# Pivot Algebra II Update Two

Another six weeks, another phase in the crazy plan finished. At the start, I didn't know how the graphing components would go, but we got some real quality conceptual work done here.

Read Part 1: The Story So Far, a summary of my equations unit

### Big Ideas

First and foremost, graphing contexts in modern Algebra II are heavily concentrated on transformations. You introduce the ideas of translation and scaling and repeat it for all the parent functions. But, it's done in such a way that gives zero context. I am guilty as charged with this. In the past, we teach the transformation rules, teach their descriptions and then move on to something else. There's no emphasis on why you want to know transformations are a thing.

### Step 1: De-emphasize Transformations

I took what is normally done in a week and condensed it to two days. After an enlightening discussion with Nowak, I issued them a challenge. Here are some un-labeled graphs, can you manipulate a parent function such that you get the exact same picture?

This was one of their first experiences with desmos, so we spent some time discussing notation and where manipulators can be placed. After letting them play for a while, we were able to tease out the transformation rules. And yes, it is weird that (x - 3)^2 seems to shift it right instead of left. Once they found a successful match, they added the equation to the picture and we discussed the description.

On Day 2, I wanted to develop some recognition of equation types with graph types, how to spot a radical graph a mile away, etc. They were issued this challenge:

The same writing component was added. For each challenge, they also had to annotate the graph with the domain and range.

### Step 2: Graphs Have a Point

Transformations out of the way, we doubled back on all the algebra we had been doing. Amazing secret kids: all those equations you were solving were really just a manipulation of transformed graphs. Take 0.5x^2 - 5 = 3 for example. We know the algebra, but did you know you can double check this? Remember how we got two answers?

Big payoff there. They felt like suckers for doing all the work by hand. Also helps justify the hand waving about multiple solutions. Going further, what if there are expressions on either side of the equal sign? Something like 0.5x^2 - 5 = -3x^2 + 7? Boom, that works too:

Now we can get nuts. At the moment the quadratic formula is unknown, so how do I determine where an un-factorable quadratic expression equals a modified linear expression?

The possibilities become endless. Absolute values intersecting with radicals, quadratics and linears and students eye-balling the solutions like it's no big deal. How fun would those be on a TI? Second question, how much time would you waste trying to graph all this by hand?

### Step 3: The Meaning of Inequality

And lastly, what about these inequality things? Ok, so the algebra isn't much different, but what does is really mean when I get x < 3 rather than x = 3?

Previously, a single point on the graph was considered the solution. Now, what's going on here? So an inequality opens it up to various sections of the graph? How do we define that? And now we have some justification for domain. For these, we redrew the graphs to better represent what was valid. If it's contained in the shaded region, that section should be solid, the rest not so much. There were some struggles with domain and range here, but it was nothing a little extra explanation couldn't fix.

In the final phase of our graphing unit, we got into complex solution regions bounded by functions of all types. I discussed the details last week. In this age of beautiful, modern graphing tools, the complexity is only limited by your imagination:

It just blows my mind how casually we can do this. It took 30 years for me to appreciate math like this, why do my students have to wait that long?

### Conclusion

In summary, the goals of the graphing unit:

• spontaneous recall of graphs of four parent functions
• justification of algebra mechanics
• make transformations just a footnote
• lower the entry level on very complex mathematics

When fall semester draws to a close, I will neatly package this material as much as possible. There were four tests during this unit. Students had access to Desmos while working on them.

I never want to graph on a TI ever again.

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