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
Monday, March 2, 2009
Sunday, March 1, 2009
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.
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!
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!
Friday, February 27, 2009
Algebra Period 3 (Tues/Wed)
Slope 7 -4
First, let's talk about what the word "slope" means in the real world:
You can think of the slope of a line as the slope of a ski mountain -
When you're climbing up, it's positive
When you're sliding down, it's negative
(if you're looking at the mountain from left to right)
The steeper the mountain, the higher the slope value
(A slope of 6 would be an expert slope because it
is much steeper than a slope of 2 which would be an intermediate's slope)
"Bunny slopes" for beginners will be lower numbers,
generally fractional slopes (like 1/2 or 2/3)
A good benchmark to know is a slope of 1 or -1 is a 45 degree angle
You can also think of slope as rise/run - read this "rise over run"
Rise is how tall the mountain is (the y value)
Run is how wide the mountain is (the x value)
VISUALIZE THE FOLLOWING 2 MOUNTAINS TO HELP YOU UNDERSTAND:
A 1000 foot high mountain (the rise) is very steep if it's only 200 feet wide (the run) (slope = 5)
Another mountain that is also 1000 feet high is not very steep if it is 2000 feet wide (slope = 1/2)
It has a much longer time to slowly reach the 1000 foot top of the mountain!
You can think of slope as a calculation using 2 coordinates:
Rise/Run
=Change in y value/Change in x value
=Difference in y value/ Difference in x value
= y2 - y1/ x2 - x1
To calculate slope you need 2 coordinates. It doesn't matter which one you start with.
Just be consistent! If you start with the y value of one point, make sure you start with the same x!
You can count the slope of a line:
1) Beginning with one point, count up to another point; however far that is, make that the numerator of your slope (because the y value of slope is the numerator)
2) Now count how far over the point is across - You'll need to either go right or left.
Make this the denominator of your slope (because the x value is the denominator of slope)
If you went to the RIGHT, the value is POSITIVE (x values going to the right or positive)
If you went to the LEFT, the value is NEGATIVE
Special slopes:
Horizontal lines in the form of y = have slopes of zero (they're flat!)
Vertical lines in the form of x = have no slope or undefined because the denominator is zero
Slope Intercept Form 7 -5
Finding the slope-intercept form of a line:
y = mx + b
where m = slope and b = y intercept
All you do is solve the equation for "y" meaning isolate the y on one side of the equal sign
(I explained this when we did Chapter 7-3 to easily find 3 coordinates in your T Chart. We just didn't call it slope intercept form at that time!)
It helps to solve the equation for y before you pick your x values, but you don't have to.
EXAMPLE from above: 2x - 3y = -6
Solve the above equation for y.
Subtract 2x from each side:
-3y = -2x - 6
Divide each side by -3:
y = 2/3 x + 2
Now pick your x values, put them on the left side of the T chart, then solve for y.
Instead of picking 0, 1, 2, it makes sense to pick x values that are multiples of 3.
Why? Because you will need to multiply the x value by 2/3 and this will keep the y value an integer:
x y
0 2
3 4
-3 0
Now graph these coordinates and join as your line
Restate Standard Form to Slope Intercept Form:
Example: 3x + 4y = 10 is the STANDARD FORM of a line
Solve for y
first subtracting 3x from both sides:
4y = -3x + 10
Now divide both sides by 4:
y = -3/4 x + 10/4 or y = -3/4 x + 5/2
The slope is the coefficient of the x
m = -3/4 (so you're sliding down at a little less than a 45 degree angle)
The y intercept is the constant
b = 5/2 (so the line crosses the y axis at 2 1/2.)
Graph when line is in Slope Intercept Form:
If you have the slope-intercept form of the equation, it's really easy to graph the line:
1) Graph the intercept on the y axis
2) "Count" the next point by using the slope or x coefficient as a FRACTION
For the equation y = 3x - 2
1) Put a dot at (0, -2)
2) From (0, -2) count up 3 and over to the right 1 to find the next coordinate (1, 1)
Remember, slope is y over x, so the numerator is the y change and the denominator is the x change
If it's positive, you're counting up (positive) and to the right (positive)
or you can count down and to the left because 2 negatives make a positive.
If it's negative, you're counting down (negative) and to the right (positive)
or you can count up and to the left because you would have a positive and negative = negative
If you're given the slope and the y intercept,
you can write the equation of any line!
Just use: y = mx + b
EXAMPLE: m = -2/3 and b = -12
The line would be y = -2/3 x - 12
First, let's talk about what the word "slope" means in the real world:
You can think of the slope of a line as the slope of a ski mountain -
When you're climbing up, it's positive
When you're sliding down, it's negative
(if you're looking at the mountain from left to right)
The steeper the mountain, the higher the slope value
(A slope of 6 would be an expert slope because it
is much steeper than a slope of 2 which would be an intermediate's slope)
"Bunny slopes" for beginners will be lower numbers,
generally fractional slopes (like 1/2 or 2/3)
A good benchmark to know is a slope of 1 or -1 is a 45 degree angle
You can also think of slope as rise/run - read this "rise over run"
Rise is how tall the mountain is (the y value)
Run is how wide the mountain is (the x value)
VISUALIZE THE FOLLOWING 2 MOUNTAINS TO HELP YOU UNDERSTAND:
A 1000 foot high mountain (the rise) is very steep if it's only 200 feet wide (the run) (slope = 5)
Another mountain that is also 1000 feet high is not very steep if it is 2000 feet wide (slope = 1/2)
It has a much longer time to slowly reach the 1000 foot top of the mountain!
You can think of slope as a calculation using 2 coordinates:
Rise/Run
=Change in y value/Change in x value
=Difference in y value/ Difference in x value
= y2 - y1/ x2 - x1
To calculate slope you need 2 coordinates. It doesn't matter which one you start with.
Just be consistent! If you start with the y value of one point, make sure you start with the same x!
You can count the slope of a line:
1) Beginning with one point, count up to another point; however far that is, make that the numerator of your slope (because the y value of slope is the numerator)
2) Now count how far over the point is across - You'll need to either go right or left.
Make this the denominator of your slope (because the x value is the denominator of slope)
If you went to the RIGHT, the value is POSITIVE (x values going to the right or positive)
If you went to the LEFT, the value is NEGATIVE
Special slopes:
Horizontal lines in the form of y = have slopes of zero (they're flat!)
Vertical lines in the form of x = have no slope or undefined because the denominator is zero
Slope Intercept Form 7 -5
Finding the slope-intercept form of a line:
y = mx + b
where m = slope and b = y intercept
All you do is solve the equation for "y" meaning isolate the y on one side of the equal sign
(I explained this when we did Chapter 7-3 to easily find 3 coordinates in your T Chart. We just didn't call it slope intercept form at that time!)
It helps to solve the equation for y before you pick your x values, but you don't have to.
EXAMPLE from above: 2x - 3y = -6
Solve the above equation for y.
Subtract 2x from each side:
-3y = -2x - 6
Divide each side by -3:
y = 2/3 x + 2
Now pick your x values, put them on the left side of the T chart, then solve for y.
Instead of picking 0, 1, 2, it makes sense to pick x values that are multiples of 3.
Why? Because you will need to multiply the x value by 2/3 and this will keep the y value an integer:
x y
0 2
3 4
-3 0
Now graph these coordinates and join as your line
Restate Standard Form to Slope Intercept Form:
Example: 3x + 4y = 10 is the STANDARD FORM of a line
Solve for y
first subtracting 3x from both sides:
4y = -3x + 10
Now divide both sides by 4:
y = -3/4 x + 10/4 or y = -3/4 x + 5/2
The slope is the coefficient of the x
m = -3/4 (so you're sliding down at a little less than a 45 degree angle)
The y intercept is the constant
b = 5/2 (so the line crosses the y axis at 2 1/2.)
Graph when line is in Slope Intercept Form:
If you have the slope-intercept form of the equation, it's really easy to graph the line:
1) Graph the intercept on the y axis
2) "Count" the next point by using the slope or x coefficient as a FRACTION
For the equation y = 3x - 2
1) Put a dot at (0, -2)
2) From (0, -2) count up 3 and over to the right 1 to find the next coordinate (1, 1)
Remember, slope is y over x, so the numerator is the y change and the denominator is the x change
If it's positive, you're counting up (positive) and to the right (positive)
or you can count down and to the left because 2 negatives make a positive.
If it's negative, you're counting down (negative) and to the right (positive)
or you can count up and to the left because you would have a positive and negative = negative
If you're given the slope and the y intercept,
you can write the equation of any line!
Just use: y = mx + b
EXAMPLE: m = -2/3 and b = -12
The line would be y = -2/3 x - 12
Math 6 H Periods 1, 6 & 7 (Tues & Wed.)
Comparing Fractions 6-4
When 2 fractions have equal denominators-- it is easy to tell which of the fractions are greater. Compare their numerators.
5/11 < 7/11 because 5 < 7
If the fractions have different denominators, find a common denominator. Using the lCM of the denominators-- the LCD-- is a surefire way of determinng the relationship between fractions.
Which is greater 5/6 or 7/9?
Since the LCM (6,9) = 18
Using equivalent fractions
5/6 = 15/18
and 7/9 = 14/18
so 5/6 > 7/9
( See Section 6-2 notes if you need to review equivalent fractions)
Another way is to use cross products to compare.
What if you needed to name a fraction between two other fractions?
FOr instance, between 7/15 and 12/ 25
FInd the LCM (15, 25) using the methods taught from chapter 5
LCM ( 15, 25) = 75
finding equivalent fractions for
7/15 = 35/75
12/25 = 36/75
If you want a fraction between, simply double the denominators and then double the numerators
35/75 = 70/150
36/75 = 72/150
So 71/150 would be a fraction that is between the two given fractions.
When 2 fractions have equal denominators-- it is easy to tell which of the fractions are greater. Compare their numerators.
5/11 < 7/11 because 5 < 7
If the fractions have different denominators, find a common denominator. Using the lCM of the denominators-- the LCD-- is a surefire way of determinng the relationship between fractions.
Which is greater 5/6 or 7/9?
Since the LCM (6,9) = 18
Using equivalent fractions
5/6 = 15/18
and 7/9 = 14/18
so 5/6 > 7/9
( See Section 6-2 notes if you need to review equivalent fractions)
Another way is to use cross products to compare.
What if you needed to name a fraction between two other fractions?
FOr instance, between 7/15 and 12/ 25
FInd the LCM (15, 25) using the methods taught from chapter 5
LCM ( 15, 25) = 75
finding equivalent fractions for
7/15 = 35/75
12/25 = 36/75
If you want a fraction between, simply double the denominators and then double the numerators
35/75 = 70/150
36/75 = 72/150
So 71/150 would be a fraction that is between the two given fractions.
Sunday, February 22, 2009
Algebra Period 3 ( 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
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... y = mx + b
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
Sunday, February 15, 2009
Algebra Period 3 ( Review)
We won't be covering Chapter 7-1 in class.
This is simple Pre-Algebra!
I have included a review below:
Graphing Ordered Pairs 7 -1
Review of x y Coordinate Plane Graphing from Pre-Algebra (ch 7-1 in your book)
Cartesian plane: Named after French mathematician Descartes.
plane: a two dimensional (across and up/down) flat surface that extends infinitely in all directions.
quadrant: 2 perpendicular lines called axes split the plane into 4 regions....quad means 4
quadrant names: begin in the top right (where you normally write your name!) and go counterclockwise in a big "C" (remember it for "C"oordinate)
They are named I, II, III, IV in Roman Numerals
The axes are NOT part of any quadrant. A point on the x-axis or the y-axis is not in a quadrant since it is on the boundary between quadrants.
coordinate - A coordinate is the position of a point in the Cartesian plane
coordinate = "co" means goes along with (COefficient, COworker, CO-president, CO-champions)
"ordinate" means in order
So coordinate means numbers that go along with each other in a certain order
The numbers are the x and y values and the order is that the x always comes first
Also called an ordered pair (x y "ordered" and they are a "pair" of numbers)
Ordered pairs are recognized by the use of ( x , y ) format
origin = (0, 0) the center of the graph (its beginning or origin)
When you count the coordinate' s position, you count from the origin.
x comes before y in the alphabet so the order is (x, y) ....
always go right or left first, then up or down
the x axis is the horizontal axis (goes across)
Remember that because the number line also is horizontal and you learn that first
(the pattern to remember is x is always first and the number line is before going up and down)
NOW LET'S GET TO WHAT YOU ACTUALLY DO!!!
1) Count your x value:
positive x, count right from origin (positive numbers are to the right of zero on number line)
negative x, count left from origin
2) Count your y value:
positive y value, count up from where your x value was (up is the positive direction)
negative y value, count down from where your x value was (down is the negative direction)
EXAMPLE:
(3, 5) Count 3 to the right from the origin, then 5 up
(3, -5) Still count 2 to the right, but now count 5 down
(-3, 5) Count 3 to the left from the origin, then count 5 up
(-3, -5) Again count 3 to the left, but now count 5 down
BUT WHAT HAPPENS WHEN
ONE OF THE VALUES IS ZERO?
If the y value is zero it means that you move right or left, but don't go up or down:
SO YOUR POINT WILL BE ON THE x AXIS........x axis is where y = 0
Example: (3, 0) is a point on the x axis, 3 places to the RIGHT
Example: (-3, 0) is a point on the x axis, 3 places to the LEFT
If the x value is zero it means that you don't move right or left, you just go up or down.
SO YOUR POINT WILL BE ON THE y AXIS...........y axis is where x = 0
Example: (0, 3) is a point on the y axis, 3 places UP
Example: (0, -3) is a point on the y axis, 3 places DOWN
Graphing Equations Section 7-2
How do you determine whether a given number is a solution?
Plug it in, plug it in, plug it in! Do this carefully. Use ( ) when you plug in a value for x and for y.
How do you find a solution to an equation yourself?
Plug in for x and find y!
You can use ANY number for x
Then plug in your number and find y
How can you graph a linear equation?
Make an x/y table of values and then graph the coordinates.
You only need 3 coordinates to make a good line!
(The 3rd coordinate serves as a "check" for the other two...in case you made a mistake!)
I always try x = zero and y = zero first because it's usually easy. Then pick another easy x value!
If this doesn't work well (you get a fraction as an answer and that's not easy to graph),
then try setting x equal to 1, then 2, then 3
Linear Equations Section 7-3
What do they look like ( and what is not a linear equation?)
The variable is to the 1 power - like x, or y, or a, or b
What is not a linear equation? the variable is not to the 1 power - like x2, x3, etc, or 1/x (x-1)
2 ways to graph:
1) 3 points using a table (like Ch 7-2)
EXAMPLE: 2x - 3y = -6
x y
0 2
3 4
-3 0
2) 2 points using the y and x intercepts (where the line intersects the y and x axis)
Standard form of a linear equation: Ax +By = C
A, B and C should not be fractions
A should be positive (y will be positive or negative)
We won't be using this form to look at the slope of the line!
This is a good format for finding the x and y intercepts!
If it's in standard form, this way works great if both the x and y coefficients are factors of the constant on the other side of the equal sign.
EXAMPLE: 2x - 3y = -6
If x = 0, y = 2
If y = 0, x = -3
Special linear equations:
Ones that are parallel to either the x or the y axis:
Lines parallel to the y axis are vertical lines:
They end up as the form x = with no y variable in the equation at all!
EXAMPLE: x = 4 ends up as a vertical line at x = 4
Still don't get this???
Pick of few points with the x value of 4:
(4, 0) (4, 2) (4, -3)
Graph those and join them in a line.
What do you get???
A vertical line!
Lines parallel to the x axis are horizontal lines:
They end up as the form y = with no x variable in the equation at all!
EXAMPLE: y = 4 ends up as a horizontal line at y = 4
Still don't get this???
Pick of few points with the y value of 4:
(0, 4) (2, 4) (-3, 4)
Graph those and join them in a line.
What do you get???
A horizontal line!
This is simple Pre-Algebra!
I have included a review below:
Graphing Ordered Pairs 7 -1
Review of x y Coordinate Plane Graphing from Pre-Algebra (ch 7-1 in your book)
Cartesian plane: Named after French mathematician Descartes.
plane: a two dimensional (across and up/down) flat surface that extends infinitely in all directions.
quadrant: 2 perpendicular lines called axes split the plane into 4 regions....quad means 4
quadrant names: begin in the top right (where you normally write your name!) and go counterclockwise in a big "C" (remember it for "C"oordinate)
They are named I, II, III, IV in Roman Numerals
The axes are NOT part of any quadrant. A point on the x-axis or the y-axis is not in a quadrant since it is on the boundary between quadrants.
coordinate - A coordinate is the position of a point in the Cartesian plane
coordinate = "co" means goes along with (COefficient, COworker, CO-president, CO-champions)
"ordinate" means in order
So coordinate means numbers that go along with each other in a certain order
The numbers are the x and y values and the order is that the x always comes first
Also called an ordered pair (x y "ordered" and they are a "pair" of numbers)
Ordered pairs are recognized by the use of ( x , y ) format
origin = (0, 0) the center of the graph (its beginning or origin)
When you count the coordinate' s position, you count from the origin.
x comes before y in the alphabet so the order is (x, y) ....
always go right or left first, then up or down
the x axis is the horizontal axis (goes across)
Remember that because the number line also is horizontal and you learn that first
(the pattern to remember is x is always first and the number line is before going up and down)
NOW LET'S GET TO WHAT YOU ACTUALLY DO!!!
1) Count your x value:
positive x, count right from origin (positive numbers are to the right of zero on number line)
negative x, count left from origin
2) Count your y value:
positive y value, count up from where your x value was (up is the positive direction)
negative y value, count down from where your x value was (down is the negative direction)
EXAMPLE:
(3, 5) Count 3 to the right from the origin, then 5 up
(3, -5) Still count 2 to the right, but now count 5 down
(-3, 5) Count 3 to the left from the origin, then count 5 up
(-3, -5) Again count 3 to the left, but now count 5 down
BUT WHAT HAPPENS WHEN
ONE OF THE VALUES IS ZERO?
If the y value is zero it means that you move right or left, but don't go up or down:
SO YOUR POINT WILL BE ON THE x AXIS........x axis is where y = 0
Example: (3, 0) is a point on the x axis, 3 places to the RIGHT
Example: (-3, 0) is a point on the x axis, 3 places to the LEFT
If the x value is zero it means that you don't move right or left, you just go up or down.
SO YOUR POINT WILL BE ON THE y AXIS...........y axis is where x = 0
Example: (0, 3) is a point on the y axis, 3 places UP
Example: (0, -3) is a point on the y axis, 3 places DOWN
Graphing Equations Section 7-2
How do you determine whether a given number is a solution?
Plug it in, plug it in, plug it in! Do this carefully. Use ( ) when you plug in a value for x and for y.
How do you find a solution to an equation yourself?
Plug in for x and find y!
You can use ANY number for x
Then plug in your number and find y
How can you graph a linear equation?
Make an x/y table of values and then graph the coordinates.
You only need 3 coordinates to make a good line!
(The 3rd coordinate serves as a "check" for the other two...in case you made a mistake!)
I always try x = zero and y = zero first because it's usually easy. Then pick another easy x value!
If this doesn't work well (you get a fraction as an answer and that's not easy to graph),
then try setting x equal to 1, then 2, then 3
Linear Equations Section 7-3
What do they look like ( and what is not a linear equation?)
The variable is to the 1 power - like x, or y, or a, or b
What is not a linear equation? the variable is not to the 1 power - like x2, x3, etc, or 1/x (x-1)
2 ways to graph:
1) 3 points using a table (like Ch 7-2)
EXAMPLE: 2x - 3y = -6
x y
0 2
3 4
-3 0
2) 2 points using the y and x intercepts (where the line intersects the y and x axis)
Standard form of a linear equation: Ax +By = C
A, B and C should not be fractions
A should be positive (y will be positive or negative)
We won't be using this form to look at the slope of the line!
This is a good format for finding the x and y intercepts!
If it's in standard form, this way works great if both the x and y coefficients are factors of the constant on the other side of the equal sign.
EXAMPLE: 2x - 3y = -6
If x = 0, y = 2
If y = 0, x = -3
Special linear equations:
Ones that are parallel to either the x or the y axis:
Lines parallel to the y axis are vertical lines:
They end up as the form x = with no y variable in the equation at all!
EXAMPLE: x = 4 ends up as a vertical line at x = 4
Still don't get this???
Pick of few points with the x value of 4:
(4, 0) (4, 2) (4, -3)
Graph those and join them in a line.
What do you get???
A vertical line!
Lines parallel to the x axis are horizontal lines:
They end up as the form y = with no x variable in the equation at all!
EXAMPLE: y = 4 ends up as a horizontal line at y = 4
Still don't get this???
Pick of few points with the y value of 4:
(0, 4) (2, 4) (-3, 4)
Graph those and join them in a line.
What do you get???
A horizontal line!
Thursday, February 12, 2009
Math 6 H Periods 1, 6 & 7 (Wednesday)
Equivalent Fractions 6-2
Fractions can be pictured on the number line.
If you multiply the numerator and the denominator of a fraction but the same nonzero number the resulting fraction is equivalent to the original fraction
1/2 (3/3) = 3/6
Properties
For any whole numbers a, b, and c, with b≠ 0 and c≠ 0
a/b = a(c)/b(c) and a/b = (a÷c) / (b÷c)
A fraction is in lowest terms if its numerator and denominator are relatively prime If their Greatest Common Factor (GCF) is 1
Fractions and Mixed Numbers 6-3
You know that ½ + ½ + ½ = 3/2
A fraction such as 3/2 (whose numerator is greater than or equal to its denominator) is called an improper fraction. Every improper fraction is greater than or equal to 1. A proper fraction is a fraction whose numerator is less than its denominator.
Proper fractions
1/4 2/3 5/9 10/12
Improper fractions
5/2, 8/3. 11/9 , 18/15
You can express any improper fractions as the sum of a whole number and a proper fraction
3/2 = 1 + ½ or 1½
A number such as 1½ (that is expressed as the sum of a whole number and a fraction) 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 simplest form.
To change an improper fraction to a mixed number in simple form, we divide the numerator by the denominator and express the remainder as a fraction
14/3 = 14 ÷ 3 = 4 2/3
30/4 = 30 ÷ 4 = 7 2/4 = 7 ½
To change a mixed number to an improper fraction, rewrite the whole number part as a fraction with the same denominator as the fractional part
1 3/8 = 8/8 + 3/8 = 11/8
Fractions can be pictured on the number line.
If you multiply the numerator and the denominator of a fraction but the same nonzero number the resulting fraction is equivalent to the original fraction
1/2 (3/3) = 3/6
Properties
For any whole numbers a, b, and c, with b≠ 0 and c≠ 0
a/b = a(c)/b(c) and a/b = (a÷c) / (b÷c)
A fraction is in lowest terms if its numerator and denominator are relatively prime If their Greatest Common Factor (GCF) is 1
Fractions and Mixed Numbers 6-3
You know that ½ + ½ + ½ = 3/2
A fraction such as 3/2 (whose numerator is greater than or equal to its denominator) is called an improper fraction. Every improper fraction is greater than or equal to 1. A proper fraction is a fraction whose numerator is less than its denominator.
Proper fractions
1/4 2/3 5/9 10/12
Improper fractions
5/2, 8/3. 11/9 , 18/15
You can express any improper fractions as the sum of a whole number and a proper fraction
3/2 = 1 + ½ or 1½
A number such as 1½ (that is expressed as the sum of a whole number and a fraction) 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 simplest form.
To change an improper fraction to a mixed number in simple form, we divide the numerator by the denominator and express the remainder as a fraction
14/3 = 14 ÷ 3 = 4 2/3
30/4 = 30 ÷ 4 = 7 2/4 = 7 ½
To change a mixed number to an improper fraction, rewrite the whole number part as a fraction with the same denominator as the fractional part
1 3/8 = 8/8 + 3/8 = 11/8
Math 6 H Periods 1, 6 & 7 (Tuesday)
Fractions 6-1
Can you think of some familiar expressions that include fractions?
Notice that the symbol ¼ can mean several things:
It means one divided by four
It represents one out of four equal parts
It is a number that has a position on a number line.
If an object is divided into 8 equal parts, each part is one eight of the whole.
1/8 means 1 divided by 8 or 1 ÷ 8.
If an object is divided into eight parts and three of these parts are being considered, then the fraction that represents the parts is 3/8
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.
Properties
For any whole numbers a, b, and c, with b ≠ 0
(a/b)(c) = ac/b
Examples:
2/7 + 2/ 7 + 2/7 = 6/7 which is really (3) 2/7 = 6/7
Can you think of some familiar expressions that include fractions?
Notice that the symbol ¼ can mean several things:
It means one divided by four
It represents one out of four equal parts
It is a number that has a position on a number line.
If an object is divided into 8 equal parts, each part is one eight of the whole.
1/8 means 1 divided by 8 or 1 ÷ 8.
If an object is divided into eight parts and three of these parts are being considered, then the fraction that represents the parts is 3/8
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.
Properties
For any whole numbers a, b, and c, with b ≠ 0
(a/b)(c) = ac/b
Examples:
2/7 + 2/ 7 + 2/7 = 6/7 which is really (3) 2/7 = 6/7
Algebra Period 3 (Wednesday)
How do you determine whether a given number is a solution?
Plug it in, plug it in, plug it in! Do this carefully. Use ( ) when you plug in a value for x and for y.
How do you find a solution to an equation yourself?
Plug in for x and find y!
You can use ANY number for x
Then plug in your number and find y
How can you graph a linear equation?
Make an x/y table of values and then graph the coordinates.
You only need 3 coordinates to make a good line!
(The 3rd coordinate serves as a "check" for the other two...in case you made a mistake!)
I always try x = zero and y = zero first because it's usually easy. Then pick another easy x value!
If this doesn't work well (you get a fraction as an answer and that's not easy to graph),
then try setting x equal to 1, then 2, then 3
Linear equations 7-3
What do they look like ( and what is not a linear equation?)
The variable is to the 1 power - like x, or y, or a, or b
What is not a linear equation? the variable is not to the 1 power - like x2, x3, etc, or 1/x (x-1)
2 ways to graph:
1) 3 points using a table (like Ch 7-2)
EXAMPLE: 2x - 3y = -6
x y
0 2
3 4
-3 0
2) 2 points using the y and x intercepts (where the line intersects the y and x axis)
Standard form of a linear equation: Ax +By = C
A, B and C should not be fractions
A should be positive (y will be positive or negative)
We won't be using this form to look at the slope of the line!
This is a good format for finding the x and y intercepts!
If it's in standard form, this way works great if both the x and y coefficients are factors of the constant on the other side of the equal sign.
EXAMPLE: 2x - 3y = -6
If x = 0, y = 2
If y = 0, x = -3
Special linear equations:
Ones that are parallel to either the x or the y axis:
Lines parallel to the y axis are vertical lines:
They end up as the form x = with no y variable in the equation at all!
EXAMPLE: x = 4 ends up as a vertical line at x = 4
Still don't get this???
Pick of few points with the x value of 4:
(4, 0) (4, 2) (4, -3)
Graph those and join them in a line.
What do you get???
A vertical line!
Lines parallel to the x axis are horizontal lines:
They end up as the form y = with no x variable in the equation at all!
EXAMPLE: y = 4 ends up as a horizontal line at y = 4
Still don't get this???
Pick of few points with the y value of 4:
(0, 4) (2, 4) (-3, 4)
Graph those and join them in a line.
What do you get???
A horizontal line!
Plug it in, plug it in, plug it in! Do this carefully. Use ( ) when you plug in a value for x and for y.
How do you find a solution to an equation yourself?
Plug in for x and find y!
You can use ANY number for x
Then plug in your number and find y
How can you graph a linear equation?
Make an x/y table of values and then graph the coordinates.
You only need 3 coordinates to make a good line!
(The 3rd coordinate serves as a "check" for the other two...in case you made a mistake!)
I always try x = zero and y = zero first because it's usually easy. Then pick another easy x value!
If this doesn't work well (you get a fraction as an answer and that's not easy to graph),
then try setting x equal to 1, then 2, then 3
Linear equations 7-3
What do they look like ( and what is not a linear equation?)
The variable is to the 1 power - like x, or y, or a, or b
What is not a linear equation? the variable is not to the 1 power - like x2, x3, etc, or 1/x (x-1)
2 ways to graph:
1) 3 points using a table (like Ch 7-2)
EXAMPLE: 2x - 3y = -6
x y
0 2
3 4
-3 0
2) 2 points using the y and x intercepts (where the line intersects the y and x axis)
Standard form of a linear equation: Ax +By = C
A, B and C should not be fractions
A should be positive (y will be positive or negative)
We won't be using this form to look at the slope of the line!
This is a good format for finding the x and y intercepts!
If it's in standard form, this way works great if both the x and y coefficients are factors of the constant on the other side of the equal sign.
EXAMPLE: 2x - 3y = -6
If x = 0, y = 2
If y = 0, x = -3
Special linear equations:
Ones that are parallel to either the x or the y axis:
Lines parallel to the y axis are vertical lines:
They end up as the form x = with no y variable in the equation at all!
EXAMPLE: x = 4 ends up as a vertical line at x = 4
Still don't get this???
Pick of few points with the x value of 4:
(4, 0) (4, 2) (4, -3)
Graph those and join them in a line.
What do you get???
A vertical line!
Lines parallel to the x axis are horizontal lines:
They end up as the form y = with no x variable in the equation at all!
EXAMPLE: y = 4 ends up as a horizontal line at y = 4
Still don't get this???
Pick of few points with the y value of 4:
(0, 4) (2, 4) (-3, 4)
Graph those and join them in a line.
What do you get???
A horizontal line!
Algebra Period 3 (Tuesday)
Graphing Ordered Pairs 7-1
Review of x y Coordinate Plane Graphing from Pre-Algebra
Cartesian plane: Named after French mathematician Descartes.
PLANE: a two dimensional (across and up/down) flat surface that extends infinitely in all directions. It’s 2-D
QUADRANT: 2 perpendicular lines called axes split the plane into 4 regions....quad means 4
quadrant names: begin in the top right (where you normally write your name!) and go counterclockwise in a big "C" (remember it for "C"oordinate)
They are named I, II, III, IV in Roman Numerals
COORDINATE - A coordinate is the position of a point in the Cartesian plane
coordinate = "co" means goes along with (COefficient, COworker, CO-president, CO-champions)
"ordinate" means in order
So coordinate means numbers that go along with each other in a certain order
The numbers are the x and y values and the order is that the x always comes first
Also called an ordered pair (x y "ordered" and they are a "pair" of numbers)
Ordered pairs are recognized by the use of ( x , y) format
origin = (0, 0) the center of the graph (its beginning or origin)
When you count the coordinate' s position, you count from the origin.
x comes before y in the alphabet so the order is (x, y)
always go right or left first, then up or down
the x axis is the horizontal axis (goes across)
Remember that because the number line also is horizontal and you learn that first
(the pattern to remember is x is always first and the number line is before going up and down)
NOW LET'S GET TO WHAT YOU ACTUALLY DO!!!
1) Count your x value:
positive x, count right from origin (positive numbers are to the right of zero on number line)
negative x, count left from origin
2) Count your y value:
positive y value, count up from where your x value was (up is the positive direction)
negative y value, count down from where your x value was (down is the negative direction)
EXAMPLE:
(3, 5) Count 3 to the right from the origin, then 5 up
(3, -5) Still count 2 to the right, but now count 5 down
(-3, 5) Count 3 to the left from the origin, then count 5 up
(-3, -5) Again count 3 to the left, but now count 5 down
BUT WHAT HAPPENS WHEN
ONE OF THE VALUES IS ZERO?
If the y value is zero it means that you move right or left, but don't go up or down:
SO YOUR POINT WILL BE ON THE x AXIS........x axis is where y = 0
Example: (3, 0) is a point on the x axis, 3 places to the RIGHT
Example: (-3, 0) is a point on the x axis, 3 places to the LEFT
If the x value is zero it means that you don't move right or left, you just go up or down.
SO YOUR POINT WILL BE ON THE y AXIS...........y axis is where x = 0
Example: (0, 3) is a point on the y axis, 3 places UP
Example: (0, -3) is a point on the y axis, 3 places DOWN
Review of x y Coordinate Plane Graphing from Pre-Algebra
Cartesian plane: Named after French mathematician Descartes.
PLANE: a two dimensional (across and up/down) flat surface that extends infinitely in all directions. It’s 2-D
QUADRANT: 2 perpendicular lines called axes split the plane into 4 regions....quad means 4
quadrant names: begin in the top right (where you normally write your name!) and go counterclockwise in a big "C" (remember it for "C"oordinate)
They are named I, II, III, IV in Roman Numerals
COORDINATE - A coordinate is the position of a point in the Cartesian plane
coordinate = "co" means goes along with (COefficient, COworker, CO-president, CO-champions)
"ordinate" means in order
So coordinate means numbers that go along with each other in a certain order
The numbers are the x and y values and the order is that the x always comes first
Also called an ordered pair (x y "ordered" and they are a "pair" of numbers)
Ordered pairs are recognized by the use of ( x , y) format
origin = (0, 0) the center of the graph (its beginning or origin)
When you count the coordinate' s position, you count from the origin.
x comes before y in the alphabet so the order is (x, y)
always go right or left first, then up or down
the x axis is the horizontal axis (goes across)
Remember that because the number line also is horizontal and you learn that first
(the pattern to remember is x is always first and the number line is before going up and down)
NOW LET'S GET TO WHAT YOU ACTUALLY DO!!!
1) Count your x value:
positive x, count right from origin (positive numbers are to the right of zero on number line)
negative x, count left from origin
2) Count your y value:
positive y value, count up from where your x value was (up is the positive direction)
negative y value, count down from where your x value was (down is the negative direction)
EXAMPLE:
(3, 5) Count 3 to the right from the origin, then 5 up
(3, -5) Still count 2 to the right, but now count 5 down
(-3, 5) Count 3 to the left from the origin, then count 5 up
(-3, -5) Again count 3 to the left, but now count 5 down
BUT WHAT HAPPENS WHEN
ONE OF THE VALUES IS ZERO?
If the y value is zero it means that you move right or left, but don't go up or down:
SO YOUR POINT WILL BE ON THE x AXIS........x axis is where y = 0
Example: (3, 0) is a point on the x axis, 3 places to the RIGHT
Example: (-3, 0) is a point on the x axis, 3 places to the LEFT
If the x value is zero it means that you don't move right or left, you just go up or down.
SO YOUR POINT WILL BE ON THE y AXIS...........y axis is where x = 0
Example: (0, 3) is a point on the y axis, 3 places UP
Example: (0, -3) is a point on the y axis, 3 places DOWN
Algebra Period 3 (Review)
FACTORING CHECKLIST
Look for a GCF of all terms
2. Binomials - look for difference of two squares
both perfect squares - double hug - one positive, one negative - square roots of both terms
2. Trinomials - look for Trinomial Square (factors as a binomial squared)
first and last must be perfect squares - middle must be double the product of the two square roots
SINGLE hug - square roots of both terms - sign is the middle sign
3. Trinomials - last sign positive - double hug with same sign as middle term - factors that multiply to last and add to middle
4. Trinomials - last sign negative - double hug with different signs, putting middle sign in first hug - factors that multiply to last and subtract to middle - middle sign will always be with the bigger factor
5. Trinomial with "a" coefficient - Use T chart - multiply first to last to get new product - then find factors that multiply to that new produce and either add or subtract to the middle term (use trinomial rules above) - replace middle term with these two factors and place appropriate signs so they will add to the original middle term - proceed as if you have factoring by grouping (see 6 below)
6. 4 term polynomial - factor by grouping - pair of the first 2 terms and then the second 2 terms by placing parentheses around them - make sure you always have a plus sign between the 2 pairs (you may need to double check) - factor out the GCF of each pair - if it factors, there should now be a new GCF - factor that out in front parentheses and place what ever is left in the second parentheses
REMEMBER:
FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!
Look for a GCF of all terms
2. Binomials - look for difference of two squares
both perfect squares - double hug - one positive, one negative - square roots of both terms
2. Trinomials - look for Trinomial Square (factors as a binomial squared)
first and last must be perfect squares - middle must be double the product of the two square roots
SINGLE hug - square roots of both terms - sign is the middle sign
3. Trinomials - last sign positive - double hug with same sign as middle term - factors that multiply to last and add to middle
4. Trinomials - last sign negative - double hug with different signs, putting middle sign in first hug - factors that multiply to last and subtract to middle - middle sign will always be with the bigger factor
5. Trinomial with "a" coefficient - Use T chart - multiply first to last to get new product - then find factors that multiply to that new produce and either add or subtract to the middle term (use trinomial rules above) - replace middle term with these two factors and place appropriate signs so they will add to the original middle term - proceed as if you have factoring by grouping (see 6 below)
6. 4 term polynomial - factor by grouping - pair of the first 2 terms and then the second 2 terms by placing parentheses around them - make sure you always have a plus sign between the 2 pairs (you may need to double check) - factor out the GCF of each pair - if it factors, there should now be a new GCF - factor that out in front parentheses and place what ever is left in the second parentheses
REMEMBER:
FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!
Sunday, February 1, 2009
Math 6 H Periods 1, 6 & 7
Graphs of Equations 11-9
An equation in two variables can produce many ordered pairs.
y = 2 – x
If we give x the value of 3, for example a corresponding value of y is determined.
y = 2- 3
y = -1
We describe this correspondence by the ordered pair (3, -1). We can created a table that shows several other ordered pairs produced by y = 2 –x
x ---> 2 - x = y ---> Ordered pair (x, y)
-1 ---> 2- -1 = 3 ---> (-1, 3)
0 ---> 2 – 0 = 2---> (0, 2)
1 ---> 2 - 1 = 1---> (1, 1)
2 ---> 2 - 2 = 0---> (2, 0)
3 ---> 2 - 3 = -1---> (3, -1)
4 ---> 2 - 4 = -2---> (4, -2)
Let’s plot those ordered pairs. The diagram suggests that if we were able to graph all of the ordered pairs produced by y = 2-x we would obtain a line. This line is the graph of the equation y = 2-x
Graph the equation y = 2x – 3
First make a table of ordered pairs and then graph the ordered pairs on a coordinate plane.
x---> 2x-3 = y---> ordered pair
-1---> 2(-1) – 3 = -5---> (-1, -5)
0---> 2(0) – 3 = -3---> (0, -3)
1---> 2(1) – 3 = -1---> (1, -1)
2---> 2(2) – 3 = 1---> (2, 1)
In the set of ordered pairs, for each value of x there is exactly one value of y. A set of ordered pairs such as this, in which no two ordered pairs have the same first component is called a function. For example, we can say that y = 2x -3 defines y as a function of x
An equation in two variables can produce many ordered pairs.
y = 2 – x
If we give x the value of 3, for example a corresponding value of y is determined.
y = 2- 3
y = -1
We describe this correspondence by the ordered pair (3, -1). We can created a table that shows several other ordered pairs produced by y = 2 –x
x ---> 2 - x = y ---> Ordered pair (x, y)
-1 ---> 2- -1 = 3 ---> (-1, 3)
0 ---> 2 – 0 = 2---> (0, 2)
1 ---> 2 - 1 = 1---> (1, 1)
2 ---> 2 - 2 = 0---> (2, 0)
3 ---> 2 - 3 = -1---> (3, -1)
4 ---> 2 - 4 = -2---> (4, -2)
Let’s plot those ordered pairs. The diagram suggests that if we were able to graph all of the ordered pairs produced by y = 2-x we would obtain a line. This line is the graph of the equation y = 2-x
Graph the equation y = 2x – 3
First make a table of ordered pairs and then graph the ordered pairs on a coordinate plane.
x---> 2x-3 = y---> ordered pair
-1---> 2(-1) – 3 = -5---> (-1, -5)
0---> 2(0) – 3 = -3---> (0, -3)
1---> 2(1) – 3 = -1---> (1, -1)
2---> 2(2) – 3 = 1---> (2, 1)
In the set of ordered pairs, for each value of x there is exactly one value of y. A set of ordered pairs such as this, in which no two ordered pairs have the same first component is called a function. For example, we can say that y = 2x -3 defines y as a function of x
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