Glenn Waddell () spent a large portion of the first day showing how he ties all of the algebra 2 functions together through a common algebraic form.
The one of these forms that most teachers likely don't use is the linear one:
y = a(x - h) + k.
Glenn has blogged about it here. I like the way this connects to the other equations which we do use. I like it, yet it bugs me and I'm not sure why. It makes a lot of sense and is a very useful form. 'a' represents the slope (or rate of change) and (h,k) is a point on the line. Slope-intercept form is great for graphing, but Glenn's form (I'm not sure what to call it) is so much more useful when trying to find the equation of a line given the slope and a point on the line or two points, etc. Once you determine the slope, substitute the point for h and k and you have finished! Here is an example using Desmos.
The more I think about it, the more I think that my discomfort with this form is simply that it is not what I am used to seeing, and perhaps, that it can be simplified (and I kinda like my functions to look tidy). But it really is a smart way of tying together so much of what we do with functions and their graphs. I think I will work with this at the beginning of my grade 10 academic class in September. They will all know y = mx + b and will be working through transformations of quadratics before long, so it seems like a natural fit. Thanks, Glenn!
I'm wondering if you think that Glenn and Jonathon's way could be merged? I like the (h, k) form and multiple representations but I like Jonathan's idea of cycling through and the algebra first. Do you think they could go together?
ReplyDeleteThey definitely could go together. Spiraling is great because you can do multiple forms of the same thing in different cycles. So if you need to do y = mx + b you could do that during the 1st cycle, then introduce y = a(x - h) + k in the next cycle and tie them together.
ReplyDeleteI know y=a(x-h)+k as "point-slope" form. I feel the same way as you: it kind of bugs me but only because I'm so used to seeing y=mx+b. I also agree that point-slope form is definitely more useful because you can choose ANY point on the line. If you choose the y-intercept, then h is zero and, voila, you have slope-intercept form. I like the idea of doing y=mx+b first (the kids are familiar with it) and then spiraling back to lines with y=a(x-h)+k later to tie in with other functions.
ReplyDeleteY=mx+b is big in Algebra 1, I have used Glenn model (vertex form, tho its really not that for several functions) the past 2 years. We have an intro functions unit and explore transformations there. I assign students different functions, they explore manipulating a, h, K and develop connections btwn eq. And graph transformation. We then jig saw, putting 3 or 4 different function people together and they share their findings, create mini-golf we on white boards of big ideas, then while class discussion. Then we stand up and model function transformations with our physical movements.
ReplyDeleteahhhh, why did i never think of this. we do y = mx+c (internationally we use c instead of b) and Ax + By = C, but not this form.
ReplyDeleteand it fits SO perfectly with what i do with my students in transformational geometry. it has always been difficult for them to understand why you use the opposite signs when you put the translation into the equation...then it doesn't really get reinforced until they do quadratics (which in our curriculum is a full year later!). this year i will reinforce it with linear equations!
now i need to rethink my pacing, because i do have linear equations before transformations, but this is too good a connection to miss. i love it. thanks for posting about it!!