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Adding and Subtracting Fractions

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Page 1: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Adding and Subtracting Fractions

Page 2: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that you may be better able to discover with accuracy the peculiar bent of the genius of each.

Plato

Page 3: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Although children learn addition of whole numbers with ease, addition of fractions — though conceptually the same as addition of whole numbers — is much harder.

It requires knowledge of fraction equivalencies.

To add two fractions, you have to know that they must be thought of in terms of like units.

We take this for granted when we add whole numbers: 3 + 5 is really 3 ones + 5 ones

— but not when we add fractions: 3 halves + 5 fourths is, for purposes of addition, 6 fourths + 5 fourths.

Why is adding fractions a difficult concept for students to grasp?

Page 4: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Equivalent fractions

In the following picture we have ½ of a cake because the whole cake is divided into two congruent parts and we have only one of those parts.

But if we cut the cake into smaller congruent pieces, we can see that

2

1 =4

2

Or we can cut the original cake into 6 congruent pieces,

a fraction can have many different appearances, these are called equivalent fractions

Page 5: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Now we have 3 pieces out of 6 equal pieces, but the total amount we have is still the same.

Therefore,

2

1 =4

2 =6

3

If you don’t like this, we can cut the original cake into 8 congruent pieces,

Equivalent fractions

a fraction can have many different appearances, these are called equivalent fractions

Page 6: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

then we have 4 pieces out of 8 equal pieces, but the total amount we have is still the same.

2

1 =4

2 =6

3 =8

4

We can generalize this to

2

1 =n

n

2

1 whenever n is not 0

Therefore,

Equivalent fractions

a fraction can have many different appearances, these are called equivalent fractions

Page 7: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Let’s Eat Pizza

The pizza is currently 8 pieces

What if I wanted to eat one eighth of the pizza?

One fourth of the pizza?

One sixteenth of the pizza?

One twelfth of the pizza?

Page 8: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Addition of Fractions

The objects must be of the same type We combine bundles with bundles and sticks

with sticks.

Addition means combining objects in two or more sets

In fractions, we can only combine pieces of the same size

In other words, the denominators must be the same

Page 9: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Click to see animation

+ = ?

Example:

Addition of Fractions

8

3

8

1

Page 10: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Example:

Addition of Fractions

+ =

8

3

8

1

Page 11: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

+ =

Example:

The answer is which can be simplified to

Addition of Fractions

8

3

8

1

8

4

8

)31(

2

1

Page 12: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Addition of Fractions with equal denominators

More examples

5

1

5

2

5

3

10

7

10

6

10

13

15

8

15

6

15

14

10

31

Page 13: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

With different denominators

In this case, we need to first convert them into equivalent fraction with the same denominator.

Example:

15

5

53

51

3

1 15

6

35

32

5

2

5

2

3

1

An easy choice for a common denominator is 3×5 = 15

Therefore,

15

11

15

6

15

5

5

2

3

1

Addition of Fractions

Page 14: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

• When the denominators are bigger, we need to find the least common denominator by factoring.

• If you do not know prime factorization yet, you have to multiply the two denominators together.

With different denominators

Addition of Fractions

Page 15: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

More Exercises:

8

1

24

23

8

1

4

3

7

2

5

3

9

4

6

5

=

=

=

57

52

75

73

69

64

96

95

=

=

=

8

1

8

6=

8

7

8

16

35

10

35

21

35

31

35

1021

=

54

24

54

45

54

151

54

69

54

2445

=

Page 16: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions Subtraction means taking objects away

Objects must be of the same type we can only take away apples from a

group of apples

In fractions, we can only take away pieces of the same size. In other words, the denominators must be the same.

Page 17: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

This means to take away

(Click to see animation)

12

11

take away

equal denominators

Page 18: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

12

11

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 19: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 20: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 21: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 22: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 23: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 24: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 25: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 26: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 27: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 28: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 29: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 30: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 31: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 32: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 33: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 34: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 35: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 36: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 37: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 38: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Now you can see that there are only 8 pieces left,therefore

12

311

12

3

12

11

3

2

12

8

Subtraction of Fractions

Example:

12

3

12

11

equal denominators

Page 39: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

More examples:

16

7

16

15

16

715

2

1

16

8

9

4

7

6

79

74

97

96

63

28

63

54

63

2854 26

63

23

11

10

7

1023

1011

2310

237

2310

1011237

230

110161

230

51

Did you get all the answers right?

Subtraction of Fractions

Page 40: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Adding/Subtracting

38 8

138 4

1= =-- 18

882=

c

ba

c

b

c

a Fraction Addition/Subtraction

Fraction Simplification

Page 41: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

2811 2

7+ ++

=

==

Common Denominator = ??????

28

2811

7

244

2811

288

28+11 8 =

2819

c

ba

c

b

c

a Fraction Addition/Subtraction

???728

211

Adding/Subtracting

Page 42: Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that

Fractions: Steps for Success

1. Know the fraction rules and how to apply them

2. Show your work and write out each step3. Check your work for errors or careless

mistakes