# Lesson 12

Solve Problems Involving Multiplication

## Warm-up: What Do You Know About 1 Year? (10 minutes)

### Narrative

The purpose of this warm-up is to elicit students’ knowledge about 1 year as a measure of time and the ways it can be represented. The reasoning and conversations here will be helpful as students solve problems that involve time in years later in the lesson.

### Launch

• Display: “1 year”
• “What do you know about 1 year?”
• 1 minute: quiet think time

### Activity

• Record responses.

### Student Facing

What do you know about 1 year?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “What does a year measure?” (time)
• “What are some aspects of time that are related to a year?” (seasons, months, weeks, days)

## Activity 1: Time Flies When We Leap Years (25 minutes)

### Narrative

In this activity, students use what they learned about multiplication of multi-digit numbers and unit conversion to solve problems involving measurements. Students may choose to represent the situations in a number of ways—concretely or visually (by drawing diagrams) or abstractly (by writing expressions and equations). While some problems can be reasoned additively, students may opt to reason multiplicatively for practical reasons.

Regardless of their chosen representations and reasoning strategy, students reason quantitatively and abstractly when they interpret and solve the questions about different units of time (MP2).

This activity uses MLR7 Compare and Connect. Advances: representing, conversing

Action and Expression: Develop Expression and Communication. Provide access to a variety of tools, including colored pencils, grid paper, base-ten blocks, and a visual display that students can use as a reference showing a variety of strategies from throughout the section.
Supports accessibility for: Conceptual Processing, Organization, Memory

### Launch

• Groups of 2–4
• Give each group tools for creating a visual display.
• “Today we’ll solve some problems that involve measurements.”
• “Before we solve any problem, let’s do some estimation.”
• Read the first problem as a class.
• “Estimate: About how many days old is the baby elephant?”
• 1 minute: quiet think time
• Display the following ranges and poll the class on their estimate. (Alternatively, post each range in a different part of the classroom and ask students to stand by the range that reflects their estimate.)
• 280 to 299
• 300 to 349
• 350 to 400
• Select a student who selected each range to explain their estimate. Then, ask students if they’d revise their estimate.

### Activity

• “Take a few quiet minutes to solve the first two problems. Then, discuss your responses with your group and work on the last problem together.”
• 5 minutes: independent work time
• 5 minutes: group work time
• Monitor for the different representations and strategies used to solve the problems.
• Assign one problem to each group.

MLR7 Compare and Connect

• “Create a visual display to show how you solved the problem assigned to your group. Organize your work so that it can be followed by others.”
• 3 minutes: group work time
• 5 minutes: gallery walk
• Prompt students to make note of the different strategies they see and any solutions that they question.

### Student Facing

1. A baby elephant was born exactly 48 weeks ago. How many days old is she?
2. A leap year has 366 days. A non-leap year (or a common year) has 365 days. How many days are in 3 leap years?
3. In our calendar system, some months are 31 days long, some are 30 days long, and one month (February) is either 28 or 29 days long.

What if the calendar system changed so that each month has 31 days? How many more days would there be in a year?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “Most of the problems can be solved by multiplication. What is the same between the solutions and multiplication strategies that you saw? What’s different?”
• 30 seconds quiet think time
• 1 minute: partner discussion
• Record students’ responses, or display students’ diagrams and representations.
• “What are some common strategies for multiplying a multi-digit number by a one-digit number (48 and 7, or 366 and 3)?”
• “What are some strategies you saw for multiplying 2 two-digit numbers (12 and 31)?”

## Activity 2: Coin Collection (10 minutes)

### Narrative

This activity offers students more practice with using multiplication to solve contextual problems (MP2), including situations in which at least one factor is four digits long, and to generate a new problem according to some parameters.

### Launch

• Groups of 2–4

### Activity

• 6–7 minutes: independent work time
• 2–3 minutes: group discussion
• Monitor for the different ways students represent the situations and solve the problems.

### Student Facing

1. Lin’s family has collected 2,074 nickels over the years. How many pennies are worth the same amount?
2. If Lin’s family saved 2,074 nickels each year for 4 years, how many nickels would her family have?
3. Create a situation that involves a problem that can be solved by finding the value of $$8 \times 1,\!049$$. Solve the problem and show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Today we used what we have learned about multiplication to solve problems involving measurement.”

Select students to share their responses to the problems in the last activity. As each student shares, ask if others in the class solved it the same way and if they approached it differently.

Prompt students to explain what their numerical solutions represent in each situation.

“How would you know if your solutions were correct?” (Sample responses: I used another strategy to see if I got the same answer. I estimated first so that I had an idea how big or small the answer would be. I checked with my groupmates.)

Consider asking: “When you had to multiply numbers, which method did you rely on the most? What made you choose that method?”

## Cool-down: Leap Year (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Section Summary

### Student Facing

In this section, we learned to multiply factors whose products are greater than 100, using different representations and strategies to do so.

When working with multi-digit factors, it helps to decompose them by place value before multiplying. For example, to find the value of $$4 \times 5,\!342$$, we can decompose the 5,342 into its expanded form, $$5,\!000 + 300 + 40 +2$$, and then use a diagram or an algorithm to help us multiply.

In both the diagram and the algorithm, the 20,000, 1,200, 160, and 8 are called the partial products. They are the result of multiplying each decomposed part of 5,342 by 4.

We can do the same to multiply a two-digit number by another two-digit number.

For example, here are two ways to find the value of $$31 \times 15$$. The 31 is decomposed into $$30 + 1$$ and 15 is decomposed into $$10 + 5$$.