6th to 8th Grade - Gateway 2

The materials reviewed for Illustrative Mathematics® v.360 Grade 6 through Grade 8 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Multiple conceptual understanding problems are embedded throughout the grade level within Warm-ups, Activities, or Cool-downs. Students have opportunities to engage with these problems both independently and with teacher support. According to the Course Guide, 7. Key Structures in This Course, Principles of IM Curriculum Design, Developing Conceptual Understanding and Fluency, “Each unit begins with Check Your Readiness, a diagnostic assessment to gauge what students know about both prerequisite and upcoming concepts and skills. Adjustments are then made accordingly. The initial lesson in a unit activates prior knowledge and provides an easy entry point to new concepts, so that students at different levels of both mathematical and English language proficiency engage productively in the work. As the unit progresses, students are systematically introduced to representations, contexts, concepts, language and notation. As their learning progresses, they make connections between different representations and strategies, consolidating their conceptual understanding, and see and understand more efficient methods of solving problems, supporting the shift toward procedural fluency. The distributed practice problems give students ongoing practice, which also supports developing procedural proficiency.” 

An example in Grade 6 includes:

• Unit 7, Rational Numbers, Lesson 4, Activity 4.3, Problem 1, students develop conceptual understanding of greater than and less than to describe order and position on a number line. Students are shown a number line with points M and N to the left of zero and points P and R to the right of zero. Student Task Statement states, “The number line shows 4 points: M, N, P, and R. Use each of the following phrases in a sentence describing or comparing the values of 2 of the points, greater than, less than, opposite of (or opposites), negative number. The Activity Synthesis states, “The goal of this discussion is to give students the opportunity to use precise language as they compare the relative positions of rational numbers. Give students 2–3 minutes to discuss their responses with a partner before a whole-class discussion. Ask students to share their partner’s reasoning, especially if it was different than their own. Here are some questions to consider: ‘Did you ever have a different answer than your partner? If so, were you both correct? If not, how did you work to reach an agreement?’ ‘How can we tell if two numbers are opposites?’ (They are the same distance from 0 but on opposite sides of 0.) ‘How can we tell if one number is greater or less than another number?’ (Numbers toward the right are considered greater, and numbers toward the left are considered less.)” (6.NS.6)

An example in Grade 7 includes:

• Unit 8, Probability and Samples, Lesson 3, Warm-up, students develop conceptual understanding of probability and possible outcomes. “Which game are you more likely to win? Explain your reasoning. Game 1: You flip a coin and win if it lands showing heads. Game 2: You roll a standard number cube and win if it lands showing a number that is divisible by 3.” The Activity Synthesis states, “Have partners share their answers and display the results for all to see. Select at least 1 student for each answer provided to give a reason for their choice. If no student mentions it, explain that the number of possible outcomes that count as a win and the number of total possible outcomes are both important to determining the likelihood of an event. That is, although there are 2 ways to win with the standard number cube and only 1 way to win on the coin, the greater number of possible outcomes in the second game makes it less likely to provide a win.” (7.SP.5)

An example in Grade 8 includes:

• Unit 2, Dilations, Similarity, and Introducing Slope, Lesson 10, Activity 10.2, Problem 1, students develop conceptual understanding of why certain triangles with one side along the same line are similar. Student Task Statement states, “The grid shows three right triangles, each with its longest side on the same line. Your teacher will assign you two of the triangles. Explain why the two triangles are similar. 2. Complete the table. 3. What do you notice about the last column in the table? Why do you think this is true?” Activity 10.3, students develop conceptual understanding of two lines with similar slopes are parallel. Student Task Statement states, “1. Draw two lines with a slope of 3. What do you notice about the two lines? 2. Draw two lines with slope \frac{1}{2}.” The Activity Synthesis states, “Ask previously selected students, as described in the Activity Narrative, to share how they drew their lines with a slope of \frac{1}{2}. Demonstrate how it does not matter if you draw a slope triangle with a vertical length of 1 and a horizontal length of 2, or a triangle with vertical and horizontal lengths of 3 and 6, or 5 and 10. Explain that the quotient of side lengths is the important feature, since any triangle drawn to match a given slope will be similar to any other triangle drawn to match the same slope. Here are some questions for discussion: ‘What did you notice about the two lines you drew with a slope of 3? With a slope of \frac{1}{2}?’ (Lines with the same slope are parallel. Slope triangles for the lines with the same slope are similar.) ‘What did you notice about the lines with a slope of 3 compared to the lines with a slope of \frac{1}{2}?’ (The lines with a slope of 3 look ‘steeper’ than the lines with a slope of \frac{1}{2}.)” Students use an applet in presentation mode. (8.EE.6)

According to Illustrative Mathematics® v.360 Course Guide, the materials are designed to provide students with opportunities to independently demonstrate conceptual understanding, when appropriate. 7. Key Structures in This Course, Principles of IM Curriculum Design, Developing Conceptual Understanding And Fluency states, “Across lessons, units, and courses, students are encouraged to use representations that make sense to them, and to make connections—between representations, as well as between representations and the concepts and procedures they show. Over time, students learn to recognize and use efficient methods of representing and solving problems, which in turn supports fluency.” 

An example in Grade 6 includes:

• Unit 3, Unit Rates and Percentages, Lesson 6, Activity 6.2, students demonstrate conceptual understanding of equivalent ratios that have the same unit rate. Student Task Statement states, “1. Two binders cost $14, and 5 binders cost $35. a. Complete the table to show the cost for 4 binders and 10 binders at that rate. Next, find the cost for a single binder in each case, and record those values in the third column. b. What do you notice about the values in this table? 2. This table shows the cost of notebooks. Complete the table. Be prepared to explain your reasoning.” A table shows the number of notebooks, total cost (dollars), and unit price (dollars per notebook). Each column has some values filled in, while others are left blank for students to complete. For example, the table provides 20 notebooks and a unit price of $3 per notebook, students must determine the total cost in dollars. (6.RP.2, 6.RP.3)

An example in Grade 7 includes:

• Unit 5, Rational Number Arithmetic, Lesson 4, Warm-up, students demonstrate conceptual understanding of positive and negative integers. “Priya wants to buy 3 tickets for a concert. Each ticket costs $50. She has earned $135. 1. What could Priya do in order to be able to buy the tickets? 2. One equation that represents this situation is 135+15=50\bullet 3. What do each of the numbers tell us about this situation? 3. Another equation that represents this situation is 135-3\bullet 50=-15. What do each of the numbers tell us in this situation?” (7.NS.1)

An example in Grade 8 includes:

• Unit 5, Functions and Volume, Lesson 8, Cool-down, students demonstrate conceptual understanding that linear functions can be represented by an equation in y = mx + b format. Student Task Statement states, “In a certain city in France, they gain 2 minutes of daylight each day after the spring equinox (usually in March), but after the autumnal equinox (usually in September), they lose 2 minutes of daylight each day. 1. Which of the graphs is most likely to represent the graph of daylight for the month after the spring equinox? 2. Which of the graphs is most likely to represent the graph of daylight for the month after the autumnal equinox? 3. Why are the other graphs not likely to represent either month?” Four graphs are pictured. (8.F.3 and 8.F.4)

The materials reviewed for Illustrative Mathematics® v.360 Grade 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.

According to the Course Guide, 7. Key Structures of This Course, Principles of IM Curriculum Design, Conceptual Understanding and Procedural Fluency, “Warm-up routines, practice problems, centers, and other built-in activities help students develop procedural fluency, which develops over time.” 

An example in Grade 6 includes: 

• Unit 5, Arithmetic in Base Ten, Lesson 11, Activity 11.2, students develop procedural skill and fluency as they complete long division problems using the standard algorithm. Problem 2 states, “Lin’s method is called long division. Use this method to find the following quotients. Check your answer by multiplying it by the divisor. a. 846\div 3 b. 1816\div 4 c. 768\div 12.” (6.NS.2)

An example in Grade 7 includes: 

• Unit 7, Angles, Triangles, and Prisms, Lesson 2 Practice Problems, students develop procedural skill and fluency as they use facts about supplementary, complementary angles to find angle pairs. Problem 1 states, “Angles A and C are supplementary. Find the measure of angle C.” Two angles are shown, with ∠BAE measuring 74°. Problem 2 states, “a. List two pairs of angles in square CDFG that are complementary. b. Name three angles whose measures add up to 180°.” (7.G.5)

An example in Grade 8 includes: 

• Unit 7, Exponents and Scientific Notation, Lesson 2, Activity 2.3, students develop procedural skill and fluency in multiplying exponents with the same base. Problem 1 states, “a. Complete the table to explore patterns in the exponents when multiplying powers of 10. You may skip a single box in the table, but if you do, be prepared to explain why you skipped it. b. If you chose to skip one entry in the table, which entry did you skip? Why?” Students fill in a table with columns labeled expression, expanded, and single power of 10. For example, 10^{2}\bullet 10^{3}=(10\bullet 10)(10\bullet 10\bullet 10)=10^{5}. (8.EE.1)

According to the Course Guide, 7. Key Structures of This Course, Principles of IM Curriculum Design, Coherent Progression, materials were designed to include opportunities for students to independently demonstrate procedural skill and fluency, when appropriate. “Each activity starts with a Launch that gives all students access to the task. Independent work time follows, allowing them to grapple with problems individually before working in small groups. In the Activity Synthesis at the end, students consolidate their learning by making connections between their work and the mathematical goals. Each activity includes carefully chosen contexts and numbers that support the coherent sequence of learning goals in the lesson.” 

An example in Grade 6 includes:

• Unit 1, Area and Surface Area, Section E, Lesson 17, Cool-down, students demonstrate procedural skill and fluency as they work with exponential expressions. “1. Which is larger, 5^{2} or 3^{3}? 2. A cube has an edge length of 21 cm. Use an exponent to express its volume.” (6.EE.1)

An example in Grade 7 includes:

• Unit 8, Probability and Sampling, Lesson 16, Activity 16.2, students demonstrate procedural skill and fluency as they convert rational numbers to decimals. Student Task Statement states, “The track coach at a high school needs a student whose reaction time is less than 0.4 seconds to help out at track meets. All the twelfth graders in the school measured their reaction times. Your teacher will give you a bag of papers that list their results. 1. Work with your partner to select a random sample of 20 reaction times, and record them in the table. 2. What proportion of your sample is less than 0.4 seconds? 3. Estimate the proportion of all twelfth graders at this school who have a reaction time of less than 0.4 seconds. Explain your reasoning. 4. There are 120 twelfth graders at this school. Estimate how many of them have a reaction time of less than 0.4 seconds.” (7.NS.2d)

An example in Grade 8 includes:

• Unit 4, Linear Equation and Linear Systems, Section C, Lesson 12, Activity 12.3, students demonstrate procedural skill and fluency as they solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Launch states, “A stack of n small cups has a height, h, in centimeters, of h = 1.5n + 6. A stack of n large cups has a height, h, in centimeters, of h = 15n + 9.” Problems 1 & 2 states, “1. Graph the equations for each cup on the same set of axes. Make sure to label the axes and decide on an appropriate scale. 2. For what number of cups will the two stacks have the same height?” (8.EE.8b)

The materials reviewed for Illustrative Mathematics® v.360 Grade 6 through 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 Activities or Cool-downs. Students have opportunities to engage with these applications both with teacher support and independently. According to the Course Guide, materials are designed to provide students with opportunities to independently demonstrate their understanding of grade-level mathematics when appropriate. 7. Key Structures in This Course, Principles of IM Curriculum Design, Coherent Progression states, “Each activity starts with a Launch that gives all students access to the task. Independent work time follows, allowing them to grapple with problems individually before working in small groups. In the Activity Synthesis at the end, students consolidate their learning by making connections between their work and the mathematical goals.” 

An example of a routine application of the math in Grade 6 includes:

• Unit 6, Expressions and Equations, Lesson 17, Activity 17.2, students engage in a routine application problem as they calculate unit rates. Student Task Statement states, “Diego, Elena, and Andre joined different teams for a robotics competition. Each team programmed a robot to travel at a constant rate. 1. Complete each table to show how far each team’s robot traveled during the competition.” Students are given three tables with missing values. Tables are labeled Diego’s Team, Elena’s Team, and Andre’s Team. Columns are titled: time in (minutes), distance traveled (meters). Problem 2 states, “Graph points to show the progress of each robot. Use a different color or symbol for each robot.” Problem 3 states, “Let’s say that d represents the distance a robot traveled in meters and t represents the time in minutes. a. Explain why d=3t relates the distance and time that Diego’s robot traveled. Part b. In this equation, which variable is independent and which one is dependent? Record your answer in the table. Be prepared to explain how you know.” (6.RP.3)

An example of a routine application of the math in Grade 7 includes:

• Unit 7, Angles, Triangles, and Prisms, Lesson 16, Activity 16.3, students engage in a routine application problem using knowledge of proportional relationships, volume, and surface area. Student Task Statement states, “The daycare has two sandboxes that are both prisms with regular hexagons as their bases. The smaller sandbox has a base area of 1,146 in² and is filled 10 inches deep with sand. 1. It took 14 bags of sand to fill the small sandbox to this depth. What volume of sand comes in one bag? (Round to the nearest whole cubic inch.) 2. The daycare manager wants to add 3 more inches to the depth of the sand in the small sandbox. How many bags of sand will they need to buy? 3. The daycare manager also wants to add 3 more inches to the depth of the sand in the large sandbox. The base of the large sandbox is a scaled copy of the base of the small sandbox, with a scale factor of 1.5. How many bags of sand will they need to buy for the large sandbox? 4. A lawn and garden store is selling 6 bags of sand for $19.50. How much will they spend to buy all the new sand for both sandboxes?” (7.RP.A)

An example of a routine application of the math in Grade 8 includes:

• Unit 5, Functions and Volume, Lesson 9, Cool-down, students independently engage in a routine application problem, determining if a single linear model can be used. Student Task Statement states, “A small company is selling a new board game, and they need to know how many to produce in the future. After 12 months, they sold 4 thousand games. After 18 months, they sold 7 thousand games. And after 36 months, they sold 15 thousand games. Could this information be reasonably estimated using a single linear model? If so, use the model to estimate the number of games sold after 48 months. If not, explain your reasoning.” (8.F.B)

An example of a non-routine application of the math in Grade 6 includes:

• Unit 2, Introducing Ratios, Lesson 15, Cool-down, students independently engage in a routine application problem as they use ratios to solve a problem. Problem 1 states, “A house has a kitchen, a playroom, and a dining room on the first floor. The areas of the kitchen, playroom, and dining room in square feet are in the ratio 4 : 3 : 2. The combined area of these three rooms is 189 square feet. What is the area of each room?” (6.RP.3)

An example of a non-routine application of the math in Grade 7 includes:

• Unit 6, Expressions, Equations, and Inequalities, Lesson 2 Practice Problems, students independently engage in a non-routine application problem using tape diagrams to explain and solve equations. Problem 3 states, “Andre wants to save $40 to buy a gift for his dad. Andre’s neighbor will pay him weekly to mow the lawn, but Andre always gives a $2 donation to the food bank in weeks when he earns money. Andre calculates that it will take him 5 weeks to earn the money for his dad’s gift. He draws a tape diagram to represent the situation. a. Explain how the parts of the tape diagram represent the story. b. How much does Andre’s neighbor pay him each week to mow the lawn?” (7.EE.3)

An example of a non-routine application of the math in grade 8 includes:

• Unit 1, Rigid Transformations and Congruence, Lesson 17, Activity 17.2, students engage in a non-routine application problem using congruence and similarity to create their own tessellation. “1. Design your own tessellation. You will need to decide which shapes you want to use and make copies. Remember that a tessellation is a repeating pattern that goes on forever to fill up the entire plane. 2. Find a partner and trade pictures. Describe a transformation of your partner’s picture that takes the pattern to itself. How many different transformations can you find that take the pattern to itself? Consider translations, reflections, and rotations. 3. If there’s time, color and decorate your tessellation.” (8.G.A)

The materials reviewed for Illustrative Mathematics® v.360 Grade 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 throughout the materials in order to develop students’ mathematical understanding of a single unit of study or topic. 

An example in Grade 6 includes:

• Unit 2, Introducing Ratios, Lesson 16, Activity 16.3, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they apply ratio reasoning to solve problems. Student Task Statement states, “Solve each problem and show your thinking. Organize it so it can be followed by others. If you get stuck, consider drawing a double number line diagram, a table, or a tape diagram. 1. A recipe for a cleaning liquid says to mix 4 parts water with 3 parts vinegar. How much water is needed to make a total of 28 tablespoons of the solution? 2. Andre and Han are moving boxes. Andre can move 4 boxes every half hour. Han can move 5 boxes every half hour. How long will it take Andre and Han to move all 72 boxes?” (6.RP.3)

An example in grade 7 includes:

• Unit 2, Introducing Proportional Relationships, Lesson 3, Activity 3.3, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they apply their knowledge of the relationship between constant of proportionality and constant speed. Student Task Statement states, “On its way from New York to San Diego, a plane flew over Pittsburgh, Saint Louis, Albuquerque, and Phoenix traveling at a constant speed. Complete the table as you answer the questions. Be prepared to explain your reasoning. 1. What is the distance between Saint Louis and Albuquerque? 2. How many minutes did it take to fly between Albuquerque and Phoenix? 3. What is the proportional relationship represented by this table? 3. Diego says the constant of proportionality is 550. Andre says the constant of proportionality is 9\frac{1}{6}. Do you agree with either of them? Explain your reasoning.” A table of time, distance, and speed is provided, with some values missing for each city segment. (7.RP.2)

An example in Grade 8 includes:

• Unit 7, Exponents and Scientific Notation, Lesson 7, Cool-down, students develop conceptual understanding alongside procedural skill and fluency as they work with the rules of exponents. “Problem 1. Rewrite each expression using a single, positive exponent: a. \frac{9^{3}}{9^{9}} b. 14^{-3}\cdot 14^{12}. Problem 2. Diego wrote 6^{4}\cdot 8^{3}=48^{7}. Explain what Diego’s mistake was and how you know the equation is not true.” (8.EE.1) 

The materials reviewed for Illustrative Mathematics® v.360 Grade 6 through Grade 8 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP1 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Grade 6 includes:

• Unit 1, Area and Surface Area, Lesson 12, Warm-up, students analyze and make sense of problems as they estimate surface area. Lesson Narrative, “Students begin by exploring surface area in concrete terms, by thinking about the number of square sticky notes it would take to cover a filing cabinet. First, they make an estimate, and then they think about what information is needed to calculate the actual number of sticky notes. Because no techniques are given, students need to make sense of the problem and persevere in solving it (MP1).” Launch states, “Arrange students in groups of 2. Show the video of a teacher beginning to cover a large cabinet with sticky notes or display the following still images for all to see. Before starting the video or displaying the image, ask students to be prepared to share one thing they notice and one thing they wonder. A man begins covering a file cabinet with post-it notes and numbers them as he goes. Give students a minute to share their observation and question with a partner. Invite a few students to share their questions with the class. If the question, ‘How many sticky notes would it take to cover the entire cabinet?’, is not mentioned, ask if anyone wondered how many sticky notes it would take to cover the entire cabinet. Give students a minute to make an estimate. Student Task Statement states, “Your teacher will show you a video about a cabinet or some pictures of it. Estimate an answer to the question: How many sticky notes would it take to cover the cabinet, excluding the bottom?” The Activity Synthesis states, “Poll the class for students' estimates, and record them for all to see. Invite a couple of students to share how they made their estimate. Explain to students that they will now think about how to answer this question.”

An example in Grade 7 includes:

• Unit 3, Measuring Circles, Lesson 11, 11.1, students apply their understanding of circumference and area to solve real-world problems. Activity Narrative, “In this activity students apply what they have learned about circles to solve a multi-step problem (MP1). Students find the area and perimeter of geometric figures whose boundaries are line segments and fractions of circles and use that information to calculate the cost of a project.” Student Task Statement states, “The students in art class are designing a stained-glass window to hang in the school entryway. The window will be 3 feet tall and 4 feet wide. Here is their design. They have raised $100 for the project. The colored glass costs $5 per square foot. The clear glass costs $2 per square foot. The material they need to join the pieces of glass together costs 10 cents per foot. The frame around the window costs $4 per foot. Do they have enough money to cover the cost of making the window?”

An example in Grade 8 includes:

• Unit 6, Associations in Data, Lesson 1, Warm-up, students work to organize data in a way that makes sense to them. Activity Narrative, “The purpose of this Warm-up is to elicit the idea that organizing data is helpful for recognizing patterns, which will be useful when students work with ways or organizing data in a later activity. While students may notice and wonder many things about this table, working towards recognizing any patterns and associations are the important discussion points. This Warm-up prompts students to familiarize themselves with the context and mathematics that might be involved by making sense of data before organizing it (MP1).” Student Task Statement states, “Here is a table of data. Each row shows 2 measurements of a triangle. What do you notice? What do you wonder?” Students are provided with a table containing data that shows the length of the short side (in inches) and the perimeter (in inches). The Activity Synthesis states, “Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the table. Next, ask students, ‘Is there anything on this list that you are wondering about now?’ Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement. If the pattern that both values increase together does not come up during the conversation, ask students to discuss this idea.”

The materials reviewed for Illustrative Mathematics® v.360 Grade 6 through Grade 8 meet expectations for supporting the intentional development of MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP2 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Overview, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Grade 7 includes:

• Unit 5, Arithmetic in Base Ten, Lesson 9, 9.3, students use base-ten diagrams to represent division of whole numbers and division of a decimal by a whole number. Student Task Statement states, “To find 53.8\div 4 using diagrams, Elena began by representing 53.8. She placed 1 ten into each group, decomposed the remaining 1 ten into 10 ones, and went on distributing the units. This diagram shows Elena’s initial placement of the units and the decomposition of 1 ten. Here’s Elena’s finished diagram, showing the quotient of 53.8\div 4.  Discuss with a partner: 1. What did Elena do after decomposing the 1 ten into 10 ones? How did she get to the last diagram? 2. Based on Elena’s work, what is the value of 53.8\div 4?” Activity Narrative, “In this activity, students analyze several diagrams that represent a decimal being divided by a whole number. They interpret and explain the presence or arrangement of base-ten units in several stages of reasoning, including in a final diagram, which shows the quotient. As students decompose and distribute base-ten pieces into equal-size groups and think about the meaning of each piece, students practice reasoning abstractly and quantitatively (MP2).”

An example in Grade 6 includes:

• Unit 1, Scale Drawings, Lesson 7, 7.2, students reason abstractly and quantitatively as they measure the lengths of scale drawings. Student Task Statement states, “Your teacher will give you a scale drawing of a basketball court. The drawing does not have any measurements labeled, but it says that 1 centimeter represents 2 meters. 1. To the nearest tenth of a centimeter, measure the distances on the scale drawing that are labeled a–d. Record your results in the first row of the table. 2. The statement ‘1 cm represents 2 m’ is the scale of the drawing. It can also be expressed as ‘1 cm to 2 m, or ‘1 cm for every 2 m.’ What do you think the scale tells us? 3. How long would each measurement from the first question be on an actual basketball court? Explain or show your reasoning. 4. On an actual basketball court, the bench area is typically 9 meters long. a. Without measuring, determine how long the bench area should be on the scale drawing. b. Check your answer by measuring the bench area on the scale drawing. Did your prediction match your measurement?” Activity Narrative, “Students measure lengths on a scale drawing and use a given scale to find corresponding lengths on a basketball court (MP2). Because students are measuring to the nearest tenth of a centimeter, some of the actual measurements they calculate will not have the precision of the official measurements. For example, the official measurement for d is 0.9 m.”

An example in Grade 8 includes:

• Unit 2, Dilations, Similarity, and Introducing Slope, Lesson 13, 13.2, students reason abstractly and quantitatively as they examine the lengths of shadows of different objects. Student Task Statement states, “Here are some measurements that were taken when the photo was taken. It was impossible to directly measure the height of the lamppost, so that cell is blank. 1. What relationships do you notice between an object’s height and the length of its shadow? 2. Make a conjecture about the height of the lamppost and explain your thinking.”A table is provided for students that shows the relationships between the heights and shadow lengths of specific people or objects. Activity Narrative, “The given measurements are real measurements rounded to the nearest inch, and therefore, the values in the table are not in a perfectly proportional relationship. The quotient of each shadow length and its corresponding object’s height is around,  but not exactly, \frac{2}{3}. Students engage in quantitative reasoning by exploring relationships with real-world measurements (MP2).”

The materials reviewed for Illustrative Mathematics® v.360 Grade 6 through Grade 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. 

Students have opportunities to engage with MP3 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Grade 6 includes:

• Unit 8, Data Sets and Distributions, Lesson 12, 12.2, students construct viable arguments and critique the reasoning of others as they use mean and MAD to make comparisons. Activity Narrative: “This activity allows students to practice calculating MAD and to build a better understanding of what it tells us. Students compare data sets with the same mean but different MADs and interpret what these differences imply in the context of the situation. During the discussion, they select a student to be on their team based on the comparison. Expect students to choose different players to be on their team, but be sure they support their preferences with a reasonable explanation (MP3).” Student Task Statement states, “1. Andre and Noah joined Elena, Jada, and Lin in recording their basketball scores. They all record their scores in the same way: the number of baskets made out of 10 attempts. Each person collects 12 data points. Andre’s mean number of baskets is 5.25, and his MAD is 2.6. Noah’s mean number of baskets is also 5.25, but his MAD is 1. Here are two dot plots that represent the two data sets. The triangle indicates the location of the mean. a. Without calculating, decide which dot plot represents Andre’s data and which represents Noah’s. Explain how you know. b. If you are the captain of a basketball team and can use 1 more player on your team, do you choose Andre or Noah? Explain your reasoning.”

An example in Grade 7 includes:

• Unit 2, Introducing Proportional Relationships, Lesson 10, 10.3, students construct viable arguments and critique the reasoning of others as they represent proportional relationships with graphs. Lesson Narrative: “In this activity, students sort different graphs and tables that represent situations they have worked with during previous activities. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7). As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).”  Student Task Statement states, “Your teacher will give you a set of cards that show representations of relationships. 1. Sort the cards into categories of your choosing. Be prepared to describe your categories. Pause for a whole-class discussion. 2. Take turns with your partner to match a table with a graph. a. For each match that you find, explain to your partner how you know it’s a match. b. For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement. 3. Which of the relationships are proportional? 4. What do you notice about the graphs of proportional relationships? Do you think this will hold true for all graphs of proportional relationships?”

An example in Grade 8 includes:

• Unit 4, Linear Equations and Linear Systems, Lesson 3, Activity 3.2, students construct viable arguments and critique the reasoning of others as they begin writing equivalent equations. Student Task Statements states, “Take turns with your partner to match a pair of equations with a description of the valid move that is used to rewrite the first equation as the second. a. For each match that you find, explain to your partner how you know it’s a match. b. For each match that your partner finds, listen carefully to your partner's explanation. If you disagree, discuss your thinking and work to reach an agreement. 2. One of the letter cards does not have a match. For this card, write two equations showing the described move.” Activity Narrative, “In this partner activity, students take turns matching equations with descriptions of valid moves between them to practice identifying and providing descriptions of moves. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).”

The materials reviewed for Illustrative Mathematics® v.360 Grade 6 through Grade 8 meet expectations for supporting the intentional development of MP4: Model with mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with MP4 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Grade 6 includes: 

• Unit 3, Unit Rates and Percentages, Lesson 17, Activity 17.3, students model with mathematics as they apply their understanding of rates and percentages to determine the amount of paint needed to paint a bedroom, the costs for buying paint in different-size containers, and how discounts affect the costs. Activity Narrative, “There are several combinations of can sizes that would be enough for two coats of paint on all the walls. Some students might consider the possibility of spills or errors and opt to buy a larger quantity. This is perfectly valid as long as students can support and explain their choices. In making assumptions and connecting calculations to real-life considerations, students practice modeling with mathematics (MP4).” Student Task Statement states, “Problem 1. An ad for paint reads: ‘Just 2 quarts covers 175 square feet!’ If you need to apply two coats of paint on all the walls, how much paint do you need to buy? Problem 2. Paint can be purchased in 1-quart or 1-gallon containers. The paint chosen for the room costs $12 a quart and $38 a gallon. Pause for a whole-class discussion. a. Which container sizes and how many could you buy to have enough paint for the room? Name at least two options. b. Which of your options would cost the least? Show your reasoning. Problem 3. The hardware store is having a sale: 20% off of quart-size paint and 30% off of gallon-size paint. a. With the sale, how much would you save with each option? b. Would the option you chose earlier still be the cheapest? Show your reasoning. Are you ready for more? Check the prices for a quart and a gallon of interior paint at a local store or online. Look for discounts or deals, but also consider possible differences in quality. Some paints may be more expensive because they are of higher quality. 1. When would it make sense to pay more for better paint? When would it make sense to buy the least expensive paint? 2. What is the best deal you can find for putting 2 coats of paint on the walls of this bedroom?”

An example in Grade 7 includes:

• Unit 4, Proportional Relationships and Percentages, Lesson 13, Activity 13.2, students model with math when they express measurement error as a percentage of the correct amount. Activity Narrative: “In this activity, students determine the amount of error in a measurement and express the error as a percentage of the correct value. They brainstorm possible sources of error and discuss how real-world limitations on humans using measuring devices can introduce measurement errors. As students interpret these results, they consider sources for error and improvement in a mathematical model (MP4).” Student Task Statement states, “A soccer field is 120 yards long. Han uses a 30-foot-long tape measure to measure the length of the field and gets a measurement of 376 feet 6 inches. Problem 1. What is the amount of the error? Problem 2. Express the error as a percentage of the actual length of the field. Explain or show your reasoning. Problem 3. What are some possible causes for this error?”

An example in Grade 8 includes:

• Unit 8, Pythagorean Theorem and Irrational Numbers, Lesson 18, Activity 18.3, students model with math as they use the Pythagorean Theorem to compare the diagonal lengths of rectangles with various dimensions. Lesson Narrative, “In this culminating lesson for the unit, students use the Pythagorean Theorem to compare the diagonal lengths of rectangles with various dimensions. They investigate different aspect ratios of items such as photographs and smartphone screens. There is an element of mathematical modeling (MP4) in the last activity, because in order to quantify the screens’ sizes to compare them, students need to refine the question that is asked.” Student Task Statement, Problem 1 states, “Before 2017, a smart phone manufacturer’s phones had a diagonal length of 5.8 inches and an aspect ratio of 16 : 9. In 2017, they released a new phone that also had a 5.8-inch diagonal length, but an aspect ratio of 18.5 : 9. Some customers complained that the new phones had a smaller screen. Were they correct? If so, how much smaller was the new screen compared to the old screen?”

The materials reviewed for Illustrative Mathematics® v.360 Grade 6 through Grade 8 meet expectations for supporting the intentional development of MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with MP5 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Grade 6 includes:

• Unit 2, Introducing Ratios, Lesson 16, Activity 16.2, students choose an appropriate strategy to solve ratio problems. Lesson Narrative, “In this lesson, students use all representations they have learned in this unit—double number lines, tables, and tape diagrams—to solve ratio problems that involve the sum of the quantities in the ratio. They consider when each tool might be useful and preferable in a given situation and why (MP5).” Student Task Statement states, “A teacher is planning a class trip to the aquarium. The aquarium requires 2 chaperones for every 15 students. The teacher plans accordingly and orders a total of 85 tickets. How many tickets are for chaperones, and how many are for students? 1. Solve this problem in one of three ways: Use a triple number line. Use a table. (Fill rows as needed.) Use a tape diagram. 2. After discussing all three representations with your class, which representation do you prefer for this problem and why? Are you ready for more? Use the digits 1 through 9 to create three equivalent ratios. Use each digit only one time.”

An example in Grade 7 includes:

• Unit 7, Angles, Triangles, and Prisms, Lesson 3, Activity 3.2, students use appropriate tools strategically to identify complementary or supplementary angles. Student Task Statement states, “Identify any pairs of angles in these figures that are complementary or supplementary.” Activity Narrative, “In this activity, students see that angles do not need to be adjacent to each other in order to be considered complementary or supplementary. Students are given two different polygons and are asked to find complementary and supplementary angles, using any tools in their geometry toolkit (MP5). The most likely approaches are: Measure each angle with a protractor and look for any that add up to 180 or 90 degrees. Trace the legs of an angle with tracing paper and align its vertex and one leg with another angle to see if the two angles, when adjacent, form a straight angle or a right angle.”

An example in Grade 8 includes:

• Unit 1, Rigid Transformations and Congruence, Lesson 7, Activity 7.2, students use appropriate tools strategically to compare the side lengths and angle measures of a figure and its image under a translation, rotation, or reflection. Activity Narrative, “The purpose of this activity is for students to see that translations, rotations, and reflections preserve side lengths and angle measures. Students can use tracing paper to help them draw the figures and make observations. While the grid helps measure lengths of horizontal and vertical segments, students may need more guidance when asked to measure diagonal lengths. It is important in the launch to elicit strategies from students to either use tracing paper or an index card to mark off unit lengths using the grid as they use tools strategically (MP5).” Student Task Statement states, “1. Translate Polygon A so point P goes to point Q. In the image, write the length of each side, in grid units, next to the side. 2. Rotate Triangle B 90° clockwise using R as the center of rotation. In the image, write the measure of each angle in its interior. 3. Reflect Pentagon C across line l. a. In the image, write the length of each side, in grid units, next to the side. You may need to make your own ruler with tracing paper or a blank index card. b. In the image, write the measure of each angle in the interior.”

The materials reviewed for Illustrative Mathematics® v.360 Grade 6 through Grade 8 meet expectations for supporting the intentional development of MP6: Attend to precision, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP6 across the year and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.”

An example in Grade 6 includes:

• Unit 7, Rational Numbers, Lesson 7, Activity 7.2, students attend to precision when they compare numbers and distance from zero. Activity Narrative, “The Information Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).” Launch, “Tell students they will be locating points on a number line. Display the Information Gap graphic that illustrates a framework for the routine for all to see. Remind students of the structure of the Information Gap routine, and consider demonstrating the protocol if students are unfamiliar with it. Arrange students in groups of 2. In each group, give a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give students the cards for a second problem, and instruct them to switch roles.” Student Task Statement states, “Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner. If your teacher gives you the problem card. 1. Silently read your card, and think about what information you need to answer the question. 2. Ask your partner for the specific information that you need. “Can you tell me ?” 3. Explain to your partner how you are using the information to solve the problem. “I need to know because . . . .” Continue to ask questions until you have enough information to solve the problem. 4. Once you have enough information, share the problem card with your partner, and solve the problem independently. 5. Read the data card, and discuss your reasoning. If your teacher gives you the data card: 1. Silently read your card. Wait for your partner to ask for information. 2. Before telling your partner any information, ask, “Why do you need to know ?” 3. Listen to your partner’s reasoning, and ask clarifying questions. Only give information that is on your card. Do not figure out anything for your partner! These steps may be repeated. 4. Once your partner says they have enough information to solve the problem, read the problem card, and solve the problem independently. 5. Share the data card, and discuss your reasoning.” Activity Synthesis, “After students have completed their work, share the correct answers, and ask students to discuss the process of solving the problems. Here are some questions for discussion: ‘What strategies were helpful in determining the location of the point?’ (Drawing a number line.) ‘Which clues were most helpful? Least helpful?’ Highlight for students the following glossary and vocabulary terms that were used in this activity and how they are related but different: ‘absolute value,’ ‘sign,’ ‘opposite,’ ‘greater than,’ ‘less than.’

An example in Grade 7 includes:

• Unit 8, Probability and Sampling, Lesson 2, Cool-down, students attend to precision when they investigate chance experiments. Student Task Statement states, “Here are some situations: According to market research, a business has a 75% chance of making money in the first three years. According to lab testing, \frac{5}{6} of a certain kind of experimental light bulb will work after three years. According to experts, the likelihood of a car needing major repairs in the first three years is 0.7. 1. Write these situations in order of likelihood from least to greatest after three years: The business makes money, the light bulb still works, and the car needs major repairs. 2. Select one of the situations. Write a chance experiment and event that has the same likelihood as the situation you selected.” Lesson Narrative, “In some cases, a value is assigned to the likelihood of an event using a fraction, decimal, or percentage chance. By comparing informal categories early and numerical quantities later, students are attending to precision (MP6) when sorting the scenarios. Later, students will connect this language to more precise numerical values on their own.”

An example in Grade 8 includes:

• Unit 5, Functions and Volume, Lesson 5, Warm-up, students attend to precision when they analyze graphs of functions. Activity Narrative: “This Warm-up prompts students to compare four graphs. It gives students a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminologies students know and how they talk about characteristics of graphs.” Launch, “Arrange students in groups of 2–4. Display the graphs for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three graphs that go together and can explain why. Next, tell students to share their response with their group and then together find as many sets of three as they can.” Student Task Statement states, “Which three go together? Why do they go together?” Activity Synthesis states, “Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure the reasons given are correct. During the discussion, prompt students to explain the meaning of any terminology they use, such as ‘continuous,’ ‘discrete,’ ‘segment,’ and to clarify their reasoning as needed. Consider asking: ‘How do you know...?’ ‘What do you mean by...?’ ‘Can you say that in another way?’ During the discussion, avoid introducing the traditional names of x and y for the axes unless students use them first. More formal vocabulary will be developed in later activities, lessons, and grades, and much of the motivation of this added vocabulary is to improve upon the somewhat clunky language we are led to use without it.”

The materials reviewed for Illustrative Mathematics® v.360 Grade 6 through Grade 8 meet expectations for supporting the intentional development of MP7: Look for and make use of structure, for students, in connection with the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP7 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Grade 6 includes:

• Unit 1, Area and Surface Area, Lesson 7, Cool-down, students look for and make use of structure by identifying patterns, recognizing relationships, and applying mathematical properties to simplify problem-solving and deepen their understanding of concepts. Lesson Narrative, “Highlighting the relationship between triangles and parallelograms is a key goal of this lesson. The activities make use of both the idea of decomposition (of a quadrilateral into triangles) and composition (of two triangles into a quadrilateral). The two-way study is designed to help students view and reason about the area of a triangle differently and to look for structure (MP7). Students see that a parallelogram can always be decomposed into two identical triangles, and that any two identical triangles can always be composed into a parallelogram.” Student Task Statement states, “1. Here are some quadrilaterals. a. Circle all quadrilaterals that you think can be decomposed into two identical triangles using only one line. b. What characteristics do the quadrilaterals that you circled have in common? 2. Here is a right triangle. Show or briefly describe how two copies of it can be composed into a parallelogram.”

An example in Grade 7 includes:

• Unit 1, Scale Drawings, Lesson 5, Warm-up, students look for and make use of structure by recognizing patterns in multiplication and division to solve equations mentally. Activity Narrative: “This Math Talk focuses on solving equations of the form px=q. It encourages students to think about the relationship between multiplication and division and to rely on properties of operations to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students examine the effects of reciprocal scale factors. To solve these equations mentally, students need to look for and make use of structure (MP7).” Student Task Statement states, “Solve each equation mentally. 8x=4, 8x=1, \frac{1}{5}x=1,  \frac{2}{5}x=1.” Activity Synthesis, “To involve more students in the conversation, consider asking: ‘Who can restate _______’s reasoning in a different way?’ ‘Did anyone use the same strategy but would explain it differently?’ ‘Did anyone solve the problem in a different way?’ ‘Does anyone want to add on to _________’s strategy?’ ‘Do you agree or disagree? Why?’ ‘What connections to previous problems do you see?’ The key takeaways are: Dividing 1 by a number gives the reciprocal of that number. Multiplying a number by its reciprocal equals 1.”

An example in Grade 8 includes:

• Unit 7, Exponents and Scientific Notation, Lesson 5, Activity 5.3, students look for and make use of structure as they look for patterns in exponents. Activity Narrative, “In this activity students make sense of negative powers of 10 as repeated multiplication by \frac{1}{10} and use this structure in order to distinguish between equivalent exponential expressions (MP7).” Activity Synthesis, “The goal of this discussion is for students to understand that the exponent rules work even with negative exponents by making a clear connection between the exponent rules and the process of multiplying repeated factors that are 10 and \frac{1}{10}. Display the expressions (10^{-2})^{3} and (10^{-2})^{3} from the first problem for all to see. Ask students what is the same and different about these two expressions. (They are both equivalent to 10^{-6}. Both expressions contain one positive exponent and one negative exponent.) Reinforce students' understanding of the exponent rules by writing out an expanded form of each expression when discussing the following questions: ‘What do the 3 and -2 in (10^{-2})^{3} mean in terms of repeated multiplication?’ (The 3 means that there are 3 factors that are each 10^{-2}, and the -2 means that there are 2 factors that are \frac{1}{10}.) So (10^{-2})^{3}=(\frac{1}{10}\cdot \frac{1}{10})(\frac{1}{10}\cdot \frac{1}{10})(\frac{1}{10}\cdot \frac{1}{10})=\frac{1}{10\cdot 10\cdot 10\cdot 10\cdot 10\cdot 10}=10^{-6}. ‘What do the 2 and -3 in (10^{-2})^{3}  mean in terms of repeated multiplication?” (The 2 means that there are 2 factors that are each 10, and the -3 means that there are 3 factors that are each \frac{1}{10\cdot 10}.) So (10^{-2})^{3}=(\frac{1}{10}\cdot \frac{1}{10})(\frac{1}{10}\cdot \frac{1}{10})(\frac{1}{10}\cdot \frac{1}{10})=\frac{1}{10\cdot 10\cdot 10\cdot 10\cdot 10\cdot 10}=10^{-6}."

The materials reviewed for Illustrative Mathematics® v.360 Grade 6 through Grade 8 meet expectations for supporting the intentional development of MP8: Look for and express regularity in repeated reasoning, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP8 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example of Grade 6 includes:

• Unit 1, Area and Surface Area, Lesson 9, Activity 9.2, students use repeated reasoning as they find areas of polygons and surface areas of polyhedra. Student Task Statement states, “For each triangle: Identify a base and a corresponding height, and record their lengths in the table. Find the area of the triangle and record it in the last column of the table.” Activity Narrative states, “Students first find the areas of several triangles given base and height measurements. They notice regularity in repeated reasoning and arrive at an expression for finding the area of any triangle (MP8).” 

An example in Grade 7 includes:

• Unit 2, Introducing Proportional Relationships, Lesson 4, Cool-down, students use repeated reasoning as they solve proportional relationships. Student Task Statement states, “Snow is falling steadily in Syracuse, New York. After 2 hours, 4 inches of snow has fallen. 1. If it continues to snow at the same rate, how many inches of snow would you expect after 6.5 hours? If you get stuck, you can use the table to help. 2. Write an equation that gives the amount of snow that has fallen after x hours at this rate. 3. How many inches of snow will fall in 24 hours if it continues to snow at this rate?” Lesson Narrative states, “As students calculate values in the tables and write equations relating the quantities, they practice looking for and expressing regularity in repeated reasoning (MP8).” Students have the opportunity to use this repeated reasoning in the Cool-down as well.

An example in Grade 8 includes:

• Unit 3, Linear Relationships, Lesson 9, Activity 9.2, students use repeated reasoning as they identify and apply the pattern in the decreasing balance of Noah's fare card. Activity Narrative states, “In this activity, students see negative slopes for the first time as they answer questions about a public transportation fare card. After computing the amount left on the card after 0, 1, and 2 rides, they express regularity in repeated reasoning to represent the amount remaining on the card after x rides (MP8).” Student Task Statement states, “Noah has $40 on his fare card. Every time he rides public transportation, $2.50 is subtracted from the amount available on his card. 1. How much money, in dollars, is available on his card after he takes: a. 0 rides? b. 1 ride? c. 2 rides? d. x rides? 2. How many rides can Noah take before the card runs out of money? Where would you see this number of rides on a graph? 3. Graph the relationship between amount of money on the card and number of rides.”