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2

If 10-11 year olds can learn programming (which some have done even before Scratch was a thing), then it's hardly a leap at all to suggest that 15 year olds can learn functions since they are a common element of programming languages. Every student is going to be different and some are going to be better at math than others, but I think this notion that ...

5

In all of these examples, I would draw a line unless that idealization led to missing something essential in the problem. These idealizations happen all the time, and they tend to be very useful. E.g., for the flow of water, we normally don't need to consider that water consists of discrete molecules. Ditto for examples like immigration, fluctuations in the ...

-1

I would do points, rather than a connected curve for A. Except if you are looking at large amounts, I wouldn't be so fussy and just draw a line. This actually has some use if you consider say different rates sold by a monopolist practicing price descrimination, for instance. (I think this is beyond your audience though.) For B, if you are interested in ...

3

Some abstraction is perfect for children. Mitsumasa Anno was a Japanese writer of children's books and he had many innovative approaches to introduction functions. This is from Anno's Math Games II. You can even see the introduction of function inverse.

11

Functions are far broader and more applicable than you give them credit for. Consider the following: Your students would be able to determine the elevation of Quito or the capital of Wyoming by interpreting that table far earlier than 15, and they're evaluating functions in order to do it. The only other thing you need to do is show them an example of ...

25

In the U.S. Common Core standards, functions are supposed to be introduced in the 8th grade, i.e., around age 13-14. So arguably age 15 is a year or two behind where they ought to be. The standard for the 8th grade says: Understand that a function is a rule that assigns to each input exactly one output. So honestly that really doesn't seem like a hugely ...

16

I think "function" is one of those notions that can be presented in different ways to people at different ages and who have different levels of ambition in math. This is similar to the notion of a "set," which I was taught in school ca. age 5 or 6 in an age-appropriate way, but then learned about at a different level in college. At a ...

2

Building off the circle example, you can actually work out the centripetal acceleration formula by implicitly differentiating twice. If your students aren't familiar with vectors you can just plug in x = 0 and y = 1: $$x^2+y^2=r^2$$ Differentiate with respect to t: $$2x\cdot x' + 2y \cdot y' = 0$$ Plugging in x = 0, y = r, you can solve $y' = 0$. ...

3

The maximum of a large number of independent, identically distributed random variables -- e.g., the highest flood observed over a long period -- has an extreme value distribution. One common case is the Gumbel distribution, whose cdf has the double-exponential form $e^{-e^{-x}}$.

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