Features of Functions is the unit and the problem focused on the following graph.
- What is f(2)?
- For what values, if any, does f(x) = 3?
- What is the x-intercept?
- 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?
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4 comments:
The definition of increasing/decreasing that I teach (and that has has been in all the calculus books I've used in the past 15+ years) is not about what's happening at a point — it's about what's happening on an interval:
A function is increasing on an interval if, for any a and b in the interval, when a < b, f(a) < f(b).
In other words, if the y-values get larger from left to right, the function is increasing on the interval. The largest such interval for the function depicted here is [-4, 6]. It's not about what's happening at any one point.
A function is increasing on an interval if, for any a and b in the interval, when a < b, f(a) < f(b).
That definition helps a lot, thanks.
I have to confess, that question/argument comes up annually in at least one of my classes, but as it's old hat to me, I often just answer and move on these days. When I first taught it, the question surprised me and I allowed the conversation to fully develop... Maybe I need to step back and revisit the old math through young eyes.
I appreciate that you have cultivated an environment where students feel free to explore and wonder. I observed a fourth grade class this week where students were invited to share their wonderings about fractions. These are some of the questions that came up...what's the smallest (largest) fraction? if numbers go on forever, do fractions go on forever? They were contemplating infinitely small, infinitely big...the basis of limits!
I see the same curiosity in the question your students posed. Takes me back to that class in college when we talked about delta epsilon and neighborhoods...it's a mind-blowing concept...at the time I wondered what does that mean about numbers and the point that depicts that number...does this mean points have an aura? I never did go much with that wondering.
Faced with the definition provided above, I now wonder how do these two ideas work together? or am i connecting dots that shouldn't be connected?
The art of teaching is finding that balance between exploring the wonderings our students bring to the table and delivering the content of the course. May this journey be fruitful David!
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