Dividing fractions mentallyDividing fractions mentally

Mostly, we divide fractions by writing.Mostly, we divide fractions by writing.

However, in some cases we can divide them However, in some cases we can divide them

mentally.mentally.

It can be useful to know how to think in such cases.It can be useful to know how to think in such cases.

Let’s investigate it...Let’s investigate it...

In this presentation we’ll practice:In this presentation we’ll practice:

1st division when the numerator is divisible by 1st division when the numerator is divisible by

numerator and the denominator by denominatornumerator and the denominator by denominator

2nd dividing a natural number by a natural number,2nd dividing a natural number by a natural number,

e.g. 4 ÷ 7,e.g. 4 ÷ 7,

3rd dividing a natural number by 2, e.g. 9 ÷ 2, 3rd dividing a natural number by 2, e.g. 9 ÷ 2,

5th division when the result is natural number,5th division when the result is natural number,

e.g. 5 ÷ 2 .e.g. 5 ÷ 2 .____1122

4th4th dividing a natural number by a proper fraction dividing a natural number by a proper fraction

with the numerator equal to 1, e.g. 4 ÷ ,with the numerator equal to 1, e.g. 4 ÷ ,____1122

Let’s gooooo…Let’s gooooo…

DDivision whenivision when the the numerator numerator

is divisible by numeratoris divisible by numerator

and and the the denominator by denominatordenominator by denominator

1. Calculate:

a) 87

__2435__ ÷ = __3

5

b) 89

__7263__ ÷ = __9

7= 1 __2

7

Can we calculate so in this case:

421__32

3__ ÷

No, because 3 is not divisible by 21 ! (21 is divisible by 3)

?

It’s not allowed to divide from right to left, but onlyfrom the left to right!In this case we should calculate in writing (not now)…

c) 29

__89__ ÷ = __4

1= 4

Let’s think: How many times does go into ?__29

__89

__29

+ __29

+ __29

+ __29

= __89

4 times

We can imagine it…

1. Calculate:

a) 87

__2435__ ÷ = __3

5

b) 89

__7263__ ÷ = __9

7= 1 __2

7

__41

= 4

1. Calculate:

a) 87

__2435__ ÷ = __3

5

b) 89

__7263__ ÷ = __9

7= 1 __2

7

c) 29

__89__ ÷ =

Let’s think: How many times does go into ?__29

__89

d) 223__ ÷ =__

1

__13

Imagine…Let’s think:

What part of the pizzawill each girl get?

1. Calculate:

__41

= 4

a) 87

__2435__ ÷ = __3

5

b) 89

__7263__ ÷ = __9

7= 1 __2

7

c) 29

__89__ ÷ =

__13

e) 489__ ÷ =

Can you say the result?

29

__

Imagine…How much of the cakewill each kid get?

1. Calculate:

__41

= 4

a) 87

__2435__ ÷ = __3

5

b) 89

__7263__ ÷ = __9

7= 1 __2

7

c) 29

__89__ ÷ =

d) 223__ ÷ =

DDividing ividing a a natural number natural number

by by a a natural numbernatural number

2. Calculate:

a) 2 ÷ 3 = __23

How much of the pizzawill each boy get?

Imagine…

b) 3 ÷ 2 = __32

= 1 __12

2. Calculate:How much of pizzawill each girl get?

Imagine…

2. Calculate:

Let’s consider last two examples:

a) 2 ÷ 3 = __23

b) 3 ÷ 2 = __32

= 1 __12

Compare the given numbers in these examples!

They swapped places!

Compare the results!

The result is the reciprocal!

In division, if numbers swap places, then we getthe reciprocal!

c) 15 ÷ 5 = 3

2. Calculate: How many candieswill each child get?

Imagine…

c) 15 ÷ 5 = 3

2. Calculate:

d) 5 ÷ 15 = __515

= __13

Imagine…

How much of pizzawill each child get?

c) 15 ÷ 5 = 3

2. Calculate:

d) 5 ÷ 15 = __515

= __13

Compare the given numbers and the results again…

Given numbers swapped places,and the result is the reciprocal!

c) 15 ÷ 5 = 3

2. Calculate:

d) 5 ÷ 15 = __515

= __13

e) 24 ÷ 4 = 6

d) 4 ÷ 24 = __424

= __16

Compare again…

c) 15 ÷ 5 = 3

2. Calculate:

d) 5 ÷ 15 = __515

= __13

e) 24 ÷ 4 = 6

d) 4 ÷ 24 = __424

= __16

e) 8 ÷ 40 = __15

Just give the final answer…

c) 15 ÷ 5 = 3

2. Calculate:

d) 5 ÷ 15 = __515

= __13

e) 24 ÷ 4 = 6

d) 4 ÷ 24 = __424

= __16

e) 8 ÷ 40 = __15

f) 9 ÷ 72 = __18

g) 3 ÷ 11 = __311

DDividing ividing a a natural number by 2natural number by 2

3. Give the final answer:

a) 7 ÷ 2 = 3 __12

Imagine…

How much strawberrieswill each boy get?

a) 7 ÷ 2 = 3 __12

b) 11 ÷ 2 = 5 __12

c) 27 ÷ 2 = 13 __12

d) 40 ÷ 2 = 20

e) 41 ÷ 2 = 20 __12

f) 203 ÷ 2 = 101 __12

3. Give the final answer:

DDividing ividing a a natural number natural number

by by a a proper fraction proper fraction

withwith the the numerator equal to 1numerator equal to 1

4. Calculate:

a) 1 ÷ =__12

Let’s think:

How many times does go into 2 ?__12

2

__12

+ __12

= 1

2 times

1st time 2nd time

4. Calculate:

b) 2 ÷ =__12 4

__12

+ __12

= 2

4 times

+ __12

+ __12

1st 2nd 3rd 4th

Let’s think:

How many times does go into 1 ?__12

4. Calculate:

c) 8 ÷ =__12

Just say the solution…

16

What is the question here?

4. Calculate:

d) 1 ÷ =__13 3

__13

+ __13

= 1

3 times

+ __13

Let’s think:

How many times does go into 1 ?__13

4. Calculate:

e) 2 ÷ =__13

Just say the solution…

6

What’s the question here?

Or…

4. Calculate:

f) 1 ÷ =__14

Just say the solution…

4

What’s the question here?

Or…

4. Calculate:

g) 3 ÷ =__15 15

Just say the solution…What’s the question here?

DDiviividing fractions and mixed numbersding fractions and mixed numbers

when the result is when the result is a a natural numbernatural number

5. Calculate:

a) 3 ÷ 1 __12 =

Let’s think:

How many times does 1 go into 3 ?__12

1st time 2nd time

2

__12

1 + __12

1 = 3

2 times

5. Calculate:

b) 4 ÷ 1 __12

__12 =

Let’s think:

How many times does 1 go into 4 ?__12

__12

1st 2nd

3rd

3

__12

1 + __12

1 + __12

1 = __12

4

3 times

5. Calculate:

c) 10 ÷ 2 __12 =

1st 2nd

3rd 4th

4

__12

2 + __12

2 + __12

2 + __12

2 = 10

4 times

Let’s think:

How many times does 2 go into 10 ?__12

Is it enough?

T H E E N D

With thanks to:

Rex BoggsRex Boggsfor support and help with the translation into fluent U.S. idiom (a.k.a. ‘American’).

Author of presentation:

Antonija HorvatekAntonija HorvatekCroatia , 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 Horvatekhttp://www.antonija-horvatek.from.hr/ ahorvatek@yahoo.com