Math 6 Honors (Period 6 and 7)

Comparing Fractions 6-4
When two fractions have equal denominators it is easy to tell which of the fractions is greater.
We simply compare their numerators.
3/11 < 5/11 since 3< 5 If the fractions have different denominators, there are a variety of methods to consider. We could find a common denominator, which we will need to do when we add or subtract fractions... but when comparing let's try other methods... Take 2/3 and 4/5 Comparing Fractions


Or compare 5/6 and 7/9

again this time you would multiply
5(9) = 45 and 7(6) = 42
so 5/6 > 7/9

Also if the numerator is the same
2/3, 2/7, 2/9, 2/11, 2/21, 2/35

The larger the denominator the smaller the fractions so to list in order from least to greatest start with the largest number in the denominator!!

and if you have fractions with the numerator just one away from the denominator
such as 3/4, 5/6, 7/8, 9/10, 23/24, 45/46
the smallest fraction will be the one with the smallest numbers
3/4 is the smallest fraction and that list is in order from least to greatest!!

What if you need to name a fraction between two fraction 1/6 and 3/8
you could find the LCD
1/6 = 4/24 and
3/8 = 9/24
so you could state
5/24, 6/24 ( but that is really 1/4), 7/24, or 8/24 ( but that is really 1/3.
There are actually an infinite number of fractions... these are only 4 of them

What if you need to find a fraction between 3/7 and 4/7
sometimes you need to change the denominators just to realize that there really are other fractions between
for instance, 3/7 = 6/14 and 4/7 = 8/ 14 so doesn't 7/14 ( or actually 1/2) work!!
... and that's just one of the fractions!!

If n > 0
Then
if a < b a/n < b/n Think about this one!! Plug in some numbers and see what happens and if a < b, then n/a > n/b
Again, plug in some numbers and see what happens!!

If a/b and c/d are fractions and if ad > bc, which fraction is greater,
a/b or c/d ?
Post your answer below in the comments for extra credit. Make sure to give your reasoning for your answer.


Ordering or comparing fractions:

Different ways:

I. Benchmarks - 0, 1/4, 1/2, 3/4, and 1 (using your gut feeling)
How do you figure out which benchmark to use?

When the numerator is close to the denominator, the fraction is approaching 1

(Ex: 9/11)

When you double the numerator and it's close to the denominator, the fraction is close to 1/2
(Ex: 4/9)

When the numerator is very far from the denominator, the fraction is approaching zero
(Ex: 1/8)

Also, if one number is improper or mixed number and other is a proper fraction,
then obviously the number greater than 1 will be bigger!

II. LCD - give them all the same denominator using the LCM as the LCD


III. Use cross multiplication when comparing two... do it several times when comparing a list of fractions

IV. Change them to decimals ( works well if you are great at decimals-- but I want you to become GREAT at fractions!!)

Math 6 Honors (Period 6 and 7)

Fractions & Mixed Numbers 6-3

1/2 + 1/2 + 1/2 = 3/2

A fraction whose numerator is greater than or equal to its denominator is called an improper fraction.
Every improper fractions is greater than 1
A proper fraction is a fraction whose numerator is less than its denominator.
Thus, a proper fraction is always between 0 and 1

1/4, 2/3, 5/9. 10/12 17/18 are all proper fractions


5/2, 8/3, 18/15, 12/5 are all improper fractions


You can express any improper fraction as the sum of a whole number and a fraction
a number such as 1 1/2 is called a mixed number

If the fractional part of a mixed number is a proper fraction in lowest terms, the mixed number is said to be in simple form.


To change an improper fraction into a mixed number in simple form, divide the numerator by the denominator and express the remainder as a fraction.
14/3 = 4 2/3
30/4 = 7 2/4 = 7 1/2


To change a mixed number to an improper fraction rewrite the whole number part as a fraction with the same denominator as the fraction part and add together.

or multiply the denominator by the whole number part and add the fractional part to that...
In class I showed the circle shortcut. If you were absent, check with a friend or ask me in class!!

2 5/6 =

(2 x 6) + 5
6
=17/6


Practice these:

785 ÷ 3

852÷ 5

3751÷ 16

98001÷231


post your answers below in the comments for extra credit !!

Math 6 Honors (Period 6 and 7)

Fractions 6-1

The symbol 1/4 can mean several things:
1) It means one divided by four
2) It represents one out of four equal parts
3) It is a number that has a position on a number line.



1/8 means 1 divided by 8 or 1 ÷ 8
A fraction consists of two numbers
The denominator tells the number of equal parts into which the whole has been divided.
The numerator tells how many of these parts are being considered.
we noted that we could abbreviate ...

denominator as denom with a line above it

and numerator as numer

we found that you could add

1/3 + 1/3 + 1/3 = 3/3 = 1
or 1/4 + 1/4 + 1/4 + 1/4 = 4/4 = 1
we also noted that 8 X 1/8 = 8/8 = 1

We also noticed that 2/7 X 3 = 6/7


So we discussed the properties
For any whole numbers a, b,and c with b not equal to zero

1/b + 1/b + 1/b ... + 1/b = b/b = 1 for b numbers added together

and we noticed that b X 1/b = b/b = 1
we also noticed that
(a/b) X c = ac/b

We talked about the parking lot problem on Page 180

A count of cars and trucks was taken at a parking lot on several different days. For each count, give the fraction of the total vehicles represented by
(a) cars

(b) trucks

Given: 8 cars and 7 trucks
We noticed that you needed to find the total vehicles or 8 + 7 = 15 vehicles
so

(a) fraction represented by cars is 8/15
(b) fraction represented by trucks is 7/15


What if the given was: 15 trucks and 32 vehicles
This time we need to find how many were cars. so 32 -15 = 17 so 17 cars

(a) fraction represented by cars is 17/32
(b) fraction represented by trucks is 15/32

Equivalent Fractions 6-2


We drew the four number lines from Page 182 and noticed that 1/2, 2/4, 3/6, and 4/8 all were at the midpoints of the segment from 0 to 1. They all denoted the same number and are called equivalent fractions.

If you multiply the numerator and the denominator by the same number the results will be a fraction that is equivalent to the original fraction

1/2 = 1 x 3/2 x 3 = 3/6

It works for division as well
4/8 = 4 ÷ 4 / 4 ÷ 8 = 1/2

So we can generalize and see the following properties
For any whole numbers a, b, c, with b not equal to zero and c not equal to zero

a/b = a x c/ b x c and
a/b = a ÷ c / b ÷c


Find a fraction equivalent to 2/3 with a denominator of 12
we want a number such that 2/3 = n/12
You could look at this and say
" What do I do to 3 to get it to be 12?
Multiply by 4
so you multiply 2 by 4 and get 8 so
8/12 is an equivalent fraction


A fraction is in lowest terms if its numerator and denominator are relatively prime-- That is if their GCF is 1

3/4, 2/7, and 3/5 are in lowest terms.
They are simplified
You can write a fraction in lowest terms by dividing the numerator and denominator by their GCF.


Write 12/18 is lowest terms
The GCF (12 and 18) = 6

so 12/18 = 12÷ 6 / 18 ÷ 6 = 2/3

Find two fractions with the same denominator that are equivalent to 7/8 and 5/12
This time you need to find the least common multiple of the denominators!! or the LCD
Using the box method from Chapter 5, we find that the LCM (8, 12 ) = 24

7/8 = 7 X 3 / 8 X 3 = 21/24
and
5/12 = 5 X 2 / 12 X 2 = 10/24


When finding equations such as
3/5 = n/15 we noticed we could multiply the numerator of the first fraction by the denominator of the second fraction and set that equal to the denominator of the first fraction times the numerator of the second... or

3(15) = 5n now we have a one step equation

If we divide both sides by 5 we can isolate the variable n and solve...
3(15)/ 5 = n
9 = n

We found we could generalize

If a/b = c/d then ad = bc