Sunday, 6 January 2013

Line Dancing in Sequence

This post comes from some inner thoughts after tweeting at Michael Pershan earlier today:

Stage Zero. The "0"th figure. The flat rate. The y-intercept. It's location is rather interesting in the (Ontario) curriculum. And it's instruction is something that's come up before, both in class and locally... at least in terms of the mathematics department at my school, and at least partially beyond. (For instance, I know there was some mention during my "Gap Closing" Professional Development last year.)

For those not versed in Ontario high schools, the setup: In Grade 8, students do patterning, starting at the 1st figure. They also create (one variable) expressions to represent such situations. Then they come to high school, and while there is some patterning in Grade 9, it's usually used as a launching point into algebraic (two-variable) equations... equations which start at the flat rate. The 0th figure.

This is always a bit of a leap.

 That transition sucks.

Of course, it has to happen sometime, and I don't necessarily have an issue with where it is. More where it ALSO is. Because what I want to talk about here is arithmetic sequences. (Sequences that follow a linear model, such as all the even numbers, or something like 12, 22, 32, 42, ...) These sequences have a DIRECT link to the sort of patterning done in Grade 8. In no small part because there is no "term zero", and the Arithmetic Formula is defined as "Tn = a + (n–1)d". That's right, no "y = mx + b" here! But where are these arithmetic sequences in the high school curriculum?

Grade 11. (And interestingly only in the University level course.)

It always feels a bit like a step backwards. After two years of getting students to work (mostly) with continuous functions on the whole Cartesian plane, for one unit, we pop out these discrete functions where n > 0, linking back to Grade 8, and using a formula that deals with the "0th figure" by using (n-1).

 Also, when did 'x' -> 'n'?
Now, it's not a completely irrational choice. Geometric sequences are dealt with at the same time, and exponential functions (their foundation) are also first taught in Grade 11. Also in Grade 11 is the notion of functions, and the link can then be made between those arithmetic sequences using (n-1) and shifting the entire sequence/pattern by one to the right.

Still, it sort of begs the question, why isn't the equation of a line:
"y = c + (x-1)m", where c = b - m?
"Because it's not simplified!" you might respond. Okay, so why isn't the formula for an arithmetic sequence:
"Tn = dn + c", where c = a + d?
(Aha, gotcha! For that matter, why do we write the slope part FIRST for lines, and LAST for sequences?!)

So where was I going with this?

I think there's some value in using arithmetic sequence patterning, all the way back in Grade 9 - whether you use "Figure 0" or not. For that matter, shifting a pattern left or right by one is a great way to illustrate parallel lines! And if students can even see that (x-1) is a movement of one right, there's a nice foundation for two years down the road. (Something to try... give students two identical patterns, A and B, but start Pattern B at A's "Figure 2"...)

By the same token, there is some value in connecting the "new" sequence formulas in Grade 11 back to more "familiar" ones - or rather, see if the students themselves can make the connections. Of course, one of the main problems with the 11U course is that there only ever seems to be about 3 days to handle all of sequences, series and recursion. @.@ That's curriculum for you.

For more on visual patterning ideas (or to submit your own), check out a recent website effort, Visual Patterns. Which is actually only the first of a few shout-outs I have planned for the week, but the others will be in my web series. (I'm never sure if I can legit call it a webcomic.) Which I bring up in part because, with semester one resuming tomorrow and exams three weeks away, I'm liable to only be using that Blog for the rest of the month. Enjoy your January!