Addition and Subtraction of Mixed Numbers 7-2
Although there are two methods discussed in our textbook, I would like students to concentrate on using the 'Stacking Method' rather than changing mixed numbers into improper fractions.
In the stacking method, we work separately with the fractional and whole- number parts of the given mixed numbers. Sometimes, we must change the form of a mixed number before we can subtract the fractional part, as with the following example
9 3/7 - 4 5/7 re write
69 3/7
-4 5/7
______
9 3/7 = 8 + 7/7 + 3/7 so re write the first line as 8 10/7
r8 10/7
-4 5/7
______ and now you can easily perform the subtraction
4 5/7
If the fractional parts of the given mixed numbers have different denominators, we must find equivalent mixed numbers whose fractional parts have the same denominator-- usually the LCD. You can use the box method for finding the LCM to find the LCD here!!
Wednesday, April 2, 2008
Tuesday, April 1, 2008
Algebra Period 3 (Wednesday)
CHAPTER 11-9: EQUATIONS WITH RADICALS
When you have an equation where the variable is under the SQRT sign,
simply square both sides to solve for the variable.
Remember to isolate the variable on one side before you square it,
unless there is a number under the SQRT sign with the variable.
Example #1 on p. 521
SQRT(x) = 5
Square both sides and you'll get x = 25
Example # 9 is much harder
3 + SQRT(x - 1) = 5
move the 3 to other side first SQRT(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?)
Math 6 Honors Periods 6 & 7 (Tuesday )
Addition and Subtraction of Fractions 7-1
The properties of addition and subtraction of whole numbers also apply to fractions.
When adding fractions with the same denominator-- simply add the numerators and place them over the denominator.
That is 5/9 + 2/9 = (5+2)/9 = 7/9
In order to add two fractions with different denominators, find two fractions with a common denominator, equivalent to the given fractions. The most convenient denominator is the least common multiple of the denominator which is called the least common denominator (LCD)
When solving equations such as
n + 1/2 = 5/6 Use the same method taught for intetergs. DO unto one side that what you do to the other
so if n + 1/2 = 5/6 You must subtract 1/2 from both sides!!
n = 5/6 - 1/2
Stack your fractions. change to a LCD and subtract
n = 1/3
The properties of addition and subtraction of whole numbers also apply to fractions.
When adding fractions with the same denominator-- simply add the numerators and place them over the denominator.
That is 5/9 + 2/9 = (5+2)/9 = 7/9
In order to add two fractions with different denominators, find two fractions with a common denominator, equivalent to the given fractions. The most convenient denominator is the least common multiple of the denominator which is called the least common denominator (LCD)
When solving equations such as
n + 1/2 = 5/6 Use the same method taught for intetergs. DO unto one side that what you do to the other
so if n + 1/2 = 5/6 You must subtract 1/2 from both sides!!
n = 5/6 - 1/2
Stack your fractions. change to a LCD and subtract
n = 1/3
Algebra Period 3 (Tuesday)
CHAPTER 11-7: DISTANCE FORMULA
(based on the Pythagorean Theorem): see p. 513 in book
Reviewed from last week:
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
The formula is:
distance = SQRT[( difference of the two x's)2 + (difference of the two y's)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
EXAMPLE: What is the distance between (3, -10) and (-7, -2)?
d = SQRT[(3 - -7)2 + (-10 - -2)2]
d = SQRT[102 +( -82)]
d = SQRT(164)
Simplifying: 2rad41
CHAPTER 11-8: USING THE PYTHAGOREAN THEOREM - WORD PROBLEMS
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.
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.
Sunday, March 30, 2008
Pre Algebra Periods 1, 2, & 4
REVIEW REAL LIFE PERCENTS
DISCOUNT = sales price This is similar to mark up , but this time you will SUBTRACT (not add)
A jacket that was $100 is 35% off. (.35)($100) = $35. Sales price = $100 - $35 = $65
DISCOUNT = % OF DECREASE
EX: original price = $200 and sales price = $50. What is the DISCOUNT %? (% of decrease?)
Decrease or discount = $200 - $50 = $150
$150 decrease/$200 original = 3/4 = 75% Discount %
MARKUP = how much you want to make on selling something Then add the markup to get what you want to sell that item for.
EX: you want to make 75% of what you paid (.75)(your cost)
You paid $100 for an ipod. (.75)($100)=$75 markup. So you want to sell it for $175!
MARKUP = a % of INCREASE
EX: Your cost = $50 and your markup = $100. What is your MARKUP %? (% of increase?)
markup = increase
$100 (increase on cost)/$50 (original) = 2 = 200% markup (selling price = $50 + $100 = $150)
CHAPTER 7-8: SIMPLE INTEREST:
Let's talk about different types of interest: credit cards, car loans, mortgages
None of them charge us simple interest, but you need to understand that first:
I =PRT (mneumonic devices "PRINT" or "PARTY")
I = Interest
P = Principal ($ deposited in savings or $ owed to credit card or bank)
R = % (Change it to a decimal or fraction before multiplying)
T = Time (based on one year so if you have 9 months, put it over 12 months or 9/12 of a year)
'
EXAMPLE: You buy an Ipod for $300 on your parent's VISA at 20% and pay it back in 2 years
I = PRT
I = ($300)(.20)(2) = $120 Interest
$300 Ipod + $120 interest to VISA = $420 total cost
EXAMPLE: You deposit $200 in your savings account at Wells Fargo for 9 months at 2%
I = PRT
I = ($200)(.02)(9/12) = $3 interest
$200 deposit + $3 = $203 in your savings account after 9 months
CHAPTER 7-8: COMPOUND INTEREST:
Same as simple, but you charge (or get) INTEREST ON THE INTEREST
So COMPOUND > SIMPLE always!
EXAMPLE: The Ipod above is still bought over 2 years on VISA, but this time interest is COMPOUNDED (added to the principal) ANNUALLY (each year)
After one year: I = ($300)(.20)(1) = $60
$300 Ipod + $60 interest = $360
Now in year 2, you're charged interest on the $360, not just the original $300!
That's why it's always more than SIMPLE interest!
Year two: ($360)(.20)(1) =$72
$360 owed at end of year one + $72 year two interest = $$432 owed at the end of year two
Notice that under SIMPLE interest, that amount was only $420
Interest can be compounded annually (once a year), semiannually (twice a year), quarterly (4 times a year), monthly (12 times a year) or even daily! (365 times a year!)
Which would give you the most interest on your savings deposit? Obviously compounding daily!
Which would give the VISA company the most? Same thing - compounding daily! (that's what they use!)
There's a formula for compound interest:
Balance owed = (principal)(1 + percent rate/# times it compounds a year)number of compoundings
So if I owe VISA $500 at 6% compounded QUARTERLY (4 times a year) and I want to know how much I will owe after 1 year, I can use this formula:
A = P(1 + r/t)n
A is the amount I'm looking for
P = $500
r = 6% or .06
t = 4 times a year
n = 4 because I want to know after a year and the interest will compound 4 times
A = 500[1 + .06/4]4
A = 500(1.015)4
A =(500)(1.06136)
A = $530.68
The other way to do this is an organized table:
500(.06)(.25) = $7.50 interest for 1st quarter + $500 = 507.50
$507.5(.06)(.25) = $7.61 interest for 2nd quarter + 507.50 = $515.11
$515.11(.06)(.25) = $7.73 interest for 3rd quarter + $515.11 = $522.84
$522.84(.06)(.25) = $7.84 interest for 4th quarter + $522.84 = $530.68
BUT YOU WOULDN'T WANT TO USE THE TABLE APPROACH FOR MORE THAN ONE YEAR BECAUSE IT JUST TAKES TOO MUCH TIME!
DISCOUNT = sales price This is similar to mark up , but this time you will SUBTRACT (not add)
A jacket that was $100 is 35% off. (.35)($100) = $35. Sales price = $100 - $35 = $65
DISCOUNT = % OF DECREASE
EX: original price = $200 and sales price = $50. What is the DISCOUNT %? (% of decrease?)
Decrease or discount = $200 - $50 = $150
$150 decrease/$200 original = 3/4 = 75% Discount %
MARKUP = how much you want to make on selling something Then add the markup to get what you want to sell that item for.
EX: you want to make 75% of what you paid (.75)(your cost)
You paid $100 for an ipod. (.75)($100)=$75 markup. So you want to sell it for $175!
MARKUP = a % of INCREASE
EX: Your cost = $50 and your markup = $100. What is your MARKUP %? (% of increase?)
markup = increase
$100 (increase on cost)/$50 (original) = 2 = 200% markup (selling price = $50 + $100 = $150)
CHAPTER 7-8: SIMPLE INTEREST:
Let's talk about different types of interest: credit cards, car loans, mortgages
None of them charge us simple interest, but you need to understand that first:
I =PRT (mneumonic devices "PRINT" or "PARTY")
I = Interest
P = Principal ($ deposited in savings or $ owed to credit card or bank)
R = % (Change it to a decimal or fraction before multiplying)
T = Time (based on one year so if you have 9 months, put it over 12 months or 9/12 of a year)
'
EXAMPLE: You buy an Ipod for $300 on your parent's VISA at 20% and pay it back in 2 years
I = PRT
I = ($300)(.20)(2) = $120 Interest
$300 Ipod + $120 interest to VISA = $420 total cost
EXAMPLE: You deposit $200 in your savings account at Wells Fargo for 9 months at 2%
I = PRT
I = ($200)(.02)(9/12) = $3 interest
$200 deposit + $3 = $203 in your savings account after 9 months
CHAPTER 7-8: COMPOUND INTEREST:
Same as simple, but you charge (or get) INTEREST ON THE INTEREST
So COMPOUND > SIMPLE always!
EXAMPLE: The Ipod above is still bought over 2 years on VISA, but this time interest is COMPOUNDED (added to the principal) ANNUALLY (each year)
After one year: I = ($300)(.20)(1) = $60
$300 Ipod + $60 interest = $360
Now in year 2, you're charged interest on the $360, not just the original $300!
That's why it's always more than SIMPLE interest!
Year two: ($360)(.20)(1) =$72
$360 owed at end of year one + $72 year two interest = $$432 owed at the end of year two
Notice that under SIMPLE interest, that amount was only $420
Interest can be compounded annually (once a year), semiannually (twice a year), quarterly (4 times a year), monthly (12 times a year) or even daily! (365 times a year!)
Which would give you the most interest on your savings deposit? Obviously compounding daily!
Which would give the VISA company the most? Same thing - compounding daily! (that's what they use!)
There's a formula for compound interest:
Balance owed = (principal)(1 + percent rate/# times it compounds a year)number of compoundings
So if I owe VISA $500 at 6% compounded QUARTERLY (4 times a year) and I want to know how much I will owe after 1 year, I can use this formula:
A = P(1 + r/t)n
A is the amount I'm looking for
P = $500
r = 6% or .06
t = 4 times a year
n = 4 because I want to know after a year and the interest will compound 4 times
A = 500[1 + .06/4]4
A = 500(1.015)4
A =(500)(1.06136)
A = $530.68
The other way to do this is an organized table:
500(.06)(.25) = $7.50 interest for 1st quarter + $500 = 507.50
$507.5(.06)(.25) = $7.61 interest for 2nd quarter + 507.50 = $515.11
$515.11(.06)(.25) = $7.73 interest for 3rd quarter + $515.11 = $522.84
$522.84(.06)(.25) = $7.84 interest for 4th quarter + $522.84 = $530.68
BUT YOU WOULDN'T WANT TO USE THE TABLE APPROACH FOR MORE THAN ONE YEAR BECAUSE IT JUST TAKES TOO MUCH TIME!
Algebra Period 3
Review Chapter 11: RADICALS
REMEMBER:
THE SQUARE ROOT OF ANYTHING SQUARED IS ITSELF!!!
TRINOMIALS UNDER THE RADICAL:
What do you think you would do if you saw x2 + 10x + 25 under the radical sign????
FACTOR IT! IT MAY BE A PERFECT SQUARE (a binomial squared!)
x2 + 10x + 25 factors to (x + 5)2 so the square root of (x + 5)2 = lx + 5l
Please read as absolute value of (x +5)
SIMPLIFYING RADICALS
SIMPLIFYING NONPERFECT NUMBERS UNDER THE RADICAL:
A simplified radical expression is one where there is no perfect squares left under the radical sign
You can factor the expression under the radical to find any perfect squares in the number:
EXAMPLE: square root of 50 = SQRT(25 * 2)
Next, simplify the sqrt of the perfect square and leave the nonperfect factor under the radical:
SQRT(25 * 2) = SQRT(25) *SQRT(2) = 5 SQRT( 2 )
HELPFUL HINTS:
When you are factoring the radicand, you're looking for the LARGEST PERFECT SQUARE that is a FACTOR of the radicand.
So start with:
Does 4 go into it?
Does 9 go into it?
Does 16 go into it?
Does 25 go into it?
etc.
HELPFUL WAYS TO ATTACK SIMPLIFYING:
Factor trees, Inverted Division, Prime Factorization
Whenever there are 2 factors that are same, it can be simplified
(Take one of the 2 factors out of the radical and leave none under)
If there is already a factor outside the front of the radical, when you bring out another factor from underneath the radical, you multiply it with what was already outside in front.
VARIABLES UNDER THE SQUARE ROOT SIGN:
An even power of a variable just needs to be divided by two to find its square root
EXAMPLE: SQRT (x10 ) = x5
We saw this already in factoring!!!
If the variable has an odd power:
If you have an odd power variable, simply express it as the even power one below that odd power times that variable to the 1 power:
Example: x5 = x4 x
so if you have the SQRT( x5 ) = SQRT (x4 x) = x2SQRT(x)
Don't forget to always:
RATIONALIZE THE DENOMINATOR
The rule is that the radical is not simplified until all radicals are removed from the denominator
Simply use the equivalent fraction approach and multiply both the numerator and the denominator by the radical.
ADDING AND SUBTRACTING RADICALS:
Radicals function like variables, so you can only COMBINE LIKE RADICALS!
CHAPTER 11-7 (an old friend) - PYTHAGOREAN THEOREM
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
REMEMBER:
THE SQUARE ROOT OF ANYTHING SQUARED IS ITSELF!!!
TRINOMIALS UNDER THE RADICAL:
What do you think you would do if you saw x2 + 10x + 25 under the radical sign????
FACTOR IT! IT MAY BE A PERFECT SQUARE (a binomial squared!)
x2 + 10x + 25 factors to (x + 5)2 so the square root of (x + 5)2 = lx + 5l
Please read as absolute value of (x +5)
SIMPLIFYING RADICALS
SIMPLIFYING NONPERFECT NUMBERS UNDER THE RADICAL:
A simplified radical expression is one where there is no perfect squares left under the radical sign
You can factor the expression under the radical to find any perfect squares in the number:
EXAMPLE: square root of 50 = SQRT(25 * 2)
Next, simplify the sqrt of the perfect square and leave the nonperfect factor under the radical:
SQRT(25 * 2) = SQRT(25) *SQRT(2) = 5 SQRT( 2 )
HELPFUL HINTS:
When you are factoring the radicand, you're looking for the LARGEST PERFECT SQUARE that is a FACTOR of the radicand.
So start with:
Does 4 go into it?
Does 9 go into it?
Does 16 go into it?
Does 25 go into it?
etc.
HELPFUL WAYS TO ATTACK SIMPLIFYING:
Factor trees, Inverted Division, Prime Factorization
Whenever there are 2 factors that are same, it can be simplified
(Take one of the 2 factors out of the radical and leave none under)
If there is already a factor outside the front of the radical, when you bring out another factor from underneath the radical, you multiply it with what was already outside in front.
VARIABLES UNDER THE SQUARE ROOT SIGN:
An even power of a variable just needs to be divided by two to find its square root
EXAMPLE: SQRT (x10 ) = x5
We saw this already in factoring!!!
If the variable has an odd power:
If you have an odd power variable, simply express it as the even power one below that odd power times that variable to the 1 power:
Example: x5 = x4 x
so if you have the SQRT( x5 ) = SQRT (x4 x) = x2SQRT(x)
Don't forget to always:
RATIONALIZE THE DENOMINATOR
The rule is that the radical is not simplified until all radicals are removed from the denominator
Simply use the equivalent fraction approach and multiply both the numerator and the denominator by the radical.
ADDING AND SUBTRACTING RADICALS:
Radicals function like variables, so you can only COMBINE LIKE RADICALS!
CHAPTER 11-7 (an old friend) - PYTHAGOREAN THEOREM
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
Subscribe to:
Posts (Atom)
