Area of Triangles

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Objective

SWBAT: • Identify and create perpendicular and parallel lines. • Identify the height and base of a triangle. • Find the area of a triangle.

Big Idea

How can you find the area of a triangle? How can you tell if two triangles have the same area? Students develop strategies for finding area of triangles.

Do Now

3 minutes

See my Do Now in my Strategy folder that explains my beginning of class routines.

Often, I create do nows that have problems that connect to the task that students will be working on that day.  Today I want students to start thinking about finding the area of a triangle.  This triangle is difficult since the grid cannot be used to identify the base or height.  Some students may initially complain that they don’t know how to find the area of a triangle.  This is okay!   I tell them that I want an estimate of how many square units the triangle covers.  Some students my count partial squares inside of the triangle.  Other students may incorrectly apply a formula they remember from 5th grade.  Some students may draw a 4 unit by 5 unit rectangle around the triangle and use the three surrounding triangles to figure out the area.  If students do this I acknowledge their strategy one-on-one.   I do not have students share this strategy at this point in the lesson.  I want to give the other students time to develop more ideas.

I call on a student to share one idea.  That student then calls on the next student to share his/her idea.  I encourage students to build on what their classmates have said by using sentence starters like, “I agree/disagree with __________ because…” and “My idea connects with ____________’s idea…”  Students are engaging in MP3: Construct viable arguments and critique the reasoning of others.

Problem

10 minutes

Notes:

  • Before this lesson make copies of the problem triangles.
  • I give each student a copy of these triangles and a pair of scissors.

I call on students to fill in the blanks for the definitions of perimeter and area.  I ask students to look at these three triangles and try to find the area.  Students work in partners.  I pass out materials. Triangle A and B are easier to work with since they are right triangles.  Some students will count the whole squares and ½ squares to come up with the area.  Other students may duplicate the triangle to create a square and figure out the area of the square is double the area of the triangle.  Other students may be familiar with the formula and use it to calculate the area.  Some students will notice the pattern and be able to express that to find the area of a triangle you multiply base times height and then divide by two.  Students are engaging in MP8: Look for and express regularity in repeated reasoning. 

Triangle C is more difficult to count partial squares.  Also the height of the triangle is on the interior of the triangle, as opposed to being one of the sides.  Some students may draw a rectangle around the triangle and see that it takes up half of the space.  Other students may duplicate the triangle and create a parallelogram.  Other students may cut the triangle in ½ and create a rectangle that is 1 unit by 3 units.  Other students may apply the formula.

I walk around to observe strategies as students work.  I strategically choose students to share their strategies under the document camera.  I have someone who counted squares and ½ squares share for triangle A or B.  I have someone who created a square or rectangle share for triangle A or B.  I have someone who created a rectangle for triangle C and someone who used the formula.  Triangle C is a great transition to talk about how counting the partial squares can get time-consuming and more difficult and using a formula can help us.  Students are engaging in MP3: Construct viable arguments and critique the reasoning of others.

Review of Lines

3 minutes

In order for students to be able to identify the base and height of a triangle, they must be able to identify perpendicular lines.  I use a few minutes to review parallel, perpendicular lines.  Parallel lines will come into play later when we classify quadrilaterals.  Students are engaging in MP6: Attend to precision.

Parts of a Triangle

5 minutes

Note:

  • I remove the words in italics on students’ notes before I print them.

I read the definitions and students fill in their notes.  Here I emphasize that in right triangles, the base and height are also sides.  With obtuse and acute triangles, the height is not a side length because the base and height must intersect at a 90 degree angle.

Students complete the base and height practice independently.  I call on students to share out their answers.  A common misconception for #3 is that the height is 15cm.  For #3 I declare, “I think the base is 22 cm and the height is 15 cm.” I call on student to share if they agree/disagree.  Students are engaging in MP3: Construct viable arguments and critique the reasoning of others and MP6: Attend to precision.  

Area of a Triangle

5 minutes

We copy the two versions of the formula.  Most of my students prefer to multiply the base and height and then divide by 2 (rather than multiply by ½).

Students complete the problems independently.  I call on students to share out their answers and how they got them.  A common mistake that students make is that they multiply the base and height and then forget to divide by 2 (or multiply by ½).  When a student makes this mistake I revisit the formula and ask, “Why do we have to divide by 2?”  We have to divide by 2 because if we didn’t we would be finding the area of a rectangle (or parallelogram) that has those dimensions.  The triangle that we have takes up ½ of the space.

The song is another way for students to remember the formula. I enjoy singing the song and watching my students wriggle in their seats.  Although they may resist it at first, I have found that many students use the song as a way to remember the formula.

Working on a Grid

15 minutes

Note:

  • Before this lesson I Post A Key.

I have a student read over the directions.  I review expectations and students start working independently.  Students are engaging in MP2: Reason abstractly and quantitatively, MP6: Attend to precision, MP7: Look for and make use of structure.

As students work I walk around to monitor student progress and behavior.  If students are struggling, I may ask them one or more of the following questions:

  • How can we identify the base and height of a triangle?
  • What are the base and height of the triangle?  How do you know?
  • How can you find the area of a triangle?  Why does that work?
  • Which of these triangles have the same area?  How can that be?

When students complete their work, they raise their hands.  I quickly scan their work.  If they are on track, I send them to check with the key.  If there are problems, I tell students what they need to revise.  If students successfully complete the chart they can work on the challenge question. 

Closure and Ticket to Go

9 minutes

For the Closure, I ask students to return to the do now question.  I ask for students to share strategies for finding the area of the triangle.  I want students to see that they cannot use the formula since they cannot use the grid to identify the base or the height.  I want students to see that they can draw a rectangle around the triangle.  This rectangle has an area of 20 square units.  The surrounding triangles have areas of 2 square units, 4 square units, and 5 square units.  Therefore triangle T has an area of 9 square units.

 

I pass out the Ticket to Go and the Homework.