PWR: Denominonsense

I have come up with a simple idea as an adjunct to teaching kids algebra. The name itself is a portmanteau of denomination and nonsense*. The idea occurred to me during my own senior cycle when I asked myself,  'what is 3 cars * 4 dinosaurs?' The answer, either 12 car-dinosaurs or 12 dinosaur-cars, is usually scoffed at until I point out the usefulness of man-hours.* We can continue with 12 car-dinosaurs / 4 houses gives us 3 car-dinosaurs per house.

* Seconds squared also seems a bit ridculous when you think about it.

The value for me came in the premature discovery of what I subsequently learned is call Dimesional Analysis. I used this to easily remember formulas. For example:

• speed is measured in km/hour (therefore km/hour = km/hour)
• km = distance
• hour = time
• speed = distance/time

I recently watched the Prof. Brian Cox lecture "A Night with the Stars". In the 44^th minute he puts an equation on the board involving Planck's Constant. If you use s (for second) instead of time and p to represent Planck's Constant, the equation is similar to:

• s   = cm*cm*g/p
• sp = cm²g
• p   = cm²g/s

Planck's Constant is actually given as Kgm²/s. So without having seen Planck's Constant before, I was able to work out its unit.

My idea is simple: teach algebra with words (which could be the NATO phonetic alphabet to increase the power-weight ratio). The units are actually called denominations because of the first rule of Denominonsense is similar to multiplying farctions:

number * number; denominantion * denomination

For fractions* that is:

numerator * numerator; denominator * denominator

Again, I think this helps increase the PWR as if a student can handle one it gives them an entry into the other.

* Furthermore for fractions 1/3 + 1/3 = 2/3 just as 1cm + 1cm = 2 cm. But for 1/3 + 1/4 or 1cm + 1km you need to find a common denomination/denominator

 

PWR: 3 * 3 Is Not A Square

Calling 3² a squared number is what I call a 'dangerous concept', as in a little, which I read as less than complete, knowledge is a dangerous thing. 3 * 3 = 9. 9 is a point on the number line. For smart people this makes no difference but I think this concept contributes to student's difficulties with units - as explained in the original power-weight ratio post.

Another way this can be seen is if someone says how silly negative - or in some case imaginary - numbers are. "How can you have -3 of somethimg?" is a variation of the popular refrain. This hints to a larger misunderstanding of numbers. When I heard this in school, I would ask my classmates, "Can you give me 3?" The key here being you can only give me three of something: 3 pencils, 3 rubbers, 3 rulers. The unit reifies the number.

And so it is with multiplication that 3units * 3units = 9units². It is the unit that becomes a square, not the numbers. If kids were thought that numbers were raised to the power of two but units were squared, wouldn't that help them to grasp the concept.

Along a similar theme, my next PWR post will be on denominonsense.

Power-Weight Ratio

Image via Wikipedia

I came across this idea in a video of a the Joshua Bloch presentation "How To Design A Good API and Why It Matters". Power is what you can do with the API. Weight is conceptual weight, the number of ideas you have to learn to use an API. We want to do more for less effort.

I have several Maths-related ideas that run along these ideas. The first concerns the terms 'plus' and 'minus'. The main thrust of my idea is to replace minus with opposite (direction) and, for the moment, to replace plus with forward* (direction).