Transcript Dividing fractions mentally

Dividing fractions mentally
Mostly, we divide fractions by writing.
However, in some cases we can divide them
mentally.
It can be useful to know how to think in such cases.
Let’s investigate it...
In this presentation we’ll practice:
1st division when the numerator is divisible by
numerator and the denominator by denominator
2nd dividing a natural number by a natural number,
e.g. 4 ÷ 7,
3rd dividing a natural number by 2, e.g. 9 ÷ 2,
4th dividing a natural number by a proper fraction
1
__
with the numerator equal to 1, e.g. 4 ÷
,
2
5th division when the result is natural number,
1
__
e.g. 5 ÷ 2
.
2
Let’s gooooo…
Division when the numerator
is divisible by numerator
and the denominator by denominator
1. Calculate:
a)
24
8
__ ÷ __
35
7
=
3
__
5
b)
72
8
__ ÷ __
63
9
=
9
__
7
= 1
2
__
7
Can we calculate so in this case:
32
4
__ ÷ __
3
21
?
No, because 3 is not divisible by 21 ! (21 is divisible by 3)
It’s not allowed to divide from right to left, but only
from the left to right!
In this case we should calculate in writing (not now)…
1. Calculate:
a)
24
8
__ ÷ __
35
7
=
3
__
5
b)
72
8
__ ÷ __
63
9
=
9
__
7
= 1
c)
8
2
__
__
÷
9
9
=
4
__
1
= 4
2
__
7
We can imagine it…
8
2
Let’s think: How many times does __ go into __ ?
9
9
2
2
2
2
8
__
+ __ + __ + __ = __
9
9
9
9
9
4 times
1. Calculate:
a)
24
8
__ ÷ __
35
7
=
3
__
5
b)
72
8
__ ÷ __
63
9
=
9
__
7
= 1
c)
8
2
__
__
÷
9
9
=
4
__
1
= 4
2
__
7
8
2
Let’s think: How many times does __ go into __ ?
9
9
1. Calculate:
a)
24
8
__ ÷ __
35
7
=
3
__
5
b)
72
8
__ ÷ __
63
9
=
9
__
7
= 1
c)
8
2
__
__
÷
9
9
=
4
__
1
= 4
2
1
__
__
2 =
÷ __
3
3
1
Let’s think:
Imagine…
What part of the pizza
will each girl get?
d)
2
__
7
1. Calculate:
a)
24
8
__ ÷ __
35
7
=
3
__
5
b)
72
8
__ ÷ __
63
9
=
9
__
7
= 1
c)
8
2
__
__
÷
9
9
=
4
__
1
= 4
d)
2
1
__
__
÷ 2 =
3
3
e)
2
8
__
__
÷ 4 =
9
9
2
__
7
HowCan
much
thethe
cake
Imagine…
youofsay
result?
will each kid get?
Dividing a natural number
by a natural number
2. Calculate:
a)
2÷3 =
2
__
3
Imagine…
How much of the pizza
will each boy get?
2. Calculate:
b)
1
3
__
__
= 1
3÷2 =
2
2
Imagine…
How much of pizza
will each girl get?
2. Calculate:
Let’s consider last two examples:
a)
2
__
2÷3 =
3
b)
1
3
__
__
=
1
3÷2 =
2
2
Compare
the givenswap
numbers
in these
examples!
In division,
if numbers
places,
then we
get
the reciprocal!
They swapped places!
Compare the results!
The result is the reciprocal!
2. Calculate:
c)
15 ÷ 5 = 3
Imagine…
How many candies
will each child get?
2. Calculate:
c)
15 ÷ 5 = 3
d)
1
5
__
__
=
5 ÷ 15 =
3
15
Imagine…
How much of pizza
will each child get?
2. Calculate:
c)
15 ÷ 5 = 3
d)
1
5
__
__
=
5 ÷ 15 =
3
15
Compare the given numbers and the results again…
Given numbers swapped places,
and the result is the reciprocal!
2. Calculate:
c)
15 ÷ 5 = 3
d)
1
5
__
__
=
5 ÷ 15 =
3
15
e)
24 ÷ 4 = 6
d)
1
4
__
__
=
4 ÷ 24 =
6
24
Compare again…
2. Calculate:
c)
15 ÷ 5 = 3
d)
1
5
__
__
=
5 ÷ 15 =
3
15
e)
24 ÷ 4 = 6
d)
1
4
__
__
=
4 ÷ 24 =
6
24
e)
1
__
8 ÷ 40 =
5
Just give the final answer…
2. Calculate:
c)
15 ÷ 5 = 3
d)
1
5
__
__
=
5 ÷ 15 =
3
15
e)
24 ÷ 4 = 6
d)
1
4
__
__
=
4 ÷ 24 =
6
24
e)
1
__
8 ÷ 40 =
5
f)
9 ÷ 72 =
1
__
8
g)
3 ÷ 11 =
3
__
11
Dividing a natural number by 2
3. Give the final answer:
a)
1
__
7÷2 = 3
2
Imagine…
How much strawberries
will each boy get?
3. Give the final answer:
a)
1
__
7÷2 = 3
2
b)
1
11 ÷ 2 = 5 __
2
c)
1
__
27 ÷ 2 = 13
2
d)
40 ÷ 2 = 20
e)
1
41 ÷ 2 = 20 __
2
f)
1
203 ÷ 2 = 101 __
2
Dividing a natural number
by a proper fraction
with the numerator equal to 1
4. Calculate:
a)
1
__
1÷
= 2
2
Let’s think:
1
__
How many times does
go into 2 ?
2
1st time
2nd time
1
1
__
__
+
=
2
2
2 times
1
4. Calculate:
b)
1
__
2÷
= 4
2
Let’s think:
1
__
How many times does
go into 1 ?
2
1st
2nd
3rd
1
1
1
1
__
__
__
__
+
+
+
2
2
2
2
4 times
4th
=
2
4. Calculate:
c)
1
__
8÷
= 16
2
Whatsay
is the
here?
Just
thequestion
solution…
4. Calculate:
d)
1
__
1÷
= 3
3
Let’s think:
1
__
How many times does
go into 1 ?
3
1
1
1
__
+ __ + __
3
3
3
3 times
=
1
4. Calculate:
e)
1
__
2÷
= 6
3
What’s
here?
Just saythe
thequestion
solution…
Or…
4. Calculate:
f)
1
__
1÷
= 4
4
What’s
here?
Just saythe
thequestion
solution…
Or…
4. Calculate:
g)
1
__
3÷
= 15
5
What’s
here?
Just saythe
thequestion
solution…
Dividing fractions and mixed numbers
when the result is a natural number
5. Calculate:
a)
1
3 ÷ 1 __ = 2
2
Let’s think:
1
__
How many times does 1
go into 3 ?
2
1st time
2nd time
1
1
1 __
+ 1 __
2
2
2 times
=
3
5. Calculate:
b)
1
1
4 __ ÷ 1 __ = 3
2
2
Let’s think:
1
1
__
__
How many times does 1
go into 4
?
2
2
1st
2nd
3rd
1
1
1
1
1 __
+ 1 __ + 1 __ = 4 __
2
2
2
2
3 times
5. Calculate:
1
c) 10 ÷ 2 __ = 4
2
Let’s think:
1
__
How many times does 2
go into 10 ?
2
1st
3rd
2nd
1
1
1
1
2 __
+ 2 __ + 2 __ + 2 __ =
2
2
2
2
4 times
4th
10
With thanks to:
Rex Boggs
for support and help with the
translation into fluent U.S. idiom
(a.k.a. ‘American’).
Author of presentation:
Antonija Horvatek
Croatia , January 2014
You are welcome to use this presentation in your teaching.
Additionally, you can change some parts of it if used solely for
teaching.
However, if you want to use it in public lectures, workshops,
websites, in writing books, articles, on CDs or any public
forum or for any commercial purpose, please ask for specific
permission from the author.
Antonija Horvatek
http://www.antonija-horvatek.from.hr/
[email protected]