4th Grade - Gateway 2

The instructional materials reviewed for JUMP Math Grade 4 partially meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single- and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. The materials include limited opportunities for students to independently engage in the application of nonroutine problems. Most problems are routine in nature and provide few opportunities for students to independently demonstrate the use of mathematics flexibly. 

The instructional materials have few opportunities for students to engage in non-routine application throughout the grade level. There is little variety in situational contexts/problem types. Engaging applications include single- and multi-step word problems presented in a context in which the mathematics is applied; however, these problems are often routine, and students have few opportunities to engage with non-routine application problems. Examples of routine application problems include:

• Teacher Resource, Part 1, Unit 4, Lesson NBT4-29, Extensions, 2, “Fred has $2347 and Marcel has $3653. Jennifer has 3 times as much money as Fred and Marchel put together. How much money does Jennifer have?”
• Teacher Resource, Part 2, Unit 3, Lesson OA4-32, Exercises, “Solve. 1a. John has 12 blue marbles. He has 9 more red marbles tha blue marbles. How many red marbles does he have? b. John also has 7 fewer green marbles than red marbles. How many green marbles does he have? c. How many red, blue, and green marbles does he have altogether?” 
• Teacher Resource, Part 2 Unit 9, Lesson G4-16, Extensions, 4, “Aputik is fixing up her basement bathroom. She is going to need a total of 28 tiles and 6 yards of wood. Tiles come in packs of 5 and cost $8 a pack. Wood is sold by the foot and costs $7 per foot. How much will it cost Auptik to fix up her basement bathroom?” Students are solving a routine multi-step problem. 
• Teacher Resource, Part 2, Unit 3,OA4-36, Exercises, Bonus, d. “A jackpine is 35 feet tall. A red pine is three times as tall as a jack pine. A giant sequoia is eight times as tall as a jack pine. How tall is each tree?”

Few opportunities for non-routine applications of mathematics are provided in the extensions and in the Assessment and Practice Books. Examples include:

• Student Resource, Assessment & Practice, Part 2, Lesson OA4-37, Item 8, “Javier is four times as old as Kong. Kong is 3 years younger than Ewa. Javier is 9 years older than Ewa. How old is each person?”
• Teacher Resource, Part 1, Unit 6, Lesson MD4-14, Extensions, Item 2, “Simon and Maria played a video game. Simon got five times as many points as Maria did in round one. Maria got three times as many points as Simaon did round two. Maria got more points overall. Give an example of Simon and Maria’s possible points.”

The instructional materials reviewed for JUMP Math Grade 4 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: Student Resource, Assessment & Practice Book, Part 2, Lesson NBT4-43, Item 2, “Divide 12 hundreds among 3 equal groups. Then finish the division equation. 12 hundreds / 3 = ___ hundreds so 1200 / 3 = _____.” Students develop conceptual understanding by using base-10 blocks to model division of a multi-digit number by a one-digit number.
• Procedural Skill and Fluency: Student Resource, Assessment & Practice Book, Part 1, Lesson MD4-6, Item 6, “Convert the mixed measurements to measurements in centimeters. a. 3 m 1 cm = 301 cm (6 Items a-f).”
• Application: Student Resource, Assessment & Practice Book, Part 1, Lesson NBT4-36, Item 2, “On average, every American uses 147 gallons of water each day. a. About how much water does each American use in a week? b. About how much water would a family of 4 use in a day?”

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

• Student Resource, Assessment & Practice Book, Part 1, Lesson NBT4-35, Item 11, “An octopus has 8 arms and 240 suckers on each arm. How many suckers does an octopus have?” Students are employ the standard algorithm to multiply in word problems that apply mathematics to real-world contexts.
• Student Resource, Assessment & Practice Book, Part 2, Lesson NF4-7, Item 14, “Quentin ate $$\frac{3}{5}$$ of a pizza and Jasmine ate $$\frac{1}{3}$$ of the pizza. Who ate more pizza?” Students engage in conceptual understanding and application as they use fraction models to compare fractions,
• Student Resource, Assessment & Practice Book, Part 2, Item 8, “Tyrell has $$\frac{4}{10}$$ of a dollar and Tania has $$\frac{7}{100}$$ of a dollar. What fraction of a dollar do they have altogether?” Students use conceptual understanding, procedural skill, and fluency as they add tenths and hundredths, base-10 models are used to develop conceptual understanding. Students also demonstrate procedural skill and application in converting between tenths and hundredths and adding tenths and hundredths.

The instructional materials reviewed for JUMP Math Grade 4 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:

• MP1: Teacher Resource, Part 1, Unit 1, Lesson OA4-5, Extension 2, “To make a sequence, Ben picks a starting number and Maria picks a number to add each time. When they write the sequence, the terms switch between odd and even numbers. a. Talk with a partner about what it means for the terms to switch between odd and even numbers. b. What can you say about Maria’s number? Explain.” Students make sense of word problems related to number patterns and plan pathways to solutions using what they know about numbers.
• MP2: Teacher Resource, Part 1, Unit 2, Lesson NBT4-23, Extensions, Item 2: “Write your answer to the question as a full sentence. What fact did you not need to use? Jay has 7 oranges. He has 8 more apples than oranges. He has 3 times as many apples as pears. How many apples does he have?” Students reason abstractly and quantitatively to decontextualize and apply mathematical operations to solve word problems, then contextualize the solutions in order to write a full sentence to answer the problem.
• MP6: In Teacher Resource, Part 1, Unit 7, G4-4, Extensions, Item 3, students make use of structures related to place value in rounding numbers: “Make up an addition where rounding to the nearest thousand is higher than the actual answer, but rounding to the nearest hundred is lower than the actual answer.”
• MP7: Teacher Resource, Part 1, Unit 2, Lesson NBT4-18, Extensions, Item 6, “Subtract mentally. (Items 6a-f include a 4 or 5 digit number - 1,010).” Students make use of structure to realize that they take one away from the thousands and tens, as well as occasions where this strategy doesn’t work (there is a zero in one of those places).
• MP8: Teacher Resource, Part 1, Unit 2, Lesson NBT4-4, Lesson Activities, “Give students ones and tens blocks. ASK: Which numbers have standard base-10 representations that can be arranged as rectangles of width at least 2? That is, if the blocks are arranged horizontally, there are at least 2 rows.” Students use repeated reasoning to notice patterns in forming rectangles with ones and tens.

For MP4, students are given models to use and have few opportunities to develop their own mathematical models. In addition, students have few opportunities to compare different models in problem contexts. Examples include:

• Teacher Resource, Part 1, Unit 2, Lesson NBT4-1, Extensions, Item 3, “A plant is 4 cm tall and grows 3 cm each week. a. How tall will the plant be after 8 weeks? Use multiplication and addition to answer the question. b. Check your answer to part a with a T-table. Use clear labels. Explain how the T-table shows your equation. c. Will the plant be an even or odd number of cm tall after 1 year (52 weeks)? Find a quick way to answer the question. Explain.” 
• Teacher Resource, Part 2, Unit 4, Lesson NBT4-35, Extensions, Item 3, “Sam has three times as much money as Megan. How much more money does Sam have than Megan? Show your work using equations. Write your answer as a complete sentence. a. Megan has $14. b. Megan has $124. c. Megan has $2,314.”
• Teacher Resource, Part 2, Unit 6, Lesson OA4-43, Extensions, Item 7, “Arsham made orange juice to have with breakfast. He squeezed 9 oranges, all the same size, to get 558 mL of juice. He shared the juice equally between himself, his sister, and his brother. After he poured the juice, his sister said, ‘That’s not enough juice!’ So he squeezed another orange and gave it all to his sister. Now how much juice does she have? Use equations to show your work. Explain what each step means in the situation.”

For MP5, students are given few opportunities to use tools strategically, as they are most often given the tools to use for a problem. Examples include: 

• Teacher Resource, Part 1 Unit 4, Lesson NBT4-35, Extensions, Item 4, “Find a shortcut way to do each problem in Extension 3 without using subtraction. Explain why the shortcut works using one of the following tools: a number line, an array, or base-10 blocks.”
• Teacher Resource, Part 2 Unit 4, Lesson NF4-5, Extensions, Item 4, “Which fraction is greater, $$\frac{3}{4}$$ or $$\frac{7}{12}$$? Use one or more of the following tools to find the answer: an array, a clock, a table, grid paper, a number line, pattern blocks.”

The instructional materials reviewed for JUMP Math Grade 4 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. 

There are few opportunities in the Teacher Resource or the Assessment & Practice for students to construct viable arguments or analyze the arguments or the work of others. MP3 is identified in the margins of the lesson. Examples of where the materials prompt students to construct viable arguments or analyze the arguments of others include, but are not limited to:

• Student Resource, Assessment & Practice Book, Part 2, Unit 4, Lesson NF4-7, Item 6, “Lina thinks $$\frac{4}{3}$$ is less than $$\frac{99}{100}$$ because the numbers are smaller. Is she right? Explain how you know?”
• Teacher Resource, Part 1, Unit 3, Lesson OA4-15, Extensions, Item 2, “Ethan says that he found another way to round a number to the nearest ten: Add 5 to the number and then replace the ones digit with zero. Does Ethan’s rule work? Explain why or why not using examples.” Then in Extension 2 a. Make a rule like Ethan’s rule from Extension 1, but for rounding to any place value. B. In pairs, explain your rule and why it works. Do you agree with each other? Discuss why or why not.” Students critique the reasoning of others.
• Teacher Resource, Part 1, Unit 3, Lesson OA4-4, Extensions, Item 4, “c. Write down a statement you think will be true for each sequence that adds 3 each time. Be as specific as you can. d. In pairs, explain why your statement is true for each sequence. Do you agree with each other? Discuss Why or why not.”

Examples where the materials miss opportunities to prompt students to construct viable arguments or analyze the arguments of others include, but are not limited to:

• Teacher Resource, Part 2, Unit 2, Lesson NBT4-43, Extensions, Item 4, “Lela reads 5 pages of a book. It takes her 37 days to finish the book. What are the possible numbers of pages in the book? Explain how you know.” Students do not construct a viable argument or analyze the arguments of others, only explain the solution.
• In Teacher Resource, Part 1 Unit 2, Lesson NBT4-8, Extensions, students write rules related to place value and operations and “agree with a partner on wording your rule.” In this problem, “agree with your partner on the wording rule” is not giving the student an opportunity to construct a viable argument or analyze the arguments of others, just agree on wording.
• In Student Resource, Assessment & Practice Book, Part 2, Lesson NF4-28, Item 4, students critique (the fictional) Sarah’s argument that 0.25 is more than 0.3 because more is shaded in on picture compared to another. However, this problem is a slight variation of a problem the teacher already covered in the group instruction, so it doesn’t indicate that the student constructed a viable argument. It verifies if the students understood the argument taught in class.

The instructional materials reviewed for JUMP Math Grade 4 partially meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Teacher guidance and questions are found in the lessons. In some lessons, teachers are given questions that prompt mathematical discussions and engage students to construct viable arguments, and in other lessons, teachers are provided questions and sentence stems to facilitate students in analyzing the arguments of others, and to justify their answers. Also, on page A49 in the “How to use the lesson plans flexibly” states, “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)”

These sentence stems are used consistently during the Lessons and Extensions. 

Examples where teachers are provided guidance to engage students in constructing viable arguments and/or analyze the arguments of others include, but are not limited to:

• Teacher Resource, Part 1 Unit 2, Lesson NBT4-2, Extensions, Item 3, “ASK: what strategy did you use to solve this problem?”
• Teacher Resource, Part 1, Unit 2, Lesson NBT4-17, Extensions, Item 4, “In part B encourage partners to ask questions to understand and challenge each other’s thinking (see p A-49 for sample sentence and question stems).” 
• Teacher Resource, Part 1 Unit 2, Lesson NBT4-21, Extensions, Item 3, “Have students discuss their strategy with a partner, then have volunteers share with the whole class.”
• Teacher Resource, Part 1 Unit 3, Lessons OA4-13 and OA4-24, “ASK: What does the pattern say you are adding to get from the top number to the bottom number? Can you explain why that happens?”
• Teacher Resource, Part 1, Unit 5, Lesson OA4-24, Extensions, “Encourage partners to ask questions to understand and challenge each other’s thinking, choice of strategy, and choice of tool. Take up several strategies. Also, discuss different tools that can be used to show why the method works.”

Within lessons, the teacher materials are not always clear about how teachers will engage and support students in constructing viable arguments or critiquing the reasoning of others. Materials identified with the MP3 standard often direct teachers to “choose a student to answer” or “have a volunteer fill in the blank.” Questions are provided but often do not encourage students to deeply engage in MP3. In addition, although answers are provided, there are no follow up questions to help redirect students who didn’t understand. Examples include:

• Teacher Resource, Part 1, Unit 2, Lesson NBT4-14, Extensions, Item 1, "a. Marko writes the number 5 hundreds + 43 tens + 6 ones as 5,436. Is Marko correct? Explain why or why not. b. Alice writes the number 54 hundreds + 3 tens + 6 ones as 5,436. Is Alice correct? Explain why or why not." Teacher instructions in the problem tell students to "explain why or why not" but don't give specific guidance for constructing viable arguments specific to the mathematics in this lesson.
• Teacher Resource, Part 2, Unit 4, Lesson NF4-3, Extensions, Item 3, "Tasha cuts a rectangle into four parts, as shown below. (picture of a rectangle with diagonal lines connecting opposite corners). Do the four parts have the same area? Explain how you know." Teacher instructions in the problem tell students to "explain how you know" but don't give specific guidance for constructing viable arguments specific to the mathematics in this lesson.
• Teacher Resource, Part 2, Unit 6, Lesson OA4-44, Extensions, Item 6 "a. Name a prime number that has 5 as a factor. b. Can a prime number have 8 as a factor? Explain." Teacher instructions in the problem tell students to "explain" but don't give specific guidance for constructing viable arguments specific to the mathematics in this lesson.

The instructional materials reviewed for JUMP Math Grade 4 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 direction 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.
• Teacher Resource, Part 2, Unit 3, Lesson OA4-36, Vocabulary, materials use “scale factor” instead of multiplicative reasoning.