8th Grade - Gateway 2

The instructional materials reviewed for JUMP Math Grade 8 partially meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics without losing focus on the major work of each grade. 

Overall, many of the application problems are routine in nature and replicate the examples given during guided practice, and problems given for independent work are heavily scaffolded. Examples include:

• Teacher Resource, Part 2, Unit 1, Lesson F8-23, Exercises, “There is some water in the bathtub. Sam pulls out the plug from the bathtub to let the water drain. After 10 seconds, there are 14 gallons of water left in the tub. After 30 seconds, there are 10 gallons of water left. a. Use two points, A (10, 14) and B (30, 10), and find the slope of the line. b. When Sam pulls out the plug, how much water was there? c). Write an equation for the number of gallons (g) left in the tub after s seconds. d. How many gallons are there after 20 seconds? e. How long does it take for all the water to drain from the tub? f. Find the y-intercept and the x-intercept of the equation and compare them to the answers to parts a and d. What did you notice? g. What does the slope represent?” (8.F.B) This problem misses the opportunity to have students apply the mathematics of using functions to model relationships between quantities. The questions include multiple prompts, guiding students along step by step for each question, rather than allowing students to attempt to apply the math and solve the problems independently. 
• Student Resource, Assessment & Practice Book, Part 2, Lesson EE8-52, Item 4, “Write the equations for the word problem. Then solve by graphing. The intersection point may have fractions or decimals. a. Two trains left Union Station at different times. Train A is 12 km from the station and is traveling 60 km/h. Train B is 27 km from the station and is traveling 50 km/h. When will Train A catch up to Train B? How far will they be from the station?” (8.EE.8c) Students solve real-world problems by writing two linear equations for the word problems and finding the solution by graphing them.
• Teacher Resource, Part 2, Unit 5, Lesson EE8-52, Exercises, “Write a formula for the word problem. a. A gravel company charges $25 per cubic yard and a delivery charge of $75. b. A yearbook company charges $500 plus $15 per yearbook.” (8.EE.8c) All questions in the lesson are structured as a “flat fee” plus a rate. The application questions follow given examples closely.

Non-routine problems are occasionally found in the materials. For example, 

• Teacher Resource, Part 2, Unit 5, Lesson EE8-55, Extensions, Item 1, “Alex is four times as old as Clara. In 5 years, Alex will only be three times as old as Clara. How old are Alex and Clara today?” Students are solving a non-routine problem leading to two linear equations in two variables. (8.EE.8c)

The instructional materials reviewed for JUMP Math Grade 8 partially meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present in the materials, but there is an over-emphasis on procedural skills and fluency.

The curriculum addresses conceptual understanding, procedural skill and fluency, and application standards, when called for, and evidence of opportunities where multiple aspects of rigor are used to support student learning and mastery of the standards. There are multiple lessons where one aspect of rigor is emphasized. The materials emphasize fluency, procedures, and algorithms. 

Examples of conceptual understanding, procedural skill and fluency, and application presented separately in the materials include:

• Conceptual Understanding: Teacher Resource, Part 2, Unit 2, Lesson G8-37, Extensions, Item 1, “Use the fact that dilations produce similar triangles to explain why a line AB and its dilation A*B*have the same slope. Hint: Draw an example. Construct a triangle ABC used to find the slope of AB. Use similarity rules and cross multiplication instead of counting grid squares.” Students develop conceptual understanding of slope.
• Application: Student Resource, Assessment and Practice, Part 2, Lesson G8-53, Item 11, “The Mayon volcano in the Philippines is cone shaped. The diameter of its base is 20km and the distance up the curved side, from the base to the apex , is 10.3km. Find the height of the volcano.” Students engage in application when solving the word problem. 
• Procedural Skills and Fluency: Teacher Resource, Part 1, Unit 4, Lesson EE8-35, Exercises, “Solve the equation. a. 3x + 3 − x = 5 b. 7x + 2 = 4x + 11.” Students use the distributive property to solve equations.

Examples of where conceptual understanding, procedural skill and fluency, and application are presented together in the materials include:

• Teacher Resource, Part 1, Unit 2, Lesson EE8-16, Exercises, “Simplify the expression to as few powers as possible by multiplying powers with the same base. a. 4$$^2$$x 7$$^2$$x 7$$^4$$x 4$$^3$$.” Conceptual understanding is developed while also practicing procedural skill.
• Teacher Resource, Part 2, Unit 3, Lesson NS8-6, Exercises, Item 2, “Alex plays baseball. Last month, he was at bat 33 times and got 19 hits. How many hits did Alex get as a percentage of the number of times he was at bat?” Students develop both procedural skill and application.

The instructional materials reviewed for JUMP Math Grade 8 partially meet expectations for carefully attending to the full meaning of each practice standard. The materials do not attend to the full meaning of MP4.

Examples of the materials carefully attending to the meaning of some MPs include:

• MP1: Teacher Resource, Part 1, Unit 3, Lesson G8-15, Extensions, Item 3, “Which is larger, 2$$^{75}$$ or 3$$^{50}$$? Do not use a calculator. Explain how you know your answer is correct.” Students would have to persevere when solving the problem as they would likely need to try multiple strategies to determine which is larger.”
• MP2: Teacher Resource, Part 1, Unit 5, Lesson EE8-44, Extensions, Item 7, “When Amy cut $$\frac{3}{4}$$ of a foot off the bottom of her curtain, the curtain became $$\frac{3}{4}$$ of the length it originally was. How long was the curtain originally? Write your answer as a full sentence.” Students  decontextualize the problem to solve the equation they write, and they put their answer back in context at the end when they write the answer as a full sentence.
• MP5: Teacher Resource, Part 2, Unit 3, Lesson NS8-1, Extensions, Item 2, “Alice buys a house for $200,000. The price of her house goes up by $8,000 every 3 years. Ken buys a house for $100,000. The price of his house doubles every 10 years. When will the two houses be worth the same amount of money? Use any tool you think will help.” Students can use any tool which will help solve the problem.
• MP6: Teacher Resource, Part 1, Unit 2, Lesson EE8-23, Extensions, Item 2, “a. If a year is 365.2422 days, and the universe is about 13.8 billion years old, about how many seconds old is the universe? Write your answer using scientific notation. b. What decimal hundredths might have been rounded to 13.8? c. What range of values might the number of seconds actually be, given the range of decimal hundredths that round to 13.8? d. What place value does it make sense to round your answer from part a to? Explain.” Students attend to precision as they work with scientific notation to solve the problem.
• MP7: Teacher Resource, Part 1, Unit 2, Lesson EE8-17, Extensions, Item 3, “Give students an easier problem: write 10$$^3$$ x 8 + 10$$^3$$ x 2 as a single power of 10. Encourage students to compute it first, and then to look back and explain why the result happened. Then encourage students to use the same technique for the harder problem.” By redirecting students in this way, students make use of the structure.
• MP8: Teacher Resource, Part 2, Unit 1, Lesson F8-19, Extensions, Item 1, “b. Describe what you are always doing the same in part a. c. Find a formula for finding the x-intercept of the line y = mx + b.” Students are given four equations in slope intercept form and calculate the x-intercept of the line. Students use repeated reasoning to find the formula as a generalization.

Examples of the materials not carefully attending to the meaning of MP4 include:

• Teacher Resource, Part 2, Unit 7, Lesson SP8-11, Extensions, Item 1, students are given a set of data about late phone charges and create a two-way table for the data. Students answer follow-up questions, “b. How many landline customers did not pay on time this year? Make a row two-way relative frequency table for the data. c. Based on the data, is there an association between the type of phone and repeated lateness? Explain.” By scaffolding the questions into a step-by-step process, students do not model with mathematics.
• Teacher Resource, Part 2, Unit 4, Performance Task, Fire Department Ladder Problems, Item 5, “Two firefighters need to reach the top of a building. The building is 20 feet high and has a stone wall around it. The stone wall is: 8 feet high, 4 feet out from the building, 1 foot thick. The firefighters need to bring a ladder (not one mounted on the truck). The firefighters have two options.” Students are shown pictures of the two options, one with the ladder between the wall and the building, and the other with the ladder outside of the wall. Students complete the following: “a. Label each picture with the distances given. b. What is the slope of the ladder in each option? c. To be safe to climb, the absolute value of the slope of the ladder has to be less than 4. Circle the option(s) that are safe.” By asking each question separately, students do not model with mathematics independently.

The instructional materials reviewed for JUMP Math Grade 8 partially meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

Students explain their thinking, compare answers with a partner, or understand the error in a problem. However, this is done sporadically within extension questions, and often the materials identify questions as MP3 when there is not an opportunity for students to analyze situations, make conjectures, and justify conclusions. At times, the materials prompt students to construct viable arguments and critique the reasoning of others. Examples that demonstrate this include:

• Teacher Resource, Part 1, Unit 4, Lesson EE8-34, Extensions, Item 5, “a. Glen says that the expression -5x - 3 is always negative because both the coefficient and the constant terms are negative. Do you agree with Glen? Why or why not? b. In pairs, discuss your answers to part a). Do you agree with each other? Discuss why or why not.”
• In Teacher Resource, Part 1, Unit 6, Lesson F8-15, Extensions, Item 1, students make an argument about where a y-intercept would be given a point and a slope. “A line passes through A (1, 2) with a negative slope. Can the y-intercept be negative? Why? Hint: Draw lines with negative slope from point A.”
• Teacher Resource, Part 2, Unit 2, Lesson G8-26, Extensions, Item 4, “A Transformation takes the point (x, y) to the point (2x, y+3). Does the transformation take the line y = 2x + 1 to another line? Explain how you know.”
• Teacher Resource, Part 1, Unit 6, Lesson F8-10, Extensions, Item 2, “Tessa says that when you find the rate of change from point A to point B, you always get the same answer as when you find the rate of change from point B to point A. a. Do you agree with Tessa? Why or why not? b. In pairs, explain your answers to part a. Do you agree with each other? Discuss why or why not.”

In questions where students must explain an answer or way of thinking, the materials identify the exercise as MP3. As a result, questions identified as MP3 are not arguments and not designed to establish conjectures and build a logical progression of a statement to explore the truth of the conjecture. Examples include:

• Teacher Resource, Part 1, Unit 4, Lesson EE8-32, Extensions, Item 5, “a. Find a positive number x that makes the equation true: 125$$^8$$ = x$$^6$$. Explain how you got your answer? b). In pairs, compare your answers. Do you agree with each other? Discuss why or why not. c. Is your answer to part a) greater or less than 125? Explain why this makes sense.” 
• Teacher Resource, Part 1, Unit 4, Lesson EE8-37, Extensions, Item 1, “Simplify the equation. Does the equation have one unique solution, no solution, or infinitely many solutions? a. 6x + 3 = 6x + 6 b. 7x + 1 = 43 c. 9(x+1) = 9x + 9” 
• Many MP3 problems in the extension sections follow a similar structure. Students are given a problem and “explain.” Then, students compare their answers with a partner and discuss if they agree or not. This one dimensional approach does not offer guidance to students on how to construct an argument or critique the reasoning of others. For example, Teacher Resource, Part 2, Unit 3, Lesson NS8-6, Extensions, Item 6, “b. Explain why, in a right triangle, the side opposite the right angle is always the longest side. Use any tool you think would help. c. In pairs, explain your answers to part b. Do you agree with each other? Discuss why or why not.”
• Students are given extension questions when they are asked to analyze the math completed by a fictional person. For example, Teacher Resource, Part 1, Unit 4, Lesson EE8-33, Extensions, Item 4, “Marta says that the expression 5x + 20 is always a multiple of 5 because 5 and 20 are both multiples of 5. Do you agree with Marta? Why or why not?” These problems begin to develop students’ ability to analyze the mathematical reasoning of others but do not fully develop this skill. Students analyze an answer given by another, but do not develop an argument or present counterexamples.

The instructional materials reviewed for JUMP Math Grade 8 partially meet expectations for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. Some guidance is provided to teachers to initiate students in constructing arguments and critiquing others; however, the guidance lacks depth and structure, and there are multiple missed opportunities to assist students to engage in constructing and critiquing mathematical arguments. 

The materials have limited support for the teacher to develop MP3. Generally, the materials encourage students to work with a partner as a way to construct arguments and critique each other. In the teacher information section, teachers are provided with the following information:

• Page A-14: “Promote communication by encouraging students to work in pairs or in small groups. Support students to organize and justify their thinking by demonstrating how to use mathematical terminology symbols, models and manipulatives as they discuss and share their ideas. Student grouping should be random and vary throughout the week.” The material provides no further guidance on thoughtful ways to group students and only limited structures that would encourage collaboration. 
• Page A-49: “Classroom discussion in the lesson plans include prompts such as SAY, ASK and PROMPT. SAY is used to provide a sample wording of an explanation, concept or definition that is at grade level, precise, and that will not lead to student misconceptions. ASK is used for probing questions, followed by sample answers in parentheses. Allow students time to think before providing a PROMPT, which can be a simple re-wording of the question or a hint to guide students in the correct direction to answer the question….You might also have students discuss their thinking and explain their reasoning with a partner, or write down their explanations individually. This opportunity to communicate thinking, either orally or in writing, helps students consolidate their learning and facilitates the assessment of many Standards for Mathematical Practice.” While this direction would help teachers facilitate discussion in the classroom, it would not help teachers to develop student’s ability to construct arguments or critique the reasoning of others.
• Page A-49: There are sentence starters that are referenced that mostly show teachers how to facilitate discussions among students. The materials state, “When students work with a partner, many of them will benefit from some guidance, such as displaying question or sentence stems on the board to encourage partners to understand and challenge each other’s thinking, use of vocabulary, or choice of tools or strategies. For example: 
• I did ___ the same way but got a different answer. Let’s compare our work. 
• What does ___ mean?
• Why is ___ true? 
• Why do you think that ___ ?
• I don’t understand ___. Can you explain it a different way?
• Why did you use ___? (a particular strategy or tool)
• How did you come up with ___? (an idea or strategy)”

Once all students have answered the ASK question, have volunteers articulate their thinking to the whole class so other students can benefit from hearing their strategies” While this direction would help teachers facilitate discussion in the classroom, it would not help them to develop student’s ability to construct arguments or critique the reasoning of others. 

• A rubric for the Mathematical Practices is provided for teachers on page K-91. For MP3, a Level 3 is stated as, “Is able to use objects, drawings, diagrams, and actions to construct an argument” and “Justifies conclusions, communicates them to others, and responds to the arguments of others.” This rubric would provide some guidance to teachers about what to look for in student answers but no further direction is provided about how to use it to coach students to improve their arguments or critiques. 
• In the Math Practices in this Unit Sections, MP3 is listed multiple times. The explanation of MP3 in the unit often consists of a general statement. For example, in Teacher Resource, Part 1, Unit 4, the MP3 portion of the section states, “In EE8-37 Extension 4, students critique an argument when they explain where the mistake is in a fictional student’s argument. Students construct an argument when they explain how to correct the mistake.” These explanations do not provide guidance to teachers in how to get students to construct arguments or critique the reasoning of others.

There are limited times when specific guidance is provided to teachers for specific problems. Examples include:

• Some guidance is provided to teachers for constructing a viable argument when teachers are provided solutions to questions labeled as MP3 in the extension questions. Some of these questions include wording that could be used as an exemplar response about what a viable argument is. For example, in Teacher Resource, Part 1, Unit 5, Lesson EE8-46, Extensions, Item 5, students are given the work of another student and asked if they agree. Teachers are provided with a sample solution, “a. I do not agree with Kyle. I solved the equation and got x = −13. In this case, 2x + 6 = − 20 and 2x + 1 = −25. It is true that 5 > 4 and −20 > −25, but it is not true that when you multiply two greater numbers, you always get a greater answer. Indeed, 5 × (−20) = −100 and 4 × (−25) = −100. Kyle would be correct if 2x + 6 and 2x + 1 were positive numbers; then it would be true that 5(2x + 6) > 5(2x + 1) > 4(2x + 1), but since 2x + 1 is negative and 5 > 4, you get 5(2x + 1) < 4(2x + 1), so the inequalities become 5(2x + 6) > 5(2x + 1) < 4(2x + 1).” In addition to the sample solution that could be used as an exemplar, teachers are also given the note, “Encourage students to not only explain why Kyle’s answer is incorrect, but why his reasoning is incorrect (when you multiply both sides of the inequality 5 > 4 by the same negative number, the inequality changes direction).” This guidance would help teachers develop students’ arguments and emphasizes the importance of not just explaining an answer but looking specifically at the mathematical reasoning. 
• In Teacher Resource, Part 2, Unit 7, Lesson SP8-6, Extensions, Item 3, students construct an argument to explain why a sphere with a given volume will or will not fit inside a box with a given, larger, volume. Students have the opportunity to critique a partner’s argument. For example, if one student argues incorrectly that the ball will fit into the cube because 11.3 < 20, their partner will need to explain that they need to compare the diameter, not the radius, to the width of the box.

Frequently, problems are listed as providing an opportunity for students to engage in MP3, but miss the opportunity to give detail on how a teacher will accomplish this. Examples include:

• In Teacher Resource, Part 2, Unit 3, Lesson NS8-6, Extension, a sample answer is provided but no support on engaging students in how to analyze the reasoning of others: “b. Explain why, in a right triangle, the side opposite the right angle is always the longest side. Use any tool you think will help. c. In pairs, explain your reasoning from part b. Do you agree with each other? Discuss why or why not.”
• In Teacher Resource, Part 1, Unit 4, Lesson EE8-31, Extensions, Item 4, students are given the question, “Without using a calculator, show that 2$$^{100}$$ has at least 31 digits.” Teachers are provided with the answer, “2$$^{10}$$ = 1,024 > 10$$^3$$, so 2$$^{100}$$ = (2$$^{10}$$)$$^{10}$$ > (10$$^3$$)$$^{10}$$= 10$$^{30}$$ , which is the smallest number with 31 digits, so 2$$^100$$ has at least 31 digits.” This sample answer does not provide any assistance for developing students ability to construct viable arguments.
• In Teacher Resource, Part 2, Unit 1, Lesson 8F-15, MP3 is identified in the section titled, “Drawing a line using the y-intercept and the slope (page L-18).” In this section, teachers are told to ask students a series of questions about drawing lines. “ASK: But how can we use the slope to find another point? SAY: As an example, let’s draw a line with y-intercept = 2 and slope = $$\frac{1}{3}$$. First, mark the y-intercept on the y-axis. SAY: The y-intercept is 2, so we mark (0, 2) on the grid. The slope is $$\frac{1}{3}$$. ASK: What does the slope fraction stand for? ($$\frac{rise}{run}$$) SAY: The fraction of rise over run is 1 over 3, so I can say the rise is 1 and the run is 3. ASK: Do the rise and run have to be 1 and 3? (no) What other numbers would work? (2 and 6, or 3 and 9).” These questions are not developing students’ abilities to construct arguments or to critique the reasoning of others.

The instructional materials reviewed for JUMP Math Grade 8 partially meet expectations for explicitly attending to the specialized language of mathematics.

Accurate mathematics vocabulary is present in the materials, however, while vocabulary is identified throughout the materials, there is no explicit directions for instruction of the vocabulary in the teacher materials of the lesson. Examples include, but are not limited to: 

• Vocabulary is identified in the Terminology section at the beginning of each unit.
• Vocabulary is identified at the beginning of each lesson.
• The vocabulary words and definitions are bold within the lesson.
• There is not a glossary.
• There is not a place for the students to practice the new vocabulary in the lessons.