It Was a Simple Question...

...until it wasn't.

Features of Functions is the unit and the problem focused on the following graph.

Typical questions like:

1. What is f(2)?
2. For what values, if any, does f(x) = 3? 
3. What is the x-intercept?
4. etc.

 Groups were working well together and I asked them to write their agreed upon answers on their easels.  As we looked around the room, everyone agreed until we got to #6. 

      6.  On what intervals is f(x) increasing?

Naturally, everyone said the function increased on the interval [-4, 6]. Everyone except Group 7.  They're always contrary like this.  Probable just stirring the pot a little.  Just pat them on the head and move on.  

Except J says, "Mr. Cox, I stand by my answer."

"Wait, what? Do you know who I... ahem, tell me more."

"Well, since the function starts at -4, it's not increasing yet.  And since it ends at 6, it stops increasing."

R chimes in, "Ok, J.  I see your point and I'd be willing to say the interval is (-4, 6] because at -4 it hasn't started increasing, but at 6 it's been increasing and then stops.  Maybe we include one and not the other."

This took us on an interesting discussion about what we really mean by rate of change, increase and decrease; how our interpretation is influenced by our left-to-right reading convention; and how many points we actually need to identify a rate of change.  We talked about instantaneous rate of change and how you can actually have a "slope" using one point.  

But I still have questions.  J was looking at the endpoints of the functions as if they were a relative maximum and minimum.  They aren't included in the increase interval because the rate of change is actually 0 at those points.  Was he correct to think this now?  Was this simply a really good wrong answer?  Should he be considered correct on the argument alone?  

What say you?

Rates of Change

Sometimes we just stumble onto stuff. I've taught slope intercept form for a lot of years, but the more I try to stay true to the concept, the more my students demonstrate innovative ways of seeing things. We just finished up linear equations and their graphs. The equations part of things usually culminates with a student taking two points and writing the standard form of the equation by way of this process:

Step 1: Find slope
Step 2: Write in point slope form
Step 3: Solve for y to get slope-intercept form
Step 4: Rewrite in Standard Form

I've usually encouraged this process as it seemed to be the efficient way to cover the majority of these skills. But this year, I kept it much more loose. We all agreed that if the two points given define a line with an integer for the y-intercept, then it's pretty easy to write in slope intercept form.

I mean, c'mon, we've got the slope and we see the y-intercept plain as day.

But what happens here?

We know the slope is 1/7 and we can estimate the y-intercept, but how can we know the y-intercept?

Rate of Change


Most of the time when we discuss Slope we talk about the path from one lattice point to another. For example, the slope between A and B is 1/7 because we go up 1 and over 7 to get from A to B.

But we have been doing more with rate of change as a unit rate which means that for every 1 unit of horizontal change, we increase 1/7. Yeah, I know that sounds elementary, but it makes a difference in how your kids see these things.

If each step increases 1/7, we increase a total of 4/7 to get to the y axis. Therefore, y-intercept is 4 4/7. We still talk about point-slope form, but this beats the heck out of the plug and chug that usually goes along with it.