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Distribute THIS

This is my answer to the question "How do we cure students of 'Universal Linearity'?" It developed over several years of teaching calculus and precalculus as I tried to prevent that disease. I started by showing the students their favorite tricks, none of which involve correct math.

The problem is that they learn the Distributive Property and
like it so much that they apply it to everything imaginable.

Pointing out the existence of another distributive property was important to drive home the point that, yes, there is more than one distributive property--exactly one more!

The order of operations provides a way to understand that the second DP (abbreviation for Distributive Property) exists because of the "niceness" of math.

The above are five more DPs, but they aren't distinct.
Each can be derived from the two discussed previously.

Below is a cute way of seeing how similar the "new" DP
is in form to the "original" DP. 

At the left is showing that "nPow" in the previous
discussion isn't far-fetched. LOTS of everyday math functions have 3- or 4-letter names.

And when you translate the symbols into the words that also can define them, you find six more "distributive-looking" equations.

NOT ONE of them is true for all x and y because none can be derived from the two DPs discussed so far. Admittedly some of the equations on the left don't resemble their literal counterparts; thought has to enter the picture. Big problem for s

Function notation looks so much like multiplication that many,
many students READ it as "f times ...". Ambiguity's a big part
of student misunderstanding and mistrust of math. But I digress.
The point is that ONE type of function DOES "distribute"
over a sum in its argument. The linear function y = x.

Thus we have proved that the identity function has the property
of "linearity"--i.e., distributing over an argument sum.
We can't prove that no other function class works this way.
But we can emphasize it.
And we can show that while there are infinitely many functions
that have the property of linearity, they are all merely dilations
of the identity function.

NO OTHER FUNCTION WORKS THIS WAY.
At least none for high school.