Numbers & Operations in Base 10 Rounding

Significant Figures

TheCommonMistake WhatsGoingOnHere

From the submitter: “I came across these answers to sig fig questions when marking a pre-algebra numeracy test last night. I’ve attached two answers, one is a very common mistake where students just can’t believe that they are asked to round up that much. The second… well I don’t have any idea what the student was thinking, so I thought your readers might be able to help.”


10 replies on “Significant Figures”

How can this be confusing? The first student—well, honestly, I wouldn’t even mark it wrong. That’s a really crappy trick to play on a kid. The second one just doesn’t quite understand what gets incremented.

Why do you see this as a trick? It seems like a good question to assess whether a student has just memorized a simple ‘turn everything to 0’ type of algorithm or not. Seems like the payoff question here is a logical one. Second student? Wow – I’d need to sit down and ask some serious questions there.

I’m usually the first person to jump on a bad problem, but I don’t take issue with this set. I don’t like it when a question is at all ambiguous or unclear. When the correct answer is very solidly obtainable but tricky or challenging, I’m OK with the problem. Just my two cents.

Actually, I think the first one might be revealing. Are they rounding to 3 significant figures, or are they rounding to HAVE 3 TRAILING ZEROS? Given 12,842 would they give 12,800 or 13,000? Kind of an important distinction. There’s also the question of whether repeating the zero counts as a “single” significant digit, meaning we need to do something with the 5.

As to the second, I’m almost seeing the 3 Zero thing again in (b) and (c) – where they doubly rounded in (b) to have the number of zeroes at 3 – but (a) is kind of baffling. 970 and 120 don’t even add to 1000. (stares longer) Nope, no idea. Sorry.

I believe the second student may rounding different parts of the number separately. For b) & c) 28 & 46 are rounded to 30 & 50 respectively as a second phase of rounding. Part a) has me puzzled.

Hi everyone, thanks for your comments so far!
@Greg – The three trailing zero thing in the first picture is something that I didn’t notice – thanks for pointing that out. I’ll keep an eye out for that in the future.
As for the second picture – maybe they are trying to round substrings of numerals, but there’s nothing consistent in it.

I think that might be it in the second one, but yeah, it’s not consistent. Like, in a, they get to a 9 and so round the one before it up, bringing the 9 down to 0. But then there’s a 6 after it, so that have to round that 0 up to 1, bringing the six down to 0. Then they’d round again, but I don’t know how they got 2. The idea definitely works for the second and third one, the only part that is out of place with that theory is the 2 near the end of a).

Here’s my analysis of the 2nd student. I think basically James Cleveland has it right but there is a twist; the student makes an error executing his or her own erroneous method.

The method seems to be to round each pair of digits from left to right, starting somewhere in the middle. This was the method in (b): 67 got rounded to 70, 02 got left alone (because it felt wrong to round 02 down to 00), 28 got rounded to 30. Of course, you don’t start all the way at the left, that would be silly. You start at a place it feels about right to start rounding.

In part (a) this method was followed but there was an error. 09 was rounded to 10, then 06 was rounded to 10, then instead of rounding 08 to 10 we got 18 rounded to 20. That previous 1 just sort of slid over next to the 8, I would guess.

In part (c), the same method was followed, but the rounding didn’t start until the last two digits. Why mess with 8005, it looks pretty round already.

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