Stage 3 - measurement – comparing areas
Students investigate the area of a triangle by comparing the area of a given triangle to the area of the rectangle of the same length and perpendicular height.
Strategy
Students can:
- explain the relationship between the area of a triangle and the area of a rectangle
- record, using words, the method for finding the area of any triangle
Activity 1 – rectangles and triangles
Divide the class into groups of five. Using grid paper, each student in the group draws and cuts out a rectangle (of any size). Have students calculate the area of each rectangle by counting the grid squares. Groups record their results in a table, e.g.
| Name | Area of rectangle | Area of triangle 1 | Area of triangle 2 |
|---|---|---|---|
| Student 1 | 24 | 12 | 12 |
| Student 2 | enter answer | enter answer | enter answer |
| Student 3 | enter answer | enter answer | enter answer |
| Student 4 | enter answer | enter answer | enter answer |
| Student 5 | enter answer | enter answer | enter answer |
Students then rule a diagonal line across the rectangle (from corner to corner) and cut along the diagonal line to make two triangles. Now have students calculate the area of each triangle and record in the table above. Have students discuss their results and establish the relationship between the rectangle and the triangles from the table – ‘The area of the rectangle is the same as the area of triangle 1 plus the area of triangle 2. The resulting triangle has half the area of the original rectangle.’
Discuss and share results with the class.
Activity 2 – the relationship between rectangles and triangles
Divide students into groups of three. The following tasks could be assigned:
- Recorder - keeps a record of all important information.
- Measurement verifier- confirms all measurements and calculations.
- Reporter -shares all pertinent information with the class.
Distribute the Squares and Rectangles Activity Sheet. Have each group calculate the area of each shape on the sheet by counting the informal uniform units (squares).
Using rulers, students draw one diagonal line from the top left hand corner to the bottom right hand corner, in each of the shapes A, B, and C, and then cut each shape along the diagonal into two parts. In their groups, have students estimate the area of each triangle formed by dividing shapes A, B, and C in half along the diagonal.
Students can estimate the areas of the newly created triangles using any methods they choose. One method is to simply count the number of squares, half-squares, and partial squares that are formed when the shapes are divided. Another method is to realise that each shape has an area equal to half the area of the original squares and rectangles. (Students can see this by placing one half over the other.)
Discuss the results with the class as a whole.
Ask students to create a formula for determining the area of a triangle. Have them explain their reasoning and prove that their formula works. To prompt discussion, you may need to ask leading questions, such as,
- "How is the area of a triangle related to the area of a rectangle?"
- "What is the formula for finding the area of a rectangle?
References
Australian curriculum
ACMMG137: Solve problems involving the comparison of areas using appropriate units.
NSW syllabus
MA3-10MG: Selects and uses the appropriate unit to calculate areas, including areas of squares, rectangles and triangles.