Grumpy chemistry teacher years later gets annoyed when students count squares, when the gridlines don’t correspond to one unit. Yes, it happens more than I’d care to admit!

@Tom, for “multimedia stuff on trig/pre-calc” I can offer no better advice than this: learn to use Geogebra really well. If you google around I’m sure you’ll find some files ready to use, but the most important thing is to be able to create files that support the way you want to teach the concepts. That assumes you have a projector in your room and/or students have access to computers.

What about this? Determine the slope using two/three different methods. This validates each methods. I try to arm my students with as many methods as possible and usually tell them to use the method they think will work best under any given situation.

@Jim — I know the feeling. I’ve spent a fair amount of time having my students figure out the slope formula by counting blocks, increasing the number of blocks to count, then eventually projecting a huge grid of teeny tiny little blocks that they could tell were blocks, but had a hard time counting because a) I wouldn’t let them out of their seats to the board to see better and b) because by the time you get to the 47th block, you probably lose count. Anyway, after all the time it took one of them to say something like “just subtract the y values and the x-values,” I had some students say, even if under their breath so I couldn’t hear…I wish he would have just showed us that in the first place, this was a waste of time. I think my head was in the right place, but after being 9 years of having math be, as you say, “knowing which formula to use and remembering that formula,” is it right of me to try to un-teach that mentality and preach understanding versus memorization? I think I just wanted to join in, too…

And I suspect a good answer to the question I posed would be yes, it is! But maybe I phrased it incorrectly. Can it be done in the amount of time that I have in a year?

I’ve done a couple of applications with rates of change in the past that I’m pretty proud of and students get a lot of buy-in from:
1. Flow rates of fire hydrants–this banks on my experiences from having spent years on our local volunteer fire department and the fact our community plays by the NFPA 291 standard pertaining to hydrant cap colors (available at http://firehydrant.org/info/hycolor.html if you want to try it yourself). I inform students that the fire engine arrives on scene with 1000 gallons in its tank, which surprises some students who think it arrives on scene empty, with just tools. Once the truck is connected to a hydrant water supply, and knowing its cap color, we can find the amount of time for the truck to dispense 10,000 or x number of gallons of water on the scene, or note that after 8 minutes there has been 10,000 gallons pushed through the truck’s pump. Either way gives them an extension of the topic at hand and gets students talking about how to account for the amount on the truck, what the output rate was from the truck, what 200 psi feels like on the end of a hose nozzle, if hydrant pressure would be considered the input or output variable in this scenario, and things like that. REAL WORLD STUFF! I especially like students taking note of the hydrant nearest their home (again, the firefighter in me appreciates the awareness) and what factors might play a role in why one student’s hydrant color differs from someone else’s, such as proximity to a water tower, the age of infrastructure in their part of town, or surrounding hills that might diminish water pressure. There is little, if any, cost behind this lesson and the link I provided earlier gives insight into the flow rates used.

2. Elevation change at access ramps–I became fascinated with all the accomodations made in building construction by the ADA (Americans with Disabilities Act) in college and used ramp slope as part of a lesson when I student taught (ramp steepness is not to exceed 10%). When I first did the lesson, we happened upon a ramp that was not up to code, which led my students to sprint to the principal’s office, singing about how school should shut down because it’s in violation and such. Principal appreciated their enthusiasm and that they grasped the concept, but asked me to find a different application at the current time. Now, a handful-and-a-half years later, I was lucky enough to use GPS devices to show students the elevation at a certain point, mark a POI (“point of interest” in GPS-speak) and measure the distance to that location. This gave us a real world “rise” and “run” behind the topic. Students liked this one as well and, now at a different school with different principal, I’ve had administrators jump in on the actvity and ask the students to explain it to them. A fun side note is how students talk about positive and negative slope in context of which way you’re going, as well as extensions into zero (an ordinary hallway or the football field) and undefined slopes (the flag pole).

Sorry this was so long, but a couple previous commenters hinted that they were looking for applications for such a topic.

If the only tool you have is a hammer, then you will hit everything like it is a nail (even if a screwdriver, level, chisel etc would be more appropriate)

Then, even if given a nail gun, which is much faster, it doesn’t guarantee that the new tool will be used or more importantly the knowledge of when it’s best to use a hammer or nailgun.

We need to give students (and teachers) a wonderful mathematical toolkit, the confidence and knowledge of which tools to use and if when applied, whether or not a correct answer is produced.

Dan, I think you left out a step (#2 in my list).

1. The teacher teaches slope using concrete examples (like actually counting the gird squares).

2. The teacher teaches the general expression for slope, but sticking to the familiar concrete examples so that the students “trust” the new method(s).

3. The teacher then opens up with many problems of slope, some that succumb to counting and some that certainly do not.

You need a decent amount of all three.

Robert,

I like your addition of step #2 “The teacher teaches the general expression for slope, but sticking to the familiar concrete examples so that the students “trust” the new method(s).”

As for intuition in students, I feel it needs to be explicitly taught to students rather than assumed, and in my own past experiences, my best teachers have tried to do that. An excellent calculus professor I once had would constantly be saying things like “…and then when you’re finished, look at the answer. Does it look right? If you get a number that’s not between 4 and 6, there’s obviously something wrong, isn’t there?”; or “…and if you don’t believe me, or you get on the exam and your mind completely blanks, try out a few easy examples and work it out. Convince yourself it’s right!”

I think for the best math students, a lot of this advice is what they’re doing anyways whether they realize it or not, but it’s not obvious for the average students.

I’m not sure whether the pic on this meme is stock footage or not, but it is definitively the same photograph used on the cover of New York State’s Teacher Professional Development Handbook.

http://www.highered.nysed.gov/tcert/certificate/maintaincert-prof.html

Take from this what you will.

Christine said
“We need to give students (and teachers) a wonderful mathematical toolkit, the confidence and knowledge of which tools to use and if when applied, whether or not a correct answer is produced.”

While I agree with the first part (about treating mathematical skills as a toolkit), as an engineering professor, it is important to me that the correct answer be produced as well. Getting the right answer by magic or cheating is useless, but getting the wrong answer (even with “understanding”) is also useless.

I’m late to this commenting party but FWIW I experienced this same phenomenon myself just before break with a colleague at school.

I was consulting with her during a class period on a project, and decided to offer some assistance checking student work on the side. I was using my tested and true plong multiplication method to check student’s work when the teacher I was consulting with looked at with almost a bemused and bewildered look on her face.

“You don’t use the partial products method?” was her shocked question at assessing my math skills.

I felt a bit ashamed that A.) I had never learned the partial products method, and that B.) I wasn’t doing it the way she did. Don’t get me wrong, it took about 15 seconds for that sense of shame to go away before confidently telling her as long as I got the answer, it didn’t concern me. I’d accomodate her students, but how I do things is just fine with me :)

I’m not sure I understand this. Doesn’t one teach about this by suggesting that slope is the common notion of how steep something is?

Why would someone need to count blocks? The key notion is that we measure slope with a number as rise/run. Do you really have to have kids count blocks to know the rise and run? Can they not see that this is a difference? Boy, if that’s the problem, we really are in trouble as kids should be doing this stuff long before we get to a discussion of slope, no?

As for the example:

“Helpful High School Teacher asks the students to find the slope between (-2, 5) and (999998, 2000005).”

Rather than see this as some confirmation that a formula is useful, it struck me as a great opportunity to talk about using estimation to get a reasonably accurate result without a pencil and paper.