The ideas in this fraction lesson are taken from the Fractions 2 ebook.
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
What is common to these problems? What 'shortcut' can you use?
Why?
10 ÷ 4 ×
4 =
20 ÷ 5 ×
5 =
28 ÷ 7 ×
7 =
10 × 4 ÷
4 =
20 × 5 ÷
5 =
28 × 7 ÷
7 =
51 ÷ 4 ×
4 =
37 ÷ 5 ×
5 =
228 ÷ 7 ×
7 =
177 × 4 ÷
4 =
1890 × 5 ÷
5 =
104056 × 7 ÷
7 =
Remember that the fraction line can also be used for division.
Let's
write the problems above using the fraction line:
10
4
× 4 = 10
20
5
× 5 = 20
28
7
× 7 = 28
10 ×
4
4
= 10
20 ×
5
5
= 20
28 ×
7
7
= 28
51
4
× 4 = 51
37
5
× 5 = 37
228
7
× 7 = 228
177 ×
4
4
= 177
1890 ×
5
5
= 1890
104056
× 7
7
= 104056
Hopefully you noticed that
If you both multiply and divide by the same number,
you have done nothing!
Examples:
4
×
7
4
= 7
5
9
× 9 = 5
Usually we indicate this by 'crossing over' the same number that
is used to both multiply and divide.
In other words, we simplify the problem BEFORE calculations and make
it easier!
Examples:
8
×
5
8
= 5
7
10
× 10 = 7
Example problem types
1. Do these problems 'the old way' according to the example, and also
by 'the new way' where you simplify beforehand.
a. "Old way":
12
7
× 7 =
12 ×
7
7
=
84
7
=
Simplifying before calculating:
12
7
× 7 =
b. "Old way":
10 ×
3
10
=
=
Simplifying before calculating:
10 ×
3
10
=
c. "Old way":
5
11
× 11 =
=
Simplifying before calculating:
5
11
× 11 =
2. Can you figure out how to simplify in these
cases? First solve the problems the old way as in the example.
a. "Old way":
4
5
×
5
9
=
4 ×
5
5 ×
9
=
20
45
=
4
9
Simplifying before calculating:
4
5
×
5
9
=
b. "Old way":
2
3
×
3
10
=
=
Simplifying before calculating:
2
3
×
3
10
=
c. "Old way":
7
10
×
3
7
=
=
Simplifying before calculating:
7
10
×
3
7
=
You can cross out the same number above the line and below
the line:
4
5
×
5
9
=
4
9
Why does this work? Compare how it is written using ÷
instead of a fraction line:
4
5
×
5
9
=
4 ÷
5×
5 ÷ 9 .
Note how there is again both division by 5 and multiplication by
5.
That is why we can simplify or "cross" those fives
out. Similarly,
8
7
×
3
8
=
8 ÷
7 × 3 ÷
8 . There is
8 and there is
division by 8, so
8
7
×
3
8
=
3
7
.
You can simplify a fraction before
multiplying. In the example here 3/6 is simplified to 1/2 before
the multiplication process, which makes it much
easier.
1 3
6 2
×
5
8
=
5
16
Why does this work? Obviously we can
write
1
2
instead of
3
6
since they are equivalent.
You can even simplify like
this: Compare to the example
above.
1 3
8
×
5
6
2
=
5
16
Why does this work? Because either way,
3
8
×
5
6
=
3 ×
5
8 ×
6
or
3
6
×
5
8
=
3 ×
5
6 ×
8
you get the same result in the end.
Actually the explanation requires talking
about factors and more talk about division/multiplication
canceling each other, and we omit the details here. In
essence,
3
8
×
5
6
=
3 ×
5
8 ×
3 ×
2
and 3 appearing above the line means we
multiply by it, and below the line
means we divide by it, so they cancel each
other.
3. Simplify before you do the
multiplication. Note: if you need to simplify the end result, it
means you could have simplified more before multiplying. Your
answer may still be right though. Simplifying before multiplying does NOT
change the final answer but makes it easier to calculate.
a.
3
10
×
1
3
=
b.
5
6
×
2
4
=
c.
4
8
×
1
3
=
d.
2
6
×
5
7
=
You can even simplify two times like this:
1 3
15
×
5
6
2
Step 1: Simplify 3 and 6.
1 3
15
3
×
1 5
6
2
=
1
6
Step 2: Simplify 5 and 15.
Or this is fine too:
1 3
15 5
×
7
14
Step 1: Simplify 3 and 15.
1 3
15
5
×
1 7
14
2
=
1
10
Step 2: Simplify 7 and 14.
4. Simplify before you multiply, but do the work in your notebook.
The ideas in this fraction lesson are taken from the Fractions 2 ebook.
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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