# Lesson 22

Solve Problems Involving Large Numbers

## Warm-up: True or False: Sums and Differences (10 minutes)

### Narrative

This warm-up prompts students to look carefully at the sum and difference of the digits in each place and to remember to compose units when needed. It also prompts students to make use of structure (MP7). For example, the last expression would be cumbersome to calculate with the standard algorithm. Recognizing that 99,999 is 1 less than 100,000 would enable students to find the difference much more quickly. Likewise, students who notice that $$300,\!000 + 99,\!999$$ is only 1 away from 400,000 would know that 311,111 is far too high and cannot be the difference between 400,000 and 99,999.

### Launch

• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategy.
• Repeat with each statement.

### Student Facing

Decide if each statement is true or false. Be prepared to explain your reasoning.

• $$7,\!000 + 3,\!000 = 10,\!000$$
• $$7,\!180 + 3,\!920 = 10,\!100$$
• $$423,\!450 - 42,\!345 = 105$$
• $$400,\!000 - 99,\!999 = 311,\!111$$

### Activity Synthesis

• “How can you explain your answer without finding the value of both sides?”

## Activity 1: The Fundraiser (15 minutes)

### Narrative

In this activity, students perform multi-digit addition and subtraction to solve problems in context and assess the reasonableness of answers. The situation can be approached in many different ways, such as:

• Arrange the numbers in some way before adding or subtracting.
• Add the largest numbers first.
• Add two numbers at a time.
• Add numbers with the same number of digits.
• Subtract each expense, one by one, from the amount collected.

Students reason abstractly and quantitatively when they make sense of the situation and decide what operations to perform with the given numbers (MP2).

### Required Materials

Materials to Gather

### Launch

• Groups of 2

• Display only the problem stem, without revealing the questions.
• “We are going to read this problem 3 times.”
• 1st Read: Read both parts of the problem including fall and spring purchases.
• “What is this story about?”
• 1 minute: partner discussion.
• Listen for and clarify any questions about the money raised and the money paid.
• 2nd Read: Read both parts of the problem including fall and spring purchases.
• “What are all the things we can count in this story?” (costs of uniforms, meet fees, competition expenses, awards, travel costs)
• 30 seconds: quiet think time
• 2 minutes: partner discussion
• Share and record all quantities.
• Reveal the questions
• “What are different ways we can solve this problem?”
• 30 seconds: quiet think time
• 1–2 minutes: partner discussion

### Activity

• 3–5 minutes: partner work time

### Student Facing

A school’s track teams raised \$41,560 from fundraisers and concession sales. In the fall, the teams paid \$3,180 for uniforms, \$1,425 in entry fees for track meets, and \$18,790 in travel costs.

In the spring, the teams paid \$10,475 in equipment replacement, \$1,160 for competition expenses, and \\$912 for awards and trophies.

1. Was the amount collected enough to cover all the payments? Explain or show how you know.
2. If the amount collected was enough, how much money did the track teams have left after paying all the expenses? If it was not enough, how much did the track teams overspend? Explain or show how you know.

### Activity Synthesis

• Invite students who solved the problem in different ways to share.

## Activity 2: The Least and the Greatest of Them All (20 minutes)

### Narrative

In this activity, students practice adding and subtracting multi-digit numbers through a game. Students draw several cards containing single-digit numbers, arrange the cards to form two numbers that would give the greatest and the least sums and differences. To meet these criteria, students look for and make use of structure in base-ten numbers (MP7).

Action and Expression: Develop Expression and Communication. Provide access to blank four-squares labeled as the greatest possible sum, the least possible sum, the greatest possible difference, and the least possible difference. Invite students to use this template to keep their work organized.
Supports accessibility for: Organization, Attention

### Required Materials

Materials to Gather

Materials to Copy

• 0-9 Digit Cards

### Launch

• Groups of 2
• A set of 10 cards (from the blackline master) for each group, each card with a single-digit number (0–9).
• “How many three-digit numbers could we form with 1, 2, and 3? Think of all of them.” (123, 132, 213, 231, 312, and 321)
• “Can you find a pair of numbers from your list that would produce the least sum? The greatest sum? The smallest difference? The greatest difference?”

### Activity

• “Take turns picking cards, one card at a time.”
• “For the first problem, pick 3 cards. Work together to make three-digit numbers that would give the greatest and the least sums and differences.”
• “For the second problem, repeat with 4 cards. Check each other's calculations.”
• 10 minutes: partner work time

### Student Facing

Your teacher will give you and your partner a set of 10 cards, each with a number between 0 and 9. Shuffle the cards and put them face down.

1. Draw 3 cards. Use all 3 cards to form two different numbers that would give:

1. the greatest possible sum

2. the least possible sum

3. the greatest possible difference

4. the least possible difference

2. Shuffle the cards and draw 4 cards. Use them to form two different numbers that would give:

1. the greatest possible sum

2. the least possible sum

3. the greatest possible difference

4. the least possible difference

### Student Response

Students may randomly place numbers and use a guess-and-check method. To draw attention to place value and possible strategies, consider asking: “When making the largest sum possible, how do you decide which digits to choose for each place?”

### Activity Synthesis

• Ask students to share their sums and differences and the rest of the class to check if they actually are the greatest and least sums and differences.
• “After playing a couple of rounds, did you figure out how to find the greatest or least sum or difference more quickly? How?” (Answers vary. Students say they estimated using the place value of the first digits of each number.)

## Lesson Synthesis

### Lesson Synthesis

Display these numbers:

• 732

• 3,005

• 8,401

• 12,475

• 218,699

“In this lesson, you added and subtracted lots of large numbers to solve problems. Suppose we’re working with these large numbers.”

“What are some ways to estimate the sum or difference of a bunch of numbers without adding them?” (Round each number to make them easier to add or subtract. Look at the digits and the place values of the numbers involved to get a sense of the sizes of the numbers.)

“If careful calculations are needed, what are some ways to organize the numbers and add or subtract them efficiently?” Some ideas:

• Start with computations that would result in multiples of 10, 100, 1,000, and so on, which would make other calculations easier. For example, in the given list of numbers, we could add 218,699 and 8,401 first because it’d give 227,100, and then add 12,475 and 3,005 because they both end in 5 and would add up to 15,480.

“How might we find two numbers that give the greatest sum or greatest difference without trying to find the sum and difference of every pair of numbers?” (Pay attention to the size of each number, based on the number of digits and their place values.)

## Student Section Summary

### Student Facing

In this section, we used our understanding of place value and expanded form to add and subtract large numbers using the standard algorithm.

We learned how to use the algorithm to keep track of addition of digits that results in a number greater than 9.

Whenever we have 10 in a unit, we make a new unit and record the new unit at the top of the column of numbers in the next place to the left.

Sometimes, it is necessary to look two or more places to the left to find a unit to decompose. For example, here is one way to decompose a ten and a thousand to find $$2,\!050 - 1,\!436$$.