If you are about to send your youngster or youngsters off to Kindergarten in the Fall you can be a great deal of help before they go. In Math they will be asked to know their numbers to 100, write them, and count from ten back to 0. Many have their reasons why it is difficult, but to me it is all about patterns. You constantly are building patterns into your children by eating at certain times, having a bed time, getting toilet trained, and knowing how to get dressed just to name a few. Math is a pattern of numbers using 10 digits starting with 0.
0 10 20 90
1 These are all the digits so what 11 Next ? 21 We continue this 91
2 is next? Our digits again with 12 22 pattern until 92
3 a 1 in front. 13 23 93
4 14 24 94
5 15 25 95
6 16 26 96
7 17 27 97
8 18 28 98
9 19 29 99
When we get to 99 we must add another place and we get 100.
Tuesday, July 17, 2012
What really happened
I do not like to step away from Math, but in this article I feel I must. In late 2008 we as a country fell into a terrible recession. The worst since our depression of 1929. Because our banks had been doing what they wanted for 8 years without anybody looking out for our well being, the loans that they had poorly made came crashing back on them. At the same time we got a new President, the first black president of our history. Color means nothing to me, this man is a Harvard graduate, and Illinois Senator. But as Bush walked away, our economy fell apart. It was placed on President Obama's shoulders to turn this country around.
Eight million people had lost their jobs in the 3 months that he waited to raise his hand no January 20th of 2009 and begin to right our sinking ship. As he took this oath to do the right thing, the head of the Republican Party had gathered his leaders together to state, "we must do everything we can to make this man a one term president". They could care less about the American people, and the recession we had fallen into. They would spend the next 3½ writing bills on abortion that did nothing to solve the problems of our country. We had so many problems, but these mean spirited yahoos only wanted to get the ball back. They would do nothing to help the President get our economy going.
Also, in 2009 we suddenly saw the rise of the "Tea Party". Their push was that the government would be spending to much money, and raising taxes. We were told that this was a grass roots party that was about taxes on middle Americans. But if Obama has decreased the middle class taxes several times during his tenure to help small business and the middle class, what was their true agenda.
I believe that these people are the racists of our country and they could not speak out so they hid behind a bogus political party that was funded by millionaires, and billionaires. At the same time Republican Governors began writing bills to make it difficult for seniors, college students, and minorities to vote. They kept telling us that it was all about "voter fraud". However, we know that this is a big lie. If a war veteran has been voting since he returned from Europe during World War II, and has been voting every 4 years for the person he wants to be President, how does he suddenly become ineligible to vote. I would ask these bigots to prove this man can not vote. When we remove the rights of a few citizens, no matter the reason, we begin the removal of those rights for all.
As we approach this 2012 election for President, my question is, will it be one that is bought by a few?,
or taken because many were denied their right to vote?, or lost to a bunch of lies played out over the television? My Grandmother once told me that this country would have a war between the races. I believe that she was almost right, however the war might be between the racists and the non-racists. To those
who have forgotten their history the first three words of the Constitution are WE THE PEOPLE. If it does not work for everyone then it works for no one.
Eight million people had lost their jobs in the 3 months that he waited to raise his hand no January 20th of 2009 and begin to right our sinking ship. As he took this oath to do the right thing, the head of the Republican Party had gathered his leaders together to state, "we must do everything we can to make this man a one term president". They could care less about the American people, and the recession we had fallen into. They would spend the next 3½ writing bills on abortion that did nothing to solve the problems of our country. We had so many problems, but these mean spirited yahoos only wanted to get the ball back. They would do nothing to help the President get our economy going.
Also, in 2009 we suddenly saw the rise of the "Tea Party". Their push was that the government would be spending to much money, and raising taxes. We were told that this was a grass roots party that was about taxes on middle Americans. But if Obama has decreased the middle class taxes several times during his tenure to help small business and the middle class, what was their true agenda.
I believe that these people are the racists of our country and they could not speak out so they hid behind a bogus political party that was funded by millionaires, and billionaires. At the same time Republican Governors began writing bills to make it difficult for seniors, college students, and minorities to vote. They kept telling us that it was all about "voter fraud". However, we know that this is a big lie. If a war veteran has been voting since he returned from Europe during World War II, and has been voting every 4 years for the person he wants to be President, how does he suddenly become ineligible to vote. I would ask these bigots to prove this man can not vote. When we remove the rights of a few citizens, no matter the reason, we begin the removal of those rights for all.
As we approach this 2012 election for President, my question is, will it be one that is bought by a few?,
or taken because many were denied their right to vote?, or lost to a bunch of lies played out over the television? My Grandmother once told me that this country would have a war between the races. I believe that she was almost right, however the war might be between the racists and the non-racists. To those
who have forgotten their history the first three words of the Constitution are WE THE PEOPLE. If it does not work for everyone then it works for no one.
Wednesday, March 7, 2012
Pythagorean Triples
I guess it was the shirt I was wearing that reads "Math is my bag" on the
front that brought this gentleman over to introduce himself today. My wife
and I were at the gym doing our workouts at the time. Although he was
not a Mathematician, numbers fascinated this man as they do me. He said
that he had noticed that if you square an odd number and use ½ of the even
numbers on each side of that square you get the three lengths of a right
triangle. He wanted to now if that was always true. I said starting with 3
and getting greater that is true for all odd numbers.
From above 9² = 81 ½(80) = 40 ½(82) = 41
Thus 9² + 40² = 41²
He then asked me about even numbers. I said, "do you mean 8–15–17 for
example". "Yes," he said. then added. "How do we get those"?
I had to remind him of a method of always getting a Pythagorean Triple.
You chose any two integers greater than 0, for example a and b where
a > b. Now enter these numbers into the following three phrases.
1. a² + b²
2. a² – b²
3. 2ab
Example: a = 7 and b = 4
a² + b² = 49 + 16 = 65
a² – b² = 49 – 16 = 33
2ab = 2(28) = 56 ➔ 33² + 56² = 65²
1089 + 3136 = 4225
4225 = 4225
Example: a = 6 and b = 4
a² + b² = 36 + 16 = 52
a² – b² = 36 – 16 = 20
2ab = 2(24) = 48 20² + 48² = 52²
400 + 2304 = 2704
2704 = 2704
also if 20 – 40 – 52 divide by 4
5 – 12 – 13
5² + 12² = 13² (our 2nd basic odd triple)
25 + 144 = 169
169 = 169
Try some on your own and find some more of those even numbered one
like 20 – 21 – 29.
front that brought this gentleman over to introduce himself today. My wife
and I were at the gym doing our workouts at the time. Although he was
not a Mathematician, numbers fascinated this man as they do me. He said
that he had noticed that if you square an odd number and use ½ of the even
numbers on each side of that square you get the three lengths of a right
triangle. He wanted to now if that was always true. I said starting with 3
and getting greater that is true for all odd numbers.
From above 9² = 81 ½(80) = 40 ½(82) = 41
Thus 9² + 40² = 41²
He then asked me about even numbers. I said, "do you mean 8–15–17 for
example". "Yes," he said. then added. "How do we get those"?
I had to remind him of a method of always getting a Pythagorean Triple.
You chose any two integers greater than 0, for example a and b where
a > b. Now enter these numbers into the following three phrases.
1. a² + b²
2. a² – b²
3. 2ab
Example: a = 7 and b = 4
a² + b² = 49 + 16 = 65
a² – b² = 49 – 16 = 33
2ab = 2(28) = 56 ➔ 33² + 56² = 65²
1089 + 3136 = 4225
4225 = 4225
Example: a = 6 and b = 4
a² + b² = 36 + 16 = 52
a² – b² = 36 – 16 = 20
2ab = 2(24) = 48 20² + 48² = 52²
400 + 2304 = 2704
2704 = 2704
also if 20 – 40 – 52 divide by 4
5 – 12 – 13
5² + 12² = 13² (our 2nd basic odd triple)
25 + 144 = 169
169 = 169
Try some on your own and find some more of those even numbered one
like 20 – 21 – 29.
Saturday, February 11, 2012
NEGATIVE EXPONENTS
One of the most bothersome concepts in Algebra or Pre–Algebra in some
places is the negative exponent. We know that the sign of the exponent
has nothing to do with the sign of the number, but tell that to a learning
Algebra I student. This is a great place for my preaching of patterns in
Math. If you have not done this already I suggest you try.
2^4 = 16 A place to start, use any base
2^3 = 8 8 is ½ of 16
2^2 = 4 4 is ½ of 8
2^1 = 2
2^0 = 1
2^–1 = ½
2^–2 = ¼ or 1÷ 2^2
2^–3 = 1/8 or 1 ÷ 2^3
As we see, having a negative exponent makes things smaller
but not negative. Keep patterns in mind when you teach.
Good luck.
a^–n = 1/a^n
Thursday, February 2, 2012
THE WARM–UP
In all my years of teaching I always felt it was important to start a
class with some sort of warm-up. As I have observed many other
colleagues I found that they also started a class with some sort of
warm–up. This need comes in many forms, but I believe the basic
idea is to get the student thinking about math in some way.
In my high school career I cannot remember ever having a
warm–up in my classes. We just started with the assignment.
Even my own Master Teacher had no need for one. In college one
of my favorite professors was Dr. Spazito, a Physics teacher.
He would come into class with the text and a new piece of
chalk. He would then write the last equation on the board
from the previous lecture, entertain some questions, and go
on from there.
I was never that great a Physics student, but I looked forward
to his class, because my notes were like the continuing of a
text we were writing. I guess sometimes we do not need to
warm-up anybody. However, that is college not junior high
or high school.
There are several great warm–ups that I used in my career, and
here I would like to mention just a few.
1. SUDOKU: We have all seen these puzzles in news-
papers, and there are books of hundreds of
them. There is a sight on line that that contain
millions of them at all levels.
In Algebra we try to get rid of all those things around the
"x" the find what the value of "x" is. I a Sudoku we remove
all the possible choices that could go into a square, so that
we know what number goes into the square. My students
loved these and some even wanted harder ones for extra
credit.
2. SMALL NUMBER OPERATIONS:
This is simply a running list of numbers with operations
in between. It is great practice of those basic operations
in small numbers that most of your students can do well
at. You can make up hundreds of these, many of which
are great patterns.
Example: Start with 2, add 2, times 2, times 6, add 3,
divide by 3.
Many times I would have to repeat the list, but at
a slower pace. So, I would say, "this time I will
do it in Southern". You need a Texas accent for
this.
Example: Start with 3, plus 3, times 3, minus 3, divide by 3
Start with 5, plus 5, times 5, minus 5, divide by 5
Start with 6, plus 6, times 6, minus 6, divide by 6
Here they may see the pattern in the solutions and
know the answer before you get there.
3. LARGE NUMBER MULTIPLICATION:
Next try multiplying 46 x 102. You want them to see that
we are really doing 46(100 + 2). Which they can do in their
heads as 4692. The key is to break the habit of always
grabbing for the calculator, and for the student to use their
own best calculator, their mind.
class with some sort of warm-up. As I have observed many other
colleagues I found that they also started a class with some sort of
warm–up. This need comes in many forms, but I believe the basic
idea is to get the student thinking about math in some way.
In my high school career I cannot remember ever having a
warm–up in my classes. We just started with the assignment.
Even my own Master Teacher had no need for one. In college one
of my favorite professors was Dr. Spazito, a Physics teacher.
He would come into class with the text and a new piece of
chalk. He would then write the last equation on the board
from the previous lecture, entertain some questions, and go
on from there.
I was never that great a Physics student, but I looked forward
to his class, because my notes were like the continuing of a
text we were writing. I guess sometimes we do not need to
warm-up anybody. However, that is college not junior high
or high school.
There are several great warm–ups that I used in my career, and
here I would like to mention just a few.
1. SUDOKU: We have all seen these puzzles in news-
papers, and there are books of hundreds of
them. There is a sight on line that that contain
millions of them at all levels.
In Algebra we try to get rid of all those things around the
"x" the find what the value of "x" is. I a Sudoku we remove
all the possible choices that could go into a square, so that
we know what number goes into the square. My students
loved these and some even wanted harder ones for extra
credit.
2. SMALL NUMBER OPERATIONS:
This is simply a running list of numbers with operations
in between. It is great practice of those basic operations
in small numbers that most of your students can do well
at. You can make up hundreds of these, many of which
are great patterns.
Example: Start with 2, add 2, times 2, times 6, add 3,
divide by 3.
Many times I would have to repeat the list, but at
a slower pace. So, I would say, "this time I will
do it in Southern". You need a Texas accent for
this.
Example: Start with 3, plus 3, times 3, minus 3, divide by 3
Start with 5, plus 5, times 5, minus 5, divide by 5
Start with 6, plus 6, times 6, minus 6, divide by 6
Here they may see the pattern in the solutions and
know the answer before you get there.
3. LARGE NUMBER MULTIPLICATION:
Next try multiplying 46 x 102. You want them to see that
we are really doing 46(100 + 2). Which they can do in their
heads as 4692. The key is to break the habit of always
grabbing for the calculator, and for the student to use their
own best calculator, their mind.
Thursday, January 26, 2012
OPPOSITES AND EXPONENTS
It was 1st period, and the class was Algebra I. I put he following on the board:
– 3² = A) –6 B) 6 C) –9 D) 9
Sadly of the 30 students 28 chose D, and 1 chose C, and 1 chose A.
I then asked 3 of them to read the problem out loud. They all said,
"negative 3 squared"
This is what they were saying (–3)², or –3 the quantity squared.
This is why 28 of them were incorrect by sign and another has no idea what
an exponent does. One of them was correct, because we know the answer is
C. This is why I strongly suggest that we teach our students to call this sign
"–", "the opposite of". Now read the problem.
"the opposite of", 3 squared
The value of 3 squared is 9, and its opposite is –9.
This was a lead in to the question what is;
– x²; x = – 5 or x = 7
the opposite of (–5)² or the opposite of (7)²
x = – 25 or x = – 49
I hope these comments will help.
– 3² = A) –6 B) 6 C) –9 D) 9
Sadly of the 30 students 28 chose D, and 1 chose C, and 1 chose A.
I then asked 3 of them to read the problem out loud. They all said,
"negative 3 squared"
This is what they were saying (–3)², or –3 the quantity squared.
This is why 28 of them were incorrect by sign and another has no idea what
an exponent does. One of them was correct, because we know the answer is
C. This is why I strongly suggest that we teach our students to call this sign
"–", "the opposite of". Now read the problem.
"the opposite of", 3 squared
The value of 3 squared is 9, and its opposite is –9.
This was a lead in to the question what is;
– x²; x = – 5 or x = 7
the opposite of (–5)² or the opposite of (7)²
x = – 25 or x = – 49
I hope these comments will help.
Monday, January 9, 2012
SQUARING A 3–DIGIT NUMBER
teachers, even Middle School, will not be interested in my last blog, however
why stop there. Let's look at squaring a 3–digit number.
Last time it was: (a + b)(a + b) = a² + 2ab + b²
This time we have: (a + b + c)(a + b + c) =
a² + ab + ac + ba + b² + bc + ca + cb + c² =
a² + 2ab + b² + 2bc + c² + 2ac
So, when we look a number like 426, we need to look at it as
400 + 20 + 6 = 426
Square it: (400 + 20 + 6)(400 + 20 + 6)
We only need to use our counting digits, 4, 2, 6, with the help
of our zero to answer our question.
a = 400, b = 20, and c = 6
Thus, 400² + 2(400)(20) + 20² + 2(20)(6) + 6² +
2(400)(6)
1. 400² = 160000
2. 2(400)(20) = 16000
3. 20² = 400
4. 2(20)(6) = 240
5. 6² = 36
6. 2(400)(6) = 4800
Grand total is 181476
You may not be able to do something like this in your head
without a great deal of practice, but what if the number you
were squaring had a zero in it, like 403, or 260.
In 403 the b value would be 0, and the only therms in the formula
you would need would be a², 2ac, and c²
400² = 160000
2(400)(3) = 2400
3² = 9 TOTAL = 162409
260² = 200² + 2(200)(60) + 60²
= 40000 + 24000 + 3600
TOTAL = 67600
These last two examples, with practice, you can do in you
head and will show you how to use zeros correctly
when multiplying. I find that these kinds of mental
practices improve the mind. These will work well as
warm–up problems in the morning, or at the beginning
of any Math class.
Friday, January 6, 2012
SQUARING A 2-DIGIT NUMBER
Remember those first days of multiplying? We memorized our
2's, 3's, 5's, 10's, and struggled with our our 7's. If you do, you
are much younger than I, but then again, except for the baby–
boomers, everybody else is.
After memorizing everything from 1 x 1 = 1 to 12 x 12 = 144
we began to multiply things like
345 x 8 = 2760
Or did we think
(345 x 10) – (345 x 2) =
3450 – 690 = 2760
I know the second method was never mentioned, however I
thought it might be a good lead into what I am about to tell
you. That chart you memorized had a diagonal. These were
the perfect squares.
Why stop at 12 x 12 = 144. What if we could square any
2–digit number in your head. You might not think that it
would come in handy, but it gives you ideas about how you
can manipulate numbers to make what you are doing easier.
What is 93² or 93 x 93.
Well it is 8649, and I did not need a calculator to know it.
I simply added 8100 + 540 + 9 in my head.
Say what?
Well, you need to look at the problem as;
(90 + 3) (90 + 3)
and multiply every number in the 1st parenthesis by each
in the 2nd parenthesis.
90 x 90 = 8100
90 x 3 = 270 (twice) which makes 540
3 x 3 = 9
Thus, 8100 + 540 + 9 = 8649
You have to know how zeros work, but otherwise you are
just multiplying 9 x 9, 3 x 9, and 3 x 3.
It all comes from Algebra, and the F–O–I–L system.
Lets try 67 x 67, or (60 + 7)(60 + 7)
60 x 60 = 3600
60 x 7 = 420
7 x 60 = 420
7 x 7 = 49/ 3600 + 840 + 49 = 4449
With practice you can do these in your head.
Sunday, January 1, 2012
THE NEW YEAR
Its January 1st, 2012. My dad died four years ago today, and he was the
last of his generation in our family. I miss him, like I miss my mom who
died 3 years before him. But as we all march towards our own end, we are
looking at a year that for many brings big questions.
What about 12-21-12 and the end of the world. The alignment of our
planets with the center of our galaxy. The Mayans said it, so we should
all believe it. It truly amazes me what we believe, and what we spend
our whole lives trusting in. Two things that I do know, on 12-22-12
1. 40% of 250 will still be 100
2. The firs derivative of 5X³ will be 15X²
Math will not change, 1 will still be less than 3, and ∏ will be a
transcendental number 3.14159265. . .. Here we are on this tiny
little planet, in a universe of billions of galaxies, in a galaxy of
billions of stars, and most of us believe that we are the most
intelligent in the universe, and that one "god" did it all. Many of
us spend our entire life worrying about the next life time, when
we need to get along in this life time, and are not doing such
a great job.
I am not saying that we should not have our beliefs, but they
should not destroy our ability to get along with each other.
So as this year begins, let us strive to get along better, and
recognize what must be true, and spend less time debating
what might be true.
Friday, December 23, 2011
ANOTHER LOOK AT INTEGERS
I think what sparked my reason to look at this again was a
resent visit to my bank. I asked them for our balance after
a deposit, and then came home. I gave that information to
my incredibly copious spouse, who has been the guardian
of our check book for the last 40 years. The gap between
their amounts was more than I could understand. Somehow,
had a positive become a negative somewhere, and if so, this
could be a big mess.
Imagine you had a deposit of $500, but in your check book
recording you subtracted. You would be off by $1000. The
simple idea of adding integers seams to be such a problem
for so many. It is all about getting back to zero. If I earn
$200, but have $200 in bills to pay, when I am done I have
no money at all.
All we need ask now is which came first, my earnings or
my bills.
$200 + (–$200) = 0 or
(–$200) + $200 = 0
Why is this confusing?
Students know that 5 + 4 = 4 + 5, don't they?
What are we not being clear about?
Perhaps we just do not explain ourselves. Let me show
you what I mean.
We start with our digits 0 – 9 and we have hundreds
of worksheets with 7 + 2 = __ and 5 + 1 = __ .
Most of our students fill in the blanks, with little or
no retention, because all they do is stare and write
a number. Where it goes in those fine minds is
anyone's guess.
Try having them write the entire problem and the
answer with the = sign. Now their minds have the
whole story to place somewhere. Even better,
7 + (+ 2 ) = __ Read as: 7 + a positive 2 = 9
Then when they get to SUBTRACTION
7 + (–2 ) = __ Read as 7 + "the opposite of" 2 = 5
On the other side of zero
–7 + (– 2) = __ Read as "the opposite of" 7
+ "the opposite of" 2 = – 9
and with SUBTRACTION
–7 – (– 2) = __ Read as "the opposite of" 7
"the opposite of the opposite of 2 = – 5
When we add we go away fro zero, and when we
subtract we head back to zero, from either side.
I believe it is truly important that students see
integers from Kindergarten. They need not be
a surprise. We do not have to work with them,
but letting them know they are there will save
a great deal of anxiety.
Friday, December 9, 2011
SUBTRACTION WITH ZEROS
We have all been in class when the following problem was put on the board.
5003
– 1468
We remember that we cannot take 8 away from 3, so we have to borrow, or
REGROUP in todays vernacular. Thus, our 0's become a 9, our 5 is reduced
to 4 and off we go. But, why do we do that? Our 0's become 9's, really?
That's cool, we get 9 out of nothing. Oh, hold on there cowboy!
Look at this a different way.
5000 + 0 + 0 + 3
– 1000 + 400 + 60 + 8
Now the 3 needs an extra 10 to be 13, but 5000 can only hand out
1000's. Thus, we will drop off 900 in the hundreds place, 90 in the
tens place and the ones place only gets 10 leaving the following.
4000 + 900 + 90 + 13
– 1000 + 400 + 60 + 8
As we see, by expanding the numbers we get a clearer understanding
of what is happening. I hope you feel better about those 9's now.
Friday, November 25, 2011
Palindromic Numbers and Companions
Many years ago I came across a method for practicing your addition
facts and having fun at the same time. It all has to do with
PALINDROMIC NUMBERS.
A Palindromic number is a numbers that looks the same backwards
and forwards; 101, 232, 5445, 63436.
If a number is Palindromic then it is its own companion, but if its
not then we need to figure it out. We start with 3–digit numbers,
and as you saw above 101 and 232 are already Palindromic. But what
about a number like 117. Sense it is not Palindromic we reverse the
number and add them together.
117
+ 711
828 Thus: 828 is the companion of 117.
What about 536
536
+ 635
1171 Not Palindromic so we do it again.
1171
+ 1711
2882 Thus: 2882 is the companion of 536
Now try 468
468
+ 864
1332
2331
+ 1332
3333 Thus: 3333 is the companion of 468
Last we will look at 579
579
+ 975
1554 Try again
1554
+ 4551
6105 Try again
6501
+ 1056
7557
We found the Companion 7557
If your class works on all of the 3–digit numbers up to 999, you will
find them being able to add faster. As they get into them, they will find
some that take as many as 15 steps. Also, when you see common
companions look back at the numbers and see how they relate.
Sunday, November 20, 2011
Those Story Problems
If you are my age, 64, or even 15 to 20 years younger, you probably learned
Algebra I from a thick red book written by Dolciani. Since then we have
had many modifications of that book as we have started to emphasize
graphing over story problems. However, we still have to be able to solve
those problems you loved so much.
Example: One plane leaves Chicago, fling east at 400
M.P.H.. A second plane leaves one hour later fling
west at 500 M.P.H.. When will they be 1800 miles
apart?
We will solve this problem in a moment, but we should first talk about
all of these types of problems. It is all about the DISTANCE. In all of
these problems we will know one of 3 things.
- The sum of the distances
- The difference of the distances
- The distances are equal
Now lets look at our problem above.
FIRST: When will they be 1800 miles apart?
SECOND: Draw a PICTURE
← ⎮→
500 M.P.H. 400 M.P.H.
.__________________________________________.
1800 Miles
DISTANCE RATE TIME
d₁ 400 T
d₂ 500 T–1
THIRD: Make an equation.
400T + 500(T–1) = 1800
400T + 500T – 500 = 1800
900T – 500 = 1800
900T = 2300
T = 23/9 Hours
T = 2 and 5/9 Hours
T = 2 hours 33 minutes and 20 seconds
The planes are 1800 miles apart in 2 hours 33 minutes
and 20 seconds.
I know it seems easy when someone else does it, but if we know patterns
just follow the pattern I created. After doing a few of them you using a
picture and a DIRT BOX it should be easy for you.
Friday, November 11, 2011
Mr. W. the numbers man: Slope of a Line
Mr. W. the numbers man: Slope of a Line: SLOPE ΔY ÷ ΔX Change in y over the change in x y₂ – y₁ ÷ x₂ – x₁ As graphing becomes a major part of our early education, th...
Slope of a Line
SLOPE
ΔY ÷ ΔX Change in y over the change in x y₂ – y₁ ÷ x₂ – x₁
As graphing becomes a major part of our early education, the understanding of
the concept of the slope of a line may become part of that understanding. Our
students already discuss the idea of a positive or negative correlation within a
data set. As they observe these correlations it gives them clues into the ideas of
how a line should look if it has a positive or negative slope.
How do we tell the story of slope? This concept, a basic concept of The Calculus,
is critical in basic Algebra. How do we teach slope? Where do we start? I think
we should start with a walk that begins at the origin of a graph. We are going to
walk to our right.
FIRST : As we stand at (0,0), the origin we are at a point. Imagine we move
six paces to the right and stand on (6,0). Now imagine these are the endpoints
of a line segment. What is the slope of that line segment? If we know this we
also know the slope of the X–axis, or the slope of any horizontal line. From
the formula above: y₂ – y₁ ÷ x₂ – x₁
0 – 0 ÷ 6 – 0
0 ÷ 6 = 0
THE SLOPE OF A HORIZONTAL
LINE IS 0!
SECOND : From the point (6,0) let's walk to the point (42,6). I hope we are
walking single file, I don't want one of us to fall of the line. Be serious, you
know we can't really walk on a line. Oh well, it was my attempt at being
humorous. Back to the slope of the line.
y₂ – y₁ ÷ x₂ – x₁
6 – 0 ÷ 42 – 6
6 ÷ 36
1 ÷ 6 or 1/6
We are looking at a positive slope, much the way we think of a positive
correlation in a set of data points.
THE SLOPE OF A LINE THAT
SLOPES FROM THE LOWER LEFT
TO THE UPPER RIGHT HAS A POSITIVE
SLOPE!
THIRD : We are now standing on the point (42,6). Next we will walk
to the point (62,2). We are walking from the upper left to the lower right.
Let's find the slope.
y₂ – y₁ ÷ x₂ – x₁
6 – 2 ÷ 42 – 62
4 ÷ –20
1 ÷ –5 or – (1/5)
Here we are looking at a segment that has negative slope much like a
negative correlation in a set of data points.
THE SLOPE OF A LINE THAT
SLOPES FROM THE UPPER LEFT TO
THE LOWER RIGHT HAS A NEGATIVE
SLOPE!
FOURTH : We now stand at (62,2). From here we can not walk, but we
will slide down the segment to (62,0). This is a vertical move so we are looking
at the slope of a vertical segment which could be extended to be a
vertical line.
y₂ – y₁ ÷ x₂ – x₁
2 – 0 ÷ 62 – 62
2 ÷ 0 is undefined, we say "no slope"
THE SLOPE OF A VERTICAL LINE
IS UNDEFINED OR HAS NO SLOPE!
We should now be able to observe a line on a graph and tell if it
has a positive slope, negative slope, a slope of 0, or no slope. If the
slope is positive or negative then we need to be able to solve for
that exact value. It many cases it will take many examples.
ΔY ÷ ΔX Change in y over the change in x y₂ – y₁ ÷ x₂ – x₁
As graphing becomes a major part of our early education, the understanding of
the concept of the slope of a line may become part of that understanding. Our
students already discuss the idea of a positive or negative correlation within a
data set. As they observe these correlations it gives them clues into the ideas of
how a line should look if it has a positive or negative slope.
How do we tell the story of slope? This concept, a basic concept of The Calculus,
is critical in basic Algebra. How do we teach slope? Where do we start? I think
we should start with a walk that begins at the origin of a graph. We are going to
walk to our right.
FIRST : As we stand at (0,0), the origin we are at a point. Imagine we move
six paces to the right and stand on (6,0). Now imagine these are the endpoints
of a line segment. What is the slope of that line segment? If we know this we
also know the slope of the X–axis, or the slope of any horizontal line. From
the formula above: y₂ – y₁ ÷ x₂ – x₁
0 – 0 ÷ 6 – 0
0 ÷ 6 = 0
THE SLOPE OF A HORIZONTAL
LINE IS 0!
SECOND : From the point (6,0) let's walk to the point (42,6). I hope we are
walking single file, I don't want one of us to fall of the line. Be serious, you
know we can't really walk on a line. Oh well, it was my attempt at being
humorous. Back to the slope of the line.
y₂ – y₁ ÷ x₂ – x₁
6 – 0 ÷ 42 – 6
6 ÷ 36
1 ÷ 6 or 1/6
We are looking at a positive slope, much the way we think of a positive
correlation in a set of data points.
THE SLOPE OF A LINE THAT
SLOPES FROM THE LOWER LEFT
TO THE UPPER RIGHT HAS A POSITIVE
SLOPE!
THIRD : We are now standing on the point (42,6). Next we will walk
to the point (62,2). We are walking from the upper left to the lower right.
Let's find the slope.
y₂ – y₁ ÷ x₂ – x₁
6 – 2 ÷ 42 – 62
4 ÷ –20
1 ÷ –5 or – (1/5)
Here we are looking at a segment that has negative slope much like a
negative correlation in a set of data points.
THE SLOPE OF A LINE THAT
SLOPES FROM THE UPPER LEFT TO
THE LOWER RIGHT HAS A NEGATIVE
SLOPE!
FOURTH : We now stand at (62,2). From here we can not walk, but we
will slide down the segment to (62,0). This is a vertical move so we are looking
at the slope of a vertical segment which could be extended to be a
vertical line.
y₂ – y₁ ÷ x₂ – x₁
2 – 0 ÷ 62 – 62
2 ÷ 0 is undefined, we say "no slope"
THE SLOPE OF A VERTICAL LINE
IS UNDEFINED OR HAS NO SLOPE!
We should now be able to observe a line on a graph and tell if it
has a positive slope, negative slope, a slope of 0, or no slope. If the
slope is positive or negative then we need to be able to solve for
that exact value. It many cases it will take many examples.
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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