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Decimal Manipulatives
by Gail Englert

A response to the question:

How would you teach multiplying decimals using only manipulatives?

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I would like to know what manipulatives you can use to teach decimals to 5th graders. How would you teach multiplying decimals using only manipulatives?



Dear Mary,

The same model pieces you use for ones, tens and hundreds can be used to represent tenths, hundredths and thousandths. When we use those models pieces in my fifth grade classroom, we refer to them as "huge cubes" (for the large cubes used most often to represent thousands), "flats" (for the large flat square pieces used most often to represent hundreds), "longs" (for the long thin pieces used most often to represent tens), and "little cubes" (for the tiny unit cubes used most often to represent ones). Then, when I wish to change the values, it is not difficult for my students, because we have been using these names, and selecting the one that would be used to represent "one whole". If you make the "huge cube" one whole, then the flat is worth a tenth, the long worth a hundredth, and the little cube worth a thousandth. You could name the flat a whole, and that would make the huge cube worth ten times that much, a ten, while the long would represent a tenth, and the little cube a hundredth.

You can also use money to represent tenths (dimes) and hundredths (pennies) pretty well, since most students that age are familiar with money. Sometimes I have my students cut a penny-model into ten equal pieces, and we refer to those as thousandths. (I relate them to the little numbers after the prices on a gas station sign) Since we live in an urban area, this is a familiar sight.

Finally, you can use grid paper (graph paper) to show the tenths and hundredths pretty well, too. Have your students draw a 10 by 10 square. That will represent one whole. If you color in just one little square in that outline, that is one hundredth, or 0.01. Two little squares, 0.02, and so on. If you color in ten of the little squares, which would be the same as coloring in one column, or one row, you have a tenth, or 0.10, or 0.1.

Now, you can use that model to show multiplication of decimals (tenths).

Suppose you have the problem 0.3 x 0.6.

Shade in three rows (going across) in one color, say, red.

Then, shade in 6 columns (going up and down) in another color, say, blue.

Where the two shadings intersect is the product of the two factors. Since 18 little squares are shaded by both blue and red, we know that the product is 18 hundredths, or 0.18.

I also have my students consider what the multiplication problem is saying to them. In the case of 0.3 x 0.6, it is saying, "you have 3/10 of 6/10". Since I know 1 of 6/10 is 6/10, and 3/10 is less than one half, I can guess that my answer will be less than half of 6/10 (which is 3/10).

So, if we are trying to multiply .48 x 3.3, I ask my students to round the amounts to compatible numbers, numbers they could work with more easily, first, to obtain an estimate. They would decide that 0.48 was about a half, and that 3.3 was about 1, so the estimation would be "half of 3", which is about 1 and a half. Then, when they do the actual multiplication, what they will discover is just like multiplying wholes, except you have to place a decimal in the product. When you are through, they would end up with 1584. To place the decimal, they would decide where to put it to get an answer that is closest to 1.5.

They could try .1584 or 1.584 or 15.84 or 158.4 or 1584

Which one is nearest to 1.5?

-Gail, for the T2T service

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