Thread by @realJ_Mitchell

Wanna play a game?
It’s called “Break Up 25”
It goes like this…break 25 up into some (positive) numbers that add to 25.
10+10+5 = 25
Change the + to *
10*10*5 = 500
That’s my score.
Can you beat my score?
#ThinkingClassroom
#OntEd
#ITeachMath

Seriously. Try it.

Then I’ll tell you how my students did.

If you are trying…stop here. Spoilers below.

You’ve been warned.
I learned this task from @AlexOverwijk (who probably learned it from someone else)
It’s one of my top five thinking tasks.
You’ll see why.

Gave this exactly as I wrote it to students. Sent them to their boards in random groups of three.
Some initial discussions.
“Let’s just do a bunch of 1s!”
I head to that group to show them the set up again.
Soon…”We got 1386!”
I make a big deal of announcing the new high score.

“Can we use more than 3 numbers?”
“I said ‘some numbers’ I didn’t say how many”
Grumbling. Thinking.
“We did it! 3125! We win”
“Neat. How did you do that?”
“We used five fives!”
“Oh cool, so you are using the power of exponential growth?”
Fist bumps are exchanged and then…

“We got 4096!”
Groups are competing now.
“We broke 25 up into as many 2s as we could.”
One group tries using only 3s (and one 1). Bang. New high score!
Some groups start mixing and matching 2s and 3s.
“Can we use decimals?”
“I said some numbers I didn’t say what kind of number”

Quickly, one group does 2.5 ten times.
The high score rockets above 9000.
Groups are busy finding other ways to break 25 up using decimal parts.

Game over.
“We got it. 9867.344”
“How did you do that?”
“I figured out an equation to graph in @Desmos”
And my heart is full because all semester I’ve been asking students to verify their work with a graphing calculator.
This is a feature not a bug!

I pull the class together at the groups board. We talk about the formula.
“You broke 25 up into 9.197 equally sized pieces. How big is each piece?”
“2.7182”
This is a big deal but they don’t know it yet.
“Ok let’s practice that formula. Tell me the high score if 36 is the number”

This goes fast.
“Hey, how big was each piece this time?”
“2.7182”
“Are you tricking us?”
“What’s happening here”
Me:

“This is weird. Try it again with a smaller number. Do…maybe a prime? Try 13 or 7!”
“It’s 2.7182 again!?”
We pause. One group has googled the number.
“What’s Euler’s Number?”
We talk.
It’s neat.
It’s been about 20 minutes since class started and it’s Calculus so I’m not done.

“Head to your boards. Graph y=e^x”
I have a plan.
Earlier the class learned how to make a moveable tangent line in @Desmos
“Use your skills to determine the slope of this new curve at various points”

“This is weird. The slope of this curve is the same as the y value of the curve?”
“Two weird things in one period?! What do you think this means?”
And just like that we’ve discovered a new derivative rule.

But I’m not done.
“What happens if we make this function more complicated? What if the exponent is 2x or 3x or the exponent is quadratic?”
Kids keep playing with Desmos and noticing more patterns.

And this was my secret goal…to find a way to introduce the Chain Rule.
I wish I grabbed an image of the classes final work. Cause we got there! Using tables and technology students found rules to derive e^x and e^(g(x)).

75 minutes.
Introduced Euler’s Number.
Introduced the Exponential Function.
Discovered the derivative rule for e^x.
Introduced the Chain Rule.

The thinking classroom pedagogy gives you the room to make big leaps.