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|
|Final 10 - one for each group|
|Getting the right length ruler|
|Finding radius of rim or tire|
|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.
|Pano of what the room looked like|
- 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
- decimal values between -6.28 and +6.28.
- between -4pi and -2pi or 2pi and 4pi with multiples of pi/4
- between 0 and 2pi with multiples of pi/8
|Getting ready to place angles|
|View from above tire|
|View from below rim|
|Checking placements of angles with radian ruler from day 1|
|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|
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|
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.