World’s Hardest Easy Geometry Problem

I spent (wasted?) several hours working this problem out years ago but saw it posted recently on Google Plus.  I soon realized that either my solution predated my blog or I never actually posted my solution.

Here’s the problem and my solution is below: (ref: worlds hardest easy geometry problem

Using only elementary geometry, determine angle x. Provide a step-by-step proof.

You may use only elementary geometry, such as the fact that the angles of a triangle add up to 180 degrees and the basic congruent triangle rules (side-angle-side, etc.). You may not use trigonometry, such as sines and cosines, the law of sines, the law of cosines, etc.

Spoiler Alert: Don’t read any further if you want to give this a shot yourself!


Continue reading

Teaching in the One iPad Classroom

ipad icon

I have the great privilege of being able to gather with some 60 K-12 educators and lead them in a workshop to learn how to incorporate technology into their classroom.

At the Region 17 Education Service Center in Lubbock on Friday, February 8, 2013, I am leading a technology workshop entitled, “Teaching in the One iPad Classroom”.

Here’s what we hope to be covering: (from the abstract on their website)

In this workshop, attendees will be introduced to strategies for using the iPad to teach in the modern classroom. Software tools and apps will be presented that allow wireless mirroring of an iPad through the projector in the classroom as well as remote control of your desktop computer while teaching. Additionally, a range of apps will be presented that offer a basis for interactivity in the classroom and a powerfully visual presentation of concepts.

Much of the content for this has been taken from the Teacher Quality Grant for which I am the lead instructor.  As part of the Teacher Quality Grant (2012-13), high school and junior high algebra teachers have learned how to integrate the iPad into their curriculum and even use it to develop their own curriculum elements.

Here is the handout that is being made available as part of the workshop: Workshop Handouts

Other Classroom Management Apps

A pastor, a doctor and a mathematician

A pastor, a doctor, and a mathematician were stuck behind a slow foursome while playing golf. The greenskeeper noticed their frustration and explained to them, “The slow group ahead of you is a bunch of blind firemen. They lost their sight saving our clubhouse from a fire last year, so we always let them play for free.”

The pastor responded, “That’s terrible! I’ll say a prayer for them.”

The doctor said, “I’ll contact my ophthalmologist friends and see if there isn’t something that can be done for them.”

And the mathematician asked, “Why can’t these guys play at night?”

- from

Penny is for Penalize

The elementary school where my boys attend is hosting a fund raising contest where students bring their change to donate. Instead of accumulating donations to win for their own grade, they are putting their change in other grades canisters. The grade with the lowest weight in donated change at the end of the fund-raising wins. Pretty innovative, I think. The kids seem motivated, so fortunately for the school they are not colluding to all bring nothing which would keep their weights down. Instead they are piling in the change especially those with siblings in other grades.

Strolling through the halls, I overheard a conversation where teachers were wondering if anyone had found out which coins weighs the most so they could give more of those.  Of course, that got me thinking.  The obvious answer would be that the larger coins like the half-dollar or presidential dollar would be the heaviest.

Although, don’t you really want to have the most weight for your money?  Sure, I could put in 10 presidential dollar coins, but which would weigh more, 10 dollars in half dollars, 10 dollars in quarters, or 10 dollars in dimes?

So, of course, I had to know the answer.  Checking out the U.S. Mint, I learned the following weights for each of the coins:

1 penny = 2.5 g  which means $1 in pennies is 250 g.
1 nickel = 5 g which means $1 in nickels is 100 g.
1 dime = 2.268 g which means $1 in dimes is 22.68 g.
1 quarter = 5.670 g which means $1 in quarters is 22.68 g.
1 half-dollar = 11.340 g which means $1 in half-dollars is 22.68 g.
1 dollar (coin) = 8.1 g which means $1 in a single coin is 8.1 g.

In spite of the larger weights for the larger coins, you are still much better off dumping in those pennies.  I did learn a pretty interesting fact though: A dollar in dimes weighs the same as a dollar in quarters which also weighs the same as a dollar in half-dollars.  Pretty cool!

Second Chance Exam

imageHave you ever heard of a “second chance exam”?  I came across the concept for the first time in an article at Faculty Focus, Revisiting Extra Credit Policies.

Here’s how the author explains it:

The instructor attaches a blank piece of paper to the back of every exam. Students may write on that sheet any exam questions they couldn’t answer or weren’t sure they answered correctly. Students then take this piece of paper with them and look up the correct answers. They can use any resource at their disposal short of asking the instructor. At the start of the next class session, they turn in their set of corrected answers which the instructor re-attaches to their original exam. Both sets of answers are graded. If students missed the question on the exam but answered it correctly on the attached sheet, half the credit lost for the wrong answer is recovered.

I currently have a standing policy in all of my classes that allow students to correct missed problems on a test after it has been graded.  They’ll receive a bonus point on their exam grade for every correctly revised problem.  Instead of a flat bonus, this gives the most reward to students who put in the most work in the corrections. 

I used to do a flat 10 point curve for corrections.  At one point I was having students hand a test notebook at the end of the term.  The notebook contained corrected versions of their tests and they were rewarded with 3 bonus points on the their final average.

I’m considering trying this new approach, the “second chance exam” because it requires students to assess what they know, put in the work of correcting a problem and it also reduces the amount of time it takes to get a final grade into the grade book.  Right now, students take a test, then get it back the next class, then turn in corrections after that, and then I eventually return their corrections.  This new way, I collect the second chance exam the next class after the exam and then return the fully graded exam after that.

Of course, a sizeable percentage of fellow faculty would probably argue that extra credit only encourages laziness and procrastination on the part of the students but if the opportunities can be manipulated into a learning experience, isn’t that better than not learning at all?

CAMT 2011 – Our Nerd Vacation


My wife, Lori, and I have come to Grapevine, TX to attend one of the largest state-level mathematics conferences in the U.S.  We are at the Conference for the Advancement of Mathematics Teaching (CAMT) which is a joint conference held by the NCTM, MAA and TASM.  (Google them if you really have to know what those acronyms stand for)

The conference is primarily for K-12 math teachers but there are few like me here that participate or supervise in the area of teacher education.  I’m here to learn two things, better techniques to teach our up and coming teachers and changes coming due to STAAR and the EOCs.  (The tests replacing the current TAKS.)

The venue is the Gaylord Texan Resort and Convention Center in Grapevine, Texas.


We have a beautiful room with an excellent view of the atrium.

IMG_1180 Stitch


Now this is more for me as a reminder than for the readers but here are the talks I have attended so far:


Day 1 (Monday, July 18)

First Timer’s Session (8:00 – 9:00)

Algebra I Activities With The TI-NspireTM Handheld – Andi Parr, Region 13 ESC (9:15 – 10:15)

The Role of Inquiry Teaching Methods in Secondary Mathematics Classrooms – Mark Daniels, University of Texas at Austin (10:30 – 11:30)

Ignite Session! – Tim Pope, Key Curriculum Press (11:45 – 12:45)
Note: This session had a unique format.  Nine speakers each had 5 minutes to present.  They each had 20 powerpoint slides that would advance ever 15 seconds automatically.  This year’s speakrs included; Pam Harris, Paula Moeller, Michelle King, James Epperson, Amber Branch, Emma Trevino, Cindy Schneider and Cindy Schimek.

Exploring AP Caluclus Activities with the TI-NspireTM – Noe Medrano (3:30 – 4:30)


Day 2 (Tuesday, July 19)

Small Group Instruction in the Secondary Classroom – Richard Yoes, Joda Mendoza, Pasadena ISD (8:00 – 9:30)

The New TI-Nspire Navigator SystemTM – Holly Larson, McKinney ISD (9:15 – 10:15)

Big Gains from Small Struggles – Cathy Seely, Charles A. Dana Center (10:30 – 11:30)

Math Curriculum Makeover – Dan Meyer, Author (11:30 – 1:00)


…more to come later

Book Review: Evolving in Monkey Town


Evolving in Monkey Town: How a Girl Who Knew All the Answers Learned to Ask the Questions

by Rachel Held Evans.

My Rating:      didn't like itit was okliked itreally liked it (my current rating)it was amazing

Status:    Read from June 25 to 30, 2011

My Review:

Based on a popular blog, this book walks through a twenty-somethings evolution of faith from a fairly fundamentalist background to a more liberal and reformed approached to Christianity. I was surprised a just how much of her experiences mirrored my own. She definitely poses the more difficult questions that a Christian must deal with today. I hoped to have her tackle the questions more directly than to just have her cherish the ability to pose them. Nevertheless, it was a good read and I would recommend it to any young adult who has struggled with doubt.