Multiplying decimals

Someone asked,
how can you use models to multiply decimals?

Learning to multiply decimals, I feel, is built on students' previous understanding of multiplying whole numbers and fractions.

So models wouldn't necessarily be the focus, but instead relating decimals to fractions first, and learning from that.

Multiply a decimal by a whole number

Of course, when multiplying a decimal by a whole number, you could use the same models as for fractions: say you have a problem

2 × 0.34

You can use little hundredths cubes, or draw something that's divided to 100 parts.

100 hundredths

BUT you can also just use fractions, and justify the calculation that way:

2 × 0.34 = 2 × 34/100 = 68/100 = 0.68.

OR you can explain it as repeated addition:

2 × 0.34 = 0.34 + 0.34 = 0.68.

I employ that idea in these lessons:

Multiply mentally decimals that have tenths and
Multiply decimals that have hundredths

Multiply a decimal by a decimal

When students are learning to multiply a decimal by a decimal, they're on 5th or 6th grade perhaps. One of the most obvious ways to teach this is to use fractions:

0.5 × 0.3 =

× 3

= 3 × 5

10 × 10


= 0.15.

With such calculations the student is made to notice that when both factors had one decimal digit (and thus had ten as denominator when written as fraction), then the result has two decimal digits (because you multiplied 10 × 10 to get 100 as denominator).

I do not know if there is any obvious pictorial model or manipulative that could be used to multiply a decimal by a decimal.

That is because it's no longer multiplying by a whole number.
It's no longer fitting the simple idea of "repeated addition".

It is instead best understood as 'taking part of' (just like multiplying a fraction by a fraction).

For example 0.1 × 45.9 is the same as taking 10th part of 45.9.

Or, 0.25 × 45.9 is taking 25/100 part of 45.9

And so on.

Understanding this is also crucial so that students can fathom how the answer number can sometimes be smaller than any of the factors... It's no longer adding where everything gets 'bigger' - it's taking a part of something.

See for example:

0.2 × 0.5 = 0.10

0.3 × 0.4 = 0.12

0.6 × 0.05 = 0.030

Here's one worksheet I've made that lets students practice the intricacies of decimal multiplication (a PDF file from my 5th grade worksheets collection).

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