Extending this notion of mathematical discourse...we tackle another set of statements. Normally what I would have done to review the distributive property and its application to expressions and equations is to generate problem sets. Students would have practiced the procedure and talked about how to do the procedure, but rarely would I have challenged them to think what lies underneath it.
What's under the hood of this math engine?
We've been using the Always, Sometimes and Never strategy for thinking about the "underneath" And this time it's varied by asking is it true for all, some or no numbers!!!
- Adding 8 to a number gives the same result as multiplying that number by 4
- x + 8 = 4x
- Subtracting 10 from a number gives the same result as adding 10 to that number
- 5n-6 = 3n + 34
And so we started into the discussion.
What's amazing is that my students now know they have to "test" each statement by picking a variety of situations. So they usually start with zero and see what happens...then they try several positive numbers and several negative numbers. Finally checking fractions and decimals.
Looking for patterns and having a generalized method of testing each statement has been one of the habits of mind we've developed. Pretty quickly students were able to tell if something was never true or not.
"There's no way that the statement Subtracting three from a number gives the same result as adding 3 to the same number, one student announced. I wasn't sure how they would handle this statement since it doesn't use the actual numbers...instead it's written out in words. Didn't seem to throw them off at all. Subtracting 5 from a number and then squaring the results gives the result 16 sometimes..."and I found 2 examples of when it works." But there weren't any other numbers that made it true. I was so impressed because most students realized that there were two successful solutions.
They really derive a sense of pride from finding these bits of evidence to support their thinking. Even if a student didn't find the "evidence", listening to the class conversation makes them strong. "I may not get it all the time, but then during the discussion I figure it out." My favorite part of the whole lesson happened near the end of the day...I'll admit that I was tired and having been through the same lesson 4 times already I might have lacked a bit of enthusiasm. But my 6th hour picked me right up...admantly suggesting that we should announce every good answer and thinking processing by saying hashtag swag. Since I hadn't a clue what this meant, the class explained it was their way of saying "Good Job" with some style.
I can do that! So lots of #swag's erupted as different students defended their ideas and announced their evidence. Did we actually Tweet these out? No. We just awarded #swag to the people for their contribution and did a little happy dance.
Discourse can be fun. It should be fun. It should be playfully serious. I'm learning and getting better with every time I try this strategy.
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