0 Ms. Hagan's Book of Exponent Rules

Confession time.  I am terrible at teaching exponent rules.  Correction.  I know how to teach them.  I am terrible at getting students to see that most of their prior knowledge of exponent rules is wrong.  A few weeks ago, I had someone ask me what superpower I would love to have.  After thinking about it for quite a while, I decided that I would choose the power to be able to erase parts of the minds of others.  If I have to take the time each year to reteach integer operations, the order of operations, and exponent rules to my Algebra 1 students, I would much prefer to teach these to them from scratch.  Because as soon as I start reteaching something that they have heard before, their minds shut down and start ignoring me.  I guess they are thinking, "I don't have to listen.  I already know this!"  But, the problem is that they don't know this.  They think that a negative exponent means that you need to change the fraction to its reciprocal to make the exponents positive.  In some cases, this works.  But, they are overgeneralizing.  They've been told that two negatives make a positive.  So, -3 + (-5) must be +8.  Again, they've taken a rule for multiplication and division and overgeneralized it.  And, don't even get me started on the order of operations.  No matter how many times I say that multiplication and division must be performed from left to right, I have a student who will argue with me that multiplication comes before division in PEMDAS so we must always do it first.

The same students who have been struggling with all of the above have been rocking our last few lessons on naming polynomials and multiplying polynomials.  Why?  My current theory is that multiplying polynomials is something they've never been exposed to before.  So, they actually found it necessary to listen to my explanation...

I know some of you will criticize me for the following.  And, I'm okay with that.  I know this isn't perfect.  It definitely isn't ideal.  My teaching of exponent rules this year relies on a lot of tricks.  I tried last year to have my students discover the rules for themselves.  I used the amazing worksheets provided by Don't Panic, The Answer is 42.  We went through each scenario by itself.  On the product rule worksheet, my students rocked the product rule.  On the quotient rule worksheet, my students rocked the quotient rule.  After a week of exploring and discovering each rule separately, I challenged my students to look at a problem and figure out which rule they were supposed to use.  They were lost.  They could do each rule in isolation, but they couldn't figure out what rule to use in a given problem.  I probably ended up spending two weeks on exponent rules, and I still had a group of students who just didn't get it.

This year, I spent three days on exponent rules.

Day 1 - We played a game that I found on Nathan Kraft's blog.  Without telling the students what we were doing, I told them all to go write their name on the dry erase board and draw four x's below.  First hour, one of my students raises their hand and asks, "Couldn't we have just written x to the fourth power below our names?"  I almost died of happiness in that moment.  I guess my continual emphasis that x squared means x times x and x cubed means x times x times x has paid off!

I put a problem on the board.  I gave students 30 seconds or so to solve it.  They held up their individual dry erase boards with their answers.  The students who got it right got to go and erase an x from under someone's name.  On the Smart Board, I demonstrated how to write out the powers in the problems as multiplication to derive the answer.  We repeated this process.  Slowly, we worked through almost all of the types of exponent problems.  Yes, there were some complainers.  "But, you've never showed us how to work out a problem that looks like this.  This isn't fair!"  To this, I told them to try their best.  I believed in them!

When a student ran out of x's, that student became a zombie.  Zombies could still take others out if they continued to get the problems right.  One of my students in third period decided from the very beginning that he wanted to be a zombie.  He was practically begging people to erase his x's.  When no one would, he started erasing his own x's.

I called this "The Game of Grudge," and my students loved it.  It sparked so many amazing conversations that wouldn't have happened otherwise.  Could we have a negative exponent?  Could we have an exponent on our exponent?  Could you raise pi to a power?  Could you raise pi to the power of pi?

Day 2 - The students wanted to know if we were going to play the game again.  They were quite devastated when I told them we would be taking notes.

I've been wanting to make one of these books since I learned about them during a professional development workshop while I was student teaching.  I've heard them called magic books and poof books.  Basically, you take a sheet of letter sized paper and fold it into a cute little book with the help of a pair of scissors and some magic.  Instructions on making the book can be found here.

Here are our notes in the form of a poof book:    

Exponent Rule Book Cover

This is my copy of the book, so it is titled "Ms. Hagan's Book of Exponent Rules."  My students titled their books with their own names.

Exponent Rules - Page 1 and Page 2

Our first two pages of the book feature some important vocabulary.  I needed to make sure that all of my students knew what we were talking about when we mentioned the exponent, base, or power.

Exponent Rules - Page 3 and Page 4

I had never seen exponent rules presented using P->M->A before Mrs. D left a comment back in February on a post I made during my student teaching.

Here's what she wrote:

"I am currently student teaching. This is what I shared with my algebra students. I write P M A down the side of a piece of paper.

Product -> (2^3)^4 = 2^(3*4) = 2^12
(draw an arrow down to multiply) "look down a line to remember what to do with exponents. I see I need to multiply them."

Multiply -> 2^3 * 2^4 = 2^(3+4) = 2^7
(draw an arrow down to add) "look down a line to remember what to do with exponents. I see I need to add them. Also keep in mind the bases need to be the same!"

Add -> 2^3 + 2^4
(draw an arrow down to... blank space) "look down a line to remember what to do with exponents. Wait, there's nothing there. I cannot do anything with the exponents.""

I changed the P to mean Power to a Power.  And, I explained it to my students like this:  The arrow tells us what to do to the exponent rules.  In a power to a power problem, the arrow points to multiply, so we multiply the exponents.  In a multiplication problem, the arrow points to add, so we add the exponents.  In an addition problem, the arrow points to nothing, so we do nothing to the exponents.

One of the things I am determined that my students will leave my classroom knowing this year is the word "vinculum."  It's one of those things that I use on a daily basis that I didn't know the name for until a year or so ago.  You know that bar you put above a repeating decimal?  It's a vinculum.  You know that bar you put between the numerator and denominator of a fraction?  It's a vinculum.  You know that top line of a radical symbol?  It's a vinculum.  I've emphasized this word so much this year, my eighth graders found it necessary to correct their science teacher for not referring to the vinculum by its proper name when learning about the density equation.  Is this word critical to my students' success?  No.  I earned a degree in pure mathematics without knowing what the word meant.  But, I do think it goes to show my students that they shouldn't be scared by new vocab words just because they sound scary.

I teach my students to remember that the vinculum looks like a giant subtraction sign.  Thus, we subtract the exponents when dividing powers with like bases.

Exponent Rules Page 5 and Page 6 

For negative exponents, I use "cross the line and change the sign of the exponent."  We didn't have time to explore why this works, but I will cover it more in depth with my students when they reach Algebra 2.  We also discussed why anything raised to the zero power is equal to 1.

Day 3 - Our last day on exponent rules was spent playing the Karuta game from Dont' Panic, The Answer is 42.  I already had the cards cut and laminated from last year, so this was an easy lesson to implement.  I started out by pairing the students up and having them match the exponent rule question cards with the exponent rule answer cards.  After checking their answers, I had them switch decks and repeat.  After each group was finished with the matching process, we played the karuta game.

Exponent Rules Karuta Cards

Basically, Karuta is a cross between Slap Jack and War.  I tell the students to lay out either the question cards or the answer cards from their decks.  Depending on which cards I had them lay out, I write either an answer or a question on the board.  The first person to slap the correct card that corresponds with it gets to keep the card.  The player with the most cards at the end wins.  This game gets very competitive and VERY violent.  

I had a lot more fun teaching exponent rules this year than last year.  Plus, I'm estimating that I saved seven days of instructional time.  I think it was a good mix of exploring the reasons behind the rules, memorizing the rules, and having fun.