Finding the Area of Triangles by Composing Parallelograms

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SWBAT find the area of right, acute, and obtuse triangles by creating a parallelogram by using a second, equally sized triangle.

Big Idea

We can determine the area of triangles by creating parallelograms.

Think About It

7 minutes

Students complete the Think About It problem independently.  

After 3 minutes of work time, I ask 3 students to share out (they can share an answer, an explanation, and agree/disagree with what has been previously stated).  If the three students share out that the area is 6 sq. inches, I will ask the class what mistake a student made if they thought the answer was 12 sq. inches.  However, if the students who share all think that the answer is 12 sq. inches, I'll say something along the lines of 'what if I tell you the answer is not 12 sq. inches?'  If there isn't consensus between the three students, I'll open the conversation up to more volunteers.

The key idea I want to emerge from our conversation is that the triangle has an area that is half as large as the area of the rectangle. 

Intro to New Material

15 minutes

In this lesson, students will use parallelograms to find the area of triangles.  Students are not provided with the formula for finding the area of a triangle in this lesson.  

To start the Intro to New Material, I review the ways to classify a triangle based on its angles.  Some of my students have learned this in the past, while others have not.  The vocabulary words come up throughout the lesson, so I provide the definitions and examples as a resource for students.

We then work together to create parallelograms by copying the given triangles.  I guide students to determine the length of the base (if not given), and then place their pencils on the highest vertex of the triangle.  They create a second base of equal length from the vertex, and then close the parallelogram by joining the new base to the bottom base.

Once we have a picture, we then find the area of the parallelogram by applying the formula.  Students learned how to do this in a previous lesson.  We then find half of this value, because the original triangle takes up half the area of the parallelogram.  I expect students to show all of their work, starting with the formula for the area of the parallelogram.

Throughout this lesson, I make sure that I refer to all of our constructed figures as parallelograms.  When we work with right triangles, a student will inevitably ask 'isn't that a rectangle?'  I use this as an opportunity to talk about the characteristics of parallelograms and rectangles, and classify the rectangle as a special parallelogram.   

Note:  if making copies of the resources with this lesson, be sure to set the copy machine to a dark setting, so that the grid lines show up clearly for students.      

Partner Practice

15 minutes

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

  • Are students correctly creating a parallelogram that accurately displays the original triangle and its copy?
  • Are students accurately finding the area of the given triangle?
  • Are students correctly showing all steps?
  • Are students including correct units?

I am asking:

  • How did you know to create a rectangle?
  • How did you know to create a parallelogram?
  • How did you know how this is the correct area of the given triangle?

After 10 minutes of work time, students complete the Check for Understanding on their own.  A CFU sample is included.  I'm looking for students to have the constructed parallelogram, the formula for finding the area of the parallelogram, the area of the parallelogram, and then the area of the triangle identified as half of the area of the parallelogram.  

Independent Practice

20 minutes

Students work on the Independent Practice problem set.  

Problem 1 can be difficult for students, as the height line can cause the misconception that the top base only measures 2 units in length.  

As students are working, some will derive the formula for finding the area of a triangle:  they'll notice that to find the area, they need to multiply the base and the height, and then divide by two.  When I circulate, if I notice that students are only showing arithmetic, rather than also constructing the rectangles, I will ask them why this works.  If they can articulate to me that a rectangle takes up half of the area of a parallelogram with the same base and height, I will allow them to continue without creating the rectangles.  We will apply the formula for finding the area of a triangle in the following lesson, and allowing them to make the jump in this lesson will set them up for success in our next class.


Closing and Exit Ticket

8 minutes

After independent work time, I have students turn and share work with a neighbor.  I have students pick the two problems they'd like to discuss.  The first partner shares the process for solving and answer to the problem of his/her choice.  The second partner then does the same.

Students independently complete the Exit Ticket to close the lesson.  Some students will continue to find the areas of the triangles with parallelograms, while other students will have derived the formula for finding the area of triangles.  Either method is acceptable at this point in my curriculum.