Exponent Rules

I've grown tired of kids blindly following rules.  Mine have the tendency to be the worst because they have always been really good at playing school.  So we went for a different approach to exponent rules this week. 

Prerequisites:  Basic understanding of exponents

Instructions:  Choose a rule that you would like to prove (read: demonstrate why it is a rule).  As you are able to demonstrate an exponent rule, move downstream to the next group and help with that rule finally working on rule #8. 

Here's the list...

...that culminates with the question:

8. What do you think x^1/2 represents?

I'll let them tell you._[1]

_[1]This conversation took about 12 minutes with the camera being shut off a couple of times. I limited the editing so as to try to capture the classroom vibe as naturally as possible. I asked a few questions that I'd like to take back, but... you live and learn.

Proper-tays

I don't usually enjoy teaching properties because they seem so math-y.  I like asking my kids to justify what they do, but for many, the properties just seem to be vocabulary that is forced upon them.  Necessary evil, I guess?  They are great for doing mental math tricks and kids use them without thinking of them, so I suppose there is no harm in giving a name to the stuff they already do. 

Raise your hand if your kids mix up associative and commutative properties? 

No more.

The Process

1.  Give example of property with respect to addition.

ex: Associative: (2 + 3) + 4 = 2 + (3 + 4)

2.  Ask students to write another example of their own. 

3.  Ask for a rule using a, b and c.

4.  Ask students to write down the key characteristics of the associative property in Tweet form. (very few words)

Now here's the kicker:

5.  Can you guess what the property for multiplication is going to look like?

This worked great for the associative, commutative and identity properties.  A great discussion on the inverse property ensued and I ended up telling them that we want a multiplication problem that equals 1. 

Done.


None of these properties are worth anything if we don't apply them. Next step is getting them to put words to all that stuff they "just do in my head."


6.

Let's synthesize this a bit more. 

7. Now write a similar problem using the multiplicative properties. 

I told the class to keep an eye out for times when we will use the inverse and identity properties--which will happen daily once we start solving equations. 

Toy Cars

Prompt:  How far would you have to pull the car back in order to get it to go 100'?

Materials:  Toy cars, meter sticks.

Hand them the cars, ask the question and get out of the way.

Question #1
But Mr. Cox, the farthest we can get the car to go is around 10'.  We can only pull it back so far until it starts clicking.

Right.  So if you could build a car that could be pulled back farther, how far would you need in order for the car to go 100'?


Question #2
Mr Cox, what do we do if our car keeps turning?

Yeah, I'm a cheapskate a father of 5 on a single teacher's income.  Give a guy a break will ya? very thrifty. So what can we do to estimate the distance the car travels?


The two groups that had problems with the car came up with two different solutions. One group decided to tie a piece of string to the spoiler and measured the amount of string the car pulled past the starting line and the other group simply estimated by breaking the curve down into short line segments. (Oh man, do you guys just realized you set me up for a lesson plan in May?  Can you say calculus?)



Question #3


Mr. Cox, if I pull the car back 3", it goes 30", but if she pulls it back 3", it goes 36".  Why?

Turns out that one kid pushed down on the car harder than the other which kept the tires from sliding.



Our Findings
I'm not sure if it is supposed to be or not, but the data was pretty linear.  One student wondered why it would be linear since the car takes time to speed up and slow down and the shorter distance it travels, the more energy it is using to simply get up to speed.



Reason #421 Twitter is awesome

I tweet some pics of the activity and Frank asks me if I'm going to have a contest to see which group can get their car closest to the line.

*ahem* Of course I'm going to have a contest at the end to see who can best predict the distance their car travels.

Three groups were able to get within 1.5". (Two of them were the groups whose cars turned).  The best was this:

Mixed Meta Four

If you haven't read about the Wolverine_[1] (here and here) or Sam's Exasperating Problem, you need to get your priorities straight go read 'em now. Sam's wondering if the problem can be scaled down so a precalculus class can handle it.  I see the problem and think, "GEOGEBRA!  I CAN USE GEOGEBRA!"  Mr. H beat me to it.  (If you haven't seen his applet, I question your dedication to the cause go ahead, we'll wait.)

I love this!  We are always looking for ways to iterate problems and extend them, but there's nothing to extend with this problem.  It's all ready for the wolverine wrangler to do his stuff.  I'm looking for the guy who can make this wolverine sit and quit bearing its teeth so my 8th graders can pet it for a second.  GeoGebra does this.  Mr. H's applet makes this problem accessible to an 8 year old.  In fact, my son was so mesmerized by the animation that I swear I heard him muttering, "Heffalumps and Woozles.  Heffalumps and Woozles."  Heck, I found a strange urge to put on some Pink Floyd myself.

Can you imagine starting a problem in middle school and finishing it with calculus?  That's how beautiful (that's right I said it!) this problem is. Why can't we let these younger kids see the beauty of the wolverine without actually having to be the one to handle it?  I can see posing the problem, setting the kids up with GeoGebra (with minimal prerequisites) and turning them loose.  They'll see the pattern, make a conjecture and inductively decide the answer.  Show the applet which demonstrates the first 360 cases and inevitably, the question will be:

Why?! 

Now, talk about storytelling. The table's been set for the sequel that the kid's gonna have to wait a couple of years to see.  precalculus kids can actually calculate the answer and the trilogy will be complete once they have the tools to actually prove that for n chords, the product is n+1.  This problem can span four years. At least. 

_[1] Apologies if I misused the metaphor.