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True or false? According to counting rules, 7+8 should be the same as 8+7.
Maria will build a rectangular greenhouse. The perimeter should be 26 metres, and the two longer sides should both be 9 metres. She calculates the shorter sides. ..and gets the answer 13 metres. Strange! The shorter sides should be shorter!
What went wrong? First thing to check is: Is the answer reasonable? Repeat the calculation using simpler numbers. She uses 30 metres instead of 26 for the perimeter, and 10 metres instead of 9 for each longer side. The two longer sides together are 10 plus 10 metres, which is 20 metres.
The other two sides are the perimeter, 30 metres, minus 20 metres. That is 10 metres. Since there are two shorter sides, she divides 10 metres by two, which equals five metres. Using simpler numbers is called an approximate calculation: Checking if the answer is reasonable, without being exactly correct. There was a huge difference between the first and the second results.
Five metres is less than half of 13 metres! To investigate what went wrong, she can calculate in reverse. Maria compares her previous calculations. In the second calculation she divided by two. To check, she multiplies by two instead.
Because multiplication is division backwards. The answer, 13, times what she divided by, two, should equal 26. And it is 26. 26 is the total of the two shorter sides. But wait.
26? That was actually the whole perimeter! Something went wrong on the way. But where? She continues to calculate backwards.
In Maria's first calculation, she subtracted nine, and then she added nine. To check, she takes the answer 26 plus nine, minus nine. Addition to check the subtraction, and subtraction to check the addition. After this calculation, the result is also 26. And that’s correct.
She has done all the calculations backwards, and now she is back at 26, which she started with. So something went wrong with the first calculation. Now Maria tries a third way of checking: using the counting rules and order of operations. Counting rules are about how to add or subtract numbers.. Five plus four is the same as four plus five.
But five minus four is not the same as four minus five. ..whereas order of operations says, among other things, that multiplication takes place before addition. The first calculation is 26 minus nine, plus nine. According to the order of operations, plus and minus have the same priority, but the calculation in parenthesis must be done first: this has higher priority. So first, the numbers inside the parentheses are added. Nine plus nine is 18.
Then calculate 26 minus 18, which is eight. The answer was eight instead of 26. Each shorter side of the greenhouse will then be eight divided by two, four metres. That is very close to the answer Maria got when she made an approximate calculation. She found the error.
Good! But could Maria have avoided this error? Maybe if she had been more careful in noting down all the calculations. Like this. Draw a figure.
The greenhouse is a rectangle with base and height. Enter the available information: The perimeter of the greenhouse and the rectangle are 26 metres. One longer side of the greenhouse, that is, the base of the rectangle is nine metres. Break down the calculations. The total length for both longer sides is nine plus nine metres, which is 18 metres.
The total length for the two shorter sides is the perimeter, 26, minus 18 metres, which equals eight metres. One shorter side of the greenhouse is eight metres divided by two, because there are two sides. The answer is four metres. So, Maria can check the answer by using: Estimation, to see if the answer is reasonable. Reverse calculation, and Checking the counting rules and order of operations.