From the Tutor's Corner

Magnitude Estimates for Products

I found the answers to some of the questions on this topic in Lesson 2.7 of the fifth grade math Journal to be very confusing. (Apparently I wasn't alone.)

Most of the answers are reasonable: \(7.6\) is estimated as \(7\), for example. But then in another problem, \(7\) is estimated as \(10\)! So why wasn't \(7.6\) estimated as \(10\)? In another problem, \(5\) is estimated as \(10\). This means that \(5*5\) is estimated as \(100\). Is \(100\) is a reasonable estimate of \(25\)? One number is in the 10's column, the other is in the 100's column. Besides, if \(7.6\) is estimated as \(7\), why can't \(5\) just be estimated as \(5\)?

One of the purposes of this exercise is to aid in locating the decimal point in the product of decimal numbers. Yet in one problem, \(0.8*0.8\) is estimated as \(1*1\), or \(1\). So a student who knows that \(8*8\) is \(64\), concludes that the answer should be \(6.4\) because this puts it in the 1's column where the estimate said it should be.

The teachers manual says that numbers should be "rounded to the nearest multiple of a power of 10". Does anybody know for sure what that means? Does it mean that numbers should be rounded to the form (\(N*10^{n}\))? Or, that numbers should simply be rounded to \(10^{n}\)? In either event, neither interpretation explains all of the answers given. It's probably a moot point because teaching students to "round to the nearest multiple of a power of 10" would be beyond their capability.

It seems to me that when estimating the distance to the nearest galaxy, or the size of the national debt, an order of magnitude estimate may be acceptable. But in most mundane cases, more accurate estimates are required. Suppose, for example, a student is given the following problem:

"A \($36\) item has been marked down \(60\%\). What is the estimated amount (in dollars) that the item has been marked down?" So the student rounds the \($36\) to \($40\), and the \(60\%\) (which has been converted to \(0.6\)) to \(1\), and concludes that the item has been marked down \(40*1\) or \($40\). In other words, the store is paying customers \($6\) to buy the item.

An even more practical problem would be: "You wish to leave a \(30\%\) tip to a particularly good waitress who has served you a \($14\) meal. Approximately how much of a tip should you leave?" So using the official rules, \($14\) rounds to \($10\), the \(30\%\) (\(0.3\)) rounds to \(0.1\), \($10*0.1=\) a tip of \($1\). The waitress may give you a dirty look, but hey, this is higher math.

It seems to me we need some higher guidance on how to teach this subject.

John Schwarz