POW #12

POW #12 - Eyeballing Game Link
Complete seven estimates with an average score below 2.60.  Submit a screenshot to ejohnson@fessenden.org and also post to your mathematics folder.

POW #11

Submissions are due by the end of the day on Monday, December 11th.  Make sure to scan your work and put it in your Mathematics/POW folder with a name of LASTNAMEWeek11.  See me with questions.  You may work with a partner but make sure you note that on your submission and you both must submit a copy for credit.  You also must explain your process and solution in sentences!

Pre-Algebra - POW11
Geometry - POW11

POW #10

Submissions are due by the end of the day on Monday, December 4th.  Make sure to scan your work and put it in your Mathematics/POW folder with a name of LASTNAMEWeek10.  See me with questions.  You may work with a partner but make sure you note that on your submission and you both must submit a copy for credit.  You also must explain your process and solution in sentences!

Pre-Algebra - POW10
Geometry - POW10

POW #9

Submissions are due by the end of the day on Monday, November 20th.  Make sure to scan your work and put it in your Mathematics/POW folder with a name of LASTNAMEWeek9.  See me with questions.  You may work with a partner but make sure you note that on your submission and you both must submit a copy for credit.  You also must explain your process and solution in sentences!

Pre-Algebra - POW9
Geometry - POW9

POW Amnesty Week

You may complete two POWs from earlier weeks for full credit.  Submission due on Monday.

POW #8

Submissions are due by the end of the day on Monday, November 6th.  Make sure to scan your work and put it in your Mathematics/POW folder with a name of LASTNAMEWeek8.  See me with questions.  You may work with a partner but make sure you note that on your submission and you both must submit a copy for credit.  You also must explain your process and solution in sentences!

Pre-Algebra - POW8
Geometry - POW8

POW #7

Submissions are due by the end of the day on Monday, October 30th.  Make sure to scan your work and put it in your Mathematics/POW folder with a name of LASTNAMEWeek7.  See me with questions.  You may work with a partner but make sure you note that on your submission and you both must submit a copy for credit.  You also must explain your process and solution in sentences!

Pre-Algebra - POW7

Geometry - POW7

POW #6

Submissions are due by the end of the day on Monday, October 23rd.  Make sure to scan your work and put it in your Mathematics/POW folder with a name of LASTNAMEWeek6.  See me with questions.  You may work with a partner but make sure you note that on your submission and you both must submit a copy for credit.  You also must explain your process and solution in sentences!

Pre-Algebra - POW6

Geometry - POW6

Global Math Project

My favorite math teacher, Dr. James Tanton, is sharing the joy of mathematics with over a million students this week with his explorations in the Global Math Project.  In all my classes, we will spend some time working with Exploding Dots!  A video description can be found here - link.  You can also use the interactive website here.  If you stick with it for about an hour, I think you will be amazed!

POW #5

Submissions are due by the end of the day on Monday, October 16th.  Make sure to scan your work and put it in your Mathematics/POW folder with a name of LASTNAMEWeek5.  See me with questions.  You may work with a partner but make sure you note that on your submission and you both must submit a copy for credit.  You also must explain your process and solution in sentences!

Pre-Algebra - POW5

Geometry - POW5

POW #4

Submissions are due by the end of the day on Tuesday, October 10th.  Make sure to scan your work and put it in your Mathematics/POW folder with a name of LASTNAMEWeek4.  See me with questions.  You may work with a partner but make sure you note that on your submission and you both must submit a copy for credit.  You also must explain your process and solution in sentences!

Pre-Algebra - POW4

Geometry - POW4

POW #3

Submissions are due by the end of the day on Monday, October 2nd.  Make sure to scan your work and put it in your Mathematics/POW folder with a name of LASTNAMEWeek3.  See me with questions.  You may work with a partner but make sure you note that on your submission and you both must submit a copy for credit.  You also must explain your process and solution in sentences!

Pre-Algebra - POW3

Geometry - POW3

Helicone - A Laser Cut Version

The Golden Ratio Project - Phyllotaxis on a Laser Cutter

In my Geometry Honors class last year, we completed a PBL on the Golden Ratio.  In the intoduction to the project, I showed my students different examples of the Golden Ratio.  One of the most popular examples was the Helicone by PlayableArt and based on the work of John Edmark.  Around the same time, I had also produced a spinner using 22mm bearings.  These two ideas seemed to have synergy so I started out on a project to produce a larger Helicone based on bearings that could be robust and interactive - ideally strong enough for student use at an all-boys K-9 school.

I went through multiple prototypes, some of which neared completion before determining that the mechanism or materials were not strong enough.  I also had to flip the rotating mechanism so that the input would be from the base and not from the top of the helicone.  Early iterations used rivets and slots to trigger and limit the rotation but these ended up bending or getting ripped from the wood after continued use.

Prototype 1 - Plywood, Rivets and Tapered Branch Lengths

This seemed like the most obvious method when I started this project.  I found hollow brass rivets on Amazon and designed the mechanism with the pin poking through the top layer of each.  The golden angle was relatively easy to calculate and designs were made on Inkscape to have a slot on the bottom of each layer to create a helix and a golden ratio "rush".  I designed many, many files so that each layer could be cut separately.  

Problems discovered:  Wood layers will warp when using wood glue.  Warped layers consequently interrupt the smooth operation of the helicone.  Also, the helicone seemed to worked best when engaged from the top layer which seemed a bit akward.

Prototype 2 - Acrylic, Rivets and Tapered Branch Lengths

Acrylic took care of the warping issue but rivets did not fit perfectly and would still get ripped out when the helicone was spun at high speeds.  Acrylic is also much more expensive and requires the use of hazardous materials to fuse together.  On the plus sides, the look of the material both the color and rastering results is impressive (may do another of these in the future).  

Prototype 3 - MDF, Rivets and Tapered Branch Lengths

MDF is hard to find but I finally found a supplier through Etsy.  Great material to work with but the edges are not particularly strong (weaker than plywood) and it became clear that rivets were not going to be the right mechanism.  I made about 12 layers with MDF before punting on this idea and almost quitting on the project.

Prototype 4 - Plywood, Bow-Tie Mechanism, Uniform Branch Lengths

I considered an idea of using wood itself for the mechanism after seeing an example of a helicone made on a 3D printer.  The benefit of this "bow-tie" design is that the mechanism has better symmetry and distributes the forces on each side of spinner.  I also liked that it did not require special order rivets and only required screws (#4 x 3/8") for construction.  During this last prototype, I also learned more about the use of adhesives.  Spray adhesive works well for large flat layers and provides a fast even coating with adequate attachment.  For the more critical pieces, such as the bow tie and pedal connectors, I still use wood glue and allow time for setting.

During this prototype I also decided to streamline the constructions process and files.  I made one file for printing 6 spinner mechanisms but did not include the pedals.  This allowed me to fine tune the design for checking the mechanism and also used the sheets of material more efficiently.  The pedals for each layer are cut separately and attached with wood glue and take advantage of a "puzzle piece" connector.  I also think this design results in less warping as each layer uses multiple pieces of wood rather than long pieces from one piece.  

After finding success with this last design, I focused on how to make it interactive.  I spent some time cutting sprockets from both wood and HDF but it became clear that bike chains exert tremendous force on the material and require the use of metal sprockets.  A quick visit to the Harris Cyclery in West Newton (and $20) and I had the front and rear sprocket as well as a matching chain.  I cut some additional plywood pieces on the laser cutter to match these sprockets and added the same 22mm bearing to allow for smooth actuation.  I also learned how to break a chain and use a connecting link to adjust the chain to the proper length.

What's next?  Another version?  Possibly.  I still like acrylic but question it's ability to withstand the forces and torque.  

POW #2

Submissions are due by the end of the day on Monday, September 25th.  Make sure to scan your work and put it in your Mathematics/POW folder with a name of LASTNAMEWeek2.  See me with questions.  You may work with a partner but make sure you note that on your submission and you both must submit a copy for credit.  You also must explain your process and solution in sentences!

Pre-Algebra - POW2
Geometry - POW2

Paradox Redux - POW #20

The birthday paradox is a great one to read and think about but here is another one...

I have two children, one of whom is a son born on a Tuesday. What is the probability that I have two boys?

Don't look up the answer just yet.  Think about it.  When you need to, here is a good write-up of the answer.  For credit for this POW, write a short paragraph summarizing the reasoning.

POW #19

Submissions are due by the end of the day on Monday, April 24th.  Make sure to scan your work and put it in your Mathematics/POW folder with a name of LASTNAMEWeek19.  See me with questions.  You may work with a partner but make sure you note that on your submission and you both must submit a copy for credit.  LET ME KNOW IF YOU PUT IT IN YOUR FOLDER!

Algebra 1 - POW19
Geometry - POW19

POW #18

Welcome back!  As a follow up to the Open Box Problem from before break, go to this link and write a paragraph response.  Submit on googledocs!  Due 4/10/17.

POW #17

Submissions are due by the end of the day on Friday, March 10th .  Make sure to scan your work and put it in your Mathematics/POW folder with a name of LASTNAMEWeek17.  See me with questions.  You may work with a partner but make sure you note that on your submission and you both must submit a copy for credit.  LET ME KNOW IF YOU PUT IT IN YOUR FOLDER!

Algebra 1 - POW17
Geometry - POW17

POW 16

Submissions are due by the end of the day on Monday, February 27th.  Make sure to scan your work and put it in your Mathematics/POW folder with a name of LASTNAMEWeek16.  See me with questions.  You may work with a partner but make sure you note that on your submission and you both must submit a copy for credit.  LET ME KNOW IF YOU PUT IT IN YOUR FOLDER!

Algebra 1 - POW16
Geometry - POW16

POW #15

POW #15

Submissions are due by the end of the day on Monday, February 13th.  Make sure to scan your work and put it in your Mathematics/POW folder with a name of LASTNAMEWeek15.  See me with questions.  You may work with a partner but make sure you note that on your submission and you both must submit a copy for credit.  LET ME KNOW IF YOU PUT IT IN YOUR FOLDER!

Algebra 1 - POW15
Geometry - POW15

POW #14

Submissions are due by the end of the day on Monday, January 23rd.  Make sure to scan your work and put it in your Mathematics/POW folder with a name of LASTNAMEWeek14.  See me with questions.  You may work with a partner but make sure you note that on your submission and you both must submit a copy for credit.  LET ME KNOW IF YOU PUT IT IN YOUR FOLDER!

Algebra 1 - POW14
Geometry - POW14

Another fun day in math class.

The day before the holiday break allowed for another fun and challenging exploration for my Fessenden students.  This time we discussed the mathematical paradox known as the Monty Hall Problem from two perspectives, one mathematical and the other psychological.

Here is the scenario (based on a game show from the 80's),

Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, (who knows what’s behind the doors), opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?

The problem has confounded many experts despite its relatively basic probability scenario.  I first heard of this paradox when many readers (some with Phd's) criticized a column written by Marlilyn vos Savant in the popular Sunday magazine, Parade.  You can read some of the harshly worded (yet wrong!) responses here.

In the class before break, I become the game show host dressed in a gaudy sports coat and stick microphone and ask players to "come on down".  We play the Pick a Door game using an online simulator and students start winning fictitious goats and cars.  The students eagerly play and also start to analyze strategies and patterns to improve their chances of winning.  At first, patterns are elusive and most students believe that "staying" or "switching" shouldn't matter as it seems like an equal probability.  We test their theories for many rounds before I show them that the simulator can also play automatically for 1000 rounds.  This large number of trials show a pattern that becomes more and more clear - Players will win 2 out of 3 times when switching but only 1 out of 3 times when sticking.  This is puzzling to pretty much everyone in the class.  My mathematical explanation to students:

- Pretend you always stick no matter what.  You will win 1 out of 3 times.  This is the theoretical probability and will closely match the experimental probability with enough trials.

- Pretend you always switch no matter what.  With this strategy, you only lose if you pick the door with the car first.  If you pick a door with a goat, the host will always show you the other goat and you will switch to the car.  The odds of picking a goat first is 2 out of 3.

The second part of our class is dedicated to the psychological analysis of how individuals play this game.  I like this part even more than the first part!  As stated before, most people don't think it matters if you stay or switch and put the probability at 50/50.  In addition, some people take it further and ironically don't want to switch as they want to "trust their gut" and stay with their original choice.  This was explored in depth by the television show Mythbusters in an episode called Wheel of Misfortune.  In class, we watch the episode and the results are astounding.  20 contestants play the Pick a Door game in an elaborate simulation and every single one of them stick with their original door choice. Every one!  Despite having twice the odds of winning, 20 people in a row make the absolute wrong choice and stick.

The conclusion I want my students to have is that you shouldn't always trust your gut.  Be informed, use math to complete an analysis and take your time and hopefully you will make better choices in life.

POW #13

POW #13

Submissions are due by the end of the day on Monday, January 9th.  Make sure to scan your work and put it in your Mathematics/POW folder with a name of LASTNAMEWeek13.  See me with questions.  You may work with a partner but make sure you note that on your submission and you both must submit a copy for credit.  LET ME KNOW IF YOU PUT IT IN YOUR FOLDER!

Algebra 1 - POW13
Geometry - POW13