6th Grade - Gateway 2

The instructional materials reviewed for JUMP Math Grade 6 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 1, Unit 1, Lesson RP6-9, Extensions, Item 2, “Liz drives 131 miles in 2 hours. It takes Mindy twice as long to drive 257 miles. Who is driving faster? How much faster?” (6RP.3) Students engage in a routine problem using ratio and rate reasoning. 
• Teacher Resource, Part 1 Unit 5, Lesson RP6-19, Exercises, “Have students find the missing percentages of other stamps in each collection: a. USA: 40% Canada: $$\frac{1}{2}$$ Other: b. Mexico: 25% USA: $$\frac{3}{5}$$ Other:” Before this problem students are guided through specific problem solving strategies, and then given problems that match the given strategy, making this a routine problem. The problem given before the Exercises was “$$\frac{2}{5}$$ of the stamps are from the United States and 36% are from Canada. What percent of Jennifer’s stamps are from neither the United States nor Canada? Solve this problem with the class. (change 2/5 to 40%, then add 40% + 36% = 76%, so the stamps from neither place make up 24% of Jennifer’s collection).” (6.RP.3) Students use ratio and rate reasoning to solve real-world problems. 
• Teacher Resource, Part 2, Unit 3, Lesson EE6-12, Exercises, “Repeat with more examples. As you give each example, ask students to first identify the smaller number, and remind them that this should be the shorter bar. a. Bethany is three times as tall as her baby brother.” (6.EE.7) Students solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. The exercises included in the lessons follow the structure of a problem presented as an example, eliminating students opportunities to apply the mathematics in a non-routine way. 
• Teacher Resource, Part 2, Unit 6, Lesson EE6-19, Extension, “Wilson has $30. For every book he reads, his mother gives him $5. a. Create a T-table that shows the number of books as input and the amount of money as output. b. Write a rule for the amount of money Wilson has after reading n books. c. How many books does Wilson need to read to get $50? $100? $1000?” (6.EE.9) Students find rules for linear graphs. 
• Teacher Resource, Part 2, Unit 6, Lesson EE6-20,Exercises, “A boat leaves port at 9:00 am and travels at a steady speed. Some time later, a man jumps into the water and starts swimming in the same direction as the boat. a. How many minutes passed between the time the boat left port and the time the man jumped into the water? (15 min) When did the man jump into the water?” Students are provided a graph to solve the problem. (6.EE.9) The lesson provides students with completed graphs and tables with which to identify the independent and dependent variables and write a rule. The graphs have limited real-world context and the tables have no real-world context.

Non-routine problems are occasionally found in the materials. Examples include: 

• Teacher Resource, Part 1, Unit 3, Lesson EE6-7, Extensions, Item 1, “Some friends bought pizza and ate 2 $$\frac{7}{10}$$ pizzas. They had $$\frac{4}{5}$$ of a pizza with pineapple left and $$\frac{1}{8}$$ of a pizza without pineapple left. 2 $$\frac{1}{2}$$ of the pizzas they ordered were vegetarian. How many pizzas did they buy? Which fact did you not need?” (6.EE.7) Students solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. However, the majority of the independent practice problems that students complete in this section, only involve pre-setup problems without real-world context, where students follow an algorithm to find the answer.

The instructional materials reviewed for JUMP Math Grade 6 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 NS6-11, Extensions, Item 2a, “Draw a number line to find the fraction halfway between -$$\frac{14}{3}$$ and -$$\frac{4}{3}$$. Repeat for the fraction halfway between -$$\frac{14}{5}$$ and -$$\frac{4}{5}$$.” Students place positive and negative fractions on a number line and use the number line to order those fractions.
• Application: Teacher Resource, Part 2, Unit 2, Lesson RP6-26, Extensions, Item 1, “A store offers you a choice between two options for fancy socks, which are usually $10 per pair. Which price option would you choose? A. 3 pairs of socks for the price of 2, or B. 30% off all pairs of socks?” Students use ratios to solve real world problems.
• Procedural Skill and Fluency: Teacher Resource, Mental Math, Skills 1, 2, 3, adn 4, Item 7, “Name the odd number that comes after the number shown. a. 37.” This section contains problems to help students maintain and develop procedural fluency with Addition, Subtraction, and Multiplication.

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

• Teacher Resource, Part 1, Unit 4, Lesson NS6-29, Extensions, Item 3, “Sarah saw four fish at different elevations: −0.025 km, −0.18 km, −0.9 km, −1.8 km. Use the information below to decide which fish was seen at which elevation. The coelacanth lives between 150 m and 400 m below sea level. The football fish lives between 200 m and 1 km below sea level. The deep sea angler lives between 250 m and 2 km below sea level. The rattail lives between 22 m and 2.2 km below sea level.” Comparing Decimal Fractions and Decimals contains both conceptual understanding and application of mathematics. Students develop conceptual understanding of comparing rational numbers by using a number line to compare both positive and negative fractions and decimals.
• Teacher Resource, Part 1, Unit 2, Lesson NS6-17, Exercises, “Find the GCF of the two numbers being added and then rewrite the sum as shown. a. 18 + 42 = __ × (__ +__ ).” This lesson contains both conceptual understanding and procedural fluency. Students develop conceptual understanding in the lesson when they model the distributive property of numbers using array models. They develop procedural fluency when they complete exercises dividing out the GCF of numbers.

The instructional materials reviewed for JUMP Math Grade 6 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 1 and 4.

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

• MP2: Teacher Resource, Part 1, Unit 5, Lesson RP6-19 Extensions, Item 2, “Mr. Bates buys: • 5 single-scoop ice cream cones for $1.45 each • 3 double-scoop ice cream cones for $2.65 each. A tax of 10% is added to the cost of the cones. Mr. Bates pays with a 20-dollar bill. How much change does he receive? Show your work.” Students reason abstractly and quantitatively to calculate the tax and change and then interpret their solution in the context of the problem.
• MP5: Teacher Resource, Part 2, Unit 6, Lesson EE6-16, Extension 2, “Write an equivalent expression without brackets. Use any tool you think will help. Explain how you got your answer. a) 3(2x +5) b) 2(4x +7) c) 5(8x -3) d) a(bx + c).” Students choose an appropriate tool to solve the problem.
• MP5: Teacher Resource, Part 2, Unit 3, Lesson EE 6-9, Extensions, Item 1, “A classroom is made up of students from Grade 6 and 7. 25% of the Grade 6 students and 70% of the Grade 7 students prefer comedy over science fiction. There are twice as many Grade 6 students as Grade 7 in the class. What percent of the class prefers comedy? Use any tool you think will help.” Students have the ability to choose an appropriate tool to solve the problem. 
• MP6: Teacher Resource, Part 2, Unit 5, Lesson G6-22, Extensions, Item 3, “Vicky bought 12 bus tickets for $9. She calculated how much 36 bus tickets cost as follows: $$\frac{12}{9}$$= $$\frac{x}{36}$$ and 4 x 9 = 36, so x =4 x 12 = 48, so 36 bus tickets cost $48. a. Do you agree with Vicky’s answer? Why or Why not? b. What did Vicky do correctly? What did she do incorrectly? c. How much would 36 bus tickets cost? Explain.” Students attend to precision in calculations as they evaluate the calculations of another student and find what is correct about how a proportion is written and what is incorrect. Students correctly set up the proportion and complete the calculations. 
• MP6: Teacher Resource, Part 2, Unit 7, Lesson SP6-2 Extensions, Item 1, “a. Find the median and the range for each set i. 2, 3, 5, 7, 9, 10; ii. 12, 16, 19, 22, 26, 26, 26, 26. b. Add 4 to each data point in i and ii. Find the new median and range. c. Why did adding 4 to each data point change the median but not the range? d. in pairs, explain your answers to part c. Do you agree with each other? Discuss why or why not.” Students attend to precision when they use the definitions of median and range to explain why adding the same number to each data point in a data set changes the median but not the range.

Examples of the materials not carefully attending to the meaning of MPs 1 and 4 include:

• MP1: Teacher Resource, Part 2, Unit 1, Lesson NS6-59, Extensions 3, “Use mental math or pencil and paper to solve. Explain your choice. a. $$\frac{5}{4}$$ ÷ $$\frac{4}{5}$$ b. $$\frac{2}{3}$$ ÷ $$\frac{4}{6}$$ c. $$\frac{3}{17}$$ ÷ $$\frac{9}{34}$$”. Students do not need to make sense of the problem or devise a strategy to solve the problem, but rather use an algorithm to solve. 
• MP1: Teacher Resource, Part 2, Unit 5, Lesson G6-20 Extensions Item a., “Plot and join the points, in order. Use the same grid for all parts. i. (-1, -2), (-1, 3), (0,4), (1,3), (1, -2), (0, -3). Join the first point to the last point. ii. (-1, -2), (-2, -3), (-3, -3), (-3, -2), (-1, 0); iii. (1, 0), (3, -2), (3, -3), (2, -3), (1, -2). b. What shape did you make? c. Find the area of the shape.” There is no opportunity to make sense of this problem as students are told how to solve the problem.
• MP4: Teacher Resource, Part 1, Unit 2, Lesson NS6-3, Extensions, Item 3, “John bikes 7km in 10 minutes and skates 900 m in 2 minutes. Does he skate or bike faster? How much faster? Write your answer in complete sentences.” Students do not model with mathematics as they do not have to interpret their solutions in the context of the problem to determine if the results make sense.
• MP4: Teacher Resource, Part 2, Unit 3, Lesson EE6-9, Extensions, Item 2, “An eBook costs $16 before taxes and $16.48 after taxes. A can of soda costs $1.60 before taxes and $1.68 after taxes. Which item was taxed at a higher rate?” Students do not model with mathematics as they do not have to interpret their solutions in the context of the problem to determine if the results make sense.

The instructional materials reviewed for JUMP Math Grade 6 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:

• In Teacher Resource, Part 2, Unit 1, Lesson NS6-56, Extensions, Item 3, students explain why a pattern emerges from a previous problem. “a. In the previous question, which positive numbers are greater than their square? Why does this make sense?”
• Teacher Resource, Part 2, Unit 7, Lesson SP6-1, Extensions, Item 4, students calculate the mean of a set of numbers and explain why the mean in part i is greater than the mean of part ii.
• In Teacher Resource, Part 1, Unit 4, Lesson NS6-27, Extensions, Item 4, “a. After the first two numbers in a sequence, each number is the sum of all the 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 your method works. Do you agree with each other? Discuss why or why not.”
• In Teacher Resource, Part 1, Unit 2, Lesson NS6-11, Extensions, Item 2, students explain their reasoning and critique the reasoning of a partner. “a. Draw a number line to find the fraction halfway between -$$\frac{14}{3}$$ and -$$\frac{4}{3}$$. Repeat for the fraction halfway between -$$\frac{14}{5}$$ and -$$\frac{4}{5}$$. b. Without drawing a number line, write the fraction halfway between −$$\frac{14}{351}$$ and -$$\frac{4}{351}$$. Explain how you know. c. In pairs, explain your answers to part b. 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 or build a logical progression of a statement to explore the truth of the conjecture. Examples include:

• In Teacher Resource, Part 1, Unit 2, Lesson NS6 - 13, Extensions, Item 3, students answer “Can a positive fraction be equivalent to a negative fraction? Explain why or why not.”
• Teacher Resource, Part 2, Unit 6, Lesson EE6-17, Extensions, Item 1, “Explain why 7y + 2y = 9y?”
• Students are given extension questions when they are asked to analyze the math completed by a fictional person. For example: Teacher Resource, Part 2, Unit 1, Lesson NS6-45, Extensions, Item 2, “Ron says 2 R 1 = 2 $$\frac{1}{4}$$ because 9 ÷ 4 = 2 R 1 and 9 ÷ 4 = 2 $$\frac{1}{4}$$ Is this reasoning correct? 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 6 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 the prompts 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.” This format does not provide any structure for constructing arguments and critiquing others, in fact, this Say, Ask, Prompt model will only lead students to learn in a step by step manner directed by the teacher. 
• Page A-49: There are sentence starters that are referenced that 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 teachers 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 I-57. 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 numerous times. Each time, the explanation of MP3 in the unit consists of a similar general statement. For example, in Teacher Resource, Part 2, Unit 5, “MP.3: In G6-21 Extension 3, students make a conjecture about what the distance will be between any number and its opposite, and construct a viable argument to explain their conjecture.” Other units all follow a similar structure in their introduction to teachers about how students will encounter MP3 in the materials. These explanations do not provide guidance to teachers in 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 to construct 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 2, Lesson NS6-20 Extensions, Item 5 the solution provided says, “I have two ways of getting from 5 to 60: multiply by 2 and then multiply by 6, or I can multiply by 6 and then multiply by something else: 5 × 2 × 6 = 5 × 6 ×? I know the two 6s will always be the same because that’s what it means to be a ratio table. Now, I see the other two numbers are also the same. That’s why switching the rows and columns gives another ratio table.”
• In Teacher Resource, Part 2, Unit 3, Lesson EE6-13, Extensions, Item 3, students are asked to add 5 to a mystery number, then double that result. Subtract 10, and then divide that result by 2. The materials state, “In pairs, explain why the trick works. Choose any tool you think will help, such as expressions with variables and brackets, bags and blocks, or a T-table and pictures.” Students are asked if they agree with each other and to discuss why or why not. Sample solutions are provided but no teacher guidance is given on engaging students in constructing viable arguments.

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:

• The opportunity is missed to provide exemplar responses for teachers when students are constructing viable arguments. For example, in Teacher Resource, Part 2, Unit 1, Lesson NS6-58, Extensions, Item 2 is labeled as MP3 and students are asked to explain their reasoning. The solution provided says, “Answers: 8.56 ÷ 0.4 = 21.4 and 8.56 ÷ 0.2 = 42.8. The second answer is double the first because it is dividing by half as much.” This provided solution would not help teachers understand how the student could construct an argument in response to this question. 
• Teacher Resource, Part 1, Unit 6, Lesson G6-4, in a section labeled as MP3, teachers are told, “ASK: Without drawing it, does this quadrilateral have any horizontal lines or vertical lines? (AB and CD are both vertical) How do you know? (A and B have the same first coordinate, as do C and D) Are AB and CD the same length? (no, AB is 3 units long, and CD is 4 units long) So is ABDC a trapezoid or a parallelogram? (a trapezoid).” These questions do not promote constructing arguments.

The instructional materials reviewed for JUMP Math Grade 6 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.