Lesson 10

This Number Talk encourages students to think about the relationship between the size of the divisor and the size of the quotient and to rely on the structure of division expressions to mentally find quotients. 

• 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.

• “Why did the quotient get bigger with each problem?” (You are making smaller groups so there are more in each of them.)

The purpose of this activity is for students to compare quotients of quantities based on the relative size of the dividend and the divisor. Students should be encouraged to use whatever strategy makes sense to them to order situations about sharing pretzels. The numbers were intentionally chosen so that students don’t have to perform any complex calculations to solve the problem which encourages them to think about the relative size of the numerator and denominator in order to compare the quotients. In upcoming lessons, students will divide a unit fraction by a whole number and a whole number by a unit fraction.

This activity uses MLR2 Collect and Display. Advances: Reading, Writing.

• Groups of 2
• Display the image of pretzels:

  

  

• “What do you notice? What do you wonder?” (I notice that there are a lot of pretzels and 3 bowls. I wonder how many pretzels there are.)

• 1–2 minutes: quiet think time
• 5–6 minutes: partner work time

MLR2 Collect and Display

• Circulate, listen for, and collect the language students use to describe the relationship between the number of students sharing and the number of pretzels being shared. Listen for: number in each group, size of the group, number of pretzels each person gets, dividend, divisor, quotient.
• Record students’ words and phrases on a visual display and update it throughout the lesson.

If students do not order the situations correctly, prompt them to draw a diagram to represent each situation and ask, “How are the diagrams the same? How are they different?”

• “Are there any other words or phrases that are important to include on our display?”
• As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.
• Display:
  3 students equally share 45 pretzels.
  3 students equally share 42 pretzels.
  3 students equally share 24 pretzels.
  3 students equally share 6 pretzels.
• “What is the same? What is different?” (The same number of students are sharing different numbers of pretzels.)
• “How does the number of pretzels each person gets change in each situation?” (It gets smaller because there are fewer pretzels to share.)
• Display:
  14 students equally share 42 pretzels.
  7 students equally share 42 pretzels.
  6 students equally share 42 pretzels.
• “What is the same? What is different?” (The number of pretzels being shared is the same. The number of students sharing is different.)
• “How does the number of pretzels each person gets change in each situation?” (When fewer people share the same number of pretzels, each person gets more pretzels.)

The purpose of this activity is for students to look for patterns in division by the same divisor. The numbers in these problems were intentionally chosen so students see that the quotient gets smaller as the dividend gets smaller (MP7). Display the poster of the language students used to describe the relationship between quotient, dividend, and divisor during the previous activity. In the last question, students think about what it means to divide a fraction by a whole number which will be the focus of upcoming lessons.

• Groups of 2

• 5 minutes: independent work time
• Monitor for students who:
  • can explain why the quotient gets smaller when the dividend gets smaller
  • describe the quotient of \(\frac {1}{3} \div 3\) as being smaller than \(\frac {1}{3}\)
  • draw a diagram to show \(\frac {1}{3}\) divided into 3 equal sections

Students may not immediately visualize the patterns in the division expressions. Encourage them to draw a tape diagram for each expression, and ask them what they notice. Consider asking, “What is happening to the size of each group as the amount being divided gets smaller?”

• Ask previously identified students to share their solutions.
• “Why does the quotient get smaller as the dividend gets smaller?” (There are a smaller number of things being split into the same number of groups, so there will be fewer in each group.)
• “Why is \(\frac {1}{3} \div 3\) going to be smaller than \(\frac {1}{3}\)?” (\(\frac {1}{3}\) is being divided into 3 equal pieces.)
• Display student diagrams like the ones in student responses.
• “How do the diagrams show \(\frac{1}{3} \div 3\)?” (They show a third divided into 3 equal pieces.)