Fractions


#1

Analogy: fractions are like having dimes, nickels, quarters – any type of coin you want. 1/17 coin (seventeenths) and so on. Thinking of an analogy that makes them more approachable to people. When you add coins, you think of their “penny value” and combine that. (nickel + dime = 15 cents). Least common denominator. For the arbitrary coin case, need to find the “penny” equivalent and use that.


#2

I somehow resolve to a cake when it comes to fraction. It is instinctive to know that 3/4 + 5/6 will be more than 1.

And to execute the calculation visually, is… to cut 1 cake to 3/4, and another to 5/6, and put them together. And with this, we can observe how much cake is there.

Using a round shape as a basic for fraction is one way. We can also use clock arm/angles instead of the cake.

Cake method :

Angle method :

For subtraction, just reverse the rotation and contnue with the same turns angle required.

  • using conventional math, you should get x/y = 7/12.

#3

Great visualization, I hadn’t seen that analogy before. I wonder if the angle method can be used with a clock analogy as well? (Makes “looping over” more clear, you are tallying up hours.)


#4

I had that thought… Yes, angle method is similar clock analogy. It can be used too.

Angle or bearing for me is something special… it actually an that open space can be defined in various ways. Eg. North (0 deg), West (270 deg), Straight-ahead (0 deg), 3 o’clock(90 deg), take a U-turn (360 deg) etc… :telescope:

As for this Fraction case, the idea to share is we should not (just) focus on the nominator over denominator typical mathematic steps. There is another way of looking at it.

Having this said, it still puzzles me why for multiplication we can just multiply both the nominator and the denominator directly and not as straightforward as fraction addition. :evergreen_tree:


#5

Yep, it’s a good question. Fractions are inherently a division (1/4), and division/multiplication ‘fit together’ very nicely. We can apply them in any order, so:

$$\frac{2}{3} \times \frac{4}{5} $$

is really

“multiply by 2, divide by 3, multiply by 4, divide by 5”

which can be re-arranged to

“multiply by 2, multiply by 4, divide by 3, divide by 5”

$$\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4 }{3 \times 5}$$

However, addition, multiplication, and division don’t “fit” in the same way: we can’t change the order around:

$$1 + 2/3$$ is not

$$(1 + 2)/3$$

or any other “simple” re-arrangement. To add fractions (i.e., to add multiplications and divisions) we have to mold them into common terms.

$$1 + 2/3 = \frac{3}{3} + \frac{2}{3} = \frac{3 + 2}{3}$$

We can use the rules of addition once the denominators are the same (“three thirds and two thirds is five thirds”). But getting the fraction ready, where all the denominators match up, takes work. I wish I had a better analogy for this though. Maybe it’s like trying to add quarters and pennies directly. You can’t say you have “2” cents because you have two coins, you have to break them into the common term (pennies) and say you really have 25 + 1 = 26 cents.