Patterns

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.

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.
    

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.

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 

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.

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.

SQUARING A 3–DIGIT NUMBER

In my last blog I talked about squaring a 2–digit number. I know that most

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.

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.

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.