1. 2.1 Basics of Fractions
A fraction shows part of something. Most of us were taught to think of fractions as:
part of a whole such as
½ means 1 out of two equal pieces

2. 2.1 Basics of Fractions
Many times we shade pictures of pies and cut-up boxes to illustrate fractions. On your homework, you will be asked to identify diagrams and their coordinating fractions.

3. 2.1 Fraction Terms The top of a fraction is called the numerator.
The bottom is called the denominator.
think downstairs=denominator
½ , ¾ , ⅓ , ⅔ , ⅛ , ⅞
½ numerator is 1; denominator is 2
¾ numerator is 3; denominator is 4

4. 2.1 Basics of Fractions
Another way to think about fractions follows: I have found this method to be more helpful as I work with fractions.
The top number tells you how many, but the bottom number tells you what they are.

5. 2.1 Basics of Fractions
½ Read one-half; means there is one and it is a “half”
¾ Read three-fourths;
means there are 3 of them and they are “fourths”

6. 2.2 Simplifying Fractions --Factors
First, let’s review the divisibility rules we learned in chapter 1
A number is divisible by:
-2 if the ones digit is even
-3 if the sum of the digits is divisible by 3
-5 if it ends in 5 or 0
-9 if the sum of the digits is divisible by 9
-10 if it ends in 0

7. 2.2 Simplifying Fractions --Factors
What is a factor?
factors are numbers that multiply resulting in a product.
We can find all the factors of a number
for example, list the factors of 12
1,2,3,4,6,12
Think of them in pairs 1,12 and 2,6 and 3,4

8. 2.2 Simplifying Fractions --Factors
What are the factors of 60? (think pairs)
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Use your divisibility rules to help you come up with the factors

9. 2.2 Simplifying Fractions -Prime and Composite
Prime numbers have only two factors-one and the number itself.
examples: 3,7,11,13,19 etc
Except for 2, all prime numbers are odd, BUT not all odd numbers are prime!!!
Composite numbers are not prime.

10. 2.2 Simplifying Fractions -Prime Factorizations
The prime factorization is what you get when you break a number down until all the factors are prime.
There are two methods for finding prime factorizations
Factor tree
Division or box method

11. 2.2 Simplifying Fractions --Factors
We’ll do some example of each on the board

12. 2.2 Simplifying Fractions
A fraction is said to be reduced, or simplified, or in lowest terms when the numerator and denominator have no factors in common except for 1.
To put a fraction in lowest terms, we divide out any common factors that exist between the top and bottom. Once we have divided out all common factors, the fraction is reduced.

13. 2.2 Simplifying Fractions -Equality of fractions
A quick trick to tell if two fractions are equal is to set them equal and then take the cross product. If the cross products are equal, then the fractions are equal as well. Take 4(28) and take 16(7) Are the products equal?

14. 2.3 Fractions in all their forms-Proper and Improper
Proper fractions have a numerator that is smaller than the denominator. Proper fractions are less than 1.
½ , ¾ , ⅓ , ⅔ , ⅛ , ⅞
These are all proper fractions.

15. 2.3 Fractions in all their forms-Proper and Improper
Improper fractions have a numerator that is greater than the denominator. Improper fractions are greater than or equal to 1.

16. 2.3 Fractions in all their Forms -Mixed Numbers
Mixed Numbers have a whole number part and a fraction part.
3 ½ three wholes and one half
5 ⅔ five wholes and two thirds
If I bought three pizzas, but ate ½ of one on the way home, I have 2 ½ pizzas left to share.

17. 2.3 Fractions in all their Forms -Mixed Numbers
Mixed Numbers are closely related to improper fractions. We can go back and forth between the two different forms.
Mixed Numbers to Improper Fractions OR
Improper Fractions to Mixed Numbers

18. Mixed to improper (circle trick)
A nice trick for changing from a mixed number to an improper fraction is:
-multiply the denominator and the whole number
-add this to the numerator
-keep the same denominator

19. Improper to mixed
The method for changing from a improper fraction to a mixed number is to divide.
Remember the fraction bar is a divide sign.
Can you see the 3 ½ ?

20. 2.4 Multiplying fractions
To multiply fractions:
Multiply the numerators, multiply the denominators and reduce
In other words, take top times top; bottom times bottom and reduce

21. a nice trick to remember
Note: you can only cross cancel across a multiplication sign-never do this across an add, subtract, or division sign.
Below you can cross cancel the two’s and then multiply. (like reducing before you multiply)

22. Multiplying by a whole number
When multiplying a fraction by a whole number, remember there is a 1 in the denominator of the whole number.

23. Multiplying by a mixed number
We cannot multiply in mixed number form. Use the circle trick to change the mixed number into an improper fraction.

24. 2.5 Dividing fractions
By definition, division is multiplying by the reciprocal
What is a reciprocal?
Two numbers are reciprocals if their product equals 1.
To find the reciprocal of a number, interchange the numerator and denominator.
In other words, flip it!
½ becomes 2 2/3 becomes 3/2

25. 2.5 Dividing fractions To divide fractions:
remember division is multipying by the reciprocal. So . . .
-leave the first fraction as is
-change the division to a multiplication
-flip the second fraction
-multiply (take top times top; bottom times bottom)
-reduce

26. 2.5 Dividing Mixed Numbers
We cannot divide mixed numbers. We must change them to improper fraction form first. Then divide as normal. Leave first fraction alone. Flip second fraction. Then multiply: top times top; bottom times bottom. Reduce.

27. 2.6 Least Common Multiples
We tend to always learn about GCF and LCM at about the same time. They tend to get confused in our brain.
In GCF ignore the G(reatest) and focus on what a factor is. Factors are things that multiply together to give you a product. The factors of 12 are:
1 and 12, 2 and 6, 3 and 4
1, 2, 3, 4, 6, 12

28. 2.6 Least Common Multiples
Somewhat like the GCF discussion we had in Chapter 2, this section about LCM will see like it has nothing to do with fractions. GCF was important because it helps us reduce fractions. LCM is important now because an LCM is the same as LCD (least common denominator) which we will need to add and subtract fractions.

29. 2.6 Least Common Multiples
Ignore the L(east) in LCM and focus on what a multiple is.
The multiples of 12 are:
12, 24, 36, 48, 60, . . .
The multiples of 10 are
10, 20, 30, 40, 50, . . .

30. 2.6 LCM by the list method Finding the LCM of 10 and 12 by list
You can list some of the multiples for the numbers involved and try to find one in common.
10, 20, 30, 40, 50, . . .
12, 24, 36, 48, 60, . . .
Do you see a LCM yet? How about 60?

31. 2.6 LCM by the list method
The list method is okay, but it can be tedious to list all the multiples and sometimes you will make a list and not go far enough, as you saw in the last slide.
It does work though and it is one option.

32. 2.6 LCM by multiplying
Finding the LCM of 10 and 12 by multiplying. This will always give you a common multiple, but it will not always give you the LEAST common multiple. It may require extra reducing at the end of the problem.
10 x 12 = 120 (remember our LCM = 60)

33. 2.6 LCM using the larger denom
Finding the LCM of 10 and 12 by counting by 12’s until you come across a number that 10 will also go into. This can be difficult if you can’t count by 12’s. 
12, 24, 36, 48, 60 there it is!

34. 2.6 LCM by Magic
Finding the LCM of 10 and 12 by magic. This means you just look at the two numbers given and you just know what the LCM is. Some people have better magic than others  but a lot of times you will look and just know.

35. 2.6 LCM by prime factorization
Finding the LCM of 10 and 12 by Prime Factorization. This is the last resort. It is the most tedious method, but it will always work. Only do this method when you have tried the other options first.

36. 2.6 LCM by prime factorization
Find the prime factorizations for 10 and 12
10 = 2 x = 2 x 2 x 3
2 x x 3
Whatever factors appear in either of our prime factorizations, they must appear in our LCM. And to the highest power that they appear.
LCM = 22 x 3 x 5 = 12 x 5 = 60

37. 2.6 Write Equivalent Fractions
The fraction ½ can take many different forms
½ is the same as 4/8
½ is the same as 6/12
½ is the same as 10/20
½ is the same as 50/100
½ is the same as 23/46
Is ½ the same as 31/63?

38. 2.6 Write Equivalent Fractions
Rewrite the given fraction with the new denominator:
Ask yourself, what would I multiply the 3 times to get a 6?
Multiply the top by that same number

39. 2.7 Adding and Subtracting Like Fractions
Like Fractions are fractions that have the same denominator.
Example: ⅔ , ⅓ OR ⅛ ,⅜ ,⅝
Unlike Fractions are fractions that have different denominators.
Example: ⅔ ,⅜

40. 2.7 Adding Like Fractions When adding like fractions:
-add the numerators
-keep the same denominator
-reduce if needed
⅔ + ⅓ = 3/3 = 1
⅛ + ⅜ = 4/8 = ½

41. 2.7 Subtracting Like Fractions
When subtracting like fractions:
-subtract the numerators
-keep the same denominator
-reduce if needed
⅝ - ⅜ = 2/8 = ¼
⅔ - ⅓ = ⅓

42. 2.7 Adding and Subtracting Like Fractions
We can only add and subtract fractions that have the same denominators? Why?
Remember when we talked about the top number tells us “how many” and the bottom number tells us “what they are”?
We cannot add 2 apples and 3 tomatoes and say we have 5 grapes.

43. 2.7 Write Equivalent Fractions
Ask yourself, what would I multiply the 3 times to get a 15?
Multiply the top by that same number

44. 2.7 Adding and subtracting unlike fractions
-find a common denominator (LCD)
-rewrite each fraction with new denominator
-add or subtract numerators as indicated
-keep new denominator
-reduce
See appendix b for more info on LCD

45. 2.7 Adding -horizontal

46. 2.7 Adding - vertical

47. 3.3 Subtracting - horizontal

48. 3.3 Subtracting - vertical

49. 2.8 ADD Mixed Numbers Regular Horizontal Method

50. 2.8 ADD Mixed Numbers Regular Vertical Method

51. 2.8 SUBTRACT Mixed Numbers Regular Horizontal Method

52. 2.8 SUBTRACT Mixed Numbers Regular Vertical Method

53. 2.8 SUBTRACT Mixed Numbers Borrowing Method
You can’t take away 7 when you only have 6 there so you have to borrow from the 2. The 1 that you borrow comes into the fraction column as 21/21 resulting in 27/21

54. 2.8 SUBTRACT Mixed Numbers Alternative to Borrowing Method

55. 2.8 Order of Operations with Fractions
Focus on your PEMDAS Take your time and breathe 

56. Comparing Fractions
When comparing numbers in any form, remember that on a number line, as we go to the left things get smaller, and as we go to the right things get bigger.
< less than
> greater than

57. Comparing Fractions
When comparing fractions, if you cannot tell just by looking at them, the easiest way to compare them is to get a common denominator and compare numerators.

58. Comparing Fractions
Since we can’t tell by looking, let’s change them both to a denominator of 12.

59. 2.9 Applications of fractions
Read the problem through once quickly
Read a second time, paying a bit more attention to detail
Make some notes
Try to come up with a plan
Do the math
Label your answer