Math Background

Lesson: Area Formulas
Introducing the Concept

Since your students have found the area of rectangles before, this is a good concept to build on to help them understand the formulas for finding the area of a triangle and a parallelogram.

Materials: an overhead transparency of a grid, a cutout of a right triangle with one side 6 units long and another side 3 units long, a cutout of a right triangle with one side 6 units long and another side 5 units long, rulers, an activity sheet with four triangles drawn on it with the height and base measures provided for 2 of the triangles.

Prerequisite Skills and Concepts: Students should know what area is and how to find the area of a rectangle. They should know that area is measured in square units.

On the overhead transparency, draw a rectangle with a length of 8 units and a width of 6 units.

  • Ask: Who can tell me how to find the area of this rectangle?
    Students should say that you need to multiply the length times the width. The product is 48 square units.
  • Say: I have a triangle here and I am going to place that triangle inside the rectangle. What can you tell me about my triangle?
  • Students should say that one of the sides of the triangle is the same as the length of the rectangle. They may also say that the vertex of the triangle is on the opposite parallel side. If they don't, point these things out to students.
    triangles

    Draw a perpendicular line from the vertex of the triangle that is opposite the side that is shared by both the triangle and rectangle.

  • Say: Look at triangles ACX and CAD. What can you say about these two triangles?
    Students should say that the two triangles are congruent. Use the right triangle you cut out of paper to lay on top of triangle ACX and show that it covers triangle CAD as well. This will help convince the students that the two triangles are congruent.
  • Ask: What do you think the relationship between triangle CBD and triangle BCY is?
    Students should say the two triangles are congruent. Use the other right triangle you cut out to lay on top of the two triangles to show they are congruent.
  • Ask: What is the relationship between the area of the original triangle and the area of the rectangle it fit into?
    Students should say that the area of the triangle is half the area of the rectangle, since you can make two triangles from the rectangle. If students don't see that, point it out to them.
  • Say: Notice that the width of the rectangle is the same as the height of the triangle and the length of the rectangle is the same as the base of the triangle. So the formula for finding the area of a triangle is A = 1/2(b x h).
    Draw a triangle on your grid that has a base of 10 units and a height of 7 units.
  • Ask: Find the area of this triangle using the formula A = 1/2(b x h).
    After the students have solved the problem, have one of them come to the board to explain how he or she did the problem. (The area is 35 square units.)

    Pass out the activity sheet for students to work on individually or in pairs.


Houghton Mifflin Math Grade 5