Chapter 12-1 and 12-2
RELATIONS: Set of ordered pairs where the x values are the DOMAIN and the y values are the RANGE.
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.
In the real world, there are excellent examples....pizza prices.
A restaurant 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.
Any line, y = mx + b, is a function.
INPUTS: x values
OUTPUTS: y values
f(x) means the value of the function at the given x value
You can think of f(x) as the y value
Finding the value of a function: Plug it in, plug it in!
f(x) = 2x + 7
Find f(3)
f(3) = 2(3) + 7 = 13
The function notation gives you more information than using y
If I tell you y = 13 you have no idea what the x value was at that point
But if I tell you f(3) = 13, you know the entire coordinate (3, 13)
Domain of a function = all possible x values (inputs) that keep the solution real
Range of a function = all possible y values (outputs) that result from the domain
EXAMPLE:
f(x) = x + 10 has the domain of all real numbers and the same range because every value will keep the answer f(x) a real number
EXAMPLE:
f(x) = x2 has the domain again of all real numbers,
BUT the range is greater than or = to zero
because when a number is squared it will never be negative!
So f(x) will always be 0 or positive
EXAMPLE:
f(x) = absolute value of x has the domain of all real numbers, but again the range will be greater than or equal to zero because absolute value will never be negative
EXAMPLE:
f(x) = 1/x has a domain of all real numbers EXCEPT FOR ZERO because it would be undefined if zero was in the denominator.
The range is all real numbers except zero as well.
This function will approach both axes but never intersect with them.
The axes are called asymptotes which means that they will get very close but never reach them
EXAMPLE:
f(x) = x - 10/x + 3
Domain is all real numbers EXCEPT -3 because -3 will turn the denominator into zero (undefined)
What is the range?
Sunday, March 29, 2009
Thursday, March 26, 2009
Math 6 H Periods 1, 6 & 7
Ratio 7-6 Word Problems
Since all of the word problems from Ratios 7-6 are excellent examples of using Ratios, I thought I would create a blog posting with the solutions. We did all of the even numbered ones in class and went over the answers to the odd numbered ones—as they were assigned for homework. Although I will list the actually word problems, you might need to turn to the textbook and page 229 for any charts included.
Read each problem carefully and set up the solutions based on the question asked
Page 229
1. What is the cost of grapes in dollars per kilogram if 4.5 kg of grapes costs $ 7.56?
Since they want $/kg you need to have 7.56/ 4.5 = 1.68; so the grapes cost $ 1.68/kg.
2. This one looked confusing, but just read it a few times: The index of refraction of a transparent substance is the ratio of the speed of light in space to the speed of light in the substance. (Read it again—it is just a ratio). Using the table on Page 229, find the index of refraction of
A. Glass
B. Water
According to the table Speed of light in space is 300,000 km/sec
Speed of light in glass is 200,000 km/ sec.
So
300,000
200,000
or 3/2
Speed of Light in Water is 225,000
so that ratio is
300,000
225,000
This takes a little more time dividing but you can simplify it to 4/3 Since it is a ratio you want to LEAVE it as an improper fraction.
3. The mechanical advantage of a simple machine is the ratio of the weight lifted by the machine to the force necessary to lift it. Now—just read that again—you don’t need to truly understand physics to get this problem. It is a RATIO again!! What is the mechanical advantage of a jack that lifts a 3200-pound car with a force of 120 pounds?
3200 =
120
80/3; so the mechanical advantage is 80/3
4. The C-String of a cello vibrates 654 times in 5 seconds, How many vibrations per second is that? (We are finding unit rate)
It’s a rate of time/sec.
654/ 5 = 130 4/5 vibrations per second
5. A four cubic foot volume of water at sea level weighs 250 lb. What is the density of water in pounds per cubic foot? (Do you need to understand density to do this problem?) No—just unit rates!! We want lbs/cubic foot. Look at the information and place it with lbs/ft and then divide to get the unit rate.
250lb/ 4 ft3. That equals 62 1/2 lb/ft3 .
6. A share of stock that costs $88 earned $16 last year. What was the price-to-earnings ratio of this stock? (Wish that was happening now!!)
Price/earnings so look carefully 88/16 = 11/2 so the Ratio is 11/2
In exercises 7 and 8 you will need to look at the diagrams on page 230. You are finding the ratio in lowest terms.
Look at both the small and large triangles. For A you are finding the ratio between segment AB of the larger triangle and DE of the smaller triangle.
4.8
3.2
Simplify ( I like to clear decimals to make it easier but in this case you could easily divide
= 3/2
B.
Perimeter of Triangle ABC
Perimeter of Triangle DEF
First you need to find the perimeter of the larger triangle by adding up the three sides
6.3 + 4.8 + 3 = 14.1
and then finding the perimeter of the smaller triangle by adding up those three sides
4.2 + 3.2 + 2 = 9.4
14.1
9.4
I like to clear decimals by multiplying by 10
141
94
= 3/2
What did you notice?
8. Again look at Page 230 for the figure
We are finding the following Ratios:
PQ:TU 2.6:6.5 or 26:65 = 2:5
QR:UV 1.2:3 = 12:30 = 2:5
Perimeter of PQRS: Perimeter of TUVW
You need to add up all the sides of each of the figures
2.2 + 2.6 + 1.2 + 3 = 9
and 5.5 + 6.5 + 3 + 7.5 = 22.5
9:22.5 = 90:225 = 2:5
What did you notice here?
For exercised 9-12 you need to use the Table on Page 230 as well
9. The population of Centerville in 1980 to its population in 1970?
44/36 = 11/9
10. The growth in the population of Easton to its 1980 population.
You need to subtract the 1970 population from the 1980 population to get the growth so
28-16 = 12 12/18 = 3/7
11. The total population of both towns in 1970 to their total population in 1980
36 + 16 = 52 and 44 + 28 = 72 52/72 = 13/ 18
12. The total growth in the population of both towns to their total 1980 population
Like number 10 you need to subtract to find the growth
(44-36) + (28-16) = 8 + 12 = 20
44 + 28 = 72 (from before)
20/72 = 5/18
13. During a season a baseball player hit safely in 135times at bat; the player struck out or was fielded out in 340 times at bat. What is the player’s ratio of hits to times at bat?
You take the 135 + 340 = 475 (the total times at bat)
135/475 = 37/95
14. A fruit drink recipe requires fruit juice and milk in the ration 3:5. What fraction of the drink is milk? What fraction is juice? Remember we need to find the total first. 3 + 5 = 8
so the fraction that is milk is 5/8 and the fraction that is fruit juice is 3/8
Since all of the word problems from Ratios 7-6 are excellent examples of using Ratios, I thought I would create a blog posting with the solutions. We did all of the even numbered ones in class and went over the answers to the odd numbered ones—as they were assigned for homework. Although I will list the actually word problems, you might need to turn to the textbook and page 229 for any charts included.
Read each problem carefully and set up the solutions based on the question asked
Page 229
1. What is the cost of grapes in dollars per kilogram if 4.5 kg of grapes costs $ 7.56?
Since they want $/kg you need to have 7.56/ 4.5 = 1.68; so the grapes cost $ 1.68/kg.
2. This one looked confusing, but just read it a few times: The index of refraction of a transparent substance is the ratio of the speed of light in space to the speed of light in the substance. (Read it again—it is just a ratio). Using the table on Page 229, find the index of refraction of
A. Glass
B. Water
According to the table Speed of light in space is 300,000 km/sec
Speed of light in glass is 200,000 km/ sec.
So
300,000
200,000
or 3/2
Speed of Light in Water is 225,000
so that ratio is
300,000
225,000
This takes a little more time dividing but you can simplify it to 4/3 Since it is a ratio you want to LEAVE it as an improper fraction.
3. The mechanical advantage of a simple machine is the ratio of the weight lifted by the machine to the force necessary to lift it. Now—just read that again—you don’t need to truly understand physics to get this problem. It is a RATIO again!! What is the mechanical advantage of a jack that lifts a 3200-pound car with a force of 120 pounds?
3200 =
120
80/3; so the mechanical advantage is 80/3
4. The C-String of a cello vibrates 654 times in 5 seconds, How many vibrations per second is that? (We are finding unit rate)
It’s a rate of time/sec.
654/ 5 = 130 4/5 vibrations per second
5. A four cubic foot volume of water at sea level weighs 250 lb. What is the density of water in pounds per cubic foot? (Do you need to understand density to do this problem?) No—just unit rates!! We want lbs/cubic foot. Look at the information and place it with lbs/ft and then divide to get the unit rate.
250lb/ 4 ft3. That equals 62 1/2 lb/ft3 .
6. A share of stock that costs $88 earned $16 last year. What was the price-to-earnings ratio of this stock? (Wish that was happening now!!)
Price/earnings so look carefully 88/16 = 11/2 so the Ratio is 11/2
In exercises 7 and 8 you will need to look at the diagrams on page 230. You are finding the ratio in lowest terms.
Look at both the small and large triangles. For A you are finding the ratio between segment AB of the larger triangle and DE of the smaller triangle.
4.8
3.2
Simplify ( I like to clear decimals to make it easier but in this case you could easily divide
= 3/2
B.
Perimeter of Triangle ABC
Perimeter of Triangle DEF
First you need to find the perimeter of the larger triangle by adding up the three sides
6.3 + 4.8 + 3 = 14.1
and then finding the perimeter of the smaller triangle by adding up those three sides
4.2 + 3.2 + 2 = 9.4
14.1
9.4
I like to clear decimals by multiplying by 10
141
94
= 3/2
What did you notice?
8. Again look at Page 230 for the figure
We are finding the following Ratios:
PQ:TU 2.6:6.5 or 26:65 = 2:5
QR:UV 1.2:3 = 12:30 = 2:5
Perimeter of PQRS: Perimeter of TUVW
You need to add up all the sides of each of the figures
2.2 + 2.6 + 1.2 + 3 = 9
and 5.5 + 6.5 + 3 + 7.5 = 22.5
9:22.5 = 90:225 = 2:5
What did you notice here?
For exercised 9-12 you need to use the Table on Page 230 as well
9. The population of Centerville in 1980 to its population in 1970?
44/36 = 11/9
10. The growth in the population of Easton to its 1980 population.
You need to subtract the 1970 population from the 1980 population to get the growth so
28-16 = 12 12/18 = 3/7
11. The total population of both towns in 1970 to their total population in 1980
36 + 16 = 52 and 44 + 28 = 72 52/72 = 13/ 18
12. The total growth in the population of both towns to their total 1980 population
Like number 10 you need to subtract to find the growth
(44-36) + (28-16) = 8 + 12 = 20
44 + 28 = 72 (from before)
20/72 = 5/18
13. During a season a baseball player hit safely in 135times at bat; the player struck out or was fielded out in 340 times at bat. What is the player’s ratio of hits to times at bat?
You take the 135 + 340 = 475 (the total times at bat)
135/475 = 37/95
14. A fruit drink recipe requires fruit juice and milk in the ration 3:5. What fraction of the drink is milk? What fraction is juice? Remember we need to find the total first. 3 + 5 = 8
so the fraction that is milk is 5/8 and the fraction that is fruit juice is 3/8
Wednesday, March 25, 2009
Pre Algebra Period 2 (Tuesday)
EXPONENTS 4-7 & 4-8
MULTIPLYING Powers with LIKE BASES:
Simply ADD THE POWERS
WITH VARIABLES:
m5m3 = m8
You can check this by EXPANDING:
(mmmmm)(mmm) = m8
WITH NUMBERS:
(25)(23) = 28
DIVIDING Powers with LIKE BASES:
Simply SUBTRACT the POWERS
m8 = m3
m5
Again, you can check this by EXPANDING:
mmmmmmmm
mmmmm
ZERO POWERS:
Anything to the zero power = 1
(except zero to the zero power is undefined or indeterminate)
Proof of this was given in class:
1 =
mmmmmmmm
mmmmmmmm
=
m8 = m0 (by power rules for division)
m8
: 1 = m0
By the transitive property of equality
MULTIPLYING Powers with LIKE BASES:
Simply ADD THE POWERS
WITH VARIABLES:
m5m3 = m8
You can check this by EXPANDING:
(mmmmm)(mmm) = m8
WITH NUMBERS:
(25)(23) = 28
DIVIDING Powers with LIKE BASES:
Simply SUBTRACT the POWERS
m8 = m3
m5
Again, you can check this by EXPANDING:
mmmmmmmm
mmmmm
ZERO POWERS:
Anything to the zero power = 1
(except zero to the zero power is undefined or indeterminate)
Proof of this was given in class:
1 =
mmmmmmmm
mmmmmmmm
=
m8 = m0 (by power rules for division)
m8
: 1 = m0
By the transitive property of equality
Tuesday, March 24, 2009
Math 6 H Periods 1, 6 & 7 (Tuesday)
Ratios 7-6
In our textbook, the example given involves the number of students --at what I called a mythical middle school --as well as the number of teachers. There are 35 teachers and 525 students. We can compare the number of teachers to the number of students by writing a quotient
=
number of teachers
number of students
35
525
1/15
The quotient of one number divided by a second number is called the ratio of the first number to the second number.
We can write a ratio in the following ways:
1/15 OR 1:15 OR 1 to 15
All of these expressions are read one to fifteen.
If the colon notation is used the first number is divided by the second. A ratio is said to be lowest terms if the two numbers are “relatively prime.”
You do not change an improper fraction to a mixed number if the improper fraction represents a ratio
There are 9 players on a baseball team. Four of these are infielders and 3 are outfielders. Find each ratio in lowest terms.
a. infielders to outfielders
b. outfields to total players
# of infielders
# of outfielders
= 4/3 or 4:3 or 4 to 3
# of outfielders
# total of players
= 3/9 = 1/3 or 1:3 or 1 to 3
Some ratios compare measurements. In these cases we must be sure the measurements are expressed in the same units
It takes Nick 4 minutes to mix some paint for his science project. It takes him 3 hours to complete painting his science project. What is the ratio of the time it takes Nick to mix the paint to the time it takes Nick to paint his project?
Use minutes as a common unit for measuring time. You must convert the hours to minutes first
3h = 3 • 60min = 180 min
The ratio is :
min. to mix
min. to paint
= 4/180 = 1/45 or 1:45
Some ratios are in the form
40 miles per hour or 5 pencils for a dollar
“ I want my… I want my…. I want my … MPG!!”
These ratios involve quantities of different kinds and are called rates. Rates may be expressed as decimals or mixed numbers. Rates should be simplified to a per unit form. When a rate is expressed in a per unit form, such a rate is often called a unit rate.
Aubrey’s dad’s car went 258miles on 12 gallons of gas. Express the rate of fuel consumption in miles per gallon.
The rate of fuel consumption is
258 miles
12 gallons
= 21 1/2 miles per gallon
Some of the most common units in which rates are given are the following:
mi/gal or mpg miles per gallon
mi/h or mph miles per hour
km/L kilometers per liter
km/h kilometers per hour
In our textbook, the example given involves the number of students --at what I called a mythical middle school --as well as the number of teachers. There are 35 teachers and 525 students. We can compare the number of teachers to the number of students by writing a quotient
=
number of teachers
number of students
35
525
1/15
The quotient of one number divided by a second number is called the ratio of the first number to the second number.
We can write a ratio in the following ways:
1/15 OR 1:15 OR 1 to 15
All of these expressions are read one to fifteen.
If the colon notation is used the first number is divided by the second. A ratio is said to be lowest terms if the two numbers are “relatively prime.”
You do not change an improper fraction to a mixed number if the improper fraction represents a ratio
There are 9 players on a baseball team. Four of these are infielders and 3 are outfielders. Find each ratio in lowest terms.
a. infielders to outfielders
b. outfields to total players
# of infielders
# of outfielders
= 4/3 or 4:3 or 4 to 3
# of outfielders
# total of players
= 3/9 = 1/3 or 1:3 or 1 to 3
Some ratios compare measurements. In these cases we must be sure the measurements are expressed in the same units
It takes Nick 4 minutes to mix some paint for his science project. It takes him 3 hours to complete painting his science project. What is the ratio of the time it takes Nick to mix the paint to the time it takes Nick to paint his project?
Use minutes as a common unit for measuring time. You must convert the hours to minutes first
3h = 3 • 60min = 180 min
The ratio is :
min. to mix
min. to paint
= 4/180 = 1/45 or 1:45
Some ratios are in the form
40 miles per hour or 5 pencils for a dollar
“ I want my… I want my…. I want my … MPG!!”
These ratios involve quantities of different kinds and are called rates. Rates may be expressed as decimals or mixed numbers. Rates should be simplified to a per unit form. When a rate is expressed in a per unit form, such a rate is often called a unit rate.
Aubrey’s dad’s car went 258miles on 12 gallons of gas. Express the rate of fuel consumption in miles per gallon.
The rate of fuel consumption is
258 miles
12 gallons
= 21 1/2 miles per gallon
Some of the most common units in which rates are given are the following:
mi/gal or mpg miles per gallon
mi/h or mph miles per hour
km/L kilometers per liter
km/h kilometers per hour
Saturday, March 21, 2009
Math 6H (Friday)
Word Problems Review
Remember to use the following plan for solving word problems.
5-Step Plan
for Solving Word Problems
1. Read the problem carefully. Make sure you understand what it says. You may need to read it more than once.
2. Use questions like these in planning the solution
a. What is asked for?
b. What facts are given?
c. Are enough facts given?
If not, what else is needed?
d. Are unnecessary facts given?
e. Will a sketch or diagram help?
3.Determine which operation or operations can be used to solve the problem.
4.Carry out the operations carefully.
5.Check your results with the facts given in the problem. Label your answer.
Remember to use the following plan for solving word problems.
5-Step Plan
for Solving Word Problems
1. Read the problem carefully. Make sure you understand what it says. You may need to read it more than once.
2. Use questions like these in planning the solution
a. What is asked for?
b. What facts are given?
c. Are enough facts given?
If not, what else is needed?
d. Are unnecessary facts given?
e. Will a sketch or diagram help?
3.Determine which operation or operations can be used to solve the problem.
4.Carry out the operations carefully.
5.Check your results with the facts given in the problem. Label your answer.
Friday, March 20, 2009
Algebra Period 3 (Thursday & Friday)
USING THE PYTHAGOREAN THEOREM - WORD PROBLEMS: 11-8
There are many real life examples where you can use the Pythagorean Theorem to find a length.
EXAMPLE: HOW HIGH A 10 FOOT LADDER REACHES ON A HOUSE
A 10 ft ladder is placed on a house 5 ft away from the base of the house.
Find how high up the house the ladder reaches.
The ladder makes a right triangle with the ground being one leg, the house the other, and the ladder is the hypotenuse ( see drawing in #1 on p. 515)
You need to find the distance on the house, so you're finding one leg.
ANOTHER EXAMPLE:
You're flying your kite for the kite project and you want to know how long the kite string must be so that it can reach a height of 13 ft in the air if you're standing 9 feet away from where the kite is in the air.
The string represents the hypotenuse.
You know one leg is the height in the air (13 ft) and the other leg is how far on the ground you are standing away from where the kite is flying (9 ft)
You need to find the hypotenuse.
EQUATIONS WITH RADICALS: 11-9
When you have an equation where the variable is under the √ sign,
simply square both sides to solve for the variable.
This is actually similar to regular equation balancing!
The goal is still to find out what the variable is.
But to find it, you need to ISOLATE the √ with the variable first
Then you square
Example #1 on p. 521
√x = 5
Square both sides and you'll get x = 25
Example # 9 is much harder
3 + √(x - 1) = 5
move the 3 to other side first √(x - 1) = 5 - 3
Now square both sides x - 1 = 22
Now add 1 to both sides: x = 4 + 1 x = 5
Look at Examples #15 & 16 ---- In both cases, there is no possible value for x because the square root of a number CAN NEVER BE NEGATIVE IN THE REAL NUMBER SYSTEM!
Try Example #17 yourself, and see what happens (you should also end up with no value, but why?)
Need to review how to estimate the value of square roots--Check out the blog post on March 16th.
There are many real life examples where you can use the Pythagorean Theorem to find a length.
EXAMPLE: HOW HIGH A 10 FOOT LADDER REACHES ON A HOUSE
A 10 ft ladder is placed on a house 5 ft away from the base of the house.
Find how high up the house the ladder reaches.
The ladder makes a right triangle with the ground being one leg, the house the other, and the ladder is the hypotenuse ( see drawing in #1 on p. 515)
You need to find the distance on the house, so you're finding one leg.
ANOTHER EXAMPLE:
You're flying your kite for the kite project and you want to know how long the kite string must be so that it can reach a height of 13 ft in the air if you're standing 9 feet away from where the kite is in the air.
The string represents the hypotenuse.
You know one leg is the height in the air (13 ft) and the other leg is how far on the ground you are standing away from where the kite is flying (9 ft)
You need to find the hypotenuse.
EQUATIONS WITH RADICALS: 11-9
When you have an equation where the variable is under the √ sign,
simply square both sides to solve for the variable.
This is actually similar to regular equation balancing!
The goal is still to find out what the variable is.
But to find it, you need to ISOLATE the √ with the variable first
Then you square
Example #1 on p. 521
√x = 5
Square both sides and you'll get x = 25
Example # 9 is much harder
3 + √(x - 1) = 5
move the 3 to other side first √(x - 1) = 5 - 3
Now square both sides x - 1 = 22
Now add 1 to both sides: x = 4 + 1 x = 5
Look at Examples #15 & 16 ---- In both cases, there is no possible value for x because the square root of a number CAN NEVER BE NEGATIVE IN THE REAL NUMBER SYSTEM!
Try Example #17 yourself, and see what happens (you should also end up with no value, but why?)
Need to review how to estimate the value of square roots--Check out the blog post on March 16th.
Thursday, March 19, 2009
Math 6 Honors Periods 6 & 7 (Thursday)
Multiplication and Division of Mixed Numbers 7-5
One method of finding the product of two mixed numbers is to first change the mixed numbers into improper fractions and then multiply
6 X 3 1/12 becomes
6/1 X 37/12
Simplify using the methods taught and reviewed this week
6/1 X 37/ 12 = 37/6 = 18 1/2
5 3/4 X 4 2/3
23/4 X 14/3 becomes
23/2 X 7/ 3 = 161/6 = 26 5/6
To divide one mixed number by another, we change the mixed numbers into improper fractions and use the methods from the previous lessons
2 2/3 ÷ 10 2/3 first change to improper fractions
8/3 ÷ 32/3 Remember. that dividing by a fraction is the same a multiplying by its reciprocal
8/3 X 3/32
Using the GCF to simplify first
1/1 X 1/4 = 1/4
What about
1 7/15 ÷ 5 1/2
22/15 ÷ 11/2
22/15 X 2/11 = 4/15
What if we have equations such as the book shows
n X 2 1/3 = 6 5/12
First rewrite it as (2 1/3)n = 6 5/12
THen change the mixed numbers to improper fractions
(7/3)n = 77/12
Now, we know to isolate the variable, we must use the inverse operation and in this case we would multiply 7/3 by its reciprocal 3/7 to both sides
(3/7)(7/3) n = (77/12)(3/7)
n = 11/4
n = 2 3/4
One method of finding the product of two mixed numbers is to first change the mixed numbers into improper fractions and then multiply
6 X 3 1/12 becomes
6/1 X 37/12
Simplify using the methods taught and reviewed this week
6/1 X 37/ 12 = 37/6 = 18 1/2
5 3/4 X 4 2/3
23/4 X 14/3 becomes
23/2 X 7/ 3 = 161/6 = 26 5/6
To divide one mixed number by another, we change the mixed numbers into improper fractions and use the methods from the previous lessons
2 2/3 ÷ 10 2/3 first change to improper fractions
8/3 ÷ 32/3 Remember. that dividing by a fraction is the same a multiplying by its reciprocal
8/3 X 3/32
Using the GCF to simplify first
1/1 X 1/4 = 1/4
What about
1 7/15 ÷ 5 1/2
22/15 ÷ 11/2
22/15 X 2/11 = 4/15
What if we have equations such as the book shows
n X 2 1/3 = 6 5/12
First rewrite it as (2 1/3)n = 6 5/12
THen change the mixed numbers to improper fractions
(7/3)n = 77/12
Now, we know to isolate the variable, we must use the inverse operation and in this case we would multiply 7/3 by its reciprocal 3/7 to both sides
(3/7)(7/3) n = (77/12)(3/7)
n = 11/4
n = 2 3/4
Wednesday, March 18, 2009
Algebra Period 3 (Wednesday)
PYTHAGOREAN THEOREM 11-7
(an old friend) -
FOR RIGHT TRIANGLES ONLY!
2 legs - make the right angle - called ‘a’ and ‘b’
(doesn't matter which is which because you will add them and adding is COMMUTATIVE!)
hypotenuse - longest side across from the right angle - called ‘c’
You can find the third side of a right triangle as long as you know the other two sides:
a2 + b2 = c2
After squaring the two sides that you know, you'll need to find the square root of that number to find the length of the missing side (that's why it's in this chapter!)
EASIEST - FIND THE HYPOTENUSE (c)
Example #1 from p. 510
82 + 152 = c2
64 + 225 = c2
289 = c2
c = 17
A LITTLE HARDER - FIND A MISSING LEG (Either a or b)
Example #5 from p. 510
52 + b2 = 132
25 + b2 = 169
b2 = 169 - 25
b2 = 144
b = 12
ONE THAT YOU WOULDN'T HAVE HAD IN PRE-ALGEBRA:
One of the legs = √5
√52 + b2 = 132
5 + b2 = 169
b2 = 169 - 5
b2 = 164
√ b2 = √164
b = √4•41
b = 2√41
DISTANCE FORMULA
(based on the Pythagorean Theorem):
see p. 513 in book
The distance between any two points on the coordinate plane (x y plane)
The distance is the hypotenuse of a right triangle that you can draw using any two points on the coordinate plane (I'll show you how to draw it in class).
The formula is:
distance = √[( difference of the two x's)2 + (difference of the two y's)2]
distance = √(x1-x2)2 + (y1- y2)2
The difference between the 2x’s is the length of the leg parallel to the y axis and
the difference between the 2y’s is the length of the leg parallel to the x axis
The difference of the two x's is length of one of the two legs
The difference of the two y's is the length of the other leg
The distance is the length of the hypotenuse
So you could actually rewrite the distance formula to look like the Pythagorean Theorem:
c = √[a2 + b2]
EXAMPLE: What is the distance between (3, -10) and (-7, -2)?
d = √[(3 - -7)2 + (-10 - -2)2]
d = √[102 +( -82)]
d = √(164)
Simplifying:
2√41
Check out the following website- with its 81 different proofs
Pythagorean Theorem and its many proofs. See if you can find the proof that is attributed to one of our US Presidents
Practice your skills with the Pythagorean Theorem using this Shodor Interactive Site at
Pythagorean Explorer
(an old friend) -
FOR RIGHT TRIANGLES ONLY!
2 legs - make the right angle - called ‘a’ and ‘b’
(doesn't matter which is which because you will add them and adding is COMMUTATIVE!)
hypotenuse - longest side across from the right angle - called ‘c’
You can find the third side of a right triangle as long as you know the other two sides:
a2 + b2 = c2
After squaring the two sides that you know, you'll need to find the square root of that number to find the length of the missing side (that's why it's in this chapter!)
EASIEST - FIND THE HYPOTENUSE (c)
Example #1 from p. 510
82 + 152 = c2
64 + 225 = c2
289 = c2
c = 17
A LITTLE HARDER - FIND A MISSING LEG (Either a or b)
Example #5 from p. 510
52 + b2 = 132
25 + b2 = 169
b2 = 169 - 25
b2 = 144
b = 12
ONE THAT YOU WOULDN'T HAVE HAD IN PRE-ALGEBRA:
One of the legs = √5
√52 + b2 = 132
5 + b2 = 169
b2 = 169 - 5
b2 = 164
√ b2 = √164
b = √4•41
b = 2√41
DISTANCE FORMULA
(based on the Pythagorean Theorem):
see p. 513 in book
The distance between any two points on the coordinate plane (x y plane)
The distance is the hypotenuse of a right triangle that you can draw using any two points on the coordinate plane (I'll show you how to draw it in class).
The formula is:
distance = √[( difference of the two x's)2 + (difference of the two y's)2]
distance = √(x1-x2)2 + (y1- y2)2
The difference between the 2x’s is the length of the leg parallel to the y axis and
the difference between the 2y’s is the length of the leg parallel to the x axis
The difference of the two x's is length of one of the two legs
The difference of the two y's is the length of the other leg
The distance is the length of the hypotenuse
So you could actually rewrite the distance formula to look like the Pythagorean Theorem:
c = √[a2 + b2]
EXAMPLE: What is the distance between (3, -10) and (-7, -2)?
d = √[(3 - -7)2 + (-10 - -2)2]
d = √[102 +( -82)]
d = √(164)
Simplifying:
2√41
Check out the following website- with its 81 different proofs
Pythagorean Theorem and its many proofs. See if you can find the proof that is attributed to one of our US Presidents
Practice your skills with the Pythagorean Theorem using this Shodor Interactive Site at
Pythagorean Explorer
Math 6 Honors Periods 6 & 7 (Tuesday)
Division of Fractions 7-4
Certain numbers when multiplied together have the product 1
5 X 1/5 = 1
3/4 X 4/3 = 1
Two numbers whose product is 1 are called reciprocals of each other.
Thus 3/4 is the reciprocal of 4/3.
Zero does not have a reciprocal
Look at the following:
We know 18 = 3 X 6 and we know 18 ÷ 6 = 3 as well as 18 X 1/6 = 3
Dividing a number by a fraction is the same as multiplying the number by the RECIPROCAL of the fraction
a/b ÷ c/d = a/b ÷ d/c
Remember- you are using the reciprocal of the divisor... that is , as students want to say "You FLIP the 2nd number!!"
42/ 55 ÷ 36/11
you must rewrite the problem using the reciprocal of the 2nd number
42/55 X 11/36
Now using your skills of observing GCF simplify before you multiply ( MUCH EASIER and FASTER)
42/ 5 X 1/36 which becomes 7/5 X 1/ 6 = 7/30
Certain numbers when multiplied together have the product 1
5 X 1/5 = 1
3/4 X 4/3 = 1
Two numbers whose product is 1 are called reciprocals of each other.
Thus 3/4 is the reciprocal of 4/3.
Zero does not have a reciprocal
Look at the following:
We know 18 = 3 X 6 and we know 18 ÷ 6 = 3 as well as 18 X 1/6 = 3
Dividing a number by a fraction is the same as multiplying the number by the RECIPROCAL of the fraction
a/b ÷ c/d = a/b ÷ d/c
Remember- you are using the reciprocal of the divisor... that is , as students want to say "You FLIP the 2nd number!!"
42/ 55 ÷ 36/11
you must rewrite the problem using the reciprocal of the 2nd number
42/55 X 11/36
Now using your skills of observing GCF simplify before you multiply ( MUCH EASIER and FASTER)
42/ 5 X 1/36 which becomes 7/5 X 1/ 6 = 7/30
Tuesday, March 17, 2009
Algebra Period 3 (Tuesday)
ADDING AND SUBTRACTING RADICALS 11-6
Radicals function like variables, so you can only COMBINE LIKE RADICALS!
You cannot add √2 to √3!!!!
However, you may add 3√2 to 5√2 and get 8√2:
(3 + 5)√2 = 8√2
Make sure you simplify all radical expressions before trying to combine them!
Sometimes, it looks like they are not like radicands, but then after simplifying they are.
EXAMPLE #34 from p. 505:
√( x2y) + √( 4x2y) + √(9y) - √( y3)
Simplify each term first!!!!!!!!
x√y + 2x√y + 3√y - y√ y
NOW THEY ARE ALL LIKE TERMS BECAUSE ALL HAVE SQRT y!
(x + 2x + 3 - y)√y
(3x - y + 3)√y (final simplified answer)
Radicals function like variables, so you can only COMBINE LIKE RADICALS!
You cannot add √2 to √3!!!!
However, you may add 3√2 to 5√2 and get 8√2:
(3 + 5)√2 = 8√2
Make sure you simplify all radical expressions before trying to combine them!
Sometimes, it looks like they are not like radicands, but then after simplifying they are.
EXAMPLE #34 from p. 505:
√( x2y) + √( 4x2y) + √(9y) - √( y3)
Simplify each term first!!!!!!!!
x√y + 2x√y + 3√y - y√ y
NOW THEY ARE ALL LIKE TERMS BECAUSE ALL HAVE SQRT y!
(x + 2x + 3 - y)√y
(3x - y + 3)√y (final simplified answer)
Math 6 H Periods 1, 6 & 7 (Tuesday)
Multiplication of Fractions 7-3
If a rectangle is divided into 4 equal parts, each part is ¼ of the whole. If each of these parts is then divided into 3 parts, that is into thirds, then there are 12 equal parts and each is 1/(3 ∙4) or 1/12 of the whole.
That is 1/3 of 1/4 is 1/(3 ∙4) or 1/12 and 1/3 ∙ 1/4 = 1/12 is
so another example 2/3 of 4/5 is 2∙4 /(3∙8) or 2/3 ∙4/5 = 8/15
Notice, that the numerator of the product, 8, is the product of the numerators 2 and 4. The denominator of the product, 15, is the product of the denominators 3 and 5
Rule
If a, b, c, and d are whole numbers with , then
a/b(c/d) = a∙c/(b∙d)
When multiplying two fractions, you can simplify the multiplication by dividing either of the numerators and either of the denominators by common factors
6/35 ( 7/3) we can simplify first because both 6 and 3 are divisible by 3
2/35 (7/1) and then both 35 and 7 are divisible by 7 so 2/5 (1(1) = 2/5
Try the following
25/6 ( 42/5) What can we do there?
7/8(20/21) How about with these two sets of fractions?
19/20 ( 25/38) … and these fractions?
If a rectangle is divided into 4 equal parts, each part is ¼ of the whole. If each of these parts is then divided into 3 parts, that is into thirds, then there are 12 equal parts and each is 1/(3 ∙4) or 1/12 of the whole.
That is 1/3 of 1/4 is 1/(3 ∙4) or 1/12 and 1/3 ∙ 1/4 = 1/12 is
so another example 2/3 of 4/5 is 2∙4 /(3∙8) or 2/3 ∙4/5 = 8/15
Notice, that the numerator of the product, 8, is the product of the numerators 2 and 4. The denominator of the product, 15, is the product of the denominators 3 and 5
Rule
If a, b, c, and d are whole numbers with , then
a/b(c/d) = a∙c/(b∙d)
When multiplying two fractions, you can simplify the multiplication by dividing either of the numerators and either of the denominators by common factors
6/35 ( 7/3) we can simplify first because both 6 and 3 are divisible by 3
2/35 (7/1) and then both 35 and 7 are divisible by 7 so 2/5 (1(1) = 2/5
Try the following
25/6 ( 42/5) What can we do there?
7/8(20/21) How about with these two sets of fractions?
19/20 ( 25/38) … and these fractions?
Monday, March 16, 2009
Algebra Period 3 (Finding SQ RTS)
FINDING SQUARE ROOTS
Many textbooks seem to think that since calculators can find square roots, that students don't need to learn how to find square roots using any pencil-and-paper method. But learning at least the "guess and check" method for finding the square root will actually help the student UNDERSTAND and remember the square root concept itself!
Practice at least the first method presented here. This method, "guess and check", actually works around what the square root is all about.
The square root of a number is just the number which when multiplied by itself gives the first number. So 2 is the square root of 4 because 2 * 2 = 4.
Method 1: Guess, Divide & Check
Start with the number you want to find the square root of. Let's use 12. There are three steps:
1. Guess
2. Divide
3. Average.
... and then just keep repeating steps 2 and 3.
First, start by guessing a square root value. It helps if your guess is a good one but it will work even if it is a terrible guess. We will guess that 2 is the square root of 12. ( Which does not really make sense because we all know that 3 * 3 = 9) However, this is a great example of how this method works—even if you pick a number that isn’t near.
In step two, we divide 12 by our guess of 2 and we get 6.
In step three, we average 6 and 2: (6+2)/2 = 4
Now we repeat step two with the new guess of 4. So 12/4 = 3
Now average 4 and 3: (4+3)/2 = 3.5
Repeat step two: 12/3.5 = 3.43
Average: (3.5 + 3.43)/2 = 3.465
We could keep going forever, getting a better and better approximation but let's stop here to see how we are doing. 3.465 * 3.465 = 12.006225
Method 2- Estimating your Square root
This method requires you to know your perfect squares. You should know them up to 400 by now. Start with the number you want to square root. Let’s say √183. We know that 183 is between two perfect squares 169 and 196 – or 132 and 142 So our SQ RT must also be between 13 and 14.
196
183
169
Next, find the difference between 196 and 169 ( 196-169) = 27
find the difference between 196 and 183 ( 196-183) = 13
and the difference between 183 and 169 ( 183-169) = 14
Looking at those three numbers, you notice that 183 is almost right in the middle or half way between the two perfect squares—so we can approximate
√183 ≈ 13.5
What happens if it isn’t quite in the middle, figure out the ratio and decide what a good approximation would be.
Years ago, teachers taught an algorithm for finding square roots, but these two methods are much easier and serve to approximate square roots accurately for middle school students.
If you have any questions or comments about these two ways, post a comment here.
Many textbooks seem to think that since calculators can find square roots, that students don't need to learn how to find square roots using any pencil-and-paper method. But learning at least the "guess and check" method for finding the square root will actually help the student UNDERSTAND and remember the square root concept itself!
Practice at least the first method presented here. This method, "guess and check", actually works around what the square root is all about.
The square root of a number is just the number which when multiplied by itself gives the first number. So 2 is the square root of 4 because 2 * 2 = 4.
Method 1: Guess, Divide & Check
Start with the number you want to find the square root of. Let's use 12. There are three steps:
1. Guess
2. Divide
3. Average.
... and then just keep repeating steps 2 and 3.
First, start by guessing a square root value. It helps if your guess is a good one but it will work even if it is a terrible guess. We will guess that 2 is the square root of 12. ( Which does not really make sense because we all know that 3 * 3 = 9) However, this is a great example of how this method works—even if you pick a number that isn’t near.
In step two, we divide 12 by our guess of 2 and we get 6.
In step three, we average 6 and 2: (6+2)/2 = 4
Now we repeat step two with the new guess of 4. So 12/4 = 3
Now average 4 and 3: (4+3)/2 = 3.5
Repeat step two: 12/3.5 = 3.43
Average: (3.5 + 3.43)/2 = 3.465
We could keep going forever, getting a better and better approximation but let's stop here to see how we are doing. 3.465 * 3.465 = 12.006225
Method 2- Estimating your Square root
This method requires you to know your perfect squares. You should know them up to 400 by now. Start with the number you want to square root. Let’s say √183. We know that 183 is between two perfect squares 169 and 196 – or 132 and 142 So our SQ RT must also be between 13 and 14.
196
183
169
Next, find the difference between 196 and 169 ( 196-169) = 27
find the difference between 196 and 183 ( 196-183) = 13
and the difference between 183 and 169 ( 183-169) = 14
Looking at those three numbers, you notice that 183 is almost right in the middle or half way between the two perfect squares—so we can approximate
√183 ≈ 13.5
What happens if it isn’t quite in the middle, figure out the ratio and decide what a good approximation would be.
Years ago, teachers taught an algorithm for finding square roots, but these two methods are much easier and serve to approximate square roots accurately for middle school students.
If you have any questions or comments about these two ways, post a comment here.
Algebra Period 3
DIVIDING RADICALS 11-5
Just as you can multiply radicals, you can also divide them by either
1) separating the numerator from the denominator,
or
2) simplifying the entire fraction underneath the radical.
HOW DO YOU KNOW WHICH METHOD TO USE?
Try both and see which one works best! (Examples below)
EXAMPLE OF TAKING THE QUOTIENT UNDER THE RADICAL APART:
Take apart fractions where either the numerator, the denominator, or both are perfect squares!
√(3/16)
Notice that the denominator is a perfect square so it makes sense to look at the denominator separately from the numerator:
√3 = √3
√16 4
√ (25/36)
Notice that both the numerator and denominator are perfect squares so it makes sense to simplify them apart:
√25= 5
√36 6
EXAMPLE OF SIMPLIFYING THE FRACTION
UNDER THE RADICAL FIRST:
Sometimes, the fraction under the radical will simplify.
If this is true, always do that first!
EXAMPLE: √(27/3)
27/3 simplifies to 9:
√9 = 3
Notice that if you took this fraction apart first and
then tried to find the square root of each part, it's much more complicated:
√(27/3)
Separate the numerator from the denominator:
√27
√ 3
Factor 27:
√(3x3)x3
√3
Simplify the numerator in pairs:
3√3
√3
Cross cancel if possible:
3
You get the same answer, but with lots more steps!!!
AGAIN, SO HOW DO YOU KNOW WHICH TO DO????
Check both ways and see which works best!!!!!
RATIONALIZING THE DENOMINATOR
THE RULE: Simplified form has
NO RADICALS IN THE DENOMINATOR.
(and you cannot change this rule even if you don't like it or think it makes sense!!!)
If you end up with a radical there, you must get rid of it by squaring whatever is under the radical.
Squaring it will result in the denominator becoming whatever was under the radical sign.
But you cannot do something to the denominator without doing the same thing to the numerator
(golden rule of fractions), so you must multiply the numerator by whatever you multiplied the denominator by.
RATIONALIZING THE DENOMINATOR EXAMPLE:
√7
√ 3
There is nothing you can simplify, whether you put it together or take it apart!
But you can't leave it this way because the rule is that
you can't leave the √3 in the denominator.
You need to multiply both numerator and denominator by √3 to get it out of there:
√7 = √7 • √3 = √(7• 3) = √21
√ 3 √3 √ 3 3 3
Note that you cannot cross cancel the 3 in denominator with 21 in numerator
because one is a square root and the other is not (they are unlike terms!)
The √21 is not 21!
It's irrational and approximately 4.58
You can't cross cancel 4.58 with 3 in the denominator!
Just as you can multiply radicals, you can also divide them by either
1) separating the numerator from the denominator,
or
2) simplifying the entire fraction underneath the radical.
HOW DO YOU KNOW WHICH METHOD TO USE?
Try both and see which one works best! (Examples below)
EXAMPLE OF TAKING THE QUOTIENT UNDER THE RADICAL APART:
Take apart fractions where either the numerator, the denominator, or both are perfect squares!
√(3/16)
Notice that the denominator is a perfect square so it makes sense to look at the denominator separately from the numerator:
√3 = √3
√16 4
√ (25/36)
Notice that both the numerator and denominator are perfect squares so it makes sense to simplify them apart:
√25= 5
√36 6
EXAMPLE OF SIMPLIFYING THE FRACTION
UNDER THE RADICAL FIRST:
Sometimes, the fraction under the radical will simplify.
If this is true, always do that first!
EXAMPLE: √(27/3)
27/3 simplifies to 9:
√9 = 3
Notice that if you took this fraction apart first and
then tried to find the square root of each part, it's much more complicated:
√(27/3)
Separate the numerator from the denominator:
√27
√ 3
Factor 27:
√(3x3)x3
√3
Simplify the numerator in pairs:
3√3
√3
Cross cancel if possible:
3
You get the same answer, but with lots more steps!!!
AGAIN, SO HOW DO YOU KNOW WHICH TO DO????
Check both ways and see which works best!!!!!
RATIONALIZING THE DENOMINATOR
THE RULE: Simplified form has
NO RADICALS IN THE DENOMINATOR.
(and you cannot change this rule even if you don't like it or think it makes sense!!!)
If you end up with a radical there, you must get rid of it by squaring whatever is under the radical.
Squaring it will result in the denominator becoming whatever was under the radical sign.
But you cannot do something to the denominator without doing the same thing to the numerator
(golden rule of fractions), so you must multiply the numerator by whatever you multiplied the denominator by.
RATIONALIZING THE DENOMINATOR EXAMPLE:
√7
√ 3
There is nothing you can simplify, whether you put it together or take it apart!
But you can't leave it this way because the rule is that
you can't leave the √3 in the denominator.
You need to multiply both numerator and denominator by √3 to get it out of there:
√7 = √7 • √3 = √(7• 3) = √21
√ 3 √3 √ 3 3 3
Note that you cannot cross cancel the 3 in denominator with 21 in numerator
because one is a square root and the other is not (they are unlike terms!)
The √21 is not 21!
It's irrational and approximately 4.58
You can't cross cancel 4.58 with 3 in the denominator!
Approximating Pi Day
Since Pi Day fell on a Saturday this year, we approximated Pi Day and celebrated on Monday instead... So What did we learn about pi?
Today we celebrate Î (Pi), a very cool number. Î is a comparison between the measurements of the circumference to the diameter of a circle—any circle. Pi is an IRRATIONAL number. That means it has no pattern and never terminates. It CANNOT be written as a fraction with an integer in the numerator and denominator. We use 3.14 and 22/7 as APPROXIMATIONS of pi. These are not the exact values. The only symbol that tells the exact value is Î . Pi is a ratio, a comparison between two numbers. You will be able to discover many interesting facts about pi—even finding it on your own.
After measuring 3 different circular objects and completing the table...
Think about your results and answer these questions:
How does the measurement of the circumference compare to the measurement of the diameter? Is it twice as large? Is it three times as large or more than three times as large?
Compare your values to 3.14 Were your calculations greater or less than the 3.14 values of pi?
Are you values of pi consistent?Explain why or why not?
What reasons do you think would account for these differences?
On your computer, read about the history of pi at this website
http://ualr.edu/lasmoller/pi.html
Record 5 new or interesting facts about pi that you learned
Finding Yourself in Pi
Pi is an irrational number. That means that it is a non-terminating, non-repeating decimal. Since the number order keeps changing you will eventually find any group of numbers in a sequence, somewhere in the never-ending list. Through the influence of high speed computers that can process large amounts of data, we can examine this aspect of pi much more easily. There are even websites that will instantly search pi for any string of numbers.
Your assignment is to locate a specific string of numbers in pi. Pick a string of 7 or more numbers that is meaningful or significant to you, such as your phone number or birthday (for example: if your birthday is February 9, 1993, then your number string would be 02091993). On your computer, use the Pi Searcher at http://www.angio.net/pi/piquery to find where in pi your number string occurs. Record the string of numbers and what position it holds in the list of numbers.
There are a number of great Pi day songs.. but I think the best is from Fort Vancouver High School .. Check out this video and rap they created...
Today we celebrate Î (Pi), a very cool number. Î is a comparison between the measurements of the circumference to the diameter of a circle—any circle. Pi is an IRRATIONAL number. That means it has no pattern and never terminates. It CANNOT be written as a fraction with an integer in the numerator and denominator. We use 3.14 and 22/7 as APPROXIMATIONS of pi. These are not the exact values. The only symbol that tells the exact value is Î . Pi is a ratio, a comparison between two numbers. You will be able to discover many interesting facts about pi—even finding it on your own.
After measuring 3 different circular objects and completing the table...
Think about your results and answer these questions:
How does the measurement of the circumference compare to the measurement of the diameter? Is it twice as large? Is it three times as large or more than three times as large?
Compare your values to 3.14 Were your calculations greater or less than the 3.14 values of pi?
Are you values of pi consistent?Explain why or why not?
What reasons do you think would account for these differences?
On your computer, read about the history of pi at this website
http://ualr.edu/lasmoller/pi.html
Record 5 new or interesting facts about pi that you learned
Finding Yourself in Pi
Pi is an irrational number. That means that it is a non-terminating, non-repeating decimal. Since the number order keeps changing you will eventually find any group of numbers in a sequence, somewhere in the never-ending list. Through the influence of high speed computers that can process large amounts of data, we can examine this aspect of pi much more easily. There are even websites that will instantly search pi for any string of numbers.
Your assignment is to locate a specific string of numbers in pi. Pick a string of 7 or more numbers that is meaningful or significant to you, such as your phone number or birthday (for example: if your birthday is February 9, 1993, then your number string would be 02091993). On your computer, use the Pi Searcher at http://www.angio.net/pi/piquery to find where in pi your number string occurs. Record the string of numbers and what position it holds in the list of numbers.
There are a number of great Pi day songs.. but I think the best is from Fort Vancouver High School .. Check out this video and rap they created...
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