8th Grade - Gateway 1

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

The curriculum consists of nine units, including one optional unit. Assessments include Cool-down Tasks, Check your Readiness, Mid-Unit Assessments, and End-of-Unit Assessments. Examples of assessment items aligned to grade-level standards include:

• Unit 2, End-of-Unit Assessment (A), Question 2, “Which pair of triangles must be similar? A. Triangles 1 and 2 each have a 35° angle. B. Triangles 3 and 4 are both isosceles. They each have a 40° angle. C. Triangle 5 has a 30° angle and a 90° angle. Triangle 6 has a 30° angle and a 70° angle. D. Triangle 7 has a 50° angle and a 25° angle. Triangle 8 has a 50°angle and a 105° angle.” (8.G.5)

• Unit 3, Lesson 8, Cool-down, “Describe how the graph of y = 2x is the same and different from the graph of y=2x-7. Explain or show your reasoning.” (8.EE.5)

• Unit 5, Mid-Unit Assessment (A), Question 1, “Select all the functions whose graphs include the point (16,4). A. y=2x; B. y=x^2; C. y=x+12; D. y=x-12; E. y=\frac{1}{4}x.” (8.F.1)

The materials reviewed for Math Nation 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 present opportunities for students to engage in extensive work and the full intent of most Grade 8 standards. Each lesson contains a Warm-Up, a minimum of one Exploration Activity, a Lesson Summary, Practice Problems, and three Check Your Understanding Questions. Each unit provides a Readiness Check and a Test Yourself! practice tool.  Examples of full intent include:

• Unit 2, Lesson 6, 2.6.5 Exploration Activity, engages students with the full intent of 8.G.4 (Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them). Students collaborate to determine multiple ways to show similarity. “Your teacher will give you a set of five cards and your partner a different set of five cards. Using only the cards you were given, find at least one way to show that triangle ABC and triangle DEF are similar. Compare your method with your partner’s method. What is the same about your methods? What is different?”

• Unit 5, Lesson 4, 5.4.2 Exploration Activity, engages students with 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). Students match three graphs to three different equations/scenarios. Students further manipulate the graphs that represent functions.  For example, after students match three graphs to three equations/scenarios, students complete the following for each graph that is a function: “2. Label each of the axes with the independent and dependent variables and the quantities they represent. 3. For each function: What is the output when the input is 1? What does this tell you about the situation? Label the corresponding point on the graph.

4. Find two more input-output pairs. What do they tell you about this situation? Label the corresponding points on the graph.”

• Unit 7, Lesson 6, 7.6.6 Practice Problems, Question 2, engages students with the full intent of 8.EE.1 (Know and apply the properties of integer exponents to generate equivalent numerical expressions). Students chose expression(s) that are equivalent to the exponent provided. “Select all the expressions that are equivalent to 4^{-3}.” The choices are the following -12; 2^{-6}; \frac{1}{4^3} ; (\frac{1}{4})\cdot(\frac{1}{4})\cdot(\frac{1}{4}); 12, (-4)\cdot(-4)\cdot(-4); \frac{8^{-1}}{2^2}.”

The materials present opportunities for students to engage with extensive work with grade-level problems. Examples of extensive work include:

• Unit 3, Lesson 3, Exploration Activity, Practice Problems, and Check Your Understanding, engage students with extensive work of 8.EE.5 (Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways). Lesson 3, 3.3.2 Exploration Activity, Question 2, students create a table of values, graph the ordered pairs, identify/calculate the constant of proportionality, and synthesize the results. “Here are two ways to represent a situation.  Description: The Origami Club is doing a carwash fundraiser to raise money for a trip. They charge the same price for every car. After 11 cars, they raised a total of $93.50. After 23 cars, they raised a total of $195.50. a. Write an equation that represents this situation. (Use c to represent number of cars and use m to represent amount raised in dollars.) b. Create a graph that represents this situation. c.  How can you see or calculate the constant of proportionality in each representation?  What does it mean?” A table is provided that gives the same information as the description in a table format. In 3.3.6 Practice Problems, Question 1, students use a graph to determine proportional relationships and write equations to complete the table. “Here is a graph of the proportional relationship between calories and grams of fish (graph given). a. Write an equation that represents this relationship using x to represent the amount of fish in grams and y to represent the number of calories. b. Use your equation to complete the table (table with either grams of fish or number of calories filled in and students find the corresponding missing value).” In 3.3.8 Check Your Understanding, Question 3, students use linear equations to determine rate of change.   “Carson is filling up his car with gas before going on a road trip. The total cost, y, has a linear relationship with the gallons of gasoline, x. The results are shown on the coordinate grid (graph is given with 3 coordinate points labeled). The rate of change for the linear relationship is ___ .”

• Unit 5, Lesson 13, Lesson 15, and Lesson 20, engage students with extensive 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). Lesson 13, 5.13.8 Practice Problems, Question 2, students use three-dimensional objects to solve real-world problems. “At a farm, animals are fed bales of hay and buckets of grain. Each bale of hay is in the shape a rectangular prism. The base has side lengths 2 feet and 3 feet, and the height is 5 feet. Each bucket of grain is a cylinder with a diameter of 3 feet. The height of the bucket is 5 feet, the same as the height of the bale. a. Which is larger in area, the rectangular base of the bale or the circular base of the bucket? Explain how you know. b. Which is larger in volume, the bale or the bucket? Explain how you know.” Lesson 15, 5.15.2 Exploration Activity, students use equations and real-world objects to find the volume of three-dimensional objects. “A cone and cylinder have the same height and their bases are congruent circles (image given). 1. If the volume of the cylinder is 90 cm^3, what is the volume of the cone? 2. If the volume of the cone is 120 cm^3 , what is the volume of the cylinder? 3. If the volume of the cylinder is V=\pi r^2 h, what is the volume of the cone? Either write an expression for the cone or explain the relationship in words.” Lesson 20, 5.20.6 Check Your Understanding, Question 3, students calculate the volume of a sphere in a real-world context. “Jaylene is purchasing golf balls for her father’s birthday. Each ball has a diameter of 42 millimeters (mm). She decides to purchase 10 golf balls for her father. Using 3.14 for , what is the total volume of all 10 golf balls? (A) 3,101,817.6 mm^3  (B) 387,727.2 mm^3   (C) 221,558.4 mm^3  (D) 55,389.6 mm^3.”

• Unit 8, Lesson 13, Exploration Activities and Practice Problems, engage students with extensive work of 8.NS.2 (Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions). In 8.13.2 Exploration Activity, students find which two whole numbers cube roots fall between. “What two whole numbers does each cube root lie between? Be prepared to explain your reasoning. 1. \sqrt[3]{5}   2. \sqrt[3]{23}   3. \sqrt[3]{81}   4. \sqrt[3]{999} “ In 8.13.3 Exploration Activity, students work on placing rational and irrational cube roots on a number line. “The numbers x, y, and z are positive, and: x^3=5, y^3=27, z^3=700 1. Plot x, y, and z on the number line. Be prepared to share your reasoning with the class. 2. Plot -\sqrt[3]{2} on the number line.” In 8.13.6 Practice Problems, Question 3, students use rational approximations of irrational numbers to order rational and irrational numbers from least to greatest. “Order the following values from least to greatest: \sqrt[3]{530}, 48, \pi, \sqrt{121}, \sqrt[3]{27}, \frac{19}{2}.”

The materials do not provide opportunities for students to meet the full intent of the following standard:

• While students engage with 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 have no opportunities to apply the Pythagorean Theorem to determine unknown side lengths in right triangles of three dimensions to meet the full intent of the grade-level standards.

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

To determine the amount of time spent on major work, the number of units, the number of lessons, and the number of days were examined. Assessment days are included. Any lesson, assessment, or unit marked optional was excluded.

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

• The approximate number of lessons devoted to major work (including assessments and supporting work connected to the major work) is 112 out of 141, which is approximately 79%. 

• The approximate number of days devoted to major work of the grade (including supporting work connected to the major work) is 112 out of 141, which is approximately 79%.

A lesson-level analysis is most representative of the materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each unit. As a result, approximately 79% of the materials focus on major work of the grade.

The materials reviewed for Math Nation 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 to connect supporting standards/clusters to the grade’s major standards/clusters. Examples of connections include:

• Unit 5, Lesson 15, 5.15.3 Exploration Activity, connects the supporting work of 8.G.9 (Know the formulas for the volume of cones, cylinders, and spheres and use them to solve real-world and mathematical problems) to the major work of 8.EE.7 (Solve linear equations in one variable). Students solve equations to find the volume of three-dimensional shapes. “1. Here is a cylinder and cone that have the same height and the same base area. What is the volume of each figure? Express your answer in terms of \pi. 2. Here is a cone. a. What is the area of the base? Express your answer in terms of \pi. b. What is the volume of the cone? Express your answer in terms of \pi. 3. A cone-shaped popcorn cup has a radius of 5 centimeters and a height of 9 centimeters. How many cubic centimeters of popcorn can the cup hold? Use 3.14 as an approximation for \pi, and give a numerical answer.” Students are provided two figures for Question 1, a cylinder and cone with a height of 4 and a diameter of 10. For Question 2, students are provided a figure of a cone with a height of 8 and a radius of 6.

• Unit 6, Lesson 6, 6.6.5 Exploration Activity, connects the supporting work of 8.SP.1 (Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association) to the major work of 8.F.4 (Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values). Students look at graphs of scatter plots to identify the association. “1. For each of the scatter plots, decide whether it makes sense to fit a linear model to the data. If it does, would the graph of the model have a positive slope, a negative slope, or a slope of zero? 2. Which of these scatter plots show evidence of positive association between the variables? Of a negative association? Which do not appear to show an association?”

• Unit 8, Lesson 4, 8.4.6 Practice Problems, Question 3, connects the supporting work of 8.NS.2 (Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions) to the major work of 8.EE.2 (Use 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. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that \sqrt{2} is irrational). Students use the solution of an equation to find the irrational number approximation. “Use the fact that \sqrt{7} is a solution to the equation x^2= 7 to find a decimal approximation of \sqrt{7} whose square is between 6.9 and 7.1.”

• Unit 8, Lesson 5, 8.5.4 Exploration Activity, connects the supporting work of 8.NS.2 (Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions) to the major work of 8.EE.2 (Use 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. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational). Students plot the solutions to square root equations on a number line.  “The numbers 𝑥, 𝑦, and 𝑧 are positive, and  x^2=3,  y^2=16, and z^2=30. 1. Plot 𝑥, 𝑦, and 𝑧 on the number line. Be prepared to share your reasoning with the class. 2. Plot -\sqrt{2} on the number line.”

The materials reviewed for Math Nation 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.

There are connections from supporting work to supporting work and/or major work to major work throughout the grade-level materials, when appropriate. Examples include:

• Unit 2, Lesson 11, 2.11.3 Exploration Activity, connects the major work of 8.EE.B (Understand the connections between proportional relationships, lines, and linear equations) with the major work of 8.G.A (understand congruence and similarity using physical models, transparencies, or geometry software). Students work with a proportional relationship stemming from similar triangles. “Here are two diagrams: 1. Complete each diagram so that all vertical and horizontal segments have expressions for their lengths. 2. Use what you know about similar triangles to find an equation for the quotient of the vertical and horizontal side lengths of \triangleDFE in each diagram.” Students are given two diagrams of lines with slope triangles drawn.

• Unit 3, Lesson 12, 3.12.2 Exploration Activity, connects the major work of 8.EE.B (Understand the connections between proportional relationships, lines, and linear equations) to the major work of 8.EE.C (Analyze and solve linear equations and pairs of simultaneous linear equations). Students identify a proportional relationship to write a linear equation and determine solutions and non-solutions to the equation. “At the corner produce market, apples cost $1 each and oranges cost $2 each. 1. Find the cost of: a. 6 apples and 3 oranges; b. 4 apples and 4 oranges; c. 5 apples and 4 oranges; d. 8 apples and 2 oranges; 2. Noah has $10 to spend at the produce market. Can he buy 7 apples and 2 oranges? Explain or show your reasoning. 3. What combinations of apples and oranges can Noah buy if he spends all of his $10? 4. Use two variables to write an equation that represents 10-combinations of apples and oranges. Be sure to say what each variable means. 5. What are 3 combinations of apples and oranges that make your equation true? What are three combinations of apples and oranges that make it false?”

• Unit 5, Lesson 14, 5.14.6 Practice Problems, Question 3, 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 use their knowledge of volume to find the radius of cylinders. “Three cylinders have a volume of 2826 cm^3. Cylinder A has a height of 900 cm. Cylinder B has a height of 225 cm. Cylinder C has a height of 100 cm. Find the radius of each cylinder. Use 3.14 as an approximation for \pi.”

• Unit 5, Lesson 17, 5.17.2 Exploration Activity, 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 write and graph an equation that shows the relationship between the side length and volume of rectangular prisms. “There are many right rectangular prisms with one side of length 5 units and another side of length 3 units. Let s represent the length of the third side and V represent the volume of these prisms. 1. Write an equation that represents the relationship between V and s. 2. Graph this equation. 3. What happens to the volume if you double the edge length, s? Where do you see this in the graph? Where do you see it algebraically?” A Desmos applet is provided for students to graph the equation on.

The following connections are entirely absent from the materials:

• No connections are made between the supporting work of 8.SP.A (Investigate patterns of association in bivariate data) and any other supporting work but this is mathematically reasonable.

The materials reviewed for Math Nation 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.

Grade-level concepts related explicitly to prior knowledge from earlier grades along with content from future grades is identified and related to grade-level work in the Teacher Edition. Generally, explicit connections are found in the Course Guide or the Full Lesson Plan.  

Examples of connections to future grades include:

• Unit 1, Lesson 11, Full Lesson Plan, Lesson Narrative, connects 8.G.2 to G-CO.B. ​​“The definition of congruence here states that two shapes are congruent if there is a sequence of translations, rotations, and reflections that matches one shape up exactly with the other…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 they will need when they study these moves (or transformations) in greater depth later.” 

• Unit 4, Lesson 15, Full Lesson Plan, 4.15.3 Activity Synthesis, connects 8.EE.8b to A-REI.C. “There are two key takeaways from this discussion. The first is to reinforce that some systems can be solved by reasoning whether its possible for a solution to one equation to also be a solution to another. The second takeaway is that there are some systems that students will only be able to solve after learning techniques in future grades. For each problem, ask students to indicate if they identified the system as least or most difficult… Tell students that this system has one solution and they will learn more sophisticated techniques for solving systems of equations in future grades.”

• Unit 7, Lesson 5, Full Lesson Plan, Lesson Narrative, connects 8.EE.1 to N-RN.1. “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.”

Examples of connections to prior knowledge include:

• Unit 2, Lesson 1, Full Lesson Plan, 2.1.2 Classroom activity, builds on 7.G.1 and connects to 8.G.A. “This activity recalls work from grade 7 on scaled copies, purposefully arranging a set of scaled copies to prepare students to understand the process of dilation…Afterward, during the discussion, the word dilation is first used in an informal way, as a way to make scaled copies (of the rectangle in this case). From this point of view, the shared vertex of each set of rectangles is the center of dilation and once we choose an original rectangle from each set, there is a scale factor associated to each copy, namely the scale factor needed to produce the copy from the original…”

• Unit 4, Lesson 2, Full Lesson Plan, Lesson Narrative, builds on 6.EE.6, 6.EE.7, 7.EE.4a and connects to 8.EE.C. “This lesson is the first in a sequence of eight lessons where students learn to work with equations that have variables on each side. In this lesson, students recall a representation they have seen in prior grades: the balanced hanger. The hanger is balanced because the total weight on each side, hanging at the same distance from the center, is equal in measure to the total weight on the other side…students encounter a hanger with an unknown weight that cannot be determined. This situation parallels the situation of an equation where the variable can take on any value and the equation will always be true, which is a topic explored in more depth in later lessons.”

• Unit 6, Lesson 2, Full Lesson Plan, Lesson Narrative, builds on 6.SP.4 and connects to 8.SP.1. “In prior grades, students represented the distribution of a single statistical variable using dot plots, histograms, and box plots. In this lesson, students review those ways of displaying data and compare them with representing the relationship between two variables in a scatter plot…They notice that scatter plots can convey information about the relationship between two variables that representations of each variable separately do not reveal. They understand that every point in a scatter plot represents two measures of a single individual in the population.”