Proving identities – what’s your philosophy?

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What happens in your classroom when you give students the following task?

Prove 1+\frac{1}{\cos{\theta}}=\frac{\tan^2{\theta}}{\sec{\theta}-1}.

Sometimes the command is Verify or Show instead of Prove, but the intent is the same.

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Two non-examples

Here are two ways that a student might work the problem.

Method 1

1+\frac{1}{\cos{\theta}}=\frac{\tan^2{\theta}}{\sec{\theta}-1}

1+\sec{\theta}=\frac{\tan^2{\theta}}{\sec{\theta}-1}

(\sec{\theta}-1)(1+\sec{\theta})=\tan^2{\theta}

\sec^2{\theta}-1=\tan^2{\theta}

\tan^2{\theta}=\tan^2{\theta}

Method 2

1+\frac{1}{\cos{\theta}}=\frac{\tan^2{\theta}}{\sec{\theta}-1}

1+\sec{\theta}=\frac{\sec^2{\theta}-1}{\sec{\theta}-1}

1+\sec{\theta}=\frac{(\sec{\theta}-1)(\sec{\theta}+1)}{\sec{\theta}-1}

\sec{\theta}+1=\sec{\theta}+1

How do you feel about these methods? In my opinion, both methods represent a fundamental misunderstanding of the prompt. Method 1 is especially grotesque, but Method 2 also leaves a lot to be desired. Let me explain. And if you think the above methods are perfectly fine, please be patient and hear me out.

This is the crux of the issue:

The prompt was to prove the statement. But if the first line of our work is the very thing we’re out to prove, then we are already assuming the thing we want to prove. We’re Begging the Question.

It’s as if someone demands,

“Prove Statement X, please!”

and we reply,

“Well, let’s first start by assuming that Statement X is true.”

This is nonsense.

What went wrong?

So what is the proper way to engage this proof? Let’s roll back a bit.

The error in these approaches seems to stem from a desire to perform algebraic operations on both sides of an equation in the same way that you might if you were solving an equation.

When we “do algebra” and write Equation B below another Equation A without any words, we always mean that Equation A implies Equation B. That is, when we write

Equation A

Equation B

Equation C

etc…

we mean that Equation C follows from Equation B, which follows from Equation A.

Some might claim that each line should be equivalent to the last. But, again, when we “do algebra” by performing algebraic manipulations to both sides of an equation to transform it from equation A into equation B, we always mean A\Rightarrow B, we don’t mean A\iff B. Take, for example, the following algebra which results in an extraneous solution:

\sqrt{x+2}=x

(\sqrt{x+2})^2=x^2

x+2=x^2

0=x^2-x-2

0=(x-2)(x+1)

x=2 \text{ or } x=-1

In this example, each line follows from the previous, however reversing the logic doesn’t work. But we accept that this is the usual way we do algebra (A\Rightarrow B\Rightarrow C\Rightarrow \cdots). Here the last line doesn’t hold because only one solution satisfies the original equation (x=2). Remember that our logic is still flawless, though. Our logic just says that IF \sqrt{x+2}=x for a given x, THEN (\sqrt{x+2})^2=x^2.

As we move through the algebra line by line, we either preserve the solution set or increase its size. In the case above, the solution set for the original equation is {2}, and as we go to line 2 and beyond, the solution set is {2,-1}.

For more, James Tanton has a nice article about extraneous solutions and why they arise, which I highly recommend.

So if this is the universal way we interpret algebraic work, which is what I argue, then it is wrong to construct an argument of the form A\Rightarrow B\Rightarrow C in order to prove statement A is true from premise C. The argument begs the question.

Both Method 1 and Method 2 make this mistake.

 

How does a proof go again?

I want to actually make a more general statement. The argument I gave above regarding how we “do algebra” is actually how we present any sort of deductive argument. We always present such an argument in order, where later statements are supported by earlier statements.

ANY time we see a sequence of statements (not just equations) A, B, C that is being put forward as a proof, if logical connectives are missing, the mathematical community agrees that “\Rightarrow” is the missing logical connection.

That is, if we see the proof A,B,C as a proof of statement C from premise A, we assume that the argument really means A\Rightarrow B\Rightarrow C.

This is usually the interpretation in the typical two-column proof, as well. We just provide the next step with a supporting theorem/definition/axiom, but we don’t also go out of our way to say “oh, and line #7 follows from the previous lines.”

Example: Given a non-empty set E with lower bound a and upper bound b, show that a\leq b.

1. E is non-empty and a and b are lower and upper bounds for E. (given)
2. Set E contains at least one element x. (definition of non-empty)
3. a\leq x and x\leq b. (definitions of lower and upper bound)
4. a\leq b. (transitive property of inequality)

Notice I never say that one line follows from the next. And also notice that it would be a mistake to interpret the logical connectives as biconditional.

The path of righteousness

I encourage my students to work with only ONE side of the expression and manipulate it independently, in its own little dark box, and when it comes out into the light, if it looks the same as the other side, you’ve proved the equivalence of the expressions.

For example, to show that \log\left(\frac{1}{t-2}\right)-\log\left(\frac{10}{t}\right)=-1+\log\left(\frac{t}{t-2}\right) for t>2, I would expect this kind of work for “full credit”:

\text{LHS }=\log\left(\frac{1}{t-2}\right)-\log\left(\frac{10}{t}\right)

=-\log(t-2)-\log(10)+\log(t)

=-\log(10)+\log(t)-\log(t-2)

= -1 + \log\left(\frac{t}{t-2}\right)

=\text{ RHS}

Interestingly, I WOULD also accept an argument of the form A\iff B\iff C as justification for conclusion A from premise C, but I would want a student to say “A is true if and only if B is true, which is true if and only if C is true.” Even though it provides a valid proof, I discourage students from using this somewhat cumbersome construction.

So let’s return to the original problem and show a few ways a student could do it correctly.

Three examples

Method A – A direct proof by manipulating only one side

\text{LHS}=\frac{\tan^2{\theta}}{\sec{\theta}-1}

=\frac{\sec^2{\theta}-1}{\sec{\theta}-1}

=\frac{(\sec{\theta}-1)(\sec{\theta}+1)}{\sec{\theta}-1}

=\sec{\theta}+1

=\text{RHS}

Method B – A proof starting with a known equality

\tan^2{\theta}=\tan^2{\theta}

\sec^2{\theta}-1=\tan^2{\theta}

(\sec{\theta}-1)(1+\sec{\theta})=\tan^2{\theta}

1+\sec{\theta}=\frac{\tan^2{\theta}}{\sec{\theta}-1}

1+\frac{1}{\cos{\theta}}=\frac{\tan^2{\theta}}{\sec{\theta}-1}

Method C – Carefully specifying biconditional implications

1+\frac{1}{\cos{\theta}}=\frac{\tan^2{\theta}}{\sec{\theta}-1}

\text{if and only if}

1+\sec{\theta}=\frac{\tan^2{\theta}}{\sec{\theta}-1}

\text{if and only if}

(\sec{\theta}-1)(1+\sec{\theta})=\tan^2{\theta}

\text{if and only if}

\sec^2{\theta}-1=\tan^2{\theta}

\text{if and only if}

\tan^2{\theta}=\tan^2{\theta}

While all of these are now technically correct, I think we all prefer Method A. The other methods are cool too. But please, please, promise me you won’t use Methods 1 or 2 which I presented in my introduction.

In conclusion

Some might argue that the heavy criticism I’ve leveled against Methods 1 and 2 is nitpicking. But I disagree. This kind of careful reasoning is exactly the business of mathematicians. It’s not good enough to just produce “answers,” our job is to produce good reasoning. Mathematics, remember, is a sense-making discipline.

Thanks for staying with me to the end of this long-winded post. Can you tell I’ve had this conversation with a lot of students over the last ten years?

Further reading

  1. Dave Richeson has a similar rant with a similar thesis here.
  2. This article was originally inspired by this recent post on Patrick Honner’s blog. A bunch of us fought about this topic in the comments, and in the end, Patrick encouraged me to write my own post on the subject. So here I am. Thanks for pushing me in the right direction, Mr. Honner!

 

What does it mean to truly prove something?

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Let me point you to the following recent blog post from Prof Keith Devlin, entitled “What is a proof, really?”

After a lifetime in professional mathematics, during which I have read a lot of proofs, created some of my own, assisted others in creating theirs, and reviewed a fair number for research journals, the one thing I am sure of is that the definition of proof you will find in a book on mathematical logic or see on the board in a college level introductory pure mathematics class doesn’t come close to the reality.

For sure, I have never in my life seen a proof that truly fits the standard definition. Nor has anyone else.

The usual maneuver by which mathematicians leverage that formal notion to capture the arguments they, and all their colleagues, regard as proofs is to say a proof is a finite sequence of assertions that could be filled in to become one of those formal structures.

It’s not a bad approach if the goal is to give someone a general idea of what a proof is. The trouble is, no one has ever carried out that filling-in process. It’s purely hypothetical. How then can anyone know that the purported proof in front of them really is a proof?

(more)

Click the link to read the rest of the article. Also read the comments below the article to see what conversation has already been generated.

I won’t be shy in saying that I disagree with Keith Devlin. Maybe I misunderstand the subtle nuance of his argument. Maybe I haven’t done enough advanced mathematics. Please help me understand.

Devlin says that proofs created by the mathematical community (on the blackboard, and in journals) are informal and non-rigorous. I think we all agree with him on this point.

But the main point of his article seems to be that these proofs are non-rigorous and can never be made rigorous. That is, he’s suggesting that there could be holes in the logic of even the most vetted & time-tested proofs. He says that these proofs need to be filled in at a granular level, from first principles. Devlin writes, “no one has ever carried out that filling-in process.”

The trouble is, there is a whole mathematical community devoted to this filling-in process. Many high-level results have been rigorously proven going all the way back to first principles. That’s the entire goal of the metamath project. If you haven’t ever stumbled on this site, it will blow your mind. Click on the previous link, but don’t get too lost. Come back and read the rest of my post!

I’ve reread his blog post multiple times, and the articles he linked to. And I just can’t figure out what he could possibly mean by this. It sounds like Devlin thoroughly understands what the metamath project is all about, and he’s very familiar with proof-checking and mathematical logic. So he definitely isn’t writing his post out of ignorance–he’s a smart guy! Again, I ask, can anyone help me understand?

I know that a statement is only proven true relative to the axioms of the formal system. If you change your axioms, different results arise (like changing Euclid’s Fifth Postulate or removing the Axiom of Choice). And I’ve read enough about Gödel to understand the limits of formal systems. As mathematicians, we choose to make our formal systems consistent at the expense of completeness.

Is Devlin referring to one of these things?

I don’t usually make posts that are so confrontational. My apologies! I didn’t really want to post this to my blog. I would have much rather had this conversation in the comments section of Devlin’s blog. I posted two comments but neither one was approved. I gather that many other comments were censored as well.

Here’s the comment I left on his blog, which still hasn’t shown up. (I also left one small comment saying something similar.)

Prof. Devlin,

You said you got a number of comments like Steven’s. Can you approve those comments for public viewing? (one of those comments was mine!)

I think Steven’s comment has less to do with computer *generated* proofs as it does with computer *checked* proofs, like those produced by the http://us.metamath.org/ community.

There’s a big difference between the proof of the Four Color Theorem, which doesn’t really pass our “elegance” test, and the proof of e^{i\pi}=-1 which can be found here: http://us.metamath.org/mpegif/efipi.html

A proof like the one I just linked to is done by humans, but is so rigorous that it can be *checked* by a computer. For me, it satisfies both my hunger for truth AND my hunger to understand *why* the statement is true.

I don’t understand how the metamath project doesn’t meet your criteria for the filling in process. I’ll quote you again, “The trouble is, no one has ever carried out that filling-in process. It’s purely hypothetical. How then can anyone know that the purported proof in front of them really is a proof?”

What is the metamath project, if not the “filling in” process?

John

If anyone wants to continue this conversation here at my blog, uncensored, please feel free to contribute below :-). Maybe Keith Devlin will even stop by!

When will she pass me for the first time? [solution]

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Recently, my dad posed the following question here:

My wife and I walk on a circular track, starting at the same point.  She does m laps in the time that it takes me to do n laps.  She walks faster than I do, so m > n.  After how many laps will she catch up with me again?

If you haven’t solved it yet, give it a crack. It’s a fun problem that has surprising depth.

Here’s my solution (in it, I refer to “mom” rather than “my wife” for obvious reasons!):

Since mom’s lap rate is m laps per unit time, and dad’s lap rate is n laps per unit time, in time t, mom goes mt laps and dad goes nt laps.

They meet whenever their distance (measured in laps) is separated by an integer number of laps k. That is, mom and dad meet when

mt=nt+k, k\in\mathbb{Z}.

This happens at time

t=\frac{k}{m-n}.

Mom will have gone

mt=\frac{mk}{m-n}

laps and dad will have gone

nt=\frac{nk}{m-n}

laps when they meet for the kth time.

And that’s it! That’s the general solution. This means that:

  • At time t=0, dad and mom “meet” because they haven’t even started walking at all (they are k=0 laps apart).
  • At time t=\frac{1}{m-n}, dad and mom meet for their first time after having started walking (they are k=1 lap apart). This is the answer to the problem as it was originally stated. Mom will have gone mt=\frac{m}{m-n} laps and dad will have gone nt=\frac{n}{m-n} laps when they meet for the first time.
  • At time t=\frac{2}{m-n}, dad and mom meet for their second time (now k=2 laps apart).
  • At time t=\frac{k}{m-n}, dad and mom meet for their kth time.

Here are two examples:

  • If mom walks 15 laps in the time it takes dad to walk 10 laps, when they meet up for the first time, mom will have gone \frac{m}{m-n}=3 laps and dad will have gone \frac{n}{m-n}=2 laps.
  • If mom walks 12 laps in the time it takes dad to walk 5 laps, when they meet up for the first time, mom will have gone \frac{m}{m-n}=1\frac{5}{7} laps and dad will have gone \frac{n}{m-n}=\frac{5}{7} laps.

Boom! Problem solved! 🙂

Catch yourself up on the world of origami

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Have you been doing other things and failed to notice the origami world evolve without you? Have you fallen asleep and been left behind? If you want to get caught up on what you’ve been missing in the world of origami, I suggest you visit Hannah’s origami blog, A Soul Made of Paper. She’ll have you caught up in no time.

I especially like giving her blog a shout-out because Hannah is a student of mine. She often comes by my classroom to show me her latest paper creations. I like origami, and I’ve dabbled in it–stuck my toe in the stream, if you will–but Hannah is like a scuba diver in the origami world. She’s loves modular origami, but she’s also great at the artsy curved creations (like origami roses), textures, and tessellations.

If you haven’t clicked over to her blog yet, here are a few more pictures to whet your appetite.

Go get lost at A Soul Made of Paper. Maybe you can join her other 3000 followers on tumblr :-).

*All of the above are Hannah’s pieces and Hannah’s photos.

Looking back on 299 random walks

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This is my 300th post and I’m feeling all nostalgic. Here are some of the popular threads that have appeared on my blog over the last few years. If you’ve missed them, now’s your chance to check them out:

Thanks for randomly walking with me over these last few years (though, some say it’s a “drunken walk” 🙂 ). Either way, I’ll raise a glass to another 300 posts!

Challenge Problems

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Want to enrich your Precalculus course with difficult problems? Look no further!

Very-Difficult-Mazes-Coloring-Page-1I teach a high-octane version of Precalculus to students in our magnet program. Our course, like most Precalculus courses, covers a very wide variety of topics. As often as possible, I like to give them more difficult problems that enrich the material from the book. These documents are a work in progress, but feel free to steal them (just email me a copy if you improve them!):

If you want solutions for any of these, shoot me an email.

These aren’t 100% polished by any means, but I’m sharing them anyway! Long live the spirit of sharing :-).

By the way, many of these problems are collected from other sources but I’m too far removed from those sources to properly attribute the problem-creator. My sincere apologies!

When will she pass me for the first time?

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[Guest post by Dr. Chase]

My wife and I walk on a circular track, starting at the same point.  She does m laps in the time that it takes me to do n laps.  She walks faster than I do, so m > n.  After how many laps will she catch up with me again?

Example:  For m = 4, and n = 3, she will catch up when I have finished 3 laps.  Reason:  When I have finished 1 lap, she finished 1 1/3 laps, so she is 1/3 of the track ahead of me.  (But hasn’t passed me yet.)  When I have finished 2 laps, she has finished 2 2/3 laps around the track, still ahead of me.  When I finish 3 laps, she has finished 3 3/3 laps, which is to say 4 laps.  So we are together for the first time since starting.  If m = 2, n = 1, she will catch up in just 1 lap.  If m = 7, n = 6, she will catch up in 6 laps.  Will she always catch up in n laps?  In how many laps will she catch up for arbitrary m and n?

What does a point on the normal distribution represent?

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Here’s another Quora answer I’m reposting here. This is the question, followed by my answer.

What does the value of a point on the normal distribution actually represent, if anything?

 

It’s important to note the difference between discrete and continuous random variables as we answer this question. Though naming conventions vary, I think most mathematicians would agree that a discrete random variable has a Probability Mass Function (PMF) and a continuous random variable has a Probability Density Function (PDF).

The words mass and density go a long way in helping to capture the difference between discrete and continuous random variables. For a discrete random variable, the PMF evaluated at a certain x gives the probability of x. For a continuous random variable, the PDF at a certain x does not give the probability at all, it gives the density. (As advertised!)

So what is the probability that a continuous random variable takes on a certain value? For example, assume a certain type of fish has length X that is normally distributed with mean 22 cm and standard deviation 1.6 cm. What is the probability of selecting a fish exactly 26 cm long? That is, what is P(X=26)?


The answer, for any continuous random variable, is zero. More formally, if X is a continuous random variable with support \mathcal{S}, then P(X=x)=0 for all x\in\mathcal{S}.

For the fish problem, this actually does make sense. Think about it. You pull a fish out of the water which you claim is 26 cm long. But is it really 26 cm long? Exactly 26 cm long? Like 26.00000… cm long? With what precision did you make that measurement? This should explain why the probability is zero.

If instead you want to ask about the probability of getting a fish between 25.995 and 26.005 cm long, that’s perfectly fine, and you’ll get a positive answer for the probability (it’s a small answer :-).

Let’s return to the words mass and density for a second. Think about what those words mean in a physics context. Imagine having a point mass–this is in an ideal case–then the mass of that point is defined by a discrete function. In reality, though, we have density functions that assign a density to each point in an object.

Think about a 1-dimmensional rod with density function \rho(x)=x, x\in (0,10). What is the mass of this rod at x=5? Of course, the answer is zero! This should make intuitive sense. Of course, we can get meaningful answers to questions like: What is the mass of the rod between x=5 and x=6? The answer is \int_5^6 xdx=5.5.

Does the physical understanding of mass vs density clear things up for you?

Guess who!

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In an effort to share more of my resources through this blog, here’s another installment.

This time I’m sharing a little worksheet that I created called Guess Who? It’s a short activity–a warm up, or an exit card–and students should be able to do it in 5 minutes or so. I do this in my Precalculus class at the beginning of the year, but depending on the timing and the context, it could be appropriate in an Algebra 2 or Calculus class as well.

The functions and the questions have a one-to-one correspondence and there is a unique solution to the worksheet.

These 12 functions might seem a bit strange, but they are the “12 basic functions” named by our Precalculus textbook authors.

Here are two additional activities that can go with a discussion of functions and their properties:

  1. I have all the functions printed out on 8.5″x11″ paper and backed with colored paper so they look nice. I get twelve volunteers to go up to the front and hold the functions. Then we can play all sorts of games. We can ask all the functions that have an asymptote to step forward. We can ask all the odd functions to step forward. Which functions are bounded? Which functions are always increasing? Which functions have a range of all real numbers? But we can also play a guessing game: A student in the audience picks a function and writes it down without telling everyone. The other students in the audience ask yes-no questions about their function, like “Is your function continuous for all real numbers?” Each time, functions that don’t qualify step back and only a few functions remain. This is repeated until the chosen function is the only one that remains.
  2. Another fun game idea comes from one of my colleagues. I love this: Have a bunch of “name tags” made up for all your students. The name tags will be one of the 12 basic functions and students will wear these on their backs, without knowing what their function is. They then have to walk around the room and ask other students yes-no questions about the features of their function until they can identify which function they are. I think I’ve played a version of this with celebrities or something. But it’s perfect for the math classroom, too!

Okay, that’s my contribution to the MTBoS for the day :-).

Haloween worksheet for Calculus

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Anyone who has been in math classes knows those corny worksheets with a joke on them. When you answer the questions, the solution to the (hilarious) joke is revealed. Did I mention these worksheets are corny? But when you get to Calculus or higher math classes, you get nostalgic for those old pre-algebra worksheets your middle school teacher gave you. I think I speak for all of us when I say this.

Not to fear, here’s a very corny joke worksheet I made just for your Calculus students. Print this on orange paper and hand it out on Halloween. When kids successfully solve the problems and discover the solution, give them candy.

Here is the solution:

Happy Halloween. Enjoy!

 

PS: I normally use my blog to share deep insights about math education or to discuss interesting higher level mathematics. But I was inspired to share more of my day-to-day activities and worksheets because of Rebecka Peterson at Epsilon-Delta. She has shared some great resources, which I’ve stolen in used in my classroom. Thanks, Rebecka!