Lesson 6

• 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.

• Display equation: \(\frac{15}{2} = 7 \frac{1}{2}\)
• “How do we know this is true?” (They are both equal to \(15 \div 2\) or 7 is \(\frac{14}{2}\) so \(7 \frac{1}{2}\) is \(\frac{15}{2}\)).

The purpose of this activity is for students to apply what they learned in earlier lessons to represent a division situation. During the synthesis, students connect what they know about division to multiplication when they see that the situation of 2 people equally sharing the distance in a 3 mile race can also be described as each person running one-half of the three mile race. Students connect the language they used to describe the situation to multiplication and division expressions. This relationship between multiplication and division is the focus of the next several lessons.

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

• Display the image from the student workbook.
• “Tell a story about this image to your partner.”
• “This image shows two children who are running in a relay race. They are on the same team. They each have to run the same distance. The boy holding the baton is finishing his turn. When he hands the baton to his teammate, his teammate will take her turn.”

• Groups of 2
• 2 minutes: independent think time
• 5–8 minutes: partner work time

MLR2 Collect and Display

• Circulate, listen for and collect the language students use to describe how to represent the situation and how far each person ran.
• Listen for:
  • 3 miles divided into 2 parts
  • half of 3 miles
  • one and one half miles
  • three halves 
• Record students’ words and phrases on a visual display and update it throughout the lesson. 

• “These are the words and phrases that you used to describe the situation. 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.
• Display: \(3 \div 2\), \(1\frac{1}{2}\), \(\frac{3}{2}\), and \(3\times \frac{1}{2}\)
• “How does each of these expressions represent the situation?” (3 miles was divided into 2 equal parts, each person ran \(1\frac{1}{2}\) miles or \(\frac{3}{2}\) miles which can be thought of as \(3\times \frac{1}{2}\).
• Consider giving students time to record their answers in their journal.
• Display: “one half of 3 miles”
• “This is another way to describe how far each person ran.”
• Display: \(\frac{1}{2}\times3\)
• “We can also use this multiplication expression to represent one-half of 3.”
• “In the next activity, you will describe how different diagrams represent each of these expressions.”

The purpose of this activity is for students to build on the previous activity to interpret diagrams as representing different expressions. Both diagrams in the activity represent solutions to the running situation in the previous activity. In this activity, the diagrams are interpreted without the running context. As students use the diagrams to interpret expressions, they begin to see relationships between the expressions \(\frac{3}{2}\), \(1\frac{1}{2}\), \(3 \times \frac{1}{2}\), \(\frac{1}{2}\times3\), and \(3 \div 2\). In this activity, students make sense of different ways to interpret a given diagram and relate the operations of division and multiplication (MP7).

• Groups of 2

• 5 minutes: independent work time
• 5–8 minutes: partner discussion
• Monitor for students who choose different diagrams to represent the same expression. 

If students do not explain where they see each expression in one of the diagrams, refer to the expressions and ask: “How are these expressions the same? How are they different?” Then, refer to one of the numbers in one of the expressions and ask students to describe how the diagrams represent the number.

• “What is the same about the diagrams?” (They both show 3 rectangles that are divided into 2 equal pieces.)
• “What is different about the diagrams?” (The shaded pieces are next to each other in one of the diagrams.)
• Ask previously selected students to share their explanations.
• Display expression: \(3 \div 2\)
• “Which diagram did you choose for the expression \(3 \div 2\)? Why?” (I chose diagram A because I see the 3 miles and the 2 equal parts. I chose diagram B because I can see the 2 equal parts that make 3 miles and can see that they make \(1 \frac{1}{2}\) miles.)
• Display expressions: \(3 \times \frac{1}{2}\), \(3 \div 2\), \(\frac{1}{2} \times 3\)
• “How do we know all the expressions are equal?” (Because the same diagram represents all of them or because they all equal \(\frac{3}{2}\) or \(1\frac{1}{2}\).)