# Lesson 5

One- and Two-step Comparison Problems

## Warm-up: Which One Doesn’t Belong: Something's Missing (10 minutes)

### Narrative

This warm-up prompts students to carefully analyze and compare ways to represent multiplicative comparison.

### Launch

• Groups of 2
• Display the image.
• “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.

$$6 \times 7 = 42$$

2.

42 is 6 times as many as _____.

### Activity Synthesis

• “How does each representation show an unknown?”
• “What is the unknown in each representation? Explain how you know.”
• Record student responses.
• Consider asking students to create a situation that can be described by these representations.

## Activity 1: The Book Fair (20 minutes)

### Narrative

The purpose of this activity is for students to represent and solve problems in context involving multiplicative comparison. The first 2 questions only require one operation, and the last one requires 2 operations. Students must interpret what values are unknown and make a plan for how to represent these values. Students may use one or more equations to represent the situation.

Students reason abstractly and quantitatively when they make sense of the given information and use it to find the different numbers of books (MP2).

### Launch

• Groups of 2
• “We are going to read this problem 3 times.”
• 1st Read: (First paragraph) “The book fair ordered 16 science experiment books and 6 times as many picture books.”
• “What is this story about?”
• 1 minute: partner discussion
• Listen for and clarify any questions about the context.
• 2nd Read: (First and second paragraphs) “For this year’s book fair, a school ordered 16 science experiment books and 6 times as many picture books.

Last year, the school ordered 4 times as many picture books and 4 times as many science experiment books as they did this year.”

• “Name the quantities we can count in this situation.” (The number of books of each kind.)
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Share and record all quantities.
• Reveal the questions.
• “What are different ways we can solve these problems?”
• 30 seconds: quiet think time
• 1 minute: partner discussion

### Activity

• “Solve the problems and share your thinking with a partner.”
• 2–5 minutes: independent work time
• 1–2 minutes: partner share time
• Monitor for students who write equations, including equations that have a symbol for the unknown value to solve each problem.

### Student Facing

For this year’s book fair, a school ordered 16 science books and 6 times as many picture books.

Last year, the school ordered 4 times as many picture books and 4 times as many science books than they did this year.

1. How many picture books were ordered this year?

2. How many picture books were ordered last year?

3. How many more science experiment books were ordered last year than this year?

### Activity Synthesis

• “How is this problem different from other problems we solved in this unit?” (We had to do more than one calculation to solve the problem.)
• Select 1–2 students share their solutions and equations that show multiple steps.

## Activity 2: More Book Fair Purchases (15 minutes)

### Narrative

The purpose of this activity is for students to apply what they learned about interpreting and representing multiplicative comparison to solve multi-step problems. They also make connections between strategies for solving problems. The synthesis of this activity makes use of the 5 Practices. As students are working, identify students who clearly represent using multiple steps to solve the problem and those who do not show every step yet share the same answers (MP6). Compare students' samples that show the same solution in different ways. Make connections between samples that clearly show multiplicative comparison more discretely and those that use more abstract representations.

Representation: Internalize Comprehension. To support understanding of the situation, provide students with a template to draw a comic strip or storyboard. For example, the template might have two empty boxes labeled Tuesday morning and Tuesday afternoon for the first problem, and two empty boxes labeled Mai's purchases and total sales for the second problem.
Supports accessibility for: Conceptual Processing, Organization, Attention

• Groups of 2

### Activity

• 5 minutes: independent work time
• “Compare strategies with a partner. You may decide to revise your thinking after talking with a partner.”
• As students are sharing with partners, identify students who noticed that they missed a step and revised their thinking to include the step after talking with a partner.
• 5 minutes: partner discussion
• “What did you hear or see that made you decide to revise your thinking?” ( I noticed that I missed a step when I heard or saw what my partners shared about Tuesday and Thursday.)
• “Would you be willing to share your revision in the class discussion? That was a great revision.”

### Student Facing

1. At the book fair, they collected \$13 Tuesday morning and 8 times as much as that in the afternoon. How much money did they collect at the book fair on Tuesday? 2. On Thursday, Mai purchased a biography for \$16 and a comic book for \\$3. That day, the book fair's amount of total sales was 9 times as much money as Mai spent.

What was the amount of total sales for the book fair on Thursday?

### Student Response

If students solve only one part of the problem, consider asking: “How can knowing the amount of money earned in the morning on Tuesday help us find the total amount of money earned?”

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Today we solved multiplicative comparison problems that involved more than one step.”

Display work samples for students identified during the last activity.

“In the last activity, how did you determine the amount of money collected on Tuesday?” (Multiply 8 and 13, and then add 13 to get the result.)

“How is this represented in the student work?” (When 13 and 8 are multiplied or with $$13 + 104 = 134$$)

Display the work of two students who solved the problem using two different approaches.

“How can we describe the strategies each student used to find out the amount collected on Thursday?” (One strategy is to add 16 and 3, and then multiply 19 and 9. The other is to multiply 16 and 9, multiply 3 and 9, and then add the two products at the end.)

Select a student who revised their thinking. Ask them to explain what they noticed during the partner discussion and how they changed their work after noticing what they missed. Consider asking a different student to rephrase this revision in their own words.