See below in Green for some corrections.
The math problem listed here has been making the rounds. It’s supposedly from the common core. If you haven’t seen it, it supposedly shows Jack using some number line to subtract 427 minus 316.
A lot of writers have been dumping on it.
I think they’re missing something — there are at least TWO errors in the work of this imaginary Jack.
The idea of trying to figure out where someone else got something wrong isn’t the worst idea in the world. However. what Jack was allegedly doing would need to be done in the head, because this method is so unwieldy if written out — as many people have pointed out. Also, if that was a carefully printed out number line, then I hope the problem is entirely imaginary, because unless we are teaching about logarithmic plots, then mathematicians take care to make sure that scales are linear (meaning that the distance from 100 to 200 equals the distance from 0 to 100, which are each exactly ten times the distance from 90 to 80 or from 57 to 67.)
As a mental exercise, number lines like this are not an entirely useless method.
Nobody seems to have pointed out that Jack made two math errors, not just one.
Since this problem asks 427-316, if you are doing this in your head, you could either count backwards from 4 by 3 units, or ask yourself how far it is from 3 to 4 — obviously 1. Writing the number line out is a lot of work, but saying silently to yourself, “427, 327, 127″ isn’t much work. So far so good.
But it’s not only in the mathematics that the imaginary Jack made an error:
I’m not sure, but it seems like the problem writer wanted Jack to confuse 16 and 60. This is not hard to do when HEARING the number, but more difficult if you are SEEING it written out. So this makes the problem a bit more, well, problematic, because nowhere in the problem is there any hint that Jack tends to hear poorly.
So this imaginary Jack mistakenly counts backwards in the tens place by tens by going 127, 107, 97, 87, 77, 67, 57 — which appears to be his final answer. Maybe. I can’t quite tell on this sheet.
So that means that “Jack” made another error in leaving out the decade 117.
Since it looks like this problem was written by some low-paid contract worker (think of “call centers” in Malaysia or India) with little scrutiny afterwards, we don’t know if the intention of the problem writer was for the student to realize that Jack is both hard of hearing and mis-counted by skipping the 117? If so, you are really asking a lot of a kid looking at the problem — and notice, if I’m right, then a whole lot of adults missed that point as well.
Did they really intend for the problem to be that difficult?
Sounds like an error on the part of the error-writer, but I could be wrong.
ADDED LATER: If the writer were aware that there were two mistakes in the problem, shouldn’t they have written “Find his error(s)” rather than “find his error”?
It turns out that I misinterpreted the problem by assuming that some of the writing was done by ‘Jack” when it was really done by the parent-engineer.
Here is more or less how the problem looked originally, minus all the blank space:
So it was the PARENT who missed the decade 117, not ‘Jack’. We see that ‘Jack’ counted back from 427 by hundreds three times to arrive at 127. Then ‘Jack’ counted backwards six times by ones from 127 to 121, which I’ve indicated by writing in the unwritten numbers below in red:
The mistake that imaginary ‘Jack’ made was neglecting to count backwards by ten one time; thus his answer was ten too large. The parent was the one who counted backwards from 127 by tens, six times, which I can sort of excuse, because the distance between the single units (127 to 126, then 126 to 127, and so on, is nothing like 100 times as small as he distance from 427 to 327).
I think the parent over-reacted, however, and made an embarrassing mistake.
My mistake was thinking that the numbers the parent wrote on the number line were ones that appeared in the original problem.
So, lots of errors all around.