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(easy hangman)
(difficult)
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The ideas in this fraction lesson are taken from the Math Mammoth Fractions 2 book.
Only a few examples of each problem type are shown; you should make more problems of each kind for
the student.
Simplifying before multiplying
Free fraction lesson plan from HomeschoolMath.net
1. Solve. You may use a calculator - but try to find a “shortcut”
so that you don't need the calculator.
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a. 20 × 5 × 5 = 28 × 7 × 7 =
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b. 5 × (37 × 5) = 7 × (228 × 7) =
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c. 1,890 × 5 ×
5 = 104,056 × 7 ×
7 =
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Let's
rewrite the problems above using the fraction line to indicate division.
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20
5 |
× 5 = 20 |
|
28
7 |
× 7 = 28 |
|
| 5 × |
37
5 |
= 37 |
| 7 × |
228
7 |
= 228 |
|
|
1,890 ×
5
5 |
= 1,890 |
|
104,056
× 7
7 |
= 104,056 |
|
|
Hopefully you noticed that
If you multiply and divide by the same number,
you have done nothing!
|
Examples:
4 |
× |
7
4 |
= 7 |
5
9 |
× 9 = 5 |
231 × 11
11 |
= 231 |
We indicate this by crossing out the number that
is used to both multiply and divide.
This crossing out of numbers is also called simplifying.
Examples:
8 |
× |
5
8
|
= 5 |
7
10
|
× 10 = 7 |
16 × 45
16
|
= 45 |
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2. Do the problems two ways:
by calculating, and by simplifying (see example).
|
a. Calculate:
|
12
7 |
× 7 = |
84
7 |
= 12 |
Simplify:
|
12
7
|
× 7 = 12 |
|
b. [available in the book]
|
c. [available in the book]
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3. Simplify.
|
a. |
82
77 |
× 77 = |
|
|
|
You can cross out any number that
appears both
above the line and below
the line... IF the only other
operation you see is multiplication. |
5 × 9
9 × 7
|
= |
5
7 |
|
|
Why does this work? Well,
the expression |
5 × 9
9 × 7 |
is how you would calculate the
fraction |
|
multiplication
problem |
5
9 |
× |
9
7 |
. But it is also the way you would calculate the
multiplication |
|
problem |
9
9 |
× |
5
7 |
. In the latter problem ,we may simplify 9/9 to 1/1 or just 1
before even |
|
calculating anything. So, |
5 × 9
9 × 7 |
equals |
5
9 |
× |
9
7 |
, which equals |
9
9 |
× |
5
7 |
, which equals |
5
7 |
. |
If you change the 9 in the above to some other
number, the same reasoning applies. So, no matter what number appears both above and below the fraction line, you can
cross it out (provided we are dealing with
multiplication, not with some
other operation)
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4. Simplify before multiplying.
|
a. |
4
5
|
× |
5
9 |
= |
|
|
|
|
|
|
|
f. |
6
11 |
× |
11
6 |
= |
|
|
You can also write another equivalent,
simpler
fraction in the place of a fraction. In the
one example on the
right, 3/6 is simplified to 1/2
before the multiplication. We
write a tiny “1” and a |
|
|
|
|
tiny “2” in place of
the “3” and “6”. In the other example, 4/10 is
simplified to 2/5 before multiplying.
| Why does this work? Obviously we can
write |
1
2 |
instead of |
3
6 |
since they are equivalent. |
|
5. Simplify one or
both fractions before multiplying. Use equivalent fractions. Look at the example.
|
a. |
3
6
10
5
|
× |
1
2
14
7
|
= |
3 × 1
5 × 7 |
= |
3
35 |
|
|
|
|
|
|
|
f. |
27
45 |
× |
21
49 |
= |
|
You may cross out the same number from the
numerator and the denominator,
IF the operation used in them is multiplication.
|
| Examples: |
|
8 ×12
7 × 8 × 3 |
= |
12
21 |
|
|
|
5 × 5 ×
2
3 × 5 |
= |
10
3 |
= 3 |
1
3 |
|
|
|
6 ×
3
5 × 3 × 6 |
= |
1
5 |
|
| |
| |
We may have more numbers in
the numerator
than in the denominator or vice versa. |
|
|
If it looks like “nothing” is left
in the numerator or denominator,
in reality there is 1. |
|
|
| Why does it work with |
6 × 3
5 × 3 × 6 |
? Well, that is equal
to |
6 × 3 × 1
5 × 3 × 6 |
, because multiplying by 1 does not |
| change a thing! And that is the way you would calculate the problem |
6
5 |
× |
3
3 |
× |
1
6 |
and |
| also the problem |
6
6 |
× |
3
3 |
× |
1
5 |
. In this last problem we can simplify 6/6 and 3/3. All that
is left is |
1
5 |
. |
|
6. Simplify by crossing out the
numbers that appear both above the fraction line and below it.
|
a. |
6 × 2
5 × 6 |
= |
|
|
|
|
|
|
|
f. |
2 × 7 × 2
7 × 2 |
= |
|
7. Simplify and multiply.
|
a. |
3
8 |
× |
8
24 |
= |
|
|
|
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Epilogue: what happens if you don't simplify before multiplying?
Let's look at two ways
to calculate a certain problem.
|
1
7
35
5
|
× |
6
8 |
= |
6
40 |
= |
3
20 |
Tina simplifies 7 and 35 into 1 and 5 first.
Lastly, she simplifies
her answer 6/40 into 3/20. |
|
1
7
35
5
|
× |
3
6
8
4
|
= |
3
20 |
Jack simplifies 7 and 35, just like Tina,
and also 6 and 8, before multiplying. |
Jack did all
of the
simplifying before
multiplying.
Tina simplified some before multiplying,
and some after. But both kids got the right answer. Simplifying before multiplying
does NOT
change the final answer - it just makes it easier to multiply, because
your numbers are smaller! |
|
You can even simplify like
this: |
1
3
8
|
× |
5
6
2
|
= |
5
16 |
or |
7
6
2
|
× |
1
3
9 |
= |
7
18 |
. |
Did you notice what happened? The 3 and 6 become
1 and 2. This is very similar to the earlier
example where the fraction 3/6
was simplified to 1/2 before multiplying. |
|
Why would this kind of thing work?
|
Notice that the problem |
3
8 |
× |
5
6 |
and the problem |
3
6 |
× |
5
8 |
have the same answer: |
3 ×
5
8 ×
6 |
= |
15
48 |
= |
5
16 |
. |
|
You know that in the problem |
3
6 |
× |
5
8 |
you may simplify 3/6 into 1/2. You can do the same |
|
in |
3
8 |
× |
5
6 |
, because the two expressions are equal. |
(They are both equal to
5/16, so they are also equal to each other.) |
If that sounded difficult, look at it this way:
| Let's write 6 as 2 × 3. The expression becomes |
3 ×
5
8 ×
3 × 2 |
. We can then cross out the “3” from both |
|
above the line and below the line: |
3 ×
5
8 ×
3 × 2 |
= |
5
8 ×
2 |
= |
5
16 |
. |
|
1. Simplify before
multiplying.
|
E. |
3
10 |
× |
1
3 |
= |
|
|
|
P. |
4
8 |
× |
1
3 |
= |
|
|
|
|
|
|
|
|
|
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I. |
7 |
× |
5
21 |
= |
|
|
|
|
These problems |
|
5
12 |
1
9 |
1
10 |
|
41
40 |
5
3 |
2
15 |
1
6 |
2
11 |
1
10 |
|
16
3 |
5
21 |
2
15 |
|
|
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|
|
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! |
|
|
You can simplify several times before multiplying. |
|
1
3
15
|
× |
5
6
2
|
First simplify 3 and 6. |
Then simplify 5 and 15. |
|
|
1
3
15
5
|
× |
7
14
|
First simplify 3 and 15. |
|
1
3
15
5
|
× |
1
7
14
2
|
= |
1
10 |
Then simplify 7 and 14. |
|
2. Simplify before you
multiply.
|
a. [available in the book] |
|
b. |
3
10 |
× |
2
18 |
|
|
|
d. |
7
21 |
× |
3
4 |
|
|
|
|
|
|
|
i. |
14
25 |
× |
35
42 |
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3. Try your skills with multiplying three fractions.
|
|
[available in the book] |
[available in the book] |
|
 [available
in the book]
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Continue to the lesson Dividing fractions by a whole number
The ideas in this fraction lesson are taken from the Math Mammoth Fractions 2 book.
Only a few examples of each problem type are shown; you should make more problems of each kind for the student.
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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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