The purpose of this activity is for students to find the area of rectangles with fractional side lengths where the scaffold of an area diagram is not provided. For the first product, the subdivision of each side into unit fractions is shown and then that is taken away. Without these divisions, students can either try to sketch them or use their understanding of how the numerator and denominator of the fractions relate to:

• The total number of shaded pieces in an area diagram.
• The total number of pieces in the unit square.

When students find products of fractions without an area diagram for support, they rely on their understanding of the meaning of the numerator and denominator and the patterns they have repeatedly observed when finding these products with area diagrams (MP8).

• Display expression:
• Invite students to share their calculations for .
• “Was a diagram helpful?” (Yes, it helped me to visualize the product and understand why is the total number of pieces and is the number of those pieces in a whole.)
• Display expression:
• “Did anyone draw a diagram for this expression?” (I started to and I got the eighths. Then I gave up for elevenths as that is a hard fraction to show.)
• “Does Diego’s diagram help to find the value of ? How?” (Yes, even though it doesn’t show any of the parts, having the numbers there helps me see that there would be 9 columns and 5 rows so 45 pieces. There would be 11 columns and 8 rows in a whole so 88 pieces in a whole.)

The purpose of this activity is for students to find missing values in equations that represent products of fractions. The numbers are complex so students will rely on their understanding of products of fractions rather than on drawing a diagram.

• “How did you find the value for the other equations?”
• “How can you adapt your strategy to find the missing factors?”

• Display:
• “How did you find the value that makes the equation true?” (I knew the numerator was 7 since and the denominator had to be 4 since .)
• Display:
• “How did you know the value that makes the equation true is ?” (I doubled .)
• Display equation:
• “How is this equation different from the others?” (It has a whole number as a factor. The others are all fractions.)
• “How did you solve this problem?” (I knew that the denominator of the missing number had to be 8 to get and then the numerator needed to be 3.)