I feel like getting up onto my little soap box. I don't know how justified my strong views of this subject are because I haven't ever talked to other math teachers about this, but in my limited experience I feel like my views are reasonable and therefore my rant will have some bearing on reality, maybe? Exponents. I love them. But when I first came to work at my current school, none of my high schoolers knew their exponent rules. I figured maybe we could just pick them up as we stumbled our way through algebra 2, pre-calculus and calculus. Boy was I wrong. Exponent rules came up all the time, in almost every topic we studied, and I tried to show the students how to use them because to me, they're pretty easy. If you expand the expressions, the rules just become clear. But the students forgot the rules, mixed them up, could never remember what a negative or rational exponent meant and for the life of me I couldn't get them to think about why the rules existed. They would not simply expand the exponent and play with the expression until it simplified itself. So when exponents came up in our pre-algebra textbook I was thrilled. A chance to nip this problem in the bud. If I could really rock exponents in pre-algebra, my students would not have these difficulties as they moved up through the classes. So I threw myself into planning the perfect exponent unit.
The interesting thing is that I couldn't find exponent rules on our state standards, nor really in other pre-algebra textbooks, and I was having a lot of trouble finding resources online. When a lot of people cover them, they cover them all at once: here's this list of 8 different rules to memorize. A lot of the resources out there did show demonstrations of why the rules work, but I kept feeling like I didn't just want to give my students the rules because they would only remember the rules, not the proofs. My high schoolers love rules. They love being able to just plug something in. The problem is that there are too many rules and they were mixing them all up and they didn't really care that they mixed them up. They had an answer that they got with a rule- presto- math is done!
I feel that teaching the exponent rules slowly and well in pre-algebra is a bridge to successful math in high school. We do this unit right after a basic algebra unit so it's a nice way to apply some of the skills they learned in algebra. Teaching them the proofs of the rules is a nice way to show them how math is built on a logical, proof based system where everything has a reason and we need to seek out that reason to fully understand the math. It cements their understanding of negative numbers because we have to play with negative exponents. And finally, they learn the exponent rules which in my experience are really REALLY important for success in high school math. This unit should do everything a math unit is supposed to do- draw in a ton of prior knowledge and show students how that stuff we studied before was useful, show them the logical foundations of math, and give them something new and cool to play with. So why did there seem to be so little emphasis on exponent rules in state standards and textbooks (except for our book of course, impact mathematics, which is awesome but really hard to teach from). Maybe it's just Oregon that doesn't give these rules the respect due to them?
Anyway, I'd like to share some of the stuff I've come up with to teach this unit. This is my third time through and I think I'm getting better, but I'm still not satisfied that I have achieved my goals of the perfect exponent unit (I know its not possible, but I still want to rock it. That is my ultimate goal.) First of all, here's the document I drew up to help the students explore all the different rules: Exponent Rules Proof Worksheets
We do one of these a day. I have different ways of presenting these to the students. Sometimes I had them pair up and work through the sheet, sometimes I had them do it silently and alone, then pair up to compare answers. Sometimes I timed them and gave them 1 minute to fill out a column, then they switched partners and had another minute to do the next column. When they finished they cut out the box and taped it into their notebooks. I think the students enjoyed it, and the best part was the sense of excitement I felt around the room as they discussed what was going to happen. They seemed to enjoy the process of creating these rules for themselves. After we do one of these worksheets, we discuss the rule, run through examples, then I give them practice problems from the book. This is the lamer part of the unit, but they do need to just practice it right?
I have another few documents to share. Before they start work on the quotient rules, I give them this worksheet I made to introduce canceling. It follows a discussion we have as a class as to why we can say 1/2 is the same as 2/4 or as 3/6. I keep pushing them until someone says that they're the same because we're multiplying by 1 and multiplying by 1 doesn't change a number. Then it's pretty easy to show them how canceling works once they've made that connection.
Here's the canceling worksheet: Canceling Worksheet
As they add more rules on, the students tend to start mixing them up, so I've made a few games to try to help them practice using the different rules and practice combining them. The first one is a simple karuta (this is the Japanese name, I don't know an English name) game. The students group into 3s or 4s and each group gets a deck of cards. First, they need to just pair the answers to the questions, and we can discuss the harder problems. Then they put the question cards to the side and spread the answer cards in front of them. I put a problem up on the board and the first student to grab the correct answer card gets to keep it. The person with the most cards at the end wins. We then play again, but this time with the question cards. I write the answer on the board and they have to find the question that matches it.
Here are the game cards: Exponent Rules Karuta
This is a game I thought the students wouldn't like that much, but I thought it could be good at helping them analyze each others mistakes and pick out common mistakes. I called it the Power Rangers game because I do it right after they learn the power rule. Surprisingly, the kids LOVED it. Every day they ask if they can play it again. Maybe they liked it so much because I did play the intro song of the power rangers TV show before we played. I had them form groups of 3. Then of those three, one had to be a power ranger. The power rangers came to the front and practiced some exponent problems. In the meantime, the group members left behind were the evil nemeses of the power rangers. Their job was to solve three exponent problems incorrectly on purpose. Furthermore, because they wanted to foil the power rangers, they needed to make their answers look correct so that the power ranger would have trouble finding the error. Once the nemeses work was done, the power rangers came back to their groups and they had to find the mistakes in the nemeses problems, then explain the to the nemeses why their answer was wrong. The nemeses got to grade their power ranger on finding the mistakes and the clarity of their explanations. I don't know how educational it was, but my students keep demanding to play it again.
Here are the documents I used to play the Power Rangers game: Power Rangers Game
Finally, I started giving them quizzes to make sure they have the rules down. From the first quiz I pulled out the problems students had the hardest time with. I put the problems, along with students' incorrect answers on note cards. I gave one note card to each student in class and asked them to tell me if the answer on the card is correct or incorrect, and if it's incorrect, explain what mistake the student made. The students rotate cards and do this for about 10 incorrect problems (I don't tell them that all the cards are incorrect.) Then they pair up and compare answers. Then the pairs pair up and compare answers. We keep doing this until students realize that all the answers were incorrect and we discuss common errors as a class.
Here's the first quiz I give: Exponents Quiz 1
Here's the second quiz I give: Exponents Quiz 2
So far, this unit has gone better than last year. My students got an 87% average on their first exponents quiz, but there are still some students below the 80% mark and I want them all above it so that our next topics- exponential growth and decay and scientific notation- go smoothly. I think I have a pretty cool project to finish the unit with, but I'll post that once we get there.
If anyone has any suggestions, I would love to have other games, resources or ideas to throw into this unit. I know it's not quite rockin' yet.