Lesson 22

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.

• 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

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

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

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).

This activity uses MLR6 Three Reads. Advances: reading, listening, representing

• 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

• 3–5 minutes: partner work time

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

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).

• 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?”

• “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

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?”

• 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.)