Finding the Area of a Triangle

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SWBAT find the area of a triangle using square units and the area formula

Big Idea

Finding area of shapes relates to Geometry and Algebra in High School


10 minutes

Part of the common core says that students should be able to find the area of shapes by decomposing and composing shapes into rectangles and triangles and then applying this to real life and mathematical problems.  In previous lessons, we have been working on developing the formulas by using square units to help us find the formula.  We haven’t done much practice with the application part.  So, I chose two problems that relate to real life to support this part of the standards.  However, the two problems I chose also require the students to understand the formula and how it works.  Students will need to work backwards to find missing side lengths.  If students struggle, they can use the square units (for example, hundredths from the base 10 blocks to model the situation) The first problem has them finding the side length of a parallelogram with an area of 52cm² and a side length of 13.  Students could model this.  They would need to know what the numbers mean (SMP2) and be able to show it using the square units or within the formula (SMP4).  To assist students, I will be asking them what the numbers mean?  For my low learners, I will have them create the parallelogram or square using their base 10 blocks (SMP5: using tools strategically).  Side note:  if the amount of manipulatives needed is too much, then change the numbers to make it more manageable.  The square problem can be 16u² instead of 144.  Students can then build the object to help them find the missing side length.

Students that are using the formula will struggle with the finding the side length of a square because they will want to divide by 2 or 4.  Remind students that squares have the same side length which makes them side x side.  Dividing the area in ½ will not work.  Ask them what it means when we say side x side or s?  Students should say that both the numbers are the same.  This should help when trying to find the missing side length.

Tools:  Finding the area of triangles power point, notes, and square unit manipulative

Finding the Area using a Manipulative

25 minutes

As we begin to find the area of a triangle, I’m going to be giving the students a manipulative to act out the formula.  I’m going to use the hundredths pieces from the base 10 blocks, but any square unit will do.  Students will first build the bottom of the triangle.  Then I will ask them to continue to build on to the bottom.  As students finish, ask them to tell you the height of the triangle?  (SMP1: students will be making sense of the problem using a manipulative) As students build upon the base of the triangle, they should notice that once again we created a rectangle. Have them tell you what they height of the triangle is and how they know.  Next, ask them what they notice about the rectangle and the triangle.  Students should say that there are two triangles or that the triangle is ½ the rectangle.

Show students the next slide to help them “see” the formula.  If we cover the triangle with square units using the base as our guide, each triangle consists of two rectangles. Therefore, to find the area of a triangle, we have divide the area in half.

As a challenge, before they write down the formula, ask them if we will be using the l x w or b x h formula?  I would like to see if students make the connection that they will have to use the formula b x h because of the slanted side lengths or because the base and height have to meet at a right angle.

Discuss the formula for a triangle and how it was derived.  Help students make a connection between the “acting it out” and the formula: b x h ÷ 2.  

Practice Makes Perfect

15 minutes

There are 3 problems for the students to try in their notes.  I’m going to do each problem together.  Students should work independently.  Once they arrive at their solution, they can discuss it with a tablemate.  Then I will call a student to the board to teach the class.  Students should speak clearly and “think out loud” so everyone can hear their thinking.

Students struggle with two items.  First, when given both side length and height, they may use the wrong number.  Give students this scenario:  when I go to the doctor and they want to measure my height, what do they ask me to do?  Students should say, “stand up straight”.  I say, “this isn’t any different when determining the height of a triangle!”  Also, when there are no units of measurement, students can use the generic term units squared or u²(SMP6: properly labeling the units of measure)

Tools:  Finding the area of a triangle power point and notes

Numbered Heads Together

25 minutes

The students will be working through 4 problems that require them to use the area formula. These problems are located within the power point.  If students are struggling, they can use the manipulative given in the beginning of class.  As students are working through the problems, I will be walking around to see that they are using the correct numbers in their formula and assist as needed.

Students will get the chance to justify their solution to a tablemate when directed to do so. (SMP3)

Tools:  Finding the area of a triangle power point


10 minutes

I’m really excited about this connect 3.  I want the students to tell me the relationship between the formula for a rectangle, triangle, and a parallelogram.  This is going to require them to understand what we have been doing over the past few days as we “acted out” how to get to the formulas.  I’m looking for students to say the following:

Rectangle/Triangle:  two triangles make up one rectangle, therefore, the formula for a triangle is the area of a rectangle divided by 2

Rectangle/Parallelogram:  rectangles can be used to help find the area of a parallelogram because the rectangle corner can be cut off and moved to form the parallelogram, therefore, the areas are the same.  We use b x h because the height needs to be straight up and down or the base and height have to meet at a right angle.

Triangle/Parallelogram: (this may be difficult for them)  Since the formula for a rectangle and parallelogram are the same thing and triangles have slanted sides, the b x h formula would be the one to use.  Then, since we have to divide the rectangle formula in half, we would have to do the same for the parallelogram.

I will be looking for these types of connections, but not in those exact words.  I will be collecting this to provide feedback to the students in the next few days.

Connect 3 uses mathematical practices:

SMP1:  Finding a starting point

SMP6:  Using mathematical language

Tools:  Connect 3 document