8th Grade - Gateway 2

The materials reviewed for STEMscopes Math Grade 8 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, Secondary, 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 12: Rate of Change, Explore, Explore 3–Interpret the Rate of Change and y-Intercept, students develop conceptual understanding about interpreting the rate of change and the y-intercept in a given situation. “Read the following scenario to the class: The Young Leaders Community Service Organization will now have an opportunity to volunteer at the local Boys and Girls Organization. The Boys and Girls Organization has created verbal descriptions of what the volunteers do. Help Javier use the verbal descriptions to write linear functions and create tables and graphs to represent the situations.Give a Student Journal to each student.Have students use the verbal descriptions to write linear functions and create tables and graphs that represent the situation. They will interpret the rate of change and y-intercept. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: DOK-1 Given a y=mx+b function, how do you identify the y-intercept? DOK-2 How can you determine the y-intercept in a situation? DOK-2 How can you determine the rate of change in a situation?” (8.F.4)

• Scope 17: Pythagorean Theorem, Explore, Explore 4–The Pythagorean Theorem on a Coordinate Grid, Procedural and Facilitation Points, students develop conceptual understanding of using the Pythagorean Theorem to solve problems. “1. Read the following scenario to students: Miniature golf course designer Yvette is adding several right-triangle- shaped holes to the existing golf course. She begins planning by using a coordinate grid to determine the layout for each hole. Let’s help Yvette by mapping out several outlines that she can use for her additions. 2. Give a Student Journal to each student. 3. Give a calculator and a number cube to each partnership. 4. Explain to students that they will create Addition 1. Instruct students to begin by plotting Point A on the coordinate grid. Then, have students use the number cube to generate the ordered pairs for Points B and C. Have students form a triangle using these three points. Have students record the measurements of each side using the table provided. Allow students to use calculators when applying the Pythagorean theorem. Then, have students repeat this process by creating Addition 2, a second right triangle. 5. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a. DOK-1: How many ordered pairs are needed to create a right triangle on a coordinate grid? b. DOK-1: How are the two perpendicular legs of a right triangle measured on a coordinate grid? c. DOK-2:  What types of numbers did you use when rolling the number cube to generate ordered pairs?” (8.G.8)

• Scope 19: Patterns in Bivariate Data, Explore, Explore 2–Lines of Fit, Procedural and Facilitation Points, students develop conceptual understanding of scatter plots and use straight lines to informally fit data. “1. Read the scenario to the class: Due to a viral outbreak, increased safety measures are imposed in Woodville Hospital. Doctors are not happy as the new shipment of latex gloves are all the wrong size. The staff at Woodville Hospital need to know the hand length and hand width of each doctor to ensure the glove order gets corrected. The hospital needs you to analyze the relationship between the length and width of the hand to help determine the missing dimensions for the glove order. 2. Project the Class Data Table on the front board, or hang a printed Class Data Table on the front board. 3. Explain to students that they will work in their groups to measure each member’s hand in centimeters. Groups will record each person’s data on the Class Data Table on the front board. 4. Give a Student Journal and a hard spaghetti noodle to each student. 5. Once all data is collected for the class, have students create a scatter plot of the data on the Student Journal. Instruct students to work with their groups to analyze the scatter plot to complete the glove order for the hospital. 6. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a. DOK-2 How would you describe the association between the two variables? b. DOK-2 Why would it be useful to draw a line through the data points? c. DOK-2 How do you determine if your line fits the data well? d. DOK-2 Will outliers affect the line that best represents the data? Explain.” (8.SP.2)

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

• Scope 7: Solving Linear Equations, Explain, Show What You Know–Part 2: One-Variable Equations, students solve the one-variable equations and determine special cases. “Solve the equations below for problems 12 to 14 to find the equation with no solution. Circle the best answer, explain your choices, and show your work. 12. 4(k – 8)= -32+4k ___; 13. -3(v+4)=2v-37 ___; 14. 36-7p=-7(p-5) ___”  (8.EE.7a)

• Scope 8: Proportional Relationships, Explain, Show What You Know–Part 3: Similar Triangles, Student Handout, students determine if the rate of change is constant using similar triangles. “Max wanted to know whether the electronic robot he bought traveled at a constant rate around the track. He gathered data on the time (in minutes) it took for the robot to make laps around the track. Plot the data points on the graph below for time (x) and laps (y). Find the unit rate (slope) by using similar triangles A and B to determine whether the robot was traveling at a constant rate. Show your work.” Students see a chart with values and a blank coordinate graph. (8.EE.6)

• Scope 12: Rate of Change, Explain, Show What You Know–Part 3: Interpret the Rate of Change and Y-Intercept, Student Handout, students represent the same function in multiple ways to interpret the slope and y-intercepts. “Interpret the following scenario, and complete the corresponding information.” Students see the following scenario: “It costs $20 to rent a bike and an additional $5 per hour.” There is a table with the x-values labeled “Hours (x)” and the y-values labeled “Cost $ (y).” Students have a blank coordinate graph labeled the same way.  They must find the equation, rate of change, and y-intercept for the scenario. Then, they must graph the equation.  These two questions are at the bottom of the page “What does the rate of change in this situation represent? What does the y-intercept in this situation represent?” (8.F.4)

The materials reviewed for STEMscopes Math Grade 8 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: Integer Exponents, Explore, Explore 4–Exponential Powers, Procedure and Facilitation Points, students develop fluency by generating equivalent expressions. “1. Read the following scenario to the students: ‘It’s the day before the test, and Ms. Taylor has provided one final review activity for you. She has given you six expressions that include an exponential term raised to a power. Not only do you have to create an equivalent expression, but you also need to explain the steps that you take to make that expression. Show Ms. Taylor that you’re ready for the test by completing her test review.’ 2. Give a Student Journal to each student. 3. Explain to students that they will collaborate with their groups to use the properties of exponents to determine equivalent expressions they can use for each of the given expressions in the table. 4. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a. DOK-1 What do the properties of integer exponents teach us about when an exponential term is raised to a power? b. DOK-1 What happens to the exponents when you multiply numbers with like bases? c. DOK-1 What happens to the exponents when you multiply numbers with unlike bases but common exponents? d. DOK-1 What happens when the bases are the same when dividing exponents? e. DOK-1 What happens when the exponents are the same when dividing exponents?” (8.EE.1)

• Scope 6: Operations with Scientific Notation, Elaborate, Fluency Builder–Operations with Scientific Notation, students develop procedural skill and fluency, with teacher support, of performing operations with numbers expressed in scientific notation. “Procedure and Facilitation Points Show students how to shuffle the cards and place them face down in a stack. Model how to play the game with a student. Shuffle the cards, and place them face down in a stack between the players. 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. Players take turns flipping over one card at a time. Players continue taking turns until all of the cards have been solved. 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.) Once all of the cards have been analyzed, students use the Fix the Mistake! Answer Key to check their answers. The player with the most correct answers is the winner. Distribute the game materials. Then, instruct students to shuffle the cards and lay them facedown in a stack between the players. Monitor students to make sure they find and record accurate responses to each card using the Fix the Mistake! Student Recording Sheet.” (8.EE.4)

• Scope 7: Solving Linear Equations, Explore, Explore 2–One-Variable Equations, Procedure and Facilitation Points, Part 2, students develop fluency by solving multistep equations. “1. Read the following scenario: ‘Tammi’s aunt wants the girls to help develop an app to determine how many solutions there are for different equations. Help the girls solve different equations to predict if the equation will have only one value for x, many values for x, or no values for x.’  2. Explain to students that will collaborate with their groups to solve each equation for x. Then each group will discuss to make a prediction if the equation will have only one solution for x, many solutions for x, or no solutions for x. 3. Monitor and assess students as they are working by asking the following guiding questions: a. DOK-1 Why do you think 4x-6=4x has no solution? b. DOK-1 Why do you think that 12x+15=3x-30 will have only one solution for x? c. DOK-1 Why do you think that x+6+2x=3x+6 will have many solutions?” (8.EE.7)

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

• Scope 2: Integer Exponents, Elaborate, Fluency Builder–Integer Exponents, Instruction Sheet, students build procedural fluency with integers as they match a problem card with an answer that includes an integer exponent. “4. Players take turns asking each other for 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! Cards example, “Create an equivalent expression using the properties of exponents. 304^37\cdot304^{62}; Answer Card, 304^{62}" (8.EE.1)

• Scope 6: Operations with Scientific Notation, Evaluate, Skills Quiz, students demonstrate procedural skill and fluency by performing operations with numbers expressed in scientific notation. “DirectionsSolve each problem and show the steps you took to get your answer. Question 1: (3.6\times10^5)+(2.7\times10^4), ___; Question 2: (9.3\times10^{-5})-(4.5\times10^{-7}), ___.”  (8.EE.4)

• Scope 10: Functions, Elaborate, Fluency Builder–Functions, Bam! Cards (Front of Page 1)  students demonstrate fluency by understanding that a function is a rule that assigns to each input exactly one output. “{(0, 0), (0, 1), (0, 2), (0, 4)} Function or not a function?” (8.F.1)

The materials reviewed for STEMscopes Math Grade 8 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.” 

Examples include:

Engaging routine applications of mathematics include:

• Scope 5: Operations with Scientific Notation, Elaborate, Fluency Builder–Operations with Scientific Notation, provides an opportunity for students to perform operations with numbers expressed in scientific notation with teacher support. Fluency Builder–Operations with Scientific Notation. “Procedure and Facilitation Points Show students how to shuffle the cards and place them face down in a stack. Model how to play the game with a student. Shuffle the cards, and place them face down in a stack between the players. 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. Players take turns flipping over one card at a time. Players continue taking turns until all of the cards have been solved. 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.) Once all of the cards have been analyzed, students use the Fix the Mistake! Answer Key to check their answers. The player with the most correct answers is the winner.Distribute the game materials. Then, instruct students to shuffle the cards and lay them facedown in a stack between the players. Monitor students to make sure they find and record accurate responses to each card using the Fix the Mistake! Student Recording Sheet.” (8.EE.4)

• Scope 7: Solving Linear Equations, Explore,  Explore 2–One-Variable Equations, students demonstrate application alongside conceptual understanding to solve linear equations with one variable and determine if the equations have one solution, infinite solutions, or no solutions. “Procedure and Facilitation Points, Part I, Read the following scenario: For week two of summer savings Erika and Tammi decided to work for Tammi’s aunt. Tammi’s aunt owns a research company. This week’s project is determining solutions to equations. Help the girls determine if given values are valid solutions to given equations. Give a Student Journal to each student. Explain to students that they will need to substitute the values of 0, 5, and 10 for x in each of the equations that Tammi and Erika are working with. Then students will shade in the boxes of the equations that are true for each value of x. Monitor and assess students as they are working by asking the following guiding questions: DOK-1 What operation did you solve first? DOK-1 What does it mean to substitute x=0? DOK-1 What does 6= -4 mean when you substitute 0 for x in the equation x+6=3x-4?” (8.EE.7a)

• Scope 12; Rate of Change, Explain, Show What You Know–Part 3: Interpret the Rate of Change and Y-Intercept, Student Handout, students independently demonstrate application of slope and rate of change in a real world problem. “It costs $20 to rent a bike and an additional $5 per hour. Equation: ___ Rate of change: ___ Y-intercept: ___ “ Students see a blank table and a blank coordinate for them to use to identify and plot points. “What does the rate of change in this situation represent? ___ What does the y-intercept in this situation represent? ___” (8.F.4)

Engaging non-routine applications of mathematics include:

• Scope 3: Square Roots and Cube Roots, Evaluate, Mathematical Modeling Task, students develop application of using cube roots with teacher support to solve real-world problems. “Rabbit Cages, Josephine has two pet rabbits, Oreo and Double Stuff. She has decided that they are in need of a bigger cage. The front door of the cage has to be at least 64 in^3 in order for the rabbits to be able to go in and out safely. Pet Mart sells three different-sized cube cages. The dimensions of those cages are shown below.” Pictures of cubes are shown labeled Cage A, Volume: 343 in^3, Cage B, Volume: 216 in^3, Cage C, Volume: 12 in^3. Part I, 1. Which of the cages would be the best size for the rabbits? Justify your answer.” (8.EE.2)

• Scope 8: Solving Pairs of Linear Equations, Engage, Hook, Procedure and Facilitation Points, Part 1–Pre-Explore, students develop application of using systems of linear equations to solve real-world problems with teacher support. “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. Lott and Mr. Marquette are neighbors who take their children to the movies the first Saturday of every month. It’s a tradition. Each dad usually buys something from the refreshment stand for their kids. One day, they went to a new movie theater and forgot to look at the prices on the menu. However, the dads thought they could figure out the cost of each item based on what each dad ordered and the total bill each dad paid. 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. Sample student answers: I notice that Mr. Lott and Mr.Marquette are solving for at least one variable. I wonder what snacks the dads bought at the refreshment counter. What did the dads buy that was different, and what did they buy that was the same? I can use math to determine the price of each snack item the dads bought by solving for the variable. 4. Project Order Up!.” Students see a table of Mr. Lotts and Mr. Marquette’s purchases at the movie theater.  Below the table, students see two linear equations in standard form that represent the purchases made by each man. “5 Explain to students that Mr. Lott and Mr. Marquette each shared what he ordered and what the total cost of his order was. Discuss the following questions: a. DOK-1 What did Mr. Lott order and what was his total cost? b. DOK-1 What did Mr. Marquette order and what was his total cost? c. DOK-1 What is similar about their orders? d. DOK-1 What is different about their orders?” (8.EE.8c)

• Scope 16: Pythagorean Theorem, Evaluate, Mathematical Modeling Task–Saving Petra’s Kitten, Student Handout, students demonstrate application of Pythagorean Theorem in a real- world problem. Students see a picture of a home with a ladder leaning against it.  The length of the ladder is labeled “x” while the height of the building is labeled “12 ft.” and the distance of the base of the ladder from the base of the building is labeled “5 ft.”. “Petra is outside of her home and sees that her kitten is stuck on the roof. The firefighters are outside of her home taking measurements in order to rescue the kitten, and Petra is trying to determine the distance from the bottom of the ladder to the top of the building. Part I What is the relationship that the firefighters can use to determine the distance from the bottom of the ladder to the top of the building? Explain. ___ Part II 1. The firefighters have ladders measuring 8 feet, 10 feet, 13 feet, and 15 feet. Use the table to help the firefighters determine the measurements at which each ladder can be used.” Students see a table and each column is titled differently.  The first column is titled “Distance from Bottom of Building to Bottom of Ladder.” The second column is titled “Distance from Bottom of Building to Top of Building.” The third column is titled “Distance from Bottom of Ladder to Top of Building.” “2. Based on the table, which ladder should the firefighters use? Explain. ___” (8.G.7)

The materials reviewed for STEMscopes Math Grade 8 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

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:

• Scope 3: Square Roots and Cube Roots, Elaborate, Fluency Builder–Square Roots and Cube Roots, Procedure and Facilitation Points, students demonstrate procedural fluency with square roots and cube roots. Students play a game of Concentration, where they match a question or problem to its solution. “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 face down. 4. Monitor students to make sure they find accurate matches.” (8.EE.2)

• Scope 6: Operations with Scientific Notation, Engage, Hook, Procedure and Facilitation Points, Part II: Post-Explore, students develop application of scientific notation to solve a real world problem. “1. Show the Phenomena Video again, and restate the problem. 2. Refer to Won’t You Be My Neighbor? and discuss the following questions: a. DOK-2 What is the process to determine how many times farther from Earth Neptune is than the Sun? b. DOK-1 What is the quotient of the decimal numbers between one and ten? c. DOK-1 What is the quotient of the powers of ten? d. DOK-1 What is the solution? How many times farther away is Neptune from Earth than the Sun is from Earth?” (8.EE.4)

• Scope 8: Proportional Relationships, Explore, Explore 1–Graph Proportional Relationships, Procedures and Facilitation Points, students develop conceptual understanding by graphing proportional data given in a table or equation. “1. Read the following scenario: Isabella and her friends are planning a trip to New York City. They thought of the costs for flights, food, tours, souvenirs, and the hotel. All of their research has different prices for a certain number of people. Help Isabella and her friends find out how much it costs for one person and for the whole group. 2. Give a Student Journal to each student. 3. Give a set of Trip Costs Scenario Cards to each group. 4. Explain to students that they will collaborate with their groups on graphing the data from the equations and tables from the Trip Costs Scenario Cards. 5. Have students use the information on the Trip Costs Scenario Cards to graph the proportional relationships that are represented in the tables or equations. Next, students will answer the questions on their Student Journals. 6. Monitor and assess students as they are working by asking the following guiding questions: a. DOK-2 How do you create a graph using an equation? b. DOK-2 How do you represent proportional relationships with equations? c. DOK-2 What relationships do you see among the numbers in the table?” (8.EE.5)

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 5: Operations with Scientific Notation, Evaluate, Skills Quiz, Question 1 and 2, students apply knowledge alongside procedural fluency to solve problems with scientific notation. “Solve each problem and show the steps you took to get your answer. Question 1: (3.6\times10^5)+(2.7\times10^4), ____; Question 2, (9.3\times10^5) - (4.5\times10^{17}, ____.” (8.EE.4)

• Scope 9: Solving Pairs of Linear Equations, Evaluate, Standards-Based Assessment, Question 1, students demonstrate procedural skill alongside conceptual understanding to solve 2 linear equations. “What statement is true about the system of linear equations? y=4x+3y, y=4x-1, A. The graph of the system of equations has no points of intersection and has no solution. B. The graph of the system of equations has no points of intersection and has more than one solution. C. The graph of the system of equations has more than 1 point of intersection and has no solution. D. The graph of the system of equations has more than 1 point of intersection and has more than one solution.” (8.EE.8a)

• Scope 18: Volume, Evaluate, Standards-Based Assessment, Question 1, students demonstrate conceptual understanding alongside application of knowledge of the formula of volume to solve problems. “The ball of a snow globe has a radius of 3 inches. What is the volume? A. 904.8 in^3, B. 113.1 in^3, C. 84.8 in^3, D. 12.6 in^3" (8.G.9)

The materials reviewed for STEMscopes Math Grade 8 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 9: Solving Pairs of Linear Equations, Explore, Explore 1–Graph Pairs of Linear Equations, Standards of Mathematical Practice, “MP.1 Make sense of problems and persevere in solving them: Systems of equations can come in many forms. The equations can be written for you, or they can be embedded inside a word problem and need to be discovered. It is important to make sense of the system-of-equations word problem by determining what the variables are representing and how they are used to develop the equations.” Exit Ticket, students develop MP1 because they must understand what the components of an equation represent to successfully solve the problem. “The equation y=-2x+6 is plotted on the graph. Plot y=2x-2 on the same graph. Based on your graph, what is the solution to the system of equations? ___”

• Scope 11: Compare Functions, Explore, Explore 1–Compare Functions in the Same Form, students make sense of problems as they dissect different forms of functions (graphs, tables, equations, etc). They determine similarities and differences between these functions, noting any relationships. Procedure and Facilitation Points, “Read the following scenario: Ethan wants to create a budget so that he can buy a new car. In order to save the money, he needs to get a second job. He has collected information on how much money different jobs will pay him per hour. He has also documented how much money he spends on certain bills and payments. Help Ethan determine which job has the highest pay per hour by discovering which function has the greatest slope. Give a Student Journal to each student. Distribute the 4 bags of Comparison Cards to groups. Explain to the students that they will compare functions of the same form to determine which function has the greatest slope. These forms will include equations, verbal descriptions, tables, and graphs. Monitor and assess students as they are working by asking the following guiding questions: DOK-2  In regard to the equations’ cards, why is the ice cream shop slope greater than the cell phone slope if 14 is greater than 7? DOK-2 In regard to the graphs cards, when looking at the graphs, do you think that a steeper line represents a greater or smaller slope? DOK-1 What does it mean if two jobs have the same slope?”

• Scope 16: Pythagorean Theorem, Explore, Explore 1–Modeling the Pythagorean Theorem and the Converse of the Pythagorean Theorem, Standards of Mathematical Practice, “MP.1 Make sense of problems and persevere in solving them: Students will be able to identify the two legs and hypotenuse, given the side lengths of a right triangle.”  In the Exit Ticket, students develop MP1 as they make sense of a real-world problem and determine how to use the Pythagorean theorem to solve it.  “Molly is hosting her birthday at the local mini-golf course. She notices two of the holes on the course are shaped like right triangles. Use the triangles to help answer the following questions.” Students see two images of a golf course.  The two images are in the shape of a right triangle and have measurements on each side.  “1. Create a formula that could be used to determine whether hole 4 is a right triangle. ___ 2. Is hole 4 a right triangle? Justify your reasoning by using the Pythagorean theorem. ___ 3. Is hole 10 a right triangle? Justify your reasoning by using the Pythagorean theorem.”

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 8: Proportional Relationships, Explore, Explore 2–Compare Proportional Relationships, Standards of Mathematical Practice, “MP.2 Reason abstractly and quantitatively: Students will reason abstractly and quantitatively as they describe the slopes of various lines as equal or not. Students will use reason to determine why slopes of horizontal and vertical lines will always have the same value.” Exit Ticket, students show development of MP2 by analyzing a graph and a table. “Isabella and her friends are looking for the cheapest flight back to Houston. Determine the slope of each graph and table to find the cheapest flight.” Students see a graph and a table with values for cost of flights. “How can the slope be interpreted as the unit rate for the cost of plane tickets on Flight 1? ___ How much does it cost to purchase 9 plane tickets on Flight 1? ___ How can the slope be interpreted as the unit rate for the cost of plane tickets on Flight 2? ___ How much does it cost to purchase 9 plane tickets on Flight 2? ___ Compare the unit rates for both days using <, >, or =. “

• Scope 10: Functions, Explore, Explore 1–Understand Functions on a Graph, students reason abstractly and quantitatively as they determine whether graphs are functions or not functions. Students will use the function to find the outputs that correspond with each input. For Example: “Procedure and Facilitation Points Part I Read the following scenario: Rosie is the sole owner of Rosie’s Boutique. As owner, she has the job of making sure that all of the store’s finances are well documented. It is almost tax season, and her accountant has asked her to gather together these important documents and send them over to him. Before she can do that, she must create monthly and quarterly graphs of her deposits and revenues. Help Rosie analyze her graphs to determine whether they are functions or not. Give a Student Journal to each student. Give a set of Monthly Deposits Cards to each group. Explain to the class that they will use the Monthly Deposits Cards Part I to graph the coordinates of the monthly deposits and analyze each graph. Discuss the following with the class: DOK-1 What could you label the x-axis? DOK-1 What could you label the y-axis? DOK-1 Would it make sense to connect the points with a line? DOK-1 When plotting coordinates on a graph, which direction does the x-value go? DOK-2 In Diagram 1, why are three different arrows pointing to $1,550? DOK-2 In Diagram 2, why are there two different arrows coming from November? DOK-1 Why do you think Rosie deposited money twice in the month of December?...” 

• Scope 12: Model Function Relationships, Explore, Explore 1–Analyzing Graphs, “MP.2 Reason abstractly and quantitatively: Students will reason abstractly and quantitatively as they analyze the graphs to determine the relationships of the data on a graph. Students use the context of each problem to explain the meaning of the data presented.” In the Exit Ticket, students use reason to analyze a graph. Students see a  graph with the x-axis labeled “Time (min.)” and the y-axis labeled “Distance From Finish Line (mi.).” “LaShawn ran a marathon during the road trip and tracked his progress. Describe the graph as linear or nonlinear and increasing or decreasing. Create a description that represents the graph. 1. How can the descriptions linear, nonlinear, constant, increasing, and decreasing be used to describe the graph? ___ 2. What is the y-intercept of the graph?  ___ Description: ___”

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

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:

• Scope 5: Scientific Notation, Standards for Mathematical Practice and Explore, Explore 2–Estimating Numbers and Scientific Notation, Procedure and Facilitation Points, “MP.3 Construct viable arguments and critique the reasoning to others: As numbers are being estimated and converted from standard form to scientific notation, students will be able to follow the direct relationship between the decimal moves and the exponential value that is attached to the power of 10. Students will be able to support their exponential values by relating them to the powers of 10 that are being collected from each movement of the decimal. Students will understand that the actual value and estimated value in both standard and scientific forms are similar but not exactly the same values.” Procedure and Facilitation Points, “1. Read the following scenario to students: Every Monday, the local news station creates a report of the top movies from the weekend, based on ticket sales. Pierre is developing the report; however, he believes that using the actual amount of money each movie made isn’t necessary to the viewer. Pierre decides to develop a reasonable estimate of each movie’s ticket sales. Let’s help Pierre create reasonable estimates of these large numbers. 2. Give a Student Journal to each student. 3. Give a set of Estimation Card Match cards to each partnership. Give a pair of scissors and a glue stick to each student. 4. Review the purpose of estimation with students to ensure they understand why we would need to estimate numbers using scientific notation: a. DOK-1 Why should we estimate numbers? b. DOK-1 How does estimation help with scientific notation? c. DOK-1 How do you estimate large numbers using scientific notation? d. DOK-1 How do you estimate small numbers using scientific notation? 5. Explain to students that they will estimate numbers using scientific notation. Each student will then cut their own set of Estimation Card Match cards. Students will use their set of matching cards to determine each movie’s reasonable estimates according to the actual values. (Note that all cards will not be used.) Students will check each other’s work before gluing the appropriate cards to the Student Journal. 6. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a. DOK-2: Is there only one way to estimate a large or small number? b. DOK-3: What makes an estimation reasonable? c. DOK-1: Does estimating a number change the actual value of the number?”

• Scope 7: Solving Linear Equations, Elaborate, Fluency Builder–Equations with Variables on Both Sides, 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 10: Functions, Explore, Explore 1–Understand Functions on a Graph, students build experience with MP3 as they collaborate with their groups to graph each of the monthly deposits on the coordinate plane on the Student Journal. Then, students will analyze their graphs to answer the questions that follow. For example “Procedure and Facilitation Points Part I Read the following scenario: Rosie is the sole owner of Rosie’s Boutique. As owner, she has the job of making sure that all of the store’s finances are well documented. It is almost tax season, and her accountant has asked her to gather together these important documents and send them over to him. Before she can do that, she must create monthly and quarterly graphs of her deposits and revenues. Help Rosie analyze her graphs to determine whether they are functions or not. Give a Student Journal to each student.Give a set of Monthly Deposits Cards to each group. Explain to the class that they will use the Monthly Deposits Cards Part I to graph the coordinates of the monthly deposits and analyze each graph. Discuss the following with the class: DOK-1 What could you label the x-axis? DOK-1 What could you label the y-axis? DOK-1 Would it make sense to connect the points with a line? DOK-1 When plotting coordinates on a graph, which direction does the x-value go? DOK-2 In Diagram 1, why are three different arrows pointing to 1,550? DOK-2 In Diagram 2, why are there two different arrows coming from November? DOK-1 Why do you think Rosie deposited money twice in the month of December?”

• Scope 19: Patterns in Bivariate Data, Explain, Show What You Know–Part 1: Bivariate Data, Student Handout, students show development of MP3 by making a conjecture about an outlier. “1. Sketch the scatter plot created by each group of data.” Students see two sets of data with two blank coordinate planes where they can graph the data. One set of data is about minutes played and baskets made.  The other set of data is about age and hair color.  “2. Does the data on the first scatter plot show a positive or negative relationship? Does that make sense? ___ 3. Does the data on the second scatter plot show a linear relationship? Does that make sense? ___ 4. Look at the data on the scatter plot below.” Students see a scatter plot relating temperature to electric bills. “Circle all of the terms that correctly describe the association between the temperature and the electricity bill. Positive, Negative, Linear, Nonlinear 5. There is one outlier on the scatter plot. Give an explanation as to what would cause that outlier.”

The materials reviewed for STEMscopes Math Grade 8 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 1: Integer Exponents, Explore, Explore 1–Properties of Integer Exponents, Procedure and Facilitation Points, Part II, students show development of MP4 by modeling expanded exponents to demonstrate combining expressions with exponents. “1. Give a set of Property Cards to each student and a pair of scissors and a glue stick to each pair of students. 2. Have students cut out the Property Cards. 3. Explain that they will use the If and But sections on the tables of their Student Journals to match the correct Property Card to the correct section. 4. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a. DOK-1 What is happening with the exponents in the first If section? b. DOK-1 What do you notice about the integers in the If section? c. DOK-1 What is changing in the If section? What is staying the same? d. DOK-2 Looking at the equation 4^2\cdot4^3=4^5, how can you tell that is a true statement? 4^2 is (4\cdot4) and 4^3 is (4\cdot4\cdot4). e. DOK-2 How is the If section different from the But section?”

• Scope 10: Compare Functions, Evaluate, Standards Based Assessment, students show development of MP4 by using different models to represent and solve real-world problems. “2. Two friends, Robert and Casey, are investing in a micro-investment account as indicated in the text and graph. Robert  He invested $400 after 10 months and invested $550 after 20 months.” Students see a linear graph that represents Casey’s investments. “Which statement is true when comparing the functions? A. The weekly investment is greater for Robert. B. The initial investment is greater for Robert. C. The amount invested for Robert will always be $5 greater than Casey. D. The amount invested for Robert will always be $20 greater than Casey.”

• Scope 12: Model Function Relationships, Explain, Show What You Know–Part 2: Sketching Graphs, students demonstrate development of MP4 by modeling real-life situations with a graph. “Last Saturday, Annette rode her bike. Listed below is how she spent the first half of her day. At 11:00 a.m., she rode 3 miles to the park. She arrived at the park at 11:30 a.m. She stayed at the park for an hour. Next, she rode home and arrived at 1:00 p.m. She stayed at home for an hour. Then, she rode 1 mile to her friend’s house and arrived at 2:10 p.m. Sketch a graph representing Annette’s distance and time between 11:00 a.m. and 2:10 p.m.” Students see the first quadrant of the coordinate plane. The x-axis is labeled “Time” and the y-axis is labeled “Distance.”

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 4: Irrational Numbers, Explain, Show What You Know–Part 3: Locate and Compare Irrational Numbers on a Number Line, Student Handout, students show development of MP5 by using a number line to estimate square roots. Using a number line will help students find the value of a square root by providing a visual model to help guide their thinking. “A development company has purchased several plots of land in a county to build attractions that will increase tourism in the area. Convert the measurements of each plot to decimal approximations without a calculator. Explain your reasoning.” Students see a table with measurements of the length and width, written as a square root, of a plot of land.

• Scope 7: Solving Linear Equations, Explain, Show What You Know–Part 1: Solve Equations with Variables on Both Sides, Student Handout, students show development of MP5 by using algebra tiles to solve equations. Using algebra tiles helps students stay focused on the procedures needed to solve multi-step problems with variables. Algebra tiles can also help to minimize errors leading to incorrect solutions. “Read the scenario. Determine an equation to represent the scenario. Then, solve by using algebra tiles and solve algebraically. Greyson earns a certain number of minutes to play video games for each chore that he completes. He loses a minute of video game time for each time he gets in trouble. On Tuesday, he completed 5 chores and got in trouble 6 times. On Wednesday, he completed 4 chores but only got in trouble once. If he earned the same amount of video game time each day, how much time does he get for each chore he completes? ___”  

• Scope 14: Transformations, Explore, Explore 1–Identifying Transformations, Procedure and Facilitation Points, students demonstrate MP5 by using transparent paper to determine if figures are congruent. Students can use the transparent paper to see how figures “move”. Students can draw the initial figure, and then determine how the figure transforms to make the second figure (slides, turning clockwise, etc.) “1. Read the following scenario: Missy and her good friend Fay are attending a summer camp for artists this week. Today they will learn about different ways to use two-dimensional figures in their paintings. The instructor, Ms. Drawert, wants to show her students that shapes can be oriented in many ways to make paintings more interesting. Help Missy and Fay determine how each of the figures moved. 2. Give a bag containing How Did it Move? Cards and 6 pieces of tracing paper to each partnership and a Student Journal to each student. 3. Explain the following to students: To determine how each figure moves, we can trace one of the figures onto tracing paper and then move the tracing paper around until the second figure matches the first figure that we traced onto the paper.Have students take turns tracing the first figure from each card onto their tracing paper. Then, have students work together to determine the movement of the first figure to create the second figure. 4. Monitor and assess students as they are working by asking the following guiding questions: a. DOK-2 What did you notice about the movement of the first figure on Card 1 to become the second figure? b. DOK-2 What did you notice about the movement of the first figure on card 5 to the second figure? c. DOK-2 What did you notice about the movement of the first figure on card 7 to create the second figure? ”

The materials reviewed for STEMscopes Math Grade 8 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 6: Operations with Scientific Notation, Explore, Explore 1–Adding and Subtracting with Scientific Notation, Procedure and Facilitation Points, students show development of MP6 while attending to precision as they solve addition and subtraction problems involving Scientific Notation. “1. Read the following scenario to the class: Plus/Minus Electronics has a special service for people who need more storage space on their computers or external hard drives. They will combine the hard drives of people’s computers and external hard drives so that they have enough room to store all of their data. They record all data sizes in bytes. Some of their new work orders are placed around the room. Help Plus/Minus know the sizes of the combined data as well as how much free space will be on the hard drive after the data is combined. 2. Give the Student Journal to each student. 3. Explain to students that they will collaborate with their groups to add hard drive 1 and hard drive 2 to find the total amount of hard-drive space used. Then students will subtract the total space used in hard drives 1 and 2 from the new hard-drive space.”

• Scope 12: Rate of Change, Explain, Show What You Know–Part : Sketching Graphs, Student Handout, students show development of MP6 as they attend to precision while creating a Distance-Time Graph and use precise vocabulary as they describe relationships within the graph. “Last Saturday, Annette rode her bike. Listed below is how she spent the first half of her day. At 11:00 a.m., she rode 3 miles to the park. She arrived at the park at 11:30 a.m. She stayed at the park for an hour. Next, she rode home and arrived at 1:00 p.m. She stayed at home for an hour. Then, she rode 1 mile to her friend’s house and arrived at 2:10 p.m. Sketch a graph representing Annette’s distance and time between 11:00 a.m. and 2:10 p.m.” Students see a blank coordinate with axes labeled appropriately for the situation. “Use terms (linear, nonlinear, increasing, decreasing, or constant) to describe relationships within this graph.”

• Scope 17: Pythagorean Theorem, Evaluate, Standards-Based Assessment, Student Handout, Question 2, students show development of MP6 while using the Pythagorean theorem to solve problems. “Right Triangle DEF is shown.” Students see an image of a right triangle with measurements labeled on each leg. “What is the length of the hypotenuse, x, of triangle DEF? Round your answer to three decimal places. Enter your answer in the box. ___”

The materials reviewed for STEMscopes Math Grade 8 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 8: Proportional Relationships, Explore, Explore 2–Compare Proportional Relationships, Procedure and Facilitation Points, students build experience with MP7 as they look for and make use of structure and as they explain how variously spread information can have the same slope. They will use the structure of the equation of a line to be able to determine the slope from an equation as well as write an equation given the slope. “Part I: Comparing Proportional Relationships Using Graphs, 1. Read the following scenario: Isabella and her friends are trying to figure out what time of the year they can fly to New York City. They want to fly out when the flight prices are cheapest. Using the graphs on the cards, help Isabella and her friends determine which season is the most expensive time and which season is the cheapest time to fly to New York City. 2. Give a Student Journal to each student. 3. Give a set of Flight Prices by Season Cards to each group. 4. Explain to students that they will analyze each graph on the cards, help Isabella and her friends compare the costs of flights from Houston to New York, and determine the best time of the year to fly to New York. 5. Have students compare the graphs on the Flight Prices by Season Cards to determine the seasons that have the highest prices and use the graphs to determine the equation and unit rate that is represented by each graph. 6. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a. DOK-2 How is each graph different? b. DOK-2 How is each graph similar? c. DOK-2 How do you determine the slope from the graph?”

• Scope 9: Solving Pairs of Linear Equations, Explore, Explore 1–Graph Pairs of Linear Equations, Exit Ticket, students build experience with MP7 as they look for structure when using the x or y equations to help solve the other equation. “The equation y=-2x+6 is plotted on the graph. Plot y=2x-2 on the same graph. Based on your graph, what is the solution to the system of equations?

• Facilitation Points, students build experience with MP7 as they look for and make use of structure as they describe the slope and y-intercept of functions in several different forms. They will recognize how to determine these relationships based on the structure of the model. “Part I: Determine the Y-Intercept in Tables and Graphs Read the following scenario: Javier is a member of the Young Leaders Community Service Organization and has been volunteering at the local hospital to get community service hours. Every department tracks the number of hours that the volunteers work. They track the hours by creating tables and graphs to show the number of hours that each volunteer works in their departments. Help Javier use the number of hours worked to determine the y value in the tables and graphs when the x value is 0. Give a Student Journal to each student. Give a set of Volunteer Hours Cards Part I, a Y-Intercept Work Mat, and a dry-erase marker to each group. Explain to students that they will analyze each Volunteer Hours Card carefully and will determine the y value on a table and a graph.Have students use the tables and the graphs to determine the y value when the x value is 0. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: DOK-1 What is the independent variable? DOK-1 What is the dependent variable? DOK-1 What are the domain and range? DOK-1 What does the y value represent in the table when the x value is 0? DOK-1 What does the y value represent in the graph when the x value is 0? DOK-1 What is the ordered pair when x is 0?”

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 4: Irrational Numbers, Explore, Explore 3–Locate and Compare Irrational Numbers on a Number Line, Math Chat, students build experience with MP8 as they use repetitive division to help identify differences between rational and irrational numbers. “Questions, Sample Student Responses, DOK-2 How does a number line help to determine the approximate decimal value of a square root that is not a perfect square? A number line can help determine the approximate value of a square root that is not a perfect square because you can determine which 2 perfect squares your square root would be between and what whole number the square root would be closer to. DOK-2 Estimate the value of \sqrt{42}. 6^2=36 and 7^2=49, so the square root of 42 is between 6 and 7. DOK-2 Would the square root of 42 be closer to 6 or closer to 7? Explain your thinking. I think it would be closer to 6 because 42 is 6 away from 36 and 7 away from 49. DOK-3 What would the value of -\sqrt{4} be? I know that \sqrt{4} is 2, so the opposite of 2 would be −2.

• Scope 11: Compare Functions, Explore, Explore 3–Linear vs. Non-Linear Functions, Procedure and Facilitation Points, students build experience with MP8 as they use repeated reasoning to find functional relationships as well as regularity in values on graphs and tables. Students use mock sales information for car sales at dealerships in graphs and tables. “Part II, 1. Read the following scenario: As Ethan visits the different dealerships, he begins to ask questions about each company and the number of cars that they sell per month. He believes that the dealership with the most consistent sales must be the best place to purchase his vehicle. Help Ethan determine whether the functions given to him are linear or nonlinear. 2. Distribute the 3 bags of Monthly Car Sales Cards to each group. 3. Explain to the students that they will compare functions of different forms to determine whether the functions are linear or nonlinear. 4. Monitor and assess students as they are working by asking the following guiding questions: a. DOK-1 In regard to the Tables Cards (point to a linear table), what type of graph do you think this table would create? b. DOK-1 In regard to the Graphs Cards, how do you determine the equation of these graphs? c. DOK-2 What if the x value had an exponent greater than 2, would it still be nonlinear?”

• Scope 14: Transformations, Explore, Explore 1–Identifying Transformations, Procedure and Facilitation Points, students build experience with MP8 as they look for regularity in repeated reasoning of transformations. Students will discover the relationships between an image and its preimage and recognize patterns in the properties of each individual transformation. “Read the following scenario: Missy and her good friend Fay are attending a summer camp for artists this week. Today they will learn about different ways to use two-dimensional figures in their paintings. The instructor, Ms. Drawert, wants to show her students that shapes can be oriented in many ways to make paintings more interesting. Help Missy and Fay determine how each of the figures moved.Give a bag containing How Did it Move? Cards and 6 pieces of tracing paper to each partnership and a Student Journal to each student.Explain the following to students: To determine how each figure moves, we can trace one of the figures onto tracing paper and then move the tracing paper around until the second figure matches the first figure that we traced onto the paper. Have students take turns tracing the first figure from each card onto their tracing paper. Then, have students work together to determine the movement of the first figure to create the second figure. Monitor and assess students as they are working by asking the following guiding questions: DOK-2 What did you notice about the movement of the first figure on Card 1 to become the second figure? DOK-2 What did you notice about the movement of the first figure on card 5 to the second figure? DOK-2 What did you notice about the movement of the first figure on card 7 to create the second figure?”