Decimals 3-3
Although decimals ( termed decimal fractions) had been used for centuries, Simon Stevin in the 16th century began using them on a daily basis and he helped establish their use in the fields of sciences and engineering.
Note that
1/10 = 1/101
1/100 = 1/102
1/1000 = 1/103
We also know that
1/10= 0/1
1/100 = 0.01
1/1000 = 0.001
1/10000 = 0.0001
and so on... these strings of digits are called decimals.
SO 1/10 = 1/101= 0.01 and it is equal to 10-1
Notice that 10-1 is NOT a negative number-- it is a small number
and 10-21 is not a negative number it is a VERY TINY number
AS with whole numbers, decimals use place values. These place values are to the RIGHT of the decimal point.
We need to be able to write decimals in words as well as expanded notation.
In class we used 0.6394 as our example
zero and six thousand three hundred ninety-four ten-thousandths.
Notice how this number when written in words begins...with "ZERO AND"
Why do we need to do that?
Also notice that there is a hyphen between ten and thousandths in ten-thousandths. It is critical to understand when you must place a hyphen.
We read the entire number to the right of the decimal point as if it represented a whole number, and then we give the place value of the digit farthest to the right.
So, although 0.400 is equivalent to 0.4
we must read 0.400 as "zero and four hundred thousandths."
Now look at the following words
"zero and four hundred-thousandths." What is the subtle difference between those two phrases above?
There is a hyphen in the last phrase-- which means that the hundred and the thousandths are attached and represent a place value so
zero and four hundred-thousandths is 0.00004 while
zero and four hundred thousandths is 0.400
Carefully see the distinction!!
Getting back to our 0.6394
to write it in decimals sums and then in exponents:
0 + 0.6 + 0.03 + 0.009 + 0.0004
0 + 6(0.1) + 3(0.01) +9(0.001) + 4(0.0001)
0(100) + 6(10-1)+ 3(10-2)+ 9(10-3)+ 4(10-4)
14.35 is read as fourteen AND thirty-five hundredths.
When reading numbers, only use the AND to indicate the decimal point
Saturday, October 15, 2011
Thursday, October 13, 2011
Algebra Honors (Period 6 & 7)
Dividing Monomials 5-2
There are 3 basic rules used to simplify fractions made up of monomials.
Property of Quotients
if a, b, c, d are real numbers with b≠0 and d ≠0
ac/bd = a/b ⋅c/d
Our example was 15/21 = (3⋅5)/(3⋅7) = 5/7
The rule for simplifying fractions follows ( when a = b)
(bc)/(bd) = c/d
This rule lets you divide both the numerator and the denominator by the same NON ZERO number.
35/42 = 5/6
-4xy/10x = -2y/5 which can also be written (-2/5)x as well as with out the (((HUGS)))
c7/c4 = c4c3/c4 = c3
another way we proved this was to write out all the c's
c⋅c⋅c⋅c⋅c⋅c⋅c⋅/c⋅c⋅c⋅c = and we realized we were left with
c⋅c⋅c = c3
In addition, we noticed that
c7/c4 = = c7-4 = c3
THen we considered
c4/c7 =
c⋅c⋅c⋅c/c⋅c⋅c⋅c⋅c⋅c⋅c = 1/c⋅c⋅c = 1/c3 = c-3
Since we all agreed that any number divided by itself was = 1
(our example was b5/b5 ), we proved the following
1 = b5/b5 = b5-5 = b0
We finally arrived at the Rule of Exponents for Division
if m > n
am/an = a m-n
If n > m
am/an = 1/a n-m
and if m = n
am/an = 1
A quotient of monomials is simplified when
1)each base appears only once in the fraction,
2) there are NO POWERS of POWERS and
3)when the numerator and denominator are relatively prime, that is, they have no common factor other than 1.
35x3yz6/ 56x5yz
5z5/8x2
Finding the missing factor when you are given the following
48x3y2z4 = (3xy2z)⋅ (______)
we find that
48x3y2z4 = (3xy2z)⋅ (16x2z3)
There are 3 basic rules used to simplify fractions made up of monomials.
Property of Quotients
if a, b, c, d are real numbers with b≠0 and d ≠0
ac/bd = a/b ⋅c/d
Our example was 15/21 = (3⋅5)/(3⋅7) = 5/7
The rule for simplifying fractions follows ( when a = b)
(bc)/(bd) = c/d
This rule lets you divide both the numerator and the denominator by the same NON ZERO number.
35/42 = 5/6
-4xy/10x = -2y/5 which can also be written (-2/5)x as well as with out the (((HUGS)))
c7/c4 = c4c3/c4 = c3
another way we proved this was to write out all the c's
c⋅c⋅c⋅c⋅c⋅c⋅c⋅/c⋅c⋅c⋅c = and we realized we were left with
c⋅c⋅c = c3
In addition, we noticed that
c7/c4 = = c7-4 = c3
THen we considered
c4/c7 =
c⋅c⋅c⋅c/c⋅c⋅c⋅c⋅c⋅c⋅c = 1/c⋅c⋅c = 1/c3 = c-3
Since we all agreed that any number divided by itself was = 1
(our example was b5/b5 ), we proved the following
1 = b5/b5 = b5-5 = b0
We finally arrived at the Rule of Exponents for Division
if m > n
am/an = a m-n
If n > m
am/an = 1/a n-m
and if m = n
am/an = 1
A quotient of monomials is simplified when
1)each base appears only once in the fraction,
2) there are NO POWERS of POWERS and
3)when the numerator and denominator are relatively prime, that is, they have no common factor other than 1.
35x3yz6/ 56x5yz
5z5/8x2
Finding the missing factor when you are given the following
48x3y2z4 = (3xy2z)⋅ (______)
we find that
48x3y2z4 = (3xy2z)⋅ (16x2z3)
Wednesday, October 12, 2011
Algebra Honors (Period 6 & 7)
Factoring Integers 5-1
When we write 56= 8⋅7 or 56 = 4⋅14 we have factored 56
to factor a number over a given set, you write it as a product of integers in that set ( the factor set).
When integers are factored over the set of integers, the factors are called integral factors.
We used the T- charts ( students learned in 6th grade) to first find the positive integer factors
56 = 1, 2, 4, 7, 8, 14, 28, 56
A prime number is an integer greater than 1 that has no positive integral factors other than itself and 1.
The first ten prime numbers are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
To find prime factorization of a positive integer, you express it as a product of primes. We used inverted division (again taught in 6th grade)
504
Try to find the primes in order as divisors.
Divide each prime as many times as possible before going on to the next prime
we found 504 - 2⋅2⋅2⋅3⋅3⋅7
which we write as 23⋅32⋅7
Exponents are generally used for prime factors
The prime factorization is unique--> and the order should be from the smallest prime to the largest.
A factor of two or more integers is called a common factor of the integers.
The greatest common factor (GCF) of two or more integers is the greatest integer that is a factor of all the given integers.
Find the GCF(882, 945)
First find the prime factorization of each integer Then form product of the smaller powers of each common prime factor.
The GCF is only the primes (and the powers) that they SHARE!!
882 = 2⋅32⋅72
945 = 33⋅5⋅7
The common factors are 3 and 7
The smaller powers of 3 and 7 are 32 and 7
You combine these as a PRODUCT and get
the GCF(882, 945) = 32⋅7 = 63
We also talked about listing ALL pairs of factors--> thus including negative integers
For example:
List all the pairs of factors of 20
(1)(20) but also (-1)(-20)
(2)(10) and (-2)(-10)
(4)(5) and (-4)(-5)
Listing all the factors of -20, we discovered
(1)(-20) but also (-1)(20)
(2)(-10) and (-2)(10)
(4)(-5) and (-4)(5)
When we write 56= 8⋅7 or 56 = 4⋅14 we have factored 56
to factor a number over a given set, you write it as a product of integers in that set ( the factor set).
When integers are factored over the set of integers, the factors are called integral factors.
We used the T- charts ( students learned in 6th grade) to first find the positive integer factors
56 = 1, 2, 4, 7, 8, 14, 28, 56
A prime number is an integer greater than 1 that has no positive integral factors other than itself and 1.
The first ten prime numbers are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
To find prime factorization of a positive integer, you express it as a product of primes. We used inverted division (again taught in 6th grade)
504
Try to find the primes in order as divisors.
Divide each prime as many times as possible before going on to the next prime
we found 504 - 2⋅2⋅2⋅3⋅3⋅7
which we write as 23⋅32⋅7
Exponents are generally used for prime factors
The prime factorization is unique--> and the order should be from the smallest prime to the largest.
A factor of two or more integers is called a common factor of the integers.
The greatest common factor (GCF) of two or more integers is the greatest integer that is a factor of all the given integers.
Find the GCF(882, 945)
First find the prime factorization of each integer Then form product of the smaller powers of each common prime factor.
The GCF is only the primes (and the powers) that they SHARE!!
882 = 2⋅32⋅72
945 = 33⋅5⋅7
The common factors are 3 and 7
The smaller powers of 3 and 7 are 32 and 7
You combine these as a PRODUCT and get
the GCF(882, 945) = 32⋅7 = 63
We also talked about listing ALL pairs of factors--> thus including negative integers
For example:
List all the pairs of factors of 20
(1)(20) but also (-1)(-20)
(2)(10) and (-2)(-10)
(4)(5) and (-4)(-5)
Listing all the factors of -20, we discovered
(1)(-20) but also (-1)(20)
(2)(-10) and (-2)(10)
(4)(-5) and (-4)(5)
Tuesday, October 11, 2011
Math 6 Honors ( Periods 1, 2, & 3)
Exponents and Powers of Ten 3-1
When two or more numbers are multiplied together--each of the numbers is called a factor of the product.
A product in which each factor is the SAME is called a power of that factor.
2 X 2 X 2 X 2 = 16. 16 is called the fourth power of 2 and we can write this as
24 = 16
The small numeral (in this case the 4) is called the exponent and represents the number of times 2 is a factor of 16.
The number two, in this case, is called the base.
When you are asked to evaluate... simplify... solve... find the answer
That is,
Evaluate
43 = 4 X 4 X 4 = 16 X 4 = 64
The second and third powers of a numeral have special names.
The second power is called the square of the number and the third power is called the cube.
We read 122 as "twelve squared" and to evaluate it
122 = 12 X 12 = 144
Powers of TEN are important in our number system.
Make sure to check out the blue sheet and glue it into your spiral notebook
First Power: 101 but the exponent is invisible = 10
Second Power: 102 = 10 X 10 = 100
Third Power 103 = 10 X 10 X 10 = 1000
Fourth Power 104 =10 X 10 X 10 X 10 = 10,000
Fifth Power 105 = 10 X 10 X 10 X 10 X 10 = 100,000
Take a look at this list carefully and you will probably see a pattern that we can turn into a general rule:
The exponent in a POWER of TEN is the same as the number of ZEROS when the number is written out.
The number of ZEROS in the product of POWERS OF TEN is the sum of the numbers of ZEROS in the factors.
For example Multiply.
100 X 1000
Since there are 2 Zeros in 100 and 3 zeros in 1000,
the product will have 2 + 3 , or 5 zeroes.
100 X 1000 = 100,000
When you need to multiply other bases:
first multiply each
For example
34 X 2 3 would be
(3 X 3 X 3X 3) X ( 2 X 2 X 2)
= 81 X 8 = 648
What happens when you multiply the same bases?
34 ⋅ 32 = 3⋅3⋅3⋅3⋅3⋅3 or 3 6
We just add the exponents if the bases are the same!!
Well then, what about (34)2 ?
Wait.. look carefully isn't that saying 34 Squared?
That would be (34)(34), right?
.. and looking at the rule above all we have to do here is then add those bases or 4 + 4 = 8 so the answer would be 38.
OR
we could have made each (34) = (3⋅3⋅3⋅3)
so (34)2 would be 3⋅3⋅3⋅3⋅3⋅3⋅3⋅3 or still 38
But wait... isn't that multiplying the two powers? So when raising a power to a power-- you multiply!!
(34)2 = 38
1 to any more is still just 1
15 = 1
0 to any power is still 0!!
Evaluate if a = 3 and b = 5
Just substitute in... but use hugs () we all love our hugs!!
a3 + b2
would be (3)3 + (5) 2
= 27 + 25 = 52
Check out this great Video on the Powers of Ten
POWERS OF TEN
When two or more numbers are multiplied together--each of the numbers is called a factor of the product.
A product in which each factor is the SAME is called a power of that factor.
2 X 2 X 2 X 2 = 16. 16 is called the fourth power of 2 and we can write this as
24 = 16
The small numeral (in this case the 4) is called the exponent and represents the number of times 2 is a factor of 16.
The number two, in this case, is called the base.
When you are asked to evaluate... simplify... solve... find the answer
That is,
Evaluate
43 = 4 X 4 X 4 = 16 X 4 = 64
The second and third powers of a numeral have special names.
The second power is called the square of the number and the third power is called the cube.
We read 122 as "twelve squared" and to evaluate it
122 = 12 X 12 = 144
Powers of TEN are important in our number system.
Make sure to check out the blue sheet and glue it into your spiral notebook
First Power: 101 but the exponent is invisible = 10
Second Power: 102 = 10 X 10 = 100
Third Power 103 = 10 X 10 X 10 = 1000
Fourth Power 104 =10 X 10 X 10 X 10 = 10,000
Fifth Power 105 = 10 X 10 X 10 X 10 X 10 = 100,000
Take a look at this list carefully and you will probably see a pattern that we can turn into a general rule:
The exponent in a POWER of TEN is the same as the number of ZEROS when the number is written out.
The number of ZEROS in the product of POWERS OF TEN is the sum of the numbers of ZEROS in the factors.
For example Multiply.
100 X 1000
Since there are 2 Zeros in 100 and 3 zeros in 1000,
the product will have 2 + 3 , or 5 zeroes.
100 X 1000 = 100,000
When you need to multiply other bases:
first multiply each
For example
34 X 2 3 would be
(3 X 3 X 3X 3) X ( 2 X 2 X 2)
= 81 X 8 = 648
What happens when you multiply the same bases?
34 ⋅ 32 = 3⋅3⋅3⋅3⋅3⋅3 or 3 6
We just add the exponents if the bases are the same!!
Well then, what about (34)2 ?
Wait.. look carefully isn't that saying 34 Squared?
That would be (34)(34), right?
.. and looking at the rule above all we have to do here is then add those bases or 4 + 4 = 8 so the answer would be 38.
OR
we could have made each (34) = (3⋅3⋅3⋅3)
so (34)2 would be 3⋅3⋅3⋅3⋅3⋅3⋅3⋅3 or still 38
But wait... isn't that multiplying the two powers? So when raising a power to a power-- you multiply!!
(34)2 = 38
1 to any more is still just 1
15 = 1
0 to any power is still 0!!
Evaluate if a = 3 and b = 5
Just substitute in... but use hugs () we all love our hugs!!
a3 + b2
would be (3)3 + (5) 2
= 27 + 25 = 52
Check out this great Video on the Powers of Ten
POWERS OF TEN
Monday, October 10, 2011
Algebra Honors (Period 6 & 7)
Problems Without Solutions 4-10
Not all word problems have solutions. We listed three of the reasons for this:
1) Not Enough Information ( NEI)
2) Unrealistic Results
3) Facts are contradictory
We used the following examples:
Aurenne drove at her normal speed for the first 2 hours of the trip--- but the road repairs slowed her down 10 mph slower than her normal speed. She made the trip in 3 hours. Find her normal speed.
Wait... just looking at this you realize you just dont have enough information.
We even made a chart with the information we had.. and it just was not enough
There is NO SOLUTION Not enough information or NEI
Liam has a beautiful lawn that is 8 m longer than it is wide... and it is surrounded by a wonderful flower bed which he and his brother maintain for his mother The flower bed is 5m wide all around. Find the dimensions of the lawn if the area of the flower bed is 140m2
When you try to solve this problem by letting the dimensions of the lawn be w and w + 8 you find the following equation
(w + 10)(w + 18) - (w)(w +8) = 140
however that leads us to
w = -2
Since the width of the lawn cannot be negative,
There is No SOLUTION and the given facts are unrealistic.
Drehan says he has equal number of dimes and quarters but that he has 3 times as many nickels as he has dimes. He also tells us that the value of his nickels and dimes is 50 cents more than the value of his quarters. How many of each kind of coin does he have?
Let d = the number of dimes
well if he has the same number of quarters as he has dimes
then d also can equal the number of quarters
and with the other information
3d = the number of nickels.
Now looking at the information he gave us
10d + 5(3d) = 25d + 50
But that simplifies to
25d = 25d + 50
which is impossible.
There is NO SOLUTION
The given facts are contradictory
Not all word problems have solutions. We listed three of the reasons for this:
1) Not Enough Information ( NEI)
2) Unrealistic Results
3) Facts are contradictory
We used the following examples:
Aurenne drove at her normal speed for the first 2 hours of the trip--- but the road repairs slowed her down 10 mph slower than her normal speed. She made the trip in 3 hours. Find her normal speed.
Wait... just looking at this you realize you just dont have enough information.
We even made a chart with the information we had.. and it just was not enough
There is NO SOLUTION Not enough information or NEI
Liam has a beautiful lawn that is 8 m longer than it is wide... and it is surrounded by a wonderful flower bed which he and his brother maintain for his mother The flower bed is 5m wide all around. Find the dimensions of the lawn if the area of the flower bed is 140m2
When you try to solve this problem by letting the dimensions of the lawn be w and w + 8 you find the following equation
(w + 10)(w + 18) - (w)(w +8) = 140
however that leads us to
w = -2
Since the width of the lawn cannot be negative,
There is No SOLUTION and the given facts are unrealistic.
Drehan says he has equal number of dimes and quarters but that he has 3 times as many nickels as he has dimes. He also tells us that the value of his nickels and dimes is 50 cents more than the value of his quarters. How many of each kind of coin does he have?
Let d = the number of dimes
well if he has the same number of quarters as he has dimes
then d also can equal the number of quarters
and with the other information
3d = the number of nickels.
Now looking at the information he gave us
10d + 5(3d) = 25d + 50
But that simplifies to
25d = 25d + 50
which is impossible.
There is NO SOLUTION
The given facts are contradictory
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