Area of a Triangle: Teacher Guide

Note to WME developers: This page is meant to be part of the page admin to guide the teacher for each lesson. Students will not be able to view this page. This page here is a draft design of what the teacher guide may look like. It is put together by Paul with much input from anyone. Thus all ideas on how to re-organize this page and what content it should contain are desperately needed.

Purpose and Goals of This Lesson

In the previous lesson students explored and developed and understanding of finding the area of rectangles and parallelograms. In this lessons students will explore the relationship between parallelograms and triangles. Encourage students to create as many different 4 sided shapes out of the two triangles as they can. Have them discuss the relationship between the 4 sided shape composed of the 2 equal triangles and one individual triangle. This should be discussed in class and have each student describe in words and pictures this relationship.

Note to WME developers: Michael says that several lessons pages share the same "Purpose and Goals" info. That is easy to do. We just put a link here to point to the shared info. RIGHT?

Related Topics

Intro to area, area units, rectangle, parallelogram, angles, parallel lines.

Annotated Lesson Page

Note to WME developers: this part is the WME lesson repeated with redish-colored ANNOTATIONS to help the teacher teach this particular lesson. And this is based on Michael's comment that current textbook supplements actually provide comments alongside the text for teachers.

You already know how to find the area of a parallogram. Now we will consider the area of a triangle.

The diagram on the right shows the base and height of a triangle. The height is the length of a perpenticular line from a vertex to the opposite side (the distance from the vertex to the side).

Take any triangle and an exact copy of it, put these two together, you will get a parallelogram of the same height and base. This is the key to finding the area of a triangle.

1. A triangle is shown in the following diagram. You can drag the points to change the triangle.

   Teacher: Please encourage students to change the triangle to their hearts content.

2. In the above diagram,

   1. Click "Copy the Triangle ABC" to make an exact copy of it.

   2. Move and rotate the copy to join with the original tirangle to form one figure with four sides (a quadrilateral). Click the button "Join" to merge the two triangles into a quadrilateral.

   3. There are two different ways to to form the quadrilateral with the same height h. Use the buttons "join" and "join the other way" to find out.

      Teacher: Please note that there is a third way to join the two identical triangles which results in a shape that is not so helpful.

Explorations

Discuss with students the reasons why the quadrilateral always form a parallelogram.

The page contains another manipulative that uses the enclosing-rectangle approach to show the area of a triangle is 1/2 of base times height. You can reveal this alternative approach for students to experiment by click on this button.

Assessment

Here is a list of items to assess student comprehension of this lesson.

• How do you know that the quadrilateral you formed have twice the area as the triangle ABC?

• Is the quadrilateral you formed this way always a parallelogram? And why or why not?

• Do you remember how to obtain the area of a rectangle? A parellelogram?