Wednesday, September 29, 2021

Unexpected Beauty

 This is really a story of the unexpected beauty of mathematics. If you know me then you know I live in a #thinkingclassroom (via Peter Liljedahl) and love to #spiral curriculum (post about it). This post really speaks to those two ideas. 

I am teaching grade 11 College math when this happened but I feel like this could be done in grade 9 or grade 10 as well.

The goal was to review Sum of Squares ( my name for Pythagorean Theorem) Post I wrote about it kinda.

Here was the plan:

Random groups #VRG on #VNPS

Give students this sequence of mixed numbers 1 1/3, 2 2/5, 3 3/7, ........

I then asked groups to extend the sequence for four more terms.

Then turn them into improper fractions.

Then I asked them to use the two numbers in the improper fraction as the shorter two sides in a right angled triangle and then find the length of the hypotenuse.

And on that note I intended to end the task and have groups practice some Sum of Squares ( pythagorean theorem)

But then I saw it.
I asked groups to list the pythagorean triples in a table with stage number, smallest side, middle side, hypotenuse.

Then I asked them to show the patterns in each column.
Then I asked them to generate the equation for each column.

Groups mostly generated the linear equation from the table and used Desmos to get the quadratic equations. One group pointed out that the equation for (c) was just the equation for (b) plus one because the numbers for any stage were just c = b+1 (clever)


And then what do those columns and equations represent????
The sides of a right angled triangle .... so a^2 + b^2 = c^2

And away some groups went.
In typical thinking classroom style not all groups got to the end as this was pretty difficult work with exponent laws, squaring linears and squaring quadratics and collecting like terms.

And it all just came to me on the fly - Unexpected Beauty!

Here are some pics of the whiteboards.

After consolidation I asked the groups to explore this sequence of fractions.

1 7/8, 2 11/12, 3 15/16.........

A few pics of this.

And yes the last picture shows a group knocking it out of the park.

This activity was exhilarating for me as well as the students. It really demonstrates the beauty of spiraling curriculum. Multiple big ideas all in one activity.



Thursday, September 24, 2020

Ever Increasing Dominos


Dominoes - Relationships between the Number of the Domino and Various Variables

This has been a great activity for me over the years. Can just do Exponential Relations or add in Logarithms. Of course Dan Meyer Rocked it as a three act task here first.
Dane Ehlert relates it the 5 practices nicely here. You should check his website out - pretty spectacular.

And then this is my take on it in my spiraled activity based #thinkingclassroom. Lot's to unpack.

Students watched this video and then were examined the relationship  between the number of the dominoe and some variable. Here were the variables I assigned.
Once groups had their variable to examine I asked them to create a poster that had the following.
Student work for Height.
Student work for Thickness.
Student work for Width.
Student work for Volume.
Student work for Face Area.
Student work for Surface Area.
Student work for Distance between dominoes.
Student work for Mass.
Here is a summary of the equations and the reverse equations.

 As a class we looked for connections between the exponential equations.


 And then connections between the reverse ( log) equations.


Monday, January 9, 2017

15 Minutes

Today marks the tenth anniversary of this video going viral. Here is the story. Enjoy:

I have a saying that I like to live by: “Everything in moderation, including moderation.” For example, perhaps you happen to like alcoholic drinks. You’ve probably had a time in your life where you’ve had one too many. In 1999, this happened to me on a Friday night, and it made for a long Saturday. I was recovering on my couch on that Saturday afternoon, channel surfing, when I came across the World Bartending Championships on ESPN. I identified with this. Being a basketball player my entire life, I enjoyed watching the tossing of the bottles behind the back and under the leg. It was cool to watch.
The following Monday I was teaching a grade twelve unit on ‘circles’ for the first time, and to illustrate a theorem, I used a wood compass with chalk in it. It was very awkward. You had to hold the middle point with one hand and swing the other end with the other hand. After about 270 degrees, you had to swap hands. It was a pain in the ass to use. I abandoned it on the Tuesday. At the start of that class I just decided to draw a circle freehand on the chalkboard. It was stunning. A young lady named Sarah (with whom I had a great rapport) said, “Wow, Sir, that is really good!”  As I took ten steps back to admire my circle, the wheels in my head started turning and before I knew it, the next words out of my mouth were, “Of course it is good! I am the world freehand circle drawing champion.” Sarah responded with “Really?”  I immediately told the story of how I go to Las Vegas every year, what the prizes were, and that I was the 1997 World Freehand Circle Drawing Champion. This was the beginning of the story of being the World Freehand Circle Drawing Champion (WFHCDC).
I continued to tell this story to every class I ever taught whenever a circle was required in the course.
“Sir, Do you have to qualify to go every year?”
My response to that question, “It is like winning the Masters. If you win it once, you are automatically invited every year.”
“Mr. O, how do they decide who wins?”
“They have a laser machine called the circleometer that creates the perfect circle closest to the one you drew. The circleometer then calculates the difference in area between the laser circle and the circle that you drew. The machine then calibrates the area difference as if you had drawn a circle with radius one meter. The person with the smallest area difference is declared the world freehand circle drawing champion.”
“So you go all the way to Vegas and only draw one circle?” is a common question I have heard over the years.
“You actually go to Vegas and only draw one circle. The rule is once your chalk touches the board you must draw a circle without having the chalk leave the board until your circle is complete. So if you are totally nervous and you accidentally touch the board with your chalk and then remove it from the board that is called a DNC – did not circle – and you are disqualified.”
In 2001 I changed schools from Lisgar Collegiate Institute to Glebe Collegiate Institute. Since these two schools were rivals, many of the students knew each other, and the story of Mr. O being the world freehand circle drawing champion followed me to my new school. On one particular day, I was covering someone else’s class when I was asked if I would draw a circle for them. I told the story and drew a circle.
A student called bullshit and said “My mom used to teach with you at Lisgar and she has never mentioned it to me.”
I responded with “Your mom probably never thought of mentioning it. Go home and ask her about it. She will tell you.” I immediately left the class and emailed his mom and asked her to go along with the story; for her to verify that I did indeed win the 1997 world freehand circle drawing championship when and if her son asked. I did not hear back from her for a couple of days, and then received an email from her verifying that she had stuck to the story despite almost losing it with laughter when he asked about it for the umpteenth time. Over the next few weeks the story spread like wildfire around the school. Everyone was asking me about it. I continued to tell the story over the next several years.
In was in June of 2006 that a young man named Luc, who was our school web-master, asked if he could take a video of me drawing the perfect circle and put it on the Glebe website. I checked with the Principal to make sure he was okay with me doing this. He gave me permission. One day Luc walked in with the video camera and proceeded to engage me in my story. It was easy for me to tell the story. I had rehearsed it for years. The video was 73 seconds long and was posted to the school website stating that I was the world freehand circle drawing champion. On the day that Luc showed up with the camera I did draw a perfect circle or as near as I could – it could not have happened on a better day. I directed many of my friends to the video over the next few months during the summer. The video was catchy and funny. People who saw it loved it.
On January third of 2007 my wife Erin and I gave birth to our first son. I like names that have meaning, so we decided to name our first born Grady Marcus Overwijk, after his mother (my wife) and my mother. Unfortunately Grady was very ill at birth. Grady was admitted to the neonatal intensive care unit. It was a difficult time as we did not know what was wrong with our child. Thanks to expert care at the Childrens’ Hospital of Eastern Ontario (CHEO) Grady has since been diagnosed with a rare genetic disorder called Congenital Myasthenic Syndrome - Rapsun variation. It is a condition that he will have to manage with medication for the rest of his life, but he is an amazing kid and I know that he will persevere and find his calling. After about 5 days at the hospital I decided to go home to check messages and email.
I had over 200 emails in my in-box all pertaining to the video that had been posted on the Glebe Collegiate website. One of the emails was from Evan Narog from Fargo, North Dakota. It turns out Evan had been collaborating with some co-workers and one of them drew a circle on a whiteboard to illustrate how everything was connected. Whoever he was, he drew a pretty good circle and one of them commented about the story of Giotto. They decided to do a google search to find the details of Giotto’s story. Giotto is generally considered the first in a line of great artists who contributed to the Italian Renaissance. According to lore, the Pope sent a messenger to Giotto, asking him to send a drawing to demonstrate his skill. Giotto drew a circle so perfect that it seemed as though it was drawn using a compass. As part of their search they eventually came across my video on the Glebe website. After watching it several times Evan decided to copy it and post it on social media. He posted it on College Humour, Break and Youtube. Within 24 hours, the video had gone viral on Youtube.
As a result of the video I began to receive attention from all sorts of sources. I got an email from a talent scout at the David Letterman Show, letters from long lost friends, inquiries from people interested in competing in the ‘World Championships’, people commenting how cool the video was, and the originators of the Rock Paper Scissors World Championships who wanted to give me some advice on how to capitalize on my new-found fame. I was also contacted by a talent scout from the George Stroumboulopoulos Show, former students, our local News stations and a Pod-Cast producer from Electric Sky. Someone had leaked the story to the Ottawa Citizen, our city’s daily newspaper, and they did a segment on the front page of the city section titled “Glebe math teacher circles the web”. I was eventually asked to do a clip for a national news story on CTV.
The George Stroumboulopoulos Show wanted to do a fun, but quirky segment on me and my circle drawing abilities. Their take on it was to portray me as someone who had been obsessed with circles his whole life. My true passion is engaging students in their learning. Pretending to be something I wasn’t felt a little bit like selling my soul. I was excited to share how I use story-telling to engage my students. They wanted to make fun of me. If you know me, you know I like fun, but this just seemed just a little too far off the mark. I declined their offer.
The Letterman show was intrigued by the video and contacted me, but they were not sure how they could include it on the show. I naturally offered “Stupid human tricks” or Letterman versus Overwijk in a circle-off. They declined and left it at “If any other major syndicated talk show offers you to go on their show let us know and we will negotiate something.”
As mentioned, I received an email from the Rock Paper Scissors Society out of Toronto. They were impressed with the video and were willing to give me some advice to stretch my 15 minutes of fame. They shared with me their action plan to have and sustain the “Rock Paper Scissors World Championships”. It was titled “Making Something from Nothing.” I found this ironic as I was trying to make something from the perfect circle, the representation of zero-nothing. During our conference call they basically communicated that the whole world was watching and I could take the 15 minutes of fame or I could stretch it. The major piece of advice they gave me was to hold a world championship and get a video on Youtube. It would validate my story from the viral video. They felt that over time no one would ever check the dates. What came first – the viral video or the WFHCDC?
                Over the next six weeks I started planning the championship. It was to take place at the Cock and Lion Pub on Sparks Street in Ottawa. The Pub was owned by a good friend of mine, Denis Hines, who lost his battle with Cancer a few years back, may he rest in peace. Ironically, we were raising money for Cancer that night. All proceeds from the World Freehand Circle Drawing Championship were going to Cancer research. That morning I had done a segment on the local morning show. The host got a real kick out of the idea and I was able to describe how the championship was going to go down. Participants would go head-to-head in single elimination. A panel of five judges decided the winner of each match.  The final match came down to a circle-off between me and a ring-maker whose studio is located here in Ottawa. The competition was tight, but in the end I came away with a victory, and a one of a kind custom ring that he designed and crafted. The night was a huge success. Fun was had by all, and we managed to raise $1 500 for the Canadian Cancer Society.
                In March of 2007, I received an email from an art school in Germany. They were hosting an artist lounge at Art Basel, Switzerland in June of that year. They were interested in having me provide three, one hour talks, over the course of the week long art festival. They also wanted me do draw circles on command and to challenge others in attendance. After working out the details and getting some time off work, I managed to swing this. Yet there were some serious details to be worked out. Clearly I was going to need a chalkboard to draw on. What could I possibly talk about for three hours? Off I went to Switzerland to give my lectures at the biggest art festival in the world. When I finally arrived and made my way to the artist lounge to check in, I realized that I was just entertainment at the bar. Basically the art school from Germany was responsible for bringing in interesting people to provide audiences with unique and talented skills. The chalkboard they had constructed for me was enormous. It went from floor to ceiling and allowed me to use my shoulder as a pivot point, instead of my elbow when I was ripping circles. It was impossible to not draw a perfect circle. This was going to be great! Over the next several days I was able to get the crowd up and interacting in what was definitely one of the most unique experiences in my life. The focus of my presentation, besides the circle drawing, was the power of social media in making the ordinary extraordinary. Twenty years ago this would have been a local story that earned a few laughs. Today, with social media as a vehicle, my name and my story garnered World-wide attention.
                A few months after I returned from Switzerland, I was contacted by Marco Martins. Marco is an award-winning writer and director from Lisbon, Portugal. Marco is best known for Alice, a film for which he won ‘Best Short Film’ at the Cannes Film Festival in 2005.  Marco had seen my video on YouTube, and it had inspired a screenplay for a new film that he wanted to call How to Draw the Perfect Circle or ComoDesenhar Um Circulo Perfeito. Marco wanted to come to Ottawa to meet with me to discuss the viral video and my thoughts on circle drawing. He visited in October of 2007. We hit it off immediately, and we spent the week forming a friendship and introducing him to Ottawa. After his visit, he invited me to go to Portugal to shoot several scenes that involved me in a classroom, drawing the perfect circle for my students. In March of 2008, my wife Erin and I flew to Lisbon, leaving Grady with his grandparents. The first 2 days of our week long stay were spent on set watching the production team in action, and meeting the cast and crew. I shot two scenes at the Escola Liceu Passos Manuel. The script referred to me as ‘Alexander Overwijk - The Master of Circularity’. Although my scenes were cut from the final version of the film, it was a great experience and one I am not likely to forget.
                The rest of the week was spent exploring Lisbon: Castelo Sao Jorge, Barrio Alto, and the Pavilhao Atlantico. On the fifth day we took a day trip up the coast visiting Pena National Palace in Sintra, the breathtaking Roca Cape, Guincho Beach, Boca do Inferno, Cascais and Estoril. Guincho and Roca Cape were especially fascinating for we landlocked Canadians. During this trip, Erin was eight months pregnant with our second child. We had a shortlist of names for our second son. Cole had made the list. While at the beach I drew several of the names in the sand. Jokingly, I wrote the word CIRCLE. I looked down for a moment, and then drew a circle around the IRC in the middle….C-O-LE. We had a winner. This pleased me as Cole’s name now had special significance. The trip was a wonderful gift. I am so glad to have been given this opportunity.
                Rude Tube is a British show that highlights the back story of viral videos. I shot a segment for this show in 2011. I have actually never seen the segment but a friend of mine came across it. This show allowed me to tell the story a little.
                In May of 2011, a Japanese TV crew came to my school to shoot a segment for a show that is equivalent to “That’s Incredible” an American reality show from the 1980’s. They had me draw circles and holler “That’s Incredible!” in Japanese. They actually shot me drawing a massive circle on the gym floor.
                Next came an email from The Today Show on NBC in September of 2012. They were inviting me to New York to do a segment where I would teach the cohosts Savannah Guthrie, Natalie Morales, Willie Geist and Al Roker how to circle draw. They asked me a few questions about my talents and then I taught them a couple of techniques live. We had the competition which Al Roker won. They wanted to know how I knew if a circle was perfect or not – unfortunately I did not bring the circleometer with me.
                My next collaboration was cool. In the summer of 2013 I received an email from Layet Johnson who wanted to do an art collaboration with me. It would show at GoodWeather Gallery in North Little Rock, USA from Nov 26th 2013 to Jan 1st 2014. Layet wanted to come to Ottawa and have me draw the circles and him doodle the insides. He planned on doing a yin and yang, an eight ball, a donut, a basketball, Saturn, peace, Volkswagon and a smiley face. This was a cool interaction. We hit it off right away. Layet enjoyed basketball as do I. We even managed to play a game of pick up while he was here. Layet brought the most beautiful baby blue blackboards with him to Ottawa. They were carefully packaged so as not to be harmed.  When the time came for me to draw the circles, I struggled as the boards were smaller than what I had been used to in my classroom. I kept looking to him for approval. Eventually he told me that this is what I do. I am the world freehand circle drawing champion. He told me to let it go. He left me alone in our makeshift studio and set me to the task. When I play a sport I believe in rhythm and being in the zone. Layet had just provided the right mindset to put me in the zone, so after he left I was done drawing eight circles in no time. After I was finished, he doodled the rest of each piece, and they turned out great. I had made a connection with Layet as I had earlier with Marco.
                 Recently I was on Outrageous Acts of Science as the number 9 superhuman in the world. Ya right!
                 I teach high school math for a living. Over the last 8 years, I have collaborated with Bruce McLaurin, a colleague of mine, to develop an idea for teaching courses that we like to call ‘cycling the curriculum’ or ‘spiraling the curriculum’. Instead of teaching courses linearly, we introduce all topics through activities. As the course evolves, activities involve more than one idea and they invoke deeper thought about the ideas in the course. Students need to make connections and understand the ideas on a deep level. Ideas always come around again in future activities. It is neat that the idea of a circle or cycle is present in my pedagogy.
There are many other small things that have happened since the video went viral. Everyone likes to hear about my circle drawing story. I like story-telling and it sure is a great story to drop at a party where people don’t really know me. People are not sure what to believe. I’m not sure what is going to happen next with my circle drawing journey but it sure has been a fun ride.
In spite of the fun, or perhaps because of it, over the past few years I have been able to spend some time focusing on what I am truly passionate about – engaging students in the study of mathematics. I have been invited to several speaking engagements and educational forums to talk about the new way we are presenting the math curriculum. Not an event goes by in which someone doesn’t recognize me as ‘that circle drawing guy’, or asks me to draw a perfect circle. My 15 minutes of fame has brought me back to my roots. I have come ‘full-circle’ and I am now enjoying teaching and learning like never before. My friends have always encouraged me to write a chronical of everything that has happened since the video first went viral. My collaboration with Layet Johnson and Haynes Riley has made this possible. To them I am forever grateful. It has been a ‘TRIP’.  A round trip.

Thursday, September 22, 2016

Analyzing a Bike Rim in 5 Days

At Glebe Collegiate we have tried hard to spiral our MHF 4U course (called advanced functions) over the past 4-5 years. It really has been a collective effort within our department. You can find a whole bunch of information about it here thanks to Janice Bernstein. I posted the whiteboard notes and an early spiraled version of the course here.

We are trying hard BUT I feel like the course is still missing that activity based feeling that I have managed to create in my grade 10 applied course (for example my lesson on T-Shirts). It just feels like we still tell them the things we would like them to discover or experience.

Also I VALUE the mathematical processes in the curriculum document. So.....I started thinking about how I could get my students to discover and experience more of the Trigonometry in the course. One of the activities we have been doing is radian rulers from plates. Here and here is what it looked like.

This is my attempt at creating an exploratory activity 1) to discover and understand radians, 2) to develop the relationship between arclength, radius and the angle measure in radians  3) to understand angular velocity and 4) to develop an understanding of the relationship between the angle rotated and the height of a point on the bicycle rim.


First I went and scavenged some bike rims from a local bike store and cut out all the spokes. Needed 10 but only found 7 so I had to use three tires as well so that each group of 3 students had their own rim or tire.
Rims I scammed
Deleted spokes 
Final 10 - one for each group
DAY 1 Creating a radian ruler for their bike rim.
Students were put into random groups of three.( I randomize students by giving them a card from 1-10 which matches the pods of three desks and whiteboards (4 by 8 ft for each group) around my room-these numbers never change all semester just the students at those desks each activity). Verbal instructions ( no paper-throughout all 5 days)

1) Tape the number of their group onto their rim or tire.  

2) Create a paper ruler that has the length of the circumference of your rim or tire. Measure that length and figure out the radius of your rim (on whiteboards). Measure the diameter roughly to check.

Getting the right length ruler
Measuring ruler
Finding radius of rim or tire
3) Mark your ruler using a red marker in radius units. HUH? If your radius was 10 cm you would put a mark every 10 cm and mark them 1, 2, 3 etc until you got to the end.

About this
As this was happening I circulated and started having discussions with groups about how many they got. Why? Students started making connections that it was a little more than 6 because the circumference is 2pi*r. Once all were at this point we had a full class discussion that we could put 2pi at the end of the ruler and that was equivalent to 6.28 (about).

4)  Next we talked about it if we folded it in half (pi) and in half again and half again. I asked groups to do this and crease their rulers ( be precise ) and then mark the creases in terms of pi. I actually hesitated with the colour and of course a student hollered " Purple for pi". Me "of course".

5) Next we talked about if we folded in half, then in thirds (like you would fold a letter for an envelope) and then in half-we would get 12 pieces - so 2pi divided into 12 parts - each part would be pi/6. Students labelled these in green. Of course some were all ready marked because they were equivalent to some of the angles from the previous folds.
Folding and creasing our paper radian rulers

6) I then had groups put a decimal value at every crease (radius units).

7) Lastly they were instructed to put the degree value in orange. Student "Orange for the degree sign". Enough said!

Here is the final ruler with the rim for one group.
Final ruler with all markings

8) Lastly we wrapped our rulers back around our rims and talked about the angle and the number of radii around the rim.

9) Clean up time-hung the rulers for tomorrow.
Two class sets

DAY 2 Circular Radian Number Lines

This idea came to me after seeing this tweet from @CHaugneland
So this required some set up. I wanted each group to have a circular number line that they could hang angles in radians on. Here is a pano of the class as they were doing this part of the activity.

Pano of what the room looked like
Obviously there were many comments about the hanging rims around the room as students entered class.

1) The class discussed that we were going to use these as circular number lines and that they would hook cards of different angles in the right spot relative to each other and then check their placements using their radian ruler from yesterday.

2) I had prepared 4 sets of cards.
  • First set had positive angles only with multiples of pi/4 between 0 and 2pi. So pi/4, pi/2, 3pi/4, pi, 5pi/4, 3pi/2, 7pi/4 and 2pi. I also included some that were equivalent with a higher denominator, for example 6pi/8.
  •  Second set had positive and negative angles with multiples of pi/4 between -2pi and +2pi. This time I included equivalent ones that were positive and negative, for example +pi/4 and -7pi/4. And again some that were equivalent with a higher denominator.
  • Third set had positive and negative angles with multiples of pi/3 between -2pi and +2pi with equivalent angles.
  • Fourth set had positive and negative angles with multiples of pi/6 between -2pi and +2pi with equivalent angles.
  • For the fifth set I had groups make a set of cards and place them based on instructions that I gave them. For example
  1. decimal values between -6.28 and +6.28.
  2. between -4pi and -2pi or 2pi and 4pi with multiples of pi/4
  3. between 0 and 2pi with multiples of pi/8
  4. etc
3) The card sets were placed around the room on the ledges of my white boards. A group would pick up a set, place them on their circular number line and then check placements with their radian ruler. Groups then checked in with me-I typically drilled them and asked lots of questions. Groups then returned the card set and picked up the next one. Repeat.

Here are some pictures.

Getting ready to place angles
View from above tire 
View from below rim
Equivalent angles
Checking placements of angles with radian ruler from day 1
Big smile
Some thoughts:

There were many groups that wanted to know where to place zero and what direction was positive? negative? 

Checking with the radian ruler was tricky but doable.

Mathematical discussions and productive struggle were through the roof.

High level of engagement.

 A colleague thought I was crazy because of the work required for just this one day. It was well worth the experience as students participated whole-heartily in rich mathematical discussions. Would I do this again-ABSOLUTELY.

Arclength = Radius *Angle in radians 

Believe it or not it took some thought for me to clue into this. Plot Arclength(y) versus Angle in radians(x). The slope of the line will be the radius of their tire, and the equation will fall out. Here is how it went down.

1) Same groups as the last two days. Get your bike rim, your radian ruler, whiteboard marker and a measuring tape.

2) Measure the arclength for a particular angle in radians and make a table of values. 
As I was demonstrating this by wrapping the radian ruler around the tire and the tape measure around the tire (which was difficult-see picture) this conversation ensued.

Student asks and states "Do we really have to wrap both around each time? Couldn't we just lay it flat and measure - it is the same thing".

Me "What do you think?"

Other student " As long as we realize that when we measure the distance on the (radian) ruler we are really measuring the arclength around the rim"

Measuring arc length for a particular angle in radians by wrapping around bike rim
Measuring arc length for a particular angle in radians using the radian ruler laid out flat
3) Make a table of values and graph the data. Find the slope of the line. Find the equation of the line using technology (or algebraically) (or both)

4) Calculate the radius of your rim again. Circumference = 2* pi * radius

5) At this point I asked if there was any connections between the slope of the line and the radius of the bike rim. We went around the room and each group yelled out their slope followed by the radius of their tire. AHA the slope of the line was the radius. And there you have it 
Arclength = Radius *Angle in radians

6) We briefly worked on an application.

Some samples of  groups whiteboards at the end of the period.
Sample 1
Sample 2
Sample 3
Sample 4
Sample 5
Sample 6
Distance A Point On The Wheel Travels

1) We grabbed a bike rim and the entire class went out into the hall. I asked some students to time how long the rim rolled for while I counted the number of revolutions. We picked a starting point. One of the students marked where we stopped the wheel. I rolled the wheel and I counted revolutions while some students timed the roll until I yelled stop.

5 revolutions of a wheel with a radius of 25.9 cm in 8.68 seconds

2) Groups were sent back into the classroom to work on whiteboards and find the total distance travelled by the point on the rim. I also asked them for the angular velocity in radians per second. I also asked them to figure out the total distance travelled by the point on the rim using the angular velocity. Here is the work of one group for the data given above.
Group work: Total distance a point travels 811 cm
For a different rim
7 revolutions of a wheel with a radius of 26.1 cm in 10.11 seconds

3) Once all groups had done the calculations we debriefed. All groups went out in the hall and measured the distance we had marked earlier. The calculations were right on.

Relationship between the ANGLE ROTATED and the HEIGHT OF A POINT ON THE BIKE RIM 

1) I did a whole class discussion on how we could collect data. Place the point at the wheel on the ground at zero on our radian ruler. Roll the rim on the ruler and stop at particular angles and then measure the height of the point. Looks like this.

2) Groups made a table of values, graph and generated the equation using technology. If they finished this I asked groups what the theoretical equation should have been. If they finished that I gave them an angle and asked for the height of the point.

DAY 5 

Here were the instructions I had written on my whiteboard for groups to do this day. We discussed average rate of change and I said I would talk to any groups about instantaneous rate of change if they got there.

Group work for 5 of the 20 groups (two classes):

Group 1
Group 2
Group 3
Group 4
Group 5
Final Thoughts

My original intent for this activity was to redo the radian plate activity and the radian war activity from this site. This is where I have grown. I am thinking what else can I do with this (thank you #MTBOS for #WCYDWT) This post reflects my creative juices in squeezing curriculum out of an activity. Hope you enjoyed. Honestly - this activity feels like what I envisioned for a spiraled course and wrote about back in 2013. #makeitstick #spiraling #activitybasedlearning #interleaving 

Also I feel like I am getting closer to a Thinking Classroom that Peter Liljedahl has been researching about.

I teach so differently now then I used to. I can't even start to describe how different my role in the class is compared to years ago when I lectured. While doing this activity I actually sent out this tweet.

Any and all feedback welcome. What else could we do with this?