This problem solving activity has a measurement focus.
Henry wants to make a rectangular chicken run at the back of the house.
He buys 12 metres of fencing wire.
What is the largest area run that he can make for his chickens?
What if Henry has 60 metres of fencing?
This problem requires students to calculate the perimeter and ares of rectangles, and to make comparisons between the area of different-sized areas. In the process, students should discover that the maximum area of a rectangle is obtained when the rectangle is a square.
The problem solving strategies employed by students might include drawing a diagram (in order to ‘see’ how the dimensions of a rectangle of fixed perimeter relate to each other), making a table (in order to compare calculations easily) and being systematic (to make the table entries easier to compare).
Henry wants to make a rectangular chicken run at the back of the house. He buys 12 metres of fencing wire. What is the largest area run that he can make for his chickens?
What if Henry has 60 metres of fencing?
Repeat the problem above with 14 metres of fencing.
Calculate the area of a number of rectangles and then choose the dimensions that give the largest area.
For example: If Henry's chicken run has these dimensions the area is 1 x 5 = 5 square metres.
If Henry's chicken run is 2 x 4 metres, the area is 8 square metres.
If the side lengths are all 3 metres, the area is 3 x 3 = 9 square metres.
Hence the maximum area is 9 square metres.
Second part of problem:
Henry must do considerably more calculations. A table is the clearest way to show these.
width | length | area |
(metres) | (metres) | (square metres) |
1 | 29 | 29 |
2 | 28 | 56 |
3 | 27 | 81 |
4 | 26 | 104 |
5 | 25 | 125 |
6 | 24 | 144 |
7 | 23 | 161 |
8 | 22 | 176 |
9 | 21 | 189 |
10 | 20 | 200 |
11 | 19 | 209 |
12 | 18 | 216 |
13 | 17 | 221 |
14 | 16 | 224 |
15 | 15 | 225 |
From the table we see that Henry’s largest chicken run has area 225 square metres.
Students might notice that the biggest area in the two examples above occurs when the rectangle is a square. Is this always the case?
Let Henry try the 14 metre fencing option.
width | length | area |
(metres) | (metres) | (square metres) |
1 | 6 | 6 |
2 | 5 | 10 |
3 | 4 | 12 |
3.5 | 3.5 | 12.25 |
? | ? | ? |
? | ? | ? |
Henry’s table now uses decimal side lengths and adds in the square case as well. Here the square is again the best option.
This might cause students to question whether decimals instead of whole numbers give a bigger area.
If so, extend Henry’s first table to include decimal lengths.
Surprisingly the square always turns out to have the biggest area. The reason behind this can be proved at Level 7 or 8.
Printed from https://nzmaths.co.nz/resource/chicken-run at 3:35pm on the 19th April 2024