# Lesson 18

Diagrams and Equations for Word Problems

## Warm-up: Notice and Wonder: Diagrams (10 minutes)

### Narrative

The purpose of this warm-up is to elicit the idea that diagrams can represent many operations, which will be useful when students connect diagrams to situations and equations in a later activity. While students may notice and wonder many things about these images, what operations the diagrams could represent is the important discussion point.

### Launch

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• 1 minute: partner discussion
• Share and record responses.

### Student Facing

What do you notice? What do you wonder?

### Activity Synthesis

• “You’ve seen the first two types of diagrams before, when you represented addition situations and multiplication situations. We are going to make sense of the last type of diagram in today's lesson.”
• “What operations do you think could be represented in the last diagram?” (It could be multiplication and addition. Like you multiply 4 times 5 and add it to 142.)
• “Could the last diagram represent addition and multiplication?”

## Activity 1: Card Sort: Situations, Equations, and Diagrams (15 minutes)

### Narrative

The purpose of this activity is for students to connect two-step word problems, diagrams, and equations with a symbol for the unknown quantity. Interpreting and relating given representations prepare students to use these as tools for reasoning when they solve two-step word problems.

As students analyze written statements and other representations and make connections among them, they reason quantitatively and abstractly (MP2).

MLR8 Discussion Supports: Students should take turns finding a match and explaining their reasoning 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.
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches.
Supports accessibility for: Visual-Spatial Processing, Attention

### Required Materials

Materials to Copy

• Card Sort: Situations, Equations, and Diagrams

### Required Preparation

• Create a set of cards from the blackline master for each group of 4.

### Launch

• Groups of 4
• Distribute one set of pre-cut cards to each group of students.

### Activity

• “This set of cards includes situations, equations, and diagrams. Work with your partner to find the cards that belong together because they represent the same situation. Be prepared to explain your decisions.”
• 8 minutes: small-group work time

### Student Facing

Your teacher will give you a set of cards showing situations, equations, and diagrams.

Sort the cards into groups so that the cards in each group represent the same situation. Be ready to explain your reasoning.

A Clare had 225 beads. A friend gave her a pack of 48 beads. Then she used 70 beads to make a necklace. How many beads does Clare have now? $$225 - (6 \times 10) = {?}$$ Elena has 7 notebooks. Each notebook has 10 paper clips in it. Elena also has a box of 225 paper clips. How many paper clips does Elena have? $$225 + (6 \times 10) = {?}$$ $${?} = 225 + 48 - 70$$ Andre has 225 crayons. He buys 6 more packs and each pack has 10 crayons. How many crayons does Andre have now? Diego has a collection of 225 baseball cards. He gets 35 more cards from a friend, then buys 72 cards. How many cards does Diego have now? Han has 225 beads. Then he makes 6 bracelets for his friends. Each bracelet has 10 beads. How many beads does Han have now? $${?} = (7 \times 10) + 225$$

### Activity Synthesis

• “Were there any cards whose placement you disagreed on? How did you come to an agreement?” (We went back and read the situation carefully together.)
• Choose a set of cards that belong together, such as B, D, and L, to discuss in detail. Ask, “How do the equation and diagram represent the situation?”
• Attend to the language that students use to describe their matches and the situations, equations, and diagrams, giving them opportunities to describe them more precisely.

## Activity 2: Makes Sense to Me: A Gallery Walk (20 minutes)

### Narrative

The purpose of this activity is for students to solve one of the problems from the card sort in the previous activity and examine their classmates' solutions to other problems. Students work in groups to create a poster of their solution. As students visit the posters, they leave comments about how they know the solution on the poster makes sense. As students make comments on the work of others, they critique the reasoning of others (MP3).

### Required Materials

Materials to Gather

### Launch

• Groups of 4
• Assign each group a situation from the previous activity.
• Give each group tools for creating a visual display and sticky notes.

### Activity

• “Now you are going to solve a problem from the card sort with your group.”
• “After you’ve solved the problem together, create a poster that shows your solution strategy.”
• 8–10 minutes: small-group work time

### Student Facing

1. Your teacher will assign a problem to your group. Work together to solve your assigned problem.
2. Create a poster of your group’s solution. Organize your work so that it can be followed by others.
3. As you visit other groups' posters, consider how each answer makes sense.

Choose one poster and make a comment on the solution. Write on your sticky note how you know the answer makes sense.

### Student Response

If students try to find exact answers to determine if their peers’ solutions make sense, consider asking:

• “What are some ways that we could determine if the solution makes sense without solving the problem?”
• “How could we use estimating to determine if the solution makes sense? How could we use rounding?”

### Activity Synthesis

• Display posters around the room. If more than one group solved the same problem, consider grouping their posters together.
• Give each student a sticky note.

## Lesson Synthesis

### Lesson Synthesis

Display a corresponding set of cards that show a diagram, situation, and equation representing the same situation, such as B, D and L:

Elena has 7 notebooks. Each notebook has 10 paper clips in it. Elena also has a box of 225 paper clips. How many paper clips does Elena have?

$${?} = (7 \times 10) + 225$$