# Lesson 19

Situations and Equations

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

### Narrative

The purpose of this warm-up is to elicit the idea that letters can be used to represent an unknown quantity in a tape diagram and an equation, which will be useful when students represent unknown quantities in word problems later in the lesson. While students may notice and wonder many things about these images, the fact that a letter can be used to represent an unknown in the same way as a question mark, line, or box is the important discussion point.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

### Launch

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

### Activity

- “Discuss your thinking with your partner.”
- 1 minute: partner discussion
- Share and record responses.

### Student Facing

What do you notice? What do you wonder?

### Student Response

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### Activity Synthesis

- “These diagrams show us that we can use a letter to represent an unknown quantity just like we have used a question mark, line, or box in previous lessons. We will explore this idea further during today’s lesson.”

## Activity 1: Mai’s Beads (20 minutes)

### Narrative

The purpose of this activity is for students to match tape diagrams, equations, and descriptions of situations and explain the connection to model with mathematics (MP4). The situations share the same context and numbers. Students consider how different unknown quantities are reflected in the diagrams, depending on what's happening in the situations.

When students relate the quantities and relationships in situations to the equations and diagrams that represent them, they reason quantitatively and abstractly (MP2).

### Launch

- Groups of 3

### Activity

- “Each member of your group should pick one of the situations to read silently. Make sense of your situation and be ready to explain to your group what is happening in your situation.”
- 2 minutes: quiet think time
- “Take turns sharing your understanding of the situation you chose.”
- “When each person has shared, work together to match each situation to a diagram. Be prepared to explain how you know they match.”
- 5–7 minutes: small-group work time
- Display the three diagrams together.
- “Turn and talk with your group. How are the diagrams alike? How are they different?” (Alike: They have the same numbers. They have the same structure with a long rectangle and 2 smaller parts. Different: A different number is missing in each one. The \(n\) moves around.)
- 2–3 minutes: small-group discussion
- Share responses.
- “How did you connect the diagrams to the situations?” (The location of the unknown shows what we are missing in the situation. The first diagram matches the situation where we don’t know how many beads Mai starts with. The second diagram matches the situation where we are missing the total number of beads. In the last diagram, we don’t know how many beads are in the 2 packs she bought.)
- 2–3 minutes: small-group discussion
- Share responses.

Part 2

- “Now you are going to match an equation to each situation from Part 1. Each letter in the equation represents the unknown quantity in the situation.”
- 3–5 minutes: small-group work time
- Monitor for groups who can articulate connections between each equation and the corresponding situation. Identify students to share during synthesis.

### Student Facing

Part 1

Match each diagram with a situation. Be ready to explain your reasoning.

- Situation 1: Mai had 104 beads. She bought two packs of beads and now she has 124 beads. How many beads were in each pack?
- Situation 2: Mai had some beads. She bought 2 more packs of beads and each pack has 10 beads in it. Now she has 124 beads. How many beads did Mai have before?
- Situation 3: Mai had 104 beads. She bought 2 more packs of beads and each pack has 10 beads in it. How many beads does she have now?

Part 2

Match each equation with a letter for the unknown quantity to a situation in Part 1.

- \(104 + 2 \times 10 = n\)
- \(104 + (2 \times n) = 124\)
- \(n + 10 + 10 = 124\)

### Student Response

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### Activity Synthesis

- For each situation, select a previously identified student to share how the matching equation makes sense.

## Activity 2: Represent, Solve, Explain (15 minutes)

### Narrative

Previously, students matched diagrams and equations to situations with an unknown quantity. Here, they generate such equations, using a letter for the unknown quantity, solve problems, and explain how they know their answers makes sense. Students should be encouraged to use any solving strategy they feel comfortable with. If not yet addressed, mention that any letter can be used for the unknown quantity in their equation.

While this activity is focused on independent practice, encourage students to discuss the problem with a partner if needed. Though the task asks students to write an equation first, students may complete the task in any order that makes sense to them.

Students reason abstractly and quantitatively when they write an equation that represents the situation (MP2). They also practice making sense of a problem and its solution in terms of the context (MP1).

*MLR5 Co-Craft Questions:*Keep books or devices closed. Display only the problem stem, without revealing the question, and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the task. Ask, “What do these questions have in common? How are they different?” Reveal the intended questions for this task and invite additional connections.

*Advances: Reading, Writing*

*Engagement: Develop Effort and Persistence.*Differentiate the degree of difficulty or complexity. Some students may benefit from starting with a familiar example or one with more accessible values before working independently on the three parts to the activity.

*Supports accessibility for: Social-Emotional Functioning, Conceptual Processing*

### Launch

- Groups of 2
- “Stones are sometimes used to play games. One of the games that can be played with stones is mancala. In mancala, stones are moved from pit to pit and players try to capture the other player's stones. The picture shows a game of mancala in progress.”
- “Now we're going to solve a problem about the stones in a game of mancala.”
- “What are some ways you will be able to determine if your answer makes sense?” (I can estimate an answer using rounding. I can think about the size of the numbers in the problem.)
- Share and record responses.

### Activity

- “Take some independent time to work on this problem. You can choose to solve the problem first or write the equation first.”
- 5–7 minutes: independent work time
- Monitor for different ways students:
- write an equation
- represent the problem, such as by using a tape diagram
- decide their answer makes sense, such as thinking about the situation or by rounding

### Student Facing

Kiran is setting up a game of mancala. He has a jar of 104 stones.

From the jar, he takes 3 stones for each of the 6 pits on his side of the board.

How many stones are in the jar now?

- Write an equation to represent the situation. Use a letter for the unknown quantity.
- Solve the problem. Explain or show your reasoning.
- Explain how you know your answer makes sense.

### Student Response

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### Advancing Student Thinking

If students don't find a solution to the problem, consider asking:

- “What is this problem about? What can be counted or measured in this situation?”
- “How could you represent the problem?”

### Activity Synthesis

- Invite students to share the equations they wrote.
- Discuss differences in equations students wrote. Consider asking: “_____ wrote _____ and _____ wrote _____. How are those equations alike and different?” (I used a different letter for my unknown. I wrote \(3\times6\) instead of \(6 \times 3\).)
- Have several students share different strategies used to solve the problem. Try to feature a student-drawn tape diagram.
- Consider asking:
- “Did anyone solve the problem in a different way?”
- “Did anyone use a tape diagram to solve?”
- “How did you know if your answer made sense?”

## Lesson Synthesis

### Lesson Synthesis

“During the last few lessons, we have represented situations with equations that have a symbol or letter for an unknown quantity. We have also used diagrams to help us solve problems.”

“What do you have to think about to represent and solve problems?” (l can draw a diagram first so that I can imagine the situation, then I can write the equation more easily. I can write the equation first so I see how the numbers are related. It helps me to round numbers and think about what the answer should be close to first.)

## Cool-down: How Many Beads? (5 minutes)

### Cool-Down

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