Planning Guide:Equations with Letter Variables

Sample Activity 1:Developing the Formula for the Area of Rectangles

Review the concept of area and build on students' prior knowledge of the concept. See the end-to-end plan for factors and multiples that connects the area of rectangles to factors and multiples.

Provide students with centicubes, 30-centimetre rulers, centimetre grid paper and a variety of rectangles.

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Ask students to predict the area of each rectangle.

Then have students measure the lengths and widths of the different rectangles using the 30-centimetre rulers and record the data in a chart, as shown below. Encourage students to use the centicubes to make the rectangles if needed (each face of the rectangular prism made with centicubes is a rectangle). Have students compare the actual areas to the predicted areas.

Ask students to look for patterns in the chart and suggest a rule or formula that could be used to find the area of any rectangle. Have students share their ideas and choose the formula that works best for them.

Rectangle / Predicted Area
(cm2) / Length (cm) / Width (cm) / Actual Area (cm2)
A
B
C
D

Through discussion, have students generalize the following formulas for the area of rectangles:

  • A = length × width
  • A = L × W.

Formula for the Area of a Square

Review that a square is a special rectangle with all sides congruent. Explain that the formulas for the area of rectangles can also be used to find the area of squares; however, the area of a square can be found more efficiently with a different formula that the students will develop.

Have students draw squares of different sizes or construct them using square tiles; e.g., the length of one side could be 2 cm, 4 cm, 6 cm, 8 cm and so on. Have students find the area of each square and record the data in a chart such as the one shown below.

Square / A / B / C / D / E
Side Length (cm) / 2 / 4 / 6 / 8
Area (cm2) / 4 / 16 / 36

Through discussion of the patterns shown in the chart, have students generalize that the formula for the area of a square can be written as A = n×n, where A = the area of the square and n = the length of one side.

Have students use the formula for the area of a square to find the area of a square that has a side length of 24 cm.

Different Rectangles with the Same Area

Have students draw on centimetre grid paper as many rectangles as possible with whole number dimensions that have an area of 20 cm2. Some students may wish to use centicubes or square tiles to create the different rectangles. Encourage them to apply the formula for the area of rectangles, use patterns and record the data in a chart.

Remind students that the values for the length and width of the rectangle are factors of 20.

Area (cm2) / Length (cm) / Width (cm)
20 / 20
10
5 / 1
2
4

(Alberta Education 1990, pp. 149–155.)

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