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Basic operations worksheet generator is updated! Make worksheets for whole numbers or integers - all four operations, including add & subtract or multiply & divide - horizontal or vertical - add a border and/or extra workspace, use a variable, and more.
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The ideas in this decimals lesson are taken from Math Mammoth Decimals 2 book ($4.00 download). The book has more problems than shown in this online lesson.
Multiply and Divide Decimals by 10, 100, and 1000
The video below shows, first of all, the common shortcut: you move the decimal point in the decimal number as many steps as there are zeros in the number 10, 100, 1000 etc. Then, I also show where this shortcut originates, using place value charts. In reality, it's not the decimal point moving (it's sort of an illusion), but the digits of the number move within the place value chart. This explanation can really help students to understand the reason behind the "trick" of moving the decimal point.
The lesson below explains the shortcut in much more detail, plus has examples of different kinds of problems and exercises for students.
When you multiply whole numbers by 10, 100, 1000 and so on, you can use this
shortcut:
Simply “tag” as many zeros to the product as there are in the factor
10, 100, 1000 etc.
There is a similar
shortcut for multiplying decimal numbers by 10, 100, 1000 and so on:
You move the decimal point to the right as many places as
there are zeros in the factors.
10 × 0
.
4
9
= 04.9 = 4.9
Move the decimal point
one step to the right.
100 × 2
.
6
5
= 2
6
5
. = 265
Move the decimal point two steps to the right.
The number 265. would be 265.0 or just 265
Why does it work this way? Let’s consider
multiplying by 10. Our number system is based on ten. Each place value
unit (ones, tens, hundreds, etc.) is 10 times the previous unit. Each
number can be broken down as a sum of the different place values. For
example 3,849 = 3,000 + 800 + 40 + 9.
When each one of these
parts is multiplied by 10, they become 30,000 + 8,000 + 400 + 90 =
38,490. The shortcut is that you simply tag a zero to the number.
It works the same way with decimals: for
example 0.429 = 0.4 + 0.02 + 0.009. When each of the parts is multiplied
by 10, the whole thing becomes 4 + 0.2 + 0.09 = 4.29. It looks like the
decimal point got moved ... but in reality the value of each digit
increased ten-fold.
Consider also! Since 100 × 2 = 200, obviously the answer to
100 × 2.105 will be a little more than 200. So you can just write the
digits
2105 and put the decimal point so that the answer is 200-something:
210.5.
On the previous page you found the shortcut for multiplying decimals
by 10, 100, and 1000.
Now it’s time to learn a similar shortcut for division by 10, 100
and 1000. Can you guess it?
Move the decimal point to the ( left / right ) for as many places (steps) as
there are zeros in the ______________.
0 0
.
4
9
÷ 10 = 0.049
Move the decimal point
one step to the ______.
You can write a zero in
front
the number to help.
0
0
5
6
.
0
÷ 1000 = 0.0560
= 0.056
Move the decimal point
three steps to the ______.
You can write zeros in front
of the number to help.
Why does it work? Division is the opposite operation of multiplication,
so it “undoes” multiplication. If we move the decimal point to the
right when multiplying by 10, 100, 1000 and so on, then it’s quite
natural that the rule for division would work the “opposite” way.
Notice
something interesting! If we use the rule of
moving the decimal point three steps,
6 ÷ 100 = 0.06. To see that, you can write 6 as 6.0 and then move the
point:
0
0
6
.
0
÷ 100 = 0.060 = 0.06
But let’s write 6 × 100 using the
fraction line:it is
6
100
or 6 hundredths, which is
written 0.06 as a decimal. So in this case you don’t need
the “shortcut,” but just think
of fractions and decimals.
These kind of “connections” make mathematics so neat!
4. Think of “fractions to
decimals,” or use the shortcut:
If we divide any whole
number by 1,000, the answer has thousandth parts or three
decimal digits. This makes it easy to divide whole numbers by 1,000. You simply make the
result have three
decimals. For example:
72
1000
= 0.072,
391
1000
= 0.391,
89,302
1000
= 89.302,
430
1000
= 0.430 = 0.43
In the last case, we can
simplify the result 0.430 to 0.43, but initially it does have three
decimals.
Similarly, if you divide any whole number by
10, the resulting decimal will have one decimal digit.
And if you divide any whole number by 100, the resulting decimal will have
two decimal digits.
Instead of thinking how the decimal point moves, you can think of how
many decimals the answer
must have.
13. Which vacuum cleaner ends up being cheaper?
Model A, with initial price $86.90, is discounted
by 3/10 of its price.
Model B costs $75 now, but you’ll get a discount
of 25/100 of its price.
An important tip
In the problem ____ ×
3.09 = 309, the number 3 becomes 300,
so obviously
the missing factor is 100.
You don’t even have to consider the decimal point.
The same works with division, too. In the problem
7,209 ÷
_____ =
7.209, the missing divisor
is one thousand, because the value of the digit 7
was first 7000,
and then it became 7.
Of course, in some problems it may be easier to think in terms of
“moving the decimal point.”
14. It’s time for some
final practice. Find the missing numbers. Match the letter of each
problem with the right answer in the boxes, and
solve the riddle. There are two
sets of boxes. The first boxes belong to the first set of exercises, and
the latter box belongs to the latter set.
Why didn’t
3.14 understand what 7 was talking about?
The ideas in this decimals lesson are taken from Math Mammoth Decimals 2 book ($4.00 download). The book has more problems than shown in this online lesson.
Basic operations worksheet generator is updated! Make worksheets for whole numbers or integers - all four operations, including add & subtract or multiply & divide - horizontal or vertical - add a border and/or extra workspace, use a variable, and more.
