Comparing Decimals

Today, Student A had a worksheet for comparing decimals. She was supposed to say which decimal in each pair was greater than the other, or if they were equal.

Because she originally completed the assignment by guessed (a pretty usual occurrence for A), I decided to give her some steps to follow in comparing decimals before we even began correcting the assignment.

First, we talked about how decimals are just like fractions – meaning they are just part of a number, instead of a whole number. With that in mind, we went on to our steps for comparing decimals:

Step 1. If there is a whole number (the part of the number that is to the left of the decimal point), compare the whole numbers first. If the whole numbers are different, then all you have to do is tell which whole number is greater. (for example, 12.3 and 14.72 – 14.72 is greater because 14 > 12, and it doesn’t matter how much “extra” you tack on to the 12, 12 is still smaller than 14). Otherwise, move on to step 2.
Step 2. Since the whole number parts are the same, we will need to compare the decimals. Remember that decimals are like fractions, so .1 = 1/10, and .01 = 1/100. Well, it’s hard to compare fractions when the bottoms are different (that is, it’s not very intuitive) and it is equally hard to compare decimals when we don’t have the same number of places, such as in 4.35 and 4.3.
To make things easier to “see” (instead of having to think too hard – mathematicians never like to think harder than they absolutely have to – that’s why we have “tricks” for everything) we are going to make sure each decimal has the same number of “places”. To do this, all we have to do is add zeros on the right hand side of the number. Going back to our example of .35 and .3, since .35 has two places and .3 only has one, we will add a zero to .3 and have .30. Now we are comparing .30 and .35 – well, that is just like comparing 30 and 35. It is clear that 35 > 30, so .35 > .30.
If you had .30 and .3, when you add a zero to .3 you will get .30 and .30 – in this case, the numbers are equal.

I was really impressed with Student A’s internalizing this concept. After five or six problems she was able to figure them out quickly, and explain her answers just as fast. She really understood the concept, and that was a huge success for us.

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