Area of Parallelograms

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SWBAT find the area of a parallelogram by moving a triangular section to form a rectangle on a grid and counting the number of square units

Big Idea

The area of a shape doesn't change if you rearrange the shape to make it into a different shape without increasing or decreasing the size of the shape

Think About It

7 minutes

Students work in pairs on the Think About It problem.  The key idea that I want to come out of our whole class discussion is that the figures have the same area, even though they've been re-arranged.  

While sharing, some students might state that the area of the figures is 15 square units.  If they think this, it signals a misunderstanding about the formula we've used to find the area of rectangles.  If students apply A = l x w to the L-figure, they don't have a firm grasp on when to use the formula.

If this does happen, I'd ask the class to unpack why we cannot use A= l x w, and then have students share out how they got to 7 square units.

Intro to New Material

15 minutes

In this lesson, students are not introduced to a formula to use to find the area of a parallelogram.  Instead, they are decomposing the parallelogram using triangles and using the triangle to compose a rectangle.  Students then either count squares or apply the formula to find the area.

After the Think About It problem, I quickly fill in the notes.  I then show the giant parallelogram from the Intro to New Material page on the document camera.  I want to be sure that the triangle that we are using to compose a rectangle is clear, so I will use a dry-erase marker in a noticeable color to really make it 'pop.'  See the model for students for a visual example.  

For lower performing students, I print off the parallelogram and physically cut the triangle off of the parallelogram and move it to create the rectangle.  I can then use this manipulative with individual students who might struggle during independent practice.  

If students seem unsure about this method, I will keep the group as a whole class and we'll try the first partner practice problem together.  

Partner Practice

15 minutes

Students work in pairs on the Partner Practice problem set. As students are working, I circulate around the room.  I am looking for:

  • Are students correctly rearranging the parallelogram to make a rectangle?
  • Are students correctly determining the area of a parallelogram by determining the area of the rectangle associated with the parallelogram?
  • Are students correctly labeling area with square units?

I am asking:

  • How did you find the area of the parallelogram?
  • How did you know the area didn’t change?
  • What does it mean to rearrange a figure?
  • Is there another way you could rearrange the figure to make a rectangle? 

After partner work time, the class goes over problems 4-7.  I cold call on students to share where they 'cut' a triangle and what the area of 4a and 4b must be.  We also quickly share out responses for the true/false probelms.


Teaching Note:  When making copies of the resources in this lesson, set the copy machine to a dark setting, so that the grid lines are visible for students!

Independent Practice

20 minutes

I provide students about 20 minutes to work on the Independent Practice problem set.  A student work sample can be seen here. 

After independent work time, we discuss problem 2b.  In this problem, the rectangle covers 15 square units, because of half units on one side.  There will be students who do not pay close attention to this, and will think that the area is 16 square units because they've counted the half unites as whole square units (there might also be students who decide to 'ignore' the half units, and record 14 units squared as the area).

I also have students share strategies for problem 7.  I expect there will be at least one student who chooses to draw a rectangle.  I always display this, along with a regular parallelogram, on the document camera.  I ask the student who used a rectangle to justify this shape.  I'm looking for the student to explain that a rectangle is a parallelogram.

Closing and Exit Ticket

8 minutes
Students work independently on the Exit Ticket to close the lesson.  An exit ticket sample is provided here. 
For this lesson, I choose to have conversation after the students complete their exit tickets.  In the next lesson, students will apply the formula to find the area of a parallelogram, and I want to be sure that they have mastered the idea that changing the shape of a parallelogram into a rectangle does not change the area of the figure.