6th Grade - Gateway 2

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), $$(0.5)\sdot(0.02)$$, (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.