# Grading a Simple Rounding Exercise

A student is given the question:

"Round off each of the following numbers, correct to two significant figures.

• 32.4892
• 8.2673
• 475.98
• 0.078625"

There are two marks for each.

• 32.0000
• 8.3000
• 480.0000
• 0.07900

I believe that $32.0000=32$ so the student has correctly rounded... although I concede they might have somewhat missed the point.

• It seems like they understand rounding and the concept of two sig figs (since all of their answers are rounded correctly and they all only contain 2 sig figs), so they got the point of the assignment. they just wrote their answers wacky and with tons of unnecessary 0's. I would deduct 1 to 2 points overall, imo. Oct 7 '16 at 12:22
• I think the issue here is with the problem statement. The student correctly solved the task you have given, I would give 100% of points for correctness. (Although I might deduce some clarity points if you have such a thing.) Oct 7 '16 at 12:52
• @celeriko 1 or 2 points out of how many? Oct 7 '16 at 13:09
• They show no understanding of the meaning of rounding. 32.4892 does not round to 32.0000. 2 or 3 off out of the 8. Oct 9 '16 at 3:44
• My husband, an engineer, has drilled into me the difference in his work of a measurement of 3.2 and 3.200. I in turn have passed this on to my students. However, many elementary school teachers don't emphasize this or even understand it. My question is, were the students taught that there is a reason to drop those extra zeroes? Oct 9 '16 at 16:10

What is the whole point of rounding values? "For brevity, we approximate long decimals by finding the nearest specified place value." (This is copied directly from the slide in my first-day college statistics lecture.) If the expression didn't get shorter, then the student really has missed the whole point, and needs correction. Since it was the same basic error repeated, I would tend to take off the value of one single item (1/4 of the total value for this sequence; i.e, 2 points per your comment).

Secondarily, while mathematically it's true that $32.0000 = 32$, in a practical scientific context that carries different, distinct information. The first expression is saying "this measurement is accurate to the ten-thousandths place", whereas the second is saying "we are only confident of the precision to the units place".

This latter issue is something that I'm not sure my students ever really understand; they may just be going through the motions by rote, which doesn't make me super happy, but we don't have extra time for investigation of instrumentation accuracy.

• Compelling arguments...rounding 475 to 500 doesn't make it shorter. Oct 7 '16 at 18:48
• @JpMcCarthy: Although for most use cases with n > 1 it makes it similarly shorter in scientific notation, e,g., rounding 475,621 to one significant digit allows us to write $5 \times 10^6$. Oct 7 '16 at 22:09
• Yes I thought of this afterwards. Of course we don't have great cause to round 475 in this fashion either. Oct 8 '16 at 7:26
• Why not? Calculations with 500 are much easier in most cases and rounding is often used in order to get a quick approximation of the final result. Oct 8 '16 at 15:23
• @Jp: 500 also has less cognitive load than 475, can be read more quickly.
– user797
Nov 7 '16 at 14:53

$32.4892$ rounded off to two significant digits is $32$. The student's answer i.e. $32.0000$ contains six significant digits and an absolute error $|\epsilon_a|=0.5×10^{-4}$