Real Mathematics – Numbers #5

The word fraction has a Latin root “fractio” which means “to break”. Let’s take a slightly different approach for fractions. Instead of breaking, we will use “folding”.

Fractions with Papers

In order to understand the four arithmetical operations in fractions, we can use a standard A4 paper. First we will fold the paper into two halves. Every half represents the fraction ½. We can continue folding one of the pieces in two halves. In the end we will have the following papers which we can use for four mathematical operations:

img_4511
a. 1
img_4522
b. 1/2+1/2
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c. 1/2+1/4+1/4
img_4526
d. 1/2+1/4+1/8+1/8
img_4527
e. 1/2+1/4+1/8+1/16+1/16

Addition

We are aware of the fact that all five of the versions are equal to one another and they each add up to 1. If we take a look at versions b and c we can conclude that ½ is equal to ¼ + ¼. If we substitute 1/2s in the version b we can find the following result:

1 = ½ + ½ = ¼ + ¼ + ¼ + ¼.

Subtraction

Let’s continue from the previous. If we subtract ¼ from both sides of ½ = ¼ + ¼, then we get:

½ – ¼ = ¼ + ¼ – ¼

½ – ¼ = ¼

Multiplication

Let’s start with another A4 paper and take its whole as 1. This time we will fold the paper and use the marks on it.

img_4511

Assume that we are trying to find ½ * ¾.

Check the first fraction and fold the paper into two halves since it is ½.

img_4514-e1535716599583

Then check the second fraction and fold the paper into four equal parts.

img_4515

Since second fraction is ¾, mark 3 parts of the paper.

img_4519

Now unfold the A4 paper completely. It shows that there are 8 equal parts and 3 of them are marked. Hence solution is 3/8.

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One wonders…

  1. Using the paper method show the connection(s) between 1/8 and 1/32.
  2. Find 1/8 – 1/32 with the paper method.

Game

Imagine a gambling game in which you don’t have to gamble your money. In the worst case scenario, you will win nothing. A dream for gamblers, isn’t it?

Let’s say you get 128.000 dollars and 6 red and black cards (3 for each). Here is how the game goes: In every hand you must bet half of your total money. When you pick a red card, you will get twice what you played. When you pick a black card, you will lose all the money you bet in that hand.

When the game is finished (after all 6 cards are played) if you end up with more than 128.000 dollars you will get that surplus.

What is your strategy to win? Explain your answer.

M. Serkan Kalaycıoğlu

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