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February 11th, 2016, 05:51 AM  #1 
Member Joined: Mar 2015 From: Los Angeles Posts: 73 Thanks: 7  helping to improve the phsyicality of the concepts
I need help in the form of teaching metaphors. Basic ideas regarding functions, rate of change, limits, continuity, asymptotes, etc. he cannot get. The thing is, when I first learned this stuff, I always imagined little men running along the graph. A steeper curve meant a greater rate of change, so obvious by considering the "greater" work to get up it. And so forth, I had many images. I'm struggling with my current calculus student. We are approaching the definition and use of the derivative. The textbook slides up toward a definition of the derivative by first discussing average rate of change. It gives examples in different units, like the growth weight of a sheep per day. Also it gives the velocity of a vehicle. In each case, the textbook gives a formula for the weight/speed at this moment and invites you to plug two different points into the formula and get the answer. It shows the plot of weight over days, or kgs/over time. Okay, first problem. I don't think my student has a good appreciation for graphs. I think he doesn't grasp that a fast moving car will have a steep slope, and a slow moving car the opposite. I have a problem explaining asymptotes to him. If I trace the line of a declining exponential, I'll say "see how this is getting closer and closer to y = zero?" He'll say something like "But it won't ever get there. I'll say, "We can get as close as we'd like." and he doesn't really understand. To me, I try to make it seem really physical, like tell a story of two long lost sisters, one travelling a track on the declining exponential, the other travelling the xaxis, and think of those trains going and going to infinity, and think how close those two sisters would get. Close enough to jump into each other's trains and set up a big party! Yup! and will they ever come apart? no. But, finally do they ever exactly touch? No. This story makes NO impression on him at all. A similar lack of intuition messes up our attempts at piecewise functions (answering limit/continuity questions) Does anyone have an idea? I don't know how to teach this guy. 
February 11th, 2016, 06:13 AM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond 
Maybe this is a jump ahead, but do you think he'd understand $\displaystyle \lim_{x\to\infty}\dfrac1x=0$ ? I am reminded of the old adage "you can lead a horse to water, but you can't make him drink". 
February 11th, 2016, 06:31 AM  #3 
Math Team Joined: Jul 2011 From: Texas Posts: 3,042 Thanks: 1626 
Recommended reading for the student. Some dated British terms may need to be explained. PDF download is free ... Calculus Made Easy by Silvanus P. Thompson  Free Ebook 
February 11th, 2016, 06:49 AM  #4 
Member Joined: Mar 2015 From: Los Angeles Posts: 73 Thanks: 7  I gave some examples to him of fractions with a fixed numerator and a denominator that gets larger and larger, and he could see that the numbers get smaller. I then asked if he could see that 1/x "goes to zero" as "x goes to infinity" and he is just vague on what "goes to" means. He nodded in agreement the first time I presented it, but then he got it mixed up a moment later in a problem set. So I never really though about how to explain something like this because it always made sense to me.

February 11th, 2016, 09:31 AM  #5 
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271 
I like skeeter's book, but it takes three chapters to get to the point and may be too much to stomach for a pupil that has struggled with pre calculus algebra. In junior high school pupils are told that letters stand for numbers and then go on to handle various expressions where these letters have to be found. Pupils are never told the difference between an equality and an identity and the equals sign is used indiscriminately for both. In particular it is never brought out that x and y each have two different meanings in different circumstances. The pupil will be fresh from solving simultaneous equations so will expect and be accustomed to x and y having specific values. I suspect your start by making sure that your pupil appreciates exactly what x and y are, viz variables. Explain about independent and dependent variables. Then use the simplest example (y = x) to demonstrate the meaning of this in relation to equations (formulae) . Then use this to ask “What is the change in y if we change x by some amount?” Show that if we double x we must double y etc. Then explore this for other formulae. Then introduce delta x and delta y and the derivative as their ratio, not the slope of a graph the pupil obviously has difficulty with. 
February 11th, 2016, 04:58 PM  #6  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,842 Thanks: 655 Math Focus: Yet to find out.  Quote:
Maybe focusing on the definitions of certain things as well as trying a few simple examples and exercises may help to get him started. Relating to the physical may become more apparent to him later. As for graphs, this website is easy to use and allows you to add sliders for different variables. Making the graphing process more interactive. https://www.desmos.com/  

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