Square Numbers & Square Roots (Continued) 5-3
Before we began---We reviewed the terminology for various numbers using 75 as our example
75---the standard form of the number
3⋅ 5⋅ 5 is it written in expanded prime factorization
3⋅ 52 is it's exponential prime factorization
If you have a prime number all three are exactly the same
For example
41 --- is the standard form of the number
but 41 is also the expanded prime factorization
and 41 ( or 411) is it written in exponential prime factorization.
[You can leave off the exponent 1 for the power of 1 because it is truly invisible!!]
Knowing the SQ's & SQRT's of numbers up to 20 is critical for finding the approximate square root of any non perfect square.
That is,
You know the √36 = 6
and √25 = 5
But what would be
√28 ?
There are several strategies to use.. one involved dividing and taking the average. This is from the yellow worksheet handed out on Friday 12/4
You know that √28 is less than 6 but more than 5
so you start with 5
divide 28 by 5
28 ÷ 5 = 5.6
Now take the average of 5 and 5.6 or
5 + 5.6
2
which equals 5.3
Divide 28 by 5.3 now
28 ÷ 5.3 = equals 5.28 or rounded 5.3
Since the divisor (5.3) and the quotient (5.3) match STOP...and say
√28≈ 5.3
That's fairly complicated... in class we showed some neat tricks about the relationship between the perfect squares...
Using the same number √28
You still think of the perfect square below and above
Stack them
36
√28
25
then take the square roots for those perfect squares
√36 = 6
√28 ≈
√25 = 5
√36 = 6
√28 ≈ 5
√25 = 5
Now, find the difference between your two perfect squares
36 -25 = 11
We discovered that the difference will always be the sum of the square roots of the perfect squares!! WOW!!
That number becomes your denominator
Now, find the difference between your number (in this case 28) and the lower of the two perfect squares (25) so 28-25 = 3
That number becomes the numerator
so you have
3/11
Now, we haven't learned about fractions but you can estimate ( since this is all about estimating... anyway)
and you know
3/10
That can be written as .3
so adding that to your estimate
we can safely estimate
√28 ≈ 5.3
Try finding the √110... with this method..
Friday, December 4, 2009
Math 6H ( Periods 3, 6, & 7)
Least Common Multiple 5-6
We looked at the first few non-zero multiples of 8 and 12
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80...
Multiples of 12: 12, 24, 36, 48, 60, 72, 84 ...
We noticed 24, 48, and 72 appear in both lists. These numbers are called common multiples of 8 and 12 and the LEAST of the multiples is 24...
It is called... the least common multiple.
We write the least common multiple of 8 and 12 as
LCM(8,12) = 24
To find the LCM of two whole numbers you could write out the lists of multiples-- and that works relatively easily with small numbers... but there are more efficient ways to find the least common multiple of two whole numbers.
1. Write out the first few multiples of the larger of the two numbers and test each multiple for divisibility by the smaller number. The first multiple of the larger number that is divisible by the smaller number is the LCM
2. You can use prime factorization to find the LCM. The LCM is EVERY factor to its GREATEST power!!
LCM(54, 60)
54 = 2⋅ 3⋅ 3⋅ 3 = 2⋅ 33
60 = 2⋅ 2⋅ 3⋅ 5 = 22⋅ 3⋅ 5
So the greatest power of 2 is 22
The greatest power of 3 is just 3
and the greatest pwoer of 5 is just 5
so the product of 22⋅ 3⋅ 5 will be the LCM
LCM(54, 60) = 540
3. You may use the BOX method as shown in class... unfortunately it does not show well here. Remember you need to create a L. The numbers on the side of the box represent the GCF!! You need to multiple them with the last row of factors.
See me before or after class if you want any review!!
We reviewed the concept of relatively prime and noticed that any two prime numbers are relatively prime. We also noticed that if two numbers are relatively prime-- neither of them must be prime....
We also found out that if one number is a factor of a second number, the GCF of the two numbers is the first number AND... if one whole number is a factor of a second whole number the LCM of the two numbers is the second number!!
GCF(12,24) = 12
LCM(12,24) = 24
WOW!!
If two whole numbers are relatively prime---
their GCF = 1
and their LCM is their product!!
GCF(8,9) =1
GCF(8,9) = 72
WOW!!
LCM & GCF Story PRoblems
1) Read the problem
2) Re-read the problem!!
3) Figure out what is being asked for!!
4) find the "magic " word... to help you determine if you are finding GCF or LCM
5) When in doubt... draw it out!!
We looked at the first few non-zero multiples of 8 and 12
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80...
Multiples of 12: 12, 24, 36, 48, 60, 72, 84 ...
We noticed 24, 48, and 72 appear in both lists. These numbers are called common multiples of 8 and 12 and the LEAST of the multiples is 24...
It is called... the least common multiple.
We write the least common multiple of 8 and 12 as
LCM(8,12) = 24
To find the LCM of two whole numbers you could write out the lists of multiples-- and that works relatively easily with small numbers... but there are more efficient ways to find the least common multiple of two whole numbers.
1. Write out the first few multiples of the larger of the two numbers and test each multiple for divisibility by the smaller number. The first multiple of the larger number that is divisible by the smaller number is the LCM
2. You can use prime factorization to find the LCM. The LCM is EVERY factor to its GREATEST power!!
LCM(54, 60)
54 = 2⋅ 3⋅ 3⋅ 3 = 2⋅ 33
60 = 2⋅ 2⋅ 3⋅ 5 = 22⋅ 3⋅ 5
So the greatest power of 2 is 22
The greatest power of 3 is just 3
and the greatest pwoer of 5 is just 5
so the product of 22⋅ 3⋅ 5 will be the LCM
LCM(54, 60) = 540
3. You may use the BOX method as shown in class... unfortunately it does not show well here. Remember you need to create a L. The numbers on the side of the box represent the GCF!! You need to multiple them with the last row of factors.
See me before or after class if you want any review!!
We reviewed the concept of relatively prime and noticed that any two prime numbers are relatively prime. We also noticed that if two numbers are relatively prime-- neither of them must be prime....
We also found out that if one number is a factor of a second number, the GCF of the two numbers is the first number AND... if one whole number is a factor of a second whole number the LCM of the two numbers is the second number!!
GCF(12,24) = 12
LCM(12,24) = 24
WOW!!
If two whole numbers are relatively prime---
their GCF = 1
and their LCM is their product!!
GCF(8,9) =1
GCF(8,9) = 72
WOW!!
LCM & GCF Story PRoblems
1) Read the problem
2) Re-read the problem!!
3) Figure out what is being asked for!!
4) find the "magic " word... to help you determine if you are finding GCF or LCM
5) When in doubt... draw it out!!
Tuesday, December 1, 2009
Algebra Period 4
Multiplying Polynomials 5-11
To multiply two polynomials, multiply each term of one polynomial by every term of the other. THEN ADD the results.
The textbook shows a column approach, please see page 249 for instructions.
In class we used the BOX method... and then combined terms.
When you multiply a trinomial by a binomial or two trinomials, it gets really tricky!
2 ways:
1) box method
2)column method (double or triple distributive with columns to combine like terms)
If you have a trinomial times a binomial, it's easier to use the Commutative Property
and make it a binomial times a trinomial:
(x2 + x - 1) (x - 1)
switch it to
(x - 1) (x2 + x - 1)
Remember the following rules
(A + B)(A + B) = (A + B) 2 = A2 + 2AB + B2
(A - B)(A - B) = (A - B)2 = A2 -2AB + B2
(A + B)(A - B)= A2- B2
You can use FOIL to multiply two binomials
remember FOIL is First Terms, Outside Terms, Inside Terms, Last Terms
You can always use FOIL-- or the BOX method but knowing these rules will make computation quicker if you know the above rules!!
To multiply two polynomials, multiply each term of one polynomial by every term of the other. THEN ADD the results.
The textbook shows a column approach, please see page 249 for instructions.
In class we used the BOX method... and then combined terms.
When you multiply a trinomial by a binomial or two trinomials, it gets really tricky!
2 ways:
1) box method
2)column method (double or triple distributive with columns to combine like terms)
If you have a trinomial times a binomial, it's easier to use the Commutative Property
and make it a binomial times a trinomial:
(x2 + x - 1) (x - 1)
switch it to
(x - 1) (x2 + x - 1)
Remember the following rules
(A + B)(A + B) = (A + B) 2 = A2 + 2AB + B2
(A - B)(A - B) = (A - B)2 = A2 -2AB + B2
(A + B)(A - B)= A2- B2
You can use FOIL to multiply two binomials
remember FOIL is First Terms, Outside Terms, Inside Terms, Last Terms
You can always use FOIL-- or the BOX method but knowing these rules will make computation quicker if you know the above rules!!
Math 6H ( Periods 3, 6, & 7)
Greatest Common Factor 5-5
If we list the factors of 30 and 42, we notice
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
We notice that 1, 2, 3, and 6 are all COMMON factors of these two numbers. The number 6 is the greatest of these and therefore is called the
GREATEST COMMON FACTOR of the two numbers. We write
GCF(30,42) = 6
Although listing the factors of two numbers and then comparing their common factors is one way to determine the greatest common factor, using prime factorization is another easy way to find the GCF
Find GCF(54, 72)
54 = 2 ⋅ 3 ⋅ 3 ⋅ 3
72 = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3
Find the greatest power of 2 that occurs IN BOTH prime factorization. The greatest power of 2 that occurs in both is just 2 1
Find the greatest power of 3 that occurs IN BOTH prime factorizations. The greatest power of 3 that occurs in both is 32
Therefore
GCF(54, 72) = 2 ⋅ 32 = 18
In class we circled the common factors and realized that
GCF(54, 72) = 2 ⋅ 3 ⋅ 3 = 18
Fin the GCF( 45, 60)
45 = 3 ⋅ 3⋅ 5
60 = 2⋅ 2⋅ 3⋅ 5
Since 2 is NOT a factor of 45-- there is NO greatest power of 2 that occurs in both prime factorizations.
The greatest power of 3 is just 31
and the greatest power of 5 is just 51
Therefore,
GCF(45,60) = 3⋅ 5 = 15
The number 1 is a common factor of any two whole numbers!! If 1 is the GCF , then the two numbers are said to be RELATIVELY PRIME. Two numbers can be relatively prime even if one or both of them are composite.
Show that 15 and 16 are relatively prime
List the factors of each number
FACTORS of 15: 1, 3, 5, 15
FACTORS of 16: 1, 2, 4, 8, 16
Since the GCF(15,16) = 1. The two numbers are relatively prime!!
If we list the factors of 30 and 42, we notice
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
We notice that 1, 2, 3, and 6 are all COMMON factors of these two numbers. The number 6 is the greatest of these and therefore is called the
GREATEST COMMON FACTOR of the two numbers. We write
GCF(30,42) = 6
Although listing the factors of two numbers and then comparing their common factors is one way to determine the greatest common factor, using prime factorization is another easy way to find the GCF
Find GCF(54, 72)
54 = 2 ⋅ 3 ⋅ 3 ⋅ 3
72 = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3
Find the greatest power of 2 that occurs IN BOTH prime factorization. The greatest power of 2 that occurs in both is just 2 1
Find the greatest power of 3 that occurs IN BOTH prime factorizations. The greatest power of 3 that occurs in both is 32
Therefore
GCF(54, 72) = 2 ⋅ 32 = 18
In class we circled the common factors and realized that
GCF(54, 72) = 2 ⋅ 3 ⋅ 3 = 18
Fin the GCF( 45, 60)
45 = 3 ⋅ 3⋅ 5
60 = 2⋅ 2⋅ 3⋅ 5
Since 2 is NOT a factor of 45-- there is NO greatest power of 2 that occurs in both prime factorizations.
The greatest power of 3 is just 31
and the greatest power of 5 is just 51
Therefore,
GCF(45,60) = 3⋅ 5 = 15
The number 1 is a common factor of any two whole numbers!! If 1 is the GCF , then the two numbers are said to be RELATIVELY PRIME. Two numbers can be relatively prime even if one or both of them are composite.
Show that 15 and 16 are relatively prime
List the factors of each number
FACTORS of 15: 1, 3, 5, 15
FACTORS of 16: 1, 2, 4, 8, 16
Since the GCF(15,16) = 1. The two numbers are relatively prime!!
Monday, November 30, 2009
Math 6H ( Periods 3, 6, & 7)
Prime Numbers & Composite Numbers 5-4
A prime number is one that has only two factors: 1 and the number itself, such as 2, 3, 5, 7, 11, 13...
A counting number that has more than two factors is called a composite number, such as 4, 6, 8, 9, 10...
Since one has exactly ONE factor, it is NEITHER PRIME NOR COMPOSITE!!
Zero is also NEITHER PRIME NOR COMPOSITE!!
Every counting number greater than 1 has at least one prime factor -- which may be the number itself.
You can factor a number into PRIME FACTORS by using a factor tree or the inverted division, as shown in class.
Using the inverted division, you also start with the smallest prime number that is a factor... and work down
give the prime factors of 42
2⎣42
3⎣21
7
When we write 42 as 2⋅3⋅7 this product of prime factors is called the prime factorization of 42.
Two is the only even prime number because all the other even numbers have two as a factor.
Explain how you know that each of the following numbers must be composite...
111; 111,111; 111,111,111; and so on....
Using your divisibility rules you notice that the sums of the digits are multiples of 3.
List all the possible digits that can be the last digit of a prime number that is greater than 10.
1, 3, 7, 9.
Choose any six digit number such that the last three digits are a repeat of the first three digits. For example
652,652. You will find that 7, 11, and 13 are all factors of that number... no matter what number you choose... why is that???? email me your response.
A prime number is one that has only two factors: 1 and the number itself, such as 2, 3, 5, 7, 11, 13...
A counting number that has more than two factors is called a composite number, such as 4, 6, 8, 9, 10...
Since one has exactly ONE factor, it is NEITHER PRIME NOR COMPOSITE!!
Zero is also NEITHER PRIME NOR COMPOSITE!!
Every counting number greater than 1 has at least one prime factor -- which may be the number itself.
You can factor a number into PRIME FACTORS by using a factor tree or the inverted division, as shown in class.
Using the inverted division, you also start with the smallest prime number that is a factor... and work down
give the prime factors of 42
2⎣42
3⎣21
7
When we write 42 as 2⋅3⋅7 this product of prime factors is called the prime factorization of 42.
Two is the only even prime number because all the other even numbers have two as a factor.
Explain how you know that each of the following numbers must be composite...
111; 111,111; 111,111,111; and so on....
Using your divisibility rules you notice that the sums of the digits are multiples of 3.
List all the possible digits that can be the last digit of a prime number that is greater than 10.
1, 3, 7, 9.
Choose any six digit number such that the last three digits are a repeat of the first three digits. For example
652,652. You will find that 7, 11, and 13 are all factors of that number... no matter what number you choose... why is that???? email me your response.
Algebra Period 4
Multiplying Binomials: Special Products 5-10
LEARN TO RECOGNIZE SOME SPECIAL PRODUCTS - IT MAKES IT EASIER!
Remember: You can FOIL these just like the other products until you remember these special patterns....but when we get to factoring next week, it will really help you to know these patterns by heart.
When you do, you actually don't need to show any work because you do it in your head!
(That should make a lot of you happy! :)
DIFFERENCE OF TWO SQUARES:
You will notice that the two factors are IDENTICAL except they have DIFFERENT SIGNS
(x + 6)(x - 6) =
x2 - 6x + 6x - 36 =
x2 - 36
This will happen every time!
The middle terms are additive inverses so they become zero.
You're left with a difference (subtraction) of two terms that are squared.
SQUARING A BINOMIAL:
When you multiply one binomial by itself (squaring it), you end up with:
First term squared + twice the product of both terms + last term squared
(x + 6)2 =
(x + 6)(x + 6) =
x2 + 2(6x) + 62 =
x2 + 12x + 36
If you foiled you would have:
x2 + 6x + 6x + 36
CAN YOU SEE THAT THE 2 MIDDLE TERMS ARE JUST DOUBLING UP???
WHY???
Another example with subtraction in the middle:
(x - 6)2 =
(x - 6)(x - 6) =
x2 + 2(-6x) + 62 =
x2 - 12x + 36
If you foiled you would have:
x2 - 6x - 6x + 36
CAN YOU SEE THAT THE 2 MIDDLE TERMS ARE JUST DOUBLING UP???
WHY???
PLEASE NOTE:
NOTICING THESE SPECIAL PRODUCTS HELPS YOU DO THESE MULTIPLICATIONS FASTER!
IF YOU EVER FORGET THEM, JUST FOIL!
(but you will need to recognize them for factoring in Chapter 6)
LEARN TO RECOGNIZE SOME SPECIAL PRODUCTS - IT MAKES IT EASIER!
Remember: You can FOIL these just like the other products until you remember these special patterns....but when we get to factoring next week, it will really help you to know these patterns by heart.
When you do, you actually don't need to show any work because you do it in your head!
(That should make a lot of you happy! :)
DIFFERENCE OF TWO SQUARES:
You will notice that the two factors are IDENTICAL except they have DIFFERENT SIGNS
(x + 6)(x - 6) =
x2 - 6x + 6x - 36 =
x2 - 36
This will happen every time!
The middle terms are additive inverses so they become zero.
You're left with a difference (subtraction) of two terms that are squared.
SQUARING A BINOMIAL:
When you multiply one binomial by itself (squaring it), you end up with:
First term squared + twice the product of both terms + last term squared
(x + 6)2 =
(x + 6)(x + 6) =
x2 + 2(6x) + 62 =
x2 + 12x + 36
If you foiled you would have:
x2 + 6x + 6x + 36
CAN YOU SEE THAT THE 2 MIDDLE TERMS ARE JUST DOUBLING UP???
WHY???
Another example with subtraction in the middle:
(x - 6)2 =
(x - 6)(x - 6) =
x2 + 2(-6x) + 62 =
x2 - 12x + 36
If you foiled you would have:
x2 - 6x - 6x + 36
CAN YOU SEE THAT THE 2 MIDDLE TERMS ARE JUST DOUBLING UP???
WHY???
PLEASE NOTE:
NOTICING THESE SPECIAL PRODUCTS HELPS YOU DO THESE MULTIPLICATIONS FASTER!
IF YOU EVER FORGET THEM, JUST FOIL!
(but you will need to recognize them for factoring in Chapter 6)
Subscribe to:
Posts (Atom)
