3act Math

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!

daum_equation_1417134353017

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!!!

IMG_2794.JPG

 

Many Many Volumes

by Jon Orr

In our senior math classes (advanced functions & calculus) we come across a problem like this….Screen Shot 2014-11-26 at 7.00.47 AM

I really like these problems, they have great potential but not really in this form. Let’s jazz it up and spend an entire class with this

Start with this video:

Ask What questions do you have about this?

Them:
What size is that rectangle?
Why are the corners cut?
Is volume always the same?

Etc,

My question:

What size of that square do we cut out so the box has the biggest volume?

Play the video again and have them yell out when they think the box has the largest volume.

Have them guess
What is too small?
What is too large?

Have them take their guess for the size of the corner and find the volume of the box

Draw a picture of the “card board” label the dimensions.

Draw the squares to cut out. Optional (Cut them out) make the boxes.

IMG_2790.JPG

What’s the new length?
What’s the new width?
What’s the height?

What’s the volume?

Is this the max?
How can we check?
Have them do another? And another.

Have them come up to your computer and enter their height and volume in the Desmos page for each box.

Screen Shot 2014-11-26 at 7.09.50 AM

 

Now, let’s generalize!
This time let your guess be x and find an expression for the volume.

What’s the new width? Take 8 and subtract twice your guess. (8-2x) Now the length? What is the height now??

Put that expression into Desmos and let them see the function, let them point to the maximum.

Screen Shot 2014-11-26 at 7.13.32 AM

For calculus: have them find the maximum using derivatives!

Show them this video to check their guesses.

From here we can solve problems like our original textbook question. The kids are invested now and they are ready to use the equation to find the value of x where the volume is say 24 cubic units.

Further reading: Jonathan Newman’s volume of a box Activity 

Credits: Algebra in motion for the Geometers Sketchpad file. Dan Meyer – this lesson mimics his Circle Square lesson.

UPDATE [Nov. 27, 2014]

Luke Walsh created a Desmos Sketch that seems super useful!

Modelling in Clash of Clans

by Jon Orr

Here is a task I’ve been playing around with lately. Let me know what you think!!!!
Thinking of using this task with my grade 11s functions class/ or more advanced functions classes. I have recently been playing Clash of Clans and if you have played you know that you have to wait for items to be built/upgraded, etc. The time to wait changes based on the your progress and cost of the item/upgrade. You do have the option to SKIP the upgrade wait time by using gems. What has me wondering is that the amount of gems needed to skip an upgrade. What’s the relationship between upgrade time and gems? Our task is to see what that relationship is.

Act 1 : I’ll show this short video to my students:

I’ll ask for any questions the students had from watching the video and settle on —How many gems would it take to upgrade the town hall? — which will take 2 days.

Get the students to make some guesses…..

Act 2 :

Then get them discussing what other info we will need. I want them to come up with the idea they need more instances of upgrade times vs. gems. I can start to show them some pics…..was thinking of revealing each “point” at a time and getting then to guess!

Clash-of-Clans-Act-2

Here comes the modelling time…….plop these down in Desmos. We’ll start to select a model based on the data we see:

Screen Shot 2014-09-24 at 11.04.31 AMpremade desmos page with some sliders built in for each type of model.

Screen Shot 2014-09-24 at 11.21.16 AM

Have them decide which model they like best for the answer…..use the model to come up with an answer. My guess is that students will assume linear and come up with an answer that is too high (I’ll update later after I use it with students). ……and then we can have the big reveal……

Act 3

Watch the video ….

Or use the image..

IMG_0909

 

Sequel:

Find the cost of upgrading immediately…..how much wait time can you skip with $20?

IMG_0890

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Minnie’s Juice Cup [3actMath]

by Jon Orr

Here is a 3-Act Math problem I’ve been working on. My first unit in the fall is measurement and I wanted something to do with volume.

Minnie’s Juice Cup!

Act 1:

Question: How many juice boxes will fit in the cup?

Act 2:

Make them guess for each of these measurements.

Minnie'sCup

 

I am open to suggestions on how to handle the two different diameters. I tried averaging them and came up with a pretty accurate answer.

JuiceBox-DImension

 

Act 3:

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