8th Grade - Gateway 2

The instructional materials reviewed for CK-12 Interactive Middle School Math 8 for CCSS partially meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics. The materials include multiple opportunities for students to engage in routine application of grade-level skills and knowledge, within instruction and independently. The materials include one non-routine application problem within instruction, but students do not demonstrate independent application of mathematics in non-routine situations.

Examples of students engaging in routine application of grade-level skills and knowledge, within instruction and independently, include:

• In Chapter 3, students write and solve linear equations in real-world contexts (8.EE.7). For example, in Lesson 3.3, Activity 3, a Supplemental Question states, “The material cost for each ingredient in the fudge at Empire fudge can be seen in the table. What are some ways to reduce the total material cost per pound from $3.10/lb to $2.83/lb?” Also, in Lesson 3.7, Review Question 8 states, “The rent for a Toyota car contains two parts, one of which is a fixed charge of 100 dollars and the other is 50 dollars for each day. Which equation correctly models the total cost of renting a Toyota car for x number of days?”
• In Chapter 5, students solve problems using systems of linear equations in two variables (8.EE.8c). For example, in Lesson 5.6, Review Question 1 states, “Terry has 25 bills. He has ten-dollar bills and fifty-dollar bills only. If he has $770, how many ten-dollar bills does he have?”, and Review Question 3 states, “Five years from now, a man’s age will be three times his son’s age and five years ago, he was seven times as old as his son. Find the present ages of father and son respectively.”
• In Chapter 7, students construct a function to model a linear relationship between two quantities (8.F.4). For example, in Lesson 7.4, Activity 2, Inline Question 1 states, “Rebecca earns $17 per hour at her new job. Which equation describes the total amount of money earned as a function of time, x?” Also, in Lesson 7.8, a Review Question states, “The cost of producing a smartphone is $42.73 per phone. Additionally, the smartphone company pays a flat rate of $175 to ship each store’s order. Write an equation to model the costs for the smartphone company, y, to produce x number of smartphones and ship them to one store.”
• In Chapter 8, students know and use volume formulas to solve problems (8.G.9). For example, in Lesson 8.8, Review Question 4 states, “A shipping box measures 16 inches by 12 inches by 8 inches. A second box has a similar shape but each dimension is $$\frac{3}{4}$$ as long. How does the volume of the second box compare to the volume of the shipping box?”

The non-routine application problem within instruction is in Chapter 7. In Lesson 7.8, Activity 1, the Discussion Question states, “Think of a real-world example for a nonlinear relationship. Explain how you know the relationship is not linear using a table and graph.”

The instructional materials reviewed for CK-12 Interactive Middle School Math 8 for CCSS partially meet expectations for identifying and using the Standards for Mathematical Practice (MPs) to enrich mathematics content within and throughout the grade-level. The materials state that teachers should use a few MPs in each lesson, but each lesson does not include guidance on which MPs to use. MPs are explicitly identified in the Teacher Notes, but MPs are identified once in Chapter 7 and twice in Chapter 5.

Examples of the materials identifying and using the MPs to enrich the mathematics content include:

• MP1: In Lesson 4.1, the Introduction includes, “In this lesson, students begin graphing relationships when given a verbal description, and then they move on to graphing from a table (MP1).” 
• MP2: Lesson 2.3 states, “In this lesson, angle measures are in the form of algebraic expressions (MP2). Students work out problems with parallel lines, triangles, and transversals. Students should use facts about all the angles formed when parallel lines are cut by a transversal. Additionally, students should use facts about angles that they learned in Grade 7, such as vertical, complementary, and supplementary angles.”
• MP4: Lesson 8.6 states, “The interactive shows that the volume of the cylinder contains the volume of the cone and the sphere. Students should already know that the volume of a cone is $$\frac{1}{3}$$ the volume of the cylinder (see Volume of Cones). After this, you can move to helping them derive the equation. From there, students use the formula to calculate the volume of spheres in real-world and mathematical problems (MP4).”
• MP5: Lesson 8.1 states, “Interactives and questions should help students derive the formula for the volume of a prism (MP5).”
• MP6: Lesson 4.8 states, “This lesson focuses on how an equation in the form y = mx+b is a translation of the equation y = mx. Furthermore, two lines with the same slope but different y-intercepts, are also translations of each other. Attend to precision when discussing and defining b; b is not the y-intercept (MP6). Rather, b is the y-coordinate of the y-intercept. Students must understand that the x-coordinate of the y-intercept is always 0.”
• MP7: Lesson 8.6 states, “In this lesson, interactives and questions help students understand the relationship between the volume of the cylinder and the sphere. They will look at a sphere inside a cylinder and a cone (all with the same radius and height). The questions help students see which dimensions (of the three shapes) are the same and get to the idea of the ratio relationship (MP7).”
• MP8: This MP was identified in one lesson. Lesson 9.8 states, “This lesson also includes some problems that use addition/subtraction as a way to revisit the preceding lesson (MP8).”

There are some instances in which the Mathematical Practices are labeled but do not enrich the content. For example, in Lesson 1.5, MP6 is identified in the Teacher Notes with, “Congruence is defined and students connect corresponding sides to congruent sides (MP6).” While congruence is defined, students are not prompted to use precise language and definitions.

The instructional materials reviewed for CK-12 Interactive Middle School Math 8 for CCSS partially meet expectations for carefully attending to the full meaning of each practice standard. The materials do not attend to the full meaning of two MPs.

Examples of the materials not attending to the full meaning of MP5 include, but are not limited to:

• In Lesson 8.1, Teacher Notes, “This lesson begins with accessing prior knowledge about volume of prisms. Students should know from past experience that volumes of prisms are found by multiplying the area of the base by the height. They should make connections from these learnings to intuiting how the volume of a cylinder would be found. Interactives and questions should help students derive the formula for the volume of a prism (MP5). Once the formula is established, students should practice finding the volume in mathematical and real examples.” Students do not select which tools to use as they are provided.
• In Lesson 8.6, Activity 1, the materials include, “As with the other shapes, you are going to have to be clever to determine how many cubic units can fit in a sphere. You can use the cube’s relation to a circumscribed cylinder to determine its volume. Circumscribed means constructed around a shape touching as many points as possible. Use the interactive below to explore this relationship.” Students do not select a tool for this investigation.

Examples of the materials not attending to the full meaning of MP8 include, but are not limited to:

• In Lesson 9.8, Teacher Notes, “This lesson is similar to the last one in that students must attend to precision with units in this lesson (MP6). They work through multi-step word problems that involve multiple operations (MP4). This lesson also includes some problems that use addition/subtraction as a way to revisit the preceding lesson (MP8).” Students do not express regularity in repeated reasoning to develop rules for multiplying and dividing numbers written in scientific notation as the rules are provided.
• In Lesson 10.4, Activity 1, students read, “As you saw above, an irrational number is a number which cannot be written as a fraction. While you cannot write a fraction or decimal to express the number, you can estimate it’s value. Use the interactive below to explore this idea.” Students also answer the following four Inline Questions: “1. The square root of 71 is between what two whole numbers?; 2. The square root of 20 is between what two whole numbers?; 3. Choose the FALSE statements. (1) The square root of 33 us closer to 5 than 6. (2) The square root of 68 is closer to 9 than 8; and 4. Fill in the Blank: Since the square root of 100 is 10, the square root of 101 will be a (little or lot) larger than 10.” In Activity 2, “Irrational numbers have non-terminating and non-repeating decimals. Even though you cannot write non-terminating decimals in their entirety, you can place them on a number line. In the case of irrational roots, you can use perfect squares to help narrow down the location of the irrational value on a number line. Example: The $$\sqrt{95}$$ is between what two whole numbers? The largest square root that is less than 95 is 81. The smallest square root that is greater than 95 is 100. Since $$\sqrt{81}=9$$ and $$\sqrt{100}=10$$, you can say that the $$\sqrt{95}$$ is between 9 and 10. Use the interactive below to practice this strategy for approximating irrational numbers.” The materials present continuous guidance, and students do not look for and express regularity in repeated reasoning.

The instructional materials reviewed for CK-12 Interactive Middle School Math 8 for CCSS partially meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

The materials provide opportunities for the students to construct arguments about the content, and examples include:

• In Lesson 1.6, Activity 2, Discussion Question, students identify if the images are congruent and construct an argument on why that was their answer. The Question states, “Is the image produced by a series of rigid motions congruent to its pre-image? How do you know?”
• In Lesson 7.8, Activity 1, Discussion Question, students use their understanding of linear relationships to construct an argument about a real-world example of a nonlinear relationship. The question states, “Think of a real-world example for a nonlinear relationship. Explain how you know the relationship is not linear using a table and graph.”

There are no opportunities for students to analyze the arguments of others, and examples include, but are not limited to:

• In Lesson 5.5, Warm Up, the materials include, “There are three methods for solving systems of equations: graphing, substitution, and elimination. Take a break from solving and discuss when each method is most efficient.” Students are not prompted to provide feedback or analyze the arguments of their classmates during the discussion.
• In Lesson 8.2, Activity 3, students discuss their design of an energy drink with the class. However, the materials do not state for students to provide feedback or analyze the arguments of others. The question states, “Discuss your design with your classmates or post in the cafe! How did you decide on your final dimensions?”
• In Lesson 10.12, Activity 1, Challenge Question, students construct an argument to answer, “Mathematically, $$0.\bar{9}=1.0$$. How could you prove this? Join the discussion in the Math Cafe!” The materials do not state for students to provide feedback or analyze the arguments of others.

The instructional materials reviewed for CK-12 Interactive Middle School Math 8 for CCSS partially meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. The materials include some examples of assisting teachers in engaging students to construct viable arguments and analyze the arguments of others, but there are also multiple instances where the materials do not assist teachers.

Examples of the materials assisting teachers to engage students in constructing and/or analyzing the arguments of others include:

• In Lesson, 1.10, Activity 1, the Teacher Notes state, “Allow students time to explore the interactive. Ask them to make a table to catalog 90°, 180°, or 270° both clockwise and counterclockwise. They should list the starting coordinate, the rotated coordinate, and what they believe the rule is. Have students share their ideas on what the rules are. The rules can be found in the next section. Why do you think a 90° clockwise rotation and a 270° counterclockwise rotation have the same rule? They both take the point to the same location going in different directions. They are equivalent rotations. What other rotations are equivalent rotations? A 90° counterclockwise rotation and a 270° clockwise rotation. Also a 180° clockwise rotation and a 180° counterclockwise rotation.”
• In Lesson 3.5, Activity 2, Discussion Question, the Teacher Notes state, “Be sure to stress the connection to combining variables on both sides. Students could simply add revenue, $15, and cost, $6.25, in this case; they would need to subtract, make the expense negative or something along these lines. How much revenue will Jared earn if he sells 216 posters? What will his profit be? His revenue will be 216⋅15 = $3,240 and his profit will be $0. He will need to sell more than 216 to make a profit. He will make $8.75 profit for each poster he sells after the 216th poster. Do you think Jared should raise the sales price, lower the sales price, or that he won’t be able to make a profit at any sales price? Answers may vary. There is no right answer. The goal is to encourage students to discuss and understand the effect of a change in sales price on the solution to the equation and its general context.”
• In Lesson 5.1, Activity 3, Take Action, the Teacher Notes state, “Now you can answer the question "What are the financial benefits of going to college?" Answers may vary. Allow students to discuss their findings (MP3, MP4). The equations are $$y = 34,700x + 138,800$$ for the high school student and $$y = 62,000x - 128,000$$ for the college student.”

Examples where the materials do not assist teachers to engage students in constructing and/or analyzing the arguments of others include:

• In Lesson 2.2, Activity 3, Discussion Question, the Teacher Notes state, “Have students discuss their observations from the interactive. Emphasize that the exterior angles of any triangle appear to form a circle. Remind students that there are 360 degrees in a circle, which means the exterior angles of the triangle always sum to 360 degrees.” The materials encourage discussion among students, but they do not assist teachers in having students analyze the arguments of others.
• In Lesson 4.9, Activity 1, Supplemental Question, the Teacher Notes indicate that the question has multiple possible answers about situations with negative slopes, but there is no assistance as to how students construct an argument. The Teacher Notes state, “What are some situations that a negative slope could represent? Answers may vary: it could represent a falling rate like the amount of water in a water tank that is being drained. Any situation where a quantity is decreasing at a constant rate should be acceptable.”
• In Lesson 8.7, Activity 3, Supplemental Questions, the Teacher Notes provide questions to ask the students with specific answers given. The Teacher Notes state, “How do you think cutting the radius in half will affect the volume of the sphere? Why? Cutting the radius in half decreased the volume by a factor of $$2^3$$ or 8. The reasoning is the same as the above questions.” The materials do not assist teachers in having students analyze the arguments of others.