3D Vector Magnitude

After watching my current IB year 2 students struggle with vectors (I took over the class this year) and it was the 1's turn to push slog death march through vectors I was determined to slow down and make sure they understood the basics... Or at least give them as much time as possible to create some conceptual understanding (certainly not the same thing).

We started vectors by talking about different notations and their meanings. We talked about the magnitude of a 2D vector. They could see the clear connection with Pythagorean Theorem, a few even suggested its use, but the 3D formula was not clear to them. Not at all. Students suggested we could use a cube root or cube the coordinates. Both potentially valid extensions of the pattern for sure. I ended class by giving them the formula with no proof no justification - time was running out and I had problems I wanted them to finish. Oops.

Realizing the obvious error of my ways, I started the next class with a model of a 3D vector (I used a wire coat-hanger as the vector and large graph paper to form the X-Y plane). My model was just to show them what I wanted them to build. Their task was to use a meter stick as their vector - which required some engineering - and find the coordinates of the vector in 3D. A few students suggested aligning the vector with a coordinate axis... I nixed that idea.

Once they had the coordinates I asked them to prove the formula given in the last class or better yet  to ignore the formula and use only the coordinates to find the length of the vector - which they already knew from the meter stick. 10-15 minutes later folks had drawn right triangles and created a solution!

Once everyone was finished, I pulled the whole class over to one of the more neatly created models so I could point and formalize the ideas.Towards the end several students commented "oh that makes sense," or "now I get it - that formula does make sense." Voila! A simple but effective way to spend 40 minutes of class.

How to Teach Math Anxiety

Maybe this one has been bumping around the interwebs for awhile... Today was my first encounter with it thanks to David Wees. Too good not to share.

Spring Break!

It has finally arrived. Two weeks to eat, sleep and ride my bike. I'm looking forward to the night I don't wake up thinking about school... I love teaching but this break is much needed and hopefully much deserved.

What got me thinking - 3/30

Yet more of what got me thinking this week.

Coaching vs. Teaching - Lessons learned coaching applied to teaching. Good stuff.

Lessons in the Medium - How we teach is more important than what we teach. - Riley Lark

Now Hiring... - More good stuff from Riley Lark.

Relativity on an Escalator - Great account of using Dan Meyer's escalator videos in a physics class.

Math Teacher Ninja - The Unit Circle

I have been trying to teach the unit circle for 5 or 6 years. Each year I think that I've tweaked the process to make it clear(er). Each year, at some point in the process, I have been met with blank stares of "what in the world are you talking about?"

This year I finally had success! In my 9th grade Geometry class of all places.

The Setup:
Earlier in the year I introduced trig through similar triangles. Despite some success, the intro had left many students with murky feelings of "so what" or "seems hard." So a few weeks ago I began planning a second trip through trig-land. I drew heavily from the Exeter Math problem sets and (contrary to Exeter's intentions) put together 3 problem sets each focused loosely around a different trig function.

• Tangent - I loved starting here to make connections to slope/gradient
• Sine
• Cosine

The students were coming around and felt better using and choosing trig ratios to solve problems. I loved the mix of the standard "how far/long is _____" combined with questions seeking deeper conceptual understanding. 

Feeling good about right angle trig I shifted the focus of class towards circles, in particular I wanted/needed to look at the equation for a circle. To start, I gave the students the equation:

In groups they found points that satisfied the equation (using any method they liked) and then used Geogebra via my computer and projector to plot the points. As a class they slowly watched the circle take form as more points were added. We then used (previously learned) geometric construction tools/concepts to find the center and the radius of the circle. I made no mention of how the radius or center were related to the equation, that would come later. 

Wanting to continue the exploration I reached out, once again, to the Exeter problems for inspiration and pulled together a bunch of problems on Circles. We spent 2-3 days working, solving and sharing solutions. Some students grabbed graph paper, some fired up Geogebra and others tackled the algebra head on... 

Today - The Unit Circle:

Then today rolled around. It's the day before spring break, we're all fried - teachers and students. I had my doubts, but I went all in and laid my cards on the table. I started my 65 minute class by putting the following image on the board or at least a hand drawn rendition of this image.

I described the circle as centered on the origin and with radius 1. As a group they all chanted the equation. I described the line segment, that it started at the origin, angled up at 30 degrees then ending when it intersected the circle. I posed the question "what are the coordinates of the point where the circle and line segment touch?" I mentioned that I could think of  3 ways to solve it and that there are probably more... I put them in groups and let them have at it. 10 minutes later 2 ways to solve the question were presented both with correct answers. 

Still, I made no overt connection to trig. At this point I emphasized that the students could use the equations for the line and circle (one group correctly pointed out that the tangent of 30 degrees is the slope of the line) and their GDC to find the intersection point. 

I put up the following spreadsheet and explained that I wanted them to repeat the process with more angles. Each group had a color that corresponded to different angles (see spreadsheet). When a group had an answer they quickly added it in on the spreadsheet. When all the results were up and the resulting patterns discussed and explained I quickly added the sin(x) and cos(x) columns to the spreadsheet. You could taste the learning that happened in those few moments. 

Now for the hard part... 

Graphing. This one took some work to explain. The idea of graphing "x" on the vertical axis made some heads spin. We talked about going around the circle more than once or going backwards and what that meant in terms of the graph... Here the GDC helped out wonderfully.

Class ended with the "non-mathy" students declaring, "I love it when you work hard and then it all makes sense" or "I really like it when I understand things." I felt like a freaking math teacher ninja! 

Reflections:

To be honest I don't know where the stage for this success was first set (what I described took 3 weeks - 8 to 9 hours in class). While having all the mathematical tools needed (circles, constructions and right angle trig) is necessary I don't think it's sufficient. Two more pieces were needed:

1. Students feeling the freedom to tackle a problem with different methods, thus allowing them to see problems in the context that is most natural to them.
2. Students being trained to solve new tough problems not simply repeating steps that the teacher has demonstrated on a whiteboard. 

What I read that got me thinking - 3/26

More bits and pieces I found on the interwebs that made me think. Some I agree with. Some I don't.

• John Swelller - Interview by Derek Muller (of Veritasium frame) discusses Cognitive Load Theory and describes why "constructivism" doesn't work. I wrote my thesis on an application of CLT. I was shocked when I read about Sweller's view of constructivism.
• The Relationship School - A shift in priorities for schools?
• Khanversations - My first read on the blog Physics First Observations. I'm looking forward to reading more.
• Minecraft Calculator - I'm convinced there are some great Design Tech type of projects that could be done with in Minecraft. Its cheap and easy to play - good potential tool for schools with tight budgets.

What got me thinking this week - 3/15

Some of this stuff I agree with some I don't, but it's all bits that have made me think.

• The White Paper - Paper out of Stanford discussing blended learning. A bit on Chromebooks in school. 
• Fun Stuff - Post on some of the coolness in the Exeter Math problem sets.
• TED - How simple ideas lead to scientific discoveries.
• Khan't Ignore How Students Learn - More good stuff from Frank Nochese.
• Quantum Progress - The importance of teaching computational methods in physics.
• Case for Guided Learning - Kirschner & Sweller. Loved their ideas about Cognitive Load Theory, based my thesis on it. Can't say I love their thoughts on inquiry.