Area of Compound Shapes, Using Decomposition

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SWBAT find the area of a compound shape by decomposing the figure into rectangles, triangles, parallelograms and/or trapezoids.

Big Idea

To determine the area of the compound shape, decompose the shape into recognizable non-overlapping shapes, such as rectangles, squares, parallelograms, trapezoids and triangles.

Think About It

7 minutes

Students work in pairs on the Think About It problem.  After 3-4 minutes of work time, I bring the class together and ask for students to share how they decided to attack this problem.  Some students may have decomposed the shape into a triangle and a trapezoid, as seen in my Think About It sample.  Other pairs may have decomposed the compound figure into two triangles and a rectangle.  I ask for all pathways that students tried for this problem, by asking 'did anyone try this differently?'  I want the class to take away from this conversation the idea that we can decompose the figures in this lesson in multiple ways, and still arrive at the same total area.  


Intro to New Material

15 minutes

This lesson builds off of the previous two lessons in this unit.  In this lesson, students will find the area of compound shapes that are composed of shapes other than just rectangles.  They will encounter triangles, trapezoids, and parallelograms.  The process students will follow - breaking the composite figure into familiar shapes, finding the area of the pieces, and then combining to find the composite area - is not new.  Therefore, the Intro to New Material section is very student centered, and does not have a large amount of new content.  

For Example One in the Intro to New Material section, students identify the shapes that make up the composite figure.  We talk through how to find the length of the base of the triangle, and then organize the work space to show the area of the rectangles and the area of the triangle.  Students then find the sum of the areas.  

For Example Two, I have students talk with their partners about how they might decompose the figure.  We then share out strategies.  Students might say that they'd create a rectangle and a triangle by drawing a vertical line to separate the two shapes. Or, they can draw one horizontal line and create two trapezoids. They also can draw two lines, one vertical and one horizontal, to make a rectangle, triangle and trapezoid.  

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 group.  I am looking for:

  • Are students decomposing the figures correctly (using a variety of methods, but making sure there are no overlapping areas)?
  • Are students correctly identifying the dimensions and area of each figure?
  • Are students keeping the work space organized?
  • Are students writing the formulas and substituting in the dimensions
  • Are students correctly identifying the area of the original figure?
  • Are students including units?


 I'm asking:

  • How did you decompose the figure?
  • Why can you decompose the figure without altering the area?
  • How did you determine the dimensions of each figure?
  • Is there another way you could have decomposed the figure?
  • How did you calculate the area of the figure?
  • Why did you use square units?


In the partner practice sample, you can see the student's arithmetic in the work space.  I decided to not give students to calculators in this lesson, so that they don't always depend on the calculators to evaluate simple expressions. 

Independent Practice

20 minutes

Students work on the Independent Practice problem set.  As I circulate, I am paying close attention to the organization of the work space.  In addition to having clear formulas and an easy-to-find final answer, students should also number the pieces of the composite figure, as seen in the independent work sample (in this sample, the student chose to use a horizontal line to decompose the compound figure.  Some students may also decompose using a vertical line, as in this student work sample). 

Problem 5 is quite involved.  Students will need to decompose the figure into multiple shapes to find the composite area.  Once they have, they'll need to determine how many sheets of wallpaper will be needed.  This requires them to interpret the quotient in a real-world situation.  Finally, they'll need to determine the cost of the wallpaper.  This type of problem, which requires students to use the answer from each previous part, is similar to the types of questions students might encounter on the SBAC and PAARC assessments. 

Closing and Exit Ticket

8 minutes

After independent work time, I have students turn and talk with their partners and share their work for Problem 6. Partners ask each other any clarifying questions they have about the problem, and then offer one piece of positive and one piece of constructive feedback.  I then have the class share out the different ways they chose to decompose the figure (there are many ways to break down the figure in this problem!).

Students independently complete the Exit Ticket to close the lesson.