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.

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.

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.

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.

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.

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.

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.
  

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.

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...

Why a negative x negative = positive

The language of Math is used incorrectly in our society by most of us.  Funny, we do not tolerate incorrect English, or improper grammar, but math seems to have its own set of rules which we seem to want to butcher.  The butchering extends from us not all understanding the definitions, so we make some up.  When we do, and it gets extended beyond us, whoops we have a bigger problem.  Lets try to correct a couple.

"Two Negatives make a Positive"

Not quite, which of the following are we talking about?

1. The opposite of a negative integer is a positive integer

2. A negative integer multiplied by a negative integer = a positive integer

3. The sum of two negative integers is a larger negative integer

The statement in red above is confusing and incorrect.  The 3 statements that follow
are correct, and keep us aware of the operation we are discussing.  We can use a
pattern to see how this happens.

#1.  – (–5) = 5  In this statement we are saying the opposite of negative 5
                           = positive 5.  It is also saying "the opposite of", "the opposite
                          of 5" which is 5.  So, we have gone from 5 to –5 and back
                          to 5, all in one statement.

#2.  (–5) (–5) = 25  So, why is this true? Lets look at a PATTERN!
         
 5 x 5 = 25         5 x (–1) = – 5
 5 x 4 = 20         5 x (–2) = –10
 5 x 3 = 15         5 x (–3) = –15
 5 x 2 = 10         5 x (–4) = –20
 5 x 1 = 5           5 x (–5) = –25 
 5 x 0 = 0    Since Multiplication is commutative!


(–5) x 5 = – 25  so   (–5) x (–1) = 5
 (–5) x 4 = – 20        (–5) x (–2) = 10
 (–5) x 3 = –15          (–5) x (–3) = 15
 (–5) x 2 = –10
 (–5) x 1 = –5
  (–5) x 0 = 0


As we see the product of a positive integer and a negative integer is a negative
integer.  However, the product of 2 negative integers is
 a positive integer.


#3. (–5) + (–5) = –10 Here we are adding to the left from 0.  Thus if I add 
                                     2 negative integers I must get a larger negative integer.
                                     We, as educators, must be clear to our students about 
                                     what is taking place, and say it correctly.


     You might say: The sum of 2 negative integers is a
                negative integer. 


    Clarify for them that a problem like, –3 – 4 = ? is a sum and not a product.
    they could rewrite it as:
  
                     +(–3) + (–4) = –7


"Lets cross cancel first"

I do not know how this improper comment became a part of our lexicon, but
it has.  People the correct word is Factor!!!
I would start a class out  by writing:

                          8 ÷ 12 ⋄ 4 ÷ 5 =        or 8/12 ⋄ 4/5 =


I would then reduce the fraction 8/12 to 2/3.  A hand would go up, and when I call
on them they say, "you can't do that its not on a cross"


My response was always the same, "gee, is the reducing of fractions
 now  illegal,  I did not get the memo".

The phrase "cross cancel" places into the minds of students the wrong information.
We should never use phrases that distort the true meaning of what we are intending
to teach.

Because of this phrase I have had students factor numbers across an = sign in a
proportion.

Example:  They would factor the 3's in ⅓ = 3/x 
Then they would say, "but its on a cross isn't it." 


They had heard the wrong phrase so often, for them it was correct.  If we are going
to teach Math as we teach any subject, use the correct vocabulary.

Mr. W. the numbers man: Factoring in Algebra

Mr. W. the numbers man: Factoring in Algebra: FACTOR THE FOLLOWING: 3X² –10X – 8 This is a common problem in in the second semester of Algebra I. Every teacher has ...

Factoring in Algebra

FACTOR THE FOLLOWING:
               3X² –10X – 8


This is a common problem in in the second semester of Algebra I.   Every teacher
has their favorite way of teaching it, but for many students it brings tears to their
eyes and pains to their stomach.


O.M.G., 2 negatives and the coefficient of x² is not 1!!


What is that now,  two negatives make a positive?


4 x 2 = 8, but 4 + 2 = 6,  3 is ½ of 6, oh darn, what
do I do anyway!!


So what's the question, "factor"?


Are you asking these questions, because you are in a Algebra I class, or you have
been there and now your children want your help.  In my last blog I talked about
problems that could go through many grades, and as the student learned more they
would be able to answer the deeper question.  This is one of those examples.


Most, if not all of us would agree that without knowing our multiplication tables
Math can be a struggle.  The process for solving the problem above can begin
with a game in the 3rd grade.  By the second part of the year most of the students
will have learned their multiplication tables through 12.  As a warm–up teachers
could start playing a game I called "I'M LOOKING FOR TWO NUMBERS".


The teacher asks: 
"I'M LOOKING FOR TWO NUMBERS WHOSE 
PRODUCT IS 24 AND WHOSE SUM IS 10."  




What is the student learning?


What words do I need to know the definitions of?

The student is learning to  "factor", break down into its divisors.
They learn and remember the following terms:

• Sum: to Add
• Difference: to subtract
• Product: to multiply

They realize there is more then one way to make 24, or 36,

or 16, or many other numbers as a product of 2 numbers .  This requires
them to further analyze the question:

6 x 4 = 24

3 x 8 = 24

2 x 12 = 24

1 x 24 = 24

But only 6 + 4 = 10

∴ the 2 numbers are 6 and 4

Now ask the same question, and end it by saying; 

•  The sum is 11
•  The sum is 14
•  The sum is 25 

Three knew questions without much thinking on the teachers part,
but great analysis for the students.  The game does not have to change
in any way, but as your students get introduced to integers we can
add knew sums.  For the above problem:

• The sum is –10
• The difference is 3
• The difference is 10 

When our students get to Algebra I the problem at the top of the
blog should not scare anyone.  We are looking for 2 numbers
whose product is –24, and whose difference is –10.

        3x² – 10x – 8  factors to
     
              (3x + 2)(x – 4)


  

  

Teaching a Language in Math

Yesterday I was lucky to be a substitute at Salmon Creek School.  I was taking over for
teachers who were being debriefed after observing a lecture to their students on a
writing program they have adopted.  This program  starts in 2nd grade and
goes through 8th grade.  Each teacher uses the same terms and ideas to teach the
concepts they have always taught.  The program has wonderful patterns and color
coding to help the students remember their writing skills.

I thought wouldn't this be great for Math.  If all teachers defined words the same way.
If all phrases that take away from learning Math could be eliminated.  If there were
problems that start in the 2nd grade and go to 8th grade.  There are problems like this!


Now as students gain more knowledge in Math they can come closer to answering the
entire question, or answer deeper problems on the same question.

Example: The Tortoise and the Hare were having a race.
                The race was 200',  100' forward and 100' back.
                The Hare takes 2 steps of 3' in the same time that
                the Tortoise takes 3 hops of 2'.
                a) Who reached 100' first?
                b) Who won the race?
                c) How far back was the loser when 
               the winner crossed the finnish line?
                d) If the loser was in the air when 
               the winner crossed the finnish line, 
               at what height was he?

We could start this problem in 2nd grade.  Here we should only look
at part "a".  But how do we do this?  First of all, how far is 100'?

Our 2nd grade teacher can use the problem as an introduction to
measurement.  Having measuring tapes of 25' they could find a
distance of 100' by simply adding 25', 4 times.

Next they could have one person be the Tortoise and one be the
Hare.  They could then see how how each would move from the
start toward the 100' mark.

 0_6'__12'__18'__24'___________________________96'_____100'

Both the Hare and the Tortoise arrive at 96' together!


What happens next?
How does the way they move effect what happens?
Are they both on the ground at 100'?


The tortoise in his next two hops lands on top of the 100' mark,
but the Hare's first step takes him to 99' and he would be in the
air going past the 100' mark as the Tortoise lands on the
100' mark.

They both get to 100' at the some time.

As our students are introduced to measurement and they find out about
feet and inches.  They may also ask about the marks between the inch
marks.

It is up to our 3rd to 5th grades to discuss the "b" of the problem.
They must decide who wins the race and why.  While they are doing
that they can work on the understanding of ½, ¼, or ¾ of an inch.  


In the 2nd grade the use of the physical measuring tape will help students
to see how being at 2¾ then adding ¾ gets you to 3½.  In the following
years they will see how 2 and 6/4 creates 3½.




They will understand how to solve "c" eventually, but we can  leave "d"
to much latter.


As the students move through the grades it is up to the teachers to use
the same terminology when solving this problem,  so there is no confusion
in the minds of our youngsters.

Probability

In the last 10 to 15 years PROBABILITY has shown up a lot more at the elementary
school level.   As a matter of fact it happens in 6th grade for most and in some places
even earlier.   It did for 2 reasons:

• It is a great example of a fraction in life.
• It is a fine way to look at percents.

For many of us as parents, when these questions come home we may be at a loss to 

help little Johnny or Mia.  But we can help if we just think of that coin being tossed 

in the air.  There are a total of 2 possibilities: HEADS OR TAILS.  However, there

is only one winner.  

The Probability of of getting heads or tails is ½, or 50% or .50

So we can think of Probability as

     P(EVENT OCCURRING)  =
                     Number of favorable outcomes 
                               Total number of outcomes

We can write probabilities as a 

• FRACTION 
• DECIMAL
• PERCENT

So lets ask some questions:

Example: You have a box containing 10 marbles.   3 are RED,
                 4 are YELLOW,  2 are GREEN, and one in WHITE.
                 They are all the same size and feel the same.  If you
                 reach in and randomly draw one out of the box, what
                 is the Probability that it will be YELLOW?

             4 Favorable and 10 Total

            ∴ 4/10 = 2/5   You have 2 chances out of 5.


Example: Given the same box and marbles you have above, what is
                the percentage chance of picking a GREEN marble?

            2 Favorable and 10 Total

            ∴ 2/10 = 1/5    1/5 = .20 = 20%

            You have a 20% chance of drawing out a GREEN marble.

What I have just discussed with you is called THEORETICAL PROBABILITY
because it is based on knowing all of the outcomes that are equally likely.




This needs to be separated from EXPERIMENTAL PROBABILITY which is
based on repeated trials of an experiment.

  EXPERIMENTAL PROBABILITY OF AN EVENT

                          =  NUMBER OF SUCCESSES 
                                NUMBER OF TRIALS


 Example:  You roll a six sided die. The chance that any one side
                  comes up is equally likely.  You roll the die 40 times
                  and note all the results.  What is the Experimental
                  Probability of getting an odd number on any role?

                    Here are your results:

     RESULT               1        2         3         4        5         6
TIMES ROLED         6        7         7        5       10         5

                      23 out of 40 are odd!   57.5% are odd
               Our P(odd) is 57.5% (Experimental)


               20/40 or 50% are odd  is the 
               Theoretical Probability.


This is only beginning of a long discussion.  In my next Blog I will discuss
the ideas of independent and dependent events.

Mr. W. the numbers man: The 3 M's of Statistics

Mr. W. the numbers man: The 3 M's of Statistics: If you are 50 or older and did not take Statistics the next 3 words were not part of your youth, and certainly not part of your every day d...

The 3 M's of Statistics

If you are 50 or older and did not take Statistics the next 3 words were not
part of your youth, and certainly not part of your every day discussions.

• MEAN
• MEDIAN
• MODE                                                                                                                                                    

The first word described the "bully" in your class that just was not nice.

The second was the line you crossed when you passed someone on the road.

The last had to do with a musical scale if you were so inclined.

But not TODAY!

Open any 5th grade math book and you find these words with slightly 

different definitions.  And this is because we are now a part of the 

               DATA GENERATION!

Our children as ourselves are bombarded with data and statistics, day-in

and day-out.  We are even supposed to understand what standard deviation

is all about.  I don't know about you, but that was not in my 5th grade
curriculum.

However, this does not mean Johnny will not be coming home and asking

us, "what's a mode, is that what you get with apple pie?"



NO, not this time.

By giving you a set of data, we can try to understand.

The following is a list of the height in closest inches of the 11 boy in
your class this year.

              50, 50, 52, 53, 54, 54, 54, 57, 59, 59, 60

MEAN This is the average of the numbers. So all we have to do 
is add them up and then divide by 11.


            The sum is 602/11 = 54.73  or     ≈ 54¾ inches


Thus: Mean ➜Average




MEDIAN This is the man in the middle.  Like the line in the two–lane
road that it is in the middle.  We are talking about the data piece in the middle.
So if we count from both ends toward the middle we land on the 6th number, 54.

              The man in the middle is 54

This is because we have an odd number of data pieces.  If we had a even
number, when we got to the middle we would have our fingers on two
different numbers.  We would then take the average of those 2 pieces.

Thus: Mean➜Man in the Middle




MODE This not the scoop of ice cream with the apple pie.  No, it is the 
pice of data that happens the most often.  Look at the list above.  There are 
three boys that are 54 inches.  Therefore it is the mode of this data.  A list
of data like the one above can have more than one mode, or it may not have 
any if all the pieces are different.


Thus: Mode➜The one that happens most often

Mr. W. the numbers man: Computer or English Language/mixed numbers or impr...

Mr. W. the numbers man: Computer or English Language/mixed numbers or impr...: The last two days I have spent trying to put higher Math symbols into my blog easily. How difficult could this be I thought. I found a fre...

Computer or English Language/mixed numbers or improper fractions

The last two days I have spent trying to put higher Math symbols into my blog easily.
How difficult could this be I thought.  I found a free down lode called Math Type.  
I did not take me much time to learn how to create the equations that I wanted.  Then
the problems began.

How do I get these equations to my blog?

Do I just move them to my desk top, open my 
blog, drag and drop?


If only it was that easy.   Here is where I ran into HTML, SEO, TeXLook.eqp,
Symbol palette, Template palette, Garamond and Times News Roman.  Although,
a couple of these were familiar I felt like I was swimming in a an ocean I had
never learned about.

It is up to us to understand the new technologies, but where do I start.  It was
recommended to me to get a book called "Blogging for Dummies".  For as much
as Math is a language that is easy for me to understand, this language of
computers still gives me a start.  I have great friends that are geeks when it comes
to this stuff.  Soon it will be second nature, but at the moment, like many of
you, I struggle.


But this I can help you with

MIXED NUMBERS TO IMPROPER FRACTIONS

One of the first concepts we learn in fractions, is changing a mixed number into
improper fraction.

          Example:   2¾ is read as 2 and ¾ = 2 + ¾
                        
                     The Process is Simple 4 x 2 = 8 and 8 + 3 = 11  therefore 11/4


                     But why!


Here is why. We need to remember that 1 = 4/4, so 2 = 8/4 or 8 ÷ 4 = 2

                    Now 8/4 + 3/4 = 11/4


Lets try another: 6⅔ = what mixed number?

                                           3 x 6 = 18 and 18 + 2 = 20, therefore 20/3

What I cannot stress enough is reading the number; 6⅔ = 6 + ⅔
The reason I mention the last item in bold is this. When the "+" is placed
between the 6 and the ⅔ students seem to think this will change the
process.  But since both of the statements are equal then the process is
the same and the result is the same.

As a matter of fact, when this concept is first introduced, it should be
shown with and without the "+" sign.  My reason is simple.  Soon the
student will be faced with the following:
                              
                        6 – ⅔ = ?

All we have to do is 3 x 6 = 18 and 18 – 2 = 16/3


Instead of adding 2 we subtract 2:  The sign tells us to do.

Now we look forward to Algebra:   x + ½ combine

We know that the x and the ½ cannot be added to make a larger number,
because we do not know what x is.  However, that does not stop us from
rewriting the sum under one denominator using the same simple process
we have used for years.


                   2 ⋄ x = 2x + 1 = 2x + 1/2


The phrase 2x + 1 is now over one denominator, 2.

Now if you were given different values of x, it would change the value of
the phrase

                      x's value                       phrases value
                     0                                      ½
    
                     1                                      3/2

                     7                                     15/2


                   –3                                    –5/2



      Example:   ⅓ – y needs to be combined into one fraction.
                 3 ⋄ (–y) = –3y and –3y + 1/3

Same process, we now have a mixed number, that in Algebra helps us,
and we are not miles out of our comfort zone.  In all cases it is important
that we read what it says, do not stare at it and hope to understand.
     
                  
         

The 3 Levels of Math

I wish to extend my thanks to my friend Mike for what I am about to write.  He and I
spend hours discussing math, physics, dark matter, string theory and alike.  We both
have similar interests that most don't.  We both even happen to be avid stamp
collectors.  Mike once said to me, "I believe there are 3 levels of Math, 1 – all the
symbols we will ever learn, 2 – the equal sign, and 3 – well the rest are just recipes."

I thought for a moment, and as a retired Math teacher I did spend most of my days
surrounded by symbols,   explaining what they meant to my students.  I also had to
agree that there is no symbol more important than the equal sign, because it allows
things to be solved in equations and alike.  Finally, we have our formulas
that solve problems in chemistry, economics, statistics, physics, etc..

Now lets take a close look at what I mean.

  level 1         SYMBOLS
+            Here are the symbols we learn, add, subtract, multiply,
—           and divide, in arithmetic.  For a lot of us this is quite 
×           enough.  But look at the key board I am touching.  
÷              Every key has a symbol and by holding down the option key WOO!!!
=           

These 5 only scratch the surface of all the symbols we need to learn and understand.
And some symbols mean the same thing, while others only hold position and do not
ask us to do anything.  It is up to us as teachers of Math or any subject to first define
clearly what a symbol means.  I believe that many of our students, when exposed to
a new symbol, become confused and often do not even ask about it.

×  ⋄  ( )  ab  All  of these symbols can be used for the operation – multiplication.
                              The x is the times sign, but so is the raised dot.  Two sets of items
                               in parenthesis (5+4)(6–3), or two variables with nothing between
                               them: rt = d.  Rate times Time = Distance


a₁, a₂, a₃, a₄, . . .   Now this is just a list.  The numbers 1 - 4  are saying 
                                these are the first four items in the pattern.  They do
                                                not tell us anything about the size of the items in the 
                                                pattern.  The three dots at the end tells us that the pattern 
                                                continues as it did.


  Example: 4, 7, 11, 14, 18, 21, 25, 28, 32, . . .       add 3, add 4, add 3, add 4, . . . 


<             Less than, 5 < 8.  Just when we thought we knew when things were the
>             the same, now we have to know when they are smaller than something 
≤             else.  Then we have less than or equal to, but not greater than.  Or we 
≥                have greater than, which is not equal to.  For heavens sake, which one 
≠              do I use. 




Sometimes symbols are a lot more than what they seem to be.  It is as though those Math
people are just lazy.  Why don't they just tell us with out the symbols?  Oh well!


{  }              {x: x ≥ 0}  The set of numbers called x, such that all values of x are 
                                          greater than or equal to zero. 


⎢  ⎥          ⎢ – 5 ⎥  the absolute value of –5.  The absolute value is simply its distance
                       away from zero.  The value here is 5. 


Then there is GEOMETRY


∠    These are just a few of my favorites that show up in 5th grade.   By this time, only 
⊥     age 10 or 11,  have been bombarded by hundreds of symbols.  And we have to 
≅     understand them all, and how they are used.  Our first stands for angle, any angle
∼      in any figure; ∠A of triangle ABC.  Our second means perpendicular, that is 
∕∕   two lines that intersect at 90º (more symbols) ∠s.  Our 3rd symbol ≅, means 
           congruent or the same size and shape.  Our 4th ∼ means similar to, or same 
           shape, but not the same size.   Our last  symbol stands for parallel lines.  Lines
           in the same plane that do not intersect.
∩       This is the symbol for intersect.


I could go on for pages more, but you get the idea, and I have just started.  
Can it be any wonder why learning is difficult if we do not repeat a symbol over and 
over.  Then we must build patterns for these symbols so that they make sense. 





    Level 2   = THE EQUAL SIGN


No sign is more important.  It allows us to solve.  It is the glue that holds all other 
ides together.  From Kindergarten on students should be writing boldly 
                           5 + 4 = 9
They must write it down, not just say it or read it.

Where would we be today without E = mc², or V = ⅓Βh, or C = πd.

These formulas could not exist without the = sign.  So with this sign we are able to
understand when things have the same value.  And when they do we can sometimes
substitute one statement in for another to get a new idea.

Certainly, one of the more difficult concepts starts in Algebra, when so many
statements = 0.    5x³ – 4x² + 3x – 30 = 0   How could so much be nothing,
but that is what it says, and there must be a truth to it.   It simply means if
we place the correct number in for the variable, if one exists, the phrase on the
left will have a value of 0.  In this case it is 2.  Far more difficult than 5 + 4 = 9,
but understanding the power of the = sign, and realizing why we use it will
make us more comfortable in our Math classes.


level 3  THE RECIPES OR FORMULAS


   Above you saw some formulas that we all are aware of.
      Remember this one:
                                             a² + b² = c²


There are not to many adults, who can't tell you this one.  If you ask them what
 it is for you may not get the right answer,  but a² + b² = c² they know for sure.  Its 
the Pythagorean Theorem, undoubtably the most elegant theorem of Math.


                                x = – b ± √ b² – 4ac  ÷  2a                                                                 


The quadratic formula, the grandaddy of all Algebra I formulas.  These are just 2
of so many of them that you learn in school.  Next comes the confidence of working
with them.  Now we have to place numbers in for the variables and get solutions.