Lesson 12

• Display one problem.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

• Record answers and strategy.
• Keep problems and work displayed.
• Repeat with each problem.

• “How does breaking apart the numbers help us find the product?” (It is easier to find the product of big numbers when we decompose the numbers and add the smaller partial products.)

The purpose of this activity is for students to find the area of a rectangle with a whole number side length and a side length that is a mixed number. Students draw diagrams to represent the area of the gardens in the problem. Students should draw diagrams and find the area in a way that makes sense to them. As students work, ask them to explain their strategy for finding the area. In the activity synthesis, students consider multiplication expressions that use the distributive property to represent decomposing the rectangle into two smaller rectangles.

• Groups of 2
• Display: Noah’s garden is 5 yards by \(6\frac{1}{4}\) yards. Priya’s garden is 6 yards by \(5\frac{1}{4}\) yards.
• “Whose garden do you think is larger? Why?”
• 1 minute: quiet think time
• 1-2 minutes: partner discussion
• “We are going to draw the diagrams of each garden and determine which garden has a larger area.”

• 1–2 minutes: independent work time
• 8–10 minutes: partner work
• Monitor for students who:
  • label the side lengths with mixed numbers
  • multiply the whole number side lengths, then add the fractional parts
  • decompose the larger rectangles into two smaller rectangles and multiply to find the area

• Select 2–3 students to share their responses and reasoning about how they determined which garden had a greater area.
•  Display diagrams in sample student responses and the expressions \((5 \times 6) + (5 \times \frac{1}{4})\) and \((6 \times 5) + (6 \times \frac{1}{4})\).
• “How are these expressions the same? How are they different?” (Each expression is a sum of two products. The first part of each expression is 30. The second part is different because one expression has a 5 and the other one has a 6.)
• “How can we determine which garden has a larger area without evaluating the expressions?” (The \(5 \times 6\) is the same as \(6 \times 5\) but one garden has 5 one-fourths and the other has 6 one-fourths.)

The purpose of this activity is for students to interpret different strategies for multiplying whole numbers and fractions greater than 1 to find the area of a rectangle. Students use a diagram to describe different ways to determine the area of a shaded region. Consider having multiple copies of the diagrams available if students want to use a separate diagram for each strategy. Encourage students to draw on the diagram to show how they decomposed the rectangle.  
Students use what they have learned about area to construct different reasonable arguments for effective calculations of the area (MP3).

• Groups of 2
• “Decide with your partner who will be Partner A and who will be Partner B. You’ll each look at how some students started a problem and how they could finish their work. Then, you’ll share your work with your partner.”

• 5–8 minutes: independent work time
• 3–5 minutes: partner discussion
• Monitor for students who:
  • complete Tyler's and Diego's work by calculating the missing or excess area

If students do not complete the steps that were started in the problem, refer to one of the diagrams and corresponding expressions and ask "How does the diagram show this expression? What else do we need to find out in order to determine the area of the shaded region?"

• Select previously identified students to share their solutions.
• “What is the same and what is different between the two diagrams?” (They have the same area and the side lengths are equal, but they are written in different ways. The expressions for the side lengths are equal, but they are written in different ways.)
• “Which student’s strategy do you prefer and why?”