What is a Mathscoach?

To: Students of All Ages
From: James F Holwell, Learning Coach

What, then, is a "learning coach" -- ?
First of all, it is a title given to me by John Seal in 1986 when he was in 9th grade, trying to learn enough math to get a good grade.

John was a member of the wrestling team and he admired his coach.
One day he stayed for extra help in math and I spent the time with him, asking him questions that brought forth his abilities.
I was not trying to pound 'material' into his head.

My reward for that was the title "Learning Coach" which John bestowed upon me.
A coach is someone who knows that the ability to achieve is present, in you, now.
A coach works with you on the basis of declarations, requests, promises, and assertions.

Declaration -- We align on a vision for your life.

Requests -- Coaching includes asking you to do things.

Promises -- The coach lets you know what benefit
you may expect to gain from everything you do.

Assertions -- Statements for which evidence is available.


Your Declaration is what You say is the purpose for your life / why you are here.
Your Promises to the coach let him know what you can be counted on for. (They are NOT what you think he wants to hear!)
The work you will do is not to be confused with 'homework for the teacher'; the assignments are requests made of you by your coach,
which you do because you said you would; you do not do them for him, but for yourself, because you remember your vision.

Your coach's Requests of you are based on what you need to know,
and what experiences are useful for you, in order for you to be, to do, and to have all that supports the purpose of your life.

If he makes an Assertion that something is so, he can back it up with evidence: proof that it is so. A coach will speak to you in a way which supports you in achieving what you said you intend to achieve!

A coach can't make you do anything -- you are always free to fire the coach.

In sending you this activity (Searching for the Golden Rectangle)
I am assuming that you are aligned with me in being willing to do things that will result in your increased understanding and ability in mathematics.


INTRODUCTION

Recently I was searching the key words [dogs friendly] for a young friend who loves dogs, and found this little wisdom from a website about dogs:

There are those who teach basic ways to survive in the human world,
and there are those who teach us the ways of love, faithfulness,
reciprocity, forgiveness, spontaneity, and a natural predisposition for joy and companionship.

One is called human wisdom, the other instinctive wisdom.
One teaches obedience (and silly tricks), while the other offers us
a window into a world that we either lost or never knew.

Excerpt from: http://www.lighthousedogs.com/index.html
.................................................................................................

Note to parents and mentors:
We are used to learning in a Sequential Curriculum modality,
with one bit of information following logically from the previous one.
This is convenient for a systems approach to education, perhaps,
but there is one problem with it:
It does not necessarily represent the way many of us learn.

The approach in my coaching, whether in Mathematics, Chess, English Grammar or whatever -- is what I have called the Chaotic Curriculum.
You get dropped into the middle of a lake and you have to learn to swim!
It is scary, but remember, a coach is not a teacher; a coach is not about to test you to find out if you have been 'working up to your ability' -- unless you ask the coach to do that for you.

So, in this method of learning, you announce happily to the coach:
"Guess what, I don't know how to do that!"

To that your coach replies:
"Guess what else, you don't know how to do that YET!" --
and then your coach takes you to that part of your mind which easily comprehends all you need to understand and to do what you intend.

If you are fearful of letting the coach know when you need more explanation or examples, that may be because you have learned in the past that you suffered in some way when someone found out you didn't know something you were supposed to know.
Maybr you got a 'Zero' or an 'F'!

YOU NEED TO UNDERSTAND THAT IT WAS NOT YOUR FAULT YOU DIDN'T KNOW IT! You need to forgive yourself totally for what you believe to be your failures. You begin again, but with a different direction, in a different context, with a coach to guide you.

Here is a simple way of looking at forgiveness:
FORGIVENESS IS GIVING UP ALL HOPE FOR A BETTER PAST.
If you feel that you have failed in the past, then for you the first activity is to forgive yourself.
You need to recognize that these 'failures' were there to help you learn what kind of teaching does not work for you.

Only then will you be able to start over and to expect success.
In this way you achieve joy and satisfaction in your life, starting today.
Like the dog teaching his master, I offer you a window into a world you "either lost or never knew".

Activity Cards are lists of things for you to do.
The purpose of activity cards is to assist you to understand, apply, analyze, synthesize and evaluate what you learn.

What you will learn while searching for a golden rectangle
will be many of the most important principles of mathematics
and how to use them to solve many kinds of problems.

That means you'll be learning why things are the way they are.
You will understand what you are doing.
You will be discovering many things for yourself.

So let's get started...


To Students:
"Searching for the Golden Rectangle"

This first unit is divided into three groups of activities:
Activity One -- Exploring the Size [area] of a rectangle
Activity Two -- Exploring the Shape of a rectangle
Activity Three -- Searching for a rectangle with the Golden Shape
____________________________________________________

Activity Card One

You need:
notebook for recording principles, activities and your discoveries
graph paper (four squares to the inch)
straightedge (may be a ruler or anything with a straight edge)
pencil, eraser
your willingness to understand by DOing!

1. Make six rectangles on the graph paper.
Their names are: A, B, C, D, E, F.
Here are the height and width of each one:
A) 1 by 1
B) 1 by 2
C) 2 by 3
D) 3 by 5
E) 5 by 8
F) 6 by 10.

2. One way to determine the 'size' of a rectangle is to count the number of 'unit squares' inside of it. [This is called the 'area'.]
The unit square is a '1 by 1', or just one box on your graph paper.
Determine the size [area] of each one of the six rectangles.

3. See if you can finish this chart by filling in the missing numbers:
Height Width Size [Area]
A) 1 1 _1_
B) 1 2 ?___
C) 2 3 _6_
D) 3 5 ?___
E) 5 8 ?___
F) 6 10 _60_

4) (Optional activity) -- not part of the Search for the Golden Rectangle -- but If you would like an interesting way to learn your multiplication facts, and to prepare yourself to be able to divide, do this:
Take a new sheet of graph paper and write the number 24 at the top.
Then, see how many different rectangles you can make --
ALL OF THEM HAVE HAVING AN AREA OF 24!

When you are finished, write down the height and width of each one.

Now you know all the numbers that make 24 when you multiply them!

You may be interested in exploring other sizes of rectangles;

What heights and widths can you find that will make a rectangle of area 48? 40? 28? 18? 19? 56? 42? 60? 64? 36? 72? 81? 100?

end of Activity Card One
[Remember to write in your maths notebook whatever you have learned so far. Insert the graph papers into the notebook.]
_______________________________________________________

Activity Card Two

You need a Pocket Calculator and the graph paper with the six rectangles you made in Activity One.

Now we ask, Is there any other way to arrange the six rectangles? [yes]
We can put them in order by Shape, from the widest to the tallest.

Let's do that now ... Which is the widest?
We would say that B (the 1 by 2) is the widest, because it is two times as wide as it is high, and none of the other rectangles has width two or more times its height.
[Check to make sure that is correct.]

The activity is to put all six rectangles in order
according to 'shape', from the widest to the tallest.
This is not easy to do just by looking at the rectangles.

Surely it is easy to see that the 'tallest' one is the square ___.
[Which letter is that?] but what about the other four?

Can you be sure of a way to arrange them in order by shape?

The good news is, there is a way to get a number with which to measure the shape.

The bad news is, these numbers are not numbers like the ones with which you count.

The good news is, you are going to find out that you can think about numbers that are not for counting; you will discover numbers that represent parts of a whole number.

More good news is, by doing one simple calculation for each rectangle, you can measure its shape.

Activities for Activity Card 2:

1. State which two of the six rectangles are the same shape.
See if you can see which ones they are before you calculate anything.
[Look and see if one of the rectangles is an exact scale model of another.]
The two rectangles which are the same shape are ___ and ___.

2. Here is how to find a number to measure the shape of any rectangle, when you know the height and width.
There is an operator button on the calculator for 'divided by'.
This button has a line and two dots, a dot above the line and a dot below the line.

So, to find the shape of a 3 by 5 rectangle, use your calculator like this:
Enter 5
Enter the operator "by"
Enter 3
Enter (=)
Read the answer: 1.6666666
We round off this number to the nearest 1000th -- 1.667
This number is the number that measures the shape of the 3 by 5 rectangle.

[Note: If you are not clear about the meaning of 1.667, you will be soon. If this is over your head for a moment, remember: You have a coach -- You won't drown!]

3. In Activity Card One you completed a chart with missing Sizes of the six rectangles.
In this activity you will find the Shape Measure:

Height Width Shape
A) 1 1 ?______
B) 1 2 ?______
C) 2 3 ?______
D) 3 5 ?______
E) 5 8 ?______
F) 6 10 ?______

Arrange the six rectangles in shape measure order, from widest to tallest.
The smaller the shape measure, the ________ the rectangle.

end of Activity Card Two
__________________________________________

Activity Card Three

You need some simple art materials -- colorful pens,
-- and of course, graph paper.
Also, you will need a ruler which can measure in centimeters.

The search for the golden rectangle begins with this activity --
but it doesn't end here!

This is the test for a Golden Rectangle:
When you remove the square from the rectangle, the shape of the smaller rectangle which is left over is the same as the shape of the rectangle with which you started.

To help you understand what this means, let's test the 3 by 5 rectangle to see if it is a golden rectangle, or not.
First, take a sheet of graph paper, and make a 3 by 5 rectangle.
Next, we mark off the largest possible square inside the rectangle.
[That would be a 3 by 3 square.]

So, with your crayon or felt-tip pen, color the 3 by 3 square, at the left, inside your 3 by 5 rectangle.
Color the 2 by 3 rectangle, at the right, with a different color.

Now we ask the question,
Is the shape of the smaller rectangle which is left over [2 by 3] the same as the shape of the rectangle with which we started [3 by 5] ?
[If you are lost then go back and read again the test for a Golden Rectangle.]

Let's see -- the shape of the smaller, left over rectangle [2 by 3] is 1.5 and the shape of the rectangle we started with [3 by 5] is 1.667
Are these shapes the same? [No.]
Is the 3 by 5 rectangle a golden rectangle? [No indeed.]

Your Activities:
1. Test the other rectangles we have worked with to see if one of them is golden.

2. Which of the rectangles is "almost golden"?
[To answer this question you will need to learn how to find the difference between numbers like 1.667 and 1.5.
You subtract to find the difference.]

3. Make several rectangles of size and shape that please you.
Use them as frames for artwork, or include them into the art itself.

4. Using a ruler which can measure in centimeters, measure the height and width of each rectangle you used in or around your artwork, and use the 'divided by' button to calculate their shapes.

5. Using graph paper, make rectangles with various heights and widths, and see if you can find a rectangle which is "almost" a golden rectangle.

end of Activity Card Three


More activities will be sent to those who go as far as they can
with these three activities, and let me know how it was for you.
[Coach me on being the best possible coach for you.]

James F. Holwell
mathscoach@gmail.com