Area of Compound Shapes with Unknown Dimensions

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SWBAT find the area of a compound shape by determining unknown dimensions and decomposing the figure into rectangles.

Big Idea

When a compound shape is decomposed into non-overlapping figures, we can can find and combine the area of each individual shape in order to determine the total area.

Think About It

7 minutes

Students work on the Think About It problem independently.  After 3 minutes of work time, I have the students clap out the correct answer - I say each letter choice and students clap when I say the option they've picked.  This gives me a way to quickly check how the class thought about this first problem.  

If the majority of claps are for the correct answer choice, I ask for 2 students to read their written responses.  If, though, there is not agreement about the correct answer choice, I ask for a student to read his/her justification for each letter (or each letter choice for which there were multiple claps).  This gives students an opportunity to share their thoughts in an authentic way.  The thinking of one student can be read in this Think About It sample.


Intro to New Material

20 minutes

This lesson differs from the previous lesson in this unit, in that the compound figures do not have all of the dimensions labeled.  

The key points in this lesson are:

  1.       Rectangles have opposite sides that are equal  in length.
  2.       A whole is the sum of its parts.

There are 4 examples in this Intro to New Material section.  

For Example 1, I start by asking students how they might decompose the compound figure.  I'm looking for students to say they can split the figure with either a horizontal or vertical line.  I ask students what is different about this problem, when compared to the work we did in the previous lesson.  I ask students to all decompose the figure all the same way, because we are working through the example together.  We decompose this figure vertically.  For organization, we number the rectangles as 1 and 2, from left to right.

I ask students to show me on their fingers which rectangle will be easier to find the area of.  I expect to see students raise one finger in the air, as rectangle #1 has both dimensions labeled.  I ask students to name the length and the width of this triangle.  I ask students to turn and talk about why the length of the rectangle isn't 10mm.  I then have a student share out.  I want students to articulate that 10mm is the length of the entire compound figure, and right now we're dealing with a part of the figure.  We talk through how to find the unknown dimension for rectangle #2, using the idea that a whole is the sum of its parts. 

Students then find the area of the parts, and then find the area of the compound figure.  

I also model for students how to find the area of the same shape by creating a horizontal dotted line as well to demonstrate that both ways will result in the same area for the entire figure.

Example 3 is different from the previous two examples in that both rectangles will have unknown dimensions, no matter which way the figure is decomposed.  

Example 4 requires students to apply the concept of compound area o a real-world situation.  Part C also asks student to find the perimeter of the compound figure, which gives them an opportunity to use the found dimensions in a different way.  




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 student.  I am looking for:

  • Are students decomposing the figures correctly?
  • Are students correctly identifying the dimensions and area of each rectangle?
  • Are students organizing work space, using the formula for area for each rectangle?
  • 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 rectangle?
  • Show me another way you could have decomposed the figure.
  • How did you calculate the area of the figure?
  • Why did you use square units?


After 10 minutes of partner practice time, students complete the Check for Understanding problem independently.  I cold call on a student to share her/his work on the document camera.  The class gives the student positive and constructive feedback on the work.  I then ask for a student who decomposed the figure in the other way (either horizontally or vertically, depending on what the first student chose to do) to also share his/her work.  The class gives this student feedback as well.

Independent Practice

15 minutes

Students work on the Independent Practice problem set.  As they work, I am checking to be sure that students are taking the time to decompose the compound figures and organizing their work to find the total area.

I do not give students access to calculators for this lesson.  I use this as an opportunity for them to build on their fluency, and practice with decimal operations.  I do, though, give calculators to a very small number of students as a support so that the arithmetic is not a barrier to concept mastery (5 of my 87 students for this lesson).

If students finish quickly, I supply them with a clean copy of problem 5, and ask them to decompose the figure into more than two rectangles (using horizontal cuts).  

Closing and Exit Ticket

8 minutes

After independent work time, I have students talk with their partners about problem 6.  This problem is an application problem, with multiple parts.  I want students to have the chance to talk through their responses and ask clarifying questions, if they have them.  I circulate around the room and listen in on conversations and clear up any misconceptions that might arise. 

Students work on the Exit Ticket to close the lesson.  Students may spit the figure vertically, as in this exit ticket sample.  They may also might solve problem two split horizontally.