Search for the Golden Shape

There are two faces of mathematics in elementary and secondary schools:

1. Calculation and manipulation of symbols

2. Using mathematical models to interpret and solve problems

Most of our time and evaluation is centered on number 1.

But which is the more essential in this age of calculators, that we know how to do long division, and to divide a mixed number by a mixed number, and deal with expressions like 2x(5a + 7) -- or that we take away from school something we can use to deal with real situations?

Here is an excerpt from something I sent to the list as a reply to someone working with black teen-agers in Chicago. I believe these considerations are relevant everywhere.

What needs to be talked about is why some students believe they are unable to succeed at doing what they 'should' do.
Isn't this at the basis of defiance of authority in nearly all cases? Are we beating them to death with 'remedial' stuff in what they are not good at, instead of expanding upon their strengths?
In math, specifically, what I see happening almost everywhere, is the holding down of a child who might excel at concepts and applications, because she or he hasn't yet memorized times tables or learned to divide.
We need to distinguish Arithmetic skills from Mathematics, just as we distinguish Grammar and Spelling from Creative Writing!


A REPLY:
Thank God someone is talking common sense about arithmetic maths!
Mathematics was ruined for me at school because I could not do the calculations quickly enough, however hard I tried, and it was assumed that this marked me indelibly as 'no good at maths'.
In later life I have seen some of the pleasures which can be obtained from mathematics as a science, and though I understand it very imperfectly, I also know that if I had had a calculator at school (they were not invented until more than a decade later), I might well have derived serious intellectual benefit from mathematics.

As it was, I stumbled along and failed miserably, as much as anything because the sheer drudgery of calculating meant that I was never really alert to what the teacher was saying.

Mathscoach responds:
Chris, that is so well put!
Here, for example, is the beginning of a unit on the Golden Mean.
I use this with children who barely know how to add, using a calculator as needed.
Of course I teach them Arithmetic also, and then we subtract, multiply, and divide with whole numbers and with 'numbers in-between numbers' whether expressed as fractions, decimal fractions, or percents.
When teaching the calculation part of math, of course calculators are not used.

HOW CAN A RECTANGLE BE "GOLDEN"?
Good, got your attention.
So here, may it lead to an even greater level of joy in your life, the joy that arises from the clear awareness that you found out something new today.

What is the Golden Rectangle?
Think of a rectangle ... the edge of a 3 by 5 index card.
Go ahead,
Get a 3 by 5 index card.
Hold it so it stands up in front of you.
Notice it is five inches wide and three inghes high.

We measure the shape of it by dividing the width by the height.
We say that idea this way: "The Ratio of Width to Height".
We agree that this ratio is the measure of the SHAPE of the rectangle.

So we say that the shape of the edge of the 3 by 5 card is measured by the VALUE OF THE RATIO 5 to 3, or 5 : 3.
This value is found by doing the division problem 5 / 3.
Find this measure as a decimal fraction.
You will need to find the result when 5 is divided by 3.
Do this division any way you like; here is how to do it with a calculator:

HOW TO DIVIDE 5 by 3 WITH A CALCULATOR
1. Enter the 5
2. Press "Divided By" (the key with two dots and a line between them)
3. Enter the 3
4. Press the Equal Sign (=)
Do you get an answer?
Depending on how your calculator is set up, it may be 1.6666666 or 1.67

HOW TO DIVIDE 5 by 3 WITH PENCIL AND PAPER
___
1. Make a division holder like this: 3 ) 5
2. Ask yourself what you can multiply 3 by to get 5.
3. Notice that 3 times 1 is less than 5, but 3 times 2 is more than 5,
so the answer starts with 1.
4. Multiply the 1 times the 3 and subtract that from the 5.
5. This gives you 5 - 3, which is 2.
6. The remainder is 2 then, and now make a fraction with the remainder over the divisor.
7. This gives you the answer 'one and two-thirds' which we can write as: 1 2/3.

Notice that all of these answers are EQUIVALENT,
that is, these are all ways of stating the value of the ratio 5 : 3.
The value of the ratio 5 : 3 is EXACTLY 1 2/3 and that is APPROXIMATELY 1.667.
(When you study decimal fractions, you learn why that is so.)

NOW WE ARE READY TO DISCOVER THE GOLD!
Is the edge of the 3 by 5 card a "Golden Rectangle"?
We are going to find out if it is, and why, or why not!

First we are going to make a smaller rectangle from the 3 by 5 card, like this:
1. Measure a 3 inch square at one end of the card.
2. Draw a line on the card so that there is a 3 by 3 square at one end of the card.
3. Cut off the square with scissors.
4. Throw away the square.
5. See the little rectangle you are holding in the other hand.

If you followed the five steps correctly, the Height and Width of this new smaller rectangle are 2 inches by 3 inches.
We call this a '2 by 3' rectangle, and we ask the question,
WHAT IS THE SHAPE of a 2 by 3 rectangle?

We will find the shape just as we did with the edge of the 3 by 5 card.
The Shape of a 2 by 3 rectangle is the value of the ratio 3 : 2.
The shape is the result when 3 is divided by 2,
so follow the steps given above, this time using the numbers 2 and 3.

This should turn out to be either 1.5 (with a calculator) or 1 1/2 when you divide 3 by 2 with a pencil and paper.

Now, here is the test to find out if the edge of a 3 by 5 card is a Golden Rectangle:
The shape of the 3 by 5 rectangle would have to be EQUIVALENT TO the shape of the new smaller rectangle (2 by 3) you had in your hand after you cut off the 3 by 3 square.

Are these two shapes EQUIVALENT?
Here is how you can tell: The value of the ratio of 5 to 3 is 1.667,
while the ratio of 3 to 2 has the value 1.5 or 1.500

Since these values are not equal we would say,
"NO, the edge of the 3 by 5 card is NOT a Golden Rectangle."

Is the edge of the 3 by 5 card NEARLY a Golden Rectangle?
We would say yes, because the DIFFERENCE between 1.667 and 1.500 (obtained by Subtracting 1.500 from 1.667) is 0.167 and that is not very large.

However, if you are IN SEARCH OF THE GOLDEN RECTANGLE,
then you will want to find one which is more nearly golden than 3 by 5, and guess what?

YOU NOW GET TO DO A DISCOVER - ACTIVITY!

Using only the numbers from 1 to 10 as the height and width of rectangles, you are going to find dimensions (height and width) of a rectangle that is MORE NEARLY GOLDEN than the edge of the 3 by 5 card!

So try the numbers from 1 to 10 as the height and width of rectangles,
and go through all the steps we did above with the 3 by 5 card.


HINT: It is easy to do this experiment if you have paper with squares already printed on it (graph paper).
(Get the graph paper with four squares to the inch.)


In other words, say you are experimenting with the numbers 4 and 7; you will then draw a 4 by 7 rectangle on the graph paper, calculate the shape of it (7 divided by 4), and then use the five steps given above to obtain a new smaller rectangle.

The only difference is that now you are not working with a card,
you are doing this on a piece of graph paper.

So here are the same five steps, adapted for use with graph paper instead of a card:
After you have made a diagram of a 4 by 7 rectangle on graph paper,
Do this:
1. Mark off 4 units along the top and bottom edges of your rectangle.
2. Draw a line on the graph paper so that there is a square (4 by 4) at one end of the rectangle.
3. Notice that you now have a 4 by 4 square and a 4 by 3 rectangle inside your 4 by 7 rectangle.
4. Put a big "X" inside the 4 by 4 square, so you will not look at it again.
5. See the new smaller rectangle and find its shape.
[Draw the 4 by 3 rectangle turned about so that it is a 3 by 4 rectangle. Its shape will then be 4 : 3, and the value of that ratio is found by dividing 4 by 3.]

You may now find the value of each shape...
the 4 by 7 rectangle you started with... 7 divided by 4 equals __________
the new 3 by 4 rectangle... 4 divided by 3 equals __________
[For convenience, show the values as a decimal fraction rounded to three decimal places.]

Are the measures of the two shapes equal?
Is the 4 by 7 rectangle "Golden"?
Is the 4 by 7 rectangle CLOSER to being golden than the edge of the 3 by 5 card?

[Hint: You will find the DIFFERENCE between the two shapes and see whether this difference is more or less than 0.167]

Your Activity now is to find the height and width (the two dimensions) of a rectangle which is CLOSER to being golden than the 3 by 5 rectangle.
As we said, there is such a pair of whole numbers, and the numbers are less than 10!

'bye for now, and Happy Searching for the Golden Shape!