1) Focus first on the "terminators," those denominators which always result in a terminating decimal. As we observed in class, we can describe this group of denominators in algebraic terms this way: 5a2b, where a and b are integers greater than or equal to zero. Explain (on a separate sheet, of course) what that notation is actually saying about the group of numbers.
2) Remember that each decimal place represents a particular fraction. The decimal .3 actually means three tenths, because there is a three in the tenths place. What does .004 mean? What does .67 mean? What does .139 mean? What do all of those fractions have in common?
3) When you look at the group of terminators, you have to wonder: why those numbers? Why 2 and 5? What's so special about them? You might try writing some proportions, such as 1/2 = 5/10.
This shows why one-half converts nicely to point 5 and terminates there. We're not talking about halfs anymore; we're talking about tenths. And tenths are a nice decimal-friendly fraction. So are hundredths, thousandths, etc., as you saw in #2.
What about one-eighth? It's on your list as something that terminates. It also can be written in the form we described in #1. It is 23 times 50. So what friendly fraction does 1/8 convert to?
Make proportions for a lot of different terminators. (You can share with your group members to get a lot of examples.) Make a chart. Include a column in the chart to show how you can write your terminator as 2a times 5b.
4) Try making some proportions for the numbers that don't terminate, such as one-third, one-sixth, etc.. How come those proportions don't work out so nicely?
5) Okay, if I am guaranteed to get a terminating decimal whenever I use factors of 2 and 5, I think I will make up this denominator: 27 times 519. Now that is a really big number. I am not going to bother multiplying it out Even if I did, my calculator would probably cut off some digits, so I wouldn't have the true value anyway. So, even though I can't express that product as a simple integer, I can still convert it into a power of ten. What would I need to multiply by? Show the proportion that helps convert 1/(27 times 519) into something/(some power of ten).
6) Try a few more examples like #5. Come up with a standard procedure that handles any problem of that type.
7) Check your standard procedure developed in #6 and see if it works for simple fractions like 1/4. How about 3/4? Does it matter what the numerator is?
8) You will find that a lot of the conclusions you arrive at in this class are difficult to prove. (That's why you're mad at me, right?) The case of the terminating decimal, however, is actually fairly easy to prove. This is an example of using algebra. You employ variables to give a general description of a group of numbers and you manipulate the symbols to show a result that must be true, no matter what the specific example is. The proof comes from the generalization. Take the general description of k/(2a times 5b) and show that any number of that form can be easily converted into something/(some power of ten).
| QR Home Page | Glossary | Previous Worksheet | Next Worksheet |