I was actually finding myself agreeing with the shortcut (sounds perfectly logical!) then spent the last 2 minutes going over in my head what might have been wrong. The key, to me, obviously falls on the ‘exponent’; and how the base can be drastically ‘increased’ once it’s doubled. Then i began to wonder: will my grade 8s be able to spot the errors in this rationale? To be honest: I am not confident that they will see the flaw in this statement. (actually, most of them, once the formula is derived, will happily accept ‘THE’ one way of doing this type of question without going further. I see this not their fault; but ours.)

I have to say that I really was struck by what you said during your TED talk about ‘conversation’ and ‘math’; too often, in school systems, we worry about covering the ‘math’ and forget about the ‘conversation’.

My challenges, and not sure if it’s shared by other educators out there, is that in an inquiry based learning environment, if the learners (collectively) do not see the flaws in above mention ‘shortcuts’ and are content to accept whatever is given, how do we move forward from here? would love to hear what you would do; such is my challenge with my kids and my staff.

Cheers!

D1*H1 > D2*H2

is the same thing as knowing that Volume 1 > Volume 2.
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To compare the volumes she can look at their ratios.

(Volume 1)/(Volume 2) = (D1/D2) * {(D1*H1)/(D2*H2)}.

She knows that the ratio in curly brackets is greater than 1 — for example it could be 3 — and she wants to know if the volume ratio is also going to be greater than 1.
_________________________

If the diameter ratio is greater than 1, then yes, the volume ratio will definitely be greater than 1. But if the diameter ratio is less than 1, then the volume ratio could be less than 1; for example, 3*1.1 is definitely greater than 1, and 3*0.9 is greater than 1, but 3*0.2 is less than 1.

Then ask the class to figure out why the volumes are different, via calculations.

“Do increases in both r and h impact the volume at the same rate? When you double r but keep h the same, what happens to V? When you double h but keep r the same, what happens to V? Does the square matter?!”

I’d start with concrete experiment –> table of numerical, sequential example values –> let the kids draw algebraic abstraction/generalization.

I think that the answer to this question has been well addressed.

What to do next?

An important idea that arises from her hypothesis is variant and invariant quantities. She correctly identified pi as an invariant quantity but may have also treated the radius as invariant or assumed that if r1 > r2 then r1^2>r2^2.

I tended to not answer “is this right” questions. Instead, I reposed the question as a hypothesis to be tested. She has made a hypothesis about this problem that can then be posed to the rest of the class. Her hypothesis works for this particular problem. The question is whether it is true in all cases. If not, then for what cases is it true and why? This extends the opportunity for the participants to problem solve, model, engage in proof, number sense, operation sense, and do mathematics.

Agreed

Showing that she is wrong with a counterexample will at least teach her that she doesn’t know algebra.:)

A useful first step before you “proceed to teach her algebra.”

And thinking Tuesday comes after Wednesday is a misconception. This was a mistake in reasoning, namely algebraic reasoning.

Without being there I would suspect that she already knew which was larger (through the discussion that was going on) and then arithmetically saw that r*h resulted in the same conclusion as r^2*h, in this case. Showing that she is wrong with a counterexample will at least teach her that she doesn’t know algebra.:)

With my science teacher hat on (rather than my maths teacher hat) I encourage students to value models based on usefulness rather than “correctness”. In this case the easier arithmetic is certainly a vote in favour of her model if it yields useful results.

Using a modeling framework, I’d ask the followup questions:
– Does it always work?
– If not, when does it work?

Most models are useful as long as we understand their limits. This model clearly works for the provided example. The fact that it doesn’t work all the time doesn’t stop the potential for it to be useful in a range of situations that may well be easily definable.

Trigonometry ratios are shortcuts that are “wrong” in the sense that they don’t work for all triangles. We use them because we understand their limits (right triangles only).

I’m stealing this one for my own classroom. The exploration potential is amazing – thanks :-)

One issue is the fact that the correct volume formula would not always correspond with the suggested radius * height as far as choosing the larger volume goes.

A different issue might be an incomplete sense of what volume is and and intuitive sense of how changes in radius & height (or other dimensions) affect volume.