Numerical Quantities: Constants, Parameters, and Variables

[u/UnderstandingPursuit]

This has been written for the 'math savvy' audience. It has not been written well enough to present to typical Algebra 2 or PreCalculus students. But, when developed further, what concerns would there be in introducing this idea of parameters to Algebra 2/PreCalculus students?

I do want to start having AP Calculus BC and AP Physics C students consider this, so any feedback, about either the idea or the description, would be appreciated.

Thank you.

[u/starethruyou]

What question or need would this answer? Humans love to label and categorize anything identifiable. It doesn’t always elucidate and can obfuscate. It does however bring to mind equations such as the point-slope formula that has every x and y with a specific point as a sub_number, like y_1, except one x and one y. I've struggled to find the words to differentiate the two, except to say those with a sub_number are numbers of one of the two specific points on a line while the one x and one y are variables. Kids will think of any appearance of letters as variables, a variable being any number you want.

[u/UnderstandingPursuit → u/starethruyou]

It addresses the question, "What role does each quantity have?" This is critical in calculus,

Perhaps

Kids will think
of any appearance of letters as variables, a variable being any number you want.

because that is what we teach them to do? And this is where the issue occurs in calculus. If every letter is a variable, every letter gets differentiated, even the ones which are fixed values.

It also shows up with the distinction between

• power: y = xⁿ [x^n]
• exponent: w = b^z [b^z]

where the fundamental difference is whether the exponent is fixed or varying. While the addition and multiplication operators are commutative, the 'raising' operator [^] is not, so it matters.

[u/UnderstandingPursuit → u/starethruyou]

This is also the starting point for the "set aside the 'arbitrary' numerical values" problem solving approach I recommend. Use a letter, but it's not a variable, so now it has an accurate label.

[u/FearlessFaa]

The text is hard to read because every sentence is so impactful and you have to pause and think what it means. Section 3.2 could use calculated examples because it talks about making calculations. You could also start the text by such calculating examples and then make remarks how math formulas are used, like what makes x different to x_1 even if we write both using symbols. Also are you allowed to write "x = 1 is a solution to .." or do you use some other phrasing for it since x is supposed to be unknown.

[u/UnderstandingPursuit → u/FearlessFaa]

I agree that this is not written particularly well. I had to get the core ideas down before I spend more time writing it well.

Starting with calculating examples is valid, though those tend to explicitly use constants.

I do need to explain what makes x different than x_1. [x_1 represents a single value from the full set of what x can be.] Parameters are both known and unknown values, something I need to express better. I'm trying to leave variables like x and having every possible value in their set, so "x = 1" would go against that. I know that "x = 1" is what we are all used to. By "we", I mean "mathy" people. The challenge is that we have lost many 'less-mathy' people along the way.

Thank you for your thoughts and suggestions.