Area of Compound Shapes

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Big Idea


Think About It

7 minutes

Students work independently on the Think About It problem.  Students need to decompose the figure and find the area.  After three minutes of work time, I ask students how this problem is different from those we worked on in the previous lesson.  I then have students share out their strategies for finding the area of the compound figure.  

One possible method has students decomposing the figure using a horizontal line.  Another strategy involves cutting vertically to decompose the figure into two rectangles.  It is important that students see that both methods result in the same area for the compound figure. 

Intro to New Material

15 minutes

There are two guided examples in the Intro to New Material section of this lesson.  

In this lesson, students need the following skills:

  1. Recognize that opposite sides of rectangles are always congruent.
  2. Recognize that you can extend line segments to create parallel opposite sides of rectangles as you decompose.
  3. Identify the length and width of a rectangle even when sides are partial.
  4. Decompose a compound figure accurately, so that we have rectangles with known dimensions versus unknown dimensions.
  5. Calculate areas and find sum of areas.

For example Number 2, I expect that I may need to give my students some context, as many may have never played mini-golf.  They may also be unfamiliar with the term, felt.

In example number two, I plan to discuss two ways to decompose the figure:  

  • Strategy 1: this approach allows us to work with all labeled sides
  • Strategy 2: requires us to find an unknown dimension.  

I will compare these strategies in order to help my students appreciate that while either strategy works, cutting this figure vertically is more efficient.


Partner Practice

15 minutes

Students work in pairs on the Partner Practice problem set. As students work, I circulate around the room and check in with each pair.  I am looking for:

  • Are students decomposing the figures correctly?
  • Are students correctly identifying the dimensions and area of each rectangle?
  • Are students correctly identifying the area of the original figure?
  • Are students including units?


I am asking:

  • How did you decompose the figure?
  • Are there other ways you can decompose the figure?
  • Why can you decompose the figure without altering the area?
  • How did you determine the dimensions of each rectangle?
  • How did you calculate the area of the figure?
  • Why did you use square units?


After 10 minutes of partner work time, students complete the Check for Understanding problem on their own.  One misconception I look for on this problem is with students who incorrectly identify the horizontal base of this figure.  If there are students who are struggling with this idea, I have them highlight each individual rectangle to make it easier for them to see each base and height.

Independent Practice

20 minutes

Students work on the Independent Practice problem set.  As I circulate, I am paying close attention to students' organization of their work.  I want their figures to be neatly decomposed and labeled.  I also expect that anyone can clearly follow the work that the students have completed to find the area of the composite figure.  

Problem 4 involves lengths of sides that are decimals.  This may be difficult for some students.  

For Problem 5, students need to carefully read the problem to realize they need to double the area of the compound figure, in order to answer this question correctly. 


Closing and Exit Ticket

8 minutes

After independent work time, I bring the class back together for discussion.  I have students share with their partners how they solved any of the Independent Practice problems.  Each student shares out.  I then cold call on a student to share work with the class on the document camera.  I ask the class to share feedback on the strategy and work shown.

Students complete the Exit Ticket independently to close the lesson.