8th Grade - Gateway 1

The materials reviewed for STEMscopes Math Grade 8 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

The curriculum is divided into 19 Scopes, and each Scope contains a Standards-Based Assessment used to assess what students have learned throughout the Scope. Examples from Standards-Based Assessments include:

• Scope 4: Irrational Numbers, Evaluate, Standards-Based Assessment, Question 5, “The diagonal length of a right triangle for a piece of fabric is \sqrt{72} inches. Part A What are the perfect squares that 72 is between? The perfect squares are ____ and ____. Part B Where is \sqrt{72} located on a number line? Plot the point on the number line below.” Students see a number line that extends from 7 to 10 with tick marks at every tenth of each number. (8.NS.2)

• Scope 7: Solving Linear Equations, Evaluate, Standards-Based Assessment, Question 3, “Eric says the correct solution to the \frac{8}{3}. Explain the error in his reasoning. Find the correct solution. Enter your answers below.” (8.EE.7b)

• Scope 12: Rate of Change, Evaluate, Standards-Based Assessment, Question 2, “The equation h=100-4t represents the height, h, from the location on a sledding hill after t seconds. What is the meaning of the rate of change and the initial value? Enter your answers below.” (8.F.4)

• Scope 14: Transformations, Evaluate, Standards-Based Assessment, Question 2, “Figure ABCDE is dilated to create figure A’B’C’D’E’ on the coordinate plane below. What is the scale factor for the dilation?” Two figures are shown on a coordinate plane. (8.G.3) 

• Scope 19: Patterns in Bivariate Data, Evaluate, Standards-Based Assessment, Question 5, “The line of fit for a scatterplot is y=-1.1x+14. Predict the value of y when x=30. -14.5; -42.9; 13.27; -19.” (8.SP.3)

The materials reviewed for STEMscopes Math Grade 8 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials provide extensive work in Grade 8 as students engage with all CCSSM standards within a consistent daily lesson structure, including Engage, Explore, Explain, Elaborate, and Evaluate. Intervention and Acceleration sections are also included in every lesson. Examples of extensive work to meet the full intent of standards include:

• Scope 2: Integer Expressions, Explore, Explore 4 - Exponential Powers, Procedure and Facilitation Points and Exit Ticket, engage students with the full intent of 8.EE.1 (Know and apply the properties of integer exponents to generate equivalent numerical expressions…) In the previous three Explores, students worked on multiplying and dividing with exponential powers. Explore 4 focuses on putting everything together. In Procedure and Facilitation Points, students work in small groups to use the properties of exponents to make equivalent and simplified equations. “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? 5. After the Explore, invite the class to a Math Chat to share their observations and learning.” On the Exit Ticket, students are given a complex equation to apply what they have learned. Students see the expression and work space to simplify and make an equivalent expression. “The test is finally here! You feel like you’ve done really well, but Ms. Taylor threw in a bonus question that encompasses everything you’ve learned so far. Find an equivalent expression for the expression below. Ms. Taylor’s Expression, (8^{5+3}\div(4\times2)^5)^3, Workspace, Equivalent Expression”

• Scope 4: Irrational Numbers, Explore, Explore 1 - Irrational Numbers vs. Rational Numbers, Procedure and Facilitation Points, engages students in extensive work to meet the full intent of 8.NS.1 (Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.) “Part I, 1. Read the scenario aloud: You are looking for an internship to improve your job skills. You’ve applied to a local builder, Team Builders, to plan and build youth sporting areas throughout the city. The hiring manager has called you in for an interview. She feels you have great potential, but you will need to prove that you can learn new math skills and apply your new knowledge before she can offer you the job. Help Team Builders sort their numbers by their classifications. 2. Discuss the following questions with the class: a. DOK–1 What types of numbers have we discovered so far? b. DOK–1 What is an example of a whole number? c. DOK–1 What is an example of an integer? d. DOK–1 What is an example of a rational number? 3. Display the Real Numbers Venn Diagram. 4. Discuss the following with the class: This is called a Venn diagram. Venn diagrams help to show information based on relationships. Ask the class the following questions: a. DOK–1 What does this Venn diagram illustrate? b. DOK–1 What set of numbers is within the integers set? c. DOK–1 What are the names of the type of numbers that include integers as subsets? … 8. Monitor and assess students as they work. Use the Real Numbers Classification Explanation Cards to provide hints and guidance as appropriate for each group. Ask students the following guiding questions: a. DOK–2 What are some key words on this classification card that were helpful? How did the key word help? b. DOK–2 What are some key numbers in the examples on this classification card that were helpful? How did the key numbers help? c. DOK–3 How can the Venn diagram help you?... Part II, 1. Read the scenario aloud: The hiring manager for Team Builders was impressed with your ability to sort each classification of real numbers! But can you apply this knowledge? Help determine if each number is rational or irrational. Then, show proof of your decision. 2. Explain to students that they will collaborate with their groups to determine if each of the numbers provided is a rational number or an irrational number. Then they will need to use the Venn diagram to help prove their decisions are correct. 3. Students will collaborate to apply their knowledge of real number classification to a variety of numbers provided. Students should use the Venn diagram completed in Part I to help prove their thinking is correct. Allow students to use a calculator to assist in determining if a number is a repeating decimal or not. 4. Monitor and assess students as they work. Ask guided questioning as needed: a. DOK–1 Why do you think 0.5 is rational? b. DOK–2 Can a decimal that does not repeat be written as a fraction?”

• Scope 5: Scientific Notation, Explain, Show What You Know-Part 3: Comparing Numbers in Scientific Notation, engages students in extensive work to meet the full intent of 8.EE.3 (Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other…) In previous Explores, students have multiplied, divided, and compared numbers with scientific notation. In Show What You Know, they apply their knowledge. Students see a table with the following information: City, Population, Duncanville, 2.4\times10^5, Lumber City, 1.2\times10^7, Parkerton, 4.8\times10^4.  “A tech company is planning to build a new manufacturing complex in a new city. The executive board needs to decide where the new complex will be built. They have narrowed their decision down to three cities. The final decision will be based on the city’s population. The population of the three cities is shown in the table below. 1. Which city has the greatest population? What is that city’s population? 2. Which city has the smallest population? What is that city’s population? 3. How many times greater is the population of Duncanville than the population of Parkerton? 4. How many times greater is the population of Lumber City than the population of Duncanville?”

• Scope 18: Volume, Explore 1 and 2, and Show What You Know–Part 3, engages students in extensive work to meet the full intent of 8.G.9 (Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.) Explore 1-Cylinders, Procedure and Facilitation Points, students work in pairs to understand the formula for the volume of a cylinder. “Part I: Understanding the Volume of a Cylinder Formula, 1. Read the following scenario: One of your friends and classmates is named Madeline Dupont. Madeline’s family owns a food processing plant. One branch of the plant involves canning foods. Because a large number of employees are absent today, Madeline’s mom has asked if she has any friends that are good at math who could help with the canning process. Your job today is to show that you understand the volume of cylinders and can determine the volume of cans (cylinders) with a variety of bases, sizes, and heights. 2. Give a Cylinder Net, a dry-erase marker, a glue stick, and a pair of scissors to each partnership. 3. Review students’ prior knowledge of finding the volume of a rectangular prism by asking the following questions: a. DOK–1 How do you find the volume of a rectangular prism? b. DOK–1 When you use the formula V=lwh, what formula is being used when you multiply the length times the width? Explain why the formula is being used. c. DOK–2 What is the formula that is used to find the volume of a rectangular prism? V=Bh where B = area of the base So V = B$$\times$$height, or length$$\times$$width$$\times$$height. d. DOK–2 What is the volume of this rectangular prism?...6. Instruct students to arrange the centimeter cubes in a single layer at the bottom of the cylinder and to fit as many cubes into the layer as possible. They will also need to find how many layers of cubes fit in the cylinder and will need to make a stack of cubes along the inside of the cylinder. Students will use page 2 of the Cylinder Net/Volume of a Cylinder Work Mat to calculate the area of the base and the volume of the cylinder. (Note that you should show the students how to use the base to determine the radius and how to determine the height of the cylinder using the formula.) Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a. DOK–1 What shapes are in a cylinder? b. DOK–1 What is the shape of the bases on a rectangular prism and cylinder? c. DOK–2 How might finding the volume of a cylinder be similar to finding the volume of a rectangular prism? d. DOK–2 How might finding the volume of a cylinder be different from finding the volume of a rectangular prism? e. DOK–2 How many layers of cubes did it take to fill the cylinder? f. DOK–2 What is the area of the cylinder’s base? Express your answer in terms of \pi. 4\pi, g. DOK–2 What is the volume of the cylinder? Express your answer in terms of \pi. 48\pi, h. DOK–2 How can you determine the approximate number of cubes that will fit in the cylinder? Why is this an approximate number of cubes? i. DOK–2 How can you calculate the volume of the cylinder?... Explore 2-Cones, Exit Ticket, students use what they have learned to find a cone’s volume. Students see a cone with a radius of 15cm and a height of 25 cm. There is also work space to complete the work to find the volume. “The Duponts want to add a new cone to their menu. It’s called the Family Sundae Cone. Look at its dimensions in the model, use the workspace to represent the area of the base expressed in terms of \pi, and then find the volume. Use 3.14 as an approximation for \pi. Round to the nearest hundredth, if necessary.” Show What You Know-Part 3: Spheres, Student Handout, students apply what they have learned about the formula for finding the volume of a sphere. “Mr. Kennedy is a physical education teacher at the neighborhood elementary school. He keeps several different kinds of balls in a bin so the students can play sports during their gym time. Use the information provided to determine the volume of each ball. Use 3.14 as an approximation for \pi. Round your answer to the nearest hundredth. There are several baseballs in Mr. Kennedy’s bin. Each baseball has a radius of 3.7 cm. What is the volume of each baseball? Volume: ___, The girls really enjoy playing volleyball during gym class. A volleyball has a diameter of 8 inches. What is the volume of a volleyball? Volume: ___, Most of the boys like to play basketball during gym class. Each basketball has a radius of 4.7 inches. What is the volume of each basketball? Volume: ___, Mr. Kennedy has several beach balls that he uses for his lessons. He keeps them deflated until he needs to use them. When inflated, each beach ball has a diameter of 11 inches. What is the volume of one beach ball? Volume: ___”

• Scope 19: Patterns in Bivariate Data, Explore, Explore 2 - Lines of Fit, Procedure and Facilitation Points and Exit Ticket, engages students in extensive work to meet the full intent of 8.SP.2 (Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.)  In Procedure and Facilitation Points, “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 hard spaghetti noodle to each student. 5. Once all data is collected for the class, have students create a scatterplot of the data on the Student Journal. Instruct students to work with their groups to analyze the scatterplot 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… Students then apply their knowledge on the Exit Ticket.  Data is provided in a graph and students select the line of three given that best represents the data given. “Lines of Fit Exit Ticket In each of the questions below, circle the label of the line that best fits each set of data.”

The materials reviewed for STEMscopes Math Grade 8 meet expectations that, when implemented as designed, the majority of the materials address the major cluster of each grade.

The instructional materials devote at least 65% of instructional time to the major clusters of the grade:

• The approximate number of scopes devoted to major work of the grade (including assessments and supporting work connected to the major work) is 15 out of 19, approximately 79%.

• The number of lesson days and review days devoted to major work of the grade (including supporting work connected to the major work) is 122 out of 154, approximately 79%.

• The number of instructional days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 139 out of 180, approximately 77%.

An instructional day analysis is most representative of the instructional materials because this comprises the total number of lesson days, all assessment days, and review days. As a result, approximately 77% of the instructional materials focus on the major work of the grade.

The materials reviewed for STEMscopes Math Grade 8 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Materials are designed so supporting standards/clusters are connected to the major standards/ clusters of the grade. Examples of connections include:

• Scope 4: Irrational Numbers, Explore, Explore 2–Decimal Expansion, Exit Ticket, connects the supporting work of 8.NS.1 (Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually…) to the major work of 8.EE.2 (Use square roots and cube root symbols to represent solutions to equations in the form x^2=p and x^3=p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that \sqrt{2} is irrational). Students convert fractions to decimals and decimals to fractions to determine if each number is rational or irrational. “Convert the decimals to fractions and the fractions to decimals. Classify each number as rational or irrational. Decimal, Fraction, Rational or Irrational?, 0.75,0.3⁻,\frac{1}{8},\pi,\frac{6}{9},\sqrt{4}”.

• Scope 17: Pythagorean Theorem, Explain, Show What You Know-Part 3: The Pythagorean Theorem in Rectangular Prisms, connects the supporting work of 8.G.9 (Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.) to the major work of 8.G.7 (Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.) Students are given cones with missing measurements.  “The Pythagorean Theorem in Rectangular Prisms, Use the measurements provided to determine the slant height of each 3-dimensional figure. Round to the nearest tenth if necessary. (1) height = 16 cm radius = 9 cm slant height =, (4) height = 13 inches diameter = 11 inches slant height = ”

• Scope 19: Patterns in Bivariate Data, Explore, Explore 3–Linear Equations, Procedure and Facilitation Points, connects the supporting work of 8.SP.2 (Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line…) to the major work of 8.EE.6 (Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y=mx for a line through the origin and the equation y=mx+b for a line intercepting the vertical axis at  b). Students work in groups to create a scatter plot and linear equation to represent that data. “Part I, 1. Read the scenario to the class: The Tanaka High School Student Council decided to do a massive 20-week fundraiser to raise money for hurricane relief for a neighboring county. Each grade level chose a different fundraiser and set the goal of raising at least $500 per class. Some of the classes started off with big donations from parents, while the others started at $0. The 9th-grade class chose to sell cookies during lunch periods, and the data for the first 8 weeks has been collected. Based on the data, are they on track to meet the goal of at least $500? 2. Give a Student Journal and ruler to each student. 3. Explain to students that they will work in their groups to create a scatter plot to represent the funds raised by the 9th-grade class over the first 8 weeks of the fundraiser. Instruct them to use the ruler as a straightedge to draw the line of fit. Have students work with their groups to answer the questions to determine whether the 9th-grade class is on track to meet their fundraising goal...” 

The materials for STEMscopes Math Grade 8 meet expectations that materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. 

Materials are coherent and consistent with the Standards. These connections are sometimes listed for teachers in one or more of the three sections of the materials: Engage, Explore and Explain. Examples of connections include:

• Scope 12: Rate of Change, Explain, Show What You Know–Part 3: Interpret the Rate of Change and Y-Intercept, connects the Expressions & Equations domain to the Functions domain. Students create equations, find rate of change and y-intercept based on the given scenario. For example, “Interpret the following scenario, and complete the corresponding information. Bike Rental RatesIt costs  $20 to rent a bike and an additional $5 per hour. Equation: ___; Rate of Change: ___; Y-intercept: ___.”

• Scope 13: Model Function Relationships, Explore, Explore 2–Sketching Graphs, Procedure and Facilitation Points connects the major work of 8.F.A (Define, evaluate, and compare functions.) to the major work of 8.F.B (Use functions to model relationships between quantities.). Students sketch a graph that shows the qualitative features of a function that can be described as linear or nonlinear and increasing or decreasing. “1. Read the following scenario: Lashawn and Diego are going to the beach for another vacation. They will use their software tracking program to keep track of the speed, distance, and time traveled using graphs. Help Lashawn and Diego sketch and analyze the graphs from their software tracking program. 2. Give a Student Journal to each student. 3. Give a set of Beach Cards to each group. 4. Explain to students that they will sketch graphs and will describe the graphs as linear or nonlinear and increasing or decreasing.”

• Scope 17: Pythagorean Theorem, Explain, Show What You Know–Part 2: Finding an Unknown Side Length in Right Triangles, connects the Expressions & Equations domain to the Geometry domain. Students are given a diagram of a right triangle with side lengths and are asked to find the length of a hypotenuse. For example, “Kevin is a carpenter and enjoys building things for his neighbors. One of his neighbors uses a wheelchair to get around. Kevin wants to help his neighbor move around the outside of his home more easily by building a few wheelchair ramps around his house. The neighbor’s home is elevated, and it has a long front porch and a porch to get to the backyard. Kevin decides to build three ramps: two for the front porch and one for the back porch. Determine the missing side length of each of Kevin’s ramps. Kevin is a carpenter and enjoys building things for his neighbors. One of his neighbors uses a wheelchair to get around. Kevin wants to help his neighbor move around the outside of his home more easily by building a few wheelchair ramps around his house. The neighbor’s home is elevated, and it has a long front porch and a porch to get to the backyard. Kevin decides to build three ramps: two for the front porch and one for the back porch. Determine the missing side length of each of Kevin’s ramps. Given a diagram of a right triangle with side lengths of legs 1.5 ft and 4.8 ft. and labeled x for the hypotenuse. ___”

• Scope 18: Volume, Explain, Show What You Know–Part 1: Cylinders, Student Handout, connects The Number System domain to the Geometry domain. Students solve real-life problems working with cylinders. “One of the main methods that a fruit company uses to distribute its fruit is in cans. Because the fruit company sells its canned fruit in different forms, it distributes its fruit in cans of different sizes. Label each can with the measurements described. Use those measurements to determine the volume. Use 3.14 as an approximation for \pi. Round to the nearest hundredth, if necessary. A can of pineapple pieces has a radius of 9 cm and a height of 4 cm. What is the volume of the can of pineapple pieces? Volume = ___, A small can of orange juice has a diameter of 3 inches and a height of 5 inches. What is the volume of the can of orange juice? Volume = ___”

The materials reviewed for STEMscopes Math Grade 8 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.

Prior and future connections are identified within materials in the Home, Content Support, Background Knowledge, as well as Coming Attractions sections. Information can also be found in the Home, Scope Overview, Teacher Guide, Background Knowledge and Future Expectations sections. 

Examples of connections to future grades include:

• Scope 6: Operations with Scientific Notation, Home, Scope Overview, Teacher Guide, Future Expectations connects 8.EE.A (Expressions and Equations Work with radicals and integer exponents.) to work that will be done in upcoming grades. “In high school, students will be working with various scales on all numerical levels including very large whole numbers and very small decimals. Students will be using numerical conversions to change the values in order to understand and guide the completion of multistep real-world or mathematical problems.”

• Scope 7: Solving Linear Equations, Home, Content Support, Background Knowledge, connects 8.EE.7b (Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.) to future grades. “Students will be continuing to solve linear equations using the distributive property and combining like terms and inverse operations throughout all levels of high school. Students will be using the various steps to evaluate systems of equations, inequalities, and linear functions, as well as factoring and solving for zeros when working with polynomials.”

• Scope 10: Functions, Home, Content Support, Background Knowledge, connect 8.F.1 (Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.) to future grades. “In the coming years, students will continue their work with linear and nonlinear functions. They will expand this thinking into quadratic, exponential, logarithmic, and other types of functions. Students will learn how to express the inputs and outputs of functions in function notation as well as how to build and interpret all types of functions and their rules.”

Examples of connections to prior grades include:

• Scope 7: Solving Linear Equations, Home, Content Support, Background Knowledge, connect 8.EE.7b (Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.) to prior grades. “In Grade 6, students began to identify and apply properties of operations in order to develop equivalent expressions that may be written in different forms. As students developed an understanding of equivalent expressions, students started to use variables to represent unknown values in real-life and mathematical problems. Students then used one inverse operation to solve for the appropriate value that the variable is representing. In Grade 7, students applied properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients in order to develop equivalent expressions. Students included more than 1 inverse operation or property in order to solve for the missing variable value when working with a single variable.”

• Scope 10: Functions, Home, Content Support, Background Knowledge, connect 8.F.1 (Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.) to prior grades. “In previous grades, students learned how to plot points on a graph. They learned about the relationship between the x- and y-coordinates and how to analyze tables and graphs. In Grade 7, they discovered proportional relationships between quantities. All of these concepts will tie together in order for students to understand the basics of functions.”

• Scope 14: Transformations, Home, Scope Overview, Teacher Guide, Background Knowledge, connects 8.G.A (Understand congruence and similarity using physical models, transparencies, or geometry software.) to learning done in prior grades. “In previous grades, students developed an understanding of congruence and similarity. They created scale drawings and discussed the relationship between geometrical figures. Being able to determine these relationships will be the foundation of transformations as students discover how images map onto one another.”