E2.5 Use the area relationships among rectangles, parallelograms, and triangles to develop the formulas for the area of a parallelogram and the area of a triangle, and solve related problems.
Activity 1: The Area of a Parallelogram
Goal
This activity develops the formula for calculating the area of the parallelogram.
Material
- Appendix 2 (parallelograms of different sizes)
- Appendix 4 (metric squared grid photocopied on a transparency)
- rulers graduated in centimetres
- scissors and tape
- interlocking cubes and small cubes from base ten materials or a set of white relational rods
Instructions
Group students into pairs.
Give each team two different cardboard parallelograms.
Ask students to use the materials available to them, including the grid paper, to find the area of the parallelogram in their hands.
Circulate to see the different strategies used.
Ask a few teams to present their strategy to the class, making sure that the following are highlighted:
- the grid;
- cutting to form a rectangle.
Ask students questions such as:
- What unit did you use to measure the area of the parallelogram?
- How many units completely cover the parallelogram?
- How many squares fit on the base of the parallelogram?
- How many squares fit on the height of the parallelogram?
- What are the dimensions of the parallelogram?
- What is the relationship between the dimensions of the parallelogram and its area?
- How would you write out the calculation of the area of the parallelogram?
Important Note: At this point, emphasize that one can calculate the area of a parallelogram by cutting one side and then making it into a rectangle of the same dimensions. This calculation is then completed by iterating the unit to get the number of units on the base, multiplied by the number of units on the height.
Example: 3 units on the base × 2 units on the height × the unit used(1 cm2) or 3 × 2 × 1 cm2 or 6 cm2.
- How do you calculate the area of a parallelogram with a base of 7 cm and a height of 3 cm?
- What is the area of a parallelogram with a base of 7 cm and a height of 3 cm?
- What is the area of a parallelogram with a base of … cm and a height of …cm?
- Complete the sentence below for a parallelogram with a base of … cm and a height of …cm.
To calculate the area of a parallelogram, you need: _______________________________.
Concepts the Student Should Understand upon Completion of This Activity:
Since a parallelogram can be transformed into a rectangle, then the formula for calculating its area is the same as for the rectangle.
The formula for the area of a parallelogram is expressed as A = b × h, where b is the length of the base and h is the height of the parallelogram.
Source: translated from L'@telier - Ressources pédagogiques en ligne (atelier.on.ca).
Activity 2: The Area of a Triangle
Goal
This activity allows the student to develop the formula to calculate the area of the triangle.
Material
- Appendix 3 (triangles of different sizes)
- Appendix 4 (metric squared grid photocopied on a transparency)
- rulers graduated in centimetres
- scissors and tape
- interlocking cubes and small cubes from base ten materials or a set of white relational rods
Instructions
Group students into pairs.
Give each student two identical right triangles.
Ask students to use the materials available to them to determine the area of one of the triangles.
Circulate to see the different strategies used.
Ask a few teams to present their strategy to the class, making sure that the following are highlighted:
- the grid;
- gluing to form a rectangle.
Ask students questions such as:
- What unit did you use to measure the area of the triangle?
- How many units completely cover two triangles put together to form a rectangle?
- How many squares fit on the base of the triangle?
- How many squares fit on the height of the triangle?
- What are the dimensions of the triangle?
- What is the relationship between the dimensions of the triangle and its area?
- How would you write out the calculation of the area of the triangle?
Important note: At this point, focus on how to calculate the area of a triangle that can easily be glued into a rectangle of the same size. This calculation is then done by iterating the unit and dividing by two, given that it takes two triangles to form a rectangle, in other words the quantity of units on the base multiplied by the quantity of units on the height multiplied by the unit chosen, all divided by two.
Example
or
\(\frac{3 \times 2 \times 1 \ cm^{2}}{2} = \frac{6 \ cm^{2}}{2}\) or \(3 \ cm^{2}\).
- How do you calculate the area of a triangle with a base of 7 cm and a height of 3 cm?
- What is the area of a triangle with a base of 7 cm and a height of 3 cm?
- What is the area of a triangle with a base of … cm and a height of …cm?
- Complete the sentence below for a triangle with a base of … cm and a height of…cm.
To calculate the area of a triangle, we need: ________________________.
Concepts the Student Should Understand upon Completion of This Activity:
Since two congruent right triangles can be transformed into a rectangle, the formula for calculating the area of a single triangle is the same as the area of the rectangle divided by two.
The formula for the area of the triangle is expressed as \(A = \frac{b \times h}{2}\) , where b is the length of the base and h is the height of the triangle.
Enrichment Activity
Have students repeat the same procedure, but this time using obtuse triangles. Using two congruent obtuse triangles, students can form a parallelogram, which can be made into a rectangle. They can then conclude that the area of any triangle is calculated using the same formula.
Source: translated from L'@telier - Ressources pédagogiques en ligne (atelier.on.ca).