Lesson 14

This Number Talk encourages students to compose or decompose multiples of 7 and to rely on properties of operations to mentally solve problems. The ability to compose and decompose numbers will be helpful when students divide multi-digit numbers. It also promotes the reasoning that is useful when finding multiples of a number, or when deciding if a number is a multiple of another number.

• 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.

• “What do the expressions have in common?” (They all involve division by 7. The dividends are all multiples of 7. The results have no remainders.)
• “How did the first three expressions help us find the value of the last expression?”
• 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 expression in a different way?”
  • “Does anyone want to add on to____’s strategy?”

This activity prompts students to use the relationship between multiplication and division and their understanding of factors and multiples to solve problems about an unknown factor (MP7). Students recognize such problems as division situations. Here the dividends are three-digit numbers and the divisors are one-digit numbers. Students use the context of factors and multiples to interpret division that results in a remainder.

One approach for solving these problems is by decomposing the dividend into familiar multiples of 10 and then dividing the remaining number (which is now smaller) by the divisor. Another is to find increasingly greater multiples of the divisor until reaching the dividend. Both are productive and appropriate. In the synthesis, help students see the connections between the two paths.

• “Who can remind the class of the meaning of ‘multiple?’”

• 4–5 minutes: quiet work time for the first set of questions
• Monitor for students who:
  • decompose 104 into a multiple of 8 and another number
  • compose 104 from increasingly larger multiples of 8
  • use multiplication or division equations to show their reasoning
• Pause for a discussion before the second set of questions.
• Select students who use different decomposition strategies to share responses. Record and display their reasoning.
• “Let’s revisit each question we just answered. What equations could we write to represent them?”
• Display equations and highlight their connection to the questions:
  1. \(8 \times {?} = 104\) or \(104 \div 8 = {?}\)
  2. \(15 \times 13 = {?}\) or \( 13 \times 15 = {?}\)
  3. \({?} \times 13 = 286\) or \(286 \div 13 = {?}\)
• 4–5 minutes: quiet work time for the second set of questions

• Invite students to share responses for the last set of questions.
• Highlight that to divide a number by a smaller number—say, divide 150 by 7, we can:
  • Use familiar multiples or multiplication facts to help us. For example, if we know \((20 \times 7) + (1 \times 7) = 147\), we know the result is 21 with a remainder of 3, or \(150 = 21 \times 7 + 3\).
  • Think of the dividend in smaller chunks. For example: We can see the 150 as \(140 + 10\) and divide each 140 and 10 by 7 separately, which gives \(20 + 1\) with a remainder of 3.

• Groups of 2

• 5 minutes: independent work time on the first question
• 2 minutes: partner discussion
• 2–3 minutes: partner work time on the second question
• Monitor for students who clearly show how they use partial products or partial quotients to answer the questions.

• “How could we use division to help us find out the mystery number?” (If the result has a remainder, then the divisor could not be the mystery number. 153 divided by 6 has 3 as a remainder. 153 divided by 9 has no remainder, so 9 is the mystery number.)