7th Grade - Gateway 2

The instructional materials reviewed for JUMP Math Grade 7 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:

• Book 2 Unit 2 has limited real-world problems for students to solve. The focus of the unit is on learning specific algorithms to solve ratio problems. 
• Teacher Resource, Part 2, Unit 2, Lesson RP7-27, Exercises, “Write a ratio with whole-number terms. a. $$\frac{3}{8}$$ of a pizza for every 3 people.” However, none of the independent student work includes real-world scenarios. (7.RP.A) Students use fractional ratios and write equivalent whole number ratios. 
• Teacher Resource, Part 2, Unit 2, Lesson RP7-28, Extensions, Item 2, “John skates $$\frac{7}{2}$$ km in 10 minutes and bikes $$\frac{19}{4}$$ km in 12 minutes. Does he skate or bike faster?” (7.RP.A) Students are using proportional relationships to solve the problem. 
• Teacher Resource, Part 1, Unit 7, Lesson NS7-28, Exercises, “Write a multiplication equation to show the amount of change. a. Ted gained $10 every hour for 5 hours.” (7.NS.3) Students are given a variety of real-world contexts and are asked to write expressions and equations for each context. Students are also asked to solve equations using multiplication and addition. 
• Teacher Resource, Part 2, Unit 1, Lesson NS7-32, Exercises, “Will the recipe turn out? a. I’m making 5 $$\frac{1}{2}$$ batches of gravy, and each batch needs $$\frac{3}{8}$$ cup of flour. I use 2 cups of flour.” (7.NS.3) Students are using the four operations with rational numbers to solve problems. However, students are presented with few opportunities to solve real-world problems involving the four operations of rational numbers. When real-world problems are given, students are encouraged to follow the given examples and the problems do not have room for multiple strategies. 
• Teacher Resource, Part 1, Unit 5, Lesson RP7-22, Exercises, “The amount of tax is 5%. Multiply the original price by 1.05 to calculate the price after taxes. a. a $30 sweater b. a $12 CD.” (7.EE.3) The application questions follow given examples closely. For example, students solve percentage increase problems by being shown the structure of the problems before this set of exercises. 
• Teacher Resource, Part 1, Unit 8, Lesson PS7-9, Exercises, “Which part in the exercises above has the same answer as the given problem? a. 40% of students in the class are boys. Students are picked at random once a week for five weeks. Estimate the probability that a boy will be chosen in at least two consecutive weeks. b. 40% of blood donors have Type O blood. What is the probability that none of the first six donors asked have Type O blood?” (7.SP.8) The application questions follow given examples closely.

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

• In Book 1, Unit 1 Lesson RP7-11, Extensions, Item 5, “Raj mixes 3 cups of white paint with 1 cup of blue paint. He meant to mix 1 cup of white paint with 3 cups of blue paint. How much blue paint does he need to add to get the color he originally wanted?” (7.RP.A) Students are shown how to use ratio tables to help them solve problems with proportional relationships.

The instructional materials reviewed for JUMP Math Grade 7 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 1, Unit 2, Lesson NS7-5, Exercises, “Add using a number line. a. (-4) + (+1).” Students are introduced to adding integers using a number line. In the guided practice of the teachers edition, cut out arrows are moved around a number line drawn on the board to show students how adding negative numbers is done on a number line.
• Application: Teacher Resource, Part 2, Unit 2, Lesson RP7-35, Exercise, “A shirt cost $25. After taxes, it costs $30. What percent of the original price are the taxes?” Students use contexts to learn to cross multiply to arrive at an equation and then solve the equation.
• Procedural Skill and Fluency: In Part 2, Unit 5, Lesson NS7-47, Extensions, Item 1, “Investigate if the estimate is more likely to be correct when the divisor is closer to the rounded number you used to make your estimate. For example, when the divisor is 31 rounded to 30, is your estimate more likely to be correct than when the divisor is 34 rounded to 30? Try these examples: 31⟌243, 31⟌249, 31⟌257, 31⟌265, 31⟌274, 34⟌243, 34⟌249, 34⟌257, 34⟌265, 34⟌274.” Students are given opportunities to develop fluency with division with rational numbers.

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

• Student Resource, Student Assessment and Practice Book, Part 2, Lesson G7-24, Item 9, “The dimensions of a cereal box are 7 $$\frac{7}{8}$$ inches by 3 $$\frac{1}{3}$$ inches by 11 $$\frac{4}{5}$$ inches. What is the volume of the cereal box in cubic inches?” This problem has students using application and procedural skill and fluency using the formula to solve the word problem. 
• Teacher Resource, Part 1, Lesson RP7-16, Exercises, “Draw models to multiply. a. 2 × 4.01 b. 3 × 3.12” develops conceptual understanding of multiplying decimals by modeling the multiplication while using procedural fluency. 
• Teacher Resource, Part 2, Lesson G7-18, Exercises, Item a, “The area of an Olympic ice rink is 1,800 m$$^2$$. A school builds an ice rink to the scale (Olympic rink) : (school rink) = 5 : 4. What is the area of the school rink?” Students develop procedural fluency when they practice calculating areas given scales while solving application problems.

The instructional materials reviewed for JUMP Math Grade 7 partially meet expectations for carefully attending to the full meaning of each practice standard. The materials do not attend to the full meaning of MPs 4 and 5.

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

• MP2: Teacher Resource, Part 1, Unit 3, Lesson EE7-6, Extensions, Item 3, “a. Sketch a circle divided into the following fractions. i. thirds, ii. fourths, iii. fifths. b. Evaluate the expression 360x for i. x = $$\frac{1}{3}$$, ii. x = $$\frac{1}{4}$$, iii. x = $$\frac{1}{5}$$, c. Use your answers to b and a protractor to check the accuracy of your sketches in a.” In this question, students take the quantitative work with the sketches of circles and connect it to the abstract work of evaluating expressions. 
• MP2: Teacher Resource, Part 2, Unit 5, Lesson NS7-46, Extensions, Item 3, “Bev made a grape drink by mixing $$\frac{1}{3}$$ cup of ginger ale with $$\frac{1}{2}$$ cup of grape juice. She used all her ginger ale, but she still has lots of grape juice. She wants to make 30 cups of the grape drink for a party. How many 355 mL cans of ginger ale does she need to buy?” In the solution, students get a remainder in their division, so they must interpret that remainder as needing to buy more cans.
• MP6: Teacher Resource, Part 2, Unit 3, Lesson EE7-15, Extensions, Item 6, “Clara’s Computer Company is making a new type of computer and Clara wants to advertise it. A 30-second commercial costs $1,500,000. Clara plans to sell the computer at a profit of $45.00. Clara determines that 8,600,000 people watched the commercial. a. What percentage of people who watched the commercial would have to buy the product to pay for the price of the commercial? Show your work using equations. Say what each equation means in the situation. b. What facts did you need to use to do part a? c. What place value did you round your answer in part a? Explain your choice. d. Do you think the commercial was a good idea for Clara? Explain.” Students attend to precision throughout the problem to determine if the commercial was a good idea. 
• MP6: Teacher Resource, Part 1, Unit 2, Lesson NS7-2, Extensions, Item 5, “Liz has red, blue, and white paint in the ratio 3:2:1. She mixes equal parts of all three colors to make light purple paint. If she uses all her white paint, what is the ratio of red to blue paint that she has leftover? Use a T-table or a tape diagram with clear labels.” MP6 is developed as students are encouraged to use clear labels in models to ensure they can understand their calculations. This would help students be precise with their ratio calculations.
• MP7: Teacher Resource, Part 1, Unit 2, Lesson NS7-3, Extensions, Item 2, “Look for shortcut ways to add the gains and losses. a. -4 - 5 - 6 +7 +8 + 9.” Students are shown how to group numbers together to make the addition easier, looking for addends that combine to make 10, and looking for opposites to cancel out. Students use structure to complete the problem.
• MP7: Teacher Resource, Part 2, Unit 1, Lesson NS7-37, Extensions 2, “Without doing the division, which do you expect to be greater? -21,317.613 ÷ $$\frac{1}{2}$$ or -21,317.613 ÷ $$\frac{3}{5}$$? Explain.” Students use the structure of dividing by fractions to help them reason about which answer would be greater.

Examples of the materials not carefully attending to the meaning of MPs 4 and 5 include:

• MP4: Teacher Resource, Part 1, Unit 5, Lesson RP7-19, Extensions, Item 4, “Ethan bought a house for $80,000. He spent $5,000 renovating it. Two years after he bought the house, the value increased by 20%. If he sells the house, what would his annual profit be, per year?” Because students are working very similar problems before this set of problems, students do not model with mathematics.
• MP4: Teacher Resource, Part 2, Unit 6, Lesson G7-25, Exercise, Item 1, “The base of a free-standing punching bag is an octagon. The area of the base is 3.5 ft$$^2$$ and the height is 3 ft. a. What is the volume of the punching bag? b. A 30 kg bag of sand fills $$\frac{2}{3}$$ ft$$^2$$. How many bags of sand do you need to fill the punching bag?” Because students are working very similar problems before this set of problems, students do not model with mathematics.
• MP5: Teacher Resource, Part 1, Unit 2, Lesson NS7-2, Extensions, Item 5, “Use a T-table or a tape diagram with clear labels. Which was faster?...What does the tape diagram show you that the T-table does not?” Students are told which tools to use.
• MP5: Teacher Resource, Part 2, Unit 6, Lesson G7-24, Exercises, “a. Find the volume of the prism in three different ways.” A picture of a rectangular prism with sides of 13 cm, 2 cm, and 5 cm is shown. “b. Which way is the easiest to calculate mentally? Solutions: a. 13 x 2 x 5 = 26 x 5 = 130, so V = 130 cm$$^3$$, 13 x 5 x 2 = 65 x 2 = 130, so V = 130 cm$$^3$$, 5 x 2 x 13 = 130, so V = 130 cm$$^3$$; b) 5 cm x 2 cm x 13 cm is the easiest to calculate because 5 x 2 = 10 and it is easy to multiply by 10.” This problem has students deciding which way to multiply the numbers is easiest, which does not require the use of tools.

The instructional materials reviewed for JUMP Math Grade 7 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 2, Unit 1, Lesson NS7-37, Extensions, Item 2, “a. Without doing the division, which do you expect to be greater, −21,317,613 ÷ $$\frac{1}{2}$$ or −21,317,613 ÷ $$\frac{3}{5}$$? Explain. b. In pairs, explain your answers to part a. Do you agree with each other? Discuss why or why not. Use math words.”
• In Teacher Resource, Part 2, Unit 3, Lesson EE7 -17, Extensions, Item 4, students complete the following problem: “a. After the first two numbers in a sequence, each number is the sum of all previous numbers in the sequence. If the 20th term is 393,216, what is the 18th term? Look for a fast way to solve the problem. b. In pairs, explain why the way you chose in part a works. Do you agree with each other? Discuss why or why not.” 
• In Teacher Resources, Part 2, Unit 5, Lesson NS7-49, Extensions, Item 3, students are told, “Eddy painted a square wall. Randi is painting a square wall that is twice as wide as the wall that Eddy painted. Eddy used 1.3 gallons of paint. Randi says she will need 2.6 gallons of paint because that is twice as much as Eddy needed and the wall she is painting is twice as big. So you agree with Randi? 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 3, Lesson EE7-2, Extensions, Item 4, “Which value for w makes the equation true? Justify your answers. a. 2 x 5 - w x 2 b. w x 6 = 6 x 3”
• Teacher Resource, Part 1, Unit 8, Lesson SP7 - 1, Extensions, Item 4, “Sam randomly picks a marble from a bag. The probability of picking a red marble is $$\frac{2}{5}$$. What is the probability of not picking red? Explain.”
• Teacher Resource, Part 2, Unit 3, Lesson EE7 -28, Extensions, Item 3, “a. The side lengths of a triangle are x, 2x +1, and 10. What can x be? Justify your answer. b. In pairs, explain your answers to part a. Do you agree with each other? Discuss why or why not.”
• 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 7, Lesson SP7-14, Extensions, Item 4, “a. What shape is the cross section of the cube? Explain how you know using math words. b. In pairs, discuss your answers to part a. 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 1, Lesson RP7-7, Extensions, Item 4, students are asked to determine if another student is correct with their reasoning. “Two whole numbers are in the ratio 1 : 3. Rob says they cannot add to an odd number. Is he right? Explain.” 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 7 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 L-71. 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 Resources, Part 1, Unit 3, the MP3 portion of the section states, “In EE7-7 Extension 3, students construct and critique arguments when they discuss in pairs the reasons why they agree or disagree with the statement 0 ÷ 0 = 1, and when they ask questions to understand and challenge each other’s thinking.” These explanations do not provide guidance to teachers to get students constructing arguments or critiquing 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, Teacher Resource, Part 1, Unit 1, Lesson RP7-10, Extensions, Item 6 students are asked to tell if the given quantities are in a proportional relationship. Teachers are provided with the sample solutions, “Sample solutions: a. The quantities are proportional. We made a table with headings “side length,” “area of square,” and “square of perimeter.” The ratio for area to square of perimeter was always 1 to 16, so the two quantities are proportional…”
• In Teacher Resource, Part 2, Unit 3, Lesson EE7-17, Extensions, Item 4 students are asked in pairs to explain why the way they chose in part a) works. Students are asked if they agree with each other and to discuss why or why not. Answers and a teacher NOTE are provided: “NOTE: In part b), encourage partners to ask questions to understand and challenge each other’s thinking (MP.3)—see page A-49 for sample sentence and question stems."

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 1, Unit 4, Lesson NS7-22, Extensions, Item 3, students are given the following problem and asked to explain: “b. Len placed a table 1.23 m long along a wall 3 m long. If his bed is 2.13 m long, will it fit along the same wall? Explain.” The answer is provided but no guidance is provided to teachers to help students explain.
• In Teacher Resource, Part 2, Unit 3, Lesson RP7-15, Extensions, Item 5, students are asked, “How would you shift the decimal point to divide by 10,000,000? Explain.” Teachers are given the sample response, “Move the decimal 7 places (because there are 7 zeros in 10,000,000) to the left (because I am dividing).” This is not facilitating the development of mathematical arguments.

The instructional materials reviewed for JUMP Math Grade 7 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.