Getting a perspective on decimals

In my years as a 6th grade teacher, I found many students who had trouble with decimal operations. They could read the numbers: "three hundred sixty-one and two hundredths," but then they'd get a problem such as 65.001 x 3.2 and come up with an answer like 2080.032, with no idea of why their answer made no sense.  Addition, subtraction, and especially division all seemed quite hard.  Their difficulty came because they didn't understand decimals enough to be able to estimate their answer to see if it made sense.  For example, in the above problem, 65.001 might as well be plain old 65 and 3.2 is pretty close to 3, so the answer should be pretty close to 65 x 3.  Even a ballpark estimate of 70 x 3 = 210 should tell the student that 2080.032 was way off.

It's easy for students to get confused about decimals, or even not understand the significance of where the decimal is placed.  After all, when a decimal number is written, all the digits appear to be the same size and that little dot is awfully small-- surely it can't be that important!  And for the many students who have trouble with whole number place value, the concept of decimal numbers can be almost meaningless.

So before introducing decimals, be sure your student has a firm grasp on place value.  For example, in the number 222, the digits are the same, but their value is completely different.  A student should understand that 222 is the same as 200 + 20 + 2.  (Writing it this way is known as expanded notation, because the number is expanded into the digits' actual values.)

If a student can write a number in expanded notation, but still seems to have trouble keeping digits lined up when adding or subtracting, it might be helpful to use manipulatives.  Here's a link to paper base ten blocks that can be cut out.

At left are some base 10 blocks (these are from the  MathUSee basic set, but others work as well) illustrating what 222 looks like.

That's all fine and dandy, but what do you do to illustrate the relative size of decimal numbers?





One way is to use base ten blocks, only this time the largest block represents "one" and each smaller segment is a decimal number.

 To match the previous set of base 10 blocks, I have made a giant "one" (out of paper and foam board) and colored it the same color as the original "one."  It's as if the original one is being seen through a magnifying glass.

Here we see 2.222; notice the large blue "decimal" that matches the scale of the blocks.










Of course, the giant blocks don't match the scale of the original whole number blocks, so when you want to model what a whole number with decimals looks like, we have to go smaller.  For this we can use modeling clay in a shade that matches the original "one" block.  (If the student performs this activity himself, it may leave more of an impression, but either way it's one he'll remember.)

Here we see a "one" block and a block made of clay, approximately the same size.

Using a knife, cut the clay block in half, and then cut each half into 5 pieces.  You have now divided one into tenths.  Roll up each tenth into a ball.

Next, take one of the tenth balls and roll it into a snake that you can cut into ten tiny pieces.  These are your hundredths.  Roll them into little balls.

At this point, the student should begin to see just how different a hundredth is from a hundred!

 It's almost impossible to divide the tiny hundredth ball into thousandths, so for the .001 place value, I use a pencil or marker of the same color to draw dots on the paper in the thousandths place.  (White boards are a good base for this!)

Here you can barely see the pencil marks, but we have 0.222 illustrated.

Now, when we put the blocks all together, it is easy to see what 222.222 really looks like.  At this point, it would be good to discuss with the student how significant the placement of the decimal point is.

And, of course, have the student practice writing and illustrating various numbers.  Can he model 256.3?  256.03?  256.003?  Which is bigger?  Can he write the correct number when you place various numbers blocks in the columns?

Most importantly, can the student see why estimating with whole numbers can help him get the decimal placement right in his final answer?  Those tiny little pieces of clay can be ignored long enough to get a ballpark idea of what the solution will be.