Whew! It's TAKS week in TEXAS high schools. Four days of altered schedules and proctoring not-your-kids and not being able to teach "real" classes for most sections because otherwise the other classes would be off whack. AND to top it all off, thank goodness, we got the go ahead to hold some AP prep sessions during TAKS, so our seniors wouldn't miss a whole week of learning right before the AP exams. AND the schedule was such that I still have my 2 calculus classes all week. AND I'm holding an AP Blitz after school every day from 4:30 to 6pm to "rah rah rah" and tutor my calculus kiddies. Whew! (a "whew" sandwich).
I also see one of my 3 precalculus preAP classes this week, so I can't teach new stuff, but I don't want to have them just sit there, so I did a fun topic. I also gave them a quiz they were sure to get 100% on at the end of class, so that the "good kids" who actually showed up to class would be rewarded for doing the right thing and not skipping class.
I taught a topic I love that can be done in any amount of time. I had about an hour, so I covered: counting numbers in different bases. Then converting between base 10 and other bases (back and forth). Then adding in different bases. Then subtracting in different bases. THEN multiplying in different bases. I liked their wide-eye understanding of what it means to "carry" the 1 now in base 10 and how to "carry" in other bases. I also love what happened with one student. She has been struggling ALL year. She is never completely comfortable with a concept. She feels stupid. She's failing, and yet she shows up to class every day and pays attention and does not quit.
Anyway. This topic started out much the same for her. She sat there all frustrated, but she kept asking questions. Then. BLING. She got it, and she was whizzing through all the things we did. Then she was helping others. Then she truly got 100% on the end-of-period quiz. She walked out a happy camper.
Anyway. I present it by talking about how we can think of numbers in base ten as filling up bins from right to left. Each "chip" or "1" in a bin means I have "1" of that type of number. Once the bin reaches 9, and I try to add one more chip, then I am overflowing in that bin, and I have to scoop all 10 chips in my hand and put a "1" or chip in the next bin over to the left. The bins are:
... 10^4, 10^3,10^2,10^1,10^0 (and so on to the left)
So in (say) base 4, the capacity of each bin is 3 (one less than the base number),
... 4^4, 4^3, 4^2, 4^1, 4^0.
For example, 231 in base 4 means you have 2 16's, 3 4's, and 1 1's, so your base 10 number that means the same thing is 2x16 + 3x4 + 1x1, or 45.
I also peaked their curiosity about what decimals would mean in different bases.