Math 6A (Periods 2 & 4)

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

Math 6A (Periods 2 & 4)

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 ∙  1/b = b/b = 1
we also noticed that

(a/b) ∙ 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

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

What if the given was: 12 trucks and 15 cars

(a) fraction represented by cars is 15/27
(b) fraction represented by trucks is 12/27

What about
GIVEN:
9 cars and    35 vehicles
This time we need to find out how many trucks there are
35 - 9 = 26
so
(a) 9/35
(b) 26/35

We aren't simplifying YET

Math 6High (Period 3)

Subtracting Integers 4.4

So to generalize the rules we discussed in class, when using a number line to show subtraction (Looking at Page 181 at the number line examples):

To subtract a positive integer , move in the negative direction

To subtract a negative integer, move in the positive direction.

ADD The Opposite!
We had a huge discussion about way too pessimistic people and how we wish we could take away a little of their negativity… and  about too positive people—even someone in our class [ :) ] whom some of us  wished was just a little less positive!! We talked about how instead of taking away a little – if we gave a little of the opposite—that might work as well… and we realized that this applies to math as well.

ADD THE OPPOSITE

That is, instead of subtracting a number—add it’s opposite. 

Then use the rules for adding integers that you have learned from the previous lessons

In general:    a - b = a + (-b)

Several examples:

6 – 8 = 6 + (-8)  In this case by adding the opposite, you are adding two integers with different signs—so you following the adding integers with DIFFERENT SIGNS RULE. Ask “Who wins?” and “by how much?”  The negative wins by 2 so the answer is -2              

6-8 = -2

-9 - 10 =  -9 + (-10) In this case you are adding two integers with the same sign. So use the SAME SIGN RULE.  -9 + (-10) = -19

What about

54- (-12) = 54 + (+12) = 54 + 12  In this case you are adding two integers with the same sign. So use the SAME SIGN RULE.  54 –(-12) = 66

We talked about  making sure to double check—that is, check, check—both  places- where you change the subtraction sign to a positive AND where you change the second numbers sign to its opposite.