The goal of this activity is for students to match expressions and diagrams and then to compare the value of each expression with one of its factors. To match the expressions with the diagrams, students likely will use the meaning of multiplication. For example, means 2 of 7 equal parts of each of 3 wholes. The area diagram shows the 7 parts, with 2 shaded, whereas the number line shows only the relative locations of and and its factor 3, requiring students to understand the relationship between and 3 in order to pick the right match. Once students have made the matches, the diagrams help them visualize that is less than 3, and the Activity Synthesis highlights this. When students match diagrams and expressions, they look for and identify structure in the number line and in area diagrams (MP7).

• Display the expression:
• “How did you decide which number line and which diagram match the expression?” (I picked the number line with 3 and a point that was less than 3. For the area diagram, I took the rectangle with length 3 and width less than 1.)
• Invite students to share how they compared with the factor 3, highlighting these strategies:
  • Reasoning about the size of .
  • Using the number line.
  • Using the area diagram.

The purpose of this activity is for students to compare a product to an unknown factor, based on the size of the other factor. In this case, students cannot calculate the values of the products to compare but instead rely on their understanding of fractions and the meaning of multiplication. Students also use a number line to help them visualize the different distances after listing the runners in order. For this part of the activity, the expectation is that students will use what they already know about the order of the distances to determine which point corresponds to which runner. Students also might reason about the quantities. For example twice Priya’s distance can be found by marking off Priya’s position on the number line a second time (MP2).

• Display:
  
  
  
  
• “What do you notice about these expressions?” (They represent the amount that each person ran, if Priya’s distance is 2 miles or 2 kilometers. All show a number multiplied by 2. They are listed in increasing order.)
• “What if Priya ran 4 kilometers? What multiplication expressions can we write to represent the distances, in kilometers, of each of the other runners?” (It would be just like the expressions above, except that the 2 would be replaced with a 4.)
• Record expressions for all to see:
  
  
  
  
• “Does the order of the distances change when Priya’s distance changes? Why or why not?” (No, the order of the products is the same as the order of the other factor, the multiple of Priya’s distance.)