Students previously may have noticed a connection between the whole number in a given multiplication expression and the numerator of the fraction that is the resulting product. In this activity, they formalize that observation. Students reason repeatedly about the product of a whole number and a unit fraction, observe regularity in the value of the product, and generalize that the numerator in the product is the same as the whole-number factor (MP8).

• “How did you use the patterns to write expressions for , and ?” (I knew that each expression had a whole number and a unit fraction. The whole number is the same as the numerator of the product.)
• Select students to share their multiplication expressions for these four fractions.
• “Can you write any fraction as a multiplication expression, using its unit fraction?” (Yes, because the numerator is the number of groups and the denominator represents the size of each group.)
• “What would it look like to write as a multiplication expression, using a whole number and a unit fraction?” ( )

This activity serves two main purposes. The first is to allow students to apply their understanding that the result of is . The second is for students to reinforce the idea that any non-unit fraction can be viewed in terms of equal groups of a unit fraction, and expressed as a product of a whole number and a unit fraction. 

The activity uses a “carousel” structure in which students complete a rotation of steps. Each student writes a non-unit fraction for their group to represent in terms of equal groups, using a diagram, and as a multiplication expression. The author of each fraction then verifies that the representations by others indeed show the written fraction. As students discuss and justify their decisions, they create viable arguments and critique one another’s reasoning (MP3).

• See the Lesson Synthesis.