# Lesson 21

Different Ways to Solve Problems

## Warm-up: Which One Doesn’t Belong: Expressions with 5 or 90 (10 minutes)

### Narrative

This warm-up prompts students to carefully analyze and compare features of expressions. In making comparisons, students practice looking for structure (MP7). The work here prepares students to reason flexibly and to use multiple strategies (including writing different expressions) to solve word problems later in the lesson.

### Launch

• Groups of 2
• Display expressions.
• “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

### Activity

• 2–3 minutes: partner discussion
• Record responses.

### Student Facing

Which one doesn’t belong?

1. $$5 \times 90$$
2. $$90 + 90 + 90 + 90 + 90$$
3. $$(4 \times 90) + (1 \times 90)$$
4. $$3 \times 3 \times 10 \times 5$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “What do all of the expressions have in common?” (They all have a value of 450.)
• “Can you write another expression that has the same value as these expressions but that doesn’t belong?”

## Activity 1: Going on a Field Trip (20 minutes)

### Narrative

In this activity, students encounter a multiplication problem that can be reasoned in a number of ways. After finding a solution, they analyze several other strategies. As they make sense of alternative solution paths and representations, students practice reasoning abstractly and quantitatively (MP2).

Before the lesson, display the posters with the following five strategies (as shown in the blackline master) around the classroom.

A. Clare:

If tickets were \$20 each, the cost would be $$45 \times 20$$ or 900. Because \$18 is \$2 less than \$20, we need to subtract $$45 \times 2$$ from $$45 \times 20$$, or subtract 90 from 900, which is 810.

B. Kiran:

$$10 \times 18 = 180\\ 20 \times 18 = 360\\40 \times 18 = 720 \\5 \times 18=90\\ 45 \times 18 = (40 \times 18) + (5 \times 18) = 720+90=810$$

C. Han:

100 tickets cost 1,800. 50 tickets is half of 1,800, which is 900. 45 tickets is less than 50 tickets, so they will have enough money.

D. Tyler:

$$2 \times 45 = 90$$
$$9 \times 2 = 18$$

This means:

$$18 \times 45\\=9 \times 2 \times 45 \\ = 9 \times 90 \\ =810$$

E. Mai:

MLR7 Compare and Connect. Synthesis: Lead a discussion comparing, contrasting, and connecting the different strategies. Ask, “How are the strategies the same?”, “How are they different?” and “How do these different strategies show the same information?”

### Required Materials

Materials to Copy

• Going on a Field Trip

• Groups of 2

### Activity

• 3 minutes: independent work time
• 1 minute: partners discuss responses to the first question
• 10 minutes: gallery walk to complete the second question, 1–2 minutes per poster
• As students answer the last question, monitor for the solution paths that students identify as sensible or understandable.

### Student Facing

1. Forty-five students are going on a field trip to a museum. Tickets for the museum are \$18 each. Teachers have \$900 to cover tickets for the trip. Will this be enough to cover tickets for every student?

If yes, will there be any leftover money and how much?

If no, how much more money is needed?

2. Your teacher will show five strategies for answering the previous question. Analyze the strategies.

1. Which strategy is closest to yours? With a partner, take turns explaining how your strategy is close to the poster you chose.
2. Discuss a different strategy with your partner. Try using this strategy to find the value of $$14 \times 35$$.

### Student Response

For access, consult one of our IM Certified Partners.

Students may not see connections between their strategies and the ones shown in posters A–E. Consider offering students an option to stand at a poster labeled “a totally different strategy” and asking: “How is your strategy different from the ones shown?” During synthesis, ask the class if they see any connections between the “totally different” strategies and a strategy they have selected.

### Activity Synthesis

• Poll the class on which strategy most closely resembles their own.
• Poll the class on which strategy that doesn’t resemble their own makes the most sense to them.
• Invite students to share their responses for the last question and why they found a particular strategy to make sense.

## Activity 2: A Trip to the Movies (15 minutes)

### Narrative

Students begin the activity by looking at the problem displayed, rather than in their books. At the end of the launch, students work on the problem. This activity prompts students to use what they know about multiplication, division, factors, and multiples to solve problems. The problem does not have a question, so students will need to make sense of the context and generate potential questions that might be answered (MP2). Students are encouraged in the task to attend to the details of the situation and to engage in genuine curiosity about the mathematics that is embedded within it.

Representation: Develop Language and Symbols. Represent the problem in multiple ways to support understanding. For example, invite students to represent the situation as a comic strip or a collage. Offer relevant images such as a movie theater, a sign showing admissions prices, a cash register, and a calendar. Alternatively, invite students to act out the situation. Consider using play money, signs with two days of the week written on them, and the action of “fast forwarding” through days and a night. Consider asking, “How might you represent the situation in a mathematical diagram?”
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing, Attention

### Launch

• Groups of 2

MLR5 Co-Craft Questions

• Read only the first two paragraphs without revealing the question(s).
• 2 minutes: independent work time
• 2–3 minutes: partner discussion
• Invite several students to share one question with the class. Record responses for use later in the task.
• “What information from the situation can be used to answer this question?” (The number of days tickets were purchased, the total number of money earned by the theater, the price of movie tickets)

### Activity

• “Choose a question from the list to answer.”
• “Work with a partner to complete the activity.”
• Remind students of the list of questions generated during the launch as a reference during the activity.

### Student Facing

Movie tickets are \$9 each. The theater sold the same number of tickets two days in a row. The theater made \$3,132 from ticket sales on the first day.

2. Use the given information about movie tickets to complete the following statement:

__________ tickets were sold on the first and second days.

3. A medium drink is \$7 and small popcorn is \$5. If each ticket holder purchases popcorn and a drink, how much money will the theater collect from the sales of popcorn and drink?

### Student Response

For access, consult one of our IM Certified Partners.

Students may think of questions that cannot be answered using the information provided. Consider asking: “What would we need to know to be able to answer this question?” and “How might we find out this information?” The result of these questions may not enable the students to answer the questions, but will support them in making sense of the problem and identifying information necessary to solve the problems.

### Activity Synthesis

• Select 1–2 students to share their reasoning and responses.
• If not clarified in students’ explanations, discuss a possible path for finding out the number of tickets sold over the two days using the given information. (For instance: Each ticket is \$9 and we know the total amount of money earned by selling tickets in one day, \$3,132. If we divide the total amount earned by the price of each ticket, we can find out how many tickets were sold on one day. $$3,\!132\div 9 = 348$$. If 348 tickets were sold on one day, then $$348 \times 2$$ or 696 tickets were sold in the two days. We can also multiply \$3,132 by 2 first then divide by \$9 to get the total number of tickets.)

## Lesson Synthesis

### Lesson Synthesis

“Today we encountered problems with more than one step that can each be solved using different strategies. For instance, we saw at least five ways to think about the product of 45 and 18. Some of the strategies involve using multiplication and division equations, or multiplying and dividing mentally.”

Display the five strategies from the first activity and students’ reasoning from the second activity.

“Look back at your work today. Can you find an example in which you solved a problem by using more than one step?”

Record strategies and discuss how strategies were used to address different steps in the multi-step problem.

## Cool-down: Big Weekend at the Movies (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.