Math 6 H Periods 1, 6 & 7 (Monday)

Changing a Fraction to a Decimal 6-5

There are two methods that can be used to change a fraction into a decimal.
The first one, we try to find an equivalent fraction whose denominator is a power of 10.

13/25 is a great example because we can easily change the denominator into 100 : multiplying 24 by 4.

So
13/25 ( 4/4) = 52/100 = .52

In the second method of changing a fraction into a decimal, we divide the numerator by the denominator.

Change 3/8




When the remainder is 0, as above, the decimal is referred to as a terminating decimal. By examining the denominator of a fraction in lowest terms, we can determine whether the fraction can be expressed as a terminating decimal. If the denominator has no prime factors of then 2 or 5, the decimal representation will terminate.. (This is so since the fraction can be written as an equivalent fraction whose denominator is a power of ten)

7/40
40 = 23 ∙ 5; since the only prime factors of the denominator are 2 and 5, the fraction can be expressed as a terminating decimal

5/12
12 = 22 ∙ 3 since 3 is a prime factor of the denominator, the fraction cannot be expressed as a terminating decimal

9/12 = 3/4
4 = 22. Since 4 has no prime factors other than 2, this fraction can be expressed as a terminating decimal


Now, what happens if the denominator of a fraction has prime factors other than 2 or 5

Change 15/22 to a decimal I know that this cannot be expressed as a terminating decimal because the denominator (22) has the prime factorization of 2 ∙ 11.


divide carefully and you will get 0.6818181….

Notice the pattern of repeating remainders of 18 and 4. They produce a repeating block of digits 81, in the quotient.

we write 15/22 = 0.681818181…. or 0.681 with a bar over the 81 where the bar, also know as the vinculum, means that the block 81 repeats without ending.


a decimal such as 0.681 , in which a block of digits continues to repeat indefinitely is called a repeating decimal.

Property

Every fraction can be expressed as either a terminating decimal or a repeating decimal..


Changing a Decimal to a Fraction 6-6
As we have seen, every fraction is equal to either a terminating decimal or a repeating decimal. It is also true that every terminating or repeating decimal is equal to a fraction.
To change a terminating decimal to a fraction in lowest terms, we write the decimal as a fraction whose denominator is a power of 10. We then write this fraction in lowest terms.

Change 0.385 to a fraction in lowest terms

.385 = 385/1000 = 77/200

Change 3.64 to a mixed number in simple form

3.64 = 3 64/100 = 3 16/25

To change a repeating decimal into a fraction follow these examples
th__
0.54

tththththh__
Let n = 0.54 = 0.54545454….

[How many numbers are under the vinculum?] 2
Multiple both sides by 102

So then, 100n = 54.54545454…


100n = 54.54545454…
n = .54545454….
We can subtract n from 100n to get 99n

100n = 54.54545454…
- n = .54545454….
99n = 54

Divide both sides by 99

99n = 54
99 99

n = 54/99 = 6/11


Let’s try
th___
0.243
theitheith___
Let n = 0.243 = .243243243243….

How many numbers are under the vinculum? 3

So multiply both sides by 103
1000n = 243.243243243243….

1000n = 243.243243…
n = 243.243243

999n = 243

Divide both sides by 999

n = 243/999 = 27/111 = 9/37

Let’s try one that is a bit more complicated
thethehtett__
Change 0.318 the vinculym is over just the 18.
[Notice this isn’t 0.318 nor is it 0.318
the__
0.318 so that means it is 03.1818181818....
How many numbers are under the vinculum? 2
So, we multiply by 102

Let n = 0.318181818…

100n = 31.818181818…
n= .318181818…

99n = 31.500000…
Divide both sides by 99

99n/99 = 31.5/99

n = 31.5/99 but that isn’t a proper fraction. What can I do to change this?

Multiply by 10

315/990 = 63/198 = 7/22

HERE ARE SOME STEPS TO FOLLOW:
Step 1 set up “ n= the repeating decimal” n = .515151…
Step 2 determine how many numbers are under the bar in this case = 2
Step 3 Use that number as a power of 10 102 = 100
Step 4 Multiply both sides of the equation in step 1 by that
power of 10 100n = 51.515151…
Step 5 Rewrite the equations so that you subtract the 1st equation FROM the 2nd equation 100n = 51.5151…
- 00n= 51.5151…
Step 6 Solve as a 1-step equation 99n = 51 so n = 51/99
Step 7 Simplify 51/99 = 17/33
**** REMEMBER- sometimes you need to get the decimal out of the numerator—so multiply by a power of 10

Math 6 H Periods 1, 6 & 7

I found this very old clip of teaching fractions-- I thought you'd enjoy seeing what they taught students about fractions and decimals-- even before my time!!

Algebra Period 3

COMPARING THE 3 FORMS OF LINEAR EQUATIONS

Standard Form
Ax + By = C
3x + 4y = 10 is the STANDARD FORM of a line
x and y are on the same side of the equation and both coefficients are integers.
This format works especially well when the coefficients are both factors of the constant.
Use the x and y intercepts to graph.

Slope Intercept Form
y = mx + b
y = -3/4 x + 5/2 is SLOPE INTERCEPT FORM of the same line
y is isolated on one side, x term is first, then the constant on the other side of the equation
The coefficient of the x term is the slope.
The constant is the y intercept.
Graph the y intercept, then count the slope to another point.
For graphing, it doesn't work well if the y intercept is a fraction!



Point Slope Form
y - y1 = m(x - x1)
y - 3 = 3/4(x - 4)
Works really well if you have the slope and a point on the line.
You can use this method, and then simplify to the point slope form.

FITTING EQUATIONS TO DATA: 7-7
(word problems)
Many real world relationships are LINEAR, meaning they can be graphed with a LINE.
For example, if candy bar costs $1.50, then 2 bars cost $3.00 etc
Think of the number of candy bars as the x value because that's what you decide
(how many candy bars you're going to buy)
The result will be how much money you owe at the register (the y value)
x y
1 --> $1.50
2 --> $3.00
3 --> $4.50
If you graph this, you'll get a line because the price is constant.
That means that the slope is constant.
What is the slope????
The price = 1.50
Think of it as the change in y ($ you owe) over the change in x (# of candy bars)
The money you owe goes up $1.50 every time you buy 1 more candy bar
This is a POSITIVE slope of 1.50
So another meaning of a positive slope is two types of data that GO IN THE SAME DIRECTION
You can reverse both directions as well:
You don't have enough money for 3 candy bars, so you decrease your purchase by 1 bar
Then your purchase price also decreases by $1.50
The data is still going in the SAME DIRECTION (both now going down!)

If we now what to find the equation of this linear relationship, simply use the slope intercept or point slope formulas of a line!

SLOPE INTERCEPT:
y = 1.5x
(the y intercept is 0 because at 0 candy bars, you owe 0)
POINT SLOPE:
y - 1.50 = 1.50(x - 1)

Y INTERCEPTS IN THE REAL WORLD THAT ARE NOT ZERO:
The scenario above has a y intercept value of 0 because you don't owe anything if you don't buy anything, But often, the y intercept value will be a number. For example, think of cell phone use. Say you are charged $.10 per minute of use, but your monthly charge is $25.
Even if you don't use your cell phone, you still owe $25!
The linear equation would be:
y = .10x + 25 where y = what you owe and x = number of minutes used
Plug in 0 for x, number of minutes, and you still owe $25 (y value)

Let's think of a real life example that will give us a NEGATIVE SLOPE...
Often, the more you buy, the smaller the unit price per item.
This happens with copying or buying things like invitations.
Companies give you a "break" if you buy more.
SEE THE EXAMPLE ON PAGE 333 IN YOUR BOOK!
(we'll go over this one in class)

Finally, sometimes real world data can be APPROXIMATED as a linear relationship.
In other words, it may not be exact, but a good way to understand the data is to look at it that way.
Think SCATTER PLOTS with POSITIVE or NEGATIVE CORRELATIONS.

2 SPECIAL LINES AND THEIR SLOPES: 7-8

PARALLEL LINES:
2 lines that are parallel to each have the SAME SLOPE!
y = 2x - 10 and y = 2x + 3/4 are parallel because they both have a slope of 2


PERPENDICULAR LINES:
2 lines that are perpendicular to each other have SLOPES that are:
OPPOSITE SIGNS and RECIPROCALS
y = 2x - 10 is perpendicular to y = -1/2 x + 3/4

THIS IS ANOTHER TWIST TO OUR MYSTERY LINE PUZZLE!!!
If you know that the mystery line is parallel or perpendicular to another given line
then you know the mystery line's slope!!!

EXAMPLE:
Your mystery line has a point of (2, -5) and is PARALLEL to line y = 2x + 3/4
So you know the mystery line's slope because it is the same as the given line ( m = 2)
Substitute the slope and the point given on the mystery line and solve for b.

EXAMPLE:
Your mystery line has a point of (2, -5) and is PERPENDICULAR to line y = 2x + 3/4
So you know the mystery line's slope because it is the opposite sign reciprocal of the given line
(since the given line's slope is 2, the mystery line's slope is -1/2)
Substitute the slope and the point given on the mystery line and solve for b.

Algebra Period 3 (Thursday)

Finding the Equation of a Line: 7-6
Review:
1) We know how to GRAPH a line by 3 points where we decide what to plug in and chug
Usually, we just try 0, 1, 2 first
2) We know how to GRAPH a line by intercepts...we plug in zero for y and x and chug
This works really well when the line is in STANDARD form and the coefficients are factors
of the constant on the other side of the equation.
3) We know how to GRAPH a line by using slope-intercept form...
We isolate y on one side
We read the y intercept (the b -> the constant on the other side)
We graph that value on the y axis
We COUNT to the next point by reading the slope, the coefficient of the x
The slope should be read what happens to the y value (+up or -down) and then what happens to the x value (+ right or - left)
If the slope is not a fraction, make it a fraction by putting the integer over 1

Oh mystery line,
What can you be?
If I could only find you,
y = mx + b

So first I find m
Then I find b.
Now put it all together
And you've found me!
y = mx + b

The rhyme has 3 steps and usually you will have 3 steps or questions to ask yourself:
1) Do I have the slope (m)? If not, find it by using the slope formula
2) Do I have the y intercept (b)? If not, find it by plugging in a point and the slope
3) Don't forget to put it all together in one equation at the end.

THERE ARE 5 CASES THAT YOUR BOOK INCLUDES:

First case:You're given the slope and the y intercept
(easiest case)
m = 3/2 b = -7/5
Just plug in to the generic slope intercept equation: y = 3/2 x - 7/5

Second case: You're given a point and the slope and need to find the intercept (b)
(3, 1) m = 2
Plug in the point and the slope and solve for b
1 = 2(3) + b
1 = 6 + b
b = -5
Now put it altogether with the given slope and the intercept you just found:
y = 2x -5

Third case: You're given a point and and the y intercept and need to find the slope
(3, 1) b = 2
Plug in the point and the y intercept and solve for slope
1 = 3m + 2
-1 = 3m
m = -1/3
Now put it altogether with the given intercept and the slope you just found:
y = -1/3 x + 2

Fourth case: You're given 2 points and need to find the slope and the intercept
(1 , 3) and (-2 , -3)
You need to first find the slope:
m = change in y / change in x = 3 - (-3)/ 1 - (-2) = 6/3 = 2
Now plug the slope in with one of the points and find the intercept b
3 = 2(1) + b
3 = 2 + b
b = 1
Finally, put it all together:
y = 2x + 1

Fifth case: You have a graph of a line and need to determine the equation
Look at the graph and find 2 easy points to use to find the slope (make sure they are integers!)
(If the y intercept is not an integer, then follow fourth case completely)
Put the information together in y = mx + b form


ANOTHER WAY TO FIND THE EQUATION
WITH 1 POINT & SLOPE: ( and it is MY favorite)
POINT SLOPE FORM OF THE EQUATION
You know one point and the slope. This is the same case as the SECOND CASE, but there is a ANOTHER WAY to solve it other using slope intercept form.
Most people use the slope intercept form for all cases.


Point-slope form of a line: You know one point and the slope. Use the following formula:
y - y1 = m (x - x1)
Using the same example from the second case above: (3 , 1) and m = 2
y - 1 = 2 (x - 3)
What you have now is point slope form of the line

If you simplify this, you will get the slope intercept form of the line!
y - 1 = 2 (x - 3)
y - 1 = 2x - 6
y = 2x - 5

If you're trying to link the slope-intercept form to the point slope form of the same line:
The point-slope version eliminates one step from using the slope intercept form.
In the slope-intercept form, you plug in the point and slope, solve for b, and then rewrite the equation using the intercept that you found.
In point-slope form, once you plug in the point and slope, you just simplify and the equation is already done!