Dividing by a Fraction

We "invert and multiply", "multiply by the reciprocal" or insist on using the fraction key because we can't remember or were never really taught the reasons or the algorithm. Is there a simple explanation for the method we old farts memorized years ago in third or fourth grade? Why does it work?
Let's start with a problem: $\frac{3}{4} \div \frac{5}{6}$ and change to a compound fraction: $\dfrac{\frac{3}{4}}{\frac{5}{6}}$

Now what? Dividing by a fraction is confusing, but dividing by one is obvious. So we turn $\frac{5}{6}$ into unity by multiplying by its reciprocal. Of course, you can't just multiply part of our problem by $\frac{6}{5}$ without changing its value, so we multiply by one: $\dfrac{\frac{6}{5}}{\frac{6}{5}}$

All in one image: $\dfrac{\dfrac{3}{4}}{\dfrac{5}{6}} \rightarrow \dfrac{\dfrac{3}{4}}{\dfrac{5}{6}} \cdot \dfrac{\dfrac{6}{5}}{\dfrac{6}{5}} \rightarrow \dfrac{\dfrac{3}{4} \cdot \dfrac{6}{5}}{\dfrac{1}{1}} \rightarrow \dfrac{3}{4} \cdot \dfrac{6}{5} \rightarrow \dfrac{18}{20} \rightarrow \dfrac{9}{10}$

Divide by one. Seems simple to me.

Always Bet on Black.

In Diamonds are Forever, James Bond bets on 17 Black. It is, of course, a European roulette wheel, with 18 red, 18 black and 1 green.

What is the probability of his getting the ball to drop in 17 Black twice in a row?

1/37 * 1/37, which equals 0.07%

 How cool is that?

Incentives and my Degree.

Joanne Jacobs has this article on Lower pay for math, science teachers.

Math and science teachers earn less than their colleagues in 19 of 30 large districts in Washington state, reports the Center on Reinventing Public Education. That’s because salary schedules reward only longevity and graduate credits. "The analysis finds that in twenty-five of the thirty largest districts, math and science teachers had fewer years of teaching experience due to higher turnover — an indication that labor market forces do indeed vary with subject matter expertise.

She comments that "Differential pay for high-demand skills would keep more math and science teachers in the classroom."

I strongly disagree. What I see as far more likely is that those math and science teachers would get paid more,and since they don't have a clue as to the particulars of teaching, they'd leave just as quickly. Five or ten thousand bucks can't overcome that. Teaching is TOUGH -- TFAs and other dilettantes aren't going to stay no matter what.  Shoveling money into the pockets of a few teachers solely based on the course they teach is not conducive to cooperation, teacher satisfaction, morale, or the work environment as a whole. 

The fact that those with a STEM degree can and do move on to other options is unfortunate but also a good motivator. "Hey, kids! Look what applying yourself in these subjects can do for you. Mr. Smith just got a job paying  ... "

Importantly, you can never pay those folks enough to keep them in the classroom, if they are chasing the dollar. TFAs are only thinking of a two year commitment and then it's off to the "Real World" of 6-figure salaries.  No school can compete with that. $100k or more -- is this what you want to pay a teacher in their first couple of years before you even know if they can teach? (What is this, some freakin' NBA rookie deal?)

Also, remember that they were not trained to be a teacher. Teaching math and doing math are different. They were trained to build machines, or solve complex systems, or write enormous amounts of code. There's nothing there about dealing with math-phobic 15-year-olds and material you learned easily twelve years ago. My biggest difficulty in the beginning was that, by and large, few of my students was as capable as I -- I had to figure out how to reach all the kids.

I'm a math teacher with an engineering degree, but I'm okay with the salary schedule paying all teachers similarly regardless of course. English is necessary, too, you know. As is art and music and history and science and computers and languages and woodshop and tech program and forestry and, and, and. Only a few students are going to specialize in math -- one could make the argument that the other teachers are more valuable to more students. How can anyone justify paying one teacher more than another based on the job offer that someone else might have gotten?

On the other hand, refusing pay increases doesn't make for a good environment either. My principal can't even visit my classroom more than once in four years and can't understand anything I teach - how is he going to fairly set my salary? I don't work for him; I work for a vague entity called the "District." It's not his money and he has no incentive to save it. This is not the classic "Boss" everyone thinks about.

The pay needs to be enough to keep money out of the conversation. The best way to do that is with the salary schedule. Then, there's no administrative BS, favoritism, stupidity, etc. You don't get worthless teachers (albeit with shiny degree) getting 5-figure signing bonuses and still skipping out after a year or two. Fairness is an issue and the evaluation process is too unclear for people to bet thousands of dollars based on it. You also don't get people comparing notes or holding out for raises in the middle of the year.

For more on motivation, go here: Dan Pink's talk on motivation.

Of course, if you want to pay me more, I won't turn it down. It's not why I teach, though. I had many choices that paid more -- industry, entrepreneur -- but I chose teaching. I'm paid well enough, I get my vacation all at once instead of having every night and weekend free and a couple weeks in August, and I enjoy what I do when I'm teaching. For me, it's a good choice.

Validating the material. Justifying the curriculum.

I'm all for the real world. I live here, too. But not everything I do is directly related to what I'll be doing tomorrow. Sometimes, I'm just guessing. Most of the time I know what's useful and what isn't.

What I refuse to do is to always justify every topic and problem that I teach as being useful to every student for every day of their future lives. I won't even guarantee that every topic will be "useful" even just once for each student. I can never do that because I know that every kid is different and will have different needs. Some kids are losers who will grow out of being losers -- what you were sure was useless last year may suddenly become very useful tomorrow.

This question arose when f(t) posted about some circle theorems and asked "What are these good for?" One answer included the following: "On the other hand, there are millions of other problems/concepts that also do that AND are useful in 'real life.' So, why do we do these that are so disconnected?"

The problem with always requiring "a real-world application or you'll dump the material" is that all of math can be reduced to this absurd level if you try hard enough. As can poetry. And chemistry. and history. and art. (and grammar and writing, if you took my principal's example). Every topic can be eliminated by somebody.

"When am I EVER going to use this?" becomes a weapon instead of a question.

Why not teach it "just because?"

Why should my short-sighted, intensely hormonal, spring-addled students have the right of refusal over anything they don't immediately see a purpose for? I can see a couple of blue-collar uses for those circle and tangent rules above. If I can't convince a student that a machinist, draftsman or custom motorcycle builder would need at least an understanding of this stuff, should it be eliminated? I don't think so.

Sometimes they just need to follow the leader. "We're doing it because I think it's neat, it's part of the course, it fits here and we have the time."

As math teachers, we need to refocus the question and answer it ourselves. We need to take the question out of the mouths of the lazy and decide what does, indeed, make for a good curriculum. If something like those circle theorems can be used in any meaningful way either later in life or later in the week, then WE should decide. If they have no purpose other than intellectual curiosity, then they have that going for them, don't they?

Geometrical theorems are rarely useful in the raw as it were, but in combination with other knowledge, make a different problem solvable. It's a polygon inscribed in a circle -- or is it a bolt-pattern for a truck wheel? Tangents and circles, central and inscribed angles, external angles come together all the time in machining and manufacture.

To answer the original question:
Every time I see these theorems, I think of the guys on Junkyard wars who recited geometrical theory when building a go-kart. They won.

Some other comments:
"My take on it is that the people behind the unified 10-12 curriculum looked at circle geometry and asked these same questions of "why?", had no answer, and got rid of them. " The commenter didn't get rid of them but the powers-that-be did.

"I don't consider "it's interesting" a good enough justification, because there are plenty of things that are both interesting and relevant to what students will later see of pure mathematics." In other words, justify or get replaced.

"it's a mathematical dead end." and "but I can't even think of a later pure-mathematics connection. Shouldn't we be suspicious of anything that's a "stub" in the curriculum?" I can't think of a reason for it, so let's be suspicious of it?

Professional Development Hell

I had to do some PD the other day and was once again stunned at the limited knowledge of so many math teachers. TI-inspire had some terrible little app running that didn't really work properly but was purportedly able to help students correct their misunderstanding of some simple math ideas. One of these: what happens to area when you double the radius? Take a 7 inch circle and drag the radius out to 14 inches and see what happens to the area.

"I don't get it. What's that mean -- area = 1.66E+2 inch² - what's the E thing?"

I guess we know why so many students have misunderstandings about math.

Go read something.

Go read Robert Talbert's comments on Calculus books, calculus reform and the best classes for freshman.

Books, Part Deux

Yesterday, I mentioned the books that I have, Part One. Those were the easy-to-read books. I got class sets of those.

The books below are harder to read, with a more limited audience so I only got 3 or 6 of each of them. They are the books a serious math student should have read, so I encourage them to try.

The "Biographies"
These biographies are written for the serious and capable 12th grade student or college student. The writing is very clear and the topic is fascinating for me. My students, not so much. An Imaginary Tale; The Story of i. A funny thing about this one was when I tried to type up the order to present it to the decider-person -- I couldn't type "i" without Word getting all angry at me.
e: The Story of a Number. Maior is a good writer, but not for the faint of brain. Also, try Zero; A Biography of a Dangerous Idea and The The Golden Ratio: The Story of PHI, the World's Most Astonishing Number .

Of course, there's John Allen Poulos:
A Mathematician Reads the Newspaper, as well as Mathematician Plays the Stockmarket and the inservice classics: Innumeracy and Beyond Innumeracy. He has some others. Try them if these do good things for you.

Damn Near Impossible for High School Students. (If that's not a challenge, I don't know what is ...)
A Brief History of Time by Hawking. You gotta throw down the gauntlet.

Math Quiz for Presidential Candidates

Darren writes that the President has become innumerate. This immediately brought to mind Paulos's proposed "Math Quiz for Presidential Candidates."

I agree completely. I think students should know most of these, too, so I gave the quiz to some of my classes when I was sick. I told the sub to try it as well, but she refused!

I think 6 is my favorite. (Below the fold)

Books on the Shelf.

What's in my classroom? Well, I like books and there was money in the budget, so ... someone had to do it. I ordered with this philosophy: if the book was an easy read, I tried to get a class set which could be lent around the department. If not, I got maybe 3 or 6 which I could lend to individual students.

Here are a couple of the easy reads:

Short Circuits.
One-Minute Mysteries: These are really easy but a good way to while away a few minutes of TA or homeroom. They are one- or two-paragraph mini-mysteries in which the criminal makes one mistake or one misstatement that shows his guilt. "How could he have known the victim was killed with a potato peeler if he wasn't there?" A similar book is Two Minute Mysteries, which is more of those "outside the box thinking" or "lateral thinking" puzzles. These can take a while because the kids have to come up with reasons that aren't readily seen or intuited. Some reasons fall into the category of "WTF? How are we supposed to know that?"

Statistics:
Huff's How to Lie with Statistics is a classic. I use at least some of the ideas in every class from consumer math through Pre-calculus. I also make the speech before the students start their science fair projects. See also "How to Lie with Maps."

From the guys who brought you Klutz's Guide to Juggling comes Cartoon Guide to Statistics. It's great. In fact, when I unpacked the books, the other math teacher instantly borrowed it. Gotta be a good sign! Witty and fun, it's still got the STUFF. The cartoons are quite descriptive and the math is all there, even if the 'toon of Galileo is goofy.

If you haven't read it ... Freakonomics (and the sequel, Super Freakonomics) is a really good collection of statistical stories and case studies. Leavitt is a good writer and he analyses his work in engaging and easy-to-read chapters. "Why Do Drug Dealers Live with Their Mothers?" and "Why is the KuKluxKlan Like a Group of Real-Estate Agents?" You'll want an entire Saturday for it.

Theory:
The Number Devil is a good book for starting kids reading about math topics without slapping them silly. Easy to read but not pandering either. Probably the best explanation for the Fibonacci sequence I've seen.

Trivia:
Loosely organized book of trivia and generally "neat stuff" is Brain Fuel. Engaging reads of one or two pages each that aren't particularly mathematical but I was able to buy them, so there! Kind of "Ripley Believe it or Not".

Geometry:
If it's a math class, it has to have Flatland & Sphereland in it. Misogynistic? Yes. Satire? Yes. Still, it's an exploration of geometry you shouldn't be without. Available online for free (long out of copyright), but you'll want tangible books. Fortunately, they're cheap.

SAT Prep
For my SAT Review class, I was able to score some $3 versions of Prince by Machiavelli and Sun Tzu's Art of War. These are used as essay topics. I'll take a paragraph almost at random (they're that good for this kind of thing) and make a question about it. Instant Essay topics.

For the same course, I got - and this is no shit - a copy of Specerian Penmanship. The students are fascinated by my ability to write cursive. Probably 50% can write poorly in cursive, the other half not at all. I, of course, had Mrs. R. to teach me this some 50ish years ago. So I lend out the workbooks with the stern instruction to use a piece of paper placed over top. SO far, no writing in the books. A couple girls have really worked on it. So funny. One texted this fact to her friend - proud she was teaching herself to write cursive. The irony escaped her.

Drafting is important.
How to Read/make Mechanical Drawings and Popular Science's The Art of Mechanical Drawing are good source books for orthographic projections and for general work. I feel it very important that students learn the basics of drafting. The art department is getting swamped and stamped and cut these days and the technical centers are becoming more exclusively for "tech kids" - where's the generally good student to learn this stuff? ME!

I can't say anything yet about 100 essential things because it's backordered.

That's a part. I'll talk about some of the rest tomorrow: the tough reads, the ones that I bought only 3 or 6 copies of.

caio for now.

Happy PI Day

Can't help it:

Manufacturing a Soccer Ball

So this used to be a truncated icosahedron. You could make one from scratch pretty easily, cut it out and Viola!, you had a ball.

Oh, how times have changed ...

Methodology

I am a collector of methods. I like finding new ways to do the simple operations and neat new ways to look at mathematics. I do not feel these methods should be shown to students.

Teach one method. I am in favor of the algorithms that I grew up with. Old fuddy-duddy? Maybe. Tough. The old ways work. The old ways are usually simpler and easier to use.

Think subtraction. 759-384. Do it your way. Now try to follow this.

Why are we bothering to re-invent this wheel?  Isn't one way sufficient for the students?  I feel we should wait until they have completely understood one method before we confuse them with "Other" possibilities.  If a student comes up with this on his own, great. 

Sheesh.  No wonder kids get confused if this is what they're taught.
h/t to parentalcation

Math Teacher Marries

Congratulations. All the best.

Combination Lock

Darren says that Combination locks are misnamed. They should be called permutation locks.

Groan.

How big is the Mersenne Prime?

Speculation about the latest Mersenne Prime:  How many digits? 
Casting Out Nines plotted the numbers and made a guess of 10.5 million digits.
God Plays Dice likes 14.5 million digits instead.
Now you can play too!

Here's my graph of the known Mersenne Primes.  Where do you think the next one will be?

But wait ... what if we eliminate the first few and concentrate on the later numbers:

 

And, possibly, more insight with this one:

What you are seeing is the M(p) in pink and lines for 11M, 12M, 13M, 14M

I'm liking the 10.5 million or 11 million guess.