Lesson 16

The purpose of this Number Talk is to elicit strategies and understandings students have for products of 4 and 6 as they relate to products of 5. These understandings help students develop fluency and will be helpful later when students consider solutions for and solve two-step word problems.

When students use products of 5 to determine products of 4 by thinking of them as one fewer group or one fewer object in each group, or work from products of 5 to determine products of 6 by thinking of them as one more group or one more object in each group, they look for and make use of structure (MP7).

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

• “How does knowing \(5\times7\) help you find some of the other products?” (I can remove a group of 7 to find \(4\times7\) or add a group of 7 to find \(6\times7\).)
• Consider asking:
  • “Who can restate _____’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone approach the problem in a different way?”
  • “Does anyone want to add on to _____’s strategy?”

The purpose of this activity is for students to apply what they learned about rounding in prior lessons to think about all the numbers that would round to a given number. Students should be encouraged to use whatever representations make sense to them. Although the number line is often used to represent rounding, it is also worth sharing other ways that students are representing or thinking about rounding.

• Groups of 2
• “Diego is thinking of a number. When you round Diego's number to the nearest ten, the answer is 40. What's a number that could be Diego's number? What's a number that could not be Diego's number?” (38 rounds to 40 so it could be his number. 34 does not round to 40 so it couldn't be his number.)
• 30 seconds: quiet think time
• Share 3–5 responses. Highlight the idea that more than one number can round to 40 and that some numbers are greater than 40 and some are less than 40.

• “Work with your partner on all these problems. Be sure to justify your reasoning.”
• 5–7 minutes: partner work time

If students don’t find all the numbers that round to the given number, consider asking:

• “How did you determine that these numbers would round to ____?”
• “How could you use a number line to find all the numbers that round to ___?”

• “How did you decide what numbers would round to 40?” (We looked at all the numbers that are closer to 40 than 50 or 30.)
• Consider asking:
  • “What does 35 round to?” (40 because it is halfway between 30 and 40)
  • “What does 45 round to?” (50 because it is halfway between 40 and 50)
• “Look at your responses for the first 2 problems. What patterns do you see in the numbers? Why is that happening?” (I see they each start with a 5 in the ones place below it because it’s halfway to the nearest ten, and the numbers end with a 4 in the ones place because that is closer than the next ten.)
• “How did you use what you learned from the first 2 problems to think about the last problem?” (Instead of thinking about fives, we thought about fifties. We looked at all the numbers that are closer to 600 than 500 or 700.)
• Consider asking:
  • “What does 550 round to?” (600 because it is halfway between 500 and 600.)
  • “What does 650 round to?” (700 because it is halfway between 600 and 700.)

The purpose of this activity is for students to apply what they’ve learned about rounding to play a game in which each student generates a mystery number with three clues. The three clues describe whether the mystery number is even or odd, what it rounds to, and two numbers that it’s between. It is possible that more than one number can fit the clues provided. In the synthesis, students reflect on which clues were most helpful for determining the mystery number.

• Each student needs an index card.

• Groups of 4
• “We’re going to play a game in which you have to guess a mystery number that someone in your group writes down.”
• Choose a mystery number and give the class three clues. Play a round of the game with the class and discuss the clues. Consider using 275 and these clues:
  • “My mystery number is odd.”
  • “My mystery number rounds to 300.”
  • “My mystery number is between 270 and 278.”
• “You’ll give your group three clues by finishing three sentences. The first clue should tell whether the number is even or odd. Take a couple minutes to choose a mystery number and write down your three clues.”
• 2 minutes: independent work time

• “Now, you’re going to play the game with your group. Everyone will get a chance to share the clues for their mystery number. If you have time, you can each create a new mystery number with three new clues.”
• 12–15 minutes: small-group work time

• “As you played the game, what clues were the most helpful and why?” (Knowing how the mystery number would round to the nearest ten was really helpful because that really narrowed it down. Knowing the numbers the mystery number was between was helpful if it was something like 150 and 160, but not if it was between 100 and 200.)