Lesson 4

The purpose of this Number Talk is for students to interpret a fraction as division of the numerator by the denominator. The strategies elicited here will be helpful later in the lesson when students match division situations, expressions, and diagrams. In this activity, students have an opportunity to notice and make use of structure (MP7) because the divisor stays the same.

• Display one problem.
• “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 problems and work displayed.
• Repeat with each problem.

• “What patterns do you notice?” (The solutions for \(35 \div 7\), \(36 \div 7\), \(37 \div 7\) increase by \(\frac {1}{7}\))
• “If we kept increasing the dividend by 1, what would be the next whole number quotient?” (6)

The purpose of this activity is for students to move back and forth between equations, situations, and diagrams. Each member of a group is assigned one of the representations to begin with and all of the representations can represent the same situation. This is the first time in this unit students have been asked to write situations that represent division equations or diagrams. As students work, ask them to explain how the diagram represents the number of objects being shared and the number of equal shares.

Students go back and forth between equations, situations, and diagrams, interpreting the diagrams and equations and creating situations that these diagrams and equations represent (MP2).

• Display image:

  

  

• “Where do you see division situations in this picture?” (The berries are split into 6 groups. )
• Groups of 3
• Assign groups. Make sure each student knows who is Partner A, Partner B, and Partner C.

• 6–8 minutes: independent work time  (problems 1, 2, 3)
• 4–5 minutes: group discussion
• Monitor for students who write division situations, for \(a \div b\), where \(a\) represents the amount to be shared and \(b\) represents the number of equal shares.

If students do not think of a situation that is represented by the equation, show them the image from the launch and ask, “How can the equation represent some people sharing some pounds of blueberries?”

• Select 2–3 students to share their problems and solutions. Show a variety of accurate student representations.
• Display equation: \(4 \div 6 = \frac{4}{6}\).
• “How did you use the equation to help determine the number of objects and equal shares in your story?” (The numerator is the number of objects. The denominator is the number of shares. This story needs 4 objects and 6 equal shares.)
• Display image:

  

  

• “How did you use the diagram to help determine the number of objects and equal shares in your story?” (The number of objects is the number of wholes or large rectangles and the number of shares is the number of equal pieces each rectangle is divided into. There are 4 objects and 6 equal shares.)
• “How were your stories the same? How were they different?” (There were 4 objects in each and there are divided into 6 equal shares, but our contexts were different.)

The purpose of this activity is for students to solve problems about the same context. The context for both problems is equally splitting some gold. There are three related quantities:

• the total amount of gold collected
• the total number of friends who collect the gold
• the amount of gold each person gets

In previous lessons, students solved problems where the unknown was the amount each person gets after some people shared something equally. In this activity, the unknowns are the amount of gold the friends share and the number of friends sharing the gold. Students may need to draw diagrams to interpret the unknown. During the synthesis, connect the meaning of division to the meaning of a fraction.

• Display first image.
• “Tell a story about what is happening in this image.”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Display second image.
• “This is gold dust.”
• “We are going to solve some problems about panning for gold.”

• 5-8 minutes: partner work time
• Monitor for students who:
  • draw a diagram to explain or show the relationship between the number of friends and the denominator.
  • use words to explain or show the relationship between the number of friends and the denominator.

If students don’t have a way to start the problem, ask: “What do you know about the problem? What are you still trying to figure out?”

• Invite previously selected students to share their responses.
• Display the first situation that describes 3 friends sharing gold and each friend gets \(\frac{4}{3}\) grams of gold.
• “What do we know about the situation?” (There are 3 friends and each friend gets \(\frac{4}{3}\) grams of gold.)
• Display: \(\frac{4}{3}\)
• “How does \(\frac{4}{3}\) represent the situation of friends sharing gold?” (It is the amount of gold each friend got.)
• “What does the 3 represent in \(\frac{4}{3}\)?” (Thirds. The amount they collected was split into 3 equal parts because there are 3 friends.)
• “What does the 4 represent in \(\frac{4}{3}\)?” (It represents how many thirds there are and also how many friends there are.)