I had thought I'd do this via a desmos activity, and I even started to make the slides, but then decided to do it live in class with everyone totally in sync, because I wanted everyone to witness something at the same time.
Here was the sequence, and the narrative, and the results in italics:
No calculators!
What do your order of operations instincts tell you to do first here: "\log_{2}\left ( 8\times 16{\color{Blue} {\color{Blue} }} \right )")
multiply the 8 and 16
No calculators!
What do your order of operations instincts tell you to do first here:
multiply the 8 and 16
Okay then do it!
Great, and how much is that log?
7
Some discussion here. How did you get that? Some recognized 128 as a power of 2, some did not. I allowed those who weren't as familiar with the powers of 2 to use their calculator ONLY to try out different powers of 2. NO log buttons. I wanted them to COUNT how many 2's were being multiplied, in order to prepare for more counting AND tallying.
Great! How about this one then?  "log_{2}(128\times 4)")
Some answered 9 very quickly, some answered log base 2 of 512. We discussed both answers. Those who got the 9 quickly were very good at expressing how they did it by using the already-known exponent for 128 and increasing the total number of 2's by the additional 2. My plan worked!
And what about "log_{2}(128\times 512)")
?
16. Lit up the board like a Christmas tree. This was what I wanted. Everyone to just count the total number of 2 factors. I pointed out that now they were seeing numbers through a new filter. Now 128 is 2^7, and 512 is 2^9.
Try these the same way - don't actually multiply the arguments, just see them through a filter:
Now comes the real challenge. Let's go back to that first example, and say it in English, along these lines:
Some answered 9 very quickly, some answered log base 2 of 512. We discussed both answers. Those who got the 9 quickly were very good at expressing how they did it by using the already-known exponent for 128 and increasing the total number of 2's by the additional 2. My plan worked!
And what about
16. Lit up the board like a Christmas tree. This was what I wanted. Everyone to just count the total number of 2 factors. I pointed out that now they were seeing numbers through a new filter. Now 128 is 2^7, and 512 is 2^9.
Try these the same way - don't actually multiply the arguments, just see them through a filter:
I'm happy to report that these were very handily answered and justified correctly, by more than just the usual one or two people.
Now comes the real challenge. Let's go back to that first example, and say it in English, along these lines:
After a bit of coaching:
And what's another way of saying "the exponent that 2 needs in order to get 8"?
After just a hint to use the word log......
Let's go back to the other ones you just worked out and say them in English:
This went super fast!
Now let's generalize:
And for the FIRST time in my career, my students told ME the additive property of logs, instead of the other way around. The third line here was supplied, for the first time, by my students:
That felt good.
What a wonderful sequencing. I love the word bubbles, and 100% agree that going with a full-class-in-sync approach was positive here. I'm not sure, but maybe discussing it as a big community instead of with small groups or 1-on-1 makes it seem more official? Something for me to ponder...
ReplyDelete(p.s., for the log LaTeX, if you slap a \ in front it will typeset the function name a little nicer [same with \ln, \sin, etc.)
Thanks Andrew! For some, I suspect the whole-group strategy does make it more official. What I was hoping for was drama - that moment when someone, or someones, would all say "OOHHHHH!!!" simultaneously. I'm into drama. Didn't quite happen, but it still felt pretty effective.
DeleteAs for the LaTeX - I used codecogs to generate it, because I don't know how to code, although I'm learning html. I'm going to try your tip though - fingers crossed.
And thanks so much for your comment!
Very helpful way to think of the log function. Enjoyed it.
ReplyDelete