So, discussion on the #MTBOS twitter page was the trying to estimate decimals… which I’d *just* shown one of my folks. She was doing 3.23 x 5 and I suggested a “sense check” was pretending the decimals weren’t there and doing 5 x 3 … she said yes, that she got that — but what about when… (pause, point, write…) what if the bottom number had decimals too? I told her how to do it and agreed that yes, that was harder to estimate…
The twitter discussion was about that — and that if you multiplied 5 by less than one, the answer would be less than one.
The reply was that knowing the answer was more or less than one could be useful.
Hello 🙂 Multiplying by less than one means the answer is smaller than …. the original number.
If we’re tripping over the language then I’d suggest if we’re discussing things with students, we be sure to have relevant, clear examples that make things concrete. (THe example did render an answer less than one. However, if it had been 5 x 0.9 it would hae rendered 4.5)
Would I have noticed this had I not spent a lot of time w/ exponential growth and decay? With compound interest computations? And saying a million times that multiplying by 1 gives us what we started with… 1.04 will be a little bigger… multiplying by less than one is a little smaller… and doing my best to connect the “.90 times is saying 90% of that, which isn’t going to be the whole thing.”
I’m thinking based on two math teachers tripping over the language that … me just chatting through things isn’t going to have made much difference, Especially since the student I was working with had not been able to find language to express “more decimal places — the second number less than 1.”
Then I pondered whether I should say “I explicitly teach this” … when actually, I at least try in the time given to not just state things explicitly… but instead, call attention to the pattern and try to get the student to get there… to ponder “oh, sometimes multiplication doesn’t make things bigger.”
I am also thinking that “explicitly teaching” math can often mean explicitly using language to explain what’s happening mathematically… but then making room and opportunity for the necessary application of that language to numbers, and that that brings its own set of discoveries.