Area of Compound Shapes on the Coordinate Grid

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SWBAT find the area of a compound shape on a coordinate grid.

Big Idea

A compound geometric shape is composed of other shapes.

Think About It

7 minutes

Students complete the Think About It problem in pairs.  After three minutes of work time, I have students share how they found the area of the figure.  I allow students to share out until we have heard three strategies:  counting the unit squares, drawing a horizontal line to create two rectangles and finding the area of each, and drawing a vertical line to create two rectangles and finding the area of each.

I tell students that this lesson focuses on finding the area of compound figures, like the one in the Think About It problem.  I ask them to turn and talk with their neighbors about what they think a compound figure is.  

Guided Example

15 minutes

In this lesson, there is one Guided Example.  Students have the knowledge they need to find the areas of these figures.

For the example, I have students plot the points independently.  I then take one student's paper and show it on the document camera.  I ask students to name the ways we can find the area of the compound figure.  I'll ask students which method is most inefficient.  While students can count up all of the unit squares, it will take longer.  Students will be better set up for success, too, for the next lesson if they practice decomposing the shapes. 

Teacher's Note:  When making copies of the materials of this lesson, be sure that the copy machine is set dark enough for the grid lines to come through for students.  The grid lines are very difficult to see, if they are not dark enough.


Partner Practice

15 minutes

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

  • Are students plotting points correctly?
  • Are students correctly creating the geometric figure?
  • Are students correctly identifying the dimensions of each rectangle?
  • Are students correctly identifying the area of the 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?
  • How did you calculate the area of the figure?
  • How did you draw the figure on the coordinate grid?
  • Why did you use square units?


Students in pairs may each want to try a different method, and compare answers.   I think this is great.  It gets students talking about multiple ways to solve a problem.  The following samples for Problem A are all from one group of three that worked together during this lesson.  One strategy was to cut the figure horizontally.   Another way involved decomposing the figure into 6 equivalent squares.  Students might also cut the figure with a vertical line.  This group had a rich conversation about which method was most efficient.  

Independent Practice

20 minutes

Students work on the Independent Practice problem set.

In this lesson, all of the compound figures are L-shaped.  If a student makes a mistake when plotting points, it is easy to spot.  I refrain from telling students what the mistake is.  Rather, I suggest that they go back and check every point.  I also ask them if they can predict which one is the outlier, based on the figures we're working with.  I want students to notice patterns and reason about errors on their own.

In Problem One, some students will be confused by the y-axis when decomposing the compound figure into rectangles.  Some students will use a base of 8 units, rather than 9 units, for the rectangle on the left because they think the y-axis is the side of the rectangle.  If this happens, have students trace over their rectangles with pen, highlighter, or colored pencil.  


Closing and Exit Ticket

8 minutes

After independent work time, I bring students back together to discuss Problem 5.  I like this problem because it requires students to apply what they know about the area of compound figures to a real-world situation.  It also asks students to find the perimeter of the same figure.  While this isn't a skill students work on throughout the lesson, it is something they know how to find.

Students work independently on the Exit Ticket to close the lesson.