The purpose of this activity is for students to learn that area is additive. Students decompose a rectangle into two smaller rectangles and find the sum of their areas in order to find the area of the large rectangle. They can find the area of the two smaller rectangles by counting or by multiplying the side lengths.

When students consider how to decompose a larger rectangle into smaller rectangles to facilitate the process of finding an area, they look for and make use of structure (MP7).

• “Tell me how you found the area of your garden.”
• “How could you use multiplication to find the area of your garden?”

• Select 2 or 3 students who created a rectangular garden that is decomposed into rectangles to share.
• Consider asking:
  • “How many square units does each part of the garden cover?”
  • “How many square units does the garden cover?”
  • “What equation would represent how you found the area of the rectangle?”

The purpose of this activity is for students to find the area of a figure by decomposing it into two non-overlapping rectangles. The Activity Synthesis should emphasize different strategies and also encourage students to directly link expressions and the use of parentheses to the way they decompose the figure. If students drew gardens in the shape of the image in the Launch, display those drawings as well during the Notice and Wonder.

Some students may partition diagonally to split the figure into what looks like 2 symmetrical parts, or cut the figure up into more than 2 parts. Both are acceptable ways of finding the area. Ask students who partition diagonally to find the area in the way they partitioned, but then encourage them to find a second way that has partitions on one of the grid lines. As students look through each others’ work, they discuss how the representations are the same and how they are different, and can defend different points of view (MP3).

When students notice that the smaller parts of the figure can be added to find the total area of the figure they are looking for and making use of structure (MP7).

This activity uses MLR7 Compare and Connect. Advances: representing, conversing.

MLR7 Compare and Connect

• Have students display their work.
• 5–7 minutes: Gallery Walk
• “What is the same and what is different about how other students found the area of the figure?” (Some students broke the rectangle apart vertically and some students broke it apart horizontally. Others imagined a part that wasn’t there. All found the same area. All broke the rectangle into smaller parts and then added the parts to find the area.)
• 30 seconds quiet think time
• 1 minute: partner discussion
• Display an expression that reflects how students found the area, such as:
• “How does this expression show how to find the area?” (The represents the rectangle on the top of the figure if we break it into 2 rectangles. The represents the bottom rectangle.)
• Add parentheses to create the expression:

• “Parentheses are grouping symbols that can be used in expressions or equations. To show how you saw the figure we can show the first rectangle with and the second rectangle with . Parentheses let us put both rectangles in the same expression like and see which part of the expression represents each rectangle.”
• Reinforce the meaning of parentheses in a way similar to other ways that students decompose the figure.