Lesson 17

• Display the number.
• “What do you know about \(\frac{60}{6} + \frac{6}{6}\)?”
• 1 minute: quiet think time

• Record responses.

• “What are some expressions that are equal to \(\frac{60}{6} + \frac{6}{6}\)?” (\(10 + 1\), 11, \((60 \div 6)+ (6 \div 6)\))

The purpose of this activity is for students to relate their understanding of fractions as representing division to think about decomposing a quotient into partial quotients in a way that simplifies the calculation.  To find the value of \(78 \div 6\), students may

• use their understanding of division.
• use multiplication and find how many groups of 6 there are in 78.
• use the fraction expressions from the first part of the problem.

• Groups of 2

• 5–8 minutes: partner work time
• Monitor for students who:
  • use multiplication to find the value of \(78\div6\).
  • use the expression \(\frac {60}{6} + \frac {18}{6}\) to find the value of \(78 \div 6\).
  • use the expression \(\frac {66}{6} + \frac {12}{6}\) to find the value of \(78 \div 6\).

If students do not identify all of the expressions whose value is equal to \(\frac{78}{6}\), ask, “How did you decide which expressions were equal to \(\frac{78}{6}\)?”

• Invite students to share the expressions that match \(78 \div 6\).
• Display: \(78 \div 6 =\frac {78}{6} \)
• “How do we know this is true?” (A fraction shows you are dividing the numerator by the denominator.)
• Display: \(78 \div 6 = \frac {60}{6} + \frac {18}{6}\)
• “How can you use this equation to find the value of \(\frac{78}{6}\)?” (I know \(\frac{60}{6}\) is 10 and \(\frac{18}{6}\) is 3 so \(\frac{78}{6}\) is 13.)
• “In the next activity we will use expressions with fractions to find values of other quotients.”

The purpose of this activity is for students to find the whole number value of quotients using sums of fractions and to think about which sums were most helpful. They may notice that it is helpful to decompose the dividend into a multiple of the divisor and multiples of 10 are particularly helpful. This is closely related to how students found quotients using partial products which requires strategically choosing the number of groups of the divisor to subtract.

This activity uses MLR1 Stronger and Clearer Each Time. Advances: Reading, Writing.

• Groups of 2
• Display:
  \(\frac{60}{6} + \frac{18}{3}\)
  \(\frac{55}{6} + \frac{13}{6}\)
• “Which of these expressions would you use to find the value of \(\frac{78}{6}\)?” (The first one because the fractions have nice whole number values.)
• 1–2 minutes: partner discussion
• “You are going to choose expressions like this one that are helpful for finding quotients.”

• 5–8 minutes: independent work time
• 1–2 minutes: partner discussion
• Monitor for students who:
  • explain that numerators that are multiples of the divisor are helpful to divide.
  • explain that numerators that are not multiples of the divisor require working with fractions.

• “Share your response as to why some expressions were helpful and others were not with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 3–5 minutes: structured partner discussion.
• Repeat with 2–3 different partners.
• (Optional) If needed, display question starters and prompts for feedback.
  • “Can you give an example to help show . . . ?”
  • “Can you use the word _____ in your explanation?”
  • “The part I understood best was . . .”
• “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time