The purpose of this activity is for students to think of different ways of using multiplication expressions to represent a non-unit fraction. Students informally use the associative property as they work toward generalizing that .

• “How did you know what fractions to use to complete each equation in the first problem?” (We looked for the numbers to multiply to get . The denominator stays the same, so we find the whole number and the numerator by knowing the factor pairs of 12.)
• Select 2–3 partners to share the following equations from the second problem:
• “Why are all these equations true?” (To get , we can multiply the whole number by the numerator and then keep the denominator the same. In all of these, multiplying the whole number and the numerator gives 6. There are different ways to multiply to get 6 as a numerator: , , , and .)

In this activity, students analyze multiplication expressions, match each to one of a given set of fractions, and explain how they know that certain expressions represent the same fraction (MP7). 

• Ask selected students to share expressions for each fraction.

• Display the possible expressions for the final equation:

  
  
  
  
  

• “How can we explain why these expressions have the value ?”