7th Grade - Gateway 2

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

STEMscopes materials develop conceptual understanding throughout the grade level. In the Teacher Toolbox, STEMscopes Math Philosophy, Conceptual Understanding and Number Sense, Secondary, STEMscopes Math Elements, this is demonstrated. “In order to reason mathematically, students must understand why different representations and processes work.” Examples include:

• Scope 2: Addition and Subtraction with Rational Numbers, Explore, Explore 5 - Using the Properties to Solve, Procedure and Facilitation points, students develop conceptual understanding of the properties of addition. “Part II: Property Management, 1. Read the following scenario: Ms. Shelley was impressed with the decoding ability of you and your partner. Now she wants to show you some of the situations she encounters in her job that require her to solve problems using the commutative, associative, additive inverse, and distributive properties. Today your job is to help the event planner by solving some of her problems. 2. Students should use their math skills to find and circle appropriate equations, identify properties, and record solutions. 3. Monitor and assess students as they are working by asking the following guiding questions: a. DOK-1 What is a rational number? b. DOK-1 Is an integer a rational number? c. DOK-1 Is a decimal an integer or a rational number? d. DOK-1 Is a fraction an integer or a rational number?” (7.NS.1)

• Scope 8: Expressions, Explore, Explore 2 - Distributive Property, Procedure and Facilitation Points, students develop conceptual understanding of the distributive property by using area models. “1. Read the scenario to the students: Joshia is eagerly wanting to determine how much he is earning for the apps he has created. He created an expression to determine his earnings. His brother, Jakobi, says he doesn’t think he has the correct expression. Help the brothers determine which expression is correct by using an area model to find equivalent expressions. 2. Give the Student Journal to each student. 3. Direct students’ attention to the area models on the Student Journal. Have students look at the area model started for Joshia. Discuss the following questions with the class: a. DOK-1 Joshia has started his area model by decomposing his expression. What step should Joshia take next? b. DOK-1 What is the value of 4 times 2x? c. DOK-1 What step should Joshia take now? d. DOK-1 What is the value of 4 times -5? e. DOK-1 How can Joshia write his equivalent expression? 4. Direct students’ attention to the area model for Jakobi’s expression. Explain to students that they should work with their groups to determine the equivalent expression for Jakobi’s expression using the area model.” (7.EE.1)

• Scope 10: Solve Equations and Inequalities, Explore, Explore 2–Solve and Compare Equations, students build connections between algebra tiles and algebra equations. “Read the scenario to the class: The Art Club has volunteered to host a booth for the school fundraiser. The Art Club will host a craft booth with several crafts for sale. Help the craft booth volunteers solve some problems related to crafts and sales in their booth.Display an Algebra Equations Mat and set of algebra tiles, or display the virtual algebra tiles.Analyze the algebra tiles and Algebra Equations Mat with the students by asking the following questions:What does a green rectangle represent? A green rectangle represents the value of one x. In algebra, what is the purpose of x. An x represents an unknown amount.What does a yellow square represent? A yellow square represents one.How can you represent 4? You can use 4 yellow squares.How could you represent negatives using the tiles? The rectangles always represent x, and the squares always represent one. If the rectangles are green and the squares are yellow, the value is positive. Negative values for x and one are represented with red rectangles and squares.What is the purpose of the algebra tiles? You use the tiles as manipulatives to represent values. You can add or remove them to solve the problem. What is the purpose of the Algebra Equations Mat? The mat helps you to set up problems that involve variables. The scale reminds you that both sides of the equations are equal. As you solve, you must keep both sides equal.Give one Student Journal to each student.Give a set of algebra tiles, an Algebra Equations Mat, and a set of colored pencils to each group. Optionally, direct students to utilize the algebra tiles virtual manipulatives.Students will work cooperatively to write and solve the equations found in the Student Journal using the algebra tiles and Algebra Equations Mat. Students will record their results in the workspace on the Student Journal and answer the reflection questions.Monitor and talk with students as they work. Check for understanding by using guiding questions. DOK-1 What should you do first when analyzing each problem? DOK-2 How can you represent 2x using algebra tiles? DOK-3 What equation did you write for the balloon animals booth? Justify why you think the equation you wrote is correct. DOK-2 Describe the steps you used to solve the balloon animal booth problem.” (7.EE.4)

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

• Scope 3: Multiplication and Division with Rational Numbers, Explain, Show What You Know– Part 2: Integer Multiplication with Number Lines, Student Handout, students use number lines to model the situation. “Nathan dove into a pool. He dove 5 feet underwater in 3 different increments. Write a multiplication sentence to represent this situation. Model the situation on the number line.” Students see a box where they should write the multiplication sentence. Under that box, there is a horizontal number line that goes from -15 to 15.  There are tick marks along the number line that are labeled in increments of 5. (7.NS.2)

• Scope 5: Proportional Relationships, Explain, Show What You Know–Part 1: Proportional vs. Non-Proportional, Student Handout, students verify proportional relations through comparing ratios and graphing. “Mr. Smith is ordering pencils for his classroom and finds the two offers below. Determine if each offer is a proportional or non-proportional relationship by circling the correct response and then answer the questions that follow each scenario.”  Students see a chart with the x column labeled “Number of Pencils” and the y column labeled “Cost.” Beside the chart, there is a coordinate graph labeled the same.  There are values in the chart. Students must use the chart to graph points to determine if the relationship is proportional. Under the chart and graph, the students see this prompt ‘List two pieces of information that helped you determine whether the situation is proportional or non-proportional.’” (7.RP.2)

• Scope 10: Solve Equations and Inequalities, Explain, Show What You Know–Part 2: Solve and Compare Equations, students write and solve an equation to represent the given situation. “Mac bought 4 shirts and used a coupon that took 3 off. His total bill was 17. Write and solve an equation to represent this situation. Record your work and solution in the workspace provided to find the price of each shirt. Algebraically (with a variable): ___; Arithmetically (using only numbers): ___; Model the problem with Algebra tiles. Be sure to include a key. ___; Each shirt is ___;  Describe the similarities and differences between solving algebraically and arithmetically. ___.” (7.EE.4)

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

STEMscopes materials develop procedural skills and fluency throughout the grade level. In the Teacher Toolbox, STEMscopes Math Philosophy, Elementary, Computational Fluency, STEMscopes Math Elements, these are demonstrated.  “In each practice opportunity, students have the flexibility to use different processes and strategies to reach a solution. Students will develop fluency as they become more efficient and accurate in solving problems.” Examples include:

• Scope 3: Multiplication and Division with Rational Numbers, Elaborate, Fluency Builder– Multiply and Divide Rational Numbers, students to develop procedural skill and fluency of multiplication and division with rational numbers, with teacher support.  “Procedure and Facilitation Points Show students how to shuffle the cards. Model how to play the game with a student. Pass out five cards to each player. Place the rest of the deck in a pile on the table. Players take turns asking each other for either the answer to match one of the problem cards or the problem card to match one of the answer cards. If the opponent has the matching card, the opponent must give it to the player. If the opponent does not have the matching card, the other player must pick a card from the deck. The winner is the player with the most matches when all of the cards are gone. Monitor students to make sure they find accurate matches.” (7.NS.2c)

• Scope 7: Percent Application, Engage, Foundation Builder, Procedure and Facilitation Points, students develop procedural skill using percent and rate to solve problems. “1. Distribute sets of Math Match Cards, one set to each pair of students. 2. Explain that they need to match real-life problems with the answer and double number line. 3. Encourage students to confer with another pair of students after they are finished matching. They should discuss their answers and justifications. 4. Ask the students the following questions: a. When you are given a total and the percent, how can you determine the numerical amounts that match up with every 10% on the double number line? Divide the total number by 10. This amount will be matched up with 10%. Add this number each time the number line is increased by 10%. b. When you are given a part and the percent, how can you determine the numerical amounts that match up with every 10%? Divide the part by the percent (i.e., divide 500 by 35 if the part is 500 and the percent is 35). Then, multiply this number by 10. This will give you 10% of the total number. Add this number each time the number line is increased by 10%.” (7.RP.3)

• Scope 12: Angle Relationships, Explore, Explore 3–Multi-Step Angle Problems, Procedure and Facilitation Points, students develop fluency by finding missing angle measures in multi- step problems. “5. Have students work cooperatively to read Gabriella and Jasmine's nature trail proposal that is found in the Student Journal. Instruct students to sketch the plan onto the current park map on the Student Journal. Students will not use protractors to determine the measurement of angles in this activity. After adding all of the plans to the map, students will use the measurements provided and their knowledge of complementary, supplementary, vertical, and adjacent angles to determine whether the nature trail proposal meets the Parks Department’s guidelines. 6. Monitor and talk with students as they work. Check for understanding by using guiding questions. a. DOK-1 Can you locate a pair of supplementary angles? How do you know they are supplementary? b. DOK-1 Can you locate a pair of complementary angles? How do you know they are complementary? c. DOK-1 Can you identify a pair of adjacent angles? How do you know they are adjacent? d. DOK-1 Can you identify a pair of vertical angles? How do you know they are vertical?” (7.G.5)

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

• Scope 3: Multiplication and Division with Rational Numbers, Evaluate, Skills Quiz, students demonstrate procedural skill and fluency of multiplication and division with rational numbers. “Question 1: Solve the expression -8\frac{1}{2}\cdot(-2\frac{3}{5})  ___; Question 2: Solve the expression -\frac{4}{5}\div5 ___.” (7.NS.2c)

• Scope 9: Equations, Evaluate, Skills Quiz, students demonstrate procedural skill and fluency by solving equations and evaluating expressions. “Question 1: Evaluate (4.5-7.5)^2  ___; Question 2: Evaluate \frac{1}{4}(\frac{1}{2})^2+\frac{1}{6} ___; Question 3: Evaluate 5.5 (\frac{1}{3}\cdot2.4) ___; Question 4: Given the equation 30(5.20+x)=276, choose the answer that makes the equation true. 156, 256, 120, 4.” (7.EE.3)

Scope 10: Solve Equations and Inequalities, Elaborate, Fluency Builder–Two-Step Equations and Inequalities, Fix the Mistake! Cards, (Front of Page 1), students demonstrate procedural skill and fluency by solving equations. “What is the solution to this equation? 2.5x+10=-25” The solution under that problem reads “ 2.5x+10=-25, -10, -10, 2.5x=-35, 2.5\div2.5, x=-14” (7.EE.4a)

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

STEMscopes materials include multiple routine and non-routine applications of mathematics throughout the grade level, both with teacher support and independently. Within the Teacher Toolbox, under STEMscopes Math Philosophy, Elementary, Computational Fluency, Research Summaries and Excerpt, it states, “One of the major issues within mathematics classrooms is the disconnect between performing procedural skills and knowing when to use them in everyday situations. Students should develop a deeper understanding of mathematics in order to reason through a situation, collect the necessary information, and use the mechanics of math to develop a reasonable answer. Providing multiple experiences within real-world contexts can help students see when certain skills are useful.” 

Engaging routine applications of mathematics include:

• Scope 2: Multiplication and Division with Rational Numbers, Elaborate, Fluency Builder–Multiply and Divide Rational Numbers, this activity provides an opportunity for students to demonstrate application through routine problems with teacher support doing multiplication and division with rational numbers. For example: “Procedure and Facilitation Points  1.Show students how to shuffle the cards. 2. Model how to play the game with a student. a. Pass out five cards to each player. b. Place the rest of the deck in a pile on the table. c. Players take turns asking each other for either the answer to match one of the problem cards or the problem card to match one of the answer cards. If the opponent has the matching card, the opponent must give it to the player. If the opponent does not have the matching card, the other player must pick a card from the deck. d. The winner is the player with the most matches when all of the cards are gone. 3. Monitor students to make sure they find accurate matches.” (7.NS.2c)

• Scope 3: Rational Number Operations, Engage, Hook, Procedure and Facilitation Points, Part II: Post-Explore, students develop application of rational number operations with teacher support. “1. Show the Phenomena Video again, and restate the problem. 2. Refer to Be a Smart Cookie, and discuss the following questions: a. DOK-1 What is a complex fraction? b. DOK-1 What was the total number of cookie cakes in the order? c. DOK-2 After determining the total number of cookies ordered, what is the next step? d. DOK-2 How is a complex fraction solved? Explain the steps. e. DOK-1 How many boxes are needed for the order?” (7.NS.3)

• Scope 8: Equations, Evaluate, Standards-Based Assessment, Question 1, students demonstrate application of equations to solve a word problem. “A band has 100 members. The band needs to raise $20,000 for travel expenses and is selling concert tickets to raise the money. The tickets are $8.75 each, and a sponsor will donate $1.50 for every ticket sold. The band sold 952 tickets. As a result, the band will need to finance the rest of the travel expenses. If each band member equally shares the remaining expense, how much does each band member need to pay? $102.42, $116.70, $130.98, $133.33” (7.EE.3)

Engaging non-routine applications of mathematics include:

• Scope 5: Ratios, Rates, and Percents, Engage, Hook, Procedure and Facilitation points, Part I: Pre-Explore, students develop application of unit rates with teacher support in a real world problem. “1. Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. 2. Explain the situation while showing the video: Baking involves many different measurements. The Bake Shop wants to produce different portion options of different baked goods that they sell. The ratio of ingredients will need to stay consistent so that the taste stays the same. Emmi finds that the recipe calls for \frac{1}{2} of a cup of flour and \frac{1}{4} of a cup of sugar. 3. Ask students, “What do you notice? What do you wonder? Where can you see math in this situation?” Allow students to share all ideas. Student answers will vary. I notice that there are different cupcakes that will require using different recipes. I wonder how the recipe would change based on the serving size. 4. Explain to students that when you are baking, you have to use the correct measurements that are required for the recipe. Discuss the following questions: a. DOK-1 Where do you see the math in baking? b. DOK-1 Where might you find ratios in baking?” (7.RP.1)

• Scope 9: Solve Equations and Inequalities, Explain, Show What You Know–Part 1: Construct Equations, this activity provides an opportunity for students to independently demonstrate application as they write and solve an equation based on the given situation and also provide a model of this equation. “Stephen is 4 times as old as Gianna. William is 5 years older than Stephen. William is 17 years old. Let a represent Gianna’s age. Use the space provided to write an equation to find Gianna's age. ___; Draw a model to find Gianna's age. ___.” (7.EE.4)

• Scope 14; Area, Surface Area, and Volume, Evaluate, Mathematical Modeling Task–Picking Planters, Procedure and Facilitation Points, students independently demonstrate application of volume and surface area to solve a word problem. “1. Distribute a Student Handout to each student. 2. Encourage students to look back at their Student Journals from the Explore activities if they need to review the skills they have learned. 3. If you notice that students are stuck, use guiding questions to help them think through the problem without telling them what steps to take. If time permits, allow students to share their solutions with the class. 4. Discuss different methods students utilized to tackle the challenge.  Ingrid chose the planter below for her garden. She found the planter at Springville Garden Center.” Students see a picture of a planter in the shape of a rectangular prism.  “Part 1, The volume of the planter is 4\frac{1}{2}in$$^3$$. What are the dimensions? Use the space below to show how the volume was calculated. ___, Part II, She decides she wants to paint the planter white to match her fence. How much paint will she need to cover the base and sides? Justify your answer. ___ Part III, If the planter was shaped like a rectangular pyramid and it had a congruent base and the same height, what would the volume of the planter be? Justify your answer. ___ “ (7.G.6)

The materials reviewed for STEMscopes Math 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 the grade. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, and application include:

• Mathematical Fluency: Operations with Integers, Dividing, Mathematical Fluency–Same Signs-Activity 1, Procedural and Facilitation Points, students demonstrate procedural fluency of division with integers to work through a maze. Students are given a maze that has division problems with integers. Students determine the pathway through the maze by identifying the correct solution to each of the problems. “1. Explain to students that each problem has at least one possible solution. Correct solutions lead to the finish line. Incorrect solutions lead to dead ends. 2. Have students start in the upper left-hand corner of the maze. 3. Have students work out solutions, using scrap paper as needed. 4. Tell students that when they have found and chosen a solution, they should trace that path on their handout. 5. Explain that if a problem does not show an accurate solution, students must go back and rework the previous problem. 6. Have students continue solving problems until they reach the finish line. 7. If time allows, have each student compare their solution pathway with a classmate’s and decide whether they found the most efficient solution pathway. 8. Monitor students as they work to ensure that they are following instructions, and assist with computation as needed. 9. Refer to the answer key, and prompt students in discovering pathways as needed.” (7.NS.2)

• Scope 3: Multiplication and Division with Rational Numbers, Elaborate, Fluency Builder–Multiply and Divide Rational Numbers, students demonstrate application of multiplication and division with rational numbers. “Procedure and Facilitation Points Show students how to shuffle the cards.Model how to play the game with a student. Pass out five cards to each player. Place the rest of the deck in a pile on the table. Players take turns asking each other for either the answer to match one of the problem cards or the problem card to match one of the answer cards. If the opponent has the matching card, the opponent must give it to the player. If the opponent does not have the matching card, the other player must pick a card from the deck. The winner is the player with the most matches when all of the cards are gone. Monitor students to make sure they find accurate matches.” (7.NS.2c)

• Scope 8: Expressions, Explore, Explore 3–Finding Equivalent Expressions Using Properties, students develop conceptual understanding of simplifying expressions and finding equivalent expressions. “Part I, Read the scenario to the students: Joshia is looking at improving some of his apps and purchasing more products to use for his app creation. He has gathered information and is ready to determine expressions he can use to find the total cost for an unknown amount of apps. Help Joshia write expressions and determine if the expressions are equivalent.Give the Student Journal to each student.Have students collaborate with their groups to read each scenario provided in Part I. Then, have students use the properties of operations on the two expressions given to determine whether the expressions are equivalent and provide an explanation. Monitor and assess students as they are working by asking the following guiding questions: DOK-1 How were you able to determine whether two given expressions were equivalent? DOK-2 What differences did you notice in some sets of equivalent expressions?” (7.EE.1)

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. Examples include:

• Scope 4: Rational Number Operations, Elaborate, Fluency Builder–Rational Number Operations, Procedure and Facilitation Points, students demonstrate procedural fluency alongside application of knowledge of multiplication with rational numbers. Students play Go Fish!, using cards that include rational number multiplication numbers and matching solutions. “1. Show students how to shuffle the cards. 2. Model how to play the game with a student. a. Pass out five cards to each player. b. Place the rest of the deck in a pile on the table. c. Players take turns asking each other for either the answer to match one of the problem cards or the problem card to match one of the answer cards. If the opponent has the matching card, the opponent must give it to the player. If the opponent does not have the matching card, the other player must pick a card from the deck. d. The winner is the player with the most matches when all of the cards are gone. 3. Monitor students to make sure they find accurate matches.” (7.NS.2a)

• Scope 10: Solve Equations and Inequalities, Evaluate, Skills Quiz, Question 1 and 3, provides an opportunity for students to demonstrate application of solving equations alongside procedural fluency of problems solving equations and inequalities. “Solve each problem. Show or explain your mathematical thinking. Question 1: Solve for x. 6x+7=22, ___; Question 3: Solve the inequality –9x+12 ≥ -78. x≤-10, x≤10, x≥10, x≥-10.” (7.EE.4)

• Scope 14; Area, Surface Area, and Volume, Evaluate, Mathematical Modeling Task–Picking Planters, Procedure and Facilitation Points, students demonstrate application of the formula for volume alongside conceptual understanding of volume and surface area to solve a word problem. “1. Distribute a Student Handout to each student. 2. Encourage students to look back at their Student Journals from the Explore activities if they need to review the skills they have learned. 3. If you notice that students are stuck, use guiding questions to help them think through the problem without telling them what steps to take. If time permits, allow students to share their solutions with the class. 4. Discuss different methods students utilized to tackle the challenge. Ingrid chose the planter below for her garden. She found the planter at Springville Garden Center.” Students see a picture of a planter in the shape of a rectangular prism.  “Part 1, The volume of the planter is 4\frac{1}{2}in ^3. What are the dimensions? Use the space below to show how the volume was calculated. ___, Part II, She decides she wants to paint the planter white to match her fence. How much paint will she need to cover the base and sides? Justify your answer. ___ Part III, If the planter was shaped like a rectangular pyramid and it had a congruent base and the same height, what would the volume of the planter be? Justify your answer. ___ “ (7.G.6)

The materials reviewed for STEMscopes Math 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 and are identified for teachers within the Standards of Mathematical Practice within the Explore sections of the scopes. MP1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with the support of the teacher and independently throughout the scopes. Examples include:

• Scope 6: Ratios, Rates, and Percents, Explore, Explore 2–Ratios of Length and Area, students will make sense of the problems and persevere in solving them as they make sense of the problem and reflect on their problem-solving strategy. Procedure and Facilitation Points, “Read the following scenario: In this week’s episode of Ready, Set, Bake, the contestants will be making beautiful cakes. The baker with the most impressive cake will win the round. Each baker has worked out their design and has some notes to help them remember the sizing that will work best with their design. Your job is to help the bakers convert their ideas into measurements to determine the exact size for each cake.Give a Student Journal to each student. Give a set of Baking Notes to each group.Review ratios and proportions with students. DOK-1 What is a ratio? DOK-1 How can you write a ratio? DOK-1 What is a proportion? DOK-1 How do you set up a proportion? Explain to students that they will be working cooperatively with their small groups to help the bakers turn their notes into measurements for their cakes. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: DOK-2 Why is it important to represent what each number represents using words when you write a ratio? DOK-1 Does it make a difference if you are solving proportions that include complex fractions? …”

• Scope 7: Percent Application, Explore, Explore 3–Tips and Commissions, Standards of Mathematical Practice, Procedure and Facilitation Points, students develop MP1.  In Standards of Mathematical Practice, “MP.1 Make sense of problems and persevere in solving them: Students will make sense of the problems by planning how to solve each question through the use of percentages and tape diagrams. Students determine whether an amount is being added to or subtracted from the original amount. Students persevere in solving these problems by strategically converting amounts to determine the final answer.” In Procedure and Facilitation Points, students make sense of a real-world problem and work to solve it. “1. Read the following scenario to students: The Yellow Rose Diner is officially open! Will is a waiter at the diner, and in addition to receiving an hourly wage, he also receives the tips left by customers after eating their meals. The tip left by his customers are percentages of their meals. Let’s help Will calculate the amount of tips left by 4 different customers. 2. Give one set of Tip Task Cards to each partnership. 3. Give the Student Journal to each student. 4. Direct students’ attention to Part I of the Student Journal. Students will work with their partners by first filling in the strip diagram using the words and phrases provided in the word bank. Students will then use the Tip Task Cards to apply their understanding of tips by calculating the tip on a meal and the total cost of the meal after the tip is applied. 5. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a. DOK-1 What operation(s) do you use when calculating a tip, and how do you use it/them? We begin by calculating the tip by changing the percent of tip to a decimal and then multiplying by the cost of the meal. b. DOK-2 What formula could be used to determine the total cost of a meal after a tip is applied? c. DOK-2 Why do you think some customers left a bigger percent tip than others?...”

• Scope 17: Compound Events, Explore, Explore 1–Simple and Compound Events, Standards of Mathematical Practice, “MP.1 Make sense of problems and persevere in solving them: Students will make sense of probability and will persevere in determining the probability of events based on predictions, experiments, and sampling.” Procedure and Facilitation Points, the teacher helps students to make sense of the problems. “1. Read the scenario to the class: The seventh-grade class at your school is going on a field trip to an amusement park. The amusement park has rides, food, shows, and games of chance. Your math class has volunteered to help everyone understand the basics of games of chance. This will make it easier to decide which games of chance they would like to spend their money on. There are two types of games of chance available at the amusement park. You will help sort out what the two types are and how they are different from each other. 2. Before beginning the activity, review basic probability with the students using the following questions: c. DOK-1 What is a game of chance? d. DOK-1 Name some games of chance. e. DOK-1 What is a game of skill? f. DOK-1 Name some games of skill. g. DOK-1 How many cards are in a standard deck of cards? 3. Give a Student Journal to each student and a set of Game Cards to each group. 4. Explain the following to the students: An event is an action that has a result, like rolling a die. A simple event is a single event. It has one outcome. Rolling a die is one action, and it results in one outcome (You will roll a 1, 2, 3, 4, 5, or 6.) A compound event is the combination of two or more simple events. A compound event has more than one outcome. Rolling a die and then rolling the die again are two simple events that together make a compound event. 5. Students will work collaboratively to read and analyze each Game Card and fill out the chart in the Student Journal. Students will also answer the reflection questions in the Student Journal. 6. Monitor and talk with students as needed to check for understanding by using guiding questions. a. DOK-1 According to your Student Journal, what is a simple event? b. DOK-2 What are some examples of simple events? c. DOK-2 According to your Student Journal, what is a compound event? d. DOK-2 What are some examples of compound events? e. DOK-2 Why do you think that game is a simple (or compound) event (referring to a specific Game Card)?”

MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with the support of the teacher and independently throughout the scopes. Examples include:

• Scope 4: Rational Number Operations, Explore, Explore 1–Convert Fractions to Decimals, students reason abstractly as they represent fractions as decimals and decimals as fractions. Students reason quantitatively through classifying decimals as either terminating or repeating. Procedure and Facilitation Points, “Read the scenario to the students: Your group wants to help Priscilla’s Pies during the county fair. Priscilla makes the best pies in the county, and she makes many different flavors and types. Based on the sizes of her pies, she cuts and sells different fractions of the pies. Before she will let you help her, she needs to make sure you understand fractions and how to convert them into decimals because she bases her pie prices on the decimal equivalents of fractions. Today you will work with your group to prove that you can convert fractions to decimals. Give a Student Journal to each student. Distribute the Fraction and Sign Cards. Have students separate the cards into two piles (Fraction Cards and Sign Cards) and place them face down. Instruct students to take turns flipping over a card from each pile. Have all students then independently convert each fraction to the equivalent decimal. Instruct students to record the card, the sign, the fraction, their computations, the solution, and whether the decimal terminates or repeats. Finally, have students check their solutions with calculators. Monitor and assess students as they are working by asking the following guiding questions: DOK-1 What is a dividend? DOK-1 What is a divisor? DOK-2 How does using a calculator help you?”

• Scope 8: Expressions, Explore, Explore 1–Combining Like Terms with Rational Coefficients, Standards of Mathematical Practice, “MP.2 Reason abstractly and quantitatively: Students make sense of quantities and their relationships by rewriting expressions in different ways. Through this abstract reasoning, students can find a new way to look at each problem.” Exit Ticket, students show development of MP2 through showing their understanding of like terms when rewriting equations. “Determine an equivalent expression for each given expression. \frac{2}{5}b-\frac{1}{6}-b+\frac{3}{10}b-\frac{3}{4}”

• Scope 13: Circles, Explore, Explore 3–Area of a Circle, Standards for Mathematical Practices, “MP.2 Reason abstractly and quantitatively:  Students will interpret real-world scenarios involving circles to reason about whether to solve for the circumference or area.” In the Exit Ticket, students must reason to determine which formula to use when solving a real-world problem. “The most popular item in Diego’s Pizzeria is the chocolate candy and marshmallow cookie pizza. Its crust is made of a sugar cookie that has a diameter of 6.4 inches. Use the information to solve the problems.1. Dominique wants to know how many square inches of cookie dough are used to make the cookie pizza. Does Dominique want to know the circumference or the area of the cookie pizza? ___, 2. What equation should Dominique use to find out how many square inches of cookie dough are used for the cookie pizza? ___, 3. How many square inches of cookie dough are used to make the cookie pizza? Round your answer to the nearest hundredth. ___”

The materials reviewed for STEMscopes Math 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.

The materials provide opportunities for student engagement with MP3 that are both connected to the mathematical content of the grade level and fully developed across the grade level. Mathematical practices are identified for teachers within the Standards of Mathematical Practice within the Explore sections of the Scopes. Students construct viable arguments and critique the reasoning of others, in connection to grade-level content, as they work with support of the teacher and independently throughout the Scopes. Examples include:

• Mathematical Fluency, Multiplying, Mathematical Fluency–Multiplication–Activity 2, Procedure and Facilitation Points, students build experience with MP3 as they justify their reasoning and compare the efficiency of solving problems with their partner. Students work their way through a maze/game and solve several decimal multiplication problems. “1. Explain to students that each problem has at least one possible solution. Correct solutions lead to the finish line. Incorrect solutions lead to dead ends. 2. Have students start in the upper left-hand corner of the maze. 3. Have students work out solutions, using scrap paper as needed. 4. Tell students that when they have found and chosen a solution, they should trace that path on their handout. 5. Explain that if a problem does not show an accurate solution, students must go back and rework the previous problem. 6. Have students continue solving problems until they reach the finish line. 7. If time allows, have each student compare their solution pathway with a classmate’s and decide whether they found the most efficient solution pathway. 8. Monitor students as they work to ensure that they are following instructions, and assist with computation as needed. 9. Refer to the answer key, and prompt students in discovering pathways as needed.” 

• Scope 3: Rational Number Operations, Evaluate, Mathematical Modeling Task–Popcorn Popper, Student Handout, students show development of MP3 by justifying their answers. “Epic Theater’s popcorn machine makes 198\frac{3}{4} pounds of popcorn per hour. Showtime Theater’s popcorn machine makes 187\frac{3}{8} pounds of popcorn per hour. The table below provides information about each theater.” Students see a table detailing each theater’s hours of operation and cost of popcorn. “Part I 1. If both theaters sell out of popcorn and have to make a new batch each hour, which theater makes more money selling popcorn? Justify your answer. ___ Part II 1. How many more batches of popcorn should the theater making less money make per hour to make the same amount of money as the theater making more money? ___ 2. How many more hours would the theater that makes less money need to stay open to make more money on popcorn? Justify your answer. ___”

• Scope 6: Percent Application, Explain, Show What You Know–Part 2: Percent Change, Student Handout, students show development of MP3 by explaining their answer. “Zola plans to buy a jacket she has been wanting. There are three different stores that have the jacket on sale, so she wants to find the best deal. Help her to find the best price for the jacket.” Students see a table that outlines details of the jacket in three different stores.  “Which store should Zola visit to buy her jacket? Explain your answer. ___”

• Scope 14: Area, Surface Area, and Volume, Elaborate, Fluency Builder–Area of Composite Figures, Procedure and Facilitation Points, students show development of MP3 by performing error analysis on worked problems. “1. Show students how to shuffle the cards and place them face down in a stack. 2. Model how to play the game with a student. a. Shuffle the cards, and place them face down in a stack between the players. b. Player 1 flips over one card. Both players analyze the problem and determine if the provided solution to the problem is correct and the student who answered it is a math expert or if the solution is incorrect and it is necessary to fix the mistake. c. Players take turns flipping over one card at a time. d. Players continue taking turns until all of the cards have been solved. e. Players should fill out the Fix the Mistake! Student Recording Sheet as they play the game. (Players should fill out the row on the Fix the Mistake! Student Recording Sheet that corresponds to each card number.) f. Once all of the cards have been analyzed, students use the Fix the Mistake! Answer Key to check their answers. g. The player with the most correct answers is the winner. 3. Distribute the game materials. Then, instruct students to shuffle the cards and lay them facedown in a stack between the players. 4. Monitor students to make sure they find and record accurate responses to each card using the Fix the Mistake! Student Recording Sheet.”

The materials reviewed for STEMscopes Math 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 and are identified for teachers within the Standards of Mathematical Practice within the Explore sections of the Scopes. MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students model with mathematics as they work with the support of the teacher and independently throughout the Scopes. Examples include:

• Scope 4: Proportional Relationships, Explore, Explore 1–Proportionality vs. Non- Proportionality, Procedure and Facilitation Points, students demonstrate development of MP4 by modeling real-life situations with proportions. “Part I 1. Read the following scenario: Jake applied for a part-time assistant coaching position at multiple organizations where he can work up to 10 hours per week. He received 4 job offers! Using the graphs that Jake created, help him determine how much money per hour he would receive at each job. 2. Give a set of Job Offer Cards to each group. 3. Discuss the following question with the class: a. DOK-1 How can I create a table using the graphs provided on the Job Offer Cards? 4. Give a Student Journal to each student. 5. Discuss the following questions with the class: a. DOK-1 What is a ratio? b. DOK-1 How do you write ratios? c. DOK-1 How can you determine if you have two equivalent ratios? d. Explain to the class: When all of the ratios in a table have the same unit rate, the relationship is proportional. If all of the ratios in the table do not have the same unit rate, then these relationships are non-proportional. 6. Explain to the class: We are going to use tables and graphs with different ratios for the four job offers Jake received. We need to determine if the ratios of money earned to hours worked form equivalent ratios (proportional relationships) or if the ratios do not form equivalent ratios (non-proportional relationships). Use the graphs on the Job Offer Cards to complete each table in your Student Journal. 7. As students collaborate, monitor their work, and use the following guiding questions to assess student understanding: a. DOK-1 What operation do you use to find the ratio of the two quantities? b. DOK-1 What do you notice about the starting amounts in the tables? c. DOK-2 How did you differentiate between proportional and nonproportional relationships in tables?”

• Scope 7: Expressions, Evaluate, Standards-Based Assessment, Student Handout, students show development of MP4 by writing expressions and equations that model real-life situations. “Read each question. Then, follow the directions to answer each question. Mark your answers by circling the correct answer choices. If a question asks you to show or explain your work, you must do so to receive full credit. 1. The cost of painting supplies at a store includes a gallon of paint for $39.98, a paintbrush for $7.99, and a roller for $11.99. What does the expression 39.98p + 19.98 represent? A. p gallons of paint, brushes, and rollers were purchased. B. 1 gallon of paint, 1 brush, and 1 roller were purchased. C. p gallons of paint, 1 brush, and 1 roller were purchased. D. 1 gallon of paint, p brushes, and p rollers were purchased. 2. A basketball team scores p points in their first game. Part A In the second game, the team scores 10% more points than they scored in their first game. Write a one-term expression to represent the number of points scored. Enter your answer in the box. ___ points Part B In the third game, the team scores 5% fewer points than they scored in their first game. Write a one-term expression to represent the number of points scored. Enter your answer in the box. ____ points 3. The area of a rectangle is 24x–30 square units. Which expressions represent the length and the width? Select all that apply. A. 6(4x-5) B. 3(8x-10) C. 6(4x-30) D. 3(8x-30) 4. A pool has a length, l, and a width, w. There is a walkway around the pool that increases each length by 2 feet and increases each width by 3 feet. What is the perimeter of the sidewalk? A. l+w+5 B. l+w+10 C. 2(l+w+5) D. 2(l+w+10)”

• Scope 15: Informal Inferences, Explore, Explore 3–Compare Data, Procedure and Facilitation Points, students show development of MP4 by comparing data sets with models (box plots, dot plots, and scatter plots). Students use these models to help interpret the data and make conclusions and observations. “Part I 1. Read the following scenario to the class: Dr. Miranda from Prime Pediatrics needs our help. Across the world, men are statistically taller than women by a global average of 7. Can the same be said of 7th-grade students? Help Dr. Miranda find the answer to this question. 2. Explain the following to the class: Think about this question, and then come up with a hypothesis. Will there be a visible difference between the height of the girls and the height of the boys when displayed in a dot plot? If so, which gender do you think will be taller? 3. After allowing students time to record their hypothesis statements, students will collaborate with their groups to measure each member’s height. 4. Project the Class Heights card on the board. Have each student record their height on the lists for girls’ heights or list for boys’ heights, as appropriate. 5. Explain to students that they will collaborate with their groups to record the collected data on the Student Journal to determine the five number summary and interquartile range to create a box plot for heights of girls and a box plot for the heights of boys. 6. Explain to students that they should look at each box plot as it compares to the other box plot. Allow groups time to review the data and discuss their observations. 7. Monitor and assess students' learning by asking the following guiding questions: a. DOK -3 Would it be reasonable for a teacher to include their height on the appropriate gender dot plot? b. DOK-2 What do you notice about the median of the girls’ data versus the boys’ data? c. DOK-3 Compare the interquartile range for each set of data and share some observations with your group. d. DOK-3 Revisit your hypothesis statement. What changes or additions would you make to your statement now that you have seen the data displayed and learned the median and interquartile range?”

MP5 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students use appropriate tools strategically as they work with the support of the teacher and independently throughout the Scopes. Examples include:

• Scope 2: Addition and Subtraction with Rational Numbers, Elaborate, Spiraled Review–Football Rival, students show development with MP5 as they review concepts previously learned making sure to use strategies and tools accurately to help solve problems. “Football Rival, Every year, the whole town of West Valley looks forward to the football game against their biggest rival, Jefferson High. As hosts this year, West Valley is hoping to bring home the win. It has been three years since they have won the rival game, but they have not yet lost hope. West Valley has a new quarterback this year. Aaron Jones, a junior, transferred from a big city school. He is the most talented quarterback that West Valley has seen in a decade. He has helped lead West Valley to an undefeated season so far, and he does not intend to break that streak for the rival game. It is Friday night, and with the whole town watching, the West Valley Lions take to the field. Aaron is ready to lead his team to victory. The referee flips the coin, and West Valley will kick off first. The crowd goes wild as the game begins. 1. The Lions are on the 50-yard line. In their first play, Aaron throws for a 5-yard advance. In the second play, he gets tackled for a loss of 5 yards. Where on the field are they now? 3. The quarterback averages 15 yards for every 3 plays. What is the unit rate of yards per play? A. 15 yards per play B. 3 yards per play C. 5 yards per play D. 45 yards per play”

• Scope 6: Ratios, Rates, and Percents, Explain, Show What You Know–Part 4: Solve Problems- Percents, Student Handout, students show development of MP5 as they use tape diagrams to solve real world percent problems. Students must accurately make, and use, the tape diagrams to find the correct solutions. “Jenna is shopping for ingredients for her Berry Surprise Smoothie. She needs to purchase 20 pieces of fruit and find the percent of the fruit that is needed for her recipe. Use a strip diagram and proportions to find the percent. 6 Strawberries Percent: ___ 5 Blueberries Percent: ___ 9 Raspberries Percent:___”

• Scope 13: Circles, Explore, Explore 1–Discovering Circumference, Procedure and Facilitation Points, Part II, students show development of MP5 while determining which tools to use to find the measurements of a circle. “1. Read the following scenario to the students: Diego’s Pizzeria will make and sell 4 different-sized pizzas: small, medium, large, and the mega pizza. Help Diego and his pizza makers measure the diameter and circumference of each pizza to determine what size each pizza should be. Record the measurements, and complete the table in your Student Journal. 2. Give a bag of Pizza Cards and string and a ruler to each group. 3. Have students work in their groups to measure and record the diameter of each pizza using the provided ruler. Students will use the provided string to measure the circumference of each circle by laying the string along the edge of each circle. Students will measure this distance around the circle using the ruler and record the length of the circumference. Students will record this data in their Student Journals. Students will then use the provided formula \frac{circumference}{diameter}to discover the value of pi (\pi). 4. Monitor and talk with students as needed to check for understanding by using guiding questions. a. DOK-1 If pi is the ratio of the circumference to the diameter, what whole number is pi (\pi) closest to? b. DOK-2 Do you think the ratio of the circumference to its diameter would be the same as the other size pizzas, if there was a pizza with a diameter smaller than the small pizza?”

The materials reviewed for STEMscopes Math 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 and are identified for teachers within the Standards of Mathematical Practice within the Explore sections of the Scopes. MP6 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students attend to precision as they work with the support of the teacher and independently throughout the Scopes. Examples include:

• Scope 8: Expressions, Explore, Explore 1–Combining Like Terms with Rational Coefficients, students build experience with MP6 as they use clear definitions and vocabulary to communicate their reasoning. Students use the equal sign consistently and appropriately when simplifying expressions and manipulating problems. Procedure and Facilitation Points Part I, “Read the scenario to the students: Josiah has created a new app! He decides that he would like to sell his app to earn extra money but isn’t sure which company to work with. Josiah looks at the first five companies he sees listed to sell apps through to determine which has the best opportunity. Help Josiah write expressions for each company to determine profit opportunities. Review how to use the algebra tiles with the students by asking the following questions: What does a green rectangle represent? A green rectangle represents the value of one x. What is x? This is a variable. An x represents an unknown amount. What does a green square represent? A green square represents one.How can you represent 2? You can use 2 green squares. How could you represent negatives using the tiles? The rectangles always represent x, and the squares always represent one. If the rectangles and squares are green, the value is positive. Negative values for x and one are represented with red rectangles and squares. What is the purpose of the algebra tiles? You use the tiles to represent values. Give a set of Expressions Cards and a set of algebra tiles to each group. Give the Student Journal to each student. Explain to the class that each Expression Card matches one company’s pay plan. Have students collaborate with their groups to find each company’s expression, model the expression with algebra tiles, and rewrite the expression to help determine profitability with the company.”

• Scope 10: Solve Equations and Inequalities, Explain, Show What You Know–Part 2: Solve and Compare Equations, students build experience with MP6 as they accurately and consistently calculate mathematical problems using correct notation and language. Students accurately write and solve an equation to represent the situation. “Mac bought 4 shirts and used a coupon that took $3 off. His total bill was $17. Write and solve an equation to represent this situation. Record your work and solution in the workspace provided to find the price of each shirt.”

• Scope 14: Area, Surface Area, and Volume, Explain, Show What You Know–Part 3: Surface Area, Student Handout, students show development of MP6 by attending to precision as they break apart larger shapes into known shapes. “Jason owns a shipping company and needs to find the most cost-effective way to make the boxes he ships. The two boxes below will hold the same volume, but require different amounts of material to make. Determine which box will be the cheapest to use by finding the total surface area of each box.” Students see an image of two different rectangular prisms with each side length labeled in inches. “Which box is the most cost effective and why? ___”

The materials reviewed for STEMscopes Math 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 the grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with the Math Practices across the year and are identified for teachers within the Standards of Mathematical Practice within the Explore sections of the Scopes. MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and make use of structure as they work with the support of the teacher and independently throughout the Scopes. Examples include:

• Scope 2: Addition and Subtraction with Rational Numbers, Explore, Explore 1–Addition of Integers with Counters, Procedure and Facilitation Points, students build experience with MP7 as they make sense of the structure of negative numbers. “1. Read the following scenario: Mr. Sanchez organized a Career Day for his class. The first speaker was a chemist doing research for a pharmaceutical company. His job is to try to find atoms that have a zero charge, which means an atom that has the same number of electrons (particles with negative charge) and protons (particles with a positive charge). He has to investigate hundreds of atoms. Today your job is to help the chemist by investigating 8 atoms to see if they have a zero charge. … 3. Explain to students that they will be working in their small groups to determine the charge of eight atoms. Students will choose an Atom Card, find the corresponding data table on the Student Journal, and investigate the charge of the atom. Explain to students that with two-color counters, the yellow side represents a proton with a +1 charge and the red side represents an electron with a −1 charge. Tell students they will prove their data through pairing counters, utilizing a number line, and writing an equation. 4. Instruct students that they may use the Horizontal Number Line or Vertical Number Line and dry-erase marker to start with protons and make jumps to the left for the electrons they are adding. They may also do the opposite and start with the electrons and make jumps to the right for the protons they are adding. They can use the number lines to check problems or to prepare for drawing the jumps on the number lines on the Student Journal. 5.Monitor and assess students as they are working by asking the following guiding questions: a. DOK-1 What does the yellow side of a two-color counter represent? b. DOK-1 What does the red side of a two-color counter represent? c. DOK-2 How can you tell with the counters if an atom has a zero charge? d. DOK-2 How do you use a number line to help you determine the charge of an atom?” 

• Scope 4: Rational Number Operations, Explore, Explore 2–Solving With Complex Fractions, Procedure and Facilitation Points, students build experience with MP7 as they look for patterns or structures in solving multi-step problems involving rational number operations. They will find that rational numbers and complex fractions can be rewritten in different forms to help perform certain operations. “Read the scenario to the students: Your group has been hired by Priscilla for your excellent math skills. As your last bit of training, you are shadowing Priscilla herself as she helps customers. She sells the pies and then packages them in different containers that hold different fractions of pies. While you train today, she wants you to listen to the orders and packaging requests. As she checks out the customers on the cash register, she expects you to pull out the correct packaging (both container and number of containers) and package the pies for the customers. With your help, she can keep the line moving, serve more customers in less time, and improve her customer satisfaction. Give a resealable bag with the Word Problem Cards to each partnership.Give the Student Journal to each student. Discuss the following with the class: The problems that we will be working with are called complex fractions. Complex fractions are fractions that have a fraction in the numerator, denominator, or both the numerator and denominator.Explain to students that they will read each scenario. Then students should work together to write the complex fraction, division expression, the corresponding multiplication expression, and the solution in the table on the Student Journal.Monitor and assess students as they are working by asking the following guiding questions: DOK-1 How can you write Word Problem Card 1 as a complex fraction expression? DOK-2 What is an inverse fraction? DOK-2 How can you be sure two fractions are inverse fractions?If we multiply them together, the product will be 1. DOK-2 How are complex fractions different from regular fractions?”

• Scope 15: Area, Surface Area, and Volume, Explore, Explore 1–Slicing 3-D Shapes, Procedure and Facilitation Points, students build experience with MP7 as they make use of structure as they find that shapes can be composed and decomposed into other shapes. This will allow students to use formulas that they know to solve bigger problems on figures that they do not recognize. An example is demonstrated by breaking down a larger figure into smaller, known prisms in order to find the total volume of the original figure. “Read the following scenario: One day, Maria looked through the storage room at her mom’s shop. She found old containers in the figures of right rectangular prisms and right pyramids. Mrs. Lopez said they were reusable shipping containers she no longer used. Maria decided to use them for an art project. She wanted to put clay inside each container to mold it to the shape of the container. Then, she would remove the clay from inside the container and cut plane sections of the clay, either vertically or horizontally. Your job is to help Maria predict and discover what two-dimensional figures the plane sections of the clay will be. Then Maria will paint and fire the two-dimensional clay figures to make colorful and unique coasters and plates. Give a Student Journal to each student and a bag with Slicing 3-D Figures Scenario Cards, a lump of clay, and a plastic knife or craft stick to each partnership. Explain to students that they will work with their partners to determine the 2-D figures of the plane sections sliced from the various right rectangular prisms and right pyramids. They will accomplish this by first building each 3-D figure from the Slicing 3-D Figures Scenario Cards and then following instructions to slice the figure correctly. Then they will draw the 2-D figure that is revealed. Monitor and assess students as they are working by asking the following guiding questions: DOK-2 Is there more than one way to slice a right pyramid? DOK-1 What two-dimensional figure is revealed with each way the right pyramid is sliced? DOK-2 Is there more than one way to slice a right rectangular prism? DOK-1 What two-dimensional figure is revealed with each way the right rectangular prism is sliced?”

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and express regularity in repeated reasoning as they work with the support of the teacher and independently throughout the Scopes. Examples include:

• Scope 9: Equations, Explore, Explore 1–Fraction, Decimal, and Percent Conversion, Procedure and Facilitation Points, students build experience with MP8 as they find opportunities in repeated reasoning to understand different algorithms and patterns in mathematics. Students find that rational numbers and complex fractions can be rewritten in different forms to help perform certain operations with more ease. “Part I: Read the following scenario: Ivy is planning her birthday party with her friends. She asks them for their preferences on the party date, location, and food, as well as whether they are available. Help Ivy change fractions into decimals and decimals into fractions using the data she gathered from her friends. Give a set of Part I of the Scenario Cards to each group. Give a Student Journal and Exit Ticket to each student.Instruct the students to read each Scenario Card carefully and answer the questions from the cards on the Student Journal. Have the students collaborate on changing fractions to decimals and changing decimals to fractions. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: DOK-1 How do you change a fraction to a decimal? DOK-1 What is the decimal place on the right of the decimal point? DOK-1 For the number 0.65, what place value is the number 5 in? DOK-2 Compare \frac{15}{16} and its equivalent decimal to 1. DOK-2 What does it mean for \frac{15}{16} of her friends to prefer the party to be on a Saturday?”

• Scope 14: Circles, Evaluate, Skills Quiz, Question 5 and 10, students build experience with MP8 as they find opportunities in repeated reasoning to understand the value of pi. “5. Explain how the area and circumference of the same circle are related. Both the area and circumference of a circle depend on the radius of the circle. You find circumference using 2\pi rand you find area using \pi r^2. 10. Explain the relationship between the circumference and diameter of the same circle. The circumference of a circle is proportional to the diameter of the circle. The relationship is known as pi.”

• Scope 16: Informal Inferences, Explore, Explore 1–Valid Generalizations, Procedure and Facilitation Points, students build experience with MP8 as they use repeated reasoning to make predictions and estimates. “Part II, 1. Give a set of Sample Data Cards to each group. 2. Explain that each card shows the results of different surveys taken by the assistant principal. The results shown are from valid sample populations of the 200 7th-grade students at North County Middle School. 3. Groups will read each card and make three generalizations, inferences, conclusions, or predictions about the entire 7th-grade population based on the results of the assistant principal’s collected data. Read and discuss the first card in the set with the whole class: “The assistant principal asked 50 7th-grade students which type of pizza they preferred.” a. DOK-1 Do you believe this sample population is large enough to represent the entire 7th grade? b. DOK-2 What recommendations would you give to the assistant principal before she orders 25 pizzas for the entire 7th-grade class? c. DOK-3 Based on the results shown in the table, what can you assume about all 200 7th-grade students at North County Middle School? d. DOK-3 Can you make any inferences about the whole group of 7th-grade students based on the results from the assistant principal’s data? e. DOK-3 How could you use this data to make decisions for all the 7th graders? 5. Explain that the students will read each of the remaining cards as a group. Students will write three statements about the entire population based on the data provided in the Student Journal. 6. While students are working together, monitor groups and ask the following questions to check for understanding:  a. DOK-1 Do you believe this sample population is large enough to represent the entire 7th grade? b. DOK-2 How could that conclusion be used to make a decision for the entire population?”