Lesson 22
Solve Problems Involving Large Numbers
Warmup: True or False: Sums and Differences (10 minutes)
Narrative
This warmup 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.
 \(7,\!000 + 3,\!000 = 10,\!000\)
 \(7,\!180 + 3,\!920 = 10,\!100\)
 \(423,\!450  42,\!345 = 105\)
 \(400,\!000  99,\!999 = 311,\!111\)
Student Response
For access, consult one of our IM Certified Partners.
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 multidigit 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).
This activity uses MLR6 Three Reads. Advances: reading, listening, representing
Required Materials
Materials to Gather
Launch
 Groups of 2
MLR6 Three Reads
 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
 3rd Read: Read the entire problem, including question(s) aloud.
 “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.
 Was the amount collected enough to cover all the payments? Explain or show how you know.
 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.
Student Response
For access, consult one of our IM Certified Partners.
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 multidigit numbers through a game. Students draw several cards containing singledigit 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 baseten numbers (MP7).
Supports accessibility for: Organization, Attention
Required Materials
Launch
 Groups of 2
 A set of 10 cards (from the blackline master) for each group, each card with a singledigit number (0–9).
 “How many threedigit 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 threedigit 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.

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

the greatest possible sum

the least possible sum

the greatest possible difference

the least possible difference


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

the greatest possible sum

the least possible sum

the greatest possible difference

the least possible difference

Student Response
For access, consult one of our IM Certified Partners.
Advancing Student Thinking
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 the largest numbers first.
 Start with the numbers with more zeros and fewer nonzeros.
 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.)
Cooldown: Populations of Three Cities (5 minutes)
CoolDown
For access, consult one of our IM Certified Partners.
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.
When we subtract numbers it may be necessary to decompose tens, hundreds, thousands or tenthousands before subtracting.
Finally, we learned that if the digit we are subtracting is a zero, we may need to decompose one unit of the digit 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 \(2,\!050  1,\!436\).