7th Grade - Gateway 2

The materials reviewed for Open Up Resources Grade 7 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

According to the Course Guide, About These Materials, Design Principles, “Each unit begins with a pre-assessment that helps teachers gauge what students know about both prerequisite and upcoming concepts and skills, so that teachers can gauge where students are and make adjustments accordingly. The initial lesson in a unit is designed to activate prior knowledge and provide an easy entry to point to new concepts, so that students at different levels of both mathematical and English language proficiency can 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 towards procedural fluency. The distributed practice problems give students ongoing practice, which also supports developing procedural proficiency.” 

Materials develop conceptual understanding throughout the grade level. Examples include:

• Unit 4: Proportional Relationships and Percentages, Section A: Proportional Relationships with Fractions, Lesson 5: Say It with Decimals, Activity 1: Repeating Decimals, Problem 1, students develop conceptual understanding of rational numbers as repeating and terminating decimals. “Use long division to express each fraction as a decimal. a. \frac{9}{25} b. \frac{11}{30} c. \frac{4}{11}.” (7.NS.2) 

• Unit 8: Probability and Samples, Section A: Probabilities of Single Step Events, Lesson 3: What are Probabilities?, Warm Up: Which Game Would You Choose? students develop conceptual understanding of probability and possible outcome. “Which game would you choose to play? Explain your reasoning. A. Game 1: You flip a coin and win if it lands showing heads. B. Game 2: You roll a standard number cube and win if it lands showing a number that is divisible by 3.” If it does not come out during student instruction, teachers are prompted to explain, “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 two ways to win with the standard number cube and only one 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)

• Unit 9: Putting It All Together, Section B: Making Connections, Lesson 7: More Expressions and Equations, Activity 2: A Souvenir Stand, Problem 5, students develop conceptual understanding of writing expressions with three unknown quantities. “The souvenir stand sells hats, postcards, and magnets. They have twice as many postcards as hats, and 100 more magnets than postcards. The souvenir stand sells all these items and makes a total profit of $953.25. a. Write an equation that represents this situation. b. How many of each item does the souvenir stand sell? Explain or show your reasoning.” (7.EE.1)

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

• Unit 5: Rational Number Arithmetic, Section B: Adding and Subtracting Rational Numbers, Lesson 4: Money and Debts, Warm Up: Concert Tickets, students develop conceptual understanding about positive and negative integers. “Priya wants to buy three tickets for a concert. She has earned $135 and each ticket costs $50. She borrows the rest of the money she needs from a bank and buys the tickets. a. How can you represent the amount of money that Priya has after buying the tickets? b. How much more money will Priya need to earn to pay back the money she borrowed from the bank? c. How much money will she have after she pays back the money she borrowed from the bank?” (7.NS.1)

• Unit 6: Expressions, Equations, and Inequalities, Section D: Writing Equivalent Expressions, Lesson 19: Expanding and Factoring, Cool Down: Equivalent Expressions, students independently develop conceptual understanding of expanding and factoring using the distributive property to write equivalent expressions. “If you get stuck, use a diagram to organize your work. a. Expand to write an equivalent expression: -$$\frac{1}{2}$$(-2x + 4y). b. Factor to write an equivalent expression: 26a - 10.” (7.EE.1)

• Unit 8: Probability and Sampling, Section B: Probabilities of Multi-step Events, Lesson 9: Multi-Step Experiments, Activity 2: Cubes and Coins, Problem 2, students independently develop conceptual understanding of probabilities for multi-step experiments. “Suppose you roll two number cubes. What is the probability of getting: a. Both cubes showing the same number? b. Exactly one cube showing an even number? c. At least one cube showing an even number? d. Two values that have a sum of 8? e. Two values that have a sum of 13?” (7.SP.8)

The materials reviewed for Open Up Resources Grade 7 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

According to the Course Guide sections “About These Materials” and “Design Principles”, “Each unit begins with a pre-assessment that helps teachers gauge what students know about both prerequisite and upcoming concepts and skills, so that teachers can gauge where students are and make adjustments accordingly. The initial lesson in a unit is designed to activate prior knowledge and provide an easy entry to point to new concepts, so that students at different levels of both mathematical and English language proficiency can 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 towards procedural fluency. The distributed practice problems give students ongoing practice, which also supports developing procedural proficiency.” Materials develop procedural skills and fluency throughout the grade level. Examples include:

• Unit 4: Proportional Relationships and Percentages, Lesson 5: Say It with Decimals, Student Work Time, students develop procedural skill and fluency as they convert rational numbers to decimals using long division. “A calculator gives the following decimal representations for some unit fractions: ½ = 0.5, ⅓ = 0.3333333, ¼ = 0.25, ⅕ = 0.2, ⅙ = 0.1666667…” Students answer, “What do you notice? What do you wonder?” Problem 1, “Use long division to express each fraction as a decimal. a. 9/25, b. 11/30, c. 4/11.” (7.NS.2d)

• Unit 6: Expression, Equations and Inequalities, Section B, Solving Equations of the Form px + q = r and p(x + q) = r, Lesson 12: Solving Problems about Percent Increase or Decrease, Activity 2: Sale on Shoes, Problem 1, students develop procedural skill and fluency as they solve equations involving percent increase or decrease. “A store is having a sale where all shoes are discounted by 20%. Diego has a coupon for $3 off of the regular price for one pair of shoes. The store first applies the coupon and then takes 20% off of the reduced price. If Diego pays $18.40 for a pair of shoes, what was their original price before the sale and without the coupon?” (7.EE.4a)

• Unit 7, Angles, Triangles, and Prisms, Section A: Angle Relationships, Lesson 2, Adjacent Angles, Practice Problems, students develop procedural skill and fluency as they use facts about supplementary, complementary angles to find angle pairs. Problem 1, “Angles A and C are supplementary. Find the measure of angle C.” Problem 2, a. List two pairs of angles in square CDFG that are complementary. Name three angles that sum to 180°.” (7.G.5)

Materials allow students to demonstrate procedural skills and fluency independently throughout the grade level. Examples include:

• Unit 6: Expressions, Equations, and Inequalities, Section B: Solving Equations of the Form px + q = r and p(x + q) = r, Lesson 10: Different Options for Solving One Equation, Cool Down: Solve Two Equations, students independently demonstrate procedural skill and fluency as they solve equations using the distributive property or by dividing each side of an equation by a number. “Solve each equation. Show or explain your method. a. 8.88 = 4.44(x - 7) b. 5(y + \frac{2}{5}) = -13.” (7.EE.4a) 

• Unit 7: Angles, Triangles, and Prisms, Section A: Angle Relationships, Lesson 3: Nonadjacent Angles, Cool Down: Finding Angle Pairs, students independently demonstrate procedural skill and fluency as they use facts about supplementary, complementary, vertical, and adjacent angles to find angle pairs. “a. Name two pairs of complementary angles in the diagram. b. Name two pairs of supplementary angles in the diagram. c. Draw another angle to make a pair of vertical angles. Label your new angle with its measure.” A diagram with angle measures is shown. (7.G.5)

• Unit 8: Probability and Sampling, Lesson 16: Estimating Population Proportions, Activity 1, Reaction Times, Problems 1 - 4, students independently demonstrate procedural skill and fluency as they convert rational numbers to decimals. “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. Problem 1, Work with your partner to select a random sample of 20 reaction times, and record them in the table. Problem 2, What proportion of your sample is less than 0.4 seconds? Problem 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. Problem 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)

The materials reviewed for Open Up Resources Grade 7 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics.

According to the Course Guide, in sections “About These Materials” and “Design Principles,” “Students have opportunities to make connections to real-world contexts throughout the materials. Frequently, carefully-chosen anchor contexts are used to motivate new mathematical concepts, and students have many opportunities to make connections between contexts and the concepts they are learning. Additionally, most units include a real-world application lesson at the end. In some cases, students spend more time developing mathematical concepts before tackling more complex application problems, and the focus is on mathematical contexts.”

Materials include multiple routine and non-routine applications of mathematics throughout the grade level. Examples include:

• Unit 5: Rational Number Arithmetic, Section F: Let’s Put It to Work, Lesson 17: The Stock Market, Activity 3: Your Own Stock Portfolio, students engage in a non-routine application problem as they apply operations with rational numbers to solve new values and changes in the stock market. “Your teacher will give you a list of stocks. a. Select a combination of stocks with a total value close to, but no more than, $100. b. Using the new list, how did the total value of your selected stocks change?” Teachers distribute “Stock Prices” and students use their completed “Changes in Stock Prices After 3 Months” page. (7.EE.3)

• Unit 7: Angles, Triangles, and Prisms, Section C: Solid Geometry, Lesson 16: Applying Volume and Surface Area, Activity 2: Filling the Sandbox, students engage in a routine application problem using knowledge of proportional relationships, volume, and surface area. “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. a. 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.) b. 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? c. 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? d. 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)

• Unit 9: Putting in all Together, Section A: Running a Restaurant, Lesson 3: More Costs of Running a Restaurant, Activity 1: Are We Making Money? Problem 1, students engage in a non-routine application problem modeling income and expenses. “Restaurants have many more expenses than just the cost of the food. a. Make a list of other items you would have to spend money on if you were running a restaurant. b. Identify which expenses on your list depend on the number of meals ordered and which are independent of the number of meals ordered. c. Identify which of the expenses that are independent of the number of meals ordered only have to be paid once and which are ongoing. d. Estimate the monthly cost for each of the ongoing expenses on your list. Next, calculate the total of these monthly expenses. (7.NS.3)

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

• Unit 4: Proportional Relationships and Percentages, Section D: Let’s Putto Work, Lesson 15: Error Intervals, Cool Down: An Anglers’ Dilemma, students independently engage in a non-routine application problem to understand of a range of possible values for measurements based on a percent error tolerance. “A fisherman weighs an ahi tuna (a very large fish) on a scale and gets a reading of 135 pounds. The reading on the scale may have an error of up to 5%. What are two possible values for the actual weight of the fish?” (7.RP.3) 

• Unit 5: Rational Number Arithmetic, Section D: Four Operations with Rational Numbers, Lesson 14: Solving Problems with Rational Numbers, Cool Down: Charges and Checks, students independently engage in a routine application problem applying operations with rational numbers. “Lin’s sister has a checking account. If the account balance ever falls below zero, the bank charges her a fee of $5.95 per day. Today, the balance in Lin’s sister’s account is -$2.67. a. If she does not make any deposits or withdrawals, what will be the balance in her account after 2 days? b. In 14 days, Lin’s sister will be paid $430 and will deposit it into her checking account. If there are no other transactions besides this deposit and the daily fee, will Lin continue to be charged $5.95 each day after this deposit is made? Explain or show your reasoning.” (7.NS.3)

• Unit 6: Expressions, Equations, and Inequalities, Section A, Representing Situations of the Form px + q = r and p(q + x) = r, Lesson 2: Reasoning About Context with Tape Diagrams (Part 1), Practice Problems, Problem 3, students independently engage in a non-routine application problem using tape diagrams to explain and solve equations. “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)

The materials reviewed for Open Up Resources Grade 7 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

All three aspects of rigor are present independently throughout each grade level. Examples include:

• Unit 2: Introducing Proportional Relationships, Section B: Representing Proportional Relationships with Equations, Lesson 6: Using Equations to Solve Problems, Practice Problems, Problem 2, students develop procedural skill and fluency as they use equations to solve problems involving a proportional relationship. “Elena has some bottles of water that each hold 17 fluid ounces. a. Write an equation that relates the number of bottles of water (b) to the total volume of water (w) in fluid ounces. b. How much water is in 51 bottles? c. How many bottles does it take to hold 51 fluid ounces of water?” (7.RP.2) 

• Unit 3: Measuring Circles, Section B: Area of a Circle, Lesson 7: Exploring the Area of a Circle, Warm-up: Estimating Areas, students demonstrate conceptual understanding as they use their knowledge of area of polygons to estimate the area of a circle. “Your teacher will show you some figures. Decide which figure has the largest area. Be prepared to explain your reasoning.” Students are shown a rectangle, circle, and parallelogram. (7.G.4)

• Unit 6: Expressions, Equations, and Inequalities, Section A: Representing Situations of the Form px + q = r and p(x + q) = r, Lesson 6: Distinguishing Between Two Types of Situations, Activity 2: Even More Situations, Diagrams, and Equations, Are you ready for more? students apply their understanding of representing and solving equations for given real-world situations. “A tutor is starting a business. In the first year, they start with 5 clients and charge $10 per week for an hour of tutoring with each client. For each year following, they double the number of clients and the number of hours each week. Each new client will be charged 150% of the charges of the clients from the previous year. a. Organize the weekly earnings for each year in a table. b. Assuming a full-time week is 40 hours per week, how many years will it take to reach full time and how many new clients will be taken in that year? c. After reaching full time, what is the tutor’s annual salary if they take 2 weeks of vacation? d. Is there another business model you’d recommend for the tutor? Explain your reasoning.” (7.EE.4)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout each grade level. Examples include:

• Unit 2: Introducing Proportional Relationships, Section A: Representing Proportional Relationships with Tables, Lesson 3: More About Constant of Proportionality, Activity 2: Pittsburgh to Phoenix, students demonstrate conceptual understanding as they apply their knowledge of the relationship between constant of proportionality and constant speed. “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. a. What is the distance between Saint Louis and Albuquerque? b. How many minutes did it take to fly between Albuquerque and Phoenix? c. What is the proportional relationship represented by this table? d. 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)

• Unit 6: Expressions, Equations, and Inequalities, Section C: Inequalities, Lesson 17: Modeling with Inequalities, Practice Problems, Problem 2, students develop procedural skill and fluency as they apply their understanding of writing and solving inequalities. “a. In the cafeteria, there is one large 10-seat table and many smaller 4-seat tables. There are enough tables to fit 200 students. Write an inequality whose solution is the possible number of 4-seat tables in the cafeteria. b. 5 barrels catch rainwater in the schoolyard. Four barrels are the same size, and the fifth barrel holds 10 liters of water. Combined, the 5 barrels can hold at least 200 liters of water. Write an inequality whose solution is the possible size of each of the 4 barrels. c. How are these two problems similar? How are they different?” (7.EE.4)

• Unit 8: Probability and Sampling, Section B: Probabilities of Multi-Step Events, Lesson 9: Multi-Step Experiments, Activity 3: Pick a Card, Problem 3, students develop conceptual understanding and procedural skill and fluency as they find probability of events. “Imagine there are 5 cards. They are colored red, yellow, green, white and black. You mix up the cards and select one of them without looking. Then, without putting that card back, you mix up the remaining cards and select another one. What is the probability that: a. You get a white card and a red card (in either order)? b. You get a black card (either time)? c. You do not get a black card (either time)? d. You get a blue card? e. You get 2 cards of the same color? f. You get 2 cards of different colors?” (7.SP.8)

The materials reviewed for Open Up Resources Grade 7 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with the Math Practices across the year. Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.” The Mathematical Practices are also identified in Lesson Preparation Narratives and Lesson Activities Narratives for some lessons.

There is intentional development of MP1 to meet its full intent in connection to grade-level content. Students make sense of problems and persevere in solving them as they work with the teacher's support and independently throughout the units. Examples include:

• Unit 1: Scale Drawings, Section A: Scaled Copies, Lesson 1: What are Scaled Copies, Lesson Narrative, “This lesson is designed to be accessible to all students regardless of prior knowledge, and to encourage students to make sense of problems and persevere in solving them (MP1) from the very beginning of the course.” Activity 2: Pairs of Scaled Polygons, students use a variety of strategies to match polygons with their scaled copies. “Your teacher will give you a set of cards that have polygons drawn on a grid. Mix up the cards and place them all face up. a. Take turns with your partner to match a pair of polygons that are scaled copies of one another. For each match you find, explain to your partner how you know it’s a match. For each match your partner finds, listen carefully to their explanation, and if you disagree, explain your thinking. b. When you agree on the matches, check your answers with the answer key. If there are any errors, discuss why and revise your matches. c. Select one pair of polygons to examine further. Draw both polygons on the grid. Explain or show how you know the one polygon is a scaled copy of the other.”

• Unit 3: Measuring Circles, Section B: Area of a Circle, Lesson 9: Applying Area of Circles, Lesson Narrative, “In previous lessons, students estimated the area of circles on a grid and explored the relationship between the circumference and the area of a circle to see that A = \pi r^2. In this lesson, students apply this formula to solve problems involving the area of circles as well as shapes made up of parts of circles (MP1 and MP2) and other shapes such as rectangles. These calculations require composition and decomposition recalling work from grade 6.” Warm Up: Still Irrigating the Field, students actively engage in problem solving as they calculate the the exact area of a circle from an estimate. “The area of this field is about 500,000 m^2. What is the field’s area to the nearest square meter? Assume that the side lengths of the square are exactly 800m. A. 502,400 m^2 B. 502,640 m^2  C. 502,655 m^2 D. 502,656 m^2 E. 502,857 m^2 ” Students are shown a circle inside of a square with an unknown radius. 

• Unit 7: Angles, Triangles, and Prisms, Section C: Solid Geometry, Lesson 13: Decomposing Bases for Area, Lesson Narrative, “In this lesson, students continue working with the volume of right prisms. They encounter prisms where the base is composed of triangles and rectangles, and decompose the base to calculate the area. They also work with shapes such as heart-shaped boxes or house-shaped figures where they have to identify the base in order to see the shape as a prism and calculate its volume (MP1).” Activity 1: A Box of Chocolates, students monitor and evaluate their progress as they find the volume of a heart-shaped box. “A box of chocolates is a prism with a base in the shape of a heart and a height of 2 inches. Here are the measurements of the base (diagram of heart base with measurements provided). To calculate the volume of the box, three different students have each drawn line segments showing how they plan on finding the area of the heart-shaped base (Lin’s, Jada’s, and Diego’s plans are shown). For each student’s plan, describe the shapes the student must find the area of and the operations they must use to calculate the total area. a. Although all three methods could work, one of them requires measurements that are not provided. Which one is it? b. Between you and your partner, decide which of you will use which of the remaining two methods. c. Using the quadrilaterals and triangles drawn in your selected plan, find the area of the base. d. Trade with a partner and check each other’s work. If you disagree, work to reach an agreement. e. Return their work. Calculate the volume of the box of chocolates.”

There is intentional development of MP2 to meet its full intent in connection to grade-level content. Students reason abstractly and quantitatively as they work with the teacher's support and independently throughout the units. Examples include:

• Unit 1: Scale Drawings, Section B: Scale Drawings, Lesson 7: Scale Drawings, Instructional Routine, “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 is 0.9 m.” Activity 1: Sizing Up a Basketball Court, students attend to the meaning of quantities as they explore scale. “Your teacher will give you a scale drawing of a basketball court. The drawing does not have any measurements labeled, but it says 1 centimeter represents 2 meters. Problem 1, measure the distances on the scale drawing that are labeled a-d to the nearest tenth of a centimeter Record your results in the first row of the table. Problem 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? Problem 3, How long would each measurement from the first question be on an actual basketball court? Explain or show your reasoning. Problem 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?”

• Unit 5: Rational Number Arithmetic, Section C: Multiplying and Dividing Rational Numbers, Lesson 9: Multiplying Rational Number, Lesson Narrative, “The purpose of this lesson is to develop the rules for multiplying two negative numbers. Students use the familiar fact that distance = velocity x time to make sense of this rule. They interpret negative time as time before a chosen starting time and then figure out what the position is of an object moving with a negative velocity at a negative time.  An object moving with a negative velocity is moving from right to left along the number line. At a negative time it has not yet reached its starting point of zero, so it is to the right of zero, and therefore its position is positive. So a negative velocity times a negative time gives a positive position. When students connect reasoning about quantities with abstract properties of numbers, they engage in MP2.” Activity 1: Backwards in Time, Problem 1, students explain the numbers and symbols in an equation as they find velocity end points. “A traffic safety engineer was studying travel patterns along a highway. She set up a camera and recorded the speed and direction of cars and trucks that passed by the camera. Positions to the east of the camera are positive, and to the west are negative.  Here are some positions and times for one car: a. In what direction is this car traveling? b.What is the velocity?” Students are given a table of values ranging from -180 to 120 for position and -3 to 2 for time. 

• Unit 6: Expressions, Equations, and Inequalities, Section C: Inequalities, Lesson 16: Interpreting Inequalities, Lesson Narrative, “In this lesson and the next, we move on to applying inequalities to solve problems. The Warm Up is a review of the work in the previous lesson about solving inequalities when no context is given. Then students interpret and solve inequalities that represent real-life situations, making sense of quantities and their relationships in the problem (MP2).”Activity 2: Club Activities Display, students understand the relationships between problem scenarios and mathematical representations. “Your teacher will assign your group one of the situations from the last task. Create a visual display about your situation. In your display: Explain what the variable and each part of the inequality represent. Write a question that can be answered by the solution to the inequality. Show how you solved the inequality. Explain what the solution means in terms of the situation.” Sample task, “They start at 12 feet and then lose 3 feet per minute. If x is the number of minutes they hike, then 3x is the change in elevation. Their elevation must be above -37 feet; perhaps this is the bottom of the cliff.”

The materials reviewed for Open Up Resources Grade 7 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 the Math Practices across the year. Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.” The Mathematical Practices are also identified in Lesson Preparation Narratives and Lesson Activities Narratives for some lessons.

Students construct viable arguments in connection to grade-level content as they work with the teacher's support and independently throughout the units. Examples include:

• Unit 1: Scale Drawings, Section A: Scaled Copies, Lesson 3: Making Scaled Copies, Lesson Narrative, “In the previous lesson, students learned that we can use scale factors to describe the relationship between corresponding lengths in scaled figures. Here they apply this idea to draw scaled copies of simple shapes on and off a grid. They also strengthen their understanding that the relationship between scaled copies is multiplicative, not additive. Students make careful arguments about the scaling process (MP3), and have opportunities to use tools like tracing paper or index cards strategically (MP5).” Activity 2: Which Operations? (Part 1), “Diego and Jada want to scale this polygon so the side that corresponds to 15 units in the original is 5 units in the scaled copy. Diego and Jada each use a different operation to find the new side lengths. Here are their finished drawings. a. What operation do you think Diego used to calculate the lengths for his drawing? b. What operation do you think Jada used to calculate the lengths for her drawing? c. Did each method produce a scaled copy of the polygon? Explain your reasoning.”

• Unit 3: Measuring Circles, Section A: Circumference of a Circle, Lesson 2: Exploring Circles, Instructional Routines, “This Warm Up prompts students to compare two figures and use the characteristics of those figures to help them sketch a possible third figure that has various characteristics of each. It invites students to explain their reasoning and hold mathematical conversations (MP3), and allows you to hear how they use terminology and talk about figures and their properties before beginning the upcoming lessons on circles. There are many good answers to the question and students should be encouraged to be creative.” Warm Up: How Do You Figure? “Here are two figures. Figure C looks more like Figure A than like Figure B. Sketch what Figure C might look like. Explain your reasoning.”

• Unit 7: Angles, Triangles and Prisms, Section B: Drawing Polygons with Given Conditions, Lesson 10: Drawing Triangles (Part 2), Launch, “Arrange students in groups of 2. Tell students that they should attempt to create a triangle with the given specifications. If they can create one, they should attempt to either create at least one more or justify to themselves why there is only one. If they cannot create any, they should show some valid attempts to include as many pieces as they can and be ready to explain why they cannot include the remaining conditions.” Activity 2, Three Angles, “Problem 1: Use the applet to draw triangles. Draw as many different triangles as you can with each of these sets of measurements: a. One angle measures 50° , one measures 60° , and one measures 70° . b. One angle measures 50° , one measures 60° , and one measures 100°. Problem 2: Did either of these sets of measurements determine one unique triangle? How do you know?”

Students critique others' reasoning concerning grade-level content as they work independently with the teacher's support throughout the units. Examples include:

• Unit 2: Introducing Proportional Relationships, Section D: Representing Proportional Relationships with Graphs, Lesson 10: Introducing Graphs of Proportional Relationships, Instructional Routines, “Students work in pairs to match tables to graphs and to practice articulating their reasoning (MP3). This task is intended to foster understanding of correspondences between tables and graphs. Students sort the graphs and justify their sorting schemes. Then, they compare the way they sorted their graphs with a different group. The purpose of this activity is to illustrate the idea that the graph of a proportional relationships is a line through the origin. Students will not have the tools for a formal explanation until grade 8. Demonstrate how the matching activity works and how to have mathematical dialogue about the decisions being made (see the instructions in the task statement). When students finish the activity, they use an answer key to check their answers. If adjustments need to be made, students discuss any errors they made.” Activity 2: Matching Tables and Graphs, “Your teacher will give you papers showing tables and graphs. a. Examine the graphs closely. What is the same and what is different about the graphs? b. Sort the graphs into categories of your choosing. Label each category. Be prepared to explain why you sorted the graphs the way you did. c. Take turns with a partner to match a table with a graph. For each match you find, explain to your partner how you know it is a match. For each match your partner finds, listen carefully to their explanation. If you disagree, work to reach an agreement. Pause here so your teacher can review your work. d. Trade places with another group. How are their categories the same as your group’s categories? How are they different? e. Return to your original place. Discuss any changes you may wish to make to your categories based on what the other group did. f. Which of the relationships are proportional? g. What have you noticed about the graphs of proportional relationships? Do you think this will hold true for all graphs of proportional relationships?” 

• Unit 3: Measuring Circles, Section A: Circumference of a Circle, Lesson 5: Circumference and Wheels, Instructional Routine, “This Warm Up reminds students of the meaning and rough value of π. They apply this reasoning to a wheel and will continue to study wheels throughout this lesson. Students critique the reasoning of others (MP3).” Warm Up: A Rope and a Wheel, “Han says that you can wrap a 5-foot rope around a wheel with a 2-foot diameter because \frac{5}{2} is less than pi. Do you agree with Han? Explain your reasoning.”

• Unit 5: Rational Number Arithmetic, Section B: Adding and Subtracting Rational Numbers, Lesson 5: Representing Subtraction, Instructional Routines, “The purpose of this activity is to apply the representation students have used while adding signed numbers, as well as the relationship between addition and subtraction, to begin subtracting signed numbers. Students are given number line diagrams showing one addend and the sum. They are asked to figure out what the other addend would be. Students examine how these addition equations with missing addends can be written using subtraction by analyzing and critiquing the reasoning of others (MP3).” Activity 1: Subtraction with Number Lines, Problem 1, “Here is an unfinished number line diagram that represents a sum of 8. a. How long should the other arrow be? b. For an equation that goes with this diagram, Mai writes 3 + ? = 8. Tyler writes 8 - 3 = ?. Do you agree with either of them? c. What is the unknown number? How do you know?”

The materials reviewed for Open Up Resources Grade 7 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate 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 the Math Practices across the year. Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.” The Mathematical Practices are also identified in “Lesson Preparation Narratives” and “Lesson Activities Narratives” for some lessons.

There is intentional development of MP4 to meet its full intent in connection to grade-level content. Students make sense of problems and persevere in solving them as they work with the teacher's support and independently throughout the units. Examples include:

• Unit 2: Introducing Proportional Relationships, Section E: Let’s Put It to Work, Lesson 14: Four Representations, Instructional Routines, “In this activity, students choose from different lists of things to define their own proportional and nonproportional relationships. Some of the things on the list will be familiar and others will be unfamiliar. This is a significant change from previous activities where students were always given two quantities and they had to decide if they were proportional or not. This new step gives students the opportunity to think about what quantities are related to some of the items on the lists, which is an important step of modeling with mathematics (MP4).” Activity 1: One Scenario, Four Representations, students use the math they know to define proportional and nonproportional relationships. “1. Select two things from different lists. Make up a situation where there is a proportional relationship between quantities that involve these things. 2. Select two other things from the lists, and make up a situation where there is a relationship between quantities that involve these things, but the relationship is not proportional. 3. Your teacher will give you two copies of the ‘One Scenario, Four Representations’ sheet. For each of your situations, describe the relationships in detail. If you get stuck, consider asking your teacher for a copy of the sample response. 1. Write one or more sentences describing the relationship between the things you chose. 2. Make a table with titles in each column and at least 6 pairs of numbers relating the two things. 3. Graph the situation and label the axes. 4. Write an equation showing the relationship and explain in your own words what each number and letter in your equation means. 5. Explain how you know whether each relationship is proportional or not proportional. Give as many reasons as you can.” Student suggestions lists contain the following categories: creatures, length, time, volume, body parts, area, weight, and substance.

• Unit 4: Proportional Relationships and Percentages, Section C: Applying Percentages, Lesson 13: Measurement Error, Lesson Narrative, “This is the first of three lessons where students encounter the idea of percent error. Unlike situations involving percent increase and percent decrease, where there is an initial amount and a final amount, situations expressed with percent error involve a measured amount and a correct amount. The measurement error is the positive difference between the measured amount and the correct amount, and the percent error is the measurement error expressed as a percentage of the correct amount. In this first lesson students see how measurement error can arise in two different ways: from the level of precision in the measurement device, and from human error. In this lesson students encounter one of the important aspects of mathematical modeling, namely that mathematical representations are usually an approximation of the real situation (MP4).” Activity 1: Measuring a Soccer Field, students identify important information that would cause measurement error to calculate percent error. “A soccer field is 120 yards long. Han measures the length of the field using a 30-foot-long tape measure and gets a measurement of 358 feet, 10 inches. a. What is the amount of the error? b. Express the error as a percentage of the actual length of the field.” 

• Unit 8: Probability and Sampling, Section A: Probabilities of Single Step Events, Lesson 6: Estimating Probabilities Using Simulation, Lesson Narrative, “Students follow a process similar to what they used in previous lessons for calculating relative frequencies (the activities in which students were rolling a 1 or 2 on a number cube or drawing a green block out of a bag). The distinction in this lesson is that the outcomes students are tracking are from an experiment designed to represent the outcome of some other experiment that would be harder to study directly. Students see that a simulation depends on the experiment used in the simulation being a reasonable stand-in for the actual experiment of interest (MP4).” Activity 1: Diego’s Walk, students model a situation using an appropriate strategy to estimate the probability of a real-world event by simulating the experience with a chance experiment. “a. Your teacher will give your group the supplies for one of the three different simulations. Follow these instructions to simulate 15 days of Diego’s walk. The first 3 days have been done for you. Simulate one day: If your group gets a bag of papers, reach into the bag, and select one paper without looking inside. If your group gets a spinner, spin the spinner, and see where it stops. If your group gets two number cubes, roll both cubes, and add the numbers that land face up. A sum of 2–8 means Diego has to wait. Record in the table whether or not Diego had to wait more than 1 minute. Calculate the total number of days and the cumulative fraction of days that Diego has had to wait so far. On the graph, plot the number of days and the fraction that Diego has had to wait. Connect each point by a line. If your group has the bag of papers, put the paper back into the bag, and shake the bag to mix up the papers.Pass the supplies to the next person in the group. b. Based on the data you have collected, do you think the fraction of days Diego has to wait after the 16th day will be closer to 0.9 or 0.7? Explain or show your reasoning. c. Continue the simulation for 10 more days. Record your results in this table and on the graph from earlier. d. What do you notice about the graph? e. Based on the graph, estimate the probability that Diego will have to wait more than 1 minute to cross the crosswalk.”

There is intentional development of MP5 to meet its full intent in connection to grade-level content. Students use appropriate tools strategically as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2: Introducing Proportional Relationships, Section B: Representing Proportional Relationships with Equations, Lesson 5: Two Equations for Each Relationship, Instructional Routines, “The theme continues by asking students to make sense of the two rates associated with a given proportional relationship. Here, students are asked to reason from an equation rather than a table, although they may find it helpful to create a table or graph (MP5). In this particular example, students work with both number of gallons per minute and number of minutes per gallon. Monitor for students who are using different ways to decide if the cooler was filling faster before or after the flow rate was changed.” Activity 2: Filling a Water Cooler, students choose appropriate tools and strategies as they determine which cooler is filling at a faster rate. “It took Priya 5 minutes to fill a cooler with 8 gallons of water from a faucet that was flowing at a steady rate. Let w be the number of gallons of water in the cooler after t minutes. a. Which of the following equations represent the relationship between w and t? Select all that apply. a. w = 1.6t b. w = 0.625t c. t = 1.6w d. t = 0.625w b. What does 1.6 tell you about the situation? c. What does 0.625 tell you about the situation? d. Priya changed the rate at which water flowed through the faucet. Write an equation that represents the relationship of w and t when it takes 3 minutes to fill the cooler with 1 gallon of water. d. Was the cooler filling faster before or after Priya changed the rate of water flow? Explain how you know.”

• Unit 3: Measuring Circles, Section B, Area of a Circle, Lesson 7: Exploring the Area of a Circle, Lesson Narrative, “This lesson is the first of two lessons that develop the formula for the area of a circle. Students start by estimating the area inside different circles, deepening their understanding of the concept of area as the number of unit squares that cover a region, and discovering that area (unlike circumference) is not proportional to diameter. Next, they investigate how the area of a circle compares to the area of a square that has side lengths equal to the circle’s radius. Students may choose tools strategically from their geometry toolkits to help them make these comparisons (MP5).” Activity 1: Estimating Areas of Circles, students choose appropriate tools and strategies as they estimate the area inside several circles. “Your teacher will assign your group two circles of different sizes. a. Set the diameter of your assigned circle and use the applet to help estimate the area of the circle. Note: to create a polygon, select the Polygon tool, and click on each vertex. End by clicking the first vertex again. For example, to draw triangle ABC, click on A-B-C-A. b. Record the diameter in column D and the corresponding area in column A for your circles and others from your classmates. c. In a previous lesson, you graphed the relationship between the diameter and circumference of a circle. How is this graph the same? How is it different?” 

• Unit 5: Rational Number Arithmetic, Section B: Adding and Subtracting Rational Numbers, Lesson 2: Changing Temperature, Instructional Routines, “In this activity, students use what they learned in the previous activity to find temperature differences and connect them to addition equations. Students who use number line diagrams are using tools strategically (MP5). Students may draw number line diagrams in a variety of ways; what matters is that they can explain how their diagrams represent the situation. Students may think of these questions in terms of subtraction; that is completely correct, but the discussion should focus on how to think of these situations in terms of addition. Students will have an opportunity to connect addition and subtraction in a future lesson.”Activity 2: Winter Temperatures, students choose appropriate tools and strategies as they find temperature differences and write appropriate addition equations to represent the situation. “One winter day, the temperature in Houston is 8° Celsius. Find the temperatures in these other cities. Explain or show your reasoning. a. In Orlando, it is 10° warmer than it is in Houston. b. In Salt Lake City, it is 8° colder than it is in Houston. c. In Minneapolis, it is 20° colder than it is in Houston. d. In Fairbanks, it is 10° colder than it is in Minneapolis. e. Use the thermometer applet to verify your answers and explore your own scenarios.”

The materials reviewed for Open Up Resources Grade 7 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with the Math Practices across the year. Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.” The Mathematical Practices are also identified in Lesson Preparation Narratives and Lesson Activities’ Narratives for some lessons.

There is intentional development of MP6 to meet its full intent in connection to grade-level content. Students attend to precision as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1: Scale Drawings, Section B: Scale Drawings, Lesson 11: Scales Without Units, Instructional Routines, “In this activity, students explore the connection between a scale with units and one without units. Students are given two equivalent scales (one with units and the other without) and are asked to make sense of how the two could yield the same scaled measurements of an actual object. They also learn to rewrite a scale with units as a scale without units. Students will need to attend to precision (MP6) as they work simultaneously with scales with units and without units. A scale of 1 inch to 16 feet is very different than a scale of 1 to 16, and students have multiple opportunities to address this subtlety in the activity. As students work, identify groups that are able to reason clearly about why the two scales produce the same scale drawing. Two different types of reasoning to expect are: Using the two scales and the given dimensions of the parking lot to calculate and verify the student calculations. Thinking about the meaning of the scales, that is, in each case, the actual measurements are 180 times the measurements on the scale drawing.” Activity 2: Same Drawing, Different Scales, students attend to precision as they scale a parking lot. “A rectangular parking lot is 120 feet long and 75 feet wide. Lin made a scale drawing of the parking lot at a scale of 1 inch to 15 feet. The drawing she produced is 8 inches by 5 inches. Diego made another scale drawing of the parking lot at a scale of 1 to 180. The drawing he produced is also 8 inches by 5 inches. a. Explain or show how each scale would produce an 8 inch by 5 inch drawing. b. Make another scale drawing of the same parking lot at a scale of 1 inch to 20 feet. Be prepared to explain your reasoning. c. Express the scale of 1 inch to 20 feet as a scale without units. Explain your reasoning.”

• Unit 3: Measuring Circles, Section A: Circumference of a Circle, Lesson 4: Applying Circumference, Lesson Narrative, “In this lesson, students use the equation C = \pid to solve problems in a variety of contexts. They compute the circumference of circles and parts of circles given diameter or radius, and vice versa. Students develop flexibility using the relationships between diameter, radius, and circumference rather than memorizing a variety of formulas. Understanding the equation C = 2\pir will help with the transition to the study of area in future lessons. Students think strategically about how to decompose and recompose complex shapes (MP7) and need to choose an appropriate level of precision for and for their final calculations (MP6).” Activity 2: Around the Running Track, students attend to precision as they compute the length of a figure that is composed of half-circles and straight line segments. “The field inside a running track is made up of a rectangle that is 84.39 m long and 73 m wide, together with a half-circle at each end. a. What is the distance around the inside of the track? Explain or show your reasoning. b. The track is 9.76 m wide all the way around. What is the distance around the outside of the track? Explain or show your reasoning.”

• Unit 4: Proportional Relationships and Percentages, Section C: Applying Percentages, Lesson 12: Finding the Percentage, Activity 2: Info Gap: Sporting Goods, Instructional Routines, “The purpose of this info gap activity is for students to identify the essential information needed to determine the total savings after various discounts are applied to different items. The info 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).” Activity 2: Info Gap: Sporting Goods, students attend to precision as they identify important information needed to find total savings after discounts are applied. “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 be able to answer the question. 2. Ask your partner for the specific information that you need. 3. Explain how you are using the information to solve the problem. Continue to ask questions until you have enough information to solve the problem. 4. Share the problem card 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. 2. Ask your partner “What specific information do you need?” and wait for them to ask for information. If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information. 3. Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions. 4. Read the problem card and solve the problem independently. 5. Share the data card and discuss your reasoning. Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.”

Students attend to the specialized language of mathematics as they work with the support of the teacher and independently throughout the units. Examples include:

• Unit 2: Introducing Proportional Relationships, Section A: Representing Proportional Relationships with Tables, Lesson 2: Introducing Proportional Relationships with Tables, Instructional Routines, “This task is designed to encourage students to use a unit rate. Its context is intended to be familiar so that students can focus on mathematical structure (MP7) and the new terms (MP6) constant of proportionality and proportional relationships. If students are having difficulty understanding the scenario, consider drawing discrete diagrams like this: This can be followed by a double number line diagram. Correspondences among the diagrams can be identified.” Activity 2: Making Bread Dough, students use the specialized language of mathematics as they find equivalent ratios and define proportional relationships for given situations. “A bakery uses 8 tablespoons of honey for every 10 cups of flour to make bread dough. Some days they bake bigger batches and some days they bake smaller batches, but they always use the same ratio of honey to flour. Complete the table as you answer the questions. Be prepared to explain your reasoning. a. How many cups of flour do they use with 20 tablespoons of honey? b. How many cups of flour do they use with 13 tablespoons of honey? c. How many tablespoons of honey do they use with 20 cups of flour? d. What is the proportional relationship represented by this table?”

• Unit 3: Measuring Circles, Section A: Circumference of a Circle, Lesson 2: Exploring Circles, Instructional Routines, “The purpose of this activity is to continue developing the idea that we can measure different attributes of a circle and to practice using the terms diameter, radius, and circumference. Students reason about these attributes when three different-sized circles are described as “measuring 24 inches” and realize that the 24 inches must measure a different attribute of each of the circles. Describing specifically which part of a circle is being measured is an opportunity for students to attend to precision (MP6).” Activity 2: Measuring Circles, students use the specialized language of mathematics as they determine which attributes of a circle are being measured. “Priya, Han, and Mai each measured one of the circular objects from earlier. Priya says that the bike wheel is 24 inches. Han says that the yo-yo trick is 24 inches. Mai says that the glow necklace is 24 inches. a. Do you think that all these circles are the same size? b. What part of the circle did each person measure? Explain your reasoning.”

• Unit 8: Probability and Sampling, Section A: Probabilities of Single Step Events, Lesson 2: Chance Experiments, Lesson Narrative, “In this lesson students investigate chance events. They use language like impossible, unlikely, equally likely as not, likely, or certain to describe a likelihood of a chance event. Students engage in MP1 by making sense of situations and sorting them into these categories. In some cases, a value is assigned to the likelihood of an event using a fraction, decimal, or percentage chance. By comparing loose 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.” Activity 1: How Likely Is It?, students use the specialized language of mathematics as they group scenarios based on their likelihood of occurring. “Label each event with one of these options: impossible, unlikely, equally likely as not, likely, and certain. a. You will win grand prize in a raffle if you purchased 2 out of the 100 tickets. b. You will wait less than 10 minutes before ordering at a fast food restaurant. c. You will get an even number when you roll a standard number cube. d. A four-year-old child is over 6 feet tall. e. No one in your class will be late to class next week. f. A flipped coin lands on heads. g. It will snow at our school on July 1. h. The sun will set today before 11:00 p.m. i. Spinning this spinner will result in green. j. Spinning this spinner will result in yellow.”

The materials reviewed for Open Up Resources Grade 7 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with the Math Practices across the year. Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.” The Mathematical Practices are also identified in Lesson Preparation Narratives and Lesson Activities Narratives for some lessons.

There is intentional development of MP7 to meet its full intent in connection to grade-level content. Students look for and use the structure as they work with the teacher's support and independently throughout the units. Examples include:

• Unit 1: Scale Drawings, Section A: Scaled Copies, Lesson 2, Corresponding Parts and Scale Factors, Lesson Narrative, “This lesson develops the vocabulary for talking about scaling and scaled copies more precisely (MP6), and identifying the structures in common between two figures (MP7). Specifically, students learn to use the term corresponding to refer to a pair of points, segments, or angles in two figures that are scaled copies. Students also begin to describe the numerical relationship between the corresponding lengths in two figures using a scale factor. They see that when two figures are scaled copies of one another, the same scale factor relates their corresponding lengths. They practice identifying scale factors. A look at the angles of scaled copies also begins here. Students use tracing paper to trace and compare angles in an original figure and its copies. They observe that in scaled copies the measures of corresponding angles are equal.” Activity 1: Corresponding Parts, students look for and explain the structure of corresponding angles in a figure and having the same measure as its scaled copy. “One road sign for railroad crossings is a circle with a large X in the middle and two R’s—with one on each side. Here is a picture with some points labeled and two copies of the picture. Drag and turn the moveable angle tool to compare the angles in the copies with the angles in the original. a. Complete this table to show corresponding parts in the three figures. b. Is either copy a scaled copy of the original figure? Explain your reasoning. c. Use the moveable angle tool to compare angle KLM with its corresponding angles in Copy 1 and Copy 2. What do you notice? d. Use the moveable angle tool to compare angle NOP with its corresponding angles in Copy 1 and Copy 2. What do you notice?”

• Unit 4: Proportional Relationships and Percentages, Section B, Percent Increase and Decrease, Lesson 9: More and Less than 1%, Lesson Narrative, “Until now, students have been working with whole number percentages when they solve percent increase and percent decrease problems. As they move towards more complex contexts such as interest rates, taxes, tips, and measurement error, they will encounter percentages that are not necessarily whole numbers. A percentage is a rate per 100, and now that students are working with ratios of fractions and their associated rates, they can work with fractional amounts per 100. In this lesson students consider situations where fractional percentages arise naturally. They also consider how to calculate a fractional percentage using a whole number percentage as a reference and dividing by 10 or 100. For example, if you know that 1% of 200 is 2, you can use the structure of the base-ten system to reason that 0.1% of 200 is 0.2 and 0.01% of 200 is 0.02 (MP7).” Activity 1: Waiting Tables, students analyze problems and look for representations to calculate the percentage of appetizers, entrees and desserts. “During one waiter’s shift, he delivered appetizers, entrées, and desserts. What percentage of the dishes were desserts? appetizers? entrées? What do your percentages add up to? a. What percentage of the dishes were desserts? appetizers? entrées? b. What do your percentages add up to?” Students use an applet to complete the problem and are shown a diagram stating 18 desserts, 13 entrées, and 9 appetizers were served. 

• Unit 8: Probability and Sampling, Section B: Probabilities of Multi-step Events, Lesson 8: Keeping Track of All Possible Outcomes, Lesson Narrative, “In this lesson, students practice listing the sample space for a compound event. They make use of the structure (MP7) of tree diagrams, tables, and organized lists as methods of organizing this information. Students notice that the total number of outcomes in the sample space for an experiment that can be thought of as being performed as a sequence of steps can be found by multiplying the number of possible outcomes for each step in the experiment (MP8).” Activity 1: Lists, Tables, and Trees, Problem 1, students look and explain the structure of probability as they write the sample spaces of multi-step experiments and explore their use in different situations. “Consider the experiment: Flip a coin, and then roll a number cube. Elena, Kiran, and Priya each use a different method for finding the sample space of this experiment. Elena carefully writes a list of all the options: Heads 1, Heads 2, Heads 3, Heads 4, Heads 5, Heads 6, Tails 1, Tails 2, Tails 3, Tails 4, Tails 5, Tails 6. Kiran makes a table. Priya draws a tree with branches in which each pathway represents a different outcome. a. Compare the three methods. What is the same about each method? What is different? Be prepared to explain why each method produces all the different outcomes without repairing any. b. Which method do you prefer for this situation?”

There is intentional development of MP8 to meet its full intent in connection to grade-level content. Students look for and express regularity in repeated reasoning as they work independently with the teacher's support throughout the units. Examples include:

• Unit 1: Scale Drawings, Section A: Scaled Copies, Lesson 5: The Size of the Scale Factor, Lesson Narrative, “In this lesson, students deepen their understanding of scale factors in two ways: 1. They classify scale factors by size (less than 1, exactly 1, and greater than 1) and notice how each class of factors affects the scaled copies (MP8), and 2. They see that the scale factor that takes an original figure to its copy and the one that takes the copy to the original are reciprocals (MP7). This means that the scaling process is reversible, and that if Figure B is a scaled copy of Figure A, then Figure A is also a scaled copy of Figure B. Students also continue to apply scale factors and what they learned about corresponding distances and angles to draw scaled copies without a grid.” Activity 1: Scaled Copies Card Sort, students make generalizations as they examine how the size of the scale factor is related to the original figure and the scaled copy. “Your teacher will give you a set of cards. On each card, Figure A is the original and Figure B is a scaled copy. a. Sort the cards based on their scale factors. Be prepared to explain your reasoning. b. Examine cards 10 and 13 more closely. What do you notice about the shapes and sizes of the figures? What do you notice about the scale factors? c. Examine cards 8 and 12 more closely. What do you notice about the figures? What do you notice about the scale factors?” 

• Unit 4: Proportional Relationships and Percentages, Section B: Percent Increase or Decrease, Lesson 8: Percent Increase or Decrease with Equations, Lesson Narrative, “In this lesson, students represent situations involving percent increase and percent decrease using equations. They write equations like y = 1.06x to represent growth of a bank account, and use the equation to answer questions about different starting amounts. They write equations like t - 0.25t = 12 or 0.75t = 12 to represent the initial price  of a T-shirt that was $12 after a 25% discount. The focus of this unit is writing equations and understanding their connection to the context. In a later unit on solving equations the focus will be more on using equations to solve problems about percent increase and percent decrease. When students repeatedly apply a percent increase to a quantity and see that this operation be expressed generally by an equation, they engage in MP8.” Activity 3: Representing Percent Increase and Decrease: Equations, students describe equations to represent situations of percent increase and decrease. “The gas tank in dad’s car holds 12 gallons. The gas tank in mom’s truck holds 50% more than that. How much gas does the truck’s tank hold? Explain why this situation can be represented by the equation (1.5) \cdot 12 = t. Make sure that you explain what t represents.”

• Unit 8: Probability and Sampling, Section B: Probabilities of Multi-Step Events, Lesson 7: Simulating Multi-Step Experiments, Instructional Routines, “In this activity, students continue to model real-life situations with simulations (MP4), but now the situations have more than one part. Finding the exact probability for these situations is advanced, but simulations are not difficult to run and an estimate of the probability can be found using the long-run results from simulations (MP8). If other simulation tools are not available, you will need the Blackline Master.” Activity 1: Alpine Zoom, students model simulations to find probabilities, “Alpine Zoom is a ski business that makes most of its money during spring break. To make money, it needs to snow at least 4 days out of the 10 days of spring break. Based on the weather forecast, there is a \frac{1}{3} chance it will snow each day for the 10 days of break. Use the applet to simulate the weather for 10 days of break to see if Alpine Zoom will make money. In each trial, spin the spinner 10 times, and then record the number of 1’s that appeared in the row. The applet reports if the Alpine Zoom will make money or not in the last column. Click Next to get the spin button back to start the next simulation. a. Describe a chance experiment that you could use to simulate whether it will snow on the first day of break. b. How could this chance experiment be used to determine whether Alpine Zoom will make money? In each trial, spin the spinner 10 times, and then record the number of 1’s that appeared in the row. The applet reports if the Alpine Zoom will make money or not in the last column. Click Next to get the spin button back to start the next simulation. c.Based on your simulations, estimate the probability that Alpine Zoom will make money.”