Dark Blue Magic

What if we were in a plane and it crashed on an island with no Calculus textbook? How much knowledge could we recover?

Dark Blue Magic is conprised of the tricks that we have developed. We call it magic and hopefully you will see that it is not a science.

What we do isn’t like analytical calculations where if you make it work once, then you are done.

Students at Mockingbird Academy are working to build a second category for magic: Black Magic. This category is intended to be the ideas at the very heart of how we think, the “hidden” stuff that is harder to “see”.

Test different examples

It is good to achieve success with two different examples before you announce to your team that you have a hypothesis.

Consistency

Look for consistency.

Look at the equation below, can you see a possible mistake?

Please note, the above equation, without the mistake, would probably be written as

You must use your powers of clairvoyance to perceive the hidden factors. This brings us to the next point:

Hidden Factors

Part of the magic is recognizing that in some examples, important things could be hiding as a factor of one or a term that is zero. For example, maybe the correct term is but in your chosen example you have a=b so for that example you would conclude, incorrectly, that the term is .

Be aware of constraints such as interchangeability.

Another part of Dark Blue magic is being aware if you are modeling something where two variables are interchangeable. If the variables a and b are interchangeable then you should see them always together like (ab) and (a+b).

Actions on Things Often Make Similar Things

For now, the last part of the magic is expecting a process to take something and produce something of the same type. Please note, we said “we expect”, we did not say “we guarantee”.

• A function takes a value and makes a value.
• A functional takes a function and makes a function.
• We expect to get a monomial when we take the derivative of a monomial.
• We expect to get a trigonometric function when we integrate a trigonometric function.

Elsewhere, Dark Blue Magic has been used by Mockingbird Academy Students to show how, on that desert island, we could recover several rules for calculating derivatives in Calculus. The Mockingbird Academy school master later realized that the Calculus studies had not covered the derivative for the following function:

It took hours, and it was necessary to develop a new notation, but in the end, all four students were confident that they had the correct answer. They went to the internet, and found a Youtube video showing how to get the answer they had in yet another way, and that was fun too.

One more Game

• Can you find the single mistake in the table below in 1 minute or less?
• Can you determine what the correct answer is for that cell, in 5 minutes or less?