The purpose of this activity is to engage students in finding the perimeter and area of rectangles. They engage in trying to find rectangles where the number measure of the perimeter equals the number measure of the area.
The background knowledge presumed for this task is outlined in the diagram below:
This activity should be used in a ‘free exploration’ way with an expectation that students will justify the solutions that they find.
The procedural approach (show more)
- The student uses trial and error approaches to find some solutions, calculating both perimeter and area correctly.
Using squared paper, the student can investigate if other rectangles are possible. They are likely to try many unsuccessful rectangles before they find one that works.Click on the image to enlarge it. Click again to close.
A more systematic approach is to constrain the length of one side and experiment with the other. Below the length is set at 10 units and various rectangles tried until the conditions are met.Click on the image to enlarge it. Click again to close.
The conceptual approach (show more)
- The student uses algebraic thinking with linear relationships by substituting values for one variable into the perimeter equals area equation.
Students may begin by defining the unknown sides as l (length) and w (width) and finding equations to work out perimeter and area.
Perimeter = l + w + l + w = 2 (l + w)
Area = l x w or lw
Since perimeter and area must be equal the single equation becomes:
2(l + w) = lw
Having established the equality students can set l and solve for w or vice versa to find solutions.Click on the image to enlarge it. Click again to close.
In only three solutions do both l and w have whole number values. Students can generate a set of solutions and organise the data in a table.
Students might also recognise that the denominator of w is always l – 2. The numerator of w is always double l.
This yields the general formula of w=2l/(l-2). The formula can be found be rearranging the equality formula though the although the algebra is about Level 6 or 7 of NZC.