# Lesson 21

Solve Problems Using the Four Operations

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

### Narrative

The purpose of this warm-up is to elicit the idea that many different questions could be asked about this situation, which will be useful when students solve problems in a later activity. While students may notice and wonder many things about this situation, the various questions that could be asked about the situation are the important discussion points.

### Launch

• Groups of 2
• Display the situation.
• “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?

A farmer picked some apples.
Some of the apples are packed into boxes and some are not.

### Activity Synthesis

• “What does it mean that some apples are packed into boxes and some are not?” (Some apples are in groups and some apples are just loose in one big group.)
• “What questions could we ask about this situation?” (How many apples did the farmer pick? How many boxes had apples in them? How many apples were in each box?)

## Activity 1: Apple Adventure (20 minutes)

### Narrative

The purpose of this activity is for students to think about what they need to know to solve two-step word problems. Students choose numbers that make sense together to complete the problem from the warm-up. They articulate relationships between the quantities in the problem to justify their number choices. If students quickly find a combination of numbers that work, encourage them to see if there are other possibilities or to write a completed situation with the numbers they have chosen.

Students who do not choose a matching set of numbers quickly make sense of and persevere in solving the problem as they consider the relationship between the different quantities and the restrictions that puts on which numbers can describe the situation (MP1).

Action and Expression: Internalize Executive Functions. Invite students to plan a strategy, including the tools they will use, for completing the chart. If time allows, invite students to share their plan with their partner before they begin.
Supports accessibility for: Conceptual Processing, Organization

### Launch

• Groups of 2 and 4
• Keep the situation from the warm-up displayed.
• “Suppose the boxes the farmer packs are all the same size.”

### Activity

• “A list of numbers is shown in the activity. Work with your partner to choose 4 numbers that would make sense together in this situation. If you find one combination of numbers that works, you can look for other combinations.”
• 8–10 minutes: partner work time
• Groups of 4
• “Share with another group of students how your number choices make sense.”
• 2–3 minutes: small-group discussion

### Student Facing

A farmer picked some apples. Some of the apples are packed into boxes and some are not.

From the list, choose 4 numbers that would make sense together in this situation. Write your choices in the table. Be ready to explain how your numbers make sense together.

total number of apples number of apples not in boxes number of boxes number of apples in each box

### Activity Synthesis

• Display this partially completed row in the table, such as:
total number of apples number of apples not in boxes number of boxes number of apples in each box
200 152 8

• “If you were given this information, how would you find the number of apples in each box?” (I could subtract 152 from the total of 200 and divide the answer by 8.)
• “What equation can we write to represent the situation in this example? Let’s use a letter for the quantity that we don’t know.”
• 1 minute: quiet think time
• Record equations that students wrote, for instance:
•  $$(200 - 152) \div 8 = n$$
• $$8 \times n + 152 = 200$$
•  $$152 + 8 \times n = 200$$

## Activity 2: Apple Days (15 minutes)

### Narrative

The purpose of this activity is for students to represent a problem with an equation using a letter for the unknown quantity and solve the problem. Students should be encouraged to use whatever strategy or representation makes sense to them.

The synthesis focuses on student thinking for the first problem. Students might represent the situation with:

• a tape diagram or an area diagram
• an equation that uses multiplication
• an equation that uses division

If students struggle to get started on a problem, encourage them to create a drawing or diagram. Students may also represent the situation or solve the problem before writing an equation if that makes more sense to them. While this activity is focused on independent practice, students can discuss with a partner if needed.

MLR8 Discussion Supports. Prior to solving the problems, invite students to make sense of the situations. Monitor and clarify any questions about the context.

### Launch

• “Now we’re going to solve some problems about an event at the apple orchard.”
• Survey the class on their familiarity with events or activities at farms or orchards.
• If students are familiar, ask: “What are some things you might see or do at a farm or an orchard?”
• If students are unfamiliar, share some activities that might take place at an orchard. Consider showing some images of a market or an event at an orchard.
• 1–2 minutes: partner discussion
• Share responses.

### Activity

• 8–10 minutes: independent work time
• Monitor for:
• a variety of strategies students use to solve or represent each problem
• lingering questions or common misconceptions and how students overcome them
• “How could you represent that?”

### Student Facing

Tyler and Clare are helping with a festival at an apple orchard.

1. Tyler is stacking apples to sell at the event. There are 85 apples for his display. He has already made 5 rows of 10 apples. How many apples are left?

1. Write an equation with a letter for the unknown quantity to represent this situation.
2. Solve the problem. Explain or show your reasoning.
2. Clare is helping sell baked goods at the event. A customer buys 8 brownies that cost \$3 each. Clare adds that money to the cash box and now there is \$125 in the cash box. How much money was in the cash box before that purchase?

1. Write an equation with a letter for the unknown quantity to represent this situation.
2. Solve the problem. Explain or show your reasoning.
3. The market at the orchard had 200 jars of applesauce for sale. At the end of the event, 184 jars had been sold. The rest of the jars were shared equally among 4 people who work there. How many jars of applesauce did each person get?

1. Write an equation with a letter for the unknown quantity to represent this situation.
2. Solve the problem. Explain or show your reasoning.

### Student Response

If students say they aren’t sure how to get started on the problem, consider asking: “What is the problem about?” and “How could you represent the problem?”

### Activity Synthesis

• Select students who used different strategies to share their responses and reasoning for each problem.

## Lesson Synthesis

### Lesson Synthesis

“What did you find most challenging about solving these problems?” (There’s a lot of information to keep track of. I have a hard time understanding how all the numbers are related to each other.)

“What ideas do you have for overcoming those challenges?” (drawing a diagram and labeling it with the numbers that we know, reading the problem carefully and acting it out, organizing what we know and don't know in a table)

“How did you know if your answer made sense?” (I put the number back into the problem and did the math to check if it makes sense. I made an estimate first so that I had an idea of about how large the answer should be.)

## Student Section Summary

### Student Facing

In this section, we divided larger numbers and solved problems that involve division.

We used base-ten blocks, diagrams, and equations to represent the numbers we divided. To help us divide, we used what we know about place value, equal groups, and the relationship between multiplication and division.

For example, here are some ways we could find the value of $$52 \div 4$$

• Put 5 tens and 2 ones into 4 equal groups.
• Think about how many groups of 4 are in 52.

10 groups of 4 make 40.
3 groups of 4 make 12.
13 groups of 4 make 52.

• Use multiplication facts and write equations.

$$4 \times 10 = 40$$
$$4 \times 3 = 12$$

$$10 + 3 = 13$$
$$4 \times 13 = 52$$

At the end of the section, we used all four operations to solve problems.