Okay, maybe I’m a bit slow but I said and thought something today that I’ve never thought about before.
We’re looking at indices & surds in my Year 11 class. Today we were looking at the rules of indices: multiplying and dividing indices was “insultingly easy”. Negative indices seemed to push it over the edge … some very confused faces. We struggled with it and I think most people managed to get an instrumental understanding but not a relational one. We’re working on that!
I was trying to explain that was equal to .
Now, in explaining this I said it was 1 divided by 6 three times. Seems fine but where does the one come from? We talked about reciprocals, we talked about the pattern that follows as we look at … and we talked about a few more bits and pieces.
I then thought about could be thought of as 1 x 6 x 6 x 6. This adds a bit of consistency to the system and made me think of an inaccuracy that I sometimes hear myself making or confirming.
“What does mean?”
“6 times by itself 3 times”
But does it? 6 times by itself three times could be 6 x (itself three times) or 6 x (6 x 6 x 6). I suddenly thought surely this could be 1 times by 6 three times. This helps with consistency in two ways.
Firstly, with our negative powers – we can say that it is 1 divided by 6 three times. This has internal consistency and I like it.
Secondly, raising something to the power of 0 makes sense – 1 times by 6 zero times is obviously 1.
I’ve never seen or heard this anywhere before – am I wrong? Does this break down somewhere that I’m not aware of? Do you think this could be a helpful way to see it?
Also, is there a richer way to introduce this apart from pattern spotting? Any ideas gladly received.