8th Grade - Gateway 2

The materials reviewed for Desmos Math 8 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 9, Screen 4, Slope, students interpret the unit rate as slope of the graph (8.EE.5). “SLOPE measures the steepness of a line. This slide forms a line with a slope of \frac{2}{3}. How do you think slope is calculated?” The image shows a slide (line) from the ground up to a person on a platform. Three triangles are formed using the line given. The smallest triangle is labeled with a base of six and height of four. The next triangle has a base of 15 and height of 10. The last triangle has a base of 24 and height of 16. 

• Unit 4, Lesson 2, Screen 3, Solve It #2, students utilize a visual representation of a balance to begin developing the concept of solving linear equations in one variable (8.EE.7). “Find the weight of the square. Press ‘Try It’ to see if the hanger is balanced.” An interactive is shown with a hanger balanced. On one side are three squares with unknown weight and 2 triangles weighing 1lb. On the other side is a square and a triangle.  Students may add or subtract triangles and squares to balance the hanger.

• Unit 7, Lesson 5, Screen 3, Patterns, students develop conceptual understanding of properties of integer exponents to generate equivalent numerical expressions as they observe patterns to surface properties of zero exponents and negative exponents (8.EE.1). “What patterns do you see in the table? Describe as many as you can.” The screen contains a table that students completed on the previous screen with the exponent form, expanded form, and value of powers of 10 descending from 10^4 to 10^{-2}. 

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

• Unit 1, Lesson 1, Practice, Screen 2,  Problem 2, students engage with semi-concrete representations to develop conceptual understanding of congruence and similarity (8.G.A). “These five frames show a shape's different positions. Describe how the shape moves to get from its position in each frame to the next.” Five boxes are shown containing the same shape in a different orientation.

• Unit 3, Quiz, Screen 4, Problem 3, students interpret diagrams or graphs of proportional relationships in context (8.EE.5). “Organic rice costs twice as much per pound as conventional rice at a bulk food store. Select ALL of the graphs that could represent the prices of rice at this store.” Four graphs are provided showing both organic rice and conventional rice graphed comparing the price (dollars) to weight (pounds). 

• Unit 5, Lesson 2, Practice Problems, Screen 4, Problem 4, students demonstrate that a function is a rule that assigns to each input exactly one output as they explain what makes a rule a function (8.F.1). “Recall this image from today's lesson. What makes a rule a function or not?” The following sentence stems are provided: “A rule is a function if… A rule is not a function if…” The students are provided with four input/output tables, three rules are examples of functions and one is not.

The materials reviewed for Desmos Math 8 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 4, Lesson 4, Practice Problems, Screen 2, Problem 2, students develop procedural skill and fluency as they solve a linear equation in one variable (8.EE.7). “Solve 3\left(x-4\right)=12x. Use paper if that helps with your thinking."

• Unit 5, Lesson 11, Screen 9, Calculate, students develop procedural skill and fluency using the formula for the volumes of cylinders to solve mathematical problems (8.G.9). “Calculate the volume of each cylinder. Enter your answers in the table. Then press ‘Check My Work.’” There are images of two cylinders on the screen with the radius and height labeled.

• Unit 6, Lesson 9, Screen 9, Are You Ready for More?, students develop procedural skill and fluency with constructing and interpreting two way frequency tables (8.SP.4). “150 Students were asked what grade they are in and whether or not they play a sport. The two way table shows the data from this survey. Fill in the missing values in the table.” A table is shown with 6th Grade, 7th Grade, 8th Grade, 9th Grade, and Total for the row labels. The column tables are Plays a Sport, Does Not Play a Sport, and Total.  Several of the values are filled in. 

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

• Unit 3, Quiz, Screen 3, Problem 2, students independently demonstrate procedural skill and fluency in deriving the equation y=mx+b for a line intercepting the vertical axis at b (8.EE.6). “Write an equation for each line.” Students are given several different lines on a coordinate plane and asked to write an equation for each line. 

• Unit 5, Quiz 2, Screen 3, Problem 2, students sketch a graph that exhibits the qualitative features of a function that has been described verbally (8.F.5). “Consider the following situation: 55 people got on an empty bus. After 30 minutes, 40 of them got off the bus. After 15 more minutes, the rest of the passengers got off the bus. Sketch a graph that represents this situation. Then enter a label for each axis in the table below.“

• Unit 8, Lesson 13, Screen 9, Cool-Down, students convert a decimal expansion which repeats eventually into a rational number (8.NS.1). “Write each decimal as a fraction.” The decimals given are 0.147 and 0.\bar{147}.

The materials reviewed for Desmos Math 8 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 3, Practice Day, Cards, Problem 4, students construct a function to model a linear relationship between two quantities (8.F.4). “A restaurant offers delivery for their sandwiches. Each sandwich costs $8 and there is a $5 delivery fee. A. What is the total cost for delivering 2 sandwiches? B. Write an equation that relates the total cost, C , to the number of sandwiches delivered, x, representing the total cost for delivering x sandwiches.” 

• Unit 5, Lesson 6, Screen 3, Tyler and the Slide, students sketch a graph of a function based on a qualitative situation (8.F.5). “Sketch a graph representing Tyler's waist height vs. time.” Students are provided a video clip that shows Tyler moving around at the playground, and an interactive graph for them to sketch their representation. 

• Unit 8, Practice Day 2, Student Worksheet, Problem 2, students apply the Pythagorean theorem to find the distance between two points in a coordinate system (8.G.8). “Find the length of the segment that joins the points (– 4, 5) and (6, –1).”

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

• Unit 3, Quiz, Screen 6, Problem 5.1, students graph proportional relationships, interpreting unit rate as the slope of the graph (8.EE.5). “Marquis started at an elevation of 3000 feet and hiked down a mountain at a constant rate. His elevation decreased 500 feet per hour. Graph the relationship between Marquis’ elevation and time as he hiked down the mountain.” There is a sample graph with Elevation (ft.) on the y-axis and Time (hr.) on the x-axis. Two points are connected by a line and can be moved on the graph by the students. 

• Unit 4, Lesson 14, Student Worksheet, Lesson Synthesis, Problems 1-3, students use their understanding of solving pairs of simultaneous linear equations to generate their own system of equations (8.EE.8). “1. Write a system of equations that you would consider difficult to solve. 2. What makes your system of equations difficult to solve? 3. What are some strategies we know for solving systems of equations that have this feature?”

• Unit 6, Lesson 10, Practice Problems, Screen 1, Problem 1, students interpret a two-way table summarizing data on two categorical variables collected from the same subjects, and use relative frequencies calculated for rows or columns to describe possible association between the two variables (8.SP.4). “A scientist wants to know if the color of water affects how much animals drink. The average amount of water each animal drinks was recorded in milliliters for a week and then graphed. Is there evidence to suggest an association between watercolor and how much animals drink?” Students are given a two-way table filled with data. Cat Intake (ml), Dog Intake (ml) and Total (ml) are the column headings and Blue Water, Green Water, and Total are the row headings. A bar graph of the data also is provided. Students have to click either Yes or No and explain their thinking  why the evidence suggests an association or why it does not.

• Unit 8, Lesson 10, Practice Problems, Screen 3, Problem 3, students apply their knowledge of the Pythagorean Theorem to determine unknown side lengths (8.G.7). “Here is an equilateral triangle. The length of each side is 2 units. A height is drawn. In an equilateral triangle, a line drawn from one corner to the center of the opposite side represents the height. 1. Find the exact height. 2. Find the area of the equilateral triangle. 3. (Challenge) Using x for the length of each side in an equilateral triangle, express its area in terms of x. Enter your answers in the table.” Students are given the picture of the equilateral triangle with one side labeled 2. The height and right angle are drawn, and the base is marked with symbols on either side of the height indicating that the parts are equivalent.

The materials reviewed for Desmos Math 8 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 8. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Unit 3, Lesson 7, Screen 5, Complete the Table, students develop procedural skill and fluency as they make connections between proportional relationships and linear equations (8.EE.B). “The table shows the amount of water remaining in the cooler after 0, 1, and 2 cups have been filled. Determine the missing values. Then continue to the next screen.” Students are given a table with the values 0, 1, and 2 filled. Students are asked to find the water left in the cooler after 3, 10 and 37 cups are filled. 

• Unit 5, Lesson 9, Screen 11, Cool Down, students apply their knowledge of functions and rates of change to solve a real-world problem (8.F.4). “Abdel ran a 100-yard dash. The red points show his distance every half-second. Draw line segments to approximately model the data. Then answer this question: When Abdel was running his fastest, approximately how fast was he running?” Students are provided with a graph of Abdel running with Distance Traveled (yd.) on the y-axis and Time (sec.) on the x-axis.

• Unit 6, Lesson 8, Screen 10, Lesson Synthesis, students attend to conceptual understanding by constructing and interpreting scatterplots to represent bivariate data to solve a problem (8.SP.1). “How could you determine whether there is an association between two variables?” A graph is shown with Brain Weight (g) on the y-axis and Body Weight (kg) on the x-axis. Several data points are provided. 

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

• Unit 2, Lesson 9, Screen 5, Slope, students engage in conceptual understanding and application as they analyze ramps (triangles) to determine if the ramps would create a smooth slide (have the same slope) (8.EE.6). “Will these ramps make a smooth slide?” Students are given three answers to select from yes, no or  I’m not sure. Once an answer is selected the students have to explain their thinking. The screen contains a picture with three adjacent triangles. The base and height of each triangle is given. One has a base of 12 and height of 20. Another has a base of 27 and height of 45. The last has a base of 21 and height of 35. This is built off of work from Slide 3, Build It, #1 where students can manipulate the height of one of the ramps to make a smooth slide. 

• Unit 4, Lesson 8, Screen 12, Cool-Down, students engage in conceptual understanding alongside procedural skill and fluency as they solve a linear equation in one variable and explain what the solution means in the context of the situation (8.EE.7). “Andrea is considering the costs of printing p pages at home and at a store. She wrote the following equation 100+0.05p=0.25p. Solve Andrea's equation. Use paper if it helps you with your thinking.” Students are prompted to submit and explain their answer.

• Unit 7, Lesson 7, Screen 3, Total Weight, students build conceptual understanding, procedural skill and fluency, and application as they solve a real-world problem using numbers expressed in the form a single digit times an integer power of 10 to estimate very large quantities (8.EE.3). “One way to represent the total weight of the plane is by using multiples of powers of 10, as shown below: 3\sdot10^5+2\sdot10^4 Enter the total weight of the plane (320000 kilograms) using a different combination of the weights shown in the diagram. Write your answer using multiples of powers of 10.” Students are given an interactive scale with a plane on one end, students have to balance the scale using their choice of the available weights.

• Unit 8, Lesson 10, Practice Problems, Screen 1, Problem 1, students engage in procedural skill and fluency alongside application as they apply the Pythagorean Theorem to a real-world problem to determine a missing length of a right triangle (8.G.7). “A man is trying to zombie proof his house. He wants to cut a length of wood that will brace the door against the wall. The wall is 4 feet away from the door, and he wants the brace to rest 2 feet up the door. About how long should he cut the brace?” Students are provided a diagram of the situation.

The materials reviewed for Desmos Math 8 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 8 Lessons and Standards section found in the Math 8 Overview. MP1 is identified and connected to grade-level content, and intentionally developed to meet its full intent. Students make sense of problems and persevere in solving them as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Lesson 6, Screen 2, Transformation Information #1, students work with transformations. “Describe a sequence of transformations that takes triangle ABC to triangle A’B’C’.” Students are shown two corresponding triangles on a coordinate grid. Students make sense of the problem and persevere as they describe a series of transformations that take triangle ABC to triangle A’B’C’. 

• Unit 2, Lesson 5, Screen 5, Describe Your Strategy, students discuss how different transformations can be useful for solving a golf challenge. Students are given a coordinate plane with a white block letter L, a shaded block letter L, and several diagonal segments.  “Describe the sequence of transformations you'll use to transform the pre-image (shaded) onto the image. Use the sketch tool if it helps you to show your thinking.” Students are expected to avoid the diagonal segments as they navigate from one image to the next. Students make sense of the problem and persevere in solving them as they test their strategy (and modify as needed) in order to complete the challenge.

• Unit 4, Lesson 8, Screen 11, Lesson Synthesis, students describe strategies to solve a problem in context. Students are given an image of a water tank and the expressions Water Tank A 30x+25 and Water Tank B -20x+1000. Students answer, “The image shows expressions that represent the amount of water, in liters, in two water tanks. Let x represent the number of seconds that pass. How could you determine when the tanks will have the same amount of water?” Students make sense of the problem in order to explain how they would determine when the tanks will have the same amount of water. 

• Unit 7, Lesson 12, Screen 2, Pick Your Power, students apply what they know about scientific notation to a context. Students are provided with an image of City A and City B along with a slider that changes the amount of power the plant produces. Students answer, “City A and City B get electricity from the same source. Here is how much electricity each city needs: City A:  5 gigawatts; City B:  3 gigawatts. You can control how much electricity is produced. Adjust the slider so that the dial says, ‘Success!’ Discuss what you think ‘success’ means in this case.” Students make sense of the problem as they discuss what success means in the context of this problem.

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 8, Screen 2, Measurements in Similar Triangles, students work with similar triangles to find unknown lengths. “Four similar triangles are shown. Examine the given side lengths. Then: 1. Enter the missing values. 2. Describe any patterns you notice.” Students reason abstractly and quantitatively to compare the triangles and discover what is similar in order to find the unknown lengths.

• Unit 3, Lesson 7, Screen 10, Are you Ready for More?, students write a scenario that represents a given graph. “Write a scenario that could be represented by this graph. In your description, be sure to mention the meaning of the slope in your scenario.” Students reason abstractly and quantitatively as they write a scenario that the graph could represent and explain the meaning of slope based on the scenario they create.

• Unit 5, Lesson 7, Group Worksheet, Activity 1: Awards, students use Context Cards to calculate calories. There are three cards each with a different situation, Card 1: a graph to represent the situation; Card 2: a table; and Card 3: an equation. “Work with the members of your group to answer the following questions: 1. Who gets the award for most calories burned overall? 2. Who gets the award for most calories burned in the first 10 minutes? 3. Who gets the award for burning the most calories per minute over any period of time?” Students reason abstractly and quantitatively as they analyze and compare different representations of contextual situations.

• Unit 6, Lesson 6, Screens 5 and 6, Measuring Turtles/Find the True Statements, students connect their understanding of slope and its association to a given context. On Screen 5, “Here is a scatter plot from the card sort. Enter a slope to fit a line to the data. (Your line will go through the red open point.)” Screen 6, “Here is the scatter plot from the previous screen. Two of these statements are true. Which are they?” Choices are:  “For these data, as turtle length increases, the turtle width tends to decrease. There is a positive association between turtle length and turtle width. If the turtle length increases by 1 centimeter, then the model predicts that the turtle width increases by \frac{2}{3} centimeter. If the turtle width increases by 1 centimeter, then the model predicts that the turtle length increases by $$\frac{2}{3}$$ centimeter.” Students reason abstractly and quantitatively as they connect the relationships between the problem scenario and mathematical representation.

The materials reviewed for Desmos Math 8 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 8 Lessons and Standards section found in the Math 8 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 6, Lesson Guide, Lesson Synthesis, provides guidance for teachers to engage students in MP3 as they construct viable arguments and critique the reasoning of others while working with similar polygons. “Display Screen 7 of the Teacher Presentation Screens. Give students one minute of quiet think-time and a few minutes to discuss with a partner and complete the lesson synthesis on their worksheet. Ask students to share and justify their responses and to critique each other's reasoning.”

• Unit 5, Lesson 6, Screen 5, Same Scenario, Different Graph, students construct viable arguments and critique the reasoning of others while working with functions. “Which graph could represent the relationship between Tyler’s distance from the right edge of the screen and time?” Students watch a video of a child who climbs a play structure and goes down a slide. Students are given two graphs, each graph shows time in seconds on the x-axis and the same interval on the y-axis. Students justify why they picked one graph over the other to their classmate.

• Unit 7, Lesson 5, Screen 8, Lesson Synthesis, students construct viable arguments as they simplify expressions with rational exponents,“How could you convince someone that 6^0=1?” 

• Unit 8, Lesson 4, Screen 4, Reflection, students identify the two whole number values that a square root is between and explain the reasoning. “Esi says that the value of z when z^2=80does not belong in either of these categories since z must be greater than 8. What whole number would be closest to?” Students are given a visual of three cards: one between 4 and 5, one between 7 and 8, and one the value of z^2=80. Teacher Moves, “Key DIscussion Screen: The purpose of this discussion is to surface strategies for estimating the value of a square root. Use snapshots or the teacher view of the dashboard to display unique answers to the class. Ask students to justify their responses and critique each other's reasoning. If time allows, consider asking students whether z is greater than 9 or less than 9, and to explain how they know. Routine (optional): Consider using the routine Decide and Defend to support students in strengthening their ability to make arguments and to critique the reasoning of others (MP3).”

The materials reviewed for Desmos Math 8 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 8 Lessons and Standards section found in the Math 8 Overview. There is intentional development of MP4 to meet its full intent in connection to grade-level content. Examples include:

• Unit 2, Lesson 8, Screen 1, Warm-Up, students use real world measurements to model with mathematics. “One triangle has side lengths 2, 3, and 4. Another triangle has side lengths 4, 5, and 6. Are the two triangles similar? Sketch them on paper if that helps with your thinking?” This attends to the intentional development of MP4, model with mathematics. 

• Unit 5, Lesson 6, Practice Problems, Screen 5, Problem 3, students model a situation by sketching a graph that exhibits the qualitative features of a function described verbally. “Deven puts a batch of cookie dough in the fridge. The dough takes 15 minutes to cool from 70°F to 40°F. Once it is cool, the dough stays in the fridge for another 30 minutes. Then Deven takes the cookie dough out and puts it into the oven. After 5 minutes in the oven, the cookies are 80°F. Sketch a graph that represents this situation.” The screen contains a graph with Temperature of Cookie Dough (°F) on the vertical axis and Time (min.) on the horizontal axis. This activity attends to the intentional development of MP4, model with mathematics.

• Unit 7, Lesson 9, Screen 2, students perform operations with numbers expressed in scientific notation. Students must select one of four scenarios to work on. Each scenario contains data and two questions that are related to the scenario. The scenarios use numbers expressed in scientific notation. Students must select one of the questions to answer and create a model to answer the question. For example, Meter Sticks to the Moon scenario, “How many meter sticks does it take to equal the mass of the Moon? If you took all those meter sticks and lined them up end to end, how many times will they reach from the Earth to the Moon? Check each item as you add it to your work: The question you selected; Important measurements (with units); Your calculations; Your answers (with units); Two important points from your work plotted and labeled on a number line.” The data presented with the scenario are the mass of a meter stick (0.2 kg), the height of a meter stick (1 m), the mass of the moon (7\sdot10^{22}kg), the distance of the moon from the earth [(3.8)\sdot10^8 meters away], the distance of Mars to the earth (8\sdot10^{10} meters away), and the length of 1 light year (10^6 meters). This activity attends to the full intent of MP4, model with mathematics.

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

• Unit 1, Lesson 4, Student Worksheet, Activity 1: Move it, students use tools to translate figures on the coordinate plane. “Use whatever tools you’d like to carry out the moves specified. Use A′, B′, and C′ to indicate vertices in the new figure that correspond to points A,  B, and C in the original figure.” Problem 1, “Translate Figure ABC 3 unites right and 1 unit down.” This activity meets the intent of MP5, use tools strategically.

• Unit 5, Lesson 8, Lesson Guide, Activity 1: Charge!, provides guidance for teachers to engage students in MP5 as they use tools to develop a linear model and assess the reasonableness of their model. In this Activity, students determine when a phone will be fully charged. They are shown images of the phone at different times with the percent charged displayed, and they use that data to create a linear model to predict when the phone will be fully charged. “As students are working, encourage them to use the tools they deem appropriate to solve the problem (MP5), such as the provided blank paper, the Desmos calculator, or any other tools that would be helpful. If students are having difficulty getting started, ask them how they might represent the information they have mathematically, such as in a table.”

• Unit 6, Lesson 2, Screen 2, Warm-Up, students use tools appropriately and strategically to construct dot plot/scatter plots and investigate patterns. “This table shows the data from your class. Discuss the following questions: 1. How could we reorganize the data to make it more useful for analyzing? 2. What are some of the questions that this data might be able to answer?” Students decide in the Lesson Synthesis which representation they want to use to organize this data, and the advantages of that representation.

The materials reviewed for Desmos Math 8 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 8 Lessons and Standards section found in the Math 8 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 2, Screen 5, Compare the Rates, “Sketch the graph of y=0.75x on each set of axes. Then explain how you decided where to sketch the lines.” Students are given two graphs to work with.  Students attend to precision as they understand the connections between proportional relationships, lines and linear equations, 

• Unit 7, Quiz, Screen 8, Problem 5.1, students explain how three expressions that have the same value are equivalent. “Here are three expressions that have the same value: A. 2^3\sdot2^3 B. 2^6 C. 4^3. Explain how you can tell that these expressions are equivalent.” Students attend to precision while knowing and applying the properties of integer exponents to generate equivalent numerical expressions. 

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 1, Lesson 4, Student Worksheet, Activity 2: Make My Transformation, Problems 1-4 students use their knowledge of transformations to describe them to a partner. Students are given a card with triangle ABC and triangle A’B’C’. “Your partner will describe the image of this triangle after a certain transformation. Sketch it here. You can only sketch (no speaking).” One partner describes the transformation while the other student sketches it. Teacher directions, “The person with the transformation card will give their partner a precise description of the transformation displayed on their card. Remind students to use the geometric language for describing reflections, rotations, and translations that were used in the previous lesson.” This problem attends to MP6 as students attend to the specialized language of mathematics as they describe transformations.

• Unit 6, Lesson 6, Screen 3, Associations, students use a previously created line of best fit  to determine the relationship between foot width and foot length. “Here is your work from the previous screen. What type of association is there between foot length and foot width? The following options are provided: Positive association, Negative association, No association, Explain your thinking.” The screen contains a scatter plot that displays bivariate data for Foot Width (cm) vs. Foot Length (cm). Teacher Moves, “Key Discussion Screen: The purpose of this discussion is to come to a consensus about what the terms positive association and negative association mean. Highlight several student responses for the class. Ask questions to help students connect concrete and abstract responses as well as formal and informal responses. Consider asking: What does it mean to have a positive association? [When one of the variables increases, the other variable tends to increase as well.]  What other pairs of things do you think would have a positive linear association?” Guidance for teachers supports students to engage  in MP6 as they attend to the specialized language of mathematics to describe patterns in data such as positive or negative association.

• Unit 8, Lesson 8, Screen 2, Highlighted Hypotenuse, students identify the hypotenuse of right triangles in different orientations. “Remind your partner of the definition of a hypotenuse. Then select all the triangles with a highlighted hypotenuse. (Select all that apply.)” Students are given four  pictures of different right triangles in different orientations, with one side of the triangle highlighted. This problem attends to MP6 as students attend to the specialized language of mathematics as they explain how to identify the hypotenuse of a right triangle.

The materials reviewed for Desmos Math 8 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 8 Lessons and Standards section found in the Math 8 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 1, Lesson 11, Student Worksheet, Activity 2: Tear It Up, Lesson Guide “Facilitate a class discussion about what this experiment ‘proves.’ Help students recognize that each individual experiment illustrates that the angles of each triangle sum to 180 degrees, and even though we have tried several different triangles, we haven’t tried them all. If it does not come up, ask: How do we know that we haven’t found one of the triangles for which this statement may not be true? Maybe it isn’t true for really large triangles, or perhaps really small triangles. We really don’t know since we can’t try them all. In the next lesson, we will justify this relationship between three angles making a line and three angles being the angles of a triangle.” The routine Compare and Connect is suggested to help students “make sense of multiple strategies and connect those strategies to their own.” This activity attends to the full intent of MP7, look for and make use of structure as students analyze a problem and look for more than one approach.

• Unit 4, Lesson 7, Screen 9, Never True, “Kiandra looked at this equation and, without writing anything, said it must never be true. Explain what she may have noticed to lead to this conclusion.” Students are shown the following equation: \frac{1}{2}+x=\frac{1}{3}=x. This activity attends to the full intent of MP7, look for and make use of structure, since students look at the structure of the equation in order to answer the question.

• Unit 8, Lesson 12, Screen 8, Lesson Synthesis, “How can you predict whether a unit fraction will terminate or repeat?” The Sample Responses gives some possible student responses that demonstrate use of MP7, “Write the denominator in factored form. If the factors consist only of 2s and 5s, then the decimal representation will terminate. Otherwise, it will repeat. If you can write an equivalent fraction with a power of 10 as the denominator, the decimal representation will terminate. If not, it will repeat.” Students make use of structure to generalize if a unit fraction will terminate or repeat when it is converted to a decimal.

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 3, Screen 2, Warm Up, “Point B is dilated using point A as the center of dilation. If the scale factor is 3.5, where should the image be? Drag point C to show your answer. Then describe your strategy.” Students are given a grid to work with the above points. This activity intentionally develops MP8 as students look for general methods and shortcuts as they practice dilating the points.  

• Unit 5, Lesson 13, Screen 5, Cylinders and Cones, “Each row of the table has information about a cylinder and cone with the same height and radius. Fill in the missing values.” Students are given a table with the volume of cylinders and the corresponding volumes of cones. This activity intentionally develops MP8 as students use repeated reasoning to fill in the missing values on the table.

• Unit 7, Lesson 6, Lesson Guide, Activity 1: Write a Rule, “Distribute a double-sided worksheet to each student. Tell students that their goal for this activity is to write their own rule for each of the groupings from the card sort. For each grouping of cards, students will write the example(s) from the cards. Then they will create their own example, write a rule, and explain or show how they know their rule will always work.” In the Warm-Up, students sorted cards with exponential equations that involved multiplying dividing powers, negative exponents, and zero exponents. Students use repeated reasoning (MP8) to develop rules for rewriting exponential expressions.