Lesson 7

This Notice and Wonder asks students to consider 2 diagrams representing a shaded region with the same side lengths. The first diagram shows the unit square and the gridlines and the second diagram just shows the side lengths of the shaded region. This prepares students to transition from the gridded diagrams they have worked with in previous lessons to the diagrams they will work with in this lesson. In the activity synthesis, students discuss different equations that represent different ways of finding the area.

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses. 

• “How do we know that the shaded regions have the same area?” (\(\frac{4}{5} \times \frac{2}{4} = \frac{8}{20}\) and \(\frac{1}{2} \times \frac{4}{5} = \frac{4}{10}\). \(\frac{8}{20} = \frac{4}{10}\) or \(\frac{1}{2} = \frac{2}{4}\), so they must be the same.)
• “How does the first diagram represent this equation: \(\frac{8}{20} = \frac{4}{10}\)?” (Each of the \(\frac{2}{20} = \frac{1}{10}\).)

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 shaded 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).

• Groups of 2
• “You are going to find some products of fractions. You will have a choice of using an area diagram to help you.”

• 5–8 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for students who:
  • fill in the first diagram which shows the division into smaller pieces
  • try to divide the second diagram into smaller pieces to show the product
  • understand that drawing all of the pieces will be difficult for the third product

If students do not represent the unknown side length correctly or find the incorrect value for the area of the shaded region, consider asking: “What do you know about the shaded region? What are you still trying to find out?”

• Display expression: \(\frac{2}{5} \times \frac{3}{4}\)
• Invite students to share their calculations for \(\frac{2}{5} \times \frac{3}{4}\).
• “Was the area diagram helpful?” (Yes, it helped me to visualize the product and understand why \(2 \times 3\) is the total number of pieces and \(5 \times 4\) is the number of those pieces in a whole.)
• Display expression: \(\frac{9}{11} \times \frac{5}{8}\)
• “Did anyone draw an area diagram for this expression?” (I started to and I got the eighths but then gave up for elevenths as that is a hard fraction to show.)
• “Does Diego’s diagram help to find the value of \(\frac{9}{11} \times \frac{5}{8}\)? 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 and that 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.

• Groups of 2

• 1–2 minutes: independent think time
• 5–8 minutes: partner work time
• Monitor for students who:
  • use what they know about whole number multiplication to determine that \(\frac{8}{9} \times \frac{7}{4} = \frac{56}{36}\)
  • use what they know about equivalent fractions to rewrite \(\frac{10}{5}\) as 2

If students do not write the correct missing factors, consider asking, “How did you find the value for the other equations? How can you adapt your strategy to find the missing factors?”

• Display: 
  \(\frac{8}{9} \times \underline{\hspace{1cm}} = \frac{56}{36}\)
• “How did you find the value that makes the equation true?” (I knew the numerator was 7 since \(7 \times 8 = 56\) and the denominator had to be 4 since \(9 \times 4 = 36\).)
• Display:
  \(\frac{10}{5} \times \frac{6}{5} = \underline{\hspace{1cm}}\)
• “How did you know the value that makes the equation true is \(\frac {12}{5}\)?” (I doubled \(\frac {6}{5}\).)
• Display equation: \(5 \times \underline{\hspace{1cm}} = \frac{15}{8}\)
• “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 \(\frac{15}{8}\) and then the numerator needed to be 3.)