Karen's Tiles
investigate how perimeter changes as area changes
see the relation between area and perimeter for similar shapes
pose questions for mathematical exploration
prove or refute mathematical conjectures
To be able to do this problem, students need to be able to explore situations and calculate the areas of rectangles and triangles. They should probably have tried Peter’s String, Level 4.
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 by following our mathematical nose.
There are seven problems in the Problem Solving section that focus on the perimeter-area relationship. These come in two sets. First there is the set 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 set 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. Throughout this Problem Solving sections of the website we are hoping to motivate students to think about what they are doing and see connections between various aspects of this. The mathematical question asked in this problem is does perimeter always increase as the area increases? This question is typical of, and fundamental to, much of mathematics, in that it looks at relations between two quantities. In general it is valuable to know how one thing varies as the result of the change in another. The whole theory relating to functions looks at this kind of relation.
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 encircle. 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 of a rectangle, we then use in a completely different application – 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. We look at polygons and their relation with circles.
Problem
Karen had discovered some old square tiles in the garage. She played around with them and made different shapes. She got to wondering whether the perimeter of these shapes increased as the area increased. Marty thought that it was obvious. Do you?
Teaching sequence
-
Introduce the problem to the class. Get them to consider how they would approach the problem.
- Students at this level may not find it useful to explore using tiles. However, don’t insist that they use them. Let them investigate the problem in any way that they want. At some stage though they will probably have to draw some diagrams. They may need some help at this point as the problem gives them no direction as to what drawings or shapes they should tackle nor in what order they should be tackled. Further they may need encouragement to try ‘unusual’ shapes.
- Move round the groups as they work to check on progress. Encourage them to draw large diagrams to show clearly what is going on. If a lot of the pairs are having problems, then you may want a brainstorming session to help them along.
- The Extension problem may not be quite as difficult as the original problem as it deals with specific shapes. Encourage as many students as possible to try this problem.
- Share the students’ answers. Get them to write up their work in their books. Make sure that they have carefully explained their arguments.
Extension problem
Maybe Karen is right if she sticks to rectangles. If the area of a rectangle increases, does its perimeter have to increase too?
What if Karen limits herself to a set of similar shapes?
Solution
Karen made a conjecture: As the area of objects increase, their perimeters also increase. We have to justify or refute this conjecture.
There are two useful strategies for dealing with investigations such as this. One is to look at simple cases. The other is to investigate extreme cases.
We’ll try the simple case first. We’ll take Karen’s basic tile and play around with some simple shapes and hope that this gives us enough information to settle the conjecture, or at least to put us on the track of a solution.
Take the square tiles and assume that they have an area of 1 unit and a perimeter of 4 units.
By combining tiles, we can make new figures and we can calculate their areas and perimeters. If we record these numbers we can begin to get a picture of how perimeter varies with area.
So the second figure that we make has an area of 2 and a perimeter of 6.
This third figure has an area of 5 and a perimeter of 12. If a new tile is added, the area goes up to 6, but the perimeter goes down to 10!
So, it’s possible to increase the area of a figure but decrease the perimeter in the process. Hence the conjecture is false.
The problem can also be looked at through extreme cases as follows. We’ll use an argument that you know of if you have looked at Peter’s String, Level 4. Join the two ends of a piece of string. Form the loop into a square. Now form it into a long very very thin rectangle. Both figures have the same perimeter, but they have very different areas. Thus it is possible to increase the area without increasing the perimeter.
Extension Solution: The same process works here as worked in the original problem. Look for an extreme case. So take a rectangle with length 1 and width 10. This will have area of 10 and a perimeter of 22. Now look for a rectangle with an area of 20 but with a smaller perimeter. Well, 4 x 5 = 20 so take the rectangle with length 4 and width 5. This has perimeter 18 and that is smaller than 20. So as area goes up perimeter may go down. (Of course it may go up too but it isn’t guaranteed to go up.)
But what if we stick to similar figures? Let’s use a rectangle as the basis for our first set of similar shapes. If we take a rectangle that is 4 units by 6 units, then a similar rectangle will have sides 4k by 6k, where k is some positive number. (This way corresponding sides are in the same ratio, a necessary condition for similarity.) For these figures, the area is 24k2 and the perimeter is 10k. So as the area increases (this is done by k increasing), the perimeter also increases.
This tends to suggest that similar figures will have the property that as the area increases then so does the perimeter. At this Level, it isn’t possible to prove this for all shapes but it is possible to do it for a range of shapes that give evidence for the conjecture. Experiment with triangles, quadrilaterals and polygons. You should find in each case, that if the basic area of the shape is A and its perimeter is P, then the area of a similar figure will be Ak2 and its perimeter will be Pk.
| Attachment | Size |
|---|---|
| KarensTiles.pdf | 36.19 KB |
Similar Resources
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
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)
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.
Karen's Second Tiles
Determine the maximum area of a regular polygon with a given perimeter
Appreciate the concept of limit as it applies to the area of regular n-gons and circle that both have the same perimeter
