Perimeter = area rectangles
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.
About this resource
This activity assumes the students have experience in the following areas:
- Find the areas and perimeters of rectangles.
- Identify linear relationships, and express those relationships using equations.
The problem is sufficiently open-ended to allow the students freedom of choice in their approach. It may be scaffolded with guidance that leads to a solution, and/or the students might be given the opportunity to solve the problem independently.
The example responses at the end of the resource give an indication of the kind of response to expect from students who approach the problem in particular ways.
Perimeter = area rectangles
Achievement objectives
GM4-3: Use side or edge lengths to find the perimeters and areas of rectangles, parallelograms, and triangles and the volumes of cuboids.
NA4-7: Form and solve simple linear equations.
Required materials
See Materials that come with this resource to download:
- Perimeter area rectangles activity (.pdf)
Activity
As you know, this is a special type of rectangle called a square.
If you walked the perimeter of this rectangle, you would travel 4 + 4 + 4 + 4 = 16 units of length.
The area of this rectangle is 4 x 4 = 16 square units.
The units for perimeter and area are different, but both measures equal 16.
- For what other rectangles are the measures for perimeter and area equal?
The following prompts illustrate how this activity can be structured around the phases of the mathematics investigation cycle.
Make sense
Introduce the problem. Allow students time to read it and discuss it in pairs or small groups.
- Do I understand the situation and the words? (Students may need support to understand the ideas of perimeter and area and the units of measurement involved. Not that there are no metric units for area used, though students may want to impose units on the rectangles.)
- What can change and what must stay fixed? (Students can vary the side lengths of the rectangles, but the area = perimeter condition must be met.)
- What will a solution look like?
- How will I know when a possible answer does or does not work?
Plan approach
Discuss ideas about how to solve the problem. Emphasise that, in the planning phase, you want students to say how they would solve the problem, not to actually solve it.
- How can I work in a systematic way?
- How will I record my mahi so I can see patterns if there are any?
- Where should I start? Which dimensions for the rectangle would be good ones to try first? Why?
- Should I fix one dimension and vary the other? How will this work?
- Which strategies should I pursue? (Drawing diagrams, creating tables, writing equations, and programming a spreadsheet are all potentially useful strategies.)
- What tools (digital or physical) could help my investigation?
Take action
Allow students time to work through their strategy and find a solution to the problem.
- Do my recordings reveal a pattern that leads to solutions?
- Can I find other solutions?
- How will I know when I have found all the possible solutions?
- Do my results look the same or different from others? Why could this be?
- Is there another possible way to find solutions? (Students might use algebra to find all solutions.)
Convince yourself and others
Allow students time to check their answers and then either have them pair share with other groups or ask for volunteers to share their solution with the class.
- What is the solution?
- Is my working clear for someone else to follow?
- How would I convince someone else that I am correct?
- Could I have solved the problem in a more efficient way?
- What connections can I see to other situations, and why would this be? Could the strategy work for other measurement problems? What kinds of problems are there?
- Have I considered all possible cases? How do I know that no more solutions exist?
- What mathematics do I need to learn more about?
Examples of work
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.
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 are tried until the conditions are met.
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 the 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 equality, students can set l and solve for w, or vice versa, to find solutions.
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.
l |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|---|---|---|---|---|---|---|---|---|---|---|
w |
NA |
NA |
6 |
4 |
10⁄3 |
3 |
14⁄5 |
16⁄6 |
18⁄7 |
20⁄8 |
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 w = 2l/(l-2). The formula can be found by rearranging the equality formula, although the algebra is about level 6 or 7 of NZC.
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