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.
Blackline Masters 25 and 27 can be downloaded from the
website http://www.ablongman.com/vandewalleseries.
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.
Look For …
Do students:
 predict the areas of rectangles
prior to measuring them?
 describe area patterns using
concrete, pictorial and
symbolic representations?
 connect area of rectangles to
arrays?
 generalize formulas by
examining area patterns for
rectangles, including arrays?
 apply their knowledge of
variables in formulating a rule
for the area of rectangles?
 demonstrate flexibility in
creating formulas for the area
of rectangles, including
squares?
 apply the formula for the area
of a rectangle to solve
problems?
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
A
B
C
D
Predicted Area
(cm2)
Length (cm)
Width (cm)
Actual Area
(cm2)
Through discussion, have students generalize the following formulas for the area of rectangles:


A = length × width
A = L × W.
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© 2011 Alberta Education
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Planning Guide: Equations with Letter Variables
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
Side Length (cm)
Area (cm2)
A
2
4
B
4
16
C
6
36
D
8
E
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)
20
Length (cm)
20
10
5
Width (cm)
1
2
4
(Alberta Education 1990, pp. 149–155.)
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© 2011 Alberta Education
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