 as they explain/justify their strategies and thinking orally or in writing using concrete models, drawings, actions, or numbers.

• Unit 2, Lesson 12, Student Worksheet, Cool-Down, students use proportional relationships to analyze a problem about water usage. “Marshall wants to buy a kitchen faucet.  Faucet A fills a 4-gallon water jug in 1 minute. Faucet B fills a 1-gallon water jug in 20 seconds. Which faucet uses less water? Explain your thinking.” Students engage with MP3 as they explain/justify their strategies and thinking orally or in writing using concrete models, drawings, actions, or numbers.

• Unit 4, Lesson 4, Screen 4, It’s All About the Money, students create different strategies to determine 10% less than 15. “In order to make more money, DesWorst Granola bars are now 10% shorter. If the original bar was 15 centimeters long, how long is the new granola bar? Use paper if it helps you with your thinking.” Teacher Move, “While students are working, monitor for different expressions students use and select them using the snapshots tool. When most students have completed this screen, facilitate a whole-class discussion. Ask students to justify their strategy and critique each other’s reasoning (MP3).”

• Unit 6, Lesson 8, Student Worksheet, Activity 2: Step by Step by Step by Step, students critique the reasoning of others as they apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. “Here is an equation and the first steps that Sadia and Amir wrote to solve it.” The worksheet shows two students’ first step in solving the equation $2(x-9)=10$. “1. Are each of their first steps correct? Explain your reasoning.” 

The materials reviewed for Desmos Math 7 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

The MPs are explicitly identified for teachers in the Math 7 Lessons and Standards section found in the Math 7 Overview.There is intentional development of MP4 to meet its full intent in connection to grade-level content. Examples include:

• Unit 4, Lesson 9, Student Worksheet, Activity 2: What’s Fair?, Problem 2, students model a real-world situation, involving multistep ratio and percent problems, with an appropriate expression and use it to solve the problem and interpret the results. In Activity 1, students are presented with information about four servers who work in four different restaurants. It tells how many hours they work, how many tables they serve in a typical week, the average bill at the restaurant, and the percentage tip they typically earn. Activity 2: What’s Fair?, Problem 2, “Consider these three approaches to paying servers that we have seen so far: A. Servers get paid $2.13 per hour, plus tips. B. Servers get paid $7.25 per hour, plus tips. C. Servers get paid $15 per hour, with no tips. Invent and describe a system to determine a server’s pay that you think is fairer than the ones above. Calculate what each of the four people would earn under your system.” This activity attends to the full intent of MP4, model with mathematics.

• Unit 6, Lesson 17, Lesson Guide, Warm-Up, provides guidance for teachers to engage students in MP4 as they solve word problems leading to inequalities. Teacher Projection Sheets, Warm-Up, “Jamal volunteers to hand out sandwiches to people who are hungry in his community. He raised $85 and i s trying to figure out how many sandwiches he can purchase for $6.25 each. He writes the inequality $6.25x≤85$. Then he solves the inequality and gets $x≤13.6$. Select all the statements that are true about this situation. A. He can order 13.6 sandwiches. B. He can order 14 sandwiches. C. He can order 12 sandwiches. D. He can order 9.5 sandwiches. E. He can order – 4 sandwiches.”Lesson Guide, “Facilitation: Display Sheet 1 of the Teacher Projection Sheets. Consider reading the story aloud as a class and asking students what connections they make to the story. Ask them what the variable x represents in Jamal’s inequality. Then give students one minute to think quietly and another minute to share their reasoning about which statements are true with a partner.” This activity intentionally develops MP4, model with mathematics.

• Unit 8, Lesson 7, Screen 1, Warm-up, students apply what they have learned about probability tools and repeated experiments to simulate multi-step real-world events. “Aniyah saw this forecast for the weekend's weather. What do you think is the probability that it will rain at least once this weekend? 1. Drag the slider to show your guess. 2. Explain your thinking”  Students are given two images depicting weather with one labled SAT (40%) and SUN (60%). This activity attends to MP4, model with mathematics.

There is intentional development of MP5 to meet its full intent in connection to grade-level content. Examples include:

• Unit 4, Lesson 3, Screen 5, Cho’s Logo, Teacher Moves, provides guidance for teachers to engage students in MP5 as they consider the effectiveness of different tools as they determine unknown values in proportional relationships. “Early Finishers: Encourage students who finish Screens 2–5 early to use different tool and to determine which tool is more effective for this problem and why. (MP5).”

• Unit 6, Lesson 3, Screen 2, Baking Cookies, Teacher Moves, provides guidance for teachers to engage students in MP5 by allowing students to consider the available tools when solving a mathematical problem. “Activity 1 Purpose: On Screens 2–5, students use equations and tape diagrams to make sense of a situation. Students answer a question about a situation involving baking cookies using any representation (MP5).”

• Unit 8, Lesson 2, Screen 7, How Many, Teacher Moves, provides guidance for teachers to engage students in MP5 as they show their familiarity with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful. Students experiment with how to organize their thinking about a complex sample space. “Facilitation: …While students are working, monitor for students who use drawings, organized lists, or other tools to help them determine the size of the sample space (MP5).”

The materials reviewed for Desmos Math 7 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

The MPs are explicitly identified for teachers in the Math 7 Lessons and Standards section found in the Math 7 Overview. Students have many opportunities to attend to precision in connection to grade-level content as they work with support of the teacher and independently throughout the modules. Examples include:

• Unit 1, Lesson 10, Student Worksheet, Activity 3: Creating Your Scaled Drawing, Problem 6, students select a unit of measure and scale to create a drawing. “On graph paper, make an accurate scale drawing using your chosen scale.” Teacher guidance, “Remind students that unlike their sketch, these scale drawings should be drawn with precision to scale.” This problem attends to MP6 as students attend to the precision of mathematics as they create a scaled drawing accurately.

• Unit 8, Lesson 2, Screen 4, Prob-bear-bility, students use a randomizer to understand sample space to determined the probability of an event. “1. Here is a randomizer. Press “Spin” to get a random creature. 2. Drag the point to show how likely you think it is to get a bear on one spin.” Students are given a number line with a range from zero to one, zero being impossible and one being certain. It is also labeled at the $\frac{1}{2}$ mark.  Students attend to precision as they understand that the probability of a chance event is a number between 0 and 1.

Students have frequent opportunities to attend to the specialized language of math in connection to grade-level content as they work with support of the teacher and independently throughout the modules. Examples include:

• Unit 2, Lesson 2, Screen 6, Proportional Relationships, “When two quantities are always in an equivalent ratio, it is called a proportional relationship. Which of these two tables represents the proportional relationship between weight and balloons? Explain your thinking.” The screen contains two tables with the data for weight (oz.) and number of balloons. Students attend to the specialized language of mathematics as they use the term proportional relationship and apply their understanding to recognize proportional relationships between quantities. 

• Unit 3, Lesson 2, Screen 6, Madison’s Circles, “Madison made a drawing using circles. Describe her drawing as precisely as you can so that someone who can't see her drawing could recreate it. Use the sketch tool if it helps you with your thinking.” Students are given an image of four circles inscribed in one another on the coordinate plane. Teacher facilaton suggests, “The purpose of this discussion is to support students in attending to precision in their language when describing circles (MP6). Use the mathematical language routine Critique, Correct, Clarify to support students in attending to precision in their language. If it does not come up naturally, consider introducing the word center.” This activity attends to MP6 as students attend to the specialized language of mathematics as they describe properties of circles. Students begin to use precise terminology to describe the parts of a circle. 

• Unit 7, Lesson 2, Screen 3, Complementary and Supplementary, “The terms complementary and supplementary describe special pairs of angles. Describe what you think these terms mean.” The screen contains pictures of examples of complementary and supplementary angles. Students attend to the specialized language of math and develop an understanding of supplementary and complementary angles.

The materials reviewed for Desmos Math 7 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. 

The MPs are explicitly identified for teachers in the Math 7 Lessons and Standards section found in the Math 7 Overview. MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities throughout the modules to look for, describe, and make use of patterns within problem-solving as they work with support of the teacher and independently. Examples include:

• Unit 2, Lesson 7, Screen 3, Are They Proportional?, “Here are the equations from the worksheet. Select all the equations that represent a proportional relationship. Then explain one way to decide if an equation represents a proportional relationship.” The screen contains 4 equations: $y=12x$, $y=500+35x$, $y=\frac{1}{2}x$, $y=x^2$. Students look for and use structure to generalize types of equations that do and do not represent proportional relationships. 

• Unit 5, Lesson 4, Student Worksheet, Activity 2: Draw Your Own Conclusion, Problem 3, “Select one of these statements. Explain whether it is always, sometimes, or never true. Use examples and number line diagrams to support your explanation.” There are 3 statements: Statement D: $x-y$ is the opposite of $y-x$, Statement E: $x$ is less than $x+y$, Statement F: $x-y$ is greater than $x+y$. Students look for and use structure to reason about whether expressions involving integer operations with variables are always, sometimes, or never true.

• Unit 7, Lesson 11, Screen 10, Lesson Synthesis,“Here are several prisms you’ve seen in this lesson. Describe a general strategy for determining the volume of any prism.” The screen shows images of three different prisms. Students look for patterns or structures to make generalizations about determining the volume of prisms.

MP8 is identified and connected to grade level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities throughout the materials, with support of the teacher or during independent practice, to use repeated reasoning in order to make generalizations and build a deeper understanding of grade-level math concepts. Examples include:

• Unit 1, Lesson 5, Screen 5, Scaling Rectangles, “Here is a rectangle and a scaled copy. The area of the original rectangle is 8 square units. What is the area of the scaled copy?” Students use repeated reasoning to explain why it is that when a figure is scaled by a scale factor, the area is not scaled by the same amount.

• Unit 4, Lesson 3, Screen 2, Different Sizes, “Here's a logo that Aditi is making into stickers. Enter the missing values so that the logo looks the same on each sticker. Then describe your strategy.” Students are given an image with four stickers. Two stickers have the length and width labeled. The other stickers have only the length or width labeled, and students must use repeated reasoning to find the other. This activity attends to the full intent of MP8, look for and express regularity in repeated reasoning as students notice repeated calculations to understand algorithms and make generalizations or create shortcuts.

• Unit 8, Lesson 4, Screen 5, Your Turn, “1. Spin Amari and Nathan’s spinner as many times as you want. 2. Discuss what is happening to the fraction of red spins as you add more spins.” Students are given a spinner and a graph to work, while they approximate the probability of how many red spins vs blue spins. This activity attends to the full intent of MP8, use repeated reasoning as students approximate the probability of a chance event by collecting data.

The materials reviewed fo Desmos Math 7 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 1, Unit Overview, “Section 1: Scaled Copies (Lessons 1-5 + (Practice Day + Quiz) Describe how scaling affects lengths, angles, and areas in scaled copies. Use scale factors to create and compare scaled copies. Section 2: Scale Drawings (Lessons 6–10 + Practice Day) Represent distances in the real world using scales and scale drawings.“

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 3, Lesson 3, Summary, the materials state, “About This Lesson The purpose of this lesson is for students to make sense of the relationship between the diameter of a circle and its circumference. By the end of the lesson, students should be able to both describe and use this relationship to calculate unknown measurements.” Lesson Summary, “Warm-Up (5 minutes) The purpose of the warm-up is for students to learn the term circumference and connect it to what they already know about perimeter. The warm-up asks students to estimate the circumference of a circle that has been unrolled into a straight line. Activity 1: Gathering Data (15 minutes) The purpose of this activity is for students to recognize that the relationship between diameter and circumference of a circle is proportional. Students use different methods to physically measure the circumference of round objects and then analyze their class's data to determine if the relationship is proportional or not. Students will learn what the constant of proportionality of this relationship is in. Activity 2. Note: Students need access to several different round objects for this activity. Activity 2: Constant of Proportionality (15 minutes) The purpose of this activity is for students to discover that the number $\pi$ is the constant of proportionality for the relationship they analyzed in Activity 1. They then use this relationship to calculate the radius, diameter, or circumference of several different circles. This activity includes a focus on using different approximations for $\pi$, such as 3.14 and $\frac{22}{7}$. Lesson Synthesis (5 minutes) The purpose of the synthesis is to surface how students used the relationships between the radius, diameter, and circumference of a circle to calculate missing measurements. Cool-Down (5 minutes)”

• Unit 6, Lesson 4, Seeing Structure, Lesson Guide, Activity 2: Write Your Own, the materials state, “Purpose: Students use the features of a situation to write their own question, then determine a solution and write an equation. Students also write their own situations. Facilitation: Consider starting this activity by asking students what they remember about how to write good questions or what they think makes a good question at the end of a situation. This builds on the work students did writing questions in Unit 4, Lesson 12. Then give students 5–10 minutes to write and answer questions for Problems 1–2. Consider consulting with pairs about their solutions to Problems 1 and 2 before they begin the ‘Are You Ready for More?’ problem. Progress Check: This is a great place to check students’ progress writing equations from situations. Offer individual support where needed, or lead a whole-class discussion if enough students are struggling. Support for Multilingual Learners Receptive/Expressive Language: Processing Time Students who benefit from extra processing time would also be aided by reading each situation aloud, either in pairs or as a class.”

• Unit 7, Lesson 6, Screen 10, Different Polygons, the materials state, “Here are four quadrilaterals that Lukas made with side lengths 3, 3, 5 and 5 units. Describe why it is possible for Lukas to create quadrilaterals that are not identical copies.”Teacher Moves, “This is a possible discussion screen.Facilitation: When students have responded, facilitate a brief discussion about why triangles with all the same side lengths are identical, but the same is not true for quadrilaterals. Consider asking questions like: Which quadrilateral was most like the one you sketched? Why do you think what we observed with triangles isn’t true for quadrilaterals? Question to push students’ thinking: Do you think there are other polygons with the same special property as triangles (that if all their side lengths are the same, then they are identical copies)?”

The materials reviewed for Desmos Math 7 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. 

All Unit Overview Videos end with an explanation and example from later grades. The presenters 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 work with rational and irrational numbers and square roots and cube roots. On the screen is a “Hit the Target #1” problem from beyond the current course, the presenter explains the problem is about asking students to enter a fraction so they can get as close as they can to the $\sqrt{13}$. Additionally, the presenter explains that the goal is to realize that students can get closer and closer, but can negative reach $\sqrt{13}$ after this screen they give a formal definition of irrational numbers.  

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, “Similarity and Dilations (8.G.A.3, 8.G.A.4) In this unit, students use scale factors to create and compare scaled copies. In Math 8, Unit 2, students will apply this to understand similar figures, which are scaled copies of one another. Figures are similar if one can fit exactly over the other after rigid transformations (translations, rotations, reflections) and dilations. Dilations are a transformation in which each point on a figure moves along a line and changes its distance from a fixed point (called the center of dilation). Each distance is multiplied by the same scale factor. In this example, each point in 𝐴𝐵'𝐶'𝐷' is twice as far from the center of dilation (𝐴) as it is in 𝐴𝐵𝐶𝐷. The scale factor from figure 𝐴𝐵𝐶𝐷 to figure 𝐴𝐵'𝐶'𝐷' is 2, and 𝐴𝐵𝐶𝐷 is similar to 𝐴𝐵'𝐶'𝐷'.“ An graph of the original figure before and after the dilation is provided.

The materials reviewed for Desmos Math 7 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

The Math 7 Overview, contains the Math Grade 7 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 2, Unit Facilitation Guide, Connections to Prior Learning, states, “The following concepts from previous grades may support students in meeting grade-level standards in this unit: Understanding and using ratio and rate language in a variety of contexts. (6.RP.A.1 , 6.RP.A.2) Finding equivalent ratios using a scale factor. (6.RP.A.2 , 6.RP.A.3) Finding unit rates in context. (6.RP.A.3.b and 6.RP.A.2) Given one value of a ratio, use the unit rate to find the other. (6.RP.A.3.b) Representing equivalent ratios in a table. (6.RP.A.1 , 6.RP.A.3.a) Graphing points in the coordinate plane. (6.RP.A.3.a)”

• Unit 7, Unit Facilitation Guide, Connections to Prior Learning, states, “The following concepts from previous grades may support students in meeting grade-level standards in this unit: Describing and estimating angle measures. (4.G.A.1 , 4.MD.C.5, 4.MD.C.6) Calculating the area of triangles and non-rectangular quadrilaterals. (6.G.A.1) Calculating the surface area and volume of right rectangular prisms. (6.G.A.2 , 6.G.A.4)”

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 2, Lesson Overview Page, Learning Goals, “Interpret a tape diagram that represents a relationship in context. Use a tape diagram to determine an unknown value in context.” Common Core State Standards: Building On: 6.EE.B.5, Addressing: 7.EE.B.3, MP.2, MP.3, Building Towards: 7.EE.B.4

The materials reviewed for Desmos Math 7 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 thinking$^4$ and offer opportunities for students to learn from each other’s responses$^5$. 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 7 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Math 7 Overview, Math 7 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 7 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, 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, **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 7 Year-At-A-Glance, Lesson-Specific Materials include: 7.1.06: Centimeter rulers, 7.2.12: 1-gallon bucket or jug, 7.3.03: Circular objects (e.g., empty toilet paper rolls), 7.7.08: Compasses, 7.8.01: Coins (e.g., a penny and a nickel); standard number cubes; a bag, bowl, or cup; paper clips; card stock, 7.8.05: Penny, standard number cube.

The materials reviewed for Desmos Math 7 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 2, End Assessment: Form B, Screen 7, Problem 6, “The equation $p=2.22x$  relates mass in pounds, p , to mass in kilograms, x. Is there a proportional relationship between p and x ? Explain or show your thinking.” Answer choices are, proportional and not proportional. The Assessment Summary and Rubric denotes the standard assesses as 7.RP.2a and MP6.

• Unit 4, End Assessment: Form A, Screen 12, Problem 7.3, “The cost of every college is expected to increase 3.5% next year. The cost to attend Faber College is currently $24000. If the percentage increase stays constant, what will the cost be in two years?” The Assessment Summary and Rubric denotes the standard assesses as 7.RP.2c, 7.RP.3, 7.EE.3, MP4 and MP6.

• Unit 7, Quiz, Screen 8, Problem 5.3, “Here are three lines that intersect at one point. Laila wrote the equation $x+18=90$. Describe the error that Laila might have made.” The Quiz Summary denotes the standard assesses as 7.G.5 and MP7.

The materials reviewed for Desmos Math 7 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 6.1, “A trail runner gets a new map of her favorite mountain. Her old map has a scale of 1 cm to 100 m. Her new map has a scale of 1 cm to 500 m. If the maps represent the same area, are the distances on the new map longer, shorter, or the same size as the old map? Explain your thinking.” Choices are, “Longer, Shorter, The same size” . The Assessment Summary and Rubric, provide the following scoring guidance: “Problem 6.1 Standard 7.G.A.1, MP6 Meeting/Exceeding 4 Student successfully answers the question and includes a logical and complete explanation. Approaching 3 Correct choice with minor flaws in explanation. Incorrect choice with logical and complete explanation. Students may not have understood the question but communicates conceptual understanding of the relationship between map scales. Developing 2 Correct choice with incomplete explanation. Incorrect choice with explanation that communicates partial understanding of the relationship between the map scales. Beginning 1 Incorrect choice with incorrect explanation or without an explanation. 0 Did not attempt.“ The Suggested Next Steps: If students struggle are, “Consider asking students how the given scales compare in size. Consider asking them how the scales can be used to determine the desired distance. Consider revisiting Lesson 8, Activity 3, Problem 1. ”

• Unit 4, End Assessment: Form A, Screen 9, Problem 6.3, “A store is offering a  20% discount on all items. The price of a hat after the discount is $18. What was the original price?” The Assessment Summary and Rubric, provides the following scoring guidance: “Problem 6.3 Standard 7.RP.A.3 Meeting/Exceeding 4 Response is complete and correct. $22.50 Approaching 3 Work shows conceptual understanding and mastery, with some errors. Students who respond with $14.40 may have solved the problem ‘What is the price after a 20% discount on $18?’ Developing 2 Work shows a developing but incomplete conceptual understanding, with significant errors. Students who respond with $3.60 may have calculated 20% of $18. Beginning 1 Weak evidence of understanding how to calculate the original price given a percent decrease. 0 Did not attempt.“  The Suggested Next Steps: If students struggle are, “Consider suggesting that students begin by completing Problem 6.3 and working backward, using their answers to help answer the other problems. Consider revisiting Lesson 6, Activity 2.”

• Unit 8, Quiz, Screen 5, Problem 3.2, “ Esi does an experiment where she picks a block out of a bag without looking 50 times, putting it back each time. She picks a green block 32 times. If the bag has 8 blocks, how many do you think are green?” The Quiz Summary, provides the following: “Problem 3 (Standards: 7.SP.C.6, 7.SP.C.7.B, MP2) This problem assesses students’ ability to use the results from a repeated experiment to make predictions about the sample space and about future events. Students reason abstractly and quantitatively when they use experimental data to make predictions about future events and unknown information. This problem corresponds most directly to the work students did in Lesson 3: Mystery Bag.”  The Suggested Next Steps: If students struggle are, “Consider asking students how they can use the results of Esi’s experiment to make predictions if the bag contained 8 blocks or if there were 200 picks. Consider revisiting Lesson 3, Activity 1.”

The materials reviewed for Desmos Math 7 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 1, End Assessment: Form A, Screen 2, Problem 1, develops the full intent of 7.G.1 as students solve problems involving scale drawings of geometric figures. “Select ALL the scaled copies of rectangle A.” Students are given six rectangles; the rectangles have varying measurements, so students have to determine which ones are scaled copies of rectangle A. 

• Unit 3, End Assessment: Form B, Screen 2, Problem 1, develops the full intent of MP6 as students attend to precision while calculating the circumference of a circle. “A circle has a radius of 40 centimeters. Which of these is closest to its circumference?” Students are given the following multiple choice items to choose from: 

  • 126 centimeters

  • 5027 centimeters

  • 1600 centimeters

  • 251 centimeters

• Unit 8, Quiz, Screen 7, Problem 4.2, attends to the full intent of 7.SP.8 as students find probability of a multistep event. “What is the probability you will spin at least one item with cheese in the name?” Students are given two pictures. One picture is labeled “Appetizers,” and the other one is “Entrees.” There are different choices under each picture. Students need to use the pictures to decide what the chances are of spinning an item with cheese.

The materials reviewed for Desmos Math 7 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 5, Screen 2, All the Time, “The car travels at a constant speed. After 6 seconds, it travels 180 meters.Write a proportional equation to find the car's distance, d, at any time, t.” Student Supports, “Students With Disabilities Conceptual Processing: Processing Time Check in with individual students, as needed, to assess for comprehension during each step of the activity.”

• Unit 5, Lesson 4, Lesson Guide, Activity 1: Draw Your Own Diagram, students make sense of number lines for expressions involving subtraction. “Support for Students With Disabilities Fine Motor Skills: Peer Tutors Allow students who struggle with fine motor skills to dictate how to create each number line.” Executive Functioning: Eliminate Barriers Chunk this activity into more manageable parts (e.g., presenting one pair of number lines at a time).”

• Unit 8, Lesson 9, Lesson Guide, Activity 1: Marco’s Mean and MAD, students use mean and MAD on a set of data to calculate which option is the fastest way to school.  “Support for Students With Disabilities Executive Functioning: Visual Aids Create an anchor chart for public display that describes how to calculate the mean and mean absolute deviation of a data set (with an example of each) for future reference.”

The materials reviewed for Desmos Math 7 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 5, Screen 8, Are You Ready for More?, “students develop fluency writing and using equations to make sense of proportional relationships in a variety of contexts.” “Here are some facts about this truck: It travels at an average rate of 50 miles per hour. It can travel 6 miles for each gallon of gas. How many hours can the truck travel without stopping if it has a full tank of 150 gallons?”  Teacher Moves, “This screen is designed as an extra challenge for students who finish Screens 4–7 before the class discussion on Screen 7. Consider inviting these students to share responses with each other in place of a whole-class discussion.”

• Unit 4, Lesson 2, Lesson Guide, Activity 1: Which Recipe?, students apply what they learned about constants of proportionality to make an argument about recipes involving fractional quantities. Student Worksheet, Activity 1: Which Recipe?, states, “Amara is making peach cobbler. She has three recipes and is deciding which one to make. Amara wants to make a recipe that isn’t too sweet. 1. She thinks Recipe C will be the least sweet because it has the least amount of sugar. Do you agree? Explain your thinking. 2. Which recipe should she make? Explain your thinking. 3. Is the relationship between number of servings and total amount of sugar proportional for each recipe? Explain your thinking.” Students are given three recipes with the number of servings and the quantity of each ingredient. The Lesson Guide, Activity 1: Which Recipe? states, “Early Finishers: Encourage students who finish Problems 1–3 early to choose one of the other recipes to adjust so that it is just as sweet as the one they chose for Amara.”

• Unit 7, Lesson 5, Screen 5, Will It Work?, students will determine whether or not three side lengths will make a triangle. “Diamond is convinced that a third side of 19 units will form a triangle. Mohamed thinks that 19 units is too long. Who is correct? Explain your thinking.” The screen contains an image with two sides of a triangle labeled seven and eleven. The Teacher Moves states the following: “Early Finishers: Encourage students who finish early to describe all the possible third side lengths that would form a triangle or to create their own sets of two side lengths that will form a triangle with a 19-unit-long side.”

The materials reviewed for Desmos Math 7 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 1, Lesson 3, Screen 6, Reflect #2, Student Supports, “Multilingual Learners MLR 7 (Compare and Connect)After students share their approaches for [calculating the total area], ask groups to discuss, ‘What is similar, and what is different?’ between the approaches. Ask students to describe what worked well with their approach and what might make an approach more complete or easier to understand.”

• Unit 6, Lesson 3, Screen 4, Liam’s Strategy, Student Support, “Multilingual Learners Expressive Language: Eliminate Barriers Provide sentence frames to help students explain their reasoning (e.g., This equation might be helpful because ________.).” 

• Unit 7, Lesson 12, Lesson Guide, Lesson Synthesis, “Support for Multilingual Learners Expressive Language: Eliminate Barriers Provide sentence frames to help students explain their strategy (e.g., First,________.Then,________.).”

The materials reviewed for Desmos Math 7 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 3, Lesson 6, Student Worksheet, Activity 2: Circle vs. Radius Square, students visualize the relationship between the radius, radius square, and area of a circle. The activity states, “For this activity, you will need circles and sets of radius squares. For each circle and set of radius squares, cut up each radius square and rearrange the pieces so that they cover just the circle. Record what you and your classmates discover in the table below. In general, how many radius squares do you think it takes to cover a circle?” Teachers provide students with a supplement sheet of each circle, a corresponding radius square, and scissors so that the students can cut out each radius square to help determine how many radius squares cover a circle.

• Unit 7, Lesson 7, Screen 6, Reflection, students use virtual segments to build triangles. The materials state, “Malik claims you will always get identical triangles if the angle is between the two sides. Is Malik Correct? Explain your thinking.” Students are given the answer choices of “Yes, No, or I’m not sure”. After they explain their thinking they can share with the class their thoughts by clicking the “Share With Class” button. Students are provided a virtual workspace where they can manipulate two sides around a given angle.