Saturday, October 29, 2011
Mr. W. the numbers man: So, what is √2 anyway?
Mr. W. the numbers man: So, what is √2 anyway?: When students, in 7th grade, encounter the Pythagorean Theorem – a² + b² = c² – they must also run into the concept of SQUARE ROOTS. So...
So, what is √2 anyway?
When students, in 7th grade, encounter the Pythagorean Theorem – a² + b² = c² –
they must also run into the concept of SQUARE ROOTS. So, lets get a historical
timeline of Math thus far.
Whole Numbers and Counting Numbers: These are good, I can count pennies
Fractions: OMG, just when I was getting used to counting you brake things into
halves, fifths, and tenths.
Decimals: Do I divide the bottom into the top or the other way?
Integers: Now you have me going "which way"?
Square Root or at times Irrational Numbers: Hay, I don't like irrational
people, why would I like irrational numbers. These numbers
cannot even be written as a fraction.
And with all of these we have the four basic operations that the student must
do with each. I know that you look at this list, and it is easy to say, "I will stick
to counting pennies". Folks it is only the 7th grade, and we have to add in
graphing, probability, statistics, equations and a few other notions.
Confused yet?
So, WHAT IS THE √2?
We know that 5² = 25, thus √25 = 5.
That means that √49 = 7. That is correct.
The square root of a perfect square is a whole number.
The √1 = 1 and the √4 = 2. That means that the √2 is between 1 and 2.
Now you understand. We are looking for a number between 1 and 2 that
when squared is the number 2.
This is when things begin to get sticky. Remember that value π or pi.
3.1415926535897932346...
Pi is a transcendental number, its decimal never repeats or terminates.
√2 is the same, 1.414213562...
If you square the value above you will get 1.99999999, close to but
not 2. Because we have this unending decimal we have no way of writing
this number as a fraction a ÷ b. Thus we call it an
IRRATIONAL NUMBER
The √3 is 1.7320508...
This is another irrational number. Now we must do our operations
with these numbers. Treat them like a variable, because you really
do not know what there true value is. I will discuss this on a separate
page.
they must also run into the concept of SQUARE ROOTS. So, lets get a historical
timeline of Math thus far.
Whole Numbers and Counting Numbers: These are good, I can count pennies
Fractions: OMG, just when I was getting used to counting you brake things into
halves, fifths, and tenths.
Decimals: Do I divide the bottom into the top or the other way?
Integers: Now you have me going "which way"?
Square Root or at times Irrational Numbers: Hay, I don't like irrational
people, why would I like irrational numbers. These numbers
cannot even be written as a fraction.
And with all of these we have the four basic operations that the student must
do with each. I know that you look at this list, and it is easy to say, "I will stick
to counting pennies". Folks it is only the 7th grade, and we have to add in
graphing, probability, statistics, equations and a few other notions.
Confused yet?
So, WHAT IS THE √2?
We know that 5² = 25, thus √25 = 5.
That means that √49 = 7. That is correct.
The square root of a perfect square is a whole number.
The √1 = 1 and the √4 = 2. That means that the √2 is between 1 and 2.
Now you understand. We are looking for a number between 1 and 2 that
when squared is the number 2.
This is when things begin to get sticky. Remember that value π or pi.
3.1415926535897932346...
Pi is a transcendental number, its decimal never repeats or terminates.
√2 is the same, 1.414213562...
If you square the value above you will get 1.99999999, close to but
not 2. Because we have this unending decimal we have no way of writing
this number as a fraction a ÷ b. Thus we call it an
IRRATIONAL NUMBER
The √3 is 1.7320508...
This is another irrational number. Now we must do our operations
with these numbers. Treat them like a variable, because you really
do not know what there true value is. I will discuss this on a separate
page.
Wednesday, October 26, 2011
Division a new way?
Hi everyone, it has been a couple of weeks, however just the other day a couple
was talking to me about their son. They told me he was having trouble with
division and they did not understand the method they were using. It has been
popular now for a few years as it leads us into Algebra easier.
EXAMPLE: X³ + 2X² – X = X³ + 2X² – X = X² + 2X – 1
X X X X
So how do we change Arithmetic to look like this?
Does it effect the way we learned to do long division?
How can we help our children?
EXAMPLE: 552 ÷ 12 or 500 + 50 + 2 = 500 + 50 + 2
12 12 12 12
We need to recognize how 0 helps us.
48 ÷ 12 = 4, now we ask about 500. 480
is close to 500, thus for our first part
we can use 40. 40 x 12 = 480, and we
have 20 left over to add to the 50. Now
70 ÷ 12, 5 times with 10 left over to add
to the 2. 12 ÷ 12 = 1. Now just add the
3 quotients. 40 + 5 + 1 = 46
552 ÷ 12 = 46
Our answer is the same as if we did it the old fashion way. However,
here the student need to understand multiplication by 10 and the
power of 0 much easier then we needed to understand. But is this being
done in the first 3 years of school. The other day I asked a 3rd grade
class why do we write 10. The best answer I heard was, "its the
number that comes after 9".
They had gotten to the tens place but had no idea why it was the 10's.
They hadn't because they thought the first digit was 1. Nobody had
talked to them about 0, and that our digits start at 0 and go to 9, 0 to 9
over and over. That never changes. When we get to 99 we have to
next have 100 and the 1 is in the hundreds place because 100 is the
1st number in that place.
EXAMPLE: 1494 ÷ 18 or 1000 + 400 + 90 + 4 =
1000 + 400 + 90 + 4 = 50 + 20 + 10 + 3 = 83
18 18 18 18
1494 ÷ 18 = 83
I know this seems new but we are still dividing into multiplying and
then subtracting until we get to our ones place. Parents need not get upset
about the process. Enjoy the process and help your children.
was talking to me about their son. They told me he was having trouble with
division and they did not understand the method they were using. It has been
popular now for a few years as it leads us into Algebra easier.
EXAMPLE: X³ + 2X² – X = X³ + 2X² – X = X² + 2X – 1
X X X X
So how do we change Arithmetic to look like this?
Does it effect the way we learned to do long division?
How can we help our children?
EXAMPLE: 552 ÷ 12 or 500 + 50 + 2 = 500 + 50 + 2
12 12 12 12
We need to recognize how 0 helps us.
48 ÷ 12 = 4, now we ask about 500. 480
is close to 500, thus for our first part
we can use 40. 40 x 12 = 480, and we
have 20 left over to add to the 50. Now
70 ÷ 12, 5 times with 10 left over to add
to the 2. 12 ÷ 12 = 1. Now just add the
3 quotients. 40 + 5 + 1 = 46
552 ÷ 12 = 46
Our answer is the same as if we did it the old fashion way. However,
here the student need to understand multiplication by 10 and the
power of 0 much easier then we needed to understand. But is this being
done in the first 3 years of school. The other day I asked a 3rd grade
class why do we write 10. The best answer I heard was, "its the
number that comes after 9".
They had gotten to the tens place but had no idea why it was the 10's.
They hadn't because they thought the first digit was 1. Nobody had
talked to them about 0, and that our digits start at 0 and go to 9, 0 to 9
over and over. That never changes. When we get to 99 we have to
next have 100 and the 1 is in the hundreds place because 100 is the
1st number in that place.
EXAMPLE: 1494 ÷ 18 or 1000 + 400 + 90 + 4 =
1000 + 400 + 90 + 4 = 50 + 20 + 10 + 3 = 83
18 18 18 18
1494 ÷ 18 = 83
I know this seems new but we are still dividing into multiplying and
then subtracting until we get to our ones place. Parents need not get upset
about the process. Enjoy the process and help your children.
Wednesday, October 5, 2011
A Sad Day
Steve Jobs died today. Age 56, founder of Apple, and developer of the computer
I am working on. He helped change the way we communicate with each other.
He brought us closer together. His Apple products are spread all over the world,
and because of that the world knows us better as a country and as individuals.
Some will say that "Arab Spring" is a direct result of the social network that has
been built on these products. We do not write any more, we "Tweet", "I Chat", or
"Skype". Imagine two bio-chemists on opposite sides of the world discussing what
each is thinking about, in order to reach a common goal. Mr Jobs created a whole
new paradigm for us to live in.
Consider what is happening outside of Wall Street, on Main Street, and across the
country. How quickly those protests were developed through the web and our ability
to connect with like minds. Regardless of the result, thank you Mr. Jobs for having
created what you did. I hope that those who follow can keep your dreams alive much
as Walt Disney's family did for him, even though he did not live to see Disney World
open in Orlando.
I am working on. He helped change the way we communicate with each other.
He brought us closer together. His Apple products are spread all over the world,
and because of that the world knows us better as a country and as individuals.
Some will say that "Arab Spring" is a direct result of the social network that has
been built on these products. We do not write any more, we "Tweet", "I Chat", or
"Skype". Imagine two bio-chemists on opposite sides of the world discussing what
each is thinking about, in order to reach a common goal. Mr Jobs created a whole
new paradigm for us to live in.
Consider what is happening outside of Wall Street, on Main Street, and across the
country. How quickly those protests were developed through the web and our ability
to connect with like minds. Regardless of the result, thank you Mr. Jobs for having
created what you did. I hope that those who follow can keep your dreams alive much
as Walt Disney's family did for him, even though he did not live to see Disney World
open in Orlando.
Tuesday, October 4, 2011
Mr. W. the numbers man: FACTORING TRINOMIALS
Mr. W. the numbers man: FACTORING TRINOMIALS: On this page I will be discussing Mr. W.'s two magic numbers, (my way of factoring Trinomials.) If you have had Algebra I or are taking i...
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