## Friday, June 19, 2009

### M4N2 a - Rounding

M4N2. Students will understand and apply the concept of rounding numbers.

a. Round numbers to the nearest ten, hundred, or thousand.

Rounding is a specific technique to approximate numbers. Some teachers in primary grades actually teach their students rounding when they want students to "estimate." However, "estimation" and "rounding" aren't the same idea. In fact, as an approximation technique, it is probably better to teach rounding when students are working with larger numbers.

As you know, rounding a number to the nearest designated place means to look at the numeral to the right of the place to which we are rounding. If the numeral is 4 or less, we will round down (i.e., simply change all places to the right of the designated place 0's) and if it is 5 or above, we round up (i.e., increase the numeral in the designated place by 1 and change all numerals to the right 0's). So, when 45,542 is rounded to the nearest thousands place, it will be 46,000, and when it is rounded to the nearest hundreds, it will be 45,500.

One question students sometime ask is why we round up with a "5" even though 5 is right in the middle (of 0, 1, ..., 9). Some teachers will simply say it's just a rule. But, is it?

Let's consider 45,542. If we want to round this number to the nearest thousands place, we are really asking is it closer to 45,000 or 46,000. According to the procedure, we will be checking the numeral in the hundreds place. So, what numbers between 45,000 and 46,000 have a 5 in the hundreds place? Well, 45,500 is definitely one. But there are a lot more: 45,501, 45,502, 45,503, ... 45, 598, 45,599. Altogether there are actually 100 numbers in this range with a 5 in the hundreds place? So, which of these numbers are closer to 45,000? 46,000? Right in the middle? Well, it's obvious that all but one of these numbers are actually closer to 46,000, and the one exception is right in the middle. If that's the case, in general, does it make sense to round a number with a 5 in the hundreds place up or down?

The problem with "5 is right in the middle" comes up only when you are rounding to the nearest tens (and only if we are looking at whole numbers). Since approximate numbers are used when we have very large numbers of very small numbers, perhaps trying to teach rounding, a specific approximation procedure, with such small numbers may not make any sense.