Tomorrow, we will be leaving piecewise functions behind for a while. The students this year are doing better overall than last year, but some students still are having significant struggles with understanding piecewise functions. When asked to graph a single line in a restricted domain, that seems to be fine. But as soon as the pieces are put together, students get very confused between the f(x) value from the equation, and the x-values from the domain condition (I haven't graded the quiz yet, so I'll have some better information about this soon).
We'll come back to it when we review for the unit test and the final exam, but for now, we need to move on. The goal for the next lesson is for students to understand vertical and horizontal shifts - why they happen, how to determine the shift, and how to generate translated graphs.
For the Do Now, students will be asked to plot out y = x^2 and y = x^2 + 3, and compare the graphs. They will do the same for y = x^2 and y = (x - 2)^2. At this point, they are working by hand so that they can, point by point, see what is happening. We will discuss their results and get some initial conjectures out. Then, they will do a graphing calculator exploration of the same ideas, which will allow them to graph more functions more quickly.
After this, we'll put the ideas together as I do some direct instruction, and define the concepts of vertical and horizontal shift. We will use what they've learned to understand why the vertex of y = a(x - h)^2 + k is (h, k) - which is, after all, a state standard! Yes! I hope that providing more scaffolding on translations will help them understand the vertex form better. I also think it will help them be more prepared for all those crazy phase-shifting, period changing trig functions they will encounter in pre-calc (along with the upcoming lesson, which will look at stretching transformations, in general and with specific attention to absolute value functions).
I graded quiz 2 and the scores were lower than I'd hoped:
9| 0 3 3
8| 0 3 3 3 5 8 8
7| 1 3 3 6 6 8
6| 3 6 8
5| 1 4 6
There were too many students who crashed on this one, although the bulk of the class was still in a good range. Many students had trouble with inequality notation - after working so heavily on interval notation, they seem to have forgotten how to use inequality notation, which is something they were already familiar with. For example, to express the numbers less than 3, we write (-infinity, 3) . Several students then wrote something like -infinity < x < 3 instead of just x < 3. It is always very interesting to me how new knowledge seems to crowd out old knowledge for a while, and then there is a process of assimilation where the mind brings them together and eventually sorts it out. The piecewise question was decent overall, although there are quite a few students who have not mastered it yet. I think that it needs some time to marinate in their brains, and we'll review these problems when we get toward the unit test. I think I'll develop a good error-checking handout for them to work on.