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Angular Velocity, Trig Whips & Elmo

by Jon Orr

The #MTBOS is an amazing group of dedicated generous teachers!! This lesson came together because teachers are happily sharing what they are doing!

Generating Curiosity!

Dan Meyer has a series of blog post on Developing the Question you need to read. In one example he uses this video below to spark student wonder and start a fight. I copied his plan on how to use the video to generate discussion on speed.
Show this video

Pause the video before the bike is revealed and have students wonder “What is going on here?, What could the dots be?” Let the video play and then ask them to rank the dots from fastest to slowest. This is where wonder will happen. Are dots B and C moving at the same speed? What do we mean by speed anyway? Enter angular velocity vs. linear velocity.
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An Example for Linear Velocity vs. Angular Velocity

Show them this video obviously fake but fun video to generate some discussion.

Main Question we looked at together:How fast is the top swimmer moving when he hits the water? How fast is his angle changing? Before we calculate any of these we’ll go and experience the difference between the two.

Experience the Change

Bob Lochel has a great activity called Trig Whips where in groups of 4 students will experience the difference between angular velocity and linear velocity. Read about it!
A few pics and videos of our class Trig Whipping!

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Whole Class
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Consolidate

We came back in and summarized our findings from Bob’s handout. We made it clear that everyone had the same angular velocity but we all had different linear velocities. We turned our attention back to the diver video and determined the angular velocity and linear velocity of the top diver.

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That’s where class ended! Tomorrow we’ll start off with….

Andrew Stadel’s Elmo Problem!

See all the resources from Andrew here
Tomorrow we’ll find Elmo’s ending position after the 1 minute, angular velocity and linear velocity.

Lollipop Lollipop oh la la Lollipop! — & Rates of Change

by Jon Orr

Last year on twitter I saw that Alex Overwijk and Janice Bernstein with their grade 12 advanced functions classes did this lollipop activity!

I knew that I wanted to give this a try for this semester! What I especially love about this activity other than students experiencing rates of change is that this is an activity that can span multi-grades!

Here is what we did,

Generating Curiosity

I found this video on YouTube and asked the class to think of great questions we could ask about what we see!

FullSizeRender-1Great questions from the kids and we all agreed to look at

  • How does the sucking time affect the radius, circumference, volume, and surface area?
  • How long will it take until the lollipop is all gone?

Let’s investigate those relationships starting with the easy to measure (circumference) and also estimate how long it will take until the lollipop is no more!

We had guesses : ranging from 10 minutes through to 35 minutes.

Gathering Data

I handed out one lollipop per pair of students, along with some dental floss for measuring circumference. We set our timer for 30 seconds and began sucking and capturing data!
We recorded the circumference every 30 seconds up to 7 minutes like Al’s and Janice’s instruct in their lesson Plan.
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They also have a great handout for tracking the circumference over the 30 second intervals. Screen Shot 2015-09-18 at 2.22.08 PM

Analyzing the Data

So we first looked at the Time vs. Circumference and Time vs. Radius relationship
Linear - Lollipop

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We discussed its linearity and why. Students predicted with more accuracy when their lollipop would run out.
Up to this point this task is great for grades 7, 8, 9, or 10!! (Just edit the file to exclude the average and instantaneous rates of change).

  • Grade 7 & 8: Practice plotting points and reading/interpreting graphs.
  • Grade 9 & 10: Find lines of best fit and first differences.

We found the average rate of change for each 30 second interval and discussed what this meant. We used the last column to talk about narrowing the interval down to estimate the instantaneous rate of change, and noticed that it’s about the same for all values. Why does this make sense???

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We moved on to looking at Time vs. Volume and Time vs. Surface Area

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Great talks around how Volume and Surface aren’t deceasing at a constant rate! It changes! Students can see these changes and see in their tables where the volume is changing the fastest.

Overall a great intro activity to get students thinking about narrowing intervals to approximate instantaneous rates of change.

Next up: We’ll relate what we did here with the tables to the graphical interpretation of rates of change (secant and tangent lines) and then on to the algebraic!

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Let’s Start with the Easy Ones

by Jon Orr

Here’s how I taught students how to solve trigonometric equations in our grade 12 advanced functions class.

Started with this Ferris wheel problem

From 101qs.com find it here

What has been working well is starting our “math” at a very low level…..like on a dial…..then we slowly turn the dial up….adding more “math” in. Read more about the Math Dial from a comment on Dan Meyer’s blog here.

Starting with this video the math on the “math dial” is very low.

I asked: What questions do you have after seeing this….

Answers:

How fast is it spinning?

What’s the radius?

What’s the period?

Where will the red dot be after 3 min?

And that last one is the question we studied.

Act 2:

From 101qs.com

Almost all kids solved this problem using proportions! They kept the dial in the low position still!  They realized that it takes 5 seconds to travel from dot to dot. Therefore it takes 40 seconds to go all the way around. They divide 3 minutes up into 40 second sections and get 4.5 rotations. The dot will end at the top of the Wheel!!  But the Trigonometry in me was screaming to get out……I asked, “Did anyone create a trig equation to model the height?” — cue crickets!

So we cranked the math dial up a tad!

I said:
When I go on a ferris wheel I always look for my house.” We talked about how high that might be in relation to Dan’s problem….we settled on about maybe 40 feet.
My question: How long will it take to get to that height?

Guesses? Will it be a nice number? No? Why not?
Crank it up a bit more …
Let’s create an equation for the height in terms of time (we had already learned how to do this and it was no problem for the class) .daum_equation_1417134251422

Now, to solve our question we have to solve this equation!

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Student: That looks super hard!
Me: It does doesn’t it!

Let’s make that our goal!
We don’t want the math dial going up too quickly!

Let’s start with the easy ones, like this:

Screen Shot 2014-11-27 at 7.27.23 PMGotta keep the math dial low for a bit more…

Screen Shot 2014-11-27 at 7.29.22 PMWe solve this as a class, then another, and another, slowly building up our skills; slowly bringing the dial up. We stop at the end of the class. I assign a few more like the ones above. “Let’s get good at these so we can do the super hard one… Practice these for homework….”

Next day:
We take up the assigned questions then get back on track! We then solve these:

Screen Shot 2014-11-27 at 7.29.45 PM

We have a discussion on how many solutions there are here… and plop down a graphical solution in Desmos

Screen Shot 2014-11-27 at 7.30.00 PMThe math dial is getting up there…

Me: “Are you ready to try the big one?”

We do it! And everyone is into it….they have been waiting two days to see the answer! And the dial is pretty far up there!
After:
One student says: “That was pretty awesome! ”
That was my highlight of the day! Best compliment for a teacher!

We then show the graphical solution in Desmos. IMG_2795.JPG

Oh…..and we started class playing Pictionary (It’s our Wednesday thing) there was a tie and we have a good o’l match of Rock, Paper, Scissors to declare the winner. It was Intense!!!

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