# Lesson 11

All About Tape Diagrams

## Warm-up: Number Talk: Addition and Subtraction (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for relating the operations of addition and subtraction. The equations \(27 + 13 = 40\), \(40 - 13 = 27\), and \(40 - 27 = 13\) each show the same relationship between numbers. Students will use this idea during the lesson when they are given a tape diagram and discuss why an addition or subtraction equation might match.

### Launch

- Display one expression.
- “Give me a signal when you have an answer and can explain how you got it.”
- 1 minute: quiet think time

### Activity

- Record answers and strategies.
- Keep expressions and work displayed.
- Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

- \(7 + 13\)

- \(27 + 13\)

- \(40 - 13\)

- \(40 - 27\)

### Student Response

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### Activity Synthesis

- “How can you use your answer for \(27 + 13\) to find the value of \(40 - 13\)?” (\(27 + 13\) is 40 so \(40 - 13\) is 27.)
- “How can you use your answer for \(27 + 13\) to find the value of \(40 - 27\)?” (\(27 + 13\) is 40 so \(40 - 27\) is 13.)

## Activity 1: Represent Story Problems (20 minutes)

### Narrative

The purpose of this activity is to match story problems with equations and tape diagrams. The numbers were selected so that there is a pair of stories, tapes, and equations using the same set of numbers. In order to determine the matching sets, students will need to think about and describe how the expression and diagrams represent the context of the story (MP2).

The goal of the activity synthesis is to discuss matching representations, highlighting how the tape diagram can help students understand the story and visually represent the equation.

*MLR8 Discussion Supports.*Students should explain their reasoning for making each match to their partner. Display the following sentence frames for all to see: “I noticed _____, so I matched . . . .” Encourage students to challenge each other when they disagree.

*Advances: Speaking, Conversing*

### Required Materials

Materials to Copy

- Represent Story Problem Cards

### Required Preparation

- Create a set of cards from the blackline master for each group of 2.

### Launch

- Groups of 2
- Give each group a set of cards.

### Activity

- “You have a set of 4 story problems, 4 diagrams, and 4 equations. Your goal is to find 4 matching sets.”
- 5 minutes: independent work time
- 5 minutes: partner discussion
- Monitor for how students distinguish the pairs of stories using the same numbers, especially those who use the tape diagrams to figure out the operation and equation.

### Student Facing

### Student Response

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### Advancing Student Thinking

- “What is the same about the cards in this group? What is different?”
- “How does the equation match the story? How does it match the tape diagram?”
- “How does the diagram match the story? How does it match the equation?”

### Activity Synthesis

- Display tape diagrams for stories A and B.

- “How are the two diagrams the same? How are they different?” (Both diagrams have a 53 and a 25. One diagram shows two separate rectangles comparing quantities. 53 is the total for one of the rectangles. In the other diagram 53 is one part and the total is unknown.)
- “Which problem does the first diagram match? How do you know?” (The baseball and tennis ball problem. The first bar shows 53 and the second bar shows 25 less than 53.)
- “How is this different from the pine cone problem?” (53 is how many pine cones are left after 25 were taken away.)
- “Why does the second diagram match the pine cone problem?” (It shows that 53 are left if you take away 25.)

## Activity 2: Write Stories (15 minutes)

### Narrative

The purpose of this activity is to write story problems that match given tape diagrams (MP2). One diagram represents a one-step Compare problem while the other two show two-step problems. For the tape diagram with three addends, students might:

- write Put Together/Take Apart or Add To stories with an unknown addend
- write Take Away stories with the change or difference unknown
- write a Compare problem that compares the full tape to the sum of the first two sections

For the tape diagrams that represent Compare problems, students may phrase the question using “How many more . . . ?” or using “How many fewer . . . ?” Although these diagrams are typically used to represent Compare problems, students may also choose to write Put Together/Take Apart or Take Away problems.

The lesson synthesis highlights different types of stories students write that match the same tape diagram.

*Action and Expression: Internalize Executive Functions.*Invite students to talk through their strategy, including the type of problem represented in the tape diagram image, the operation that is needed, and a possible context for the story that will represent the tape diagram. If time allows, invite students to share their plan with a partner before they begin writing.

*Supports accessibility for: Attention, Organization*

### Launch

- Groups of 3
- “We are now going to write math stories. What are some things in our school that you could count and write stories about?” (kids, teachers, chairs, tables, classrooms, sports equipment, posters, windows)
- Make a list of student ideas on a chart and save the chart for the next day’s lesson.

### Activity

- “Now you are going to look at a few tape diagrams and choose one diagram to write a story problem that matches it. Use the chart for ideas if you’d like.”
- “In your group of 3, make sure everyone is working on a different diagram. You will have 5 minutes to work on your own, then you will have time to share your story problem with your group and get feedback on it.”
- “Before we come back together as a whole group, take time to make any revisions to your stories after you receive feedback.”
- 5 minutes: independent work time
- 5 minutes: group work time
- Monitor for students who write different types of stories for the same tape diagram.

### Student Facing

Choose one of the diagrams. Write and solve a story problem that the diagram could represent.

Problem ______

### Student Response

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### Activity Synthesis

- Invite previously identified students to share their stories for the same tape diagram or share one story and have the class generate a different story after discussing the student work.
- “How did you know that the second diagram represents a comparison?” (It has two different rectangles. We know the larger amount, but it shows a question mark for the smaller amount.)
- “How did you solve the problem?” (I subtracted 20 from 63 and then subtracted 6. Or, I added 4 to 26 to make 30, then added 3, and then added 30 more.)

## Lesson Synthesis

### Lesson Synthesis

“Today we connected story problems to diagrams and equations. We also used diagrams to create our own story problems.”

“What did you look for in the diagrams to help you match it to a story or an equation?”

“How did the diagram help you write a story problem?”

## Cool-down: What’s the Story? (5 minutes)

### Cool-Down

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