Language is a funny thing.  I have talked about it before but the use of language and our ability to reason numerically is so interesting.  I had a conversation with a student today where he told me about his dogs. It was one of those off topic conversations.  He was describing the size of his dogs and he said "I have two hundred ten pound dogs."  Now I have intentionally left out any dashes, because I want to let you in on what I understood.  When he said that I thought of this massive group of hundreds of these ten-pound puppies.  He meant he had two dogs that were 110 pounds.

Which brings me to my thought.  In one sentence we can have three different meanings, the likes of which are such.

I have:

1. I have 210 pound dogs
2. I have 200 10 pound dogs
3. I have 2 110 pound dogs*

What does this tell us about the nature of quantity?  They all sound the same but all produce different quantities.  In scenario 1 we have 210x pounds of dogs. We do not know how many dogs I have, but they are all around the same size.  In scenario 2 I have 2000 pounds of dogginess, and in scenario 3 I have 220 pounds of dogs.  In some weird linguistic sense, these seem like they should all be similar in some sense, but they all produce different images, and different quantities entirely.

I do not know why this particular quantity pun amuses me so much, but I feel there is something here.

* Ya, I realise that mathematically we should say two one-hundred-ten pound dogs, but conversationally we rarely say that.

**Another fun quantity pun to ask kids especially is would rather have one and a half million dollars or one million and a half dollars? Something seems eerily the same about those, but they are screamingly different.

Well Geoff and Michael have been bugging me, and I  was also really inspired by Stadel's recent post on exponents that I wanted to quickly share my little intro to exponent rules for my grade eights this year. They (and I) really enjoyed it, and I think it touches a bit on what Dan Meyer is trying to get at with Tiny Math games.  In Grade 8 students do not need to know exponents, but our school wants them to be familiar with them, before grade nine, so I concocted this little activity.

Nothing too fancy here, but basically I printed out this sheet multiple times

I sliced each section, and put them in envelopes at the front of the board titled "Product Rule," "Quotient Rule," "Power Rule," and a^0=?  I told students a few simple directions

1. Follow the directions on the first slip of paper.
2. Complete all directions on your large whiteboard.
3. When you think you have it figured have me come and take a picture.
4. Practice that rule with some basic worksheets.
5. Move to the next rule.

If you can't tell or don't recognize the questions, I was inspired by Exeter.  Reading their problems I realise they have a knack for assessing, and teaching within the context of a clear and concise question.  The question itself pushes students to work through the problem solving in a natural way (even if it is pure math not applied). I worked on these questions for awhile, to make sure that they put the students through the algebraic clarity but they also had a chance to play and punch numbers in their calculator.

(aside) This in my mind is also a less rigorous version of David Cox's fantastic lesson.

Students had a concept of repeated multiplication.  They knew how to evaluate exponents within whole numbers and could hit the "equals" button on their calculator a repeated number of times.  They were not instructed how to find the exponential form of a number (for example they can find 2^3=8 but they had never tried 8=2^3).

Since this only needed to be an overview, and we just needed to touch on these concepts I felt that I didn't need to make sure that everyone learned every rule.  Most students were very proficient with the product rule and quotient rule, and figured that the power rule would somehow involve multiplication of the exponents.

What students LOVED is that they could work through it "at their own pace" (KA buzzwords I KNOW, but still!).  Students struggled through it, but it was set up in a way that was just out of their intuitive reach!  The students who arrived at a^0=1 were very easily convinced of that fact, and because of that, I knew I had a winner.

Sure they know the rules, and that is fine and dandy, but do they know the reasons behind the rules?  I do not teach associative property in grade eight, and therefore reasoning through WHY these concepts work (at this stage, rather than because my teacher said so, it is kind of because my calculator said so), but I think this is only a small drawback, because students are convinced of these rules, and they feel they are natural because they found them.  The rigour can come later.

I will do this again with students, especially as review.  I think it puts them in a state of problem solving that is not to laborious, but also not rote procedural notes.  Tell me what y'all think!