The materials reviewed for Desmos Math 6 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 4, Screen 7, Balancing Act, students understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities (6.RP.1). “The table shows some ratios of limes to lychees that balance the scale. Dyani says 22 limes will balance with 55 lychees. Will the $22:55$ ratio balance?” A table is provided with the headings Limes and Lychees. The Limes column contains 2 and 20. The Lychees column contains 5 and 50.

• Unit 4, Lesson 3, Screen 6, Sirnee, students interpret and compute quotients of fractions to solve word problems (6.NS.1). “Emmanuel needs 2 cups of flour to make a sirnee, a sweet dish that is often made as part of Islamic celebratory feasts. He only has a $\frac{1}{3}$-cup measuring scoop. How many scoops does he need?”  

• Unit 6, Lesson 5, Student Worksheet, Cool-Down, students use variables to represent numbers and write expressions when solving a real-world or mathematical problem (6.EE.6). “Here is an equation: $x+2.5=10$.  Write a situation to match this equation. 1. Explain what x represents in your situation. 2. Determine the solution to the equation. 3. Explain what the solution means in your situation. ”

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

• Unit 2, Lesson 5, Screen 7, How Many Lemons, students use a double number line to generate equivalent ratios (6.RP.3). “Here is a new double number line. The scale balances with a ratio of 4 lemons to 6 limes. How many lemons will balance with 12 limes?” A scale is shown with 12 lines on one side and a question mark on the other. A double number line is given under the graphic, one is labeled lemons and the other limes.

• Unit 6, Lesson 2, Practice Problems, Screen 3, Problem 2.1-2.2, students connect tape diagrams and equations to solve a situation in context (6.EE.7). “Aaliyah filled a water bottle with 24 ounces of water before school. They drank 15 ounces at lunch. There are x ounces of water left. 1. Draw a tape diagram to represent the situation. 2. Select all of the equations that could represent this situation.” Choices are $24-15=x$, $24+15=x$, $x+15=24$, $15x=24$, $24\div15=x$.

• Unit 7, Lesson 5, Practice Problems, Screen 12, Explore, students complete a problem in which they compare integer and absolute values (6.NS.7). “1. Drag the cards so that each number sentence is true. (You will have one card left over.) 2. Describe your thinking.” A graphic is provided with three number sentences: An absolute value equal to a number, an absolute value greater than another absolute value, and a number that is less than an absolute value.  The card choices are: -3, -2, -1, 0, 1, 2, 3.

The materials reviewed for Desmos Math 6 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 2, Lesson 6, Screen 6, Apricots and Limes, students understand the concept of a ratio and use ratio language to describe a ratio relationship (6.RP.1). “The scale balances with a ratio of 10 apricots to 6 limes. Select all of the equivalent ratios.” Answer choices: 20 apricots to 16 limes, 50 apricots to 30 limes, 7 apricots to 3 limes, or 5 apricots to 3 limes.

• Unit 4, Lesson 13, Screen 8, Four Challenges, students apply the formulas $V=lwh$ and $V=bh$ to find volumes of right rectangular prisms with fractional edge lengths (6.G.2). “Use paper to calculate the volume of each prism.” The screen contains a table with two columns Dimensions (units), Volume (cubic units) and an image of the prism with the dimensions shown. Students enter their answer to the volume and click the “Check My Work” button to submit. The prism then begins to fill with their answer, if their answer is correct, a “Try Another'' button appears, if they are incorrect they can click “Try again.” Each new prism is accompanied by a new image with dimensions and the dimensions are entered into the table as well. Students are challenged to find the volume of four prisms.

• Unit 6, Lesson 10, Screen 5, Not Equivalent, students write and evaluate numerical expressions involving whole-number exponents (6.EE.1). “Victor put one card in this group that is not equivalent to the others. Which card is not equivalent in this group?” Choices include: $2^5$, $2\sdot2\sdot2\sdot2\sdot2$, $2+2+2+2+2$, $2^4\sdot2$. 

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

• Unit 3, Lesson 4, Practice Problems, Screen 4, Problem 2, students solve unit rate problems involving unit price (6.RP.3b). “The cost of 5 cans of pinto beans is $3.35. At this rate, how much do 11 cans of pinto beans cost?”

• Unit 5, Lesson 5, Practice Problems, Screen 5, Problems 4.1–4.3, students fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation (6.NS.3). “Determine the value of each expression.” Expressions include: $(0.3)\sdot(0.2)$, , $(1.2)\sdot5$. 

• Unit 7, Lesson 10, Screen 12, Cool-Down, students graph points in all four quadrants of the coordinate plane (6.NS.6). “Drag the points to these locations: (-10,4), (-10,-4), (2,-6).” There is a coordinate plane that ranges from -16 to 16 on both the x and y axis with three draggable points.

The materials reviewed for Desmos Math 6 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 13, Screen 9, Putting It All Together, students use ratio and rate reasoning to solve real-world problems (6.RP.3). “Overall, Evergreen requires a $4:1:3$ ratio of market-rate housing to affordable housing to green space. Here are 24 units of land. Design a neighborhood that meets Evergreen City's requirements.” There is a grid that is $4\times6$ (24 units) on the screen. There are icons for Market-Rate, Affordable, and Green above the grid. Students must place the appropriate quantities of each icon on the grid to maintain the required ratio.

• Unit 4, Lesson 8, Screen 4, How Many Bags, students interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions (6.NS.1). “It takes $\frac{2}{3}$ of a bag of soil to fill $\frac{1}{4}$ of this planter. How many bags does it take to fill 1 planter?” There is an image of the rectangular planter divided into fourths. It is shown that $\frac{1}{4}$ of the planter represents $\frac{2}{3}$ of a bag of soil.

• Unit 6, Lesson 15, Student Worksheet, Activity 1: What’s Missing, Problems 1-3, students analyze the relationship between the dependent and independent variables by matching a table, graph, and equation to a given situation (6.EE.9). Students are provided three situations and asked to tape or glue the corresponding table, graph and equation for each situation. Students are given the following situations to match a table, graph, and equation: “1. Amanda sells paletas for $2 each. What is the total amount of money she can earn? 2. Tameeka sells paletas for $2.50 each. What is the total amount of money she can earn? 3. Esteban sells piraquas for $3.50 each. What is the total amount of money he can earn?” Problem 1 task students to, “Choose one row above. Circle or highlight the price per item in each representation.” In Problem 2, students make comparisons to a graph from a previous situation, “Angel sells piraguas for $4.50 each. How will Angel’s graph be different from Esteban’s?” 

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

• Unit 1, Lesson 8, Screen 6, Jasmine and Callen, students find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes (6.G.1). “Here is how Jasmine and Callen calculated the area for this polygon. 1. Discuss with a partner how Jasmine’s and Callen’s methods are similar and how they are different. 2. Describe how you could help these students revise their work.” Students are shown cards with rectangles on them (one for Jasmine and one for Callen) and the methods they used to find their answers. 

• Unit 3, Lesson 9, Screen 12, Cool-Down, students solve problems involving percentages (6.RP.3). “Callen bought new sneakers for $60. Miko bought sneakers that cost 80% of that price. How much did Miko pay for his sneakers?”

• Unit 6, Quiz, Screen 10, Problem 5.1, students solve real-world and mathematical problems by writing and solving equations (6.EE.7). “Cho has $10 to buy tacos that cost $2.50 each. Cho can buy x tacos in total. Which equation represents this situation?” Students are given the following answer choices: $x+10=2.50$, $10x=2.50$, $x+2.50=10$, and $2.50x=10$.

• Unit 7, Lesson 12, Student Worksheet, Activity 2: Graph Telephone, students solve real-world problems by graphing points in all four quadrants of the coordinate plane (6.NS.8). “For this activity, you will need one Story Card.” The student group will be given one of four story cards (A, B, C, and D), a space for Round 1, Round 2, Round 3, and Round 4. Similar to the game of telephone, only one student in the group will see the original story, and graphs that information in Round 1, all the other students in the group will independently, either write a story based on a previous Round graph, or make a graph based on a previous Round story. At the end of the activity, students “unfold the paper and look at how the story changed throughout the rounds.”

The materials reviewed for Desmos Math 6 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 6. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Unit 3, Lesson 9, Practice Problems, Screen 7, Problem 3.1, students develop procedural skill and fluency while solving problems including finding the whole, given a part and the percent (6.RP.3c). “Leonardo works as a server in a restaurant. He gets a 20% tip on the food cost for every order. What tip will he get when the food costs $60?” 

• Unit 4, Lesson 3, Screen 10, Lesson Synthesis, students develop conceptual understanding while interpreting and computing quotients of fractions (6.NS.1). “How can you use an equation or a diagram to figure out how many $\frac{1}{2}$-cup scoops you need to make 6 cups?” A diagram is included which shows four cards one with $6\div\frac{1}{2}=?$, $\frac{1}{2}\times?=6$, a picture of six measuring cups with a line through the half and a tape diagram with 12 equal parts each labeled $\frac{1}{2}$ and the entire diagram measures 6. 

• Unit 6, Lesson 5, Practice Problems, Screen 4, Problem 2.1, students apply their understanding of solving an equation as a process of answering a question (6.EE.5). “Here is an equation: $\frac{1}{2}+x=4$. Write a situation that the equation could represent. Describe the meaning of x in your situation.”

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

• Unit 1, Lesson 6, Screen 5 and 6, Make your own & Areas, students build conceptual understanding alongside application as they find the area of right triangles (6.G.1). “Can you always combine two copies of a triangle to form a parallelogram? Let’s test it. 1. Create a triangle. Then press “Copy”. 2. Discuss with a partner how many different parallelograms you can create. Here is the triangle you created and a parallelogram. What is the area of the triangle? What is the area of the parallelogram?”

• Unit 3, Quiz, Screen 6, Problem 4.1-4.2, students develop procedural skill and fluency and apply their knowledge as they solve unit rate problems (6.RP.3b). “A strawberry milk recipe uses 3 teaspoons of strawberry syrup for every 8 ounces of milk. How many teaspoons of strawberry syrup per ounce of milk does this recipe use? How many ounces of milk are needed per teaspoon of strawberry syrup?” Students are given the option of using a sketch tool if it helps them with their thinking.

• Unit 5, Lesson 8, Screen 4, Reflect, students develop procedural skill and fluency alongside conceptual understanding while they fluently add, subtract, multiply and divide multi-digit decimals (6.NS.3). “Diamond claims that $2\div0.04$ has the same value as $200\div4$. Explain why this makes sense.” Students are given a diagram of two 100 square grids, one grid has four of its squares colored blue.

• Unit 7, Quiz, Screen 2, Problem 1, students develop conceptual understanding and application while understanding a rational number as a point on the number line (6.NS.6). “If these numbers were plotted on a number line, which would be farthest to the left?” Students are given the following numbers: $-1\frac{3}{4}$, $-\frac{11}{4}$, $-2$, and $\frac{11}{4}$.

The materials reviewed for Desmos Math 6 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 6 Lessons and Standards section found in the Math 6 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 2, Lesson 12, Screen 2, How Much of Each ?, students solve problems involving part-part-whole ratios. “Tyrone makes a green paint by mixing 3 cups of blue with 2 cups of yellow. He now needs 20 more cups of green paint to finish painting a mural. How much of each color should he mix?” The screen is interactive and allows the student to enter different numbers for cups of blue and yellow paint and then check their work. The Teacher Moves, Facilitation, “If students are having trouble getting started , encourage them to enter values and ask: Does this make a blue to yellow ratio of 3 : 2 ? Does this make 20 cups total? Monitor for different strategies, both correct and incorrect. Some students may pay more attention to 20 cups total while others will pay more attention to the 3 : 2 ratio (MP1).” Students make sense of the problem as they look for different strategies that will equal the correct ratio and total.

• Unit 4, Lesson 10, Lesson Guide, Activity 1: Match and Solve, “Students use expressions to represent questions in context, then use fraction operations (subtraction, multiplication, and division) to answer each question.” The activity, facilitation section, “Encourage students to justify each card placement before putting it on the worksheet. If students are struggling, consider inviting them to try solving a simpler problem first by substituting the fractions with integers before selecting an expression (MP1).” Students make sense of the problems and persevere in solving them as they solve the simpler problem(s) before writing their own expression and solution.

• Unit 7, Lesson 5, Screen 7, Puzzle #1, students work with inequalities to compare and order rational numbers and absolute values. “1. Make a true inequality by dragging the cards. 2. Explain how you know your inequality is true.” Students are given an interactive where they can drag the following numbers to create a true statement of inequality: -2, -1, 1, 2. The Teacher Moves, Launch, “Consider starting the activity paused and dragging the cards to create a false inequality, like $\vert-1\rvert>1$. Give students one minute to share with a partner how they know this inequality is false and then create a true inequality (MP1).” Students make sense of the problem as they explain why the false inequality is false, in order to create a true inequality.

• Unit 8, Lesson 14, Lesson Guide, Activity 1: Car or Plane ? and Activity 2: Bus or Train?, students interpret information in box plots and make connections between data and various plots (i.e. dot and box). Purpose, “This lesson introduces a new way to visualize data (as a box plot) and two new ways to measure spread (range and IQR). Students make sense of a box plot for a data set, then interpret information on a different box plot (MP1). Students also make connections between data sets, box plots, interquartile range, and range. In this lesson, students informally compare box plots as they decide which method of transportation they would recommend.” 

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 2, Lesson 7, Screen 8, Darkest to Lightest, students develop and share strategies for comparing ratios by considering different ratios of blue and white paint. “Order these ratios from darkest blue to lightest blue. Use paper to support your thinking.” The Teacher Moves, Launch and Facilitation, “Launch: Consider sharing that students can order the paint colors using any strategies they’ve heard or new ones that they come up with (MP2). Facilitation: Give students several minutes to order the colors. Encourage students to use paper to support their thinking. If students are struggling to get started, invite them to select any two paint colors and compare those, then to compare a third color to the ones they’ve already ordered.” Students reason quantitatively as they compare the ratios using equivalent ratios and other strategies.

• Unit 3, Lesson 9, Screen 7, Two Strategies, students examine different strategies to solve a problem with percentages. Students are given a ratio table and a double number line diagram, “Here are two different strategies for calculating the goal when 20% is 8 kilometers. Discuss how each student used ratios to calculate the goal.” Students reason quantitatively as they understand the relationships between problem scenarios and mathematical representations.

• Unit 6, Lesson 5, Student Worksheet, Activity 1: Stronger and Clearer Each Time, students connect equations to situations by writing their own situations to match equations, and then trade situations with classmates. “1. Select an equation from the list your teacher shared and determine the solution. 2. Write a situation to match this equation. 3. Explain what the variable represents in your situation.” Students are given boxes to write the equation and the solution. Students are provided space to write notes from their conversation and describe what the solution to the equation means in this situation. The mathematical content is enriched by students making sense of quantities and their relationships in problem situations.

• Unit 8, Lesson 4, Screen 8, Settle a Dispute, students compare and contrast dot plots, focusing on the center and spread of each data set. “Axel and Zoe studied the minimum wages from 2010. Axel said: I think $7.25 is the center of the data because it represents most of the states. Zoe disagreed: I think $7.25 is too low because there are states that are more than $7.25. Who do you agree with? Explain your thinking.” Students are given two dot plots to compare, one for Minimum Hourly Wage in 2010 and one for Minimum Hourly Wage in 2020. The Teacher Moves, Facilitation, “Consider displaying the distribution of responses using the dashboard’s teacher view, calling attention to any conflict or consensus you see. Highlight students who make connections between the dot plot representation and Axel's and Zoe’s statements about minimum wage (MP2).” This activity attends to the full intent of MP2, reason abstractly and quantitatively as students understand the relationships between the dot plot and Axel and Zoe statements.

The materials reviewed for Desmos Math 6 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 6 Lessons and Standards section found in the Math 6 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 2, Lesson 10, Screen 6, What Will Happen?, students construct viable arguments as they solve a unitless ratio problem. “Red balloons float purple marbles at a ratio of 12:4. What will happen to the marbles if we add 1 balloon and 1 marble. Answer choices: Sink down, Float in place, Fly up.” Teacher Moves, “Display the distribution of responses. Invite students to justify their responses (MP3). If it does not come up naturally, consider asking: How many balloons would you need to float 1 marble?” 

• Unit 5, Lesson 6, Screen 7, Help Diego, students multiply decimals using an area model. “Diego made an error while multiplying $4.5\sdot2.9$. 1. Circle the error in Diego's work. 2. What would you say to help him understand his mistake?” Students critique the reasoning of others as they perform an error analysis of provided student work/solutions.

• Unit 7, Lesson 1, Screen 2, Settle a Dispute, students critique the reasoning of others as they analyze student work involving positive and negative numbers to represent quantities in real-world contexts. The screen contains an image of a crab above a number line with a sand dollar 4 units to the left and another sand dollar 3 units to the right. On the previous screen, students were shown a similar picture and had to write clues to help the crab find sand dollars. In this screen, students look at other students' clues. “Here are Juliana's and Kai's clues: Juliana The sand dollars are at positive 3 and negative 4. Kai Go 3 steps to the right to find the first sand dollar, then 7 steps to the left to find the other one. Whose clue is correct?” Students can choose Juliana, Kai, Both or Neither. They get a different next question depending on how they respond. If Juliana is chosen, “How would you change Kai’s clue to be correct?” If Kai is chosen, “How would you change Juliana’s clue to be correct?” If both are chosen, “What does Juliana mean when she says negative 4?” If Neither is chosen, “How would you change each clue to be correct?”

• Unit 8, Lesson 13, Screen 10, How Many Pumpkins?, students critique the reasoning of others and construct viable arguments as they work with measures of center and variability. “A store has 80 pumpkins for sale. Here are the values of the quartiles. About how many of the 80 pumpkins would you expect to weigh less than 15.5 pounds?” Students are given a number line with a range from 8 to 18. The three quartiles marked are: 10.5, 13, and 15.5. Once students submit their answers they have to justify their thinking and come to consensus as a class.

The materials reviewed for Desmos Math 6 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 6 Lessons and Standards section found in the Math 6 Overview. There is intentional development of MP4 to meet its full intent in connection to grade-level content. Examples include:

• Unit 2, Lesson 9, Supplement, Activity 2: FEMA Poster, students use tables of equivalent ratios to determine the number of specific supplies that a town would need in the event of a natural disaster. “1. Use FEMA’s guidance to make recommendations for preparing 3 cities for a disaster. 2. Is there anything you disagree with? If yes, explain which numbers you think should change and why. If no, explain why not.” Students create a poster, “Choose a city or town that is meaningful to you and look up its population. Make recommendations to the city or town. Choose at least four different supplies from the list. Determine how many of each item the city should have on hand in case of a disaster. Explain or show how you determined the amount of each item your city will need. Explain at least two changes or additions you think FEMA should make to its guidance.” Teacher Moves, “Give students several minutes to read FEMA’s recommendations aloud as a group and ensure everyone understands them. Note: Students can either count the number of cotton balls or the number of bags of cotton balls. This is left intentionally ambiguous. Once a group has confirmed they understand the recommendations, give them several minutes to make recommendations and to analyze FEMA’s guidelines. When most students have finished Problems 1 and 2, consider facilitating a brief discussion or sharing an answer key and inviting groups to revise their recommendations and discuss the reasoning. Give students 5–10 minutes to choose a city and create a poster with their recommendations (MP4). If time allows, invite students who have completed their posters to do a gallery walk to compare the strategies they used with those of their classmates. Consider using the mathematical language routine Compare and Connect.” This activity attends to the full intent of MP4, model with mathematics.

• Unit 4, Lesson 14, Student Worksheet, Activity 2: Build Your Own, students interpret and compute quotients of fractions while modeling with mathematics to build a planter. “You are planning a planter for a school greenhouse. 1. Select at least three types of plants from the supplement to grow in your planter. 2. Select a planter to grow your plants in. 3. Figure out how many of each type of plant you can fit. Be sure each plant has enough space to grow.” This activity intentionally develops MP4 as students model with mathematics.

• Unit 8, Lesson 10, Student Worksheet, Activity 1: Hollywood Salaries, students use data from top actor salaries, as well as mean and mean absolute deviation (MAD) to settle Tay’s and Cho’s dispute. “Tay and Cho used 2019 salary data to help them settle their dispute. Use the supplement to help you gather data to answer the questions below. 1.1 One data set has a mean of 39.4 million dollars. Discuss: Which data set is it? 1.2 Calculate the mean of the other data set. Record both means on your supplement. 1.3 Describe what the mean you calculated tells us about the data set. 2.1 The MAD of the salaries of the actors who are women is 8.48 million dollars. Calculate the MAD for the actors who are men. 2.2 Tay says that since the MADs are similar, the salaries of the men and women in this data are also similar. What would you say to help Tay understand their mistake?”

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

• Unit 1, Lesson 5, Lesson Guide, Activity 1: Area Strategies, provides guidance for teachers to engage students in MP5 as they consider which tools they could use to determine area of triangles. Facilitation Guide: “For students who are having difficulty getting started, consider asking: How is this triangle similar to one you know how to find the area of? Which tools could be helpful? The grid? Scissors? What else? (MP5) Circulate to select several students to share their strategies. Monitor for students who use strategies similar to the two shown on Screen 3, as well as others. Invite several students to share their strategies for determining the area of triangle 𝐵.”

• Unit 4, Lesson 10, Lesson Guide, Activity 2: Write, Trade, Solve!, provides guidance for teachers to engage students in MP5 as they consider which tools they could use to represent division of fractions. Facilitation: “Encourage students to use any tool or strategy (e.g., tape diagram, calculator, paper) that would be helpful. (MP5)”

• Unit 6, Lesson 4, Screen 6, Solutions, Teacher Moves, provides guidance for teachers to engage students in MP5 as they consider which tools they could use to determine how to solve an equation. “Facilitation: Monitor for students who describe a variety of tools and strategies they would use, including but not limited to creating tape diagrams or hangers, using undoing steps, or reasoning about the solution. (MP5)”

The materials reviewed for Desmos Math 6 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 6 Lessons and Standards section found in the Math 6 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 3, Lesson 13, Screen 2, Brazil’s Population, “The population of Brazil is about 214 million people. How many people in Brazil have each of these characteristics?” Students are shown cards with the following information: Population of Brazil: 214 million, 65 out of every 100 people are Catholic, 93% of people can read and write, 146 million people have access to the internet, 1 out of every 5 people are under 15 years old.” Students use this information to fill out a table and convert the information to the number of people in the millions. Students attend to precision as they solve problems involving finding the whole, given a part and the percent.

• Unit 5, Lesson 7, Student Worksheet, Activity 1: Multiplication Methods, Problem 4.1, “Select all of the expressions that have a product of 0.024. $0.06\sdot0.4$, $0.6\sdot0.04$, $0.04\sdot0.06$, $2\sdot0.012$, $1.2\sdot0.02$. Students attend to precision as they multiply, and divide multi-digit decimals using the standard algorithm.

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, Student Worksheet, Lesson Synthesis, Problem 1, “Describe the ratio between moons and stars in as many different ways as you can.” There is an image with a string of repeating stars and moons. The pattern of two stars and one moon repeated three times. Unit 2, Lesson 2, Lesson Guide, “The purpose of this discussion is to surface the three different ways of describing ratios.” Students attend the specialized language of mathematics as they use ratio language to describe the relationship between two quantities.

• Unit 6, Lesson 12, Screen 6, Which Prism?, students answer, “Which prism has a volume of $(2x)^3$ cubic units?”  Students are given an image consisting of two Prisms, C and D, with different dimensions. Answer choices: Prism C, Prism D, Both, or Neither. Teacher facilitation suggests, “If it does not come up naturally, consider asking how the parentheses affect which prism is represented and why.” Discussion questions include: What part of $(2x)^3$ do you think tells us we should be thinking about prisms and not areas like in Activity 1? Why are the parentheses important in the expression $(2x)^3$? What do they mean? What expression represents the volume of the other prism?” This problem attends to MP6 as students attend to the specialized language of mathematics as they examine how the placement of parentheses affect a mathematical situation. 

• Unit 8, Lesson 9, Screen 11, Lesson Synthesis, ”How does the mean absolute deviation (MAD) help you compare data sets?” Dot plots are provided for the Number of Baskets three players made. Each plot has the mean and the mean absolute deviation provided.  Students attend to the specialized language of mathematics as they recognize that a measure of variation in data describes how its values vary with a single number.

The materials reviewed for Desmos Math 6 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 6 Lessons and Standards section found in the Math 6 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 3, Lesson 6, Screen 8, Write Instructions, “Here is one student’s table from the previous screen.” A table of values for the number of gallons of paint are required to paint different quantities of robots. On the previous screen, they were informed that six robots need two gallons of paint. “Write instructions for how you could determine the amount of paint needed for any number of robots.” Students look for and use structure (MP7) to generalize how to generate equivalent ratios.

• Unit 5, Quiz 2, Screen 2, Problem 1, “Determine the product of $0.03\sdot0.08$.” They are given the choices: 2.4, 0.24, 0.024, 0.0024. Students look for and use structure (MP7) of place value to multiply decimals. 

• Unit 7, Lesson 9, Screen 11, Lesson Synthesis, “Explain what you know about the coordinates of this sand dollar.” The screen contains an image of a sand dollar located in Quadrant II of the coordinate plane. There are no numbers labeled on the coordinate plane. Students have to  look for and make use of structure (MP7) to write down what they know about the sand dollar; the graph does not have numbers. 

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 2, Lesson 10, Screen 13, Repeated Challenges and Screen 14, Lesson Synthesis, Screen 13, “Red balloons float purple marbles at a ratio of 6:2. How many purple marbles will 12 red balloons float?” There is an animation on the screen with balloons and marbles in the given ratio and another set of balloons with no marbles. When students enter their answer, they click Try It and the animation models their solution and shows them if they are correct or incorrect. They can select Try Another and complete as many problems of the same type as they wish to. Screen 14, “Describe a strategy for determining missing values in equivalent ratios, like an unknown number of balloons or marbles. Use the sketch tool if it helps you show your thinking.” Students use repeated reasoning (MP8) to generate equivalent ratios.

• Unit 4, Lesson 8, Screen 7, Card Sort, “Match each diagram with at least one equation. You should have one card left over.” This activity intentionally develops MP8, as students use repeated reasoning to match each diagram and equation. 

• Unit 6, Lesson 9, Student Worksheet, Activity 1: Card Sort, “1. Sort the expression cards into two or more groups according to similarities you see. 2. Match each area model with two expressions for its area. You will have two leftover cards.” The worksheet shows a table with 3 columns (Area Model, Product, and Sum). The image of the area model with numbers and variables is included. The Expression Cards contain expressions that match the area models. For example, Row A has an area model with a height of 3 and a width of $x+6$. The Expression Cards include the cards $3(x+6)$ and $3x+18$. Students use repeated reasoning (MP8) to apply the properties of operations to generate equivalent expressions.

The materials reviewed for Desmos Math 6 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

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

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

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

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

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

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

Examples include:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Examples from the Unit Facilitation Guide includes:

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

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

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

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

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

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

For example:

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

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

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

Works Cited include:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The materials reviewed for Desmos Math 6 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

Formative assessments include Quizzes and End Assessments. All assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types such as multiple choice, select all, short answer/fill in the blank, extended response prompts, graphing, mistake analysis, matching, constructed response and technology-enhanced items. Examples Include:

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

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

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

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

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

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

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

• Opportunities for extension and support are provided when appropriate.

• Most digital activities are screen reader friendly.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• Intentional Space for Informal Language

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

• Math and Language in Context 

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

• Embedded Mathematical Language Routines 

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

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

Examples of these supports within the materials include the following:

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

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

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

The materials reviewed for Desmos Math 6 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods. 

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

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

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