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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 as½ means 1 out of two equal pieces

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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.

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2.1 Fraction TermsThe 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

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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.

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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”

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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

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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 numberfor 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

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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

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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.

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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

1) Factor tree2) Division or box method

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2.2 Simplifying Fractions --Factors

We’ll do some example of each on the board

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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.

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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?

287

164

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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.

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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.

27

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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 half5 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.

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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 FractionsOR

Improper Fractions to Mixed Numbers

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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

213 632

716 27

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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 ?

372

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2.4 Multiplying fractionsTo multiply fractions:Multiply the numerators, multiply the denominators

and reduceIn other words, take top times top; bottom times

bottom and reduce

bdac

dc

ba

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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)

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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.

223 1

223

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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.

522

23

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2.5 Dividing fractionsBy definition, division is multiplying by the

reciprocalWhat 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

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2.5 Dividing fractionsTo 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

bcad

cd

ba

dc

ba

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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.

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 41, 2, 3, 4, 6, 12

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.

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 are10, 20, 30, 40, 50, . . .

2.6 LCM by the list method

Finding the LCM of 10 and 12 by listYou 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?

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.

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)

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!

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.

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.

2.6 LCM by prime factorization

Find the prime factorizations for 10 and 1210 = 2 x 5 12 = 2 x 2 x 3

2 x 5 22 x 3Whatever 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

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 10/20½ is the same as 50/100½ is the same as 23/46Is ½ the same as 31/63?

2.6 Write Equivalent Fractions

Rewrite the given fraction with the new denominator:

6?

32

64

32

Ask yourself, what would I multiply the 3 times to get a 6? Multiply the top by that same number

2.7 Adding and SubtractingLike 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

2.7 Adding Like Fractions

When adding like fractions:-add the numerators-keep the same denominator-reduce if needed

⅔ + ⅓ = 3/3 = 1⅛ + ⅜ = 4/8 = ½

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

2.7 Adding and SubtractingLike 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.

2.7 Write Equivalent Fractions

15?

32

1510

32

Ask yourself, what would I multiply the 3 times to get a 15? Multiply the top by that same number

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-reduceSee appendix b for more info on

LCD

dc

ba

dc

ba

2.7 Adding -horizontal

12?

12?

43

32

129

128

1298

1217

1251

2.7 Adding - vertical

43

32

12?

12?

129

128

1217

1251

3.3 Subtracting - horizontal

12?

12?

41

32

123

128

1238

125

3.3 Subtracting - vertical

41

32

12?

12?

123

128

125

2.8 ADD Mixed Numbers Regular Horizontal Method

21?1

21?2

311

722

2171

2162

217612

21133

2.8 ADD Mixed Numbers Regular Vertical Method

311

722

21?1

21?2

2171

2162

2176

12

21133

2.8 SUBTRACT Mixed Numbers Regular Horizontal Method

21?1

21?2

311

732

2171

2192

217912

2121

2.8 SUBTRACT Mixed Numbers Regular Vertical Method

311

732

21?1

21?2

2171

2192

2179

12

2121

2.8 SUBTRACT Mixed Numbers Borrowing Method

21?1

21?2

311

722

2171

2162

2120

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

2171

21271

2.8 SUBTRACT Mixed Numbers Alternative to Borrowing Method

311

722

34

716

21?

21?

2128

2148

212848

2120

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2.8 Order of Operations with Fractions

Focus on your PEMDASTake your time and breathe

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

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.

34

21

35?

411

Comparing Fractions

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

1220

1233

35?

411

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2.9 Applications of fractions

Read the problem through once quicklyRead a second time, paying a bit more

attention to detailMake some notesTry to come up with a planDo the mathLabel your answer