Self-Directed Course: Transitional Math Module 2: … 2 - Fractions.pdf · Self-Directed Course:...

Self-Directed Course: Transitional Math

Module 2: Fractions

Property of: Portage la Prairie School Division

Lesson #1: Comparing Fractions

Comparing fractions means finding out which fraction is larger or smaller than the other. To compare

fractions, use the following inequality and equal signs:

- greater than is > *If the fractions are not equal, the

- less than is < inequality sign always points to the

- equal to is = smallest fractions.

A fraction can be made up of three parts:

- a denominator (bottom number)

- a numerator (top number)

- a whole number (number in front of the denominator and numerator)

numerator

whole number 5�

�

denominator

Before you can compare fractions, the denominators have to be the same. If the denominators are

different, you will need to come up with a “common denominator”. Let’s compare the following

fraction:

�

� ___

�

�

Since the denominators are different, we have to find a common denominator to make them the same.

Both 4 and 5 go into 20, so our new denominator will be 20.

��

___ ��

Now we need to figure out the numerator. To do that, we multiply the numerator by whatever we

multiplied the denominator by to get 20.

�

�x �

� = �

�� ___

�

� x �

� =

��

Our new fractions are �

�� ___

��. Now we can compare them because the denominators are the same.

Since 15 is smaller than 16, we use the less than sign: �

�� <

��. Therefore,

�

� <

�

�


Module 2: Fractions


Compare the following fraction:

�

___

�

Step #1: Find a common denominator

�

___ �

Step #2: Find the new numerators

�

x �

� =�

� ___

�x �

� =

�

�

Step #3: Compare new fractions

�

� >

�

�

Step #4: Write final answer

�

>

�

*Remember, the inequality sign always points to the smallest fractions.


Module 2: Fractions


Assignment #1: Comparing Fractions

Compare the following fractions by using an inequality or equal sign.

1)�

�

�

�

2)

�

�

�

3)�

�

�

4)

�

�

5)�

�

�

6)�

�

�

7)�

�

8)

�

�

9)�

�

10)

�

�


Module 2: Fractions


Lesson #2: Converting Mixed Numbers to Improper Fractions

There are three types of fractions:

1) Proper Fractions – the numerator is smaller than the denominator: �

�

2) Improper Fractions – the numerator is larger than the denominator: �

�

3) Mixed Numbers – a proper fraction next to a whole number: 3�

Example #1

To change 3�

(a mixed number) into an improper fraction, use the following steps:

1) Multiply the denominator by the whole number 6 x 3 = 18

2) Add the numerator to your answer 18 + 4 = 22

3) Place this value over the original denominator ��

Example #2

Change 2�

� into an improper fraction:

1) Multiply the denominator by the whole number 7 x 2 = 14

2) Add the numerator to your answer 14 + 3 = 17

3) Place this value over the original denominator �

�

Example #3

To change �

� (an improper fraction) into a mixed number, use the following steps:

1) Divide the numerator by the denominator 7 ÷ 3 = 2.333

2) The number in front of the decimal is your whole number 2

3) Multiply you whole number by your denominator 2 x 3 = 6

4) Subtract this number from your numerator 7 – 6 = 1

5) Place this value over the denominator and place it after the 2

�

Number you found in step2.


Module 2: Fractions


Assignment #2: Converting Mixed Numbers to Improper Fractions

Write each mixed number as an improper fraction.

1) 8

�

2) 7�

�

3) 1�

�

4) 10�

�

5) 7�

�

6) 4

�

7) 9�

�

8) 1�

�

9) 5

�

10) 4�

�

11) 7�

�

12) 3

�

13) 4

�

14) 4

�

15) 2

�

16) 6�

�

17) 1�

�

18) 3�

19) 5�

�

20) 9

21) 5�

�

22) 7�

23) 7

�

24) 9�

25) 8�

�

26) 4

�

27) 8�

�

28) 10


Module 2: Fractions


Lesson #3: Multiply and Reduce Fractions

When multiplying fractions you need to multiply the numerators together and the denominators

together. If your answer is an improper fraction, be sure to convert it to a mixed numbers. Make sure

you reduce your answers to lowest terms by dividing the numerator and denominator by the same

number. If the numerator and denominator can’t be divided by the same number, then you have lowest

terms. For example:

�

� x

�

� =

�� ÷ �

� =

�

�

�

� x

�

� =

�

�� ÷ �

� =

�÷ �

� =

�

�

�

� x

�

� =

�

� ÷ �

� =�

� = 1

�


Module 2: Fractions


Assignment #3: Multiply and Reduce Fractions

Multiply the following fractions and reduce them to lowest terms when possible.

1) �

� x

�

�

2) �

x

�

�

3) �

x

�

4) �

� x

�

5)

� x

6) �

� x

�

�

7)

� x

�

�

8) �

� x

�

�

9) �

x

�

10) �

� x

�

11) �

x

�

12)

� x

�

13) 2 x

14)

� x 3


Module 2: Fractions


15) �

� x 8

16) �

� x 7

17) �

� x 6

18) �

� x 4

19) 7 x �

�

20)

� x 9

21) 8 x �

22) 2 x �

�

23) 5 x �

24) �

� x 3

25) 9

� x

�

26) 7�

� x

�

�

27) 1�

� x

�

28) 2�

� x

�

29) 6�

� x

�

30) 7�

� x

�


Module 2: Fractions


31) 2

� x

�

�

32) 5

� x

�

33) 3�

� x

�

�

34) 2�

� x

�

35) 1

� x 4

�

36) 3

� x 1

�

�

37) 6�

� x 4

�

�

38) 8�

x 5

�

�

39) 2�

� x 3

�

40) 7

� x 5

�

41) 7�

� x 4

�

42) 1�

� x 3

�

�

43) 9�

� x 1

�

44) 9

� x 5

�

�


Module 2: Fractions


Lesson #4: Dividing Fractions

Dividing fractions is very similar to multiplying fractions. Instead of dividing, you multiply the fractions

together by flipping the second fraction upside-down. This is called multiplying by the “reciprocal”.

After you have flipped the second fraction, follow the same steps for multiplying fraction together. Be

sure to reduce your answers to lowest terms and to convert all improper fractions to mixed numbers.

For example:

�

� ÷ �

� =

�

� x �

� =

��

�

� ÷ �

� =

� x �

� =

��

� ÷ �

� = �

� = 1

�

�

÷ �

� =

�

x �

� = �

�� ÷ �

� =

�

�

Sometimes you might see a question like this: �

� ÷ 6. To do this simply change the 6 (a whole number) to

a fraction by placing a 1 under it and carry out the normal steps to solve it.

�

� ÷ 6

�

� ÷

= �

� x

=

�

�� ÷ �

� =

�


Module 2: Fractions


Assignment #4: Dividing Fractions

Divide the following fractions.

1)

� ÷

�

�

2) �

� ÷

�

3)

� ÷

�

4) �

� ÷

�

�

5) �

� ÷

�

6) �

� ÷

�

7) �

� ÷

�

8)

� ÷

�

9)

� ÷

�

10)

� ÷

�

�

11) 2

� ÷

�

12) 3�

� ÷

�

�

13) 1�

� ÷

�

�

14) 7�

÷

�


Module 2: Fractions


15) 8

� ÷

�

�

16) 7

� ÷

�

�

17) 5

� ÷

�

�

18) 2�

� ÷

�

�

19) 9�

� ÷

�

20) 1

� ÷

�

21) �

÷ 6

22) �

� ÷ 7

23)

� ÷ 5

24) �

� ÷ 6

25)

� ÷ 4

26)

� ÷ 6

27) �

÷ 3

28)

� ÷ 6

29)

� ÷ 4

30) �

� ÷ 7


Module 2: Fractions


31) 8

� ÷ 2

�

32) 3�

÷ 9

�

33) 2�

� ÷ 7

�

�

34) 2

� ÷ 1

�

35) 1�

� ÷ 5

�

�

36) 5�

� ÷ 8

�

37) 5�

� ÷ 2

�

�

38) 1�

� ÷ 4

�

�


Module 2: Fractions


Lesson #4: Adding Fractions

When adding fractions, make sure that the denominators are the same. If they are different, you have

to find a common denominator. Be sure to reduce your answers to lowest terms and to convert all

improper fractions to mixed numbers. For example:

Example #1 �

� +

�

Step #1: Find a common denominator �

+ �

Step #2: Find the new numerators �

� x �

� =

� +

�

x �

� = �

�

Step #3: Re-write new addition statement

� + �

�

Step #4: Add fractions together

� + �

� =

� ÷ �

� = �

� = 1

�

Example #2 �

� +

�

�

Step #1: Find a common denominator ��

+ ��


� x �

� =��

�� +

�

�x �

� = ��

��

Step #3: Re-write new addition statement ��

�� + ��

��

Step #4: Add fractions together ��

�� + ��

�� = ��

�� = 1

�

��


Module 2: Fractions


Assignment #4: Adding Fractions

Add the following fractions.

1)

� +

�

2) �

� +

�

3)

� +

�

4) �

+

�

�

5)

� +

�

�

6) �

� +

�

�

7) �

+

�

�

8) �

� +

�

�

9) �

� +

�

10) ��

+

��


Module 2: Fractions


11) ��

� +

��

�

12) �

� +

��

�

13) �

+

��

�

14) ��

� +

��

�

15) �

� +

�

�

16) ��

�� +

��

17)

+

��

18) ��

� +

��

�


Module 2: Fractions


Lesson #5: Adding Mixed Numbers

When adding mixed numbers, you follow the same steps as you would when adding proper fractions.

The only difference is there is an additional step because you have to add the whole numbers. Make

sure the denominators are the same when adding and reduce your answers to lowest terms.

Example #1 3

� + 5

�


+ �

Step #2: Find the new numerators 3

� x �

� = 3

�

� + 5

�

x �

� = 5

�

�

Step #3: Re-write new addition statement 3�

� + 5

�

�

Step #4: Add the whole numbers 3 + 5 = 8

Step #4: Add numerators �

� +

�

� =

�

Step #5: Place whole number in front of fraction 8

�

Example #2 6�

� + 2

�

�


+ ��

Step #2: Find the new numerators 6�

� x �

� = 6

�� + 2

�

�x �

� = 2

�

��

Step #3: Re-write new addition statement 6

�� + 2

�

��

Step #4: Add the whole numbers 6 + 2 = 8

Step #4: Add numerators

�� + �

�� = �

�� = 1

��

Step #5: Add whole numbers 8 + 1 = 9

�


Module 2: Fractions


Assignment #5: Adding Mixed Numbers

Add the following mixed numbers.

1) 4

� + 3

�

2) 1�

� + 1

3) 2 + 4�

�

4) 4�

� + 1

�

5) 3�

� + 4

�

�

6)

� + 4

�

�

7) 2�

+ 4

�

8) 3�

� 2

�

9) 2

� + 4

�

�

10) 1 + 3�

�


Module 2: Fractions


11) 1

� + 6

�

�

12) 3

� + 7

�

13) 8�

� + 10

�

��

14) 3�

� + 10

�

��

15) 3�

� + 10

�

�

16) 1�

�� + 7

�

�

17) 10 + 12�

�

18) 5�

+ 11

�


Module 2: Fractions


Lesson #6: Subtracting Improper Fractions

When subtracting fractions, you follow some of the same steps as you would when adding fractions.

You have to make sure the denominators are the same and reduce your answers to lowest terms.

Example #1 �

� –

�

�


+ ��


� x �

� = �

�� –

�

�x �

� = ��

��

Step #3: Re-write new subtraction statement �

�� –

��

��

Step #4: Subtract numerators �

�� –

��

�� =

��

Example #2 6

� – 2

�

�


– �

Step #2: Find the new numerators 6

� x �

� = 6

�

� – 2

�

�x �

� = 2

�

�

Step #3: Re-write new subtraction statement 6�

� – 2

�

�

Step #4: Subtract the whole numbers 6 – 2 = 4

Step #4: subtract numerators �

� –

�

� =

�

�

Step #5: Place whole numbers in front 4�

�


Module 2: Fractions


Assignment #6: Subtracting Improper Fractions

Subtract the following Improper Fractions.

1) �

� –

�

�

2) 1 –

�

3) �

� –

�

4) �

� –

�

5) �

� –

�

�

6) �

� –

�

7) 1 – �

�

8) �

� –

9) �

� –

�

�

10) �

� –

�

�


Module 2: Fractions


11) 2 – �

�

12) 19 –

�

13) �

� –

�

�

14) 2 –

�

15) 1 –

�

16) �

� –

�

17) ��

�� –

�

�

18) 20 – �

19)

–

��

��

20) �

� –

�

�

21) 4�

–

�

22) 1�

� – 1

�


Module 2: Fractions


23) 6 – 1

�

24) 3

� – 1

�

�

25) 4�

� –

�

�

26) 3�

� – 2

�

27) 4

� – 2

�

�

28) 4

� – 4

�

�

29) 2�

� – 2

�

30) 7�

� – 1

�

�

31) 7�

� – 2

�

�

32) 6

� – 2

�

�

33) 9

� – 6

�

34) 9�

�� –

�

�


Module 2: Fractions


35) 20 – 5�

�

36) 4�

�� –

�

37) 9�

– 8

�

38) 12�

� – 6

�

�


Module 2: Fractions


Assignment #7: Adding and Subtracting Fractions and Mixed Numbers

Add or Subtract the following Fractions and Mixed Numbers.

1) 2 + �

�

2) �

� +

�

3) �

� –

�

4)

� +

�

�

5) �

� +

�

6) �

� +

�

�

7) �

� +

�

�

8) �

� +

�

�

9) 6 – �

�

10) �

+

�

�


Module 2: Fractions


11) 3�

� – 2

�

�

12) �

–

�

13) 2�

� –

�

14) 6 – 2�

�

15) 2�

� + 1

�

�

16) 4

� + 4

�

�

17) 4�

� – 3

�

�

18) 4�

+ 2

�

19) 5�

� + 6

20) 3�

� + 3

�

21) 8�

�� +

�

� –

�

�

22) 2�

� + 3

�

� + 8

�


Module 2: Fractions


23) 9�

� –

� + 2

24) 6

� + 9

� –

�

��

25) 8�

+ 4

�

� +

�

�

26) �

� –

�

� +

�

�

27) 6�

� –

�

� – 2

�

�

28)

+ 1 + 2

�

��

29)

� +

��

� + 3

�

�

30) 10�

� + 2

�

� – 5

�

Self-Directed Course: Transitional Math Module 2: … 2 - Fractions.pdf · Self-Directed Course:...

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