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.