More STAR Review
Person-Hours
This little concept is almost always on the STAR
The number of hours it takes to complete a job
This job can be done by 1 person or multiple people.
So if a job requires 24 person-hours and you have 3 people working, each would need to work 8 hours a piece.
If you have 8 people working, each would only need to work 3 hours a piece.
Square Roots & Irrational Numbers: 11-1
Square root undoes squaring!
So you're looking for the the number/variable that was squared to get the radicand
1) RADICAL sign: The root sign, which looks like a check mark.
If there is no little number on the radical, you assume it's the square root
But many times there will be a number there and then you are finding the root that the number says.
For example, if there is a 3 in the "check mark," you are finding the cubed root.
One more example:
The square root of 64 is 8.
The cubed root of 64 is 3.
The 6th root of 64 is 2.
2)RADICAND : Whatever is under the RADICAL sign
In the example above, 64 was the radicand in every case.
3) ROOT (the answer): the number/variable that was squared (cubed, raised to a power)
to get the RADICAND (whatever is under the radical sign)
4) SQUARE ROOTS: The number that is squared to get to the radicand.
Every POSITIVE number has 2 square roots - one positive and one negative.
Example: The square root of 25 means what number squared = 25
Answer: Either positive 5 squared OR negative 5 squared
You can estimate non-perfect square roots by guess and check
Find the 2 numbers that it is between
Example: Square root of 52
It's between the 2 perfect squares: 49 and 64
So the square root is between 7 and 8
Since 52 is only 3 away from 49 and 12 away from 64, the square root will be closer to 7
Guess: 7.2 Square this: (7.2)(7.2) = 51.84 (adjust your estimate as necessary)
Real Number System 9-2
2 PARTS: RATIONAL AND IRRATIONAL (both real)
Review LEAP FROG number systems
I, RATIONAL NUMBERS (definitions of different number systems):
Natural = counting = 1, 2, 3, . . .
Whole = natural + 0 = 0, 1, 2, 3, . . .
Integers = whole + opposites = -3, -2, -1, 0, 1, 2, 3, . . .
Rational = integers and all the fractions/decimals in between - terminating and repeating decimals
II. IRRATIONAL - numbers like pi and square root of 3 - never repeat or terminate - round!
Friday, April 16, 2010
Algebra (Period 4)
Simplify, Multiply, and Divide RATIONAL EXPRESSIONS 10-1, 10-2, and 10-3
Rational Expressions = Expressions in fraction format (division) with a variable in the denominator
You have already been simplifying, multiplying and dividing these throughout this year!
SIMPLIFY: 10-1
You will need to FACTOR (Chapter 6) both the numerator and denominator and "cross out" common factors in both (their quotient is 1!)
EXAMPLE: Simplify
y2 + 3y + 2 =
y2 - 1
(y + 2)(y + 1) =
(y - 1)(y + 1)
y + 2
y - 1
MULTIPLY: 10-2
FACTOR if possible, cross cancel if possible, multiply numerators, then denominators, simplify
EXAMPLE:
(y + 4)3[y2 + 4y + 4] =
[(y + 2) 3(y2 + 8y + 16)]
(y + 4) 3][ (y + 2) 2 =
(y + 2) 3(y + 4) 2
y + 4
y + 2
DIVIDE: 10-3
Same as the previous example, only this time you will need to
“FLIP the SECOND” fraction,
FACTOR then
MULTIPLY!!!!!
EXAMPLE:
x + 1 ÷ x + 1 =
x2 - 1 x2 - 2x + 1
( x + 1) ( x2 - 2x + 1) =
(x + 1)( x - 1) (x + 1)
( x + 1)( x - 1)(x - 1) =
(x + 1)( x - 1)(x + 1)
x - 1
x + 1
Rational Expressions = Expressions in fraction format (division) with a variable in the denominator
You have already been simplifying, multiplying and dividing these throughout this year!
SIMPLIFY: 10-1
You will need to FACTOR (Chapter 6) both the numerator and denominator and "cross out" common factors in both (their quotient is 1!)
EXAMPLE: Simplify
y2 + 3y + 2 =
y2 - 1
(y + 2)(y + 1) =
(y - 1)(y + 1)
y + 2
y - 1
MULTIPLY: 10-2
FACTOR if possible, cross cancel if possible, multiply numerators, then denominators, simplify
EXAMPLE:
(y + 4)3[y2 + 4y + 4] =
[(y + 2) 3(y2 + 8y + 16)]
(y + 4) 3][ (y + 2) 2 =
(y + 2) 3(y + 4) 2
y + 4
y + 2
DIVIDE: 10-3
Same as the previous example, only this time you will need to
“FLIP the SECOND” fraction,
FACTOR then
MULTIPLY!!!!!
EXAMPLE:
x + 1 ÷ x + 1 =
x2 - 1 x2 - 2x + 1
( x + 1) ( x2 - 2x + 1) =
(x + 1)( x - 1) (x + 1)
( x + 1)( x - 1)(x - 1) =
(x + 1)( x - 1)(x + 1)
x - 1
x + 1
Thursday, April 15, 2010
Pre Algebra (Period 1)
Relations & Functions 8-1
Look at p. 384-5 together - graphs of real world relationships
RELATIONS: Set of ordered pairs where the x values are the DOMAIN and
the y values are the RANGE.
(You can remember which is which because it's alphabetic... D comes before R just as x comes before y!)
FUNCTIONS: Relations where there is just one y value for each x value
IN OTHER WORDS----YOU CAN'T HAVE TWO y VALUES for the SAME x value!!!
If you see x repeated twice, it's still a relation, but it's not a function.
So, KEEP YOUR EYES ON THE x's!
AS LONG AS YOU DON'T SEE AN x REPEATED, YOU'VE GOT A FUNCTION!
In the real world, I have a good example...pizza prices. You can't have two different prices for the same size cheese pizza.
If you charge $10 and $12 on the same day for the same pizza, you don't have a function.
But, you certainly can charge $10 for a cheese pizza and $12 for a pepperoni pizza.
VERTICAL LINE TEST: When you graph a function, if you draw a vertical line anywhere on the graph, that line will only intersect the function at one point!!!!
If it intersects at 2 or more, it's a relation, but not a function.
So a horizontal line function, y = 4, is a function,
but a vertical line function, x = 4 is not.
INPUTS: x values (also the Domain)
OUTPUTS: y values (also the Range)
Graphing A Line 8-2
You can graph a line by making an x y table
Pick an x value
Plug in the equation to find the y
Graph the (x, y) for 3 points and connect
Slope of a Line 8-3
Slope is a formula. The slope is a constant along the entire line!
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
(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)
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)
You can think of slope as a calculation:
Rise = Change in y value = Difference in y value = y2 - y1
Run=Change in x value Difference in x value 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!
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
Y INTERCEPT
Where the line crosses the y axis
y = mx +b
the m represents the slope and
the b represents the y intercept-- or where the line crosses the y axis!!
CHAPTER 13-2: GRAPHING PARABOLAS
You will do this in Algebra
For the STAR you just need to know a couple of things to make an educated guess:
you will have a PARABOLA when the x term is SQUARED
A PARABOLA looks like a smile when the a coefficient is positive
or looks like a frown when the a coefficient is negative.
Example:
y = 3x2 -2
The coefficient of the x2 term is POSITIVE 3 so it's a smile.
The NEGATIVE 2 means the smile starts at -2
Example:
y = -2x2 +7 the coefficient of the x2 term is NEGATIVE 2 so it's a frown.
The POSITIVE 7 means the frown starts at +7
Look at p. 384-5 together - graphs of real world relationships
RELATIONS: Set of ordered pairs where the x values are the DOMAIN and
the y values are the RANGE.
(You can remember which is which because it's alphabetic... D comes before R just as x comes before y!)
FUNCTIONS: Relations where there is just one y value for each x value
IN OTHER WORDS----YOU CAN'T HAVE TWO y VALUES for the SAME x value!!!
If you see x repeated twice, it's still a relation, but it's not a function.
So, KEEP YOUR EYES ON THE x's!
AS LONG AS YOU DON'T SEE AN x REPEATED, YOU'VE GOT A FUNCTION!
In the real world, I have a good example...pizza prices. You can't have two different prices for the same size cheese pizza.
If you charge $10 and $12 on the same day for the same pizza, you don't have a function.
But, you certainly can charge $10 for a cheese pizza and $12 for a pepperoni pizza.
VERTICAL LINE TEST: When you graph a function, if you draw a vertical line anywhere on the graph, that line will only intersect the function at one point!!!!
If it intersects at 2 or more, it's a relation, but not a function.
So a horizontal line function, y = 4, is a function,
but a vertical line function, x = 4 is not.
INPUTS: x values (also the Domain)
OUTPUTS: y values (also the Range)
Graphing A Line 8-2
You can graph a line by making an x y table
Pick an x value
Plug in the equation to find the y
Graph the (x, y) for 3 points and connect
Slope of a Line 8-3
Slope is a formula. The slope is a constant along the entire line!
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
(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)
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)
You can think of slope as a calculation:
Rise = Change in y value = Difference in y value = y2 - y1
Run=Change in x value Difference in x value 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!
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
Y INTERCEPT
Where the line crosses the y axis
y = mx +b
the m represents the slope and
the b represents the y intercept-- or where the line crosses the y axis!!
CHAPTER 13-2: GRAPHING PARABOLAS
You will do this in Algebra
For the STAR you just need to know a couple of things to make an educated guess:
you will have a PARABOLA when the x term is SQUARED
A PARABOLA looks like a smile when the a coefficient is positive
or looks like a frown when the a coefficient is negative.
Example:
y = 3x2 -2
The coefficient of the x2 term is POSITIVE 3 so it's a smile.
The NEGATIVE 2 means the smile starts at -2
Example:
y = -2x2 +7 the coefficient of the x2 term is NEGATIVE 2 so it's a frown.
The POSITIVE 7 means the frown starts at +7
Math 6H ( Period 3, 6 & 7)
Percents and Decimals 9-2
By looking at the following examples, you will be able to see a general relationship between decimals and percents
57% = 57/100
0.79 = 79/100 = 79%
113% = 113/100 = 1 13/100
0.06 = 6/100 = 6%
Rules
To express a percent as a decimal, move the decimal point two places to the left and remove the percent sign
57% = 0.57
113% = 1.13
To express a decimal as a percent, move the decimal point two places to the right and add a percent sign
0.79 = 79%
0.06 = 6%
In 9-1 you learned one method of changing a fraction into a percent. Here is an alternative method
Rule
To express a fraction as a percent, first express the fraction as a decimal
and then as a percent
Express 7/8 as a percent
Divide 7 by 8
7/8 = 0.875 = 87.5%
Express 1/3 as a percent
divide 1 by 3
0.33333….. it’s a repeating decimal
express the decimal as a percent 0.333… = 33 1/3%
so, to the nearest tenth of a percent = 33.3% but it is much more accurate to keep the 1/3 and write 33 1/3%
By looking at the following examples, you will be able to see a general relationship between decimals and percents
57% = 57/100
0.79 = 79/100 = 79%
113% = 113/100 = 1 13/100
0.06 = 6/100 = 6%
Rules
To express a percent as a decimal, move the decimal point two places to the left and remove the percent sign
57% = 0.57
113% = 1.13
To express a decimal as a percent, move the decimal point two places to the right and add a percent sign
0.79 = 79%
0.06 = 6%
In 9-1 you learned one method of changing a fraction into a percent. Here is an alternative method
Rule
To express a fraction as a percent, first express the fraction as a decimal
and then as a percent
Express 7/8 as a percent
Divide 7 by 8
7/8 = 0.875 = 87.5%
Express 1/3 as a percent
divide 1 by 3
0.33333….. it’s a repeating decimal
express the decimal as a percent 0.333… = 33 1/3%
so, to the nearest tenth of a percent = 33.3% but it is much more accurate to keep the 1/3 and write 33 1/3%
Wednesday, April 14, 2010
Math 6H ( Period 3, 6 & 7)
Percent and Fractions 9-1
The word “percent” is derived from the Latin “per centum” meaning “per hundred” or “out of one hundred” so 28% means 28 out of 100
A percent is a ratio that compares a number to 100. Therefore you can write a percent as a fraction with a denominator of 100, so 28% is also 28/100
Our book’s example is as follows;
During basketball season, Alice made 17 out of 25 free throws, while Nina made 7 out of 10. To see who did better, we compare the fractions representing each girl’s successful free throws. 17/25 or 7/10
We have calculated this type of problem before.. this time when we compare fractions use the common denominator 100, even if 100 is not the LCD of the fractions.
17/25 = 68/100 and
7/10 = 70/100
Since Alice makes 68 free throws per 100 and Nina makes 70 per hundred, Nina is the better free throw shooter.
the ratio of a number to 100 is called a percent. We write percents by using the symbol %
so
17/25 =68% and
7/10= 70%
Rule
To express the fraction a/b
as a percent, solve the equation
a/b = n/100
for the variable n and write n%
Express 17/40 as a percent
n/100 = 17/40 multiply both sides by 100 100 ( n/100) = 17(100)/40
n = 17(100)/40 n = 85/2 n= 42½
Therefore, 17/40 = 42 1/2 %
Rule
To express n% as a fraction, write the fraction
n/100 in lowest terms
Express 7 ½ % as a fraction in lowest terms
7 ½ % = 7.5% = 7.5/100 How do we get rid of the decimal?
multiply the numerator and the denominator by 10
7.5(10)/100(10) simplify
Similarly, you could change a mixed numebr into an improper fraction
5 3/8% becomes 43/8 % and to change that to a fraction simple divide by 100
That looks messy but if you remember that to divide by 100 you are actually multiplying by 1/100
(43/8) (1/100) = 43/800 and you are finished with your calculations!! EASY!!
Since a percent is the ratio of a number to 100, we can have percents that are greater than or equal to 100%
1 = 100/100 = 100%
165/100 = 165 %
Write 250% as a mixed number in simple form
250% = 250/100
250/100 = 2 50/100 = 2 1/2
The town of Wonderful spends 42% of its budget on education. What percent is used for other purposes?
the whole budget is represented by 100%. Therefore, the part used for other purposes is
100 - 42 or 58%
The word “percent” is derived from the Latin “per centum” meaning “per hundred” or “out of one hundred” so 28% means 28 out of 100
A percent is a ratio that compares a number to 100. Therefore you can write a percent as a fraction with a denominator of 100, so 28% is also 28/100
Our book’s example is as follows;
During basketball season, Alice made 17 out of 25 free throws, while Nina made 7 out of 10. To see who did better, we compare the fractions representing each girl’s successful free throws. 17/25 or 7/10
We have calculated this type of problem before.. this time when we compare fractions use the common denominator 100, even if 100 is not the LCD of the fractions.
17/25 = 68/100 and
7/10 = 70/100
Since Alice makes 68 free throws per 100 and Nina makes 70 per hundred, Nina is the better free throw shooter.
the ratio of a number to 100 is called a percent. We write percents by using the symbol %
so
17/25 =68% and
7/10= 70%
Rule
To express the fraction a/b
as a percent, solve the equation
a/b = n/100
for the variable n and write n%
Express 17/40 as a percent
n/100 = 17/40 multiply both sides by 100 100 ( n/100) = 17(100)/40
n = 17(100)/40 n = 85/2 n= 42½
Therefore, 17/40 = 42 1/2 %
Rule
To express n% as a fraction, write the fraction
n/100 in lowest terms
Express 7 ½ % as a fraction in lowest terms
7 ½ % = 7.5% = 7.5/100 How do we get rid of the decimal?
multiply the numerator and the denominator by 10
7.5(10)/100(10) simplify
Similarly, you could change a mixed numebr into an improper fraction
5 3/8% becomes 43/8 % and to change that to a fraction simple divide by 100
That looks messy but if you remember that to divide by 100 you are actually multiplying by 1/100
(43/8) (1/100) = 43/800 and you are finished with your calculations!! EASY!!
Since a percent is the ratio of a number to 100, we can have percents that are greater than or equal to 100%
1 = 100/100 = 100%
165/100 = 165 %
Write 250% as a mixed number in simple form
250% = 250/100
250/100 = 2 50/100 = 2 1/2
The town of Wonderful spends 42% of its budget on education. What percent is used for other purposes?
the whole budget is represented by 100%. Therefore, the part used for other purposes is
100 - 42 or 58%
Tuesday, April 13, 2010
Pre Algebra ( Period 1)
Scatter Plots 8-5 & 8-6
CORRELATION GRAPHS: SCATTER PLOTS USES SOMETHING YOU ALREADY KNOW: (x, y) graphing
Scatter plots are graphs that show the RELATIONSHIP between 2 SETS OF DATA
This type of graph is used for surveys
For example, you survey students about how long they study and what grade they received
You survey people about how many hours they use their heaters in relation to the temperature outside.
CORRELATION = relationship
There are 3 types of correlations:
1) POSITIVE - the two sets of data travel in the SAME DIRECTION
The scatter graph looks like it's going up from left to right (as one increases, so does the other or as one decreases, so does the other)
Ex) The amount of time spent studying and their grades
2)NEGATIVE - the two sets of data travel in OPPOSITE DIRECTIONS
The scatter graph looks like it's going down from left to right (as one increases, the other decreases or vice versa)
Ex) As the temperature increases, the use of heaters decreases
3) NO CORRELATION - the two sets have no relationship to each other
The scatter graph is all over the place
Ex) Your math grade and the temperature outside
LINE OF BEST FIT: If the data has a correlation (either positive or negative), you will be able to draw a line that "fits" the data. It may not actually go through any of the specific data points, but it will be in the center of most of the data.
This line is a TREND LINE and can be used to make projections or analytical statements
CORRELATION GRAPHS: SCATTER PLOTS USES SOMETHING YOU ALREADY KNOW: (x, y) graphing
Scatter plots are graphs that show the RELATIONSHIP between 2 SETS OF DATA
This type of graph is used for surveys
For example, you survey students about how long they study and what grade they received
You survey people about how many hours they use their heaters in relation to the temperature outside.
CORRELATION = relationship
There are 3 types of correlations:
1) POSITIVE - the two sets of data travel in the SAME DIRECTION
The scatter graph looks like it's going up from left to right (as one increases, so does the other or as one decreases, so does the other)
Ex) The amount of time spent studying and their grades
2)NEGATIVE - the two sets of data travel in OPPOSITE DIRECTIONS
The scatter graph looks like it's going down from left to right (as one increases, the other decreases or vice versa)
Ex) As the temperature increases, the use of heaters decreases
3) NO CORRELATION - the two sets have no relationship to each other
The scatter graph is all over the place
Ex) Your math grade and the temperature outside
LINE OF BEST FIT: If the data has a correlation (either positive or negative), you will be able to draw a line that "fits" the data. It may not actually go through any of the specific data points, but it will be in the center of most of the data.
This line is a TREND LINE and can be used to make projections or analytical statements
Monday, April 12, 2010
Pre Algebra ( period 1)
Box & Whiskers 12-2
Statistics Unit
(We have covered: Mean, Median & Mode: Chapter 3-3
Frequency Tables and Line Plots: Chapter 12-1
Stem and Leaf Plots: p. 618-619)
Box & Whiskers
Uses 3 medians:
(1)the true median,
(2)the median of the upper data and
(3)the median of the lower data
The median divides the data into 2 parts:
The upper data half The lower data half
Now we will take the median of each of these halves.
The data is now separated into 4 quarters which are called QUARTILES
The median of the lower half is called the LOWER QUARTILE
The median of the upper half is called the UPPER QUARTILE
The lower median and the upper median will form the left side and right side of the BOX
The true median will be somewhere inside the box
The lowest value forms the lower WHISKER and the highest value forms the higher WHISKER
These are called the lower extreme and upper extreme
The box represents the MIDDLE 1/2 (50%) of the data
Circle Graphs
Page 471
Find the percent of the total data using the intervals from the histogram
Draw a circle and approximate the percentages of the total circle
Show you how to do it exactly with a protractor and 360 degrees
set up a proportion !!
SOMETHING A LITTLE DIFFERENT...
Trend Over Time & Line Graphs
LINE GRAPHS: p. 98-99
Data that takes place OVER TIME is represented well on a Line Graph.
For example, a student's math grade over the course of a year, or over middle school, or even over a single month.
The length of the time interval depends on what you're trying to show.
If you have more than one set of data over the same time period, you can make MULTIPLE LINES on the graph.
For example, a student's math grade, along with his science grade.
BAR GRAPHS: Data over time can also be shown as a Bar Graph, but Bar Graphs are especially useful to show data that compares either non-numeric data or comparing multiple types of the same data at the same time.
Non-numeric data: Really a Line Plot but without the "X's"
Allows you to represent very large data sets easily because you can label the y axis (vertical axis) in any way that you need to.
Numeric data: Going back to a student's grades on the Line Graph...Say you wanted to compare multiple students' math grades and science grades in 8th grade, you could make a DOUBLE BAR GRAPH with one bar color representing math grades and another bar color representing their science grades
Statistics Unit
(We have covered: Mean, Median & Mode: Chapter 3-3
Frequency Tables and Line Plots: Chapter 12-1
Stem and Leaf Plots: p. 618-619)
Box & Whiskers
Uses 3 medians:
(1)the true median,
(2)the median of the upper data and
(3)the median of the lower data
The median divides the data into 2 parts:
The upper data half The lower data half
Now we will take the median of each of these halves.
The data is now separated into 4 quarters which are called QUARTILES
The median of the lower half is called the LOWER QUARTILE
The median of the upper half is called the UPPER QUARTILE
The lower median and the upper median will form the left side and right side of the BOX
The true median will be somewhere inside the box
The lowest value forms the lower WHISKER and the highest value forms the higher WHISKER
These are called the lower extreme and upper extreme
The box represents the MIDDLE 1/2 (50%) of the data
Circle Graphs
Page 471
Find the percent of the total data using the intervals from the histogram
Draw a circle and approximate the percentages of the total circle
Show you how to do it exactly with a protractor and 360 degrees
set up a proportion !!
SOMETHING A LITTLE DIFFERENT...
Trend Over Time & Line Graphs
LINE GRAPHS: p. 98-99
Data that takes place OVER TIME is represented well on a Line Graph.
For example, a student's math grade over the course of a year, or over middle school, or even over a single month.
The length of the time interval depends on what you're trying to show.
If you have more than one set of data over the same time period, you can make MULTIPLE LINES on the graph.
For example, a student's math grade, along with his science grade.
BAR GRAPHS: Data over time can also be shown as a Bar Graph, but Bar Graphs are especially useful to show data that compares either non-numeric data or comparing multiple types of the same data at the same time.
Non-numeric data: Really a Line Plot but without the "X's"
Allows you to represent very large data sets easily because you can label the y axis (vertical axis) in any way that you need to.
Numeric data: Going back to a student's grades on the Line Graph...Say you wanted to compare multiple students' math grades and science grades in 8th grade, you could make a DOUBLE BAR GRAPH with one bar color representing math grades and another bar color representing their science grades
Algebra ( Period 4)
How about one more video..
let me know if you find one you think everyone in our class would like
Solving Rational Expressions: 13-5
YOU NEED TO GET RID OF ALL DENOMINATORS!
After you do that, you may end up with a Quadratic that you can solve using any of your methods.
Easy one first
(a+1)/2 = 1/a
You could find the LCM of the denominators which would be 2a but what about using cross products—as you did with proportions—last year.
then the above becomes
a(a + 1) = 2(1)
0r
a2 + a = 2
a2 + a - 2= 0 using ZERO products property
Factor
(a + 2)(a-1) = 0
so a = -2 and a= 1
When you MUST Find the LCM of the denominators:
MULTIPLY EACH & EVERY TERM ON BOTH SIDES BY THE LCM!
You should end up with NO DENOMINATORS!
(Otherwise, you don't have the right LCM!)
EXAMPLE:
4/x – 4/(x+2) = 1
In this case the LCM is discovered just by multiplying both of the existing denominators. It is as if they are relatively prime!!
The LCM is x(x + 2)
Multiply each term on BOTH sides by x(x + 2)
[x(x +2)4]/x – [4 x(x +2)]/(x+2) = 1 [x(x +2)]
Simplify the denominators with the numerators:
(x + 2) 4 - x(4) = x(x + 2)
SIMPLIFY more:
4x + 8 - 4x = x2 + 2x
USE THE ZERO PRODUCTS PROPERTY:
x2 + 2x - 8 = 0
FACTOR OR QUADRATIC FORMULA:
(x + 4)(x - 2) = 0
x = -4 and x = 2
Solving Radical Equations 13-6
THIS IS A REVIEW OF CHAPTER 11
You square both sides to get rid of the radical sign
For these problems, you'll get a quadratic on one side after you square
BE SURE TO CHECK BOTH ANSWERS TO MAKE SURE THEY BOTH CHECK!
x – 5 = SQRT (x+ 7)
squaring both sides gives us
(x -5)2 = [SQRT(x + 7)]2
x2 - 10x + 25 = x + 7
x2- 11x + 18 = 0
(x – 9)(x -2) = 0
or x = 9 and x = 2
BUT when you check you discover
x -5 = SQRT ( x + 7)
9 – 5 =?= SQRT (9 + 7)
4 = 4
BUT
x -5 = SQRT ( x + 7)
2 – 5 =?= SQRT (2 + 7)
-3 DOES NOT EQUAL 3
so only solution is x = 9
Solve
[SQRT ( 27 - 3x)] + 3 = x
move the 3 to the other side using transformations
[SQRT ( 27 - 3x)] = x -3
NOW squaring both sides
[SQRT ( 27 - 3x)]2 = (x -3)2
27 – 3x = x2 - 6x + 9
move everything to the right side – to use the ZERO PRODUCTS PROPERTY
0 = x2 - 3x – 18
0 = ( x – 6) (x +3)
so x = 6 or x = -3
WE MUST CHECK again:
Start with the ORIGINAL equation
[SQRT ( 27 - 3x)] + 3 = x
Plug in for x = 6
[SQRT ( 27 – 3(6))] + 3 =? = 6
[SQRT ( 27 - 18)] + 3 =? = 6
[SQRT ( 9)] + 3 =?= 6
3 + 3 = 6 YES
But we discover
[SQRT ( 27 - 3x)] + 3 = x
Plug in for x = -3
[SQRT ( 27 – 3(-3))] + 3 =?= -3
[SQRT ( 27 +9)] + 3 =?= -3
[SQRT ( 36)] + 3 =?= -3
6 + 3 DOES NOT EQUAL -3
so only x = 6 is the solution
let me know if you find one you think everyone in our class would like
Solving Rational Expressions: 13-5
YOU NEED TO GET RID OF ALL DENOMINATORS!
After you do that, you may end up with a Quadratic that you can solve using any of your methods.
Easy one first
(a+1)/2 = 1/a
You could find the LCM of the denominators which would be 2a but what about using cross products—as you did with proportions—last year.
then the above becomes
a(a + 1) = 2(1)
0r
a2 + a = 2
a2 + a - 2= 0 using ZERO products property
Factor
(a + 2)(a-1) = 0
so a = -2 and a= 1
When you MUST Find the LCM of the denominators:
MULTIPLY EACH & EVERY TERM ON BOTH SIDES BY THE LCM!
You should end up with NO DENOMINATORS!
(Otherwise, you don't have the right LCM!)
EXAMPLE:
4/x – 4/(x+2) = 1
In this case the LCM is discovered just by multiplying both of the existing denominators. It is as if they are relatively prime!!
The LCM is x(x + 2)
Multiply each term on BOTH sides by x(x + 2)
[x(x +2)4]/x – [4 x(x +2)]/(x+2) = 1 [x(x +2)]
Simplify the denominators with the numerators:
(x + 2) 4 - x(4) = x(x + 2)
SIMPLIFY more:
4x + 8 - 4x = x2 + 2x
USE THE ZERO PRODUCTS PROPERTY:
x2 + 2x - 8 = 0
FACTOR OR QUADRATIC FORMULA:
(x + 4)(x - 2) = 0
x = -4 and x = 2
Solving Radical Equations 13-6
THIS IS A REVIEW OF CHAPTER 11
You square both sides to get rid of the radical sign
For these problems, you'll get a quadratic on one side after you square
BE SURE TO CHECK BOTH ANSWERS TO MAKE SURE THEY BOTH CHECK!
x – 5 = SQRT (x+ 7)
squaring both sides gives us
(x -5)2 = [SQRT(x + 7)]2
x2 - 10x + 25 = x + 7
x2- 11x + 18 = 0
(x – 9)(x -2) = 0
or x = 9 and x = 2
BUT when you check you discover
x -5 = SQRT ( x + 7)
9 – 5 =?= SQRT (9 + 7)
4 = 4
BUT
x -5 = SQRT ( x + 7)
2 – 5 =?= SQRT (2 + 7)
-3 DOES NOT EQUAL 3
so only solution is x = 9
Solve
[SQRT ( 27 - 3x)] + 3 = x
move the 3 to the other side using transformations
[SQRT ( 27 - 3x)] = x -3
NOW squaring both sides
[SQRT ( 27 - 3x)]2 = (x -3)2
27 – 3x = x2 - 6x + 9
move everything to the right side – to use the ZERO PRODUCTS PROPERTY
0 = x2 - 3x – 18
0 = ( x – 6) (x +3)
so x = 6 or x = -3
WE MUST CHECK again:
Start with the ORIGINAL equation
[SQRT ( 27 - 3x)] + 3 = x
Plug in for x = 6
[SQRT ( 27 – 3(6))] + 3 =? = 6
[SQRT ( 27 - 18)] + 3 =? = 6
[SQRT ( 9)] + 3 =?= 6
3 + 3 = 6 YES
But we discover
[SQRT ( 27 - 3x)] + 3 = x
Plug in for x = -3
[SQRT ( 27 – 3(-3))] + 3 =?= -3
[SQRT ( 27 +9)] + 3 =?= -3
[SQRT ( 36)] + 3 =?= -3
6 + 3 DOES NOT EQUAL -3
so only x = 6 is the solution
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