The purpose of this activity is for students to find the area of rectangles with a side length that is a non-unit fraction. Students may use a variety of approaches to find the areas of the shaded region. Students may notice and use the structure of the rectangle and expressions to determine the area (MP7).

Monitor for and select students with the following approaches to share in the Activity Synthesis:

• Group the shaded pieces with a fractional area to make whole unit squares.
• Reason that each piece is a unit fraction and count the number of pieces.
• Multiply the whole number of units by the numerator of the fractional side length and divide the result by the denominator.

The approaches are sequenced from more concrete to more abstract to help students connect how the number of shaded pieces and the unit fraction relate to the area when a side length is a non-unit fraction. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven't shared recently.

• Invite previously selected students to share in the given order. Record or display their work for all to see.
• Connect students’ approaches by asking:
  • “What are different ways to find the area when we have a non-unit fraction as a side length?” (They all use the diagram and side lengths. They have different ways to think about the shaded pieces using the whole number and the fraction that are the side lengths. Some approaches show the whole number times the numerator and then divide by the denominator.)
• Connect students’ approaches to the learning goal by asking:
  • “How are the second and third shaded regions the same? How are they different?” (They are each 4 units long. There are 12 shaded pieces in each. The 12 pieces in the second example are of a unit square. The 12 shaded pieces in the third shaded region are of a unit square.)
  • Display the expression:
  • “How does this expression relate to the second and third shaded regions?” (In both of them, the number of shaded pieces is .)
  • “Why are the areas of these two shaded regions different?” (The small pieces are not the same size. They are of a unit square in one and in the other.)
  • Display the expressions: and
  • “The expressions for the area show that there are 12 shaded pieces in both but they are different sizes.”

The purpose of this activity is for students to find the area of rectangles with a fractional side length. The side lengths are not labeled, so students will have to determine them by considering the number and size of the shaded pieces.

In the second problem, students are given expressions and they determine whether or not each expression represents the shaded area in a given diagram. The numbers in the expressions are similar, so students need to consider the structure of the expressions and the shaded regions. While they may match the expressions with the diagrams based on their value, students are encouraged to look for and clearly express how the diagrams represent the different expressions (MP6, MP7).

• “How did you choose that expression?”
• Show them an expression that matches the area of the shaded region. “How does this expression represent the shaded region?”

• “How did you determine the side lengths for the shaded region in Diagram X?” (I looked at the side lengths of the squares and the partitions, then I counted how many are shaded.)
• Display the expression:
• “How does this expression represent the shaded region in Diagram X?” (The vertical rectangle has 3 rows of 4 shaded pieces, so that’s  shaded pieces. Each one has an area of square unit.)