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Slippery Slopes Barney Ricca AMTRA 2012

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**Why Can’t Students Do _____?**

Let’s look at the steps from elementary school through Calculus, to see where the problem comes from. We’ll follow one thread, which includes: Proportional Reasoning Slopes Derivatives Caveat: What we do here this morning is not sufficient, but it includes the right steps.

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Some pictures

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Mr. Big & Mr. Small Use the paper clips and buttons to measure Mr. Small. Give me the paper clips Use the buttons to measure Mr. Big Question: How many paper clips tall is Mr. Big? How do you know? The most common high school answer is eight! Why?

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**Proportional Reasoning**

Most students reason additively rather than proportionally unless they are explicitly told to do the latter. Students must learn how to reason proportionally AND when to do which type of reasoning the way they learn to do it is to have experiences to guide them: Measure lots of stuff Predict other stuff What changes students is discourse, and NOTHING ELSE!

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Adding Cable A cable is just long enough to go around the equator of the earth. (Assume the earth’s equator is a circle.) I want people to be able to walk under the cable, so I want to raise the cable to stand everywhere 15 feet above the earth’s surface. How much more cable is needed? You have the tools to do this problem without me giving you any other numbers…so don’t ask me!

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**My way C = circumference, d = diameter: Why didn’t you do it that way?**

(C + ΔC) = π (d + Δd) But, C = πd, so… ΔC = π Δd ≈ 100 feet Why didn’t you do it that way?

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Change, Rate & Slopes Most students think of change as involving the difference between two points. They DO NOT think of change as an entity in itself. Then, we go to rates and slopes, and so they DO NOT ever get the “per unit” thing… This is where the graphing calculator explorations of y=mx+b come in handy: Adjust m and look create tables from graphs(!) find slopes from graphs Play green globs

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Another problem You have a block of wood with a total mass of 540 kg. This type of wood has 0.85 g in each cubic centimeter. Suppose you were to add 38 g of wood to the block. By how much would you increase the volume of the entire block? Do not substitute into a formula; explain the relevant arithmetical reasoning in your own words. (Hint: Be sure to think about change in volume rather than entire volume.)

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The Frog Puzzle Professor Thistlebush, an ecologist. conducted an experiment to determine the number of frogs that live in a pond near the field station. Since he could not catch all of the frogs he caught as many as he could, put a white band around their left hind legs, and then put them back in the pond. A week later he returned to the pond and again caught as many frogs as he could. Here are the Professor's data. First trip to the pond 55 frogs caught and banded Second trip to the pond 72 frogs caught; of those 72 frogs, 12 were found to be banded. The Professor assumed that the banded frogs had mixed thoroughly with the unbanded frogs, and from his data he was able to approximate the number of frogs that live in the pond. If you can compute this number, please do so. Explain in words how you calculated your results.

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**The Frog Puzzle – part 2 Check out the video…**

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Calculus Now, going to dy/dx without any one (or more) or these* understandings is, quite literally, impossible: Proportional Reasoning Change as an entity Per unit *There’s really more, like thinking about f(x) as an entity rather than a collection of points, etc.

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**How do we develop these understandings?**

Practice alone won’t do it Instead, we develop the understanding by using these types of problems this way Start with something concrete (and manipulible) – e.g., measure Force students into a cognitive conflict (e.g., 8 buttons vs. 9) or other surprise (don’t need lots of numbers) Students must socially construct an understanding (e.g., justify to one another) Students must reflect on the situation – this is very important! Students must bridge the idea to something else

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