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Peace through Constructions

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Illustration from Euclids Elements.pngCompass and straightedge constructions have been in the geometry curriculum for a loooonng time. These constructions were around before Euclid included them in his textbook, The Elements, 2300 years ago. The painting shown here is an illumination from a translation of that book 1600 years later. I had to do constructions in junior high; I don’t remember being quite as unhappy about the endeavor as these monks or the woman teaching them. And now constructions are in the Common Core, too. Why are we eternally requiring students to construct things with these arcane and arbitrary tools?

Here is one answer: Plutarch tells a story of Plato and his buddy Simmias of Thebes coming back from a trip to Egypt when they encountered a delegation from the isle of Delos. The famous oracle at their temple had told them that the Delians and the other Greeks could put an end to their current troubles if they could figure out how to double the temple’s altar, which was a cube. In other words, they had to make a new cubical altar that was twice the volume. They told Plato that they had tried to double all the dimensions, but they ended up with an altar that was 8 times the volume, not double the volume.

Plato told them that this was not a problem for estimation but for precise construction. (It can be solved by finding two lengths in mean proportion, in other words, construct a length of the cube root of 2.) Although a few Athenians knew how to do it, the point was that the Delians should figure it out themselves. The oracle’s motive was that the Greeks would only end their troubles when they focused on cultivating the art of mathematical problem solving, including making logical arguments for their constructions. This would give them the ability to sort out their differences through reason rather than force. 

This classic problem of doubling the cube turns out to be impossible with only a compass and straightedge, but you can do it with other Greek tools (and you can do it with paper folding, too!). The work that occurred on this problem led to entirely new fields of mathematical study in ancient Greece.

I think maybe the reason we keep doing constructions is that it allows us to engage in at least two of the Common Core’s Standards for Mathematical Practice (SMP 1: Make sense of problems and persevere in solving them and SMP 3: Construct viable arguments and critique the reasoning of others). By forcing ourselves to figure out how to do something with a limited set of tools, we refine our problem-solving skills. By forcing ourselves to justify why we know our constructions will always work, we improve our ability to argue rationally, to show how ideas logically rely on other ideas.

Maybe the oracle of Delos was onto something. . . . 


Rob Ely, MTMS Blogger Rob Ely is an associate professor in the Department of Mathematics at the University of Idaho.  He likes researching student reasoning and how it relates to historical reasoning, particularly with the infinite, infinitesimal, and algebra.  He likes playing hammered dulcimer, bridge, and fetch. 

 

 


 

 

Bootstrapping Revisited: A Regular 3 1/2-gon?

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This is another weird example of the bootstrapping I mentioned in my first post. We were brainstorming a geometry class a few months ago when my friend Ian asked me, “Can you draw a regular 3 1/2-gon?”

After I mused about this Zen math koan for awhile, he handed me a ruler and a protractor and said, “Suppose you had only these and you needed to draw a regular 9-gon? What would you do?”

I told him I’d first figure out what each angle would have to be. The total angle sum in a 9-gon is (9 - 2) x 180° (because if you draw diagonals from one vertex of a 9-gon, you end up dividing it into 7 triangles, and each triangle contributes 180°). That’s 1260°. If the 9-gon is regular, then its 9 angles are all congruent, so they are each 1260/9 = 140°. To construct it, I would pick my favorite length, go 140°, and copy the same length. I would do this 8 times.

Then Ian said, “Okay, just do the same thing for a 3.5-gon.”

I said, “So. . . . How would I figure out each vertex angle when I don’t even know what it would look like?”

“Just use the same formula you used for the 9-gon: (n -2) x 180°, then divide by n to get each angle measure,” he said.

When I did this, I got 77.14°. Then I drew an inch-long side, measured 77.14°, drew another inch-long side, measured another 77.14°, and just kept going. Eventually I got this:

 MTMSPost3Pic 

Using this same method, for what values of n would you get a closed figure for an n-gon?

Here is the bootstrapping:

  1. We started with a definition of a regular n-gon: a polygon with n congruent sides and vertex angles.
  2. We discovered a rule and justified that it had to work: We proved that each vertex angle had to be (n - 2) x 180°/n, by putting in all the triangles.
  3. Then we generalized the rule to a new domain, namely, for fractional n. The only way was to extend the formula, which works fine if n is a fraction. But the formula can only be proved for whole-number values of n.
  4. This created new mathematical objects. Again, they didn’t have to be defined this way, but it nicely continued the rule we had gotten fond of.

Another example of the justification structure of the definition and the proven rule getting switched around and bootstrapped when we want to generalize to a new domain. This phenomenon occurs when we give ourselves the chance to express regularity in repeated reasoning (practice 8), giving us formulas that can then be generalized, weirdly, to new domains.

Want to try a bunch of other fractional gons?

RobElyRob Ely is an associate professor in the Department of Mathematics at the University of Idaho.  He likes researching student reasoning and how it relates to historical reasoning, particularly with the infinite, infinitesimal, and algebra.  He likes playing hammered dulcimer, bridge, and fetch. 

Apples and Remainders

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ApplesHere’s a classic question:  

Out of the 6000 apples we harvested, every third apple was too small, every fourth apple was spotted, and every tenth apple was bruised. The remaining apples were good. How many good apples were there?

What is the purpose of constructing a viable mathematical argument or proof? Many people would say that the purpose is to convince another person or yourself that the position being taken is true, beyond a shadow of a doubt. But I think we have focused too much on conviction as the purpose of mathematical argumentation. I think the primary benefit of constructing a viable argument is that it uncovers the way mathematical ideas relate to one another. When you make a viable argument for a claim, it reveals what mathematics the claim relies on. Here is an example:

Students usually start by saying, “Subtract from 6000 the 2000 apples that are too small, the 1500 that are spotted, and the 600 that are bruised to get 1900 good ones left.” This explanation isn’t correct, because the students have subtracted some of the apples twice (e.g., the ones that are spotted and bruised).

After realizing this, students sometimes take a different tack. They might list the first, say, 100 apples and find out the good ones by brute force (there are 47, I think). Then they say that if there are 47 in the first 100, there will be 47 x 60 in the first 6000. But this argument is not sound because it assumes that the second 100 apples (101-200) behave exactly like the first 100, which they don’t.

But if this is done with the first 60 apples instead of the first 100, it does work because every subsequent batch of 60 in fact does behave like the first 60. Why is this?

Good numbers here are precisely the ones with a nonzero remainder when you divide them by 3, 4, and 10. But if any number n is good, then n + 60 will be good, too. The reason is that 60 has no remainder when divided by 3, 4, or 10. This means that when you divide n + 60 by 3, 4, or 10, you get the same remainder as when you divide n by 3, 4, or 10 because the 60 contributes nothing to the remainder. This is the same for n + 120, n + 180, etc. This means that apple n + 60, n + 120, etc., are all good precisely when n is good.

Notice the benefit of pushing through to the viable argument here in that it uncovers an important piece of mathematics that our claim relies on: If I take a number n and divide it by some other number p, I will get the same remainder if I divide n + m by p, provided m is divisible by p. This is a big rule in number theory and is the basis for modular arithmetic.

This mathematical idea doesn’t become apparent if you just stop at the answer or even just stop at how the answer was arrived at.

RobElyRob Ely is an associate professor in the Department of Mathematics at the University of Idaho.  He likes researching student reasoning and how it relates to historical reasoning, particularly with the infinite, infinitesimal, and algebra.  He likes playing hammered dulcimer, bridge, and fetch. 

The 2° th post

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MTMSPicCaptionWelcome to the Blogarithm! We’re exploring ideas raised by the Standards for Mathematical Practice in the Common Core State Standards for the middle grades.

I noticed something while watching an eighth-grade class. The teacher was getting the students to notice the rule about adding exponents, asking them to figure out 24 x 23 and other similar cases. When they saw it was 27, and that you could, in general, just add the exponents with the same base, she asked them why it is always true. One student said, “Because you are just multiplying 2 times itself 3 times and then 5 more times, so it’s 8 times total.” They then did it for division (e.g., 27 ÷ 23 = 24), which was just a rearrangement of the same thing.

Later, the teacher was talking about 20 and negative exponents like 2-3. A student asked why 20 is 1 because shouldn’t 2 times itself 0 times be 0? Hmm. The teacher fielded this pretty well by asking the class what 23 ÷ 23 was, based on the subtraction rule. They said that it was 20 and noticed that it also had to be 1 because it was 8 ÷ 8. So it makes sense to say 20 is 1. Crisis averted.

This weird bootstrapping thing happened. Students—

  1. started with a definition (exponentiation as repeated multiplication)
  2. discovered a rule and justified that it had to always work, based on this definition (addition/subtraction rule of exponents)
  3. generalized the rule to a new domain on which the definition, and hence the justification, does not even make sense (0 and negative integer exponents)
  4. created new mathematical objects (0 and negative integer exponents), defining the old thing on this new domain. The new definition didn’t logically have to be that way; it’s just nice for it to be that way because it continues the rule we’ve gotten fond of.

This bootstrapping thing shows a big difference between two kinds of arguments that can show up in the math classroom as students engage in Practice 3: “This claim must be true based on this other thing we know” vs. “We ought to define this new thing in this way because then it allows us to generalize a rule we like.”

I also think the bootstrapping reveals some underlying structure to the math curriculum that emerges as we alternately generalize and justify. I think it might happen all over the place. For instance, in elementary school, teachers justify the distributive property (2 + 5) x 6 = 2 x 6 + 5 x 6 based on the definition of multiplication as grouping or repeated addition (2 groups of 6 and 5 groups of 6 is just 2 + 5 groups of 6). To generalize the distributive property to (2 + 5) x  3 , we need a new definition of multiplication as, say, scaling (because what the heck is  3  groups?). Can anyone think of other examples?

RobElyRob Ely is an associate professor in the Department of Mathematics at the University of Idaho.  He likes researching student reasoning and how it relates to historical reasoning, particularly with the infinite, infinitesimal, and algebra.  He likes playing hammered dulcimer, bridge, and fetch. 

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