During the 2nd week of school I had to get a sub for Friday to attend a 2-day math workshop. Oh my, whine, whine, whine: too early in the school year, too hard to leave students with meaningful work when we just started, sets a bad precedent, .....
Okay. It was a great workshop. I learned so many things that will be BIG payoffs in the long run, so can I take back my whining? (except for the fact that now in 2 weeks time I'm basically mandated to miss another day to go to another workshop).
Here are some things I learned:
* a way one teacher gets her kids to remember the divide by zero and zero divided by something rule: 0/K = 0 means it's "okay to have a zero in the numerator". N/0 = undefined means "NO! you cannot have a zero in the denominator".
* which reminded me of the way I learned a long time ago to help kids remember the no slope and the zero slope of vertical and horizontal lines: "No" starts with a capital "n" and when you're writing "N", the first line is a vertical line, so vertical lines have "no slope". "Zero" starts with a capital "z" and when you're writing "Z", the first line is a horizontal line, so horizontal lines have "zero slope".
* with calculator usage for calculus and showing a function and its derivative function at the same time ..... put one in y1 and one in y2 and in MODE change the graph style to "simultaneous", and then you can start and stop the graphs at will and have a rich discussion about positive and negative slopes and + & - derivative values throughout the whole graph.
* I never remember which one of the "ON" or "ENTER" buttons on the TI=84 stops or pauses the graph, so I always end up pushing both. Well! The "ON" button stops the graph, so you can get "ON" with your life.
* Which reminds me of what I shared in the same session regarding calculators and limits to infinity. I have my kids look at just one of the polynomials of the numerator or the denominator in a rational function, say 4x^3 - 2x^2 + 100. I ask if you plug in larger and larger numbers, will that evaluate to basically the same or different value than if you plugged in the same large number to 4x^3 (the dominant term). They are convinced they'll be different. So we put 4x^3 - 2x^2 + 100 into y1, and in the main window do "function notation" of y1(5) and compare it to 4(5)^3. Then we start comparing the 2 things by typing in larger and larger numbers. They're eventually the "same" on the calculator. I tell them that the highest power term is like the ocean and all the other stuff is like spit, and if you spit in the ocean, it doesn't change the volume much, so you can basically ignore it when you are calculating limits at infinity.