As you said, it’s a passage, not a leap. Having students look for similarities between the two points might help smooth that transition. (e.g. “They’re always opposite each other”, “They’re on the same height”, “They’ve got the same y-coordinate”, etc). If students still struggle, why not have students write down the coordinates of some red-blue pairs, then see how they differ. Or you ask students to write down a rule that MIGHT be right (no pressure) and one rule they know is wrong. Then put them up and discuss why some of them don’t work. I haven’t tried it yet, but I’m fairly confident students will arrive at the formal end eventually.

What do you do when you have to handle the unavoidable passage from informal to formal?

I mean, there is a time when a student “has” to make this leap. You can’t keep going in high school doing arithmetic all the time, or can you?

So how do you deal with the reluctant student who refuses to abandon the soft spot of informal for the more formal?

Great ideas again.

It’s putting numbers in to see how something works. In some cases I might even do a similar thing with a class. (The bit at the bottom about students making mistakes with variables that they wouldn’t with numbers is certainly spot on the money.)

But the statement posed was “Before I ask for a solution, I ask students to guess and check”. That’s a whole different kettle of fish.

At least where I come from, guess and check means going through a cycle of numerical calculations to get closer and closer to the actual answer. That process, to me, is inimical to getting students to move past arithmetic solutions and on to algebraic ones.

3^4=80 too high

2^5=32 < 50 < 2^6=64

3^3=27 too low

Goldilocks Diamond might also be apropos.

I sometimes play a variation on this called “Guess my rule & Guess my mistake!”

A couple of preservice teachers in one of my classes were doing a standard presentation on “Guess my rule” as a way to broach linear functions, but inadvertently miscalculated.

I thought it was a great way to explore mistake making; for example, when you want to guess linear functions, any 2 input/output combos uniquely define the rule (function). I would consider this game “pretty well understood” if folks realize that any two points uniquely determine a line, but play using 0 as the first input to find the y-intercept, and 1 as the second input to compute the slope. (What about 0 and -1? And how is the equation for a line being written? And…)

Okay: Suppose I may make exactly one error. How many input/output combos do you now need to be sure that you’ve guessed the right linear function?

(It’s no longer 2. Is it 3? 4? More?)

And yes, for those who favor generalization: One may ask analogously about an nth degree polynomial for which up to k errors are made. But I think the line game alone (with 1 or 2 errors) is already a pretty rewarding activity!

Before I ask for rules, I ask for patterns, before I ask for patterns I ask for details.

I assume you are familiar with the Van Hiele model for learning geometry (see https://en.wikipedia.org/wiki/Van_Hiele_model).

Based on that list, if it is correct, then there is a step before conjectures.

Before I ask for conjectures, I ask what do you notice?

Also, based on this list there is a distinction made between deductive reasoning made at the high school level and the formalization required by mathematicians. One could argue that at the high school level, assumptions being made are not always being made explicitly, and that this is a key difference with mathematicians who have designed different sets of axioms on which everything else can be built up.

Before I ask for rigorous proofs, I ask what assumptions are we making here?