6th to 8th Grade - Gateway 2

The materials reviewed for Math & YOU Grades 6 through Grade 8 meet expectations for supporting the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

Multiple conceptual understanding problems are embedded throughout the grade level within the Investigate, Discussions, In-class examples and Big Ideas Tasks. Students have opportunities to engage with these problems both independently and with teacher support. 

According to the Implementation Handbook, Foundational Beliefs, “Each lesson begins with opportunities for students to engage in investigation resulting in observations, conjectures, and discovery of informal strategies. These opportunities support students in developing an understanding of the mathematical concepts grounded in meaningful experience and connected to prior learning, building a sturdy conceptual foundation. From here, the focus turns to formalizing these ideas through explicit instruction of new mathematical terminology, formal strategies, and key concepts, while connecting back to students’ experiences during their investigation.”

Examples include:

• Grade 6, Chapter 3, Lesson 2, Investigate, students develop conceptual understanding of ratios as they use ratio reasoning to solve problems. Students apply visual models and ratio relationships to determine the lengths of various snowboarding trails. Teacher Notes provide suggestions for facilitating discussion about the ratio relationships that help students reason and find unknown trail lengths. The Teacher Guide states, "The tape diagram is purposely shown without numbers. You want students to use quantitative reasoning to see the model as a 1 : 4 ratio. Exercise 1: Listen and look for explanations of a multiplicative relationship. The length of the Corkscrew Trail is 4 times the length of the Sidewinder Trail, or the Sidewinder Trail is \frac{1}{4} as long as the Corkscrew Trail. Students should also realize that they do not know the actual length of either trail. Exercise 2: Make connections to the Motivate activity, if necessary. 'Each rectangular part (or bar) in the diagram represents the same value.' Have a summary discussion to compare and connect solution strategies used in Exercises 3 and 4. Talk About It: 'How does the tape diagram help you make sense of the quantities and the relationship between the quantities?’" Student Edition states, “Work with a partner. The tape diagram models the lengths of two snowboarding trails. 1. Compare the lengths of the two trails. 2. The Sidewinder Trail is 300 yards long. How long is the Corkscrew Trail? How do you know? 3. The Northface Trail is 5 times longer than the Southside Trail. Use a tape diagram to compare the lengths of the trails. 4. The combined length of the Northface Trail and the Southside Trail is 1,500 yards. How long is each trail?”(6.RP.A)

• Grade 7, Chapter 8, Big Idea Tasks, students demonstrate conceptual understanding by using a sample to generalize information about a population. Exercise 2 states, “Design a survey that compares two populations. Describe the populations you would sample. a. Describe an unbiased sample of each population. Explain your reasoning. Then determine if the results of the survey will be valid or not. Justify your reasoning. b. Describe a biased sample of each population. Explain your reasoning. Then determine if the results of the survey will be valid or not. Justify your reasoning. c. What are some possible survey responses? What conclusions can you draw from the responses? d. How would you display the data from the survey? Explain your reasoning.” (7.SP.1)

• Grade 8, Chapter 2, Lesson 1, students develop conceptual understanding as the teacher highlights the differences between a translation and a transformation and explains how Examples 1 and 2 demonstrate translations. Lesson 1 states, “A transformation changes a figure into another figure. The new figure is called the image. A translation is a transformation in which a figure slides but does not turn.” Nick’s Notes, Lesson Insights, directs the teacher to, “Point out the difference between a transformation and a translation. A translation is a transformation, but a transformation may not be a translation. You will study other transformations later in this chapter. A translation does not have to be in a horizontal or vertical direction. It can also be in a diagonal direction.” Lesson Insights, Example 2 states, ”In Example 2, explain that translating the triangle on a diagonal is equivalent to translating the triangle horizontally and then vertically. The two steps focus on what happens to each of the coordinates in an ordered pair. Reinforce the concept of identical figures having the same size and same shape by discussing the lengths of the corresponding sides, the measures of the corresponding angles, and the perimeters and areas of the two triangles.” Students apply their understanding of translations to independently graph a translation in a coordinate plane with all four quadrants. Student Edition, Practice, Exercise 10 states, “Translate the triangle 4 units right and 3 units down.” (8.G.1)

The materials reviewed for Math & YOU Grades 6 through Grade 8 meet expectations for providing intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.

Multiple procedural skill and fluency problems are embedded throughout the grade levels within In-Class Practice, which extends their learning from the Key Concept. The formal assessments offered at the lesson, chapter, multi-chapter, and course levels provide opportunities for students to independently demonstrate their procedural skills and fluency. 

According to the Implementation Handbook, Foundational Beliefs, “Students have opportunities to develop procedural fluency through targeted practice supported by question prompts designed to encourage reflection on the accuracy and efficiency of their strategy. Practice opportunities support students in solving tasks that incorporate procedures with connections, requiring students to think meaningfully about which strategies they are using and how they apply in the problem context, and to reason about the meaning of the resulting solution. Students regularly apply their learning in new real-world or mathematical contexts, focusing on how strategies extend to these contexts and interpreting the meaning of the solution in light of the situational context.”

For example,

• Grade 6, Chapter 2, Lesson 7, students develop procedural skill and fluency by fluently dividing multi-digit numbers using the standard algorithm. Example 2 states, “a. Find 6,084\div12. b. Find the quotient of 9,216 and 150.” The teaching notes include a prompt for a Turn and Talk, in which students "Describe the procedure used to divide the multi-digit numbers in Example 2(a). How do you know that the quotient is reasonable?" For Example 2(b), fluency is built by anticipating and reflecting on answers. ’Should the quotient in Example 2(b) be greater than or less than 9,216? Explain." (6.NS.2)

• Grade 7, Chapter 4, Lesson 3, In-Class Practice Exercise 7, students develop procedural skill and fluency by comparing an algebraic solution to an arithmetic solution and identifying the sequence of operations used in each approach. Example 4 states, “You install 500 feet of invisible fencing along the perimeter of a rectangular yard. The width of the yard is 100 feet. What is the length of the yard? Use the formula for the perimeter of a rectangle. Check Use a different form of the formula for the perimeter of a rectangle, P=2(l+w).” Nick’s Notes, Lesson Insights states, “‘Do you know anyone who uses invisible fencing? Why do you think people use invisible fencing instead of panel fencing?’ Have students read the problem to themselves. Then have them explain the problem in their own words to a partner. ‘What do you know? What are you being asked to find? What is a good first step?’ Diagrams play an important role in problem solving. Encourage students to draw and label a diagram whenever possible. Students may not remember the formula for the perimeter of a rectangle, P = 2l + 2w. Discuss how the Distributive Property can be used to factor out the 2 in the formula to get P = 2(l + w).” In-Class Practice Exercise 7 states, “You must scuba dive to the entrance of your room at the Jules’ Undersea Lodge in Key Largo, Florida. You are 1 foot deeper than \frac{2}{3} of the elevation of the entrance. What is the elevation of the entrance?” (7.EE.4a)

• Grade 8, Chapter 8, Chapter Test, students demonstrate procedural skill and fluency by applying the properties of integer exponents to generate equivalent numerical expressions. Exercise 7 states, “Simply the expression. Write your answer as a power. \frac{2^7\times2^3\times2^4}{2^8}=_______.” (8.EE.1)

The materials reviewed for Math & YOU Grades 6 though Grade 8 meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters.

Multiple routine and non-routine applications of mathematics are included throughout the grade level, with single and multi-step application problems embedded within lessons, including Investigate and Connecting to Real Life. Students engage with these applications both with teacher support and independently through Examples and Practice. Materials are designed to provide opportunities for students to demonstrate their understanding of grade-level mathematics when appropriate. 

For example:

• Grade 6, Chapter 6, Lesson 4, Practice, students apply their understanding of writing and solving equations with two variables to find the answer to a word problem. Exercise 14 states, “It costs 1,200 to create a video advertisement, and it costs 0.50 per day to display the advertisement on a website. Write and graph an equation that represents the total cost (in dollars) to create and display the advertisement.” (6.EE.7)

• Grade 7, Chapter 6, Lesson 3, Practice, students apply their understanding as they create a survey of their classmates, collect data, and create a graphic that displays the results in percentages. Exercise 15 states, “Survey students in your school about their favorite sport. Your survey should include at least 2 different sports and an “other” category. a. Create a diagram that shows the percent of students who chose each sport listed in your survey and the number of students who chose ‘other’. b. Exchange diagrams with a classmate and determine the numbers of students who chose each category in your classmate’s survey.” (7.RP.3, 7.EE.3)

• Grade 8, Unit 7, Lesson 5, Nick’s Notes Motivate, students apply their understanding as they sketch graphs for a non-routine, real-life problem that exhibits the qualitative features of a function. The materials states, “Students will sketch graphs to represent different paces of walking. Ask three volunteers to be walkers. Each walker should stand 5 feet from the front wall and do the following when you tell them to go. Walker A: Walk slowly to the back wall at a constant rate. Walker B: Walk quickly to the back wall at the constant rate. Walker C: Stay where you are. Do not walk. Have the rest of the class sketch a graph representing the distance of each walker from the front wall over time. Turn and Talk: ‘How are the graphs alike? How are they different?’ After partners compare their graphs, have students compare and discuss their graphs in small groups.” (8.F.5)

The materials reviewed for Math & YOU Grades 6 through Grade 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 as reflected by the standards.

Multiple aspects of rigor are engaged simultaneously across the materials to develop students' mathematical understanding of individual lessons. Each lesson within the curriculum supports a variety of instructional approaches that incorporate conceptual understanding, procedural fluency, and application in a balanced way. 

For Example:

• Grade 6, Chapter 1, Lesson 2, Practice, students demonstrate conceptual understanding and application as they reason about expressions. They explain what the parts of an expression represent in a word problem and then solve the problem. Exercise 14 states, “You and 7 other people are in a theater. Five groups of 4 people enter, and then three groups of 5 people leave. Explain what each part of the expression 5(4) − 3(5) + 8 represents in this context. Then determine how many people are in the theater.” (6.EE.1)

• Grade 7, Chapter 8, Digital Learning Path, Connecting Big Ideas, students demonstrate conceptual understanding and application as they interpret a data display about social media shopping trends for different age groups and describe the percent decrease. Exercise 4 states, “For each category, describe the percent decrease from the youngest age group to the oldest age group. Then interpret each percent of decrease and explain why your answers make sense.” Nick’s Notes Connecting Big Ideas, teachers are urged to support these aspects of rigor with insights for facilitating class discussion. The materials state, “Provide students silent think time to engage with the data. ‘What mathematical statements can you make about the data display? Is there any information that surprises you in the display? Explain. Which professions may be interested in this data? Explain how these professions could use the data. What mathematical tools can you use to organize and represent your data? What is the problem asking you to find? How will you begin to solve?'” (7.RP.3, 7.SP.1)

• Grade 8, Chapter 4, Lesson 3, Practice, students demonstrate conceptual understanding and application as they graph proportional relationships and interpret the unit rate as the slope of the graph. Students compare two different proportional relationships represented in different ways. Exercise 7 states, “You are a financial analyst building an investment portfolio. You recommend that your client purchase 2 bonds for every 4 stocks. a. Complete the ratio table. b. Write an equation that represents the situation. c. Choose two points from the ratio table and plot them in a coordinate plane. How does the slope of the line that passes through the two points relate to the ratio table?” A table is provided for students to complete, with stocks given as 4 and 18, and bonds purchased given as 3 and 6. (8.EE.5)

The materials reviewed for Math & YOU Grades 6 through Grade 8 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students across the 6-8 grade band engage with MP1 throughout the year. MP1 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

Across the grades, students engage in open-ended tasks that support key components of MP1, including sense-making, strategy development, and perseverance. These tasks prompt them to make sense of mathematical situations, develop and revise strategies, and persist through challenges. They are encouraged to analyze problems by engaging with the information and questions presented, use strategies that make sense to them, monitor and evaluate their progress, determine whether their answers are reasonable, reflect on and revise their approaches, and increasingly devise strategies independently. 

An example in Grade 6 includes:

• Chapter 4, Lesson 2, Student Edition, students calculate multi-step percentages with real-world numbers to determine the percent of home team fans who are under 18 and wearing jerseys, and then explain their reasoning. Practice, Exercise 18 states, “A football game is attended by 68,000 spectators, 64% of whom are fans of the home team. Three-fourths of the fans of the home team at the game are wearing jerseys and 9,792 of those fans are under 18. What percent of fans of the home team are under 18 and wearing jerseys? Explain.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide, states, “Provide appropriate time for students to engage in the rich problems throughout this lesson. Encourage students to draw pictures, diagrams, or tables to make sense of the problem (SMP.1). Encourage students to check their answers and make sure solutions are reasonable and make sense. Use Practice Exercise 18 to encourage making sense of problems and persevering in solving them. Suggested prompts: ‘What information do you have? What do you need to find out? Can you draw a picture or a model?’ ‘How can you check your answer? Did you discuss your answer with a partner? Did you agree?’”

An example in Grade 7 includes:

• Chapter 4, Lesson 4, Student Edition, students make sense of problems and persevere as they determine strategies they will use to solve a problem.  Then they check its effectiveness before and as they work through the problem to find possible dimensions for a box and the corresponding volumes. Practice Exercise 19 states, “The girth of a package is the distance around the perimeter of a face. A mailing service says that a rectangular package can have a maximum combined length and girth of 108 inches. Find three different sets of allowable dimensions that are reasonable for the package. Then find the volume of each package.” Digital Teaching Experience, Supporting the Mathematical Practice: Facilitation Guide, states, “Making sense of problems and persevering in solving them (SMP.1) comes into play as students translate inequalities from verbal statements to graphical and symbolic representations. It is essential that students take time to understand a problem scenario before being able to plan a solution pathway and solve As students complete Practice Exercise 19, encourage class discussion about students’ problem-solving approaches. Suggested prompts: ‘Can you restate in your own words what the problem is asking you to do?’; ‘What helped you? What confused you? How did you persevere?’ ‘Did everyone solve using the same method? How are the various methods similar? How are they different?’  ‘Did you check your work? Is your answer reasonable?’”

An example in Grade 8 includes:

• Chapter 1, Lesson 3, Student Edition, students analyze a geometric situation by reasoning about area, adjusting stripe widths to change proportions, and explaining how the new design creates twice as much blue paint as gray paint. They engage with both the diagram and the written description, decide on a strategy to determine if the design is possible, and explain their reasoning. Students monitor and evaluate their progress, checking that their solution makes sense mathematically and in the real-world context. Practice Exercise 26 states, “The floor of a six-foot-wide hallway is painted as shown, using equal amounts of gray and blue paint. Can you change the widths of the stripes so that you use twice as much blue paint as gray paint? Explain.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide, states, “Students will expand their equation-solving toolbox to incorporate variables on both sides of an equation. Being the first time students will engage with equations having no solution or infinitely many solutions, the lesson provides timely moments to model ‘grit’ as students must embrace SMP.1 and persist even if they become frustrated. You will likely field questions about which solution paths to take. Encourage discourse among students here so they might observe and recognize the value in discovering multiple paths to solving an equation with variables on both sides.”

The materials reviewed for Math & YOU Grades 6 through Grade 8 meet expectations for supporting the intentional development of MP2: Reason abstractly and quantitatively, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students across the 6-8 grade band engage with MP2 throughout the year. MP2 is found in lessons where they focus on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

Across the grades, students participate in tasks that support key components of MP2, including reasoning with quantities, representing situations symbolically, and interpreting the meaning of numbers and symbols in context. These tasks encourage students to consider the units involved in a problem, analyze the relationships between quantities, and connect real-world scenarios to mathematical representations. Teachers are guided to support this development by modeling the use of mathematical notation, asking clarifying and probing questions, and facilitating conversations that help students make connections between multiple representations.

An example in Grade 6 includes:

• Chapter 6, Lesson 4, Student Edition, students engage in abstract reasoning to recognize and analyze the relationship between dependent and independent variables in real-world contexts. Investigate states, “Work with a partner. In Section 3.4, you used a graph to represent the speed of an airplane. Below is one possible graph.” Students are given a line graph that represents distance in kilometers over time in hours. “1. Describe the relationship between the two quantities, time and distance. Which quantity depends on the other quantity? 2. Write an equation that represents the relationship between the time and the distance. 3. The airplane is 2,000 kilometers away from its destination. Write and graph an equation that represents the relationship between time and distance from the destination.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide, states, “Students will now write and graph equations in two variables. As students model real-life situations mathematically, ask them to look for relationships among multiple representations. It is important for students to look for patterns within tables, graphs, and equations that represent relationships between two quantities. They will reason abstractly and quantitatively (SMP.2) to help them understand the relationship between the independent and dependent variables. Throughout the learning activities in this lesson, look for ways to support students in mathematical practices as they learn. For example: In the Investigate, students will reason abstractly and quantitatively to explore relationships between two variables. Suggested prompts: ‘What are the quantities in this problem? How are the quantities related? How do you know?’ ‘What does the ordered pair (3, 1,200) represent in the context of the situation?’ ‘Which quantity depends on the other quantity?’”

An example in Grade 7 includes:

• Chapter 4, Lesson 5, Student Edition, students write and solve inequalities with addition and subtraction, reasoning abstractly about the reasons their solutions are logical for the problem. Investigate states, “Work with a partner. Use one die on which the odd numbers are negative and a second die on which the even numbers are negative. Roll the dice. Write an inequality that compares the numbers. Roll one of the dice. Add the number to each side of the inequality and record your result. Repeat the previous two steps five more times. 1. Is an inequality still true when you add the same number to each side? 2. Is an inequality still true when you subtract the same number from each side? 3. Use your results in Exercises 1 and 2 to explain how to solve an inequality of the form x+a<b for x.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “As students write and solve inequalities using addition or subtraction, use this opportunity to encourage SMP.2 by having students call attention to the meaning of the quantities in scenarios, how to apply the properties to solve, and then how to contextualize their answers back into the problem context. In the Investigate, students will reason abstractly and quantitatively about the dice-rolling scenarios. Suggested prompts: ‘What are the quantities? How are the quantities related? How do you know? Is there language that supports this relationship? How can these quantities be related using symbols and numbers?’ ‘Work with a partner to describe a real-life situation that could be modeled by x+3<8. Solve and explain its meaning in the situation.’ ‘How is writing an inequality the same as or different than writing an equation?’”

An example in Grade 8 includes:

• Chapter 4, Lesson 1, Student Edition, students create graphs to represent the snowfall in several towns and use the graphs to compare the snowfall in each town. Investigate states, “1. It started snowing at midnight in two towns. The snow fell at a rate of 1.5 inches per hour. a. In Town A, there was no snow on the ground at midnight. How deep was the snow at each hour between midnight and 6 a.m.? Make a graph that represents this situation. Town B had 4 inches of snow at midnight. Draw a graph for Town B. c. The equations C: y=2x+3 and D: y=8 represent the depths y (in inches) of snow x hours after midnight in Town C and Town D. Graph each equation. d. Use your graphs to compare the snowfall in each town.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students will engage in SMP.2 as they use tables of values to plot points and create graphs of linear equations. As students encounter these relationships, they will reason about the meaning of the quantities to create graphical representations. They should also be able to consider the context when interpreting information from their graphs. Throughout the learning activities of this lesson, look for ways to support students in the mathematical practices as they learn. For example: In the Investigate, ask probing questions to ensure students are reasoning abstractly and quantitatively. Suggested prompts: ‘What ordered pair is associated with midnight for Town A?’ Listen to see if students are able to represent the problem scenario symbolically. ‘What real-life quantity does the x-coordinate of this point represent?’ If students just say ‘time’, press for more detail. ‘Time since when? And what are the units of measure?’ ‘In part (d), use your graphs in your explanations of how the snowfall amounts compare. How do the graphs help you to compare?’ ‘Do any of the towns ever have the same depth of snow? How do you know?’”

The materials reviewed for Math & YOU Grades 6 through Grade 8 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 6-8 grade band engage with MP3 throughout the year. MP3 is found in lessons where they focus on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

Across the grades, students participate in tasks that support key components of MP3. These include constructing mathematical arguments, analyzing errors in sample student work, and explaining or justifying their thinking orally or in writing using concrete models, drawings, numbers, or actions. Students are encouraged to listen to or read the arguments of others, evaluate their reasoning, and ask clarifying questions to strengthen or improve the argument. They also have opportunities to make and test conjectures as they solve problems. Teachers are guided to support this development by providing opportunities for students to engage in mathematical discourse, set clear expectations for explanation and justification, and compare different strategies or solutions. Teachers are prompted to ask clarifying and probing questions, support students in presenting their solutions as arguments, and facilitate discussions that help students reflect on and refine their reasoning.

An example in Grade 6 includes:

• Chapter 3, Lesson 1, Student Edition, students evaluate provided work to determine ratio equivalence and justify their conclusions with clear reasoning. Practice Exercise 18 states, “Your friend says that the two ratios are equivalent. Is your friend correct? Explain your reasoning.” Students examine the ratios, 4:8 and 8:12. An explanation is then provided to illustrate the student’s reasoning. “Because you can add 4 to each number in the first ratio to obtain the numbers in the second ratio, the ratios are equivalent.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “The student will understand the concepts of ratios and equivalent ratios. This is a great opportunity to emphasize constructing viable arguments and critiquing the reasoning of others (SMP.3) during learning and problem solving. Require students to explain their reasoning and solicit several explanations. Model for and encourage students to ask questions to help clarify or improve their mathematical understanding of ratios. Provide opportunities for students to discourse in pairs or groups.”

An example in Grade 7 includes:

• Chapter 8, Lesson 1, Student Edition, students examine data samples related to why students leave school for lunch and develop a supported argument based on their analysis. In-Class Practice Exercise 1 states, “You want to estimate the number of students who leave school for lunch. Are any of the samples unbiased? Explain.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “An important idea in this lesson is that a sample must be randomly selected from equally likely choices to make a valid conclusion. Engage students in SMP.3 by actively prompting them to articulate their thinking and reasons why a sample is biased or unbiased. Students should also listen to the ideas of others and ask questions to clarify their own mathematical thinking. Use In-Class Practice Exercises 1 and 4 to generate class discussion and reveal students' mathematical thinking about what makes a sample biased or unbiased. Suggested prompts In Exercise 1, talk about each choice and why you think it is / is not an unbiased sample.’ Have several volunteers share. ‘Do you agree with ___'s reasoning? Explain.”

An example in Grade 8 includes:

• Chapter 10, Lesson 4, Student Edition, students reason about similarity and volume relationships between two cones and use mathematical arguments to justify or refute a claim. Practice Exercise 17 states, “You and a friend make paper cones to collect beach glass. You cut out the largest possible three-fourths circle from each piece of paper. a. Are the cones similar? Explain your reasoning. b. Your friend says that because your sheet of paper is twice as large, your cone will hold exactly twice the volume of beach glass. Is this true? Explain.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students will find surface areas and volumes of similar solids. Engaging in SMP.3 will reveal student thinking and help you gauge understanding as they work to determine surface area or volume based on similarity. Ask probing questions to ensure they have conceptual understanding and are not just mimicking a procedure. For Practice Exercise 17, understanding the problem scenario is an important step. Suggested prompts: ‘What is the problem situation? Describe it in your own words. What do you know from the problem statement? What do you know from the diagram?’ ‘In part (a), what makes two cones similar figures? What information is given that you can use to work with? How does knowing if the figures are similar help with this problem?’ ‘What is the friend's reasoning based upon? If you agree, explain why using mathematics to justify your thinking. If you disagree, provide a mathematical argument. How can you check your answer? Turn and talk with a partner to discuss your solution pathways.’”

The materials reviewed for Math & YOU Grades 6 through Grade 8 meet expectations for supporting the intentional development of MP4: Model with Mathematics, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 6-8 grade band engage with MP4 throughout the year. MP4 is found in lessons where the lessons focus on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

Across the grades, students participate in tasks that support the intentional development of MP4, including putting problems or situations in their own words and identifying important information, using the math they know to solve problems and everyday situations, modeling the situation with appropriate representations and strategies, describing how their model relates to the problem situation, and checking whether their answer makes sense, revising the model when necessary.

An example in Grade 6 includes:

• Chapter 8, Lesson 1, Student Edition, students use a model to solve problems involving integers and temperature. In-Class Practice Exercise 10 states, “The freezing point of water is 32\degree F. One cup of water is 25\degree F. and another cup is 68\degree F. Write integers that represent how the temperature of each cup of water must change to start changing from liquid to solid or from solid to liquid. Justify your answers using a number line.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Throughout the lesson, students will use SMP.4 to help make connections to familiar real-life situations. Students will develop a model to represent positive and negative numbers within the context of a problem and use it to find a solution and explain their reasoning. A vertical or horizontal number line is a powerful model students will use in this lesson. Use the In-Class Practice Exercise 10 to have students model with mathematics. Suggested prompts: ‘How can you model the quantities in this problem to answer the question? How will your model help you to find the answer?’ ‘How do you know whether to write your answer as a positive or a negative integer? What does that represent about the real-life situation?’ ‘Do your answers seem reasonable? How can you use your model to check your answers?’”

An example in Grade 7 includes:

• Chapter 1, Lesson 5, Student Edition, students use a mathematical model to solve problems involving money and bank accounts. In-Class Practice Exercise 11 states, “You withdrew 55 from an account, and then made a deposit the next day. a. Find and interpret the distance between the points. b. What was the account balance before making the withdrawal? Use a number line to justify your answer.” Students examine a vertical number line labeled with the following information: After Deposit: 6.18 and After Withdrawal: –$25.32. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “As in earlier lessons, students will continue to model with mathematics (SMP.4). Students will develop models for solving real-life problems by using expressions and equations. These models will help students attend to the relationships between quantities. It is important that students use their model to interpret the results and check for reasonableness. Use real-life contexts to help students represent word problems. Use In-Class Practice Exercise 11 to engage students in stages of modeling with mathematics. Suggested prompts: ‘What relationships can you identify in the problem? How can you model those relationships?’ ‘How can you use your model to answer the problem question?’ ‘What does your solution mean in the context of the problem?’ ‘Does your answer make sense in the real-life context? How can you use your model to check the reasonableness of your answer?’”

An example in Grade 8 includes:

• Chapter 6, Lesson 2, Student Edition, students work with lines of best fit and consider mathematical models that could be used to solve the problems. Practice Exercise 4 states, “The table shows the weights and the prescribed dosages of medicine for six patients. a. Find an equation of the line of best fit. Identify and interpret the slope and correlation coefficient. b. A patient who weighs 140 pounds is prescribed 135 milligrams of medicine. How does this affect the line of best fit?” Students see a table with patient weights and the corresponding dosages for each patient. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “This lesson combines students’ understanding of scatter plots with their previous knowledge of using various forms of linear equations to graph linear relationships. They will engage in SMP.4 by finding a linear equation to fit a set of real-life data and then interpret the graph and draw conclusions about the strength of correlation. For Practice Exercise 4, encourage students in the stages of modeling. Suggested prompts: ‘In part (a), what relationship between the data do you suspect might exist? Why? After you model the data and find a line of fit, do you still think that?’ ‘Turn and talk with a partner to share your interpretations of the slope and correlation coefficient. Explain them in terms of the real-life context.’ ‘For part (b), describe in your own words what this new data point represents in the real-life problem scenario. Do you need to redraw a line of fit with this new point included? Discuss your conclusions.”

The materials reviewed for Math & YOU Grades 6 through Grade 8 meet expectations for supporting the intentional development of MP5: Use appropriate tools strategically, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 6-8 grade band engage with MP5 throughout the year. MP5 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

Across the grades, students participate in tasks that support key components of MP5. These include selecting and using appropriate tools and strategies to explore mathematical ideas, solve problems, and communicate their thinking. Students are encouraged to consider the advantages and limitations of various tools such as manipulatives, drawings, measuring devices, and digital technologies, and to choose tools that best support their understanding and reasoning. They have opportunities to use tools flexibly for investigation, calculation, representation, and sense-making. Teachers are guided to support this development by making a variety of tools available, modeling their effective use, and encouraging students to make strategic decisions about when and how to use tools. Teacher materials prompt opportunities for student choice in tool selection, promote discussion around tool effectiveness, and support the comparison of multiple tools or representations. Teachers are also encouraged to highlight how different tools can yield different insights and to help students reflect on their tool choices as part of the problem-solving process.

An example in Grade 6 includes:

• Chapter 3, Lesson 6, Student Edition, students determine the unit rate relationship between quantities expressed in both standard and metric units of measure. Investigate states, “Work with a partner. 1. You have 4 one-liter bottles and a one-gallon jug. a. You empty the one-gallon jug into multiple one-liter bottles, as shown. Write a unit rate that estimates the number of liters per gallon. b. You empty a one-liter bottle into the one-gallon jug, as shown. Write a unit rate that estimates the number of gallons per liter. c. How many liters are in 5.5 gallons? How many gallons are in 12 liters? Explain your method. 2. How can you use rulers to convert a rate given in inches per minute to centimeters per second?” Nick’s Notes-Investigate states, “‘When would rulers not be a useful tool for converting between centimeters and inches?’” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students will use ratio reasoning to convert units of measure. Students will use appropriate tools strategically (SMP.5), which can include physical tools as well as mathematical methods and strategies. Make rulers readily accessible for students to use in this lesson as appropriate. Encourage tools from previous lessons to convert units such as a double number line or ratio table. Discuss the benefits and limitations of tools and strategies. The Investigate and In-Class Practice Exercise 1 provide opportunities for students to use appropriate tools strategically. Suggested prompts: ‘When are physical measuring tools helpful and when are they not helpful? Give examples.’ ‘What tools or strategies can help you when you are solving measurement converting problems? Why is your strategy helpful?’”

An example in Grade 7 includes:

• Chapter 9, Lesson 4, Student Edition, students find the area of two triangular patios, determine which tools will help them solve the problem, and identify which patio is larger. In-Class Practice Exercise 8 states, “Two rooftops have triangular patios with the given side lengths. Which has the greater area? Rooftop A: 9 m, 10 m, 11 m, Rooftop B: 6 m, 10 m, 15 m.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students will construct polygons with given measures using a variety of methods and tools. This lends itself naturally to encouraging them to use appropriate tools strategically (SMP.5). Make a variety of tools readily accessible to students and encourage student choice. Some tools students may use in this section include protractors, rulers, and geometry software. Use In-Class Practice 8 to continue to encourage student choice about tools. Suggested prompts: ‘What tool can help you solve? Explain your choice of tool(s) to a partner. What are the advantages/disadvantages of this tool?’ ‘When is technology more helpful than handheld methods? When do you prefer handheld tools? Why?’”

An example in Grade 8 includes:

• Chapter 8, Lesson 5, Student Edition, students use a mathematical tool (such as a calculator, technology, or other resource) to verify the simplified equation they created to represent a situation. Investigate states, “Work with a partner. 2. Your friend says that the properties of exponents apply to powers of quotients. a. Copy and complete the table to see if your friend is correct. Rewrite each expression as a single power. State the property you used. b. Use technology to see if your friend is correct. Do you think the properties of exponents apply to rational numbers?” Students examine a table with the headings Expression, Simplified Expression, and Property. The expressions in the table are: 0.4^{2}\cdot0.4^{2}, [(\frac{1}{4})^{2}]^{3}, and \frac{1.5^{9}}{1.5^{4}}. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Since rational numbers raised to powers are less intuitive for mental math, you can elevate SMP.5 to students' minds as they consider tools for checking their answers. Encourage them to choose a calculator, software, or a mobile app if students wish to check their numeric answers. Insist that students always still provide their original work via by-hand methods. In the Investigate, students will develop the Power of a Quotient Property. Suggested prompts: ‘In Exercise 1, what general rule did you write? Explain how you obtained it. Try a few more cases to check that your rule works.’ ‘In Exercise 2, what tools can you use to check each simplified expression in your table?" Students can use a calculator, technology, or another tool. Encourage them to explain their tool choice and both the benefits and limitations of their selection. ‘Why is it important to be skilled in using pencil-and-paper methods first before using a calculator or technology as a mathematical tool?’”

The materials reviewed for Math & YOU Grades 6 through Grade 8 meet expectations for supporting the intentional development of MP6: Attend to precision, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 6-8 grade band engage with MP6 throughout the year. MP6 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

Across the grades, students engage in tasks that support key components of MP6. These include formulating clear explanations, using grade-level appropriate vocabulary and conventions, applying definitions and symbols accurately, calculating efficiently, and specifying units of measure. Students are also expected to label tables, graphs, and other representations appropriately, and to use precise language and notation when presenting mathematical ideas. Teachers support this development by modeling accurate mathematical language, ensuring students understand and apply precise definitions, and providing feedback on usage. Lessons include opportunities for students to clarify the meaning of symbols, evaluate the accuracy of their own and others’ work, and refine their communication. Teachers are encouraged to facilitate discussions and structure tasks that promote attention to detail in both reasoning and representation.

An example in Grade 6 includes:

• Chapter 6, Lesson 1, Student Edition, students write one-variable equations to represent real-world situations. Investigate states, “Work with a partner. 1. The equation 6.75s=20.25 represents one customer’s purchase from the menu. Interpret the equation.” Students are shown a menu listing a roast beef sandwich priced at 6.75. “2. Four customers buy multiple sandwiches of the same type. For each customer, write an equation that represents the situation. How many sandwiches does each customer buy?” Students examine a table with the headings Customer, Sandwich Ordered, Amount Paid, and Change Received. The data are as follows: Customer A, Reuben, 20, 0.65, Customer B, Chicken salad, 10, 0.10, Customer C, BLT, 30, 9.00, Customer D, Egg salad, 50, $26.75. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Encourage students to attend to precision (SMP.6) throughout the lesson. They have many opportunities to use precise units in problem solving. Encourage and model the use of key mathematical words and phases that indicate equality (is, the same as, equals). Use the Investigate to give students practice attending to precision. Suggested prompts: ‘What does the variable represent in this exercise? Why is a variable needed? What does it help you to find?’ Encourage students to use math language as they explain. ‘What units are involved in the equation? How do those units impact the units of the answer?’ Allow students time to discuss. ‘Can you explain what you did to solve the problem? Did you use the most efficient way?’”

An example in Grade 7 includes:

• Chapter 5, Lesson 2, Student Edition, students use unit rates in fraction form to solve real-world word problems and explain their solutions using precise language and appropriate mathematical vocabulary. In-Class Practice Exercise 5 states, “You traveled \frac{1}{10} mile every 15 seconds in a five-mile go-kart race. Your friend traveled \frac{3}{8} mile every 48 seconds. Who won the race? What was the difference of the finish time?” Digital Teaching Experience, Supporting the Mathematical Standards: Facilitation Guide states, “This lesson provides opportunity to encourage precision (SMP.6) as students apply their understanding of solving rate problems to using unit rates for rates involving fractions to solve real-life problems. Students need to communicate precisely using clear language and accurate mathematics vocabulary. Insist on the use of academic language from students. Vocabulary terms used throughout this section include ratio, rate, unit rate, and equivalent rate. During In-Class Practice Exercise 5, students will attend to precision. Suggested prompts: ‘Can you explain what you did to solve the problem?’ ‘How do you know that your answers are reasonable? Use key vocabulary terms to explain.’”

An example in Grade 8 includes:

• Chapter 10, Lesson 3, Student Edition, students use a formula to find the volume of a sphere and determine the radius when the volume is given. They are reminded to apply precision in using the formula and to verify that the correct values have been substituted. Practice Exercise 1-4 states, “Find the volume of the sphere. 1. Sphere with a radius of 5 in. 2. Sphere with a diameter of 28 m. Find the radius of the object. 3. Sphere with a volume of 121.5\pi cm^{3} 4. Sphere with a volume of 29 in^{3}.” Digital Teaching Experience, Supporting the Mathematical Standards: Facilitation Guide states, “In this lesson, as students learn about volumes of spheres, they must engage in SMP.6 and pay attention to given details and be careful when representing the quantities in equations. Encourage students to use mathematical vocabulary during their explanations throughout the lesson. For Practice Exercises 1 and 2, remind students to attend to precision. Suggested prompts: (After solving) ‘Were you careful to pay attention to the given information on the spheres? (one is a radius; one is a diameter) ’Did you provide units with your answers? Are units needed?’ ‘What units would you provide if no specific unit of measurement is known for the solid?’” 

The materials reviewed for Math & YOU Grades 6 through Grade 8 meet expectations for supporting the intentional development of MP7: look for and make use of structure, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 6-8 grade band engage with MP7 throughout the year. MP7 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

Across the grades, students engage in tasks that support key components of MP7. These include looking for and describing patterns, identifying structure within mathematical representations, and decomposing complex problems into simpler, more manageable parts. Students are encouraged to analyze problems for underlying structure and to consider multiple solution strategies. They make generalizations based on repeated reasoning and use those generalizations to solve problems efficiently. Teachers support this development by selecting tasks that highlight mathematical structure and prompting students to attend to and describe patterns they notice. Lessons provide opportunities for students to compare approaches, justify their reasoning, and reflect on how structure helps deepen their understanding. Teachers are encouraged to facilitate discussions and structure lessons in ways that promote the recognition and use of patterns and structure in problem-solving.

An example in Grade 6 includes:

• Chapter 2, Lesson 4, Student Edition, students solve division problems with mixed numbers using both models and equations. As they work, they look for and apply patterns to support their problem solving. Investigate states, “Work with a partner. Write a real-life context for each question. Then use a model to answer each question. 1. How many three-fourths are in four and one-half? 2. How many seven-sixths are in three and one-third? 3. How many three and one-halves are in two and one-half? 4. How many one and one-halves are in six? 5. How many four and one-halves are in one and one-half” Students use a model to represent each problem and solution. Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students will look for and make use of structure (SMP.7) in their problem solving. They will see patterns in the division using models. The patterns will help them see complicated computation and begin to use this structure to problem solve. It is important that students be given time to find the patterns themselves. In the Investigate, students look for and make use of structure. Suggested prompts: ‘What do you already know about fraction division that can help you with mixed number division?’ ‘What about the structure of the quantities \frac{3}{4} and 4\frac{1}{2} can help you to analyze the model?’ ‘What patterns do you find in using a model to divide by a fraction to help you solve a future problem? Explain.’”

An example in Grade 7 includes:

• Chapter 9, Lesson 2, Student Edition, students estimate the area of a circle and then calculate the exact area using the formula. Investigate states, “Work with a partner. 1. Each grid contains a circle with a diameter of 4 centimeters. Use each grid to estimate the area of the circle. Which estimate do you think is closest to the actual area? Explain.” Students are shown three circles, each drawn on a different size of grid paper. “You divide a circle with radius r into several equal sections. You then arrange the sections to form a shape that resembles a parallelogram. Use the diagram to write a formula for the area of a circle in terms of the radius. Then use the formula to find the area of the circle in Exercise 1. Compare the answer with your estimates.” Students examine a circle partitioned into triangles and a parallelogram partitioned into triangles. Teaching Edition, Talk About It states, “Explain the relationship between the circumference and the area of a circle.” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students will look for and make use of structure (SMP.7) as they explore areas of circles to develop conceptual understanding of a formula for the area of a circle. They will analyze the relationship between the circumference and the area of a circle and apply their knowledge of the area of a parallelogram from previous grades. Engage students in discussions emphasizing the relationship between these attributes. In the Investigate, students will look for and make use of structure. Suggested prompts: ‘In Exercise 1, how does the size of the grid impact your estimate of the area? Which is most accurate?’ ‘In Exercise 2, how does the arrangement of pieces help you to think about the area? What would be true if the circle could be divided into even smaller sections?’”

An example in Grade 8 includes: 

• Chapter 2, Lesson 3, Student Edition, students identify a rotation and use coordinates to rotate a figure about the origin. They look for structural cues within the shape and the data to guide their solution. Practice Exercise 17 states, “You rotate a triangle 90° counterclockwise about the origin. Then you translate its image 1 unit left and 2 units down. The vertices of the final image are (−5, 0), (−2, 2), and (−2, −1). What are the vertices of the original triangle?” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “The practice SMP.7 comes into play as students recognize and use the structure of figures to help them visualize rotations and understand patterns in the coordinates of transformed points. As students encounter combinations of transformations, they make use of structure by analyzing the problem and understanding the individual stages within the overall transformation. Use Practice Exercise 17 to generate conversation about structure within rotations. Suggested prompts: ‘What approach did you use to solve this problem? How did the structure of the triangle help inform your decision?’ Listen for students to reason that because one side of the final figure was vertical, it must have been horizontal in the original orientation. ‘How did you break down the parts of this transformation into smaller, more manageable parts?’”

The materials reviewed for Math & YOU Grades 6 through Grade 8 meet expectations for supporting the intentional development of MP8: Look for and express regularity in repeated reasoning, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 6-8 grade band engage with MP8 throughout the year. MP8 is found in lessons as each lesson focuses on one or more of the Standards for Mathematical Practice. It is evident in both the Student Experience and the Teaching Experience. Notes in the Teaching Experience remind teachers to foster student engagement with the practice through prompts and questions that promote participation. In addition, course, chapter, and lesson level resources for teachers emphasize an ‘over time’ approach to developing students’ use of the math practices.

Across the grades, students engage in tasks that support key components of MP8. These include notice and use repeated reasoning to make sense of problems, recognize patterns, and develop efficient, mathematically sound strategies. Students are encouraged to create, describe, and explain general methods, formulas, or processes based on patterns they identify. Lessons provide opportunities for students to evaluate the reasonableness of their answers and refine their approaches through discussion and reflection. Teachers support this development by structuring tasks that highlight repeated reasoning, prompting students to make generalizations, and guiding them to build conceptual understanding, distinct from relying on memorized tricks. Instructional guidance also encourages teachers to model and elicit strategies that build toward formal algorithms or representations through consistent reasoning.

An example in Grade 6 includes:

• Chapter 6, Lesson 2, Student Edition, students write and solve equations involving addition and subtraction, looking for repeated reasoning as they substitute values into the equation. Investigate states, “Work with a partner. 1. When two sides of a scale weigh the same, the scale will balance. a. How is an equation like a balanced scale? b. When you add or subtract weight on one side of a balanced scale, what can you do to balance the scale? How does this relate to solving an equation? 2. You can use algebra tiles to model and solve equations.” Students see visual representations using algebra tiles: a yellow plus cube represents +1, a red minus cube represents –1, and a green plus rectangle represents a variable. “a. Write the equations modeled below. Use algebra tiles to solve each equation. Explain. b. How can you solve each equation without using algebra tiles?” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, allow students to look for and express regularity in repeated reasoning (SMP.8) to write and solve equations using addition or subtraction. Enable their own discovery in generalizing procedures for adding or subtracting to solve for the variable. Allow students to discover and define the relationship between x, p, and q rather than you doing the work for them. In the Investigate, students explore balancing an equation with various models to generalize a procedure. Suggested prompts: ‘How does the balance scale help represent an equation? How does that impact how you use algebra tiles to solve an equation?’ ‘What algebra tile procedure did you use? What procedure with numbers and variables did you discover to solve any equation of the form x+p=q?”

An example in Grade 7 includes:

• Chapter 9, Lesson 4, Student Edition, students draw polygons with specified angles and side lengths, using repeated reasoning to identify patterns in the construction of the polygons. Investigation states, “Work with a partner. 1. Use geometry software to complete the table.” Students see a table with several side lengths or angle measures. They are expected to fill in the column labeled, How Many Polygons are Possible? “2. Without constructing, how can you tell whether it is possible to draw a triangle given three angle measures? three side lengths? 3. Without constructing, how can you tell whether it is possible to draw a quadrilateral given four angle measures? four side lengths?” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “Students will look for and express regularity in repeated reasoning (SMP.8). They will work to develop generalizations when finding the sums of the angle measures that resulted in triangles and quadrilaterals. Allow students to do the work of finding the patterns that exist in the sums of the angle measures. Drawing pictures and diagrams and creating tables may help students make sense of the problem. Provide ample sense making experiences for students to explore constructing triangles to solve real-life problems. Use the Investigate to have students look for and express regularity in repeated reasoning. Suggested prompts: ‘As you worked through each row in the table, were you able to form a polygon? If not, why wasn’t it working? What patterns did you notice whenever this happened?’ Keep in mind that the goal of this Investigate is for students to gain an informal understanding of angle sum rules and the Triangle Inequality Theorem. ‘What do you notice about the sums of the angle measures that resulted in each polygon? What generalizations can you make?’”

An example in Grade 8 includes:

• Chapter 3, Lesson 3, Student Edition, students find the interior angles of a regular polygon using an equation. In addition, they explain how the sum of the interior angles can be used to determine unknown angles. Investigation states, “Work with a partner. 1. Find the sum of the interior angle measures by dividing each polygon into triangles.” Students are shown shapes with 4 through 9 sides in items a–f. “2. Complete the table. Then write a formula that gives the sum S of the interior angle measures of a polygon with n sides.” The table lists the number of sides, 𝑛 (3-9), across the top. The left-hand side of the table is labeled Number of Triangles and Interior Angle Sum, s. Teaching Edition, Talk About It states, “Examine the values in the table. What do you notice? What conclusions can you make?’” Digital Teaching Experience, Supporting the Mathematical Practices: Facilitation Guide states, “In this lesson, students will discover patterns and write a formula for the interior angle measures of a polygon. SMP.8 is engaged as students use repeated reasoning to generalize and develop a rule to find the sum of any polygon’s interior angle measures. In the Investigate, students will explore the structure of polygons and use repeated reasoning to find the sum of the interior angles. ‘In Exercise 1, how is the structure useful in finding the total sum of the interior angles within each polygon? How can you use what you know to find the sum of all interior angle measures?’ ‘In Exercise 2, what patterns do you notice as you complete your table? Can you make a generalization? Do you think it will always work? Why or why not?’ If students struggle, have them first look at the first two rows. This should help them see the first part of the formula, (n - 2).”