8th Grade - Gateway 1

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

Program assessments include Pre-Unit Diagnostic Assessments, Cool Downs, Mid-Unit Assessments, Performance Tasks, and End-of-Unit Assessments which are summative. According to the Course Guide, “At the end of each unit is the end-of-unit assessment. These assessments have a specific length and breadth, with problem types that are intended to gauge students’ understanding of the key concepts of the unit while also preparing students for new-generation standardized exams. Problem types include multiple-choice, multiple-response, short answer, restricted constructed response, and extended response. Problems vary in difficulty and depth of knowledge.” Examples of summative End-of-Unit Assessment problems that assess grade-level standards include:

• Unit 2: Dilations, Similarity, and Introducing Slope, End-of-Unit Assessment: Version A, Problem 1, “Select all the true statements. A. Dilations always increase the lengths of line segments. B. Dilations take perpendicular lines to perpendicular lines. C. Dilations of an angle are congruent to the original angle. D. Dilations increase the measure of angles. E. Dilations of a triangle are congruent to the original triangle. F. Dilations of a triangle are similar to the original triangle.” (8.G.A)

• Unit 3: Linear Relationships, End-of-Unit Assessment: Version B, Problem 6, “A sandwich store charges a $10 delivery fee, and $4.50 for each sandwich. A. What is the total cost (sandwiches and delivery charge) if an office orders 6 sandwiches? B. What is the total cost for x sandwiches? C. Graph the total cost of sandwiches and delivery based on the number of sandwiches ordered. Be sure to label your axes and scale them by labeling each gridline with a number (graph provided) D. Is there a proportional relationship between number of sandwiches and the cost of the order? Explain how you know. E. At a different sandwich shop, there is a $5 delivery fee, and each sandwich costs $4.50. On the same grid, graph the total cost of sandwiches and deliver based on number of sandwiches ordered for this new shop. Describe how the two graphs are the same and how they are different.” (8.EE.6)

• Unit 5: Functions and Volume, End-of-Unit Assessment: Version A, Problem 4, “For cones with radius 6 units, the equation V = 12$$\pi$$h relates the height h of the cone, in units, and the volume V of the cone, in cubic units. a. Sketch the graph of this equation on the axes (graph provided). b. Is there a linear relationship between height and volume? Explain how you know.” (8.F.1 and 8.F.4)

• Unit 6: Associations in Data, End-of-unit Assessment: Version B, Problem 4, “A. Draw a scatter plot that shows a negative, linear association and has one clear outlier. Circle the outlier. B. Draw a scatter plot that shows a positive association that is not linear.” (8.SP.1)

• Unit 8: Pythagorean Theorem and Irrational Numbers, End-of-Unit Assessment: Version B, Problem 4, “What is the decimal expansion of \frac{16}{9}?” (8.NS.1)

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

Each lesson is structured into four phases: Warm Up, Instructional Activities, Lesson Synthesis, and Cool Down. This structure ensures thorough engagement with grade-level problems and alignment with educational standards.

The Warm Up phase starts each lesson, helping students prepare for the day’s content and enhancing their number sense or procedural fluency. Following the Warm Up, students participate in one to three instructional activities focusing on learning standards. These activities, described in the Activity Narrative, form the lesson's core.

After completing the activities, students synthesize their learning, integrating new knowledge with prior understanding. The lesson concludes with a Cool Down phase, a formative assessment to measure student understanding. Additionally, each lesson includes Independent Practice Problems to reinforce the concepts.

Instructional materials engage all students in extensive work with grade-level problems. Examples include:

• Unit 4: Linear Equations and Linear Systems, Section B: Linear Equations in One Variable, Lesson 9: When are They the Same? students write and interpret one variable equations to represent situations with two conditions. Activity 2: Elevators, “A building has two elevators that both go above and below ground. At a certain time of day, the travel it takes elevator A to reach height h in meters is 0.8h + 16 seconds. The travel time it takes elevator B to reach h in meters is -0.8h + 12 seconds. a. What is the height of each elevator at this time? b. How long would it take each elevator to reach ground level at this time? c. If the two elevators travel toward one another, at what height do they pass each other? How long would it take? d. If you are on an underground parking level 14 meters below ground, which elevator would reach you first?” Are You Ready for More? Problem 1, “In a two-digit number, the ones digit is twice the tens digit. If the digits are reversed, the new number is 36 more than the original number. Find the number.” Cool Down: Printers and Ink, “To own and operate a home printer, it costs $100 for the printer and an additional $0.05 per page for ink. To print out pages at an office store, it costs $0.25 per page. Let p represent number of pages. a. What does the equation 100 + 0.05p = 0.25p represent? b. The solution to that equation is p = 500. What does the solution mean?” Practice Problems, Problem 4, “For what value of x do the expressions \frac{2}{3}x + 2 and \frac{4}{3}x - 6 have the same value?” Materials present students with extensive work with grade-level problems of 8.EE.7 (Solve linear equations in one variable.)

• Unit 5: Functions and Volume, Section A: Inputs and Outputs, Lesson 1: Inputs and Outputs, students identify a rule that describes a relationship between input-output pairs and the strategy used. Activity 1: Guess My Rule, “Try to figure out what’s happening in the ‘black box’. Note: You must hit enter or return before you click GO.” Students enter numbers into column A and a value is given in column B. Activity 2: Making Tables, “For each input-output rule, fill in the table with the outputs that go with a given input. Add two more input-output pairs to the table. a. \frac{3}{4} \rArradd 1 then multiply by 4 \rArr 7 b. \frac{3}{4} \rArrname the digit in the tenths place \rArr 7 c. \frac{3}{4} \rArr write 7 \rArr 7 d. x \rArr divide 1 by the input \rArr \frac{1}{x}." Practice Problems, Problem 2, “Here is an input-output rule. Complete the table for the input-output rule: write 1 if the input is odd; write 0 if the input is even.” Input values given are: -3, -2, -1, 0, 1, 2, 3. Materials present students with extensive work with grade-level problems of 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.)

• Unit 8: Pythagorean Theorem and Irrational Numbers, Section A: Side Lengths and Area of Squares, Lesson 3: Rational and Irrational Numbers, students determine if numbers are rational or irrational. Activity 3: Looking for \sqrt{2}, “A rational number is a fraction or its opposite (or any number equivalent to a fraction or its opposite). a. Find some more rational numbers that are close to \sqrt{2}. b. Can you find a rational number that is exactly \sqrt{2}?” Cool Down: Types of Solutions, “a. In your own words, say what a rational number is. Give at least three different examples of rational numbers. b. In your own words, say what an irrational number is. Give at least two examples.” Practice Problems, Problem 1, “Decide whether each number in this list is rational or irrational. \frac{-13}{3}, 0.1234, \sqrt{37}, -77, -\sqrt{100}, -\sqrt{12}.” Materials present students with extensive work with grade-level problems of 8.NS.1 (Know that numbers that are not rational are called irrational…)

Instructional materials provide opportunities for all students to engage with the full intent of grade-level standards. Examples include: 

• Unit 5: Functions and Volume, Section A: Inputs and Outputs, Lesson 2: Introduction to Functions, students describe functions and identify their rules. Activity 2: Using Function Language, “Here are the questions from the previous activity. For the ones you said yes to, write a statement like, ‘The height a rubber ball bounces depends on the height it was dropped from’ or ‘Bounce height is a function of drop height.’ For all of the ones you said no to, write a statement like, ‘The day of the week does not determine the temperature that day’ or ‘The temperature that day is not a function of the day of the week.’ a. A person is 5.5 feet tall. Do you know their height in inches? b. A number is 5. Do you know its square? c. The square of a number is 16. Do you know the number? d. A square has a perimeter of 12 cm. Do you know its area? e. A rectangle has an area of 16 cm². Do you know its length? f. You are given a number. Do you know the number that is as big? g. You are given a number. Do you know its reciprocal?” Cool Down: Wait Time, “For each statement, if you answer yes, draw an input-output diagram and write a statement that describes the way one quantity depends on another. If you answer no, give an example of 2 outputs that are possible for the same input. You are told that you will have to wait for 5 hours in a line with a group of other people. Determine whether: a. You know the number of minutes you have to wait. b. You know how many people have to wait.” Practice Problems, Problem 2, “A group of students is timed while sprinting 100 meters. Each student’s speed can be found by dividing 100 m by their time. Is each statement true or false? Explain your reasoning. a. Speed is a function of time. b. Time is a function of distance. c. Speed is a function of number of students racing. d. Time is a function of speed.” The materials meet the full intent of 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.)

• Unit 6: Associations in Data, Section B: Associations in Numerical Data, Lesson 5: Describing Trends in Scatter Plots, students draw a linear model to fit data in a scatter plot and describe features of the line. Warm-Up: Which One Doesn’t Belong: Scatter Plots, “Which one doesn’t belong?” Students are shown four scatter plots and justify their reasoning of which one does not belong. Activity 1: Fitting Lines, “Experiment with finding lines to fit the data. Drag the points to move the line. You can close the expressions list by clicking on the double arrow. a. Here is a scatter plot. Experiment with different lines to fit the data. Pick the line that you think best fits the data. Compare it with a partner’s. b. Here is a different scatter plot. Experiment with drawing lines to fit the data. Pick the line that you think best fits the data. Compare it with a partner’s. c. In your own words, describe what makes a line fit a data set well. Activity 3: Practice Fitting Lines, Problems 1 and 2, “1. Is this line a good fit for the data? Explain your reasoning. 2. Draw a line that fits the data better.” The materials meet the full intent of 8.SP.2 (Know that straight lines are widely used to model relationships between two quantitative variables…)

• Unit 8: Pythagorean Theorem and Irrational Numbers, Section B: The Pythagorean Theorem, Lesson 7: A Proof of the Pythagorean Theorem, students calculate the unknown side length of a triangle using the Pythagorean Theorem. Activity 2: Let’s Take It for a Spin, students find the missing lengths of a side of a right triangle. “Find the unknown side lengths in these right triangles.” Students are given a diagram of 2 right triangles. The first triangle has side lengths of 2 and 5. The second triangle has a hypotenuse of 4 and a side length of \sqrt{8}. Activity 3: A Transformational Proof, “Use the applets to explore the relationship between areas. Consider Squares A and B. Check the box to see the area divided into five pieces with a pair of segments. Check the box to see the pieces. Arrange the five pieces to fit inside Square C. Check the box to see the right triangle. a. Arrange the figures so the squares are adjacent to the sides of the triangle. b. If the right triangle has legs a and b and hypotenuse c, what have you just demonstrated to be true? c. Try it again with different squares. Estimate the areas of the new Squares, A, B, and C and explain what you observe. d. Estimate the areas of these new Squares, A, B, and C, and then explain what you observe as you complete the activity. e. What do you think we may be able to conclude?” Cool Down: When Is It True?, “The Pythagorean Theorem is A. True for all triangles B. True for all right triangles C. True for some right triangles D. Never true.” The materials meet the full intent 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.)

The materials reviewed for Open Up Resources Grade 8 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade. 

When implemented as designed, the majority (at least 65%) of the materials, when implemented as designed, address the major clusters of the grade. Examples include:

• The approximate number of units devoted to major work of the grade (including assessments and supporting work connected to the major work) is 7 out of 8, approximately 88%.

• The number of lessons devoted to major work of the grade, including supporting work connected to major work is 101 out of 114, approximately 89%. 

• The number of instructional days devoted to major work of the grade and supporting work connected to major work (includes required lessons and assessments) is 108 out of 122, approximately 89%.

An instructional day analysis is most representative of the materials, including the required lessons and End-of-Unit Assessments from the required Units. As a result, approximately 89% of materials focus on major work of the grade.

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

Each lesson contains Learning Targets that provide descriptions of what students should be able to do after completing the lesson. Standards being addressed are identified and defined. Materials connect learning of supporting and major work to enhance focus on major work. Examples include:

• Unit 5: Functions and Volume, Section E: Dimensions and Spheres, Lesson 17: Scaling One Dimension, Activity 2: Halve the Height, connects the supporting work of 8.G.9 (Know the formulas for the volumes of cones, cylinders and spheres) to the major work of 8.F.1 (Understand that a function is a rule that assigns to each input exactly one output). Students continue their work with functions to investigate what happens to the volume of a cylinder when you halve the height, “There are many cylinders with radius 5 units. Let h represent the height and V represent volume of these cylinders. a. Write an equation that represents the relationship between V and h. Use 3.14 as an approximation of π. b. Graph this equation and label the axes (use applet in presentation mode). c. What happens to the volume if you halve the height, h? Where can you see this in the graph? How can you see it algebraically?”

• Unit 8: Pythagorean Theorem and Irrational Numbers, Section A: Side Lengths and Areas of Squares, Lesson 2: Side Lengths and Areas, Activity 1: One Square, connects the supporting work 8.NS.A (Know that there are numbers that are not rational, and approximate them by rational numbers) to the major work of 8.EE.2 (Uses square root and cube root symbols to represent solutions to equations of the form x^2 = p, and x^3 = p where p is a positive rational number) and 8.F.B (Use functions to model relationships between quantities). Students estimate the side length of a square using a geometric construction that relates the side length of the square to a point on the number line, “Use the circle to estimate the area of the square shown here.” Students are shown a coordinate plane with a circle. Covering part of the circle is a square with one of the vertices on the origin.  

• Unit 9: Putting It All Together, Section B: The Weather, Lesson 4: What Influences Temperature, Activity 3: Is There an Association Between Latitude and Temperature? connects the supporting work of 8.SP.A (Investigate patterns of association in bivariate data)to the major work of 8.F.1 (Understand that a function is a rule that assigns to each input exactly one output) and 8.F.B (Use functions to model relationships between quantities). Students collect and analyze data to determine if there is a relationship between latitude and temperature, “Lin and Andre decided that modeling temperatures as a function of latitude doesn’t really make sense. They realized that they can ask whether there is an association between latitude and temperature. a. What information could they gather to determine whether temperature is related to latitude? b. What should they do with that information to answer the question?”

The materials reviewed for Open Up Resources Grade 8 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

Each lesson contains Learning Targets that describe what the students should be able to do after completing the lesson. Standards being addressed are identified and defined.

Materials connect major work to major work throughout the grade level when appropriate. Examples include:

• Unit 2: Dilations, Similarity, and Introducing Slope, Section C: Slope, Lesson 11: Writing Equations for Lines, Activity 1: What We Mean By an Equation of a Line, connects the major work of 8.EE.B (Understand the connections between proportional relationships, lines, and linear equations.) to the major work of 8.G.A (Understand congruence and similarity using physical models, transparencies, or geometry software.) Students use the structure of a right triangle to examine the coordinates of points lying on a particular line and produce a linear equation, “Line j is shown in the coordinate plane. a. What are the coordinates of B and D? b. Is point (20, 15) on line j? Explain how you know. c. Is point (100, 75) on line j? Explain how you know. d. Is point (90, 68) on line j? Explain how you know. e. Suppose you know the x and y coordinates of a point. Write a rule that would allow you to test whether the point is on line j.” 

• Unit 3: Linear Relationships, Section B: Representing Linear Relationships, Lesson 8: Translating to y = mx + b, Activity 1: Increased Savings, Problem 4, connects the major work of 8.EE.B (Understand the connections between proportional relationships, lines, and linear equations.) to the major work of 8.G.A (Understand congruence and similarity using physical models, transparencies, or geometry software.) Students find slopes and y-intercepts and write equations for lines using y = mx + b. “Write an equation for each line. a. Diego earns $10 per hour babysitting. Assume that he has no money saved before he starts babysitting and plans to save all of his earnings. Graph how much money, y, he has after x hours of babysitting. b. Now imagine that Diego started with $30 saved before he starts babysitting. On the same set of axes, graph how much money, y, he would have after x hours of babysitting. c. Compare the second line with the first line. How much more money does Diego have after 1 hour of babysitting? 2 hours? 5 hours? x hours? d. Write an equation for each line.” Students use an applet in presentation mode.

• Unit 8: Pythagorean Theorem and Irrational Numbers, Section A: Side Lengths and Areas of Squares, Lesson 2: Side Lengths and Areas, Activity 2: The Sides and Areas of Tilted Squares, Problem 1, connects the major work of 8.EE.A (Work with radicals and integer exponents.) to the major work of 8.F.B (Use functions to model relationships between quantities.) Students find the areas of three squares, estimate and find the exact side lengths, make a table of side-area pairs, and graph the ordered pairs. “Find the area of each square and estimate the side lengths using your geometry toolkit. Then write the exact lengths for the sides of each square.” Three squares on grid paper are shown.

Materials provide connections from supporting work to supporting work throughout the grade-level when appropriate. Examples include:

• Unit 5: Functions and Volume, Section D: Cylinders and Cones, Lesson 16: Finding Cone Dimensions, Activity 3: Popcorn Deals, connects the supporting work of 8.NS.A (Know that there are numbers that are not rational, and approximate them by rational numbers.) to the supporting work of 8.G.C (Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.) Students reason about the volume of popcorn that a cone-shaped and cylinder-shaped popcorn cup holds and the price they pay for it, “A movie theater offers two containers: Which container is the better value? Use 3.14 as an approximation for π.” Pictured is a cone-shaped cup with a diameter of 12 cm and height of 19 cm costing $6.75 and a cylinder-shaped cup with a diameter of 8 cm and height of 15 cm costing $6.25.

• Unit 5: Functions and Volume, Section E: Dimensions and Spheres, Lesson 20: The Volume of Spheres, Cool Down: Volume of Spheres, connects the supporting work of 8.NS.A (Know that there are numbers that are not rational, and approximate them by rational numbers.) to the supporting work of 8.G.C (Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.) Students calculate the volume of a sphere, “Recall that the volume of a sphere is given by the formula V = \frac{4}{3}π$$r^3$$. a. Here is a sphere with a radius 4 feet. What is the volume of the sphere? Express your answer in terms of π. b. A spherical balloon has a diameter of 4 feet. Approximate how many cubic feet of air this balloon holds. Use 3.14 as an approximation for π, and give a numerical answer.”

The materials reviewed for Open-Up Resources 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.

The Course Guide contains a Scope and Sequence explaining content standard connections. Some Unit Overviews, Lesson Narratives, and Activity Syntheses describe the progression of standards for the concept being taught. Each Lesson contains a Preparation section identifying learning standards (Building on, Addressing, or Building toward). Content from future grades is identified and related to grade-level work. Examples include:

• Unit 1: Rigid Transformations and Congruence, Section C: Congruence, Lesson 11: What Is the Same?, “The material treated here will be taken up again in high school (G-CO.B) from a more abstract point of view. In grade 8, it is essential for students to gain experience executing rigid motions with a variety of tools (tracing paper, coordinates, technology) to develop the intuition that they will need when they study these moves (or transformations) in greater depth later.”

• Unit 2: Dilations, Similarity, and Introducing Slope, Section B: Similarity, Lesson 8: Similar Triangles, Lesson Narrative, “Students will use the similarity criterion in future lessons to understand the concept of the slope of a line. Later on in high school, they will learn that three proportional sides (but not two) is also enough to deduce that two triangles are similar.”

• Unit 7: Exponents and Scientific Notation, Section B: Exponent Rules, Lesson 5: Negative Exponents with Powers of 10, Lesson Narrative, “Sometimes in mathematics, extending existing theories to areas outside of the original definition leads to new insights and new ways of thinking. Students practice this here by extending the rules they have developed for working with powers to a new situation with negative exponents. The challenge then becomes to make sense of what negative exponents might mean. This type of reasoning appears again in high school when students extend the rules of exponents to make sense of exponents that are not integers.”

Materials relate grade-level concepts explicitly to prior knowledge from earlier grades. Examples include:

• Course Guide, Scope and Sequence, Unit 2: Dilations, Similarity, and Introducing Slope, “Work with transformations of plane figures in grade 8 builds on earlier work with geometry and geometric measurement, using students’ familiarity with geometric figures, their knowledge of formulas for the areas of rectangles, parallelograms, and triangles, and their abilities to use rulers and protractors. Grade 7 work with scaled copies is especially relevant. This work was limited to pairs of figures with the same rotation and mirror orientations (i.e. that are not rotations or reflections of each other). In grade 8, students study pairs of scaled copies that have different rotations or mirror orientations, examining how one member of the pair can be transformed into the other, and describing these transformations. Initially, they view transformations as moving one figure in the plane onto another figure in the plane. As the unit progresses, they come to view transformations as moving the entire plane.”

• Unit 6: Associations in Data, Section A: Does This Predict That?, Lesson 2: Plotting Data, Building on, students use previous learning in 6th grade “6.SP.B.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots, ”as they represented the distribution of single statistical variables using dot plots, histograms, and box plots to now construct and interpret scatter plots in 8th grade. Previous learning includes 

• Unit 8: Pythagorean and Irrational Numbers, Section A: Side Lengths and Areas of Squares, Lesson 1: The Areas of Squares and Their Side Lengths, Lesson Narrative, “Students know from work in previous grades how to find the area of a square given the side length. In this lesson, we lay the groundwork for thinking in the other direction: if we know the area of the square, what is the side length? Before students define this relationship formally in the next lesson, they estimate side lengths of squares with known areas using tools such as rulers and tracing paper (MP5). They also review key strategies for finding area that they encountered in earlier grades that they will use to understand and explain informal proofs of the Pythagorean Theorem.”