```
```Thoughts on this division mistake? #mathmistakes http://t.co/TZ1dxsNFBH

— Chris Robinson (@absvalteaching) January 31, 2014

Or did I get this wrong?

Categories

- Post author By mpershan
- Post date February 4, 2014
- 6 Comments on File This Under “Calculator Mistake”

```
```Thoughts on this division mistake? #mathmistakes http://t.co/TZ1dxsNFBH

— Chris Robinson (@absvalteaching) January 31, 2014

Or did I get this wrong?

## 6 replies on “File This Under “Calculator Mistake””

Looks to me like confusion between 10/45 and 45/10, indeed, and with no work shown on the long division, almost surely a calculator entry error. A sanity check on the answer (should be more than $4 per pound, not less than $1) would certainly help here!

My theory is that this is a case of fumbled fingers on the calculator keyboard. The student intended to calculate 45/10 but instead of 10 entered 200 by accidentally pressing the 2 key instead of the 1 key and accidentally pressing the 0 key twice. (Possibly the second press of the 0 key was intended as a press of the decimal key.) So the result was 45/200=0.225 instead of 45/10=4.5.

If I’m right, there are several contributory mistakes here: 1) the student failed to check the calculator display to verify that the correct data was entered; 2) the student failed to check the reasonableness of the result by recognizing that since 45 is greater than 10, 45/10 must be greater than 1; 3) the student failed to check the result by reversing the operation and finding 0.22*10 = 2.2, not 45.

Finally, and most importantly, the student should not be using a calculator to divide by 10.

I think the 10/45 error is a more parsimonious explanation than the 45/200 as a path to 0.22. I agree with point (2) from Zeno, that the estimation skill to recognize we want something greater than 1 would be a useful direction to focus on as a way of catching this category of mistake.

The simple transposition error Joshua suggests seems likely. It’s probably attributable to the fact that the long division bracket format the student used to indicate the division puts the divisor before the dividend. If the student had indicated the division with a division sign (obelus) or slash (virgule) then the operands would have appeared in calculator entry order and the error would not have occurred.

Could it be they reversed it? 4 goes in to 10 twice with 2 left over hence 0.22. Or maybe 4 goes into 10 twice, 5 goes into 10 twice? Alternative theories if no calculator was used.

The correct answer depends on how much cheese I want.

If I don’t like cheese, I want to spend as little money as possible. Either I don’t buy any cheese, or I buy just 10 pounds for $ 45.

If I only need 10 pounds of cheese, I buy 10 pounds for $ 45, at an average price of $ 4.50 per pound.

If I only need 15 pounds of cheese, I buy 15 pounds for $ 71.25 — even though I am paying $ 5.25 per pound for the incremental 5 pounds. (The incremental cost is $ 71.25 – $ 45 = $ 26.25, for an incremental gain of 15 pounds – 10 pounds = 5 pounds. The incremental price per pound is $ 26.25 / 5 pounds = $ 5.25 per pound.) This costs less than two packages of 10 pounds each ($ 45 + $ 45 = $ 90).

If I need 20 pounds of cheese, I spend $ 90 (on two packages of 10 pounds each).

If I need 25 pounds of cheese, I spend $ 116.25 (on a package of 10 pounds, and a package of 15 pounds).

If I need 30 pounds of cheese, I spend $ 135 (on three packages of 10 pounds each). This costs less than two packages of 15 pounds each. ($ 71.25 + $ 71.25 = $ 142.50)