I am planning this multivariable calculus class that has taken me down this strange and circuitous rabbit hole of mathematics, (which I will not outline because zzzzzz), but here’s one thing that just popped up in this rabbit hole:

I’m having my kids think about alternative questions they can ask about a system of lines, other than “here is a system of lines… find the solution.” And I was brainstorming things they might come up with. And I came up with this:

Write a system of lines with solution (2,5).

I haven’t taught Algebra II in years, so I’m unsure if I’ve asked this before [1]… but I kinda love it. My favorite types of questions tend to be the “backwards question” (given the conclusion, come up with the givens). And I suspect that this kind of question for kids in Algebra I or Algebra II would generate some awesome discussions and help the teacher figure out misconceptions, and draw some neat graphical connections.

(For example: I imagine the idea of having one equation fixed, and changing the second equation shows the graphical connection that the point of intersection remains constant… and vice versa.)

[1] I know I’ve asked students to write a system of lines with no solution before, but I’m not sure about this one!

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Wow. Teaching this in Alg 2 THIS WEEK! I love this question! Thanks for making systems more interesting for my classes! I’ll tell you how it goes.

❤️, Julie

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Here is a related idea; changes and adaptations are welcomed!

Well before multivariable Calculus, students sometimes play a game in which they volunteer an input, and the instructor (though a student could certainly take on this role…) provides the output according to some

rule.In particular, the rule is a function; and, early on, the functions may be limited to linear functions (lines).

Question: How many inputs do you need in order to determine an equation for the line?

This is not a “trick” question; the answer is 2. (Perhaps use 0 to find the y-intercept, and 1 to find the slope.) In fact, any 2 inputs will suffice (which meshes well with the notion that 2 points uniquely define a line).

The answer to the next question is not provided.

Question: Suppose the instructor makes an error. Just one error, i.e., records one output incorrectly. How many inputs do you need in order to determine an equation for the line? (Why?)

In the process of asking this question, let us notice at least two attributes:

(1) We are using a degree 1 polynomial (i.e., a line) as our rule.

(2) The number of errors is 1.

And so we can rather quickly complexify the problem by varying (1) and (2). In a general setting more befitting of a multivariable Calculus course, one might ask:

How many guesses do you need to determine the function if

(1) it is a degree n polynomial, and

(2) the number of output errors is k?

For example, if n = 2 and k = 3, we are saying:

Suppose you are trying to figure out the equation for a parabola, f(x). To do so, you give inputs, x, and hear back their respective outputs, f(x). If the latter are reported with as many as 3 errors, then how many inputs must you provide before you know with surety what function corresponds to f(x)? Why?

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Yeah, I like this question because I think it has tremendous learning potential. How many students really understand that solving a system of equations (algebraically) involves various transformations of equations with the goal being to end up with x=a and y=b. I like the idea of graphing each transformation as they are created and eventually ending up with a horizontal and vertical line that intersect at (a,b). Asking your question begins to uncover all of that. I’m not sure how many students would start with x=2 and y=5, but if no one does, maybe a prompt along the lines of “what is the simplest system of equations that has solution (2,5)?” might get them there. (I wonder how many students connect (2,5) to x=2 and y=5 as equations.) From x=2 and y=5, we are ready to start talking equivalence and building other systems.

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