6th Grade - Gateway 2

The materials reviewed for STEMscopes Math Grade 6 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, Elementary, Conceptual Understanding and Number Sense, STEMscopes Math Elements, this is demonstrated. “In order to reason mathematically, students must understand why different representations and processes work.” Examples include:

• Scope 6: Positive Rational Number Operations, Explore, Explore 2–Modeling Fraction Division, Procedure and Facilitation Points, students develop conceptual understanding as they model fraction division. “1. Read the following scenario to students: Each of the grade levels at Sally’s school will be in charge of some part of the fundraiser. Sally’s grade level is in charge of partitioning baked goods into individual packages to be sold at the dessert booth. The sixth-grade teachers prepared Fundraiser Dessert Cards with instructions on how each dessert should be partitioned. Determine how many individual packages of each dessert item can be made with the information provided on the Fundraiser Dessert Cards. 2. Assign each group a station at which to begin working. 3. Give a Student Journal to each student. 4. Students will read the information on each Fundraiser Dessert Card at their stations. 5. Have students discuss in their groups what they could do to find how many individual packages can be made with the total amount of dessert. a. Students should determine that they can divide the total amount of dessert into fractional pieces based on the information that is provided to them. 6. Encourage students to use the provided Cuisenaire Rods to build a concrete model to solve. 7. As students work, move from group to group and scaffold as needed. Remind students to draw a pictorial model of the concrete model, write a solution equation, and write a solution statement in the Student Journal. Optionally, have students use their colored pencils to show each different-colored rod that was used. a. Students may struggle to determine which pieces to use for each part. Guide students to using the following rods: i. Station 1– Use the blue rod to equal one whole. ii. Station 2–Use the light green rods to model eighths. iii. Station 3–Use the dark green rod to equal one whole. 8. Monitor and assess students’ understanding as they collaborate by asking the following guiding questions: a. DOK-1 How can you partition the chocolate cake into thirds? b. DOK-1 What do you do with extra Cuisenaire Rods if you do not have one whole? c. DOK-1 Can you use a fraction in the quotient? 9. Have students rotate to the remaining stations, continue to build concrete models to solve, and record their thinking on their Student Journals.” (6.NS.1)

• Scope 7: Equivalent Numerical Expressions, Explore, Explore 2–Prime Factorization, students develop conceptual understanding about factors, prime number, composite number, and prime factors. An example is “Part I, Display the Factor Tree Discussion Card for students. Give a Student Journal to each student. Discuss the following concepts and questions with the class: Look at the diagram shown on the board. DOK-1 What are two factors that will result in 120? DOK-1 What is a prime number? DOK-1 What is a composite number? DOK-1 Are twelve and ten prime numbers or composite numbers? DOK-1 Can twelve and ten be decomposed into more factors? DOK-1 What are two factors of 12? DOK-1 Are either of the factors six or two prime numbers? DOK-1 Should we continue to decompose prime numbers like two? Prime numbers can only be decomposed into one and itself. Therefore, once we reach a prime number, we do not need to decompose that factor again. Explain to students that once we get a prime number, we will circle it so we know that we do not need to continue to find factors for that number. DOK-1 What are two factors of 10? DOK-1 Are either of the factors two or five prime numbers? DOK-1 What are two factors of 6? DOK-1 Are either of these factors prime? DOK-1 What do you notice about the factors that are circled? DOK-1 How can you write these prime numbers as an expression? Explain to class: Mathematicians call this expression the prime factorization. Mathematicians write prime factorization expressions from least to greatest. … Give one Factor Tree Diagram Card and one dry-erase marker to each group. Discuss the following items with the class: Think about what the prime factorization expression would be if we started the factor tree with different factors than twelve and ten. DOK-1 What are two other factors of 120? Discuss the following question with the class: DOK-1 Is the prime factorization expression for this factor tree the same or different than the first one? Explain.” (6.NS.4)

• Scope 10: Ratios, Rates, and Unit Rates, Engage, Hook, Procedure and Facilitation Points, students develop conceptual understanding of ratios and use proper terminology. “Part I: Pre-Explore, 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 behind you. Mr. Smith is everyone’s favorite custodian at Lincoln Middle School. Every few weeks he gets out his ladder and climbs on top of the school roof to collect the balls that have gotten stuck up there during PE, recess, and post-lunch games and sports. Mr. Smith has decided to evaluate the data he collected to determine which balls get stuck most often. Once he knows, he will put the data in the form of a ratio, demonstrating which type of ball makes up the biggest share of total balls stuck on the roof. Then, he will recommend that students play sports and games that use those balls farther from the school building at locations such as the field or track. 3. Ask students the following questions: 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 Mr. Smith is using ratios. I wonder what balls will be found on the roof. What ball will Mr. Smith find to be most common on the roof? How many total balls will be found on the roof? I can use math to compare quantities of balls found on the roof in the form of ratios. 4. Project He’s a Baller! 5. Explain to students that Mr. Smith found 12 balls on the roof. The types of balls include soccer balls, footballs, and tennis balls. Each type of ball was found in a different quantity. a. DOK-1 What is a ratio? b. DOK-2 What ratios could be created from the balls visible on the slide? c. DOK-1 How many balls were found in all? d. DOK-1 How many of each type of ball were found?” (6.RP.1)

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

• Scope 2: Integers, Explain, Show What You Know–Part 1: A Number and Its Opposite, Student Handout, students plot numbers on a number line. “Locate and plot the integers that are 2 units above and 2 units below zero.” Students see a vertical number line with a 0 in the middle and tick marks above and below.  To the right of the number line, there is a box for students to write the positive integer and the negative integer that they have just plotted.  Below those boxes are two more boxes with the title “What is the distance of each integer from zero?” (6.NS.5)

• Scope 7: Equivalent Numerical Expressions, Explain, Show What You Know–Part 2: Prime Factorization, students work to understand the concept of factor tree and prime factorization. An example is “Complete the missing information. Number18, Factor Tree ___; Prime Factorization ___; Number100, Factor Tree ___; Prime Factorization ___.” (6.NS.4)

• Scope 8: Algebraic Expressions, Elaborate, Fluency Builder-Match Equivalent Algebraic Expressions, Procedure and Facilitation Points and Instruction Sheet, students solve algebraic expressions as well as finding equivalent expressions. Procedure and Facilitation Points example is  “1. Show students how to shuffle the cards and place them face down in a 4\times6 array. 2. Model how to play the game with a student. a. Player 1 flips over 2 cards to try to find a match. A match is a problem with its correct answer. Problems will need to be solved in order to determine matching answers. b. If player 1 matches a problem with the correct answer, then player 1 keeps the matched set and takes another turn. c. If player 1 does not find a match, then they place the cards face down again, and it is the next player’s turn. d. Players continue taking turns until all of the matches have been found. e. The player who collects the most cards wins. 3. Distribute materials. Then, instruct students to shuffle the cards and lay them facedown. 4. Monitor students to make sure they find accurate matches.” The Concentration Cards have algebraic expressions with a matching pair for each.  Instruction Sheet, “Concentration Instruction Sheet Play this game with a partner. You Will Need 1 Set of Concentration Cards (per pair) How to Play 1. Shuffle the cards, and place them face down to form a 4 × 6 array. Player 1 flips over two cards to try to find a match. A match is a problem with its correct answer. Problems will need to be solved in order to determine matching answers. 3. If player 1 matches a problem with the correct answer, then player 1 keeps the matched set and takes another turn. 4. If player 1 does not find a match, then they place the cards face down again, and it is the next player's turn. 5. Players continue taking turns until all of the matches have been found. 6. The player who collects the most cards wins.” (6.EE.4)

The materials reviewed for STEMscopes Math Grade 6 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 2: Integers, Elaborate, Fluency Builder–Integers, students develop procedural skill and fluency, with teacher support, understanding that positive and negative numbers are used together to describe quantities. “Procedure and Facilitation PointsShow 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.” (6.NS.5)

• Scope 6: Positive Rational Number Operations, Elaborate, Fluency Builder–Multiply and Divide Decimals, students develop procedural skill and fluency of multiplying and dividing decimals, with teacher support.  “Procedure and Facilitation PointsShow students how to fold the cards and place them in the jar. Model how to play the game with a student. Player 1 pulls out a card from the jar and hands it to player 2. Player 2 will read the question aloud for player 1 to solve. Player 2 can check the answer from player 1 at the bottom of the card. If a player gets a problem correct, they keep the card. If they are incorrect, the other player keeps the card. Note: If the card contains an image such as a graph or a number line, the player asking the question can show the image while covering up the answer with their hand.If a player pulls out a Bam! picture card, all of that player’s cards go back into the jar. Players take turns pulling cards from the jar and answering questions until time is up. Players must try to get as many cards as they can before time is up. The player with the most cards wins. Set a time limit. When time is up, the student with the most cards wins.  Distribute materials, and instruct students to begin when the timer starts. Monitor students to make sure they solve problems correctly.” (6.NS.3)

• Scope 7: Equivalent Numerical Expressions, Explore, Explore 1–Exponents, Procedure and Facilitation Points, Part 1, students develop procedural skill by rewriting expressions using exponents. “1. Read the following scenario: BEST Party Planners are ready to turn in their order forms for cupcakes to their boss. Their boss has asked them to write each order as an exponential expression. Use the information on the Cupcake Order Form to write the prime factorization expression as the order’s expanded form and then rewrite it as an exponential expression to turn into the boss. 2. Give one copy of the Student Journal to each student. 3. Project the Cupcake Order Form for the class. Model simplifying expressions using exponents with students. Discuss the following items with the class: a. BEST Party Planners needs to order two red velvet cupcakes. b. DOK-2 What is the prime factorization of 2? c. DOK-1 How many twos are in this prime factorization expression? d. BEST Party Planners needs to order 4 fudge cupcakes. e. DOK-2 What is the prime factorization of 4? f. DOK-1 How many twos are in this prime factorization expression? h. DOK-2 What is the prime factorization of 8? i. DOK-1 How many twos are in this prime factorization expression? j. DOK-2 What do you notice about the number of twos that are in the prime factorization expression each time? 4. Allow students time to complete the second and third columns for the last three cupcake types. 5. Monitor and talk with students as needed to check for understanding by using the following guiding questions: a. DOK-1 What is the prime factorization of 32? b. DOK-1 How many twos are in this prime factorization? 6. Discuss the following questions with the class: a. DOK-2 What number is used in every prime factorization? b. Each number of cupcakes was decomposed into an expression involving some number of 2s. DOK-1 For the confetti cupcakes, how many twos need to be multiplied together to get 8? c. Explain to class: Mathematicians call the number that we are repeatedly multiplying together the base. d. We write the base 2 in our cupcakes example in the last column. Model for students writing a 2 in the last column for the confetti cupcakes. e. Look at the confetti cupcakes row again. DOK-1 How many twos are in the prime factorization expression? f. Write a smaller 3 at the top right side of the base 2. Model for students writing the exponent. g. Explain to class: Mathematicians call this smaller number on the top right the exponent or power. h. DOK-1 How many twos are multiplied together in the prime factorization expression for 2? i. DOK-2 What will be the base for red velvet cupcakes? j. DOK-2 What will be the exponent for red velvet cupcakes? 7. Allow students time to work with their partners to write the exponential expression for each of the remaining cupcake orders on the Student Journal. 8. Monitor and talk with students as needed to check for understanding by using the following guiding questions: a. DOK-1 How many twos are in the prime factorization expression for strawberry cupcakes? b. DOK-1 What is the exponent for vanilla cupcakes?” (6.EE.1)

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

• Scope 2: Integers, Elaborate, Fluency Builder–Integers, Instruction Sheet, students independently demonstrate procedural skill and fluency as they match problem cards to answer cards involving integers. “4. 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.” Go Fish! Card example, “Write the integer to represent borrowed $20.” Answer, “-$20.” (6.NS.5)

• Scope 6: Positive Rational Number Operations, Explain, Show What You Know - Part 1: Divide Multi-Digit Numbers, Student Handout, Questions 1-3, students demonstrate procedural skill and fluency by dividing multi-digit rational numbers. “Each batch of cotton candy uses 12 cups of sugar. How many batches can be made from 1,488 cups of sugar?” (6.NS.2)

• Scope 10: Ratios, Rates, and Unit Rates, Elaborate, Fluency Builder–Ratios and Rates in Various Representations, students demonstrate fluency by finding unit rates in a card game. One card says, “6 teachers to 15 students on a field trip.” The matching card says, “2.5 students per teacher.” (6.RP.3b)

The materials reviewed for STEMscopes Math Grade 6 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 5: Positive Rational Number Operations, Engage, Hook, Procedure and Facilitation Points, Part II: Post-Explore, students develop application of dividing fractions. “1. Show the Phenomena Video again, and restate the problem. 2. Refer to That Takes the Cake! and discuss the following questions: a. DOK-1 How can you determine the number of pieces of cake that Stephan will have if he cuts each cake into tenths? b. DOK-1 What model can help you solve this problem? c. DOK-1 What is the equation to find how many pieces of cake Stephan will have if he divides the cakes into tenths? d. DOK-1 If Stephan divides 1\frac{1}{2} cakes into pieces \frac{1}{10} the size of a cake, will he have enough pieces to serve 13 people?” (6.NS.1) 

• Scope 8: Equations and Inequalities, Explain, Show What You Know–Part 2: Write and Solve Equations, Student Handout, students show application of writing and solving equations through routine problems. “Roberta earns $60 for working 4 hours. What is her hourly rate?” (6.EE.7)

• Scope14: Volume and Area, Engage, Hook, Procedure and Facilitation Points, students develop application of area and volume formulas with teacher support. “Part II: Post-Explore, 1. Show the Phenomena Video again, and restate the problem. 2. Refer to the Garden Plots slide, and discuss the following questions: a. DOK-1 How can you determine the area of the unique garden? b. DOK-1 What is the area of the unique garden? c. DOK-1 How can you determine the volume of the planter box? d. DOK-1 What is the volume of the planter box?” (6.G.1)

Engaging non-routine applications of mathematics include:

• Scope 2: Compare and Order Rational Numbers, Explain, Show What You Know–Part 1: Compare Rational Numbers, this activity provides an opportunity for students to independently demonstrate application by putting numbers on the number line and writing inequalities to compare numbers. “The following temperatures were recorded last February in Boston, Massachusetts. Day 1: 0\degreeC Day 2: -2\frac{1}{5}\degreeC Day 3: =1.5\degree Day 4: 1\frac{3}{10}\degreeC Day 5: 2.2\degreeC; Question 1: Plot each temperature on the number line. Given a number line; Question 2: Write an inequality to compare the temperatures on days 3 and 4. ___; Question 3: A temperature of 0℃ or below allows water to freeze. On which days did water reach the freezing point? ___; Question 4: Between which two consecutive days was the temperature change the greatest? Justify your reasoning in relation to the number line. ___. ” (6.NS.6c)

• Scope 12: Measurement Conversions, Evaluate, Mathematical Modeling Task–Time for a Vacation, Student Handout, students demonstrate application of ratio reasoning to convert unit measurements with teacher support. “Time for a Vacation! Summer break is here, and your family is taking you on a camping trip with your friends. The trip will be one week long. Your parents have given you a planning guide so the trip can be exactly how you want. It will be a road trip, so you need to make sure to pack all of your necessities. You should also know how many people will be coming on the trip. Part I: Destination Your family and friends live in San Diego, CA, and have decided to go camping at a campsite. Choose your destination below: Austin, TX: Your map shows that you and your family would travel 2,108 kilometers to reach your destination. Phoenix, AZ: Your map shows that you and your family would travel 571.3 kilometers to reach your destination. Los Angeles, CA: Your map shows that you and your family would travel 180 kilometers to reach your destination. Destination: ___, Distance in Meters: ___, Part II: Water Next, you need to pack water. Each person needs to drink 64 fluid ounces of water each day. Your family will be buying gallon jugs of water to supply everyone with enough to drink. Make sure enough water is packed for everybody on the trip. 1. How many people are going on the trip? 2. How many total fluid ounces do you need for one day? 3. How many gallons do you need for one day? 4. What is the total number of gallons needed? Part III: Camping Gear The final items you need to pack are the tents and sleeping bags. Fill in the quantity and the total weight in ounces in the table below: Item, Weight, Quantity, Total Weight in Ounces, Tent, 3lb., Sleeping Bag, 2lb., What process was used to perform the measurement conversion for your destination, water, and camping gear? How did the quantity impact your total amounts? Justify your answer.” (6.RP.3d)

• Scope 13: Dependent and Independent Variables, Evaluation, Mathematical Modeling Task– Settling on the Moon, Student Handout, students independently show application of representing and analyzing quantitative relationships between dependent and independent variables. “The time has finally come to establish a human colony on the Moon, and your team will help plan the first one! Part I: Homes You anticipate fast growth in this moon colony, so you will build many homes as you invite families. Each home will cost 2 million dollars. Represent the cost of building homes with a table, a graph, and an equation. Part II: Transportation Settlers will get around town both on moon bikes and in moon mobiles. You and your team will need to design a moon bike that costs $8,000 and a moon mobile that costs $9,000. Both modes of transportation need to be designed for the terrain of the Moon. The settlers will need enough moon bikes and moon mobiles for each family. Select one of your choice to represent with a table, a graph, and an equation. Part III: Food Settlers will spend 1 thousand dollars a month on food. Each family will need to purchase meal plans that will include the nutrition that they will need to survive on the Moon. Represent the cost of meal plans with a table, graph, and equation.” (6.EE.9)

The materials reviewed for STEMscopes Math Grade 6 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, Activities, Mathematical Fluency–Different Signs-Activity 1, Procedure and Facilitation Points, students demonstrate procedural fluency to solve problems using the four operations with integers. Students use a fraction puzzle. Each problem has a solution match the student needs to find. “1. Have students work together to cut out all of the triangle puzzle pieces and arrange them faceup on the table. 2. Have students work together to solve expressions and find a matched solution. 3. Each expression is matched with a solution on a different puzzle piece. When a match is found, the triangles are arranged so that the matched expression is opposite and upside down from its solution. 4. Some pieces may share the same solution, but a match is only formed if all 3 sides of each piece correspond to the adjacent sides. 5. When both students agree that all of the pieces have been matched, have students tape or glue the pieces onto the blank paper. Students should glue each piece separately to ensure proper orientation and alignment. 6. Monitor students as they work to ensure that they are following instructions, and assist with computation as needed. 7. Refer to the answer key to check student answers.” (6.NS.5)

• Scope 2: Integers, Evaluate, Skills Quiz, students demonstrate conceptual understanding using positive and negative numbers to represent quantities in real-world contexts. “Question 1: A diver can swim up to 130 feet below sea level. Write a negative integer to represent the depth that a diver could possibly dive and explain your reasoning. ____.” (6.NS.5)

• Scope 8: Equations and Inequalities, Explore, Explore 2–Write and Solve Equation, students demonstrate application as students write and solve equations using properties to find the value of the variable. “Procedure and Facilitation Points Read the following scenario: The local amusement park offers parking and pizza deals every Tuesday through Friday. Montrell’s family wants to see which day has the best deals for parking and pizza. Help Montrell’s family determine which day has the better deal by writing and solving equations.If needed, revisit the following Math Chat discussion questions from Explore 1 to review using inverse operations to solve equations. DOK-2 For any multiplication equation, px=q, how can you find the value of x? DOK-2 For any addition equation, x+p=q, how can you find the value of x? … Project Tuesday’s parking price scenario for the class to see. Have a class discussion about using fractions and decimals in equations. Model writing and solving equations with students as you ask the class the following questions: DOK-1 How can I write an equation to represent the price of parking for Tuesday? DOK-1 How can I get the variable h by itself? DOK-1 Does it matter which order I write the division equation on the other side of the equal sign? DOK-1 How do you solve 5\div\frac{1}{4}.  4. DOK-1 What is the price for one hour of parking? DOK-1 How can I write an equation to represent the price for pizza on Tuesday? DOK-1 What step should we take to get the variable p by itself? DOK-1 Does it matter which order I write the subtraction equation on the other side of the equal sign? DOK-1 What is 15.42-2.15? DOK-1 What is the price for the pizza on Tuesday? Give a set of Daily Deals Cards to each group. Students will work collaboratively to write and solve an equation for each of the remaining daily deals. As students are working together, monitor their learning, and ask the following questions to check for understanding: DOK-1 What operation do you use to solve the parking price deals? DOK-1 What operation do you use to solve the pizza price deals? DOK-2 What process would you use when your coefficient is a fraction to get the variable by itself? …” (6.EE.7)

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 2: Integers, Explain, Show What You Know–Part 1: A Number and Its Opposite, Student Handout, students show conceptual understanding alongside application of knowledge of number lines as they plot numbers on a number line. “Locate and plot the integers that are 2 units above and 2 units below zero.” Students see a vertical number line with a 0 in the middle and tick marks above and below. To the right of the number line, there is a box for students to write the positive integer and the negative integer that they have just plotted. Below those boxes are two more boxes with the title “What is the distance of each integer from zero?” (6.NS.5)

• Scope 3: Compare and Order Rational Numbers, Evaluate, Skills Quiz, Question 1,  students demonstrate application of rational number knowledge alongside procedural skills by ordering rational numbers. “Question 1:  Order the following rational numbers from least to greatest: - 4,25, -3.5, -2.5, -2.0, -2.75, 3.5, ___, ___, ___, ___, ___, ___.” (6.NS.6c)

• Scope 6: Positive Rational Number Operations, Explain, Show What You Know–Part 3: Division of Fractions, Student Handout, students demonstrate application of division of fraction skills alongside procedural fluency of dividing fractions. “Complete the missing information. There are 3\frac{1}{2} sandwiches that are cut into fourths and placed on a platter. How many pieces are on the platter? Divide sandwiches into groups of ___. Equation: Strategy: Use a model or multiply by the reciprocal. Solution statement: Complete the missing information. There is 3 feet of string left on a spool. Packages are being tied with string, and each package requires \frac{3}{4} of a foot of string. How many packages can be tied?  Divide feet into groups of ___. Equation: Strategy: Use a model or multiply by the reciprocal. Solution statement:” (6.NS.1)

The materials reviewed for STEMscopes Math Grade 6 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 4: Coordinate Plane Problem Solving, Explore, Explore 1–Distances Between Points, Standards of Mathematical Practice, “MP.1 Make sense of problems and persevere in solving them: Students reason about lengths of line segments on the coordinate plane and know the solution for length must be positive (using absolute value) in order for the solution to make sense in real-world applications.” Exit Ticket, students determine the location and distance on a coordinate plane that makes sense in a real-world problem. “The Violet Team has joined in on the group showcase for the rectangular egg toss. The locations of three members of the Violet Team are represented by ordered pairs. Vihaan is located at (7, −3), Gabriel is located at (7, −8), and Hanna is located at (10, −3). 1. Where should Jahzara stand to complete the rectangle? ___, 2. Vihaan is tossing the egg to Gabriel. What is the distance between them in units?___”

• Scope 6: Positive Rational Number Operations, Explore, Explore 2–Modeling Fraction Division, students seek the meaning of a problem and work to find efficient ways to solve it. Students check the reasonableness of their answers and find alternative strategies as needed. Procedure and Facilitation Points, “Read the following scenario to students: Each of the grade levels at Sally’s school will be in charge of some part of the fundraiser. Sally’s grade level is in charge of partitioning baked goods into individual packages to be sold at the dessert booth. The sixth-grade teachers prepared Fundraiser Dessert Cards with instructions on how each dessert should be partitioned. Determine how many individual packages of each dessert item can be made with the information provided on the Fundraiser Dessert Cards. Assign each group a station at which to begin working.Give a Student Journal to each student. Students will read the information on each Fundraiser Dessert Card at their stations.Have students discuss in their groups what they could do to find how many individual packages can be made with the total amount of dessert. Students should determine that they can divide the total amount of dessert into fractional pieces based on the information that is provided to them. Encourage students to use the provided Cuisenaire Rods to build a concrete model to solve. As students work, move from group to group and scaffold as needed. Remind students to draw a pictorial model of the concrete model, write a solution equation, and write a solution statement on the Student Journal. Optionally, have students use their colored pencils to show each different-colored rod that was used.Students may struggle to determine which pieces to use for each part. Guide students to using the following rods: Station 1–Use the blue rod to equal one whole. Station 2–Use the light green rods to model eighths. Station 3–Use the dark green rod to equal one whole. Monitor and assess students’ understanding as they collaborate by asking the following guiding questions: DOK-1 How can you partition the chocolate cake into thirds? DOK-1 What do you do with extra Cuisenaire Rods if you do not have one whole? DOK-1 Can you use a fraction in the quotient?...”

• Scope 10: Percents, Explore, Explore 1–Represent Percents Using a Hundreds Grid, Standards of Mathematical Practice, “MP.1 Make sense of problems and persevere in solving them: Students interpret models of fractions and decimals to determine the percent shown.” Procedure and Facilitation Points, “Part I: Understanding Percents Using a Hundreds Grid 1. Read the following scenario to the class: David just started a lawn-mowing business. He schedules 100 customers each week in various neighborhoods with some neighborhoods having a higher percentage of lawns to mow than others. David needs your help determining how many lawns he is mowing in each neighborhood. 2. Give the Percent Work Mat, a dry-erase marker, and a set of colored pencils to each group. 3. Explain to students that a percent is a special ratio that means “out of 100.” It is measured by the number of units as compared with 100. A percent is just another way to show a part-to-whole ratio. A percent can be represented as a fraction, decimal, or ratio since all of these represent a part-to-whole. We will be using the hundreds grid on the Percent Work Mat to represent percents. 4. Guide students in representing a fraction and decimal as a percent. Have students represent \frac{1}{4} and 0.03 on the Percent Work Mat. 5. Actively monitor and assess students’ understanding as they collaborate with their groups by asking the following questions: a. DOK-2 What percent is equal to the value of one unit in the grid? Explain. b. DOK-2 How did you represent \frac{1}{4} on the Percent Work Mat? c. DOK-2 What percent of the hundreds grid was shaded? d. DOK-2 How did you represent 0.03 on the Percent Work Mat? e. DOK-2  What percent of the hundreds grid was shaded? 7. Give a set of Part I Mowing Cards to each group. 8. Explain to the class that they will be working with their groups to read each Mowing Card and will use the information on the card to create a model using a hundreds grid. 9. Encourage students to use the Percent Work Mat to assist them with determining the percent that is represented for each scenario. They will then collaborate with their groups to determine how the percent is written as a part-to-whole ratio in fraction and decimal form. Once they shade their models on the Percent Work Mat, they will show their work by shading the models found on the Student Journal. They will then use their models to complete the corresponding table of information based on the Mowing Card. 10. Instruct students to simplify each fraction into an equivalent fraction, if possible. 11. Actively monitor and assess student understanding as they collaborate with their groups by asking the following questions: a. DOK-1 What percent is being represented in this scenario? b. DOK-2 How can we represent the lawn care company’s percent of lawns mowed using the hundreds grid? c. DOK-2 How can we represent the lawn care companies amount of lawns mowed as a part-to-whole ratio in fraction form? DOK-2 Can you simplify the fraction into an equivalent fraction? Explain. d. DOK-2 How can we represent the percent of lawns mowed as a part-to-whole ratio in decimal form? …”

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 2: Integers, Explore, Explore 1–A Number and Its Opposite, students represent a wide variety of real-world contexts to understand the values of positive and negative rational numbers. Procedure and Facilitation Points, “Read the following scenario to the class: You and your group work for the Top-Notch Rock Wall Company. It’s the top rock wall designers and creators in the country. They have hired your group to assemble different parts of multiple rock wall designs based on customer instructions. It is important to get the placement of each rock wall stone in the correct spot so that these designs accurately match the customers’ requests. Give a Student Journal to each student. Give the Horizontal and Vertical Number Lines, Rock Wall Scenario Cards, and a dry-erase marker to each group. Have students quickly analyze the number lines, and ask the following questions: DOK-1 What is the starting point or middle point on each number line? DOK-1 What is the greatest number for each number line? DOK-1 What is the smallest number for each number line? DOK-1 On a vertical number line, do the numbers increase or decrease as you move farther above zero? DOK-1 On a vertical number line, do the numbers increase or decrease as you move farther below zero? DOK-1 On a horizontal number line, do the numbers increase or decrease as you move to the right of the zero?DOK-1 On a horizontal number line, do the numbers increase or decrease as you move to the left of zero? Explain to students that they will be working with their groups to read each Rock Wall Scenario Card to determine the positive and negative numbers where the rock wall stones will be placed. Mathematicians call positive and negative whole numbers integers.Instruct students to use the number lines to represent their integers. Monitor and assess students as they collaborate by asking the following guiding questions: DOK-2 How can you determine the location of ___ (6 feet above 0, 3 feet to the left of 0, etc.) on the number line? DOK-2 How can you use that location to determine the opposite location? DOK-1 What location do you always start at in order to determine the correct numbers on the number line? DOK-1 What do you notice about the two points on the number line? DOK-3 How would you describe a positive integer? DOK-3 How would you describe a negative integer? A negative integer has a negative symbol in front of it. Allow students enough time to complete all the work for their scenario cards. Explain the following concepts to the class: The numbers you were working with today are known as integers. An integer is any whole number that is positive or negative, including zero. A negative integer is any whole number to the left of zero or less than zero and includes a negative sign in front of the number to represent that it is negative. A positive integer is any whole number to the right of zero or greater than zero…” 

• Scope 11: Measurement Conversions, Explore, Explore 1–One-Step Measurement Conversions, Standards of Mathematical Practice, “MP.2 Reason abstractly and quantitatively: Students conceptualize relative sizes of units in order to reason about unit conversions.” Exit Ticket, students create a model to use ratio reasoning to convert measurements. “The horns of a bull can weigh approximately 720 grams. If 1 gram is equal to 1,000 milligrams, how many milligrams will the bull’s horns weigh? Model: ___ Equivalent Measurement: ___”

• Scope 15: Understand Variability, Explore, Explore 1–Variability in Data, Standards of Mathematical Practice, Exit Ticket, Standards of Mathematical Practice, “MP.2 Reason abstractly and quantitatively: Students determine whether a question will have a response with various data or a single data response.” Exit Ticket, students must use reason to determine if a question will have multiple responses. “Survey Tiger was collecting data from a local library. They asked the head librarian the following survey questions. Circle whether the survey questions are statistical questions or non-statistical questions, and explain your response.

The materials reviewed for STEMscopes Math Grade 6 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: Operations with Integers, Dividing, Mathematical Fluency–Different Signs-Activity 1, Procedure and Facilitation Points, students build experience with MP3 as they justify their reasoning for domino placement in a game working on operations with integers. “Model playing the game with a student according to the rules outlined below. a. Players place the dominoes facedown and mix them up. b. Each player chooses three dominoes from the pile and holds them in their hand. c. The extra three dominoes remain facedown. d. The first player places one domino from their hand faceup in the middle of the table. e. The second player chooses a domino with an expression or fraction equal to one side, and places the match together, either in line with the first domino or perpendicular to it. Dominoes cannot be placed side by side. The player must justify their reasoning. If there is an error, and the dominoes are not a match, the player must pick their domino back up and will not get to place another domino until the next turn. f. If a player does not have a matching domino, the player chooses a domino from the extra set and may play that domino if it matches. If it does not match, the player adds the domino to their hand. If there are no more extra dominoes to choose from, the player passes. g. If a match has been made, the player pauses to record the match on the Student Recording Sheet. Players only record their own matches. h. Players continue playing until one person runs out of dominoes, all of the dominoes are used, or they run out of time. i. When the game is over, the student with the most recorded matches wins. 2. Monitor students as they work to ensure that they are following instructions, and assist with computation as needed.”

• Scope 2: Integers, Evaluate, Mathematical Modeling Task–Stock Values, Part II, students show development of MP3 by justifying how numbers they have chosen meet set criteria. “On Monday, the stock values at Reliable Chemical included numbers based on the following criteria in the table below. Question 1 Input eight possible numbers into the appropriate categories on the table.” Students see a table with three columns. The first column is labeled, “Type of Number.” The second column is labeled “Range.” The third column is labeled “Number.” Students must input numbers in the third column that meet the constraints in the first two columns. “Question 2 Plot each of the numbers on the number line.” Students are provided with a blank number line. “Question 3 Justify how you know that each of the eight numbers meets the criteria from the table.”

• 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.” 

• Scope 11: Measurement Conversions, Evaluate, Mathematical Modeling Task–Time for a Vacation! Part III, students show development of MP3 by explaining the methods they used to convert measurements and justifying their answer. “The final items you need to pack are the tents and sleeping bags. Fill in the quantity and the total weight in ounces in the table below:” Students see a four column table that they will use to determine the quantity and weight of items to take with them. “What process was used to perform the measurement conversion for your destination, water, and camping gear? How did the quantity impact your total amounts? Justify your answer.”

The materials reviewed for STEMscopes Math Grade 6 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 5: Positive Rational Number Operations, Explore, Explore 2–Modeling Fraction Division, Procedure and Facilitation Points, students show development of MP4 as they use Cuisenaire Rods and draw models to represent fraction division. “1. Read the following scenario to students: Each of the grade levels at Sally’s school will be in charge of some part of the fundraiser. Sally’s grade level is in charge of partitioning baked goods into individual packages to be sold at the dessert booth. The sixth-grade teachers prepared Fundraiser Dessert Cards with instructions on how each dessert should be partitioned. Determine how many individual packages of each dessert item can be made with the information provided on the Fundraiser Dessert Cards. 2. Assign each group a station at which to begin working. 3. Give a Student Journal to each student. 4. Students will read the information on each Fundraiser Dessert Card at their stations. 5. Have students discuss in their groups what they could do to find how many individual packages can be made with the total amount of dessert. a. Students should determine that they can divide the total amount of dessert into fractional pieces based on the information that is provided to them. 6. Encourage students to use the provided Cuisenaire Rods to build a concrete model to solve. 7. As students work, move from group to group and scaffold as needed. Remind students to draw a pictorial model of the concrete model, write a solution equation, and write a solution statement in the Student Journal. Optionally, have students use their colored pencils to show each different-colored rod that was used. a. Students may struggle to determine which pieces to use for each part. Guide students to using the following rods: i. Station 1–Use the blue rod to equal one whole. ii. Station 2–Use the light green rods to model eighths. iii. Station 3–Use the dark green rod to equal one whole. 8. Monitor and assess students’ understanding as they collaborate by asking the following guiding questions: a. DOK-1 How can you partition the chocolate cake into thirds? b. DOK-1 What do you do with extra Cuisenaire Rods if you do not have one whole? c. DOK-1 Can you use a fraction in the quotient? 9. Have students rotate to the remaining stations, continue to build concrete models to solve, and record their thinking on their Student Journals.”

• Scope 13: Area and Volume, Explore, Explore 4–Finding the Area of Composite Figures, Procedure and Facilitation Points, students show development of MP4 as they model decomposing and composing figures to find the area. “1. Read the following scenario to students: X-traordinary Landscaping Company just got some recent requests for gardens that are composite figures. This means the gardens are each in a shape that is a combination of rectangles, triangles, and/or parallelograms. They need you to use your knowledge of area formulas for each of these polygons to determine the area of the gardens shaped like composite figures. 2. Give a bag of Garden Cards to each group. 3. Have students discuss with their groups their thoughts about the definition of the term composite figure. Explain to students that a composite figure is a figure that consists of two or more geometric shapes. 4. Give a Student Journal to each student. 5. Explain to students that they will be working with their groups to determine the area of each of the eight customers’ gardens. They will decompose and rearrange the gardens into a variety of figures to help them determine the area. 6. Have students use the Garden Cards to determine the area of each customer’s garden. If the Garden cards are laminated, students can draw on the cards with dry-erase markers to show their decomposition and rearrangement of the composite figure. They will write how many of each 2-D figure the composite figure can be decomposed into. They will also use the area formulas to determine the area of each customer’s garden. 7. As students are working together, monitor their learning, and ask the following questions to check for understanding: a. DOK-2 How can you decompose this garden? b. DOK-2 Why do you need to divide by 2 to find the area for a triangle? For a trapezoid?”

• Scope 14: Surface Area, Explore, Explore 1–Nets, Procedure and Facilitation Points, students show development of MP4 by modeling real-life situations with nets. “Part I: Nets 1. Read the following scenario to students: Tent-tastic is looking into making all of its tents out of new polyester fabric. Before the company can find out how much this will cost, Tent-tastic must create the patterns for each of the tents. Tent-tastic is challenging you to help match the pattern for each tent to its correct tent model. 2. Distribute Tent Model Cards, Pattern Cards, scissors, and tape to each group. 3. Have students work with their groups to cut out each Pattern Card and build their tents by folding and using tape. Then students will match each Tent Model Card to its three-dimensional model that was created from each Pattern Card. Note: Instruct each student in the group to cut and tape a different Pattern Card so all tents are created in a timely manner. 4. Discuss with students that these patterns are called nets. Explain that nets are the two-dimensional shapes used to create three-dimensional figures. 5. Give a Student Journal and a set of Student Journal Cutouts to each student. 6. Have students cut out the Student Journal Cutouts. Students should match the Student Journal Cutouts to their models of the tent and pattern (net). Once the cards are matched, have students glue the Student Journal Cutouts in the Student Journal. 7. While students are working, actively monitor their progress. Ask the following guiding questions: a. DOK-2 What are the attributes of this figure? b. DOK-1 What two-dimensional figures do you see on this Pattern Card? c. DOK-2 What is the relationship between the faces on each tent and the two-dimensional figures you see on its matching Pattern Card?”

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: Compare and Order Rational Numbers, Explore, Explore 1–Compare Rational Numbers, Procedure and Facilitation Points, students show development of MP5 by determining when it is more useful to use a vertical versus a horizontal number line. “1. Read the following scenario to students: Jack loved seeing all the animals on his vacation! He decided to watch another animal show and record as much information as he could about it. Help Jack understand the information that he recorded by creating number lines and using comparison symbols and absolute value to understand the information. 2. Distribute the Animal Show Information Cards and one bag of index cards to each group. 3. Have students look at the Jumping Contest card. Discuss the following questions with the class: a. DOK-1 What comparison symbols can we use to compare the height that each animal jumped? b. DOK-1 What would you write to compare Sally’s jumping height to Sindy’s jumping height? c. Explain to the class about inequalities: Mathematicians call this an inequality. We will use inequalities and words to compare Jack’s recordings at the animal show. 4. Explain to students that we have previously found absolute value. In this Explore we will need to find the magnitude. Magnitude is found by taking the absolute value of a number. a. DOK-1 Jack has a debt of $5. What is this value as a rational number? b. DOK-1 Magnitude is found by taking the absolute value of a number. What is the magnitude of -5? 5. Give a Student Journal to each student. 6. Students will work collaboratively with their groups and their giant number lines on the floor to create number lines using the information from each Animal Show Information Card. Students will need to determine the intervals for each number line and write the rational numbers on the index cards. Some cards will need to be reused for different sets; for example, write 0 on one index card and use it for every number line created. Students will then use the giant number lines to determine the location of each point on the number lines on the Student Journal. Then students will write inequalities and answer the questions for each scenario. 7. As students are working, actively monitor each group. For students who are struggling, you can provide the Blank Number Lines to help them. Ask the following guiding questions: a. DOK-1 Where are negative numbers on a horizontal number line? On a vertical number line? b. DOK-1 Where are positive numbers on a horizontal number line? c. DOK-1 How can you determine the interval to use for each scenario? d. DOK-1 What should you do to plot both fractions and decimal numbers on the number line?”

• Scope 4: Coordinate Plane Problem Solving, Explore, Explore 1–Distances Between Points, Procedure and Facilitation Points, students show development of MP5 as they use the Coordinate Plane to solve real-world problems. Students understand the limits of the coordinate grid as distance cannot be measured with a negative number. “Part I: Partner Competition 1. Read the following scenario to students: Pecan Park was holding its annual Spring Egg-Toss Competition. In the partner competition, teams consisted of a pair of participants who continued to move apart until they dropped their egg. The coordinates of each member of four different teams were recorded in a table. The coordinates reflected the positions of the partners during their last successful egg toss. The team with the greatest distance between partners was the winner of the competition. Use this information to plot each team on the coordinate plane provided and calculate the distance between the team members to determine the winner of the competition. 2. Give a Student Journal to each student. 3. Students will work in their groups to plot the location of each partner on the coordinate plane by using their colored pencils to connect each team’s coordinates with its corresponding team color. Then, students will calculate the distance between each set of partners on the map and record the distance in the table. 4. Monitor student collaboration, and use the following guiding questions to assess understanding: a. DOK-2 What do you notice about the coordinates of the Orange Team and the Green Team? b. DOK-1 What do these coordinates look like when graphed? c. DOK-1 Why are none of the distances negative? d. DOK-1 What is a mathematical term that represents a number’s distance from 0?”

• Scope 16: Represent and Interpret Data, Explore, Explore 3–Box Plots, Procedure and Facilitation points, students show development of MP5 by using box plots to analyze a data set. Students explain their reasoning as to which type of data reporting works best based on the situation they encounter. “1. Read the following scenario: You are reporting on the Straw Tower Challenge for your school newspaper. One sixth-grade class that is participating in the Straw Tower Challenge has given you their data. Every pair of students in the class recorded the height of their straw tower on a class data chart. You will need to analyze the data given to you so that you can correctly report the results for the school newspaper. 2. Discuss the following concepts and questions with the class: a. DOK- 1 What are some of the types of graphs we have worked with so far? b. DOK-2 How were these graphs similar and different? c. Display the Class A Box Plot Card for the class. d. Explain to class: This is another type of graph that we can use called a box plot. This graph also has a unique way of organizing data. You will be examining how a box plot is set up and how it specializes in communicating measures of data in ways that the other two graphs do not. 3. Give the Student Journal and a Class A Box Plot Card to each student. 4. Students will glue the box plot in the top box on the Student Journal. Students will collaborate with their teams to explore how box plots are set up and what measures of data we can find in a box plot. Each group will collaborate to make predictions about the data from the box plot and record their thinking on the Student Journal. 5. As students are working, actively monitor the students. Ask guiding questions such as the following: a. DOK-1 What is the lowest data value? b. DOK-1 What is the highest highest data value? c. DOK-1 What do you predict is the middle value of this data? 6. Allow students enough time to complete the predictions about the data from the box plot. 7. Read the following scenario: Class A’s teacher found the data from the Straw Tower Challenge! Use these data points to determine the values of different measures of data for the Straw Tower Challenge for your report. 8. Give the Class A Data Card to each student. 9. Have students glue the Class A Data Card in the box on the Student Journal. 10. Instruct students to collaborate with their teams to determine the measures of data that they previously made predictions about. Have students circle each measure of data on the data card. a. DOK-1 What is the height of the shortest straw tower? b. DOK-1 What is the height of the tallest straw tower? c. DOK- 1 What is the height of the straw tower that is in the middle of the data?”

The materials reviewed for STEMscopes Math Grade 6 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 3: Compare and Order Rational Numbers, Explain, Show What You Know–Part 1: Compare Rational Numbers, students build experience with MP6 as they attend to precision when labeling rational numbers on vertical and horizontal number lines to help determine comparison statements. “The following temperatures were recorded last February in Boston, Massachusetts. Day 1: 0\degreeC; Day 2: -2\frac{1}{5}\degreeC ; Day 3: -1.5\degreeC; Day 4: 1\frac{3}{10}\degreeC; Day 5: 2.2\degreeC. Question 1: Plot each temperature on the number line. Question 2: Write an inequality to compare the temperatures on days 3 and 4. Question 3: A temperature of 0℃ or below allows water to freeze. On which days did water reach the freezing point? Question 4: Between which two consecutive days was the temperature change the greatest? Justify your reasoning in relation to the number line.”

• Scope 10: Percents, Evaluate, Mathematical Modeling Task–Savvy Shopper, Student Handout, students show development of MP6 as they attend to detail in solving real-life problems and use precise vocabulary to explain their thinking. “During the summer, Jessica saved $300 to buy herself a new school wardrobe. Her favorite store at the mall is having a back-to-school sale, and she has chosen to purchase the items below.” Students see an image of various clothing items each labeled with their cost and discount amount. “Part I 1. Will Jessica have enough money to purchase all of the items? Justify your answer.”

• Scope 16: Represent and Interpret Data, Elaborate, Fluency Builder–Represent and Interpret Data, Procedure and Facilitation Points, students show development of MP6 as they use clear and precise language in discussion with others. “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.”

The materials reviewed for STEMscopes Math Grade 6 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 6: Positive Rational Number Operations, Explore, Explore 4–Add and Subtract Multi-Digit Decimal Numbers, Procedure and Facilitation Points, students build experience with MP7 as they look for patterns or structures to model and solve problems while working with operations involving whole numbers, fractions, and decimals. “Part I: Open the Explore with the following class discussion where students will discover when adding decimal numbers, you first need to line up the place values. DOK-1 What is the sum of 2,412+176? Accept all answers. Students should recall lining up place values before they begin adding. Allow students to revise their answers if needed. Come to a class consensus that 2,412+176=2,588. DOK-1 What is the sum of 241.2+176? Accept all answers. Students should recall lining up place values before they begin adding. Allow students to revise their answers if needed. Come to a class consensus that 241.2+176=417.2. DOK-1 What is the sum of 241.2+17.6? Accept all answers. Students should recall lining up place values before they begin adding. Allow students to revise their answers if needed. Come to a class consensus that 241.2+17.6=258.8. DOK-1 What do you notice about the digits of the addends? …Read the following scenario to students: Sally’s homeroom is divided into four teams. Each team will provide a different item for the bake sale portion of the fundraiser. They are responsible for making one batch of their assigned item. Using a list of ingredients and their prices, determine the cost of each bake sale item.Give a Student Journal to each student. Have students collaborate with their groups to determine the amount that each team will need to spend in order to purchase items for the bake sale. Students will use the Bake Sale Card for the ingredients list and the Grocery Ad for pricing. As students are working, actively monitor students and provide support as needed. DOK- 1 What steps should we take to add decimal numbers together? We must first line up the decimal points and each place value. We may need to add placeholder zeros when adding two decimal numbers that have different place values. DOK- 1 What should we do when we line up decimal numbers with different place values? DOK- 1 What strategy could we use to check our answers? DOK- 1 How can you use graph paper to help you when adding decimal numbers?”

• Scope 7: Equivalent Numerical Expressions, Evaluate, Skills Quiz, Question 1-5, students build experience with MP7 as they look for patterns and use structure to solve problems involving factors. “Solve each problem. Show or explain your mathematical thinking. 1. Find the prime factorization of 84. Write 84 as a product of its prime factors. 2. What are the first two common multiples of 4 and 5? 3. Find the greatest common factor of 52 and 68. 4. Name all common factors of 33 and 99. 5. Find the least common multiple of 6 and 8.”

• Scope 10: Ratios, Rates, and Unit Rates, Explore, Explore 1–Ratios, Procedure and Facilitation Points, students build experience with MP7 as they look closely to discern a pattern or structure. “Read the following scenario: Trixie is picking fruit from her fruit farm to sell at the local farmers’ market fruit stand. Each week, she picks different fruits, depending on that week’s demand. She wants to keep a log of each week’s demand by representing the numbers of each type of fruit in the form of a ratio. Help her determine the ratio between each type of fruit that she brings to the market each week. Display the Fruit Stand Card for week 1. Discuss the following questions with the students:How many blueberries are there for week 1? There are 6 blueberries. How many strawberries are there for week 1? There are 8 strawberries.We want to show the relationship of strawberries to blueberries in week 1. (Write out, “There are ___ strawberries for every ___ blueberries.”) To model the relationship between strawberries and blueberries, what would we need to fill in the blanks with? In the first blank we would need to put 8 because there are 8 strawberries. In the second blank we will need to write 6 because there are 6 blueberries. Explain to class: Mathematicians call this relationship a ratio. Mathematicians write ratios in the order that the ratio asks. In this scenario, we want to represent strawberries to blueberries, therefore we write the number for strawberries in the first spot and the number for blueberries in the second spot. We can also write the ratio for the information shown on week 1s Fruit Stand Card in the following ways: “For every 8 strawberries, there are 6 blueberries”; “8:6”; “8 to 6”; or “There are 8 strawberries for every 6 blueberries.” How can we draw a tape diagram to represent the relationship between strawberries and blueberries? Draw one tape diagram with 8 spaces to represent the strawberries. Below draw a tape diagram that has only 6 spaces to represent the 6 blueberries. Model for students how to draw the tape diagram model for 8 strawberries to 6 blueberries. As you work through each week’s scenario, we will learn other ways to represent ratio relationships as well. Give one Student Journal to each student and one set of Fruit Stand Cards to each group. Students will work collaboratively to represent the information on each Fruit Stand Card. They will record the number of each type of fruit, draw a tape diagram to show the number of each type of fruit, and then write the ratio describing their models in two different ways. Provide linking cubes to students who are struggling. Have students model the ratio with linking cubes by linking the correct amount together for each fruit in the scenario. Then students can draw a model of their linking cubes as a tape diagram on the Student Journal. As students are working together, monitor their learning, and ask the following questions to check for understanding: DOK-1 What does each number in your ratio represent? DOK-1 How would the ratio of fruit for week 3 change if I added one more strawberry to the picture? DOK-1 Does the larger number always have to go second in a ratio? DOK-1 How do you know which number to put first in the ___:___ ratio? DOK-2 Is your ratio comparing part to part or part to whole? Explain how you know. DOK-2 How would this ratio change if you were comparing part to whole instead of part to part?”

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 12: Measurement Conversions, Evaluate, Skills Quiz, Question 2, 4, and 5, students build experience with MP8 as they use repeated reasoning when converting units of measurement. “Directions: Solve each problem. Show or explain your mathematical thinking. 2. Francisco drove 537.6 kilometers on his road trip across Texas. One mile is approximately 1.6 kilometers. Approximately how many miles did Francisco drive? 4. Convert 200 gallons to liters. (1 gallon is approximately 3.8 liters.) 5. If 1 inch is approximately 2.5 centimeters, approximately how many inches is 80 centimeters?”

• Scope 16: Understand Variability, Explore, Explore 1–Variability in Data, Procedure and Facilitation Points, students build experience with MP8 as they use repeated reasoning to determine if a question is statistical or non-statistical. “Part II: Understanding Statistical Questions, 1. Read the following scenario to the class: The company called Survey Tiger also has a business that contacts small and big businesses to ask them questions and track the feedback they receive to these questions. Some of these questions are statistical questions, and their data and responses can vary. Other questions are non-statistical questions, and their data and responses do not vary. Survey Tiger needs your help in determining which questions they asked are statistical and non-statistical questions. 2. Quickly review the definition of a statistical question by asking the following questions: a. DOK-1 What is a statistical question? b. DOK-1 What is a survey question? c. DOK-1 Describe how data can vary. 3. Explain to the class that a statistical question can be answered with numerical data or categorical data, meaning the data can vary with numbers or different categories as a response to the question. a. DOK-2 What are some examples of data that can have various categories? b. DOK-2 What are some examples of data that can have various numbers as a response? 4. Assign each group to a station card around the room. Instruct the students to read each scenario and the questions that belong with that scenario. 5. Encourage each group to collaborate together in deciding how to sort the questions and whether the question is a statistical question and has a response with various data, or a non-statistical question and has a response of only one answer. 6. Monitor and check each group for understanding by asking the following questions: a. DOK-2 Does this question require a response with one answer or various answers? b. DOK-1 What do we call a question that requires a response with various data? c. DOK-1 What do we call a question that requires a response with only one piece of data? d. DOK-2 Can a statistical question only be answered with various numbers? Explain. e. DOK-1 What does it mean to find the average of something?”

• Scope 18: Summarize Numerical Data, Explore, Explore 1–Mean and Median, Procedure and Facilitation Points, students build experience with MP8 as they make connections between measures of center and spread and recognize trends when interpreting graphical data representations. “Part I: Mean as Balance PointRead the following scenario to the class: José is tracking his total number of home runs and total number of hits for several series of baseball games he has played in. José wants to determine how many home runs and how many total hits he would have to have in each series to have balanced out his hits and home runs evenly between each series. Help José find the mean as a balance point to determine the number of home runs and the number of total hits he would have to have in each series to balance the home runs and total hits. Give a Student Journal to each student. Give a set of José’s Baseball Scenario Cards and linking cubes to each group.Have students discuss what the term balance point means. Explain to students that the balance point is the mean of a data set.Have students use José’s Home Runs scenario card to record the number of home runs hit in each series on the table in the Student Journal. Students will also create a dot plot to represent the number of home runs hit in each series. They will use the linking cubes to represent the data points for José’s home runs. NoteOne linking cube represents one data point (table). For example, in series 1 José has 3 home runs = 3 linking cubes. Have students stack the linking cubes to represent each series. NoteSeries 1 should have 3 cubes, series 2 should have 1 cube, series 3 should have 5 cubes, series 4 should have 5 cubes, and series 5 should have 1 cube. Once the correct number of linking cubes is stacked for each series, students should figure out how many cubes each series should have to have an equal number of home runs hit by redistributing the stacks so that each stack has an equal amount of cubes. Students will only use cubes for José’s Home Runs scenario card. Actively monitor students as they are working with their groups using the cubes to understand mean as the balance point. Ask questions such as the following: DOK-1 How can you use the linking cubes to determine the balance point? DOK-2 How do you think the balance point would be affected if we added more data points above the balance point? DOK-2 How can you interpret the mean as the balance point using a dot plot?”