Peter's String
Determine the area of a rectangle with a given perimeter;
Pose questions for mathematical exploration
Make conjectures in a mathematical context
Prove or refute mathematical conjectures
To be able to do this problem students need to be able to measure lengths and calculate the perimeters and areas of rectangles using the formulae: perimeter = twice length plus twice width and area = length x width. They should also know how to use a table.
Apparently in some areas of New Guinea they measure the area of land by its perimeter. When you think about it this isn’t such a good idea. A piece of land can have a relatively large perimeter and only a small area. This sequence of problems is built up from this simple bad idea.
There are seven problems in the Problem Solving section that focus on the perimeter-area relationship. First there is the string of Peter’s String problems. These are Peters’ String, Measurement, Level 4, Peters’ Second String, Measurement, Level 5, Peters’ Third String, Algebra, Level 6, The Old Chicken Run Problem, Algebra, Level 6 and the Polygonal String Problem, Algebra, Level 6. These follow through on the non-link between rectangles’ areas and perimeters, going as far as showing that among all quadrilaterals with a fixed perimeter, the square has the largest area. In the second last of these five problems we are able to use an idea that has been developed to look at the old problem of maximising the area of a chicken run. This is often given as an early application of calculus but doesn’t need more than an elementary knowledge of parabolas. The final problem looks at the areas of regular polygons with a fixed perimeter. We show that they are ‘bounded above’ by the circle with the same perimeter.
The second string of lessons looks at the problem from the other side: does area have anything to say about perimeter? This leads to questions about the maximum and minimum perimeters for a given area. The lessons here are Karen’s Tiles, Measurement, Level 5 and Karen’s Second Tiles, Algebra, Level 6.
Mathematics is more than doing calculations or following routine instructions. Thinking and creating are at the heart of the subject. Though there are some problems that have a set procedure or a formula that can be used to solve them, most worthwhile problems require the use of known mathematics in a novel way.
Throughout this web site we are hoping to motivate students to think about what they are doing and see connections between various aspects of what they are doing. The mathematical question asked here is what effect does the perimeter of a rectangle have on its area? This question is typical of a lot of mathematical ones that attempt to see what effect one quantity has on another quantity.
The ideas in this sequence of problems further help to develop the student’s concept of mathematics, the thought structure underlying the subject, and the way the subject develops. We start off with a piece of string and use this to realise that there is no direct relation between the perimeter of a rectangle and its area. This leads us to thinking about what areas are possible. A natural consequence of this is to try to find the largest and smallest areas that a given perimeter can encompass. We end up solving both these problems. The largest area comes from a square and the smallest area is as small as we like to make it.
Some of the techniques we have used to produce the largest area are then applied in a completely different situation – the chicken run. This positive offshoot of what is really a very pure piece of mathematics initially, is the kind of thing that frequently happens in maths. Somehow, sanitised bits of mathematics, produced in a pure mathematician’s head, can often be applied to real situations.
The next direction that the problem takes is to turn the original question around. Don’t ask given perimeter what do we know about area, ask given area what do we know about perimeter. Again there seems to be no direct link.
But having spent time with rectangles, the obvious thing to do is to look at other shapes.
Problem
Peter had kept a piece of string that had been on a parcel that had come for his birthday. It was 30 cm long. He played with it and made different shapes out of it. Then he got stuck on rectangles. He wasn’t sure but he thought that all the rectangles he made had had the same area. His sister Veronica said that was crazy but she didn’t have a good reason for saying that. Who was right and why?
Teaching sequence
- Introduce the problem to the class. Get them time to consider how they would approach the problem.
- Give each pair of students a piece of string. It’s probably a good idea to give different groups different lengths. Let them investigate Peter’s conjecture in any way that they want. If they prefer to draw diagrams rather than to measure the sides of rectangles produced with their string, that is OK.
- Move round the groups as they work to check on progress. The first problem is reasonably straightforward so they will move on to the Extension Problem reasonably quickly. Look to see how they attack it. If a lot of the pairs are having difficulties, then you may want to have a brainstorming session to give them some ideas to follow.
- Share the students’ answers in a class discussion. Get them to write up their results in their books.
Extension problem
Show that Peter can make a rectangle with area 24 cm2. Veronica got hold of a piece of string too. It was only 20 cm long though. Can she make a rectangle with a bigger area than that?
Solution to the problem
We would expect students to take a practical approach to this problem and to spread out the string in a shape of a rectangle, measure the length and width of the rectangle and calculate its area. It would help to draw up a table so that the students’ explorations are recorded systematically. We do that below for some sizes of rectangles.
|
length |
width |
area |
|
1 |
14 |
14 |
|
2 |
13 |
26 |
|
3 |
12 |
36 |
Table 1
It is clear that Peter’s conjecture that the areas he could make were all the same is false.
Extension Solution:
If Peter is to make an area of 24 cm2 with his string, then he has to find L and W so that 2L + 2W = 28 and LW = 24. Using quadratic equations can solve this but we don’t expect students to be able to do that until at least Level 6. (This problem might be used again at that Level to encourage the practical use of quadratic equations.) But we can do it using an approximation method. This could be via a graphics calculator but we will use a table here. We will guess L and W so that L + W = 15 (half of 30) and keep adjusting our guess until we get LW equal, or very close to, 24. (The answer we give here is sufficiently close for our purposes. Accuracy to more decimal places can be obtained if required.)
We first point out that the previous table (Table 1) gives us a place to start. L is clearly just under 2 and W just above 13.
|
L |
W |
L + W |
LW |
|
1.9 |
13.1 |
15 |
24.89 |
|
1.8 |
13.2 |
15 |
23.76 |
|
1.85 |
13.15 |
15 |
24.33 |
|
1.84 |
13.16 |
15 |
24.21 |
|
1.83 |
13.17 |
15 |
24.10 |
|
1.82 |
13.18 |
15 |
24.00 |
Table 2
Using a table similar to Table 1, we can see that Veronica can, in fact, produce an area of 25 cm2. (See Peter’s Third String, Level 6, for a justification of the fact that the rectangle of largest area and given perimeter is a square.) So even though Veronica has a smaller string than Peter, she can produce a bigger area than a particular rectangle that he can make.
| Attachment | Size |
|---|---|
| PetersString.pdf | 40.43 KB |
| PetersStringMaori.pdf | 50.36 KB |
Similar Resources
The Old Chicken Run Problem
use algebraic equations to determine the maximum area of a rectangle with a given partial perimeter.
Polygonal Strings
This is a problem from the number and algebra strand.
Peter's Third String
Determine the maximum area of a rectangle with a given perimeter
Consider how the area of a quadrilateral changes as its shape changes
Interpret a relationship from a graph
Peter's Second String
determine the maximum area of a rectangle with a given perimeter
determine the range of areas of a rectangle with a given perimeter
pose questions for mathematical exploration
prove or refute mathematical conjectures
The Chicken Run
explain the relationship between the area and perimeter of rectangles
use a table to solve a problem
devise and use problem solving strategies to explore situations mathematically (guess and check, be systematic, make a table, make a drawing)
