Read Part One
For years my presentation of Algebra II has a matching activity towards the beginning. You line up parent function equations with the vocabulary word and graph.
My first two years this activity was a bit of a throwaway. The students would dutifully cut out the squares, match everything up, and paste it to a colored sheet of paper. It would promptly disappear and be lost for the ages. After a transition to notebooks it would at least stick around a little longer. But we never really referenced it again.
That bothered me though. This shouldn't be a throwaway activity at the beginning of the year. It should be THE activity. We should constantly come back to this to make comparisons between the different functions we are studying. In the old method of attacking one specific function at a time, it could be 6 or 7 months before we get to some of these.
How do we solve this? What does it look like to teach Algebra II in bigger, more universal strokes instead of what comes across as eleven special cases?
I've arranged the major functions into levels based on familiarity. The main players of Algebra I come first.
Level 1: Linear, Quadratic, Absolute Value, Radical
Students in my Algebra II classes have vague recollections of these four parent functions. Absolute Value and Radicals have been introduced as concepts by now, even if the students didn't do anything more advanced.
To help build what will eventually be my testing standards, I created a guide sheet for this level. This is not something I would give the students, but could be the foundation for a milestone exam. Problem sets would evolve out of this as well.
The guiding principle in my approach is to impart a universality to all this algebra stuff. Abstractly, solving a quadratic is just an added layer to linear. Radicals can be an added layer on linear or quadratic. Same with absolute value. I want the students to see equations as layers they peel back to get to the tasty x at the center. Each layer has a different consideration, but it all falls under the universal concept of isolating a variable.
Other than a brief introduction to each parent function and how you "peel back its layer," I don't see the need to isolate these four types of problems. For all the criticism I gave [rational functions] the other day, I would like to build students who would be comfortable unpacking something like that. Spotting linear components, quadratic components, etc at a distance is the first step.
After perhaps 3 or 4 weeks of tackling the algebraic natures of these functions, we would move on to graphs.
Level 1 Graphs: Linear, Quadratic, Absolute Value, Radical
A big connection lost of most students is that a graph always represents an equation and an equation can always be represented on a graph. A "solution" is really a point we want to include on the graph. As a recall/introduction of function notation, we would graph some of the same equations we were just manipulating algebraically. Being able to identify f(3) on a graph is something I have had a lot of trouble with for unknown reasons. Often the nuance of f(x) standing for y is a bit strange. To the student they're like "uh, wait, but that's an x, right?"
Next we would establish transformation properties through two activities:
The first establishes transformation rules through plotting. The second matches descriptions with equations and pictures. As we progress up to higher leveled functions, we can expand on these activities. Transformation rules are actually not much of a problem after you fight with them early on. The main change would be to stick with the guiding principal from the equations section, multiple parent functions simultaneously. Many math behaviors are universal.
Last we would incorporate graphs of inequalities in the mix. It's possible to introduce graphical solutions to systems at this point. I think algebraic system solutions are going to wait until later.
After 3 to 4 weeks of graph related topics, the process starts over again. We go back to equation and inequality procedures but with a new batch of functions. Possibly with quadratic and radical elements layered underneath.
In Part Three I will show off the full curriculum layout. There are bits and pieces of the curriculum that are in new, exciting places. Systems being pushed to the back, conic-section style parabolas pushed up with quadratics where it belongs, etc. The idea with this work is to improve the flow of topics and focus not on parent function silos, but more universal concepts of algebra.