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(easy hangman)
(difficult)
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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.
Multiplying Decimals by Decimals
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You have learned to think of multiplication by a whole number,
such as 3 × 4 or 8 × 0.6, as repeated
addition. However, this concept doesn’t work when neither of the factors is a whole number, as in 0.83 × 1.43 or 2/3 × 7/11.
Instead, when you multiply decimals or fractions, think of it
as finding “a certain part of” the other factor.
In this sense, the symbol “×” translates to “of.”
For example, think of 0.4 × 80 as finding four-tenths
“of” 80. Since 0.1 of 80 is 8, then 0.4 of 80 is 32.
Similarly, think of 0.21 × 700 as finding 21/100 of 700. Since 0.01 of 700 is 7, then 0.21
of 700
is 147.
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1. Solve and compare the
questions in each box:
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a. Find 0.1 × 30.
b. Find 0.4
× 30. |
(More problems available
in the book.) |
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Scaling a “stick”
Scaling means expanding or shrinking
something by some factor.
Scaling is a useful model for multiplication. Let’s look at scaling a
“stick” (a line segment). |
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This red stick
 is 40 pixels
long.
Let’s scale it to be four times as long:
→
We can write a multiplication “equation”:
4 ×
=
Using pixels, 4 × 40 px = 160 px.
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Now let’s scale the red stick to be
0.4 times as long as
it is at first:
→
And we write the multiplication equation:
0.4 ×
=
In pixels, 0.4 × 40 px = 16 px.
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The number we multiply by
(4 and 0.4 above) is called the scaling factor.
If the scaling factor is more than 1, the
resulting stick is longer than the original one.
If the scaling factor is less than 1, such as 0.5 or 0.66, the resulting
stick is shorter.
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2. The red stick
is 40 pixels
long. It is being scaled. Complete the corresponding multiplication
sentence:
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a. 0.1 ×
→
0.1 × 40 px = ______ px
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(More problems available
in the book.) |
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b. 0.2 ×
→
0.2 × 40 px = ______ px
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Towards a shortcut
Half of 5 is 2.5, or
0.5 × 5 = 2.5. This resembles the familiar multiplication 5 × 5 = 25!
One-tenth of 6 is 0.6, or 0.1 ×
6 = 0.6. Therefore, seven-tenths of 0.6 is 4.2,
or 0.7 × 6 = 4.2.
This resembles the
familiar multiplication 7 × 6 = 42!
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The shortcut to decimal
multiplication is:
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1) Multiply as if there
were no decimal points.
2) Place the decimal point in the answer.
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But where? We will explore that in the next exercise. |
3. Multiply first as if there was
no decimal point. Then add the decimal point to the number.
Think how BIG the answer should be.
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Examples:
0.8 × 0.8 has to be
slightly smaller than 0.8,
because scaling anything by 0.8 is close to
the original, but somewhat smaller.
So, 0.8 × 0.8 can’t be 64,
and it can’t be 6.4,
but it is 0.64!
0.1 × 5.6 has to be 1/10 of the size of 5.6.
So, 0.1 × 5.6 = 0.56.
0.06 × 0.4 has to be very
much smaller than 0.4.
It also has to be smaller than 0.06!
It can’t be 24, nor 2.4, nor 0.24. So it is 0.024.
0.9 × 0.04 would be just
slightly less than 0.04.
So 0.9 × 0.04 = 0.036. |
a.
0.5 × 0.3 =
b.
0.9 × 0.6 =
c.
0.4 × 0.7 =
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d.
0.4 × 0.08 =
e.
0.5 × 0.09 =
f.
0.7 × 0.02 =
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(More problems available
in the book.)
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4. a. Check your answers
above with a calculator or the answer key.
b. Look carefully at the
problems you did in Exercise #3. We’re still thinking about where to
put the decimal point. Look at the number
of decimal digits (decimal places) in the factors,
and the number of decimal digits in the answer. Do you notice a pattern?
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The shortcut to decimal multiplication
1) Multiply
as if there were no decimal points.
2) Place the decimal point in the answer.
The number of decimal digits in the answer
is exactly the sum of the number of decimal digits in each of the factors. |
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Examples:
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0.05 × 0.7
5 × 7 is 35. The factors 0.05
and 0.7 have 2 and 1
decimal digits. The answer
has to have 3, so
the answer is 0.035. |
0.1 × 1.2 × 1.1
1 × 12 × 11 = 132. The
factors have 1 and 1 and 1
decimal digits. The answer
has to have 3, so
so the answer is 0.132. |
31 ×
0.03 × 2
31 × 3 × 2 = 186.
The factors have have 0,
2, and 0 decimal digits
The answer has to have 2,
so the answer is 1.86. |
5. Solve.
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a.
0.4 × 0.8 =
b.
0.7 × 1.1 =
c.
0.02 × 0.9 =
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(More problems available
in the book.) |
When you first multiply the numbers, ignoring the decimal point, the
“answer”
to that multiplication may end in one or more zeros. That is no problem.
Apply the rule
to the number ending in zeros as you would to any other number.
However, after placing the decimal point, you may simplify
the final answer
by dropping the ending decimal zeros. |
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50
× 0.006
50 × 6 = 300. The factors have 0 and
3
decimal digits, so the answer also has to
have 3.
Therefore, the
answer is 0.300.
This simplifies to 0.3. |
400
× 0.05
400 × 5 = 2000. The factors have 0
and 2
decimal digits, so the answer has to have 2.
Therefore,
the answer is 20.00.
You can simplify that to 20. |
6. Solve.
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a.
0.4 × 0.5 =
b.
20 × 0.06 =
c.
0.9 × 0.5
× 0.2 = d.
40 × 0.05 =
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(More problems available
in the book.) |
See also: Multiply and divide by 10, 100 and 1000
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.
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New! Times Tales is now on DVD!
The fast, FUN, and easy way to learn multiplication. Learn the upper times tales in two sittings using mnemonic stories.
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