Aug 152015

At #TMC15 I shared my favorite of the “High 5”. Richard Villanueva is awesome enough to record them all and post, so I will just share the video of what I said. It is short and sweet:

There is the video. I want to stress a few points.

  1. Giving high fives to my learners absolutely changed me. I got 150 high fives every day. How can you NOT be in a great mood getting 35 high fives several times a day, every day.
  2. I am serious. I didn’t teach math. I taught people the subject of math. The high fives was just one step that demonstrated this philosophy.
  3. This was an evolution of my approach that on the first day of class scared me to death. I was freaked out and thinking that it was going to be a massive failure.
  4. I was wrong.
  5. It was the single thing I did all year long that had the greatest impact on my classroom environment, my relationship with my learners, and my own personal attitude.

I wrote about it last year as it occurred:

Before school started: August 10th: School started on the 11th.

After 1 week of school: August 20th

After 1 month of school: August 27th

I finally EARNED a high five from my one holdout: September 10th  : This is the one high five I am most proud of.

That was last year. Then #TMC happened. After #TMC15, several teachers told me they were going to try it. We had several Twitter conversations about it at different times with different teachers. A sample is below. And this is ONLY a small sample of the more relevant tweets.






And here are some captured images from @misscalcul8. Elissastartpic2Elissa2 elissa3  Elissa1 And finally: pic1Elissa    

Let’s pause and reflect a moment.    What effort did it take me to give a high five? Very little. I had to get over my introvertedness. I had to fight my impulse to just stand there and say hi, and I had to make the effort to actually acknowledge each learner one at a time with the motion. I had to grab some hand sanitizer afterwards as I was walking into class. …. 

Yea, that is really what it cost me. That’s it.

Not to diminish the fright / frustration / and uncomfortableness that the introversion creates, but getting over it did not damage me in any way.  

What did I gain? My learners received the one on one acknowledgement from me every day. They walked into my classroom looking forward to the personal contact that went beyond the subject and touched them personally. Learners who were just standing in the hallway saw it and started asking for a high five every day. They recognized that it was something to get and feel good about themselves.

It changed my outlook on the class period. Every period be came a 1st period of the day. Every period was a “good morning” because every period started with 30 to 35 high fives. How could every period NOT be a fresh start, a clean slate, and a new beginning. It changed the class outlook towards me. I wasn’t just that weird math teacher (and I was) who wore strange socks everyday (because I did). I was also the math teacher who treated them like human beings. I also was the math teacher who acknowledged they were weird learners (because they were) who struggled with the ideas (because they did) and who needed the reassurance that if they kept trying they would get it (because they absolutely DID.)  

The cost / benefit analysis there is pretty clear. What it cost me was very little. What I gained was huge. What my learners gained was even greater.  


I am not teaching high school anymore. I am teaching college and the standards are different, the expectations are different, and the stakes are different. Guess what I am NOT going to give up. I think these outcomes are too valuable. It will definitely be a radical departure for the college setting. It is worth it.  


Edit: Some research to back up why it works:   More teachers on board! Yay!    








These next three go together. Wow, the power in these three tweets.




Amy posted on her blog the following paragraph.

High fives at the door. Glenn’s “my favorite” has been popular for good reason. It is simple, but we have already discovered it is powerful. My colleagues and I decided to make it a department thing, and also roped in the two non-math teachers in our hallway. So the 200 hallway is officially the “high-five hallway” at our school. I am surprised by how something so small has already helped me feel more connected to my students, and how the classroom atmosphere gets an immediate boost. You just can’t be too grumpy after a high-five.

Chris Shore said:

High 5’sGlenn Waddell (@gwaddellnvhs): Glenn was right. Offering the High 5’s at the door does more for my mood and mental preparation for the class than it did for the kids.

Bob, on his blog, said:

GREETING STUDENTS WITH HIGH FIVES – Intertwined with all of the mathy goodness of Twitter Math Camp this past July was a simple and powerful device for student engagement from my friend Glenn Waddell – the High Five.

Each day last year, Glenn met his students at the door to give them a high five – a simple, caring gesture to establish a positive tone for class. I often meet students at the door before class or linger in the hallway for informal chat, but I love the tradition and rapport Glenn establishes here and hope to emulate it.

Lisa, on her blog, was even more positive about the effects:

After five days of being at the door and high fiving students, students are positioning their books to be ready to give me a high five as they approach my class. I have had students high five me in the hall when I am not at my door and walking in the hallway (when I don’t have a class). It makes me smile.

This is only one paragraph of a much longer post by Lisa, but you get the sense right there something amazing is happening.

Stephanie Bower tried it too. Her post says so much about it.

Most of the time, the high-fives give me a chance to gauge the moods of each student in a split-second. (Glenn pointed this out too.) I can tell by the tone of their high-five, the way they return my verbal greeting, and their body language if something is “off” that day.


[And yes, that graphic is golden, and will be stolen and reused. Forever.]

highfiveclub Thank you @conniehamilton.

Aug 042014

2014-07-25 12.48.04

My Favorites are some of the best part of the TwitterMathCamp experience, and this year was no different. I know one favorite I had was walking into this building and seeing that even a public high school could afford to build a dedicated Science & Math Center!

But inside the building, we were offering our own My Favorites. I had one my favorite that I offered, which is a cheap (free) way to record your class so you can observe yourself.

Take an old smartphone and remove all the apps. It is best to use a phone that has a SD card, but if you can clean enough space off of an internal memory phone that works too. Once you have at least 4 gig free, then you have enough space to record 45 minutes of video at 720p or 30 minutes at 1080p. Ideally you would want at least 8 gigs so there is extra space.

Once you have this phone ready, you can use whatever you have on hand to construct your own stand. Lego’s work great, a coffee cup, or even a paperclip. Learners will freak out at a tripod and video camera set up in the back of the room (I know, take it from personal experience) while they will not even think of the phone sitting on a shelf recording them.

Two other My Favorites that were offered by others that I really liked are Plickers and a very interesting and annoying problem that has incredible extensions.

Plickers are “Paper Clickers” and it is genius. Using a cell phone or tablet with camera and the paper funky symbols you can poll the class on a question and have the responses immediately tracked and recorded. You can show the class results in bar graphs, and later can use the results for data tracking and demonstrating what you are doing for your admin. Great discussion and engagement in class and  data tracking for later. It is a win-win.

Finally is this problem. IT is tricky, fun, amazing, and all around a well designed problem.


What proportion of the triangles is shaded?

That is it, just find the shaded area. The solution has extensions all over the place and is a great problem to try and work through.

I hope you Enjoy!

May 232013

The AP exam is over, finals are in two weeks, and my learners have been busy bees constructing knowledge for the community.

That is their goal in this post-AP Exam stage of the class. Their charge and challenge is to do something that constructs, creates, or consolidates knowledge for the school or the community. Below are the projects that have come out of this as examples, in no particular order.

Group 1:  This group took last year’s project of finding out which is the cheaper grocery store (Smith’s, Walmart, Raleys and Scolari’s – they found out Smith’s & Walmart were the same) and are extending it. This year they are seeing if last years results are still true, AND then extending the matched pair to a two sample to compare last year’s prices to this year’s prices.  The manager of Smith’s is very interested in the outcome of this.

Group 2: Does Walmart change their prices based on the income level of the zip code the store is located in? So far the results show no, but that hasn’t been finalized yet.

Group 3: Are high school students prepared for disasters. This group wanted to do something related to the zombie apocalypse. I used my veto power. Now they are back down to earth doing a multistage cluster & stratified sample of the school.

Group 4: What percent of our seniors go out of state for college and how does this compare to previous years? The trouble is, our counselors have never kept track of this! There is no data base or spreadsheet for previous years. So, this group had to collect the information (census) from all the seniors, build a spreadsheet to house the information and then compare vs. the numbers we were able to get from our local university.  When they are done they are giving the results and database to the counselors to maintain.

Group 5: This group of 1 person originally proposed a nationwide survey. I talked them out of it. Then they proposed a city wide survey. Again, talked them out of it. There wasn’t enough time to accomplish such aggressive projects. In the end, the person narrowed the idea down to “what impacts graduation rates the most, income or poverty in Washoe County School District?” This is achievable. The group is collecting information from census bureau data and grad rate data and then doing a regression test on the data. There are some interesting results so far.

Group 6:  Are freshmen aware of or comfortable with basic science based on their middle school and freshmen biology class? They took questions from the science standards and talked to science teachers to create a short quiz based on the standards. They did a random sample of freshmen classes and then sampled randomly 5 from those classes using a deck of cards. The sample is VERY small, but the process is good.

Group 7: Do gasoline prices fluctuate together in an area at the gas pump AND do the prices move with the price of light sweet crude on the international market? Whew, big project dealing with daily checking at 7 gas stations and market prices. They have a very interesting topic, and the mechanics of it will require a regression test.

Group 8: Another one person group, but this time the learner is interested if course “difficulty” can be quantified. The learner was upset at being told that certain classes are harder than others and therefore should not be taken because it would hurt the GPA. When the learner asked “how do you know?” to the counselor the learner was not given a straight answer. To help, I received a complete grade distribution of every class in the school, removed the teacher names and replaced with a random word that starts with either “d” or “g” (for no reason, just to make it difficult) and this is what the learner is using. So far, so good. Although I am not sure the question can be achieved, it is worth doing.

Group 9: This group is interested in whether the plans of the senior class changed from when they were freshmen. A cluster sample is being done on the government classes to get senior viewpoints.

and finally,

Group 10: A multistage sample to see if being involved in sports helps or hurts GPA. They are all athletes and they all have different opinions on what the outcome will be.


I am excited to see the results of many of these, and I will be sharing the results with store managers in the area, the local newspaper in several cases, as well as next year’s group of AP learners.

Any suggestions or questions for them (or me)?

Feb 012013



Yea, that’s right. I just had 2 classes in a row teach themselves and others how to complete the square with circles and ellipses. How did I accomplish this miracle, because I really do consider it to be a miracle.

In my Advanced Algebra Class we have the following problem called, creatively enough, The Lost Hiker.

Suppose you need to find yet another lost hiker.  Fortunately you have information from 3 different radio transmitters.  From this information you know that he is:

25 miles from Transmitter A

5 miles from Transmitter B and

13 miles from Transmitter C

These radio transmitters are at the following coordinates:

Transmitter A – (25.5, 7.5)

Transmitter B – (-2.5, 3.5)

Transmitter C – (6.5, -11.5)

Use this information to find the coordinates of the lost hiker using the intersections of the theoretical circles.  Show your work below.

There are 4 major steps here; 1. Write the equation of 3 circles, 2. Expand the circle equations, 3. “Collapse” the circles through polynomial subtraction to 2 lines, and 4. Solve the system of lines.

We worked on these for 3 days. These problems are huge and complex, and one single minus sign incorrect means the whole thing works out wrong. But they persevered for three periods in class, until every single learner could do 3 unique problems on their own, and get the correct answer.

I have an excel spreadsheet programmed to calculate an infinite number of nice problems with solutions. The solutions are posted on the board w/ a magnet, and I walk around with a bunch of problems in my hand. Get one right, here is another.

Well, at the end of 3 days, I wrote the following equation on the board and asked them to turn it back into a circle.

x2+ y2 – 6x + 8y = 12

It took them all of 1 minute to have the entire class finished. One learner who failed Alg2 taught the rest how to do it because it was, “Easy, just take the -6, divide by two, because when you square it you have two of them.”

Then I wrote 5 problems on the board.

x2+ y2 – 4x + 12y = 10

x2+ y2 + 2x – 10y = -8

x2+ y2 – 4x + 7y = -20

2x2+ 3y2 + 6x + 15y = 0

4x2+ 5y2 + 16x – 100y = 27

The first two were just some more practice. The question was asked, “Can we have a decimal?” When the answer was yes, they didn’t bat an eye.

The last two did make them think, but they had the factoring done correctly on the left side, they just needed a little hinting for the right side of the equals.

Why did it work so nicely? I think because they really were engaged with the lost hiker problem and honestly worked them. And they worked. They agonized over why they didn’t get the correct answer. They gave me dirty looks, and when we found the minus sign they missed or the extra number they wrote in, they were angry at themselves.

But they persisted! It was a thing of beauty and I loved it.

When they did the completing the square on their own, with only some gentle nudging from me, I told them how proud of them I am. They needed to hear it.

Heck, they EARNED it.

Jan 102013


It has been a while since I have done a #MyFavFriday, but I realized this week that I absolutely have a favorite I would like to share.

My pens. I love pens.

2013-01-10 18.00.40

I have a ritual I do every morning to make sure I have my wallet, room keys, cellphone and yes, by Rotring Quad-point pen. My pen is similar to the linked pen at Amazon, but has 4 tips instead of 3 and is graphite instead of silver (The bottom pen pictured). I love this pen. Currently it has blue, black, purple and a .5mm pencil. The learners in my class know it and recognize that it is “my” pen because they see it every day in my pocket, and I use it every day working with them. I broke it once, and sent it in for repairs. It cost me $25.00 but was worth every penny and I stressed every day it was gone.

I got hooked on the multi-pen idea from the middle pen above. It is a black Niji Tri-point pen. “Niji” is Japanese for rainbow, and this pen is definitely a rainbow pen. I have retired this pen from daily use, but as you can tell from the black chipping off it was used for many years (I count 13 years, 1987-2000) and loved a great deal. My father gave it to me when I graduated high school, and I cherished it and used it through every college course I took. The physics and math equations that pen solved, let me tell you.

These multi-point pens are impossible to find anymore. Rotring doesn’t make this model, and Niji is long out of business. The current crop of multi-pens is very different shaped. I have the Fisher Multi-pen (not pictured) and it is okay, but I don’t like the grip on it. I prefer the smooth grip instead of a rubber grip. I also prefer the heft of a brass pen instead of the lighter plastic and aluminum pens.

For refills, I settled a while ago on the Monteverde brand refill. They come in many colors and are the smoothest writing ballpoint I have found for these pens. [It is really sad that I have tried enough refills to know what writes best! But I guess after using the same type of pen for 26 years I should have learned that by now.]

Finally, my third favorite pen is a Levenger brand fountain pen. It is heavy, and writes like a dream. There is nothing like putting the purple ink cartridge in it and writing letters. The words just flow out of the pen onto the paper. I really do enjoy writing with this pen because of the weight and smoothness of the ink.

I counted for this article and I have 13 pens in my collection right now. All of them get used to some degree, but the three above are actually loved. The wooden pens; hand carved, polished and assembled are beautiful, the eagle carved from wood is amazing, the Cross and all the rest are nice, but my Rotring, Niji and Levenger are the pens I actually truly enjoy.

Sep 212012

It has been a while since I did a #MyFavFriday post, but I have to share this because it has been making my life so much easier this school year.

Dropbox.  Yes, that service. I know, so many other people have written about it in the past, and so have I, but this year I made the move to put 100% of my teaching files* into dropbox and I haven’t looked back.

All dropbox is, is a small program on your machine that monitors a particular folder, named “Dropbox”. Anything in that folder is synced with the web portal and any other computer that is signed in. It is a folder, nothing more. If you can save a file in “My Documents” you can use dropbox. Just save it to “Dropbox/Algebra 1” instead of “My Documents/Algebra 1”.

These are some of the things I no longer have to carry around with me when I leave and return to school because of dropbox.

  1. Laptop
  2. power cords
  3. flashdrives

That just removed 7 lbs from my backpack, and turned my motorcycle commute into a much more pleasant experience. In fact, I only take my backpack home when I have to take dead tree materials home or to school.

But that is not all I use it for. I have dropbox installed on 6 different computers. 2 at school, 4 at home. Dropbox allows me to streamline my workflow and be more efficient. Let me explain.

At school, one computer is in the front of the room attached to an LCD projector and a smartpanel, the second computer is at the back of the room at my desk.  I will work on a document for class at my desk, and immediately upon saving it will be updated on the front computer. This means I can pull something out of my email at the back of the room, walk to the front, and show it to the class. I can have software installed on the back computer, do a screen shot and save, then show. No flash drives, no futzing with anything.

At home, I have it on all my computers. When I am working on a project at school, I can save the document and go home. When I arrive home, boot my computer, and my full project is sitting there ready for me. It is on all my computers, even the old clunker that I boot once a week. That old clunker downloads and syncs all the files, so I have a weekly backup of everything.

My wife also has dropbox, and we have a shared folder between us. That allows us to connect our dropboxes and have files shared between our computers. We now have 1 shared, encrypted password file instead of 2.

If you want dropbox and don’t have it yet, you get 2 Gigs of storage for free. If you sign up with this link you get and extra 500 megs and I get an extra 500 megs for free. Because of that I have 13.4 Gigs of free storage right now.

When you include picture uploading automatically (through the smartphone app) which saves you from needing to connect your phone to your computer or emailing the pics, the accessibility of dropbox on the internet through the web portal, and the seamless syncing of the docs, it is a win win win for teachers.

If you are on the fence, do it. It takes the process of managing your files and turns it into a non-entity.

Aug 172012

My #myfavfri post this week is on 2 ways to graph equations online easy and simply. I struggled with showing graphs large and in charge in a way that learners could duplicate at home, and then I came across these 2 methods.

1. Desmos’ “A Better Calculator” which is found at Some nice features. You can login and save your graphs, graph many equations and then “hide” them to reveal them when you need, graph conic sections, etc.

Really nice for the classroom is the fact that it has a projector mode, as well as a “points of interest” button. Here is a screen shot.


I honestly use this in class more than anything else.

2. Google. That’s right, our friend Google. Just type into the search box the following:

“graph y=3x^2 +5x – 6” It is the graph of the equation above. Try it. It will look like this:


and before you even have the equation typed in, the graph is there.

Move your mouse around the graph, and you get the (x,y) coordinates up in the right corner, and you can click the graph and move the axes around.

You don’t need a fancy graphing calculator at home, all you need is internet access and a brower that supports HTML5 (which means not Internet Explorer 7 or below.)

And those are my favorites for today!

Aug 102012

I was having a tough time writing this post this week because I have been at Exeter Math training for the last 2 days, and will be for 2 more days. This training has been a huge time suck, with homework given every night that I have spent a lot of time on thinking as I work.

And then I remembered that I don’t have to write the post on Friday, I can write it Tuesday night and SCHEDULE the post to post on Friday morning at 4:00 AM Pacific time so that all the East coasters will see it by 7:00 AM.

Scheduling posts is really how I have been doing my posts for the last week. I have been busy (as everyone always is) and writing when I have time and then scheduling the post for a certain time allows me to meet the deadlines that are holding me accountable, yet still getting posts done.

Great idea, and I wanted to spread it.

And then Shelli goes and says this on Tuesday night.

image Arg. She stole my thunder! Oh well. Good stuff is meant to be shared, not kept back. Look on the right side of the screen in both WordPress and Blogger and you will see an option to schedule the post. As it stands, I am writing this post on Tuesday, but it will post on Friday morning.

My second Favorite right now is a Foray Journal notebook that I bought at Office Depot in the clearance section for a single buck. I go to the clearance section in Office Depot every time I am in there, and usually don’t find a lot there. It is mostly junk or old ipod / iphone cases. But I found a journal that just looked awesome. Mine is blue with no “Journal” across the front, but it does have the ribbon and the silver edges. It is a very high quality, large journal.

This is my Exeter Journal, and I am using it as a modified Interactive Notebook. Modified because I am not using the left / right structure, but instead using a color / pencil approach. It looks like this:

Essentially, it is my favorite because I have been writing in it every day for several weeks and reflecting on the Exeter materials. Pencil for my work, and purple for reflections. If I had started this after Megan’s INB presentation at #TMC12, I would have done the left / right side division.

Working in this notebook has just made me see how powerful they can be for my learners as well as myself.




And I will leave you with a poem:

Math is beauty, poetry and fun, and it all begins with number One.
As you go from One to Two, you will learn what math can do for you.
So if a number isn’t your hero, you’re nothing but a big fat zero.
-Dan Carter, math teacher Reno, NV

Aug 032012

This year I am trying to make my classroom more personal and less institutional. So many math classrooms are can be described and “lacking a personal touch,” I think because as math teachers we don’t realize just how much that personal touch is missed.

I know when I was in college, we spent so much time working problems and figuring out methods to simplify a complex issue into a simpler issue. In grad school taking my teaching classes, the courses were about methods of teaching and learning, not about classroom management and work flow.

To that end, I need to make my classroom more …. Me.

So, my short little My Favorite Friday post today is on a couple of things I am doing to make my classroom about me and about math (And those in the peanut gallery keep down about how that last sentence is a tautology and doesn’t make sense. 🙂

First off. I have many Seniors. They are under the impression that you go to college once, and never again. They also are under the impression that community colleges are somehow not as good as universities. To show just how wrong both impressions are, I hunted down a banner from every college / university I have attended and did this:

2012-07-30 12.02.47

From left to right, it is the banner from Mesa Community College, where I did not graduate from, but did attend for 2 years before transferring credits to Knox College. I then attended the University of Iowa and later the University of Reno, NV. To the right I have my degrees and my license in frames.

The goal isn’t to brag, but so show the seniors (and juniors) that learning is lifelong and you should never stop, as well as where you start does not matter. What matters is the effort you put into your learning.

The other MyFavFri is a new company I discovered that does some pretty amazing things. It is called Moo. Yes that is right, Moo. They are a full color printer of cards, stationary, etc. How that differs from a company like Vistprint is in the full color. VistaPrint usually does 4 color work very cheaply, but Moo is doing full color work.

They end up being a bit pricier, but the results are better. For instance, I ordered the following “Mini-Cards” from them. Minicards are about 1/2 the size of a business card.

Einstein Cards  Gauss Cards

The top teal part is the front of the card, while the bottom part is the back of the cards. You can tell these images are from money (I found the images here) and they will be printed in full color with all the details of the original image. I recommend you order, for FREE, a sample pack. I did, the quality is beyond compare. I am proud to have my name and website on the quality of these materials. I can’t say that about the school provided cards.

Again, the idea is to personalize the materials in my classroom to be uniquely me and high quality, not just “boring math.”

Now I just need to paint some walls, I think. hmm…..

Jul 272012

For today’s #myfavfriday I am presenting an idea that has been percolating in my head for a while. If you want to know what a #myfavfriday is, then see Druinok’s blog here.


Learners have a devil of a time with quadratics. Afterall, there can be 2 solutions, 1 solution, or no solutions in Algebra 1, and then in Algebra 2, we come at them with the fact that those equations with no solution really do have 2 solutions after all, they are just “imaginary” (could there be a worse name for them, really? Thanks a lot Descarte.”)

But I came across a picture on some site one day, and it has stuck with me. I never bookmarked it, or wrote down the site, so it is lost to me (and I have searched hard for it) but the work blew my mind, and as I have shown it to learners, they have at least gotten a sense that the “imaginary” really does have meaning.

Let’s begin with 2 equations and graphs that are simple, straight-forward and make sense. [all images are clickable to see full size]

graph1and graph2

The equations are y = x^2 – 4x + 3 and y = x^2 – 4x + 4.

A simple change of one number changes the number of solutions from 2 distinct to 2 repeating solutions, and learners don’t have a problem with that idea, generally. Then comes this bad boy.

graph3 y = x^2 –4x + 6

Now they have to do the whole Quadratic formula on it to get the solution, and the solution has those i thingies in it, which makes them all confused and irritated until they wrap their heads around it. And why does it still have 2 solutions? It doesn’t touch at all!

But wait! We can play a game with this quadratic function. What if we reflect the parabola around the vertex in the downward direction? Then we end up with something that looks like this:

graph4a To do this reflection, we first had to complete the square on the original equation to get y = (x-2)^2 + 2. Now, with this equation, we can put the – sign into the equation and get the reflection, y = -(x-2)^2 + 2.

But hold on, see those 2 points where it crosses the X-axis? And see the Axis of Symmetry that goes through both equations? If we use those three points as definitions for a circle, we get the following graph and equation.

graph5 (x-2)^2 + y^2 = 2

Guess what the solution to the quadratic equation y = x^2 –4x + 6 is. If you guessed 2 + root(2)i and 2 – root(2)i  then you are absolutely correct.

The real number part of the complex solution of a quadratic with two imaginary roots is the X value of the Axis of Symmetry, and the imaginary part of the solution is the radius of the circle created by the center and endpoints created when the inverted parabola crosses the X-Axis!

Okay, mind blown. Why? How could I prove this?

Aha! now come into play the hours I spend on a motorcycle every summer. How could I PROVE that this will always work? I have the proof. I am working it up, but it is a pain to type. That, I think, will be the focus of a future, #myfavfriday!

[And I really need to look in to a LaTex module for my blog if I am going to do math. The equations look horrible.]

Edit: 29 July 2012: I proved this assertion, at least to my satisfaction in a followup post:

Edit: 4 August 2012: I found, stuffed in the bottom of my backpack, a rumpled piece of paper with this link on it. I think I did this page justice with my treatment. I wish I had found the page before I spent hours thinking about how to prove it, it gives the suggestion right there at the bottom!

Edit: 18 Dec 2012: @Mythagon posted this picture on Twitter. It is a great visual of what is discussed above, and clearly shows why the rotation is so important.

From: Teaching Mathematics, 2nd edition by m. Sobel and E. Maletsky

Edit: 27 Sep 2015: Wow, a long time since the original post, however I still come back to this every year. Love it. Now, Luke Walsh, aka @LukeSelfwalker added this to the mix. Love it. Click it for the live Desmos file.