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?