Exponent Rules GAME

I've posted on this topic a bunch of times here, here and here, but I'm not tired yet of hammering more nails into this coffin.  I think that correctly mastering exponent rules is a gateway skill.  Maybe one of the most important gateway skills in algebra.  Exponent rules:

• Formalize the meaning of multiplication and division for algebra.
• Provide the first forum for students to effectively use reducing in an algebraic context.
• Introduces students for the first time to how simple algebra definitions (i.e. the definition of an exponent) can be used to prove a multitude of other cool rules that make doing math easier.  In other words exponent rules formalize the structure of mathematical logic and proof for students.
• Is often the first time students see and manipulate algebraic rules represented purely with variables.  If students can understand and use exponent rules, it prepares them for using and understanding other rules represented with abstract mathematical language. 
• Set the foundation for a student's understanding of polynomial functions, radical functions, exponential functions, and logarithmic functions.  Without a solid understanding of exponents and their properties students will struggle with all of these types of functions later.

And I think we can teach the exponent rules well because they're just not that hard to derive, but the level of abstraction is what makes it difficult for students.  So in teaching exponent rules, I believe we should focus on teaching students the abstraction, and the rules get learned along the way.  

That being said, I don't know how to do it but I keep trying.  I've updated my lesson on developing the exponent rules.  You can find that here and also, I've developed a simple game that I hope helps students cement the rules and learn to play with them.  The game involves both strategy, luck and understanding of exponents so I think it's pretty good but it's only had a few trial runs.

Materials: You'll need a set of blue cards and a set of green cards.  You can download the cards and the rules here.  Lay the cards out like so:

Object: Combine your starting expression with green cards to create the target expression.

Rules:

(1) You may use as many green cards as you wish.

(2) Cards that look like this: (   )^2 must be applied to your whole expression so far.  So if you start with the card “ab" and you grab the green card (   )^2  you will end up with a^2 b^2

(3) If neither player can find the right cards to create the target expression, three more green cards can be put down. 

(4) Once the target expression is reached by a player, that player gets the blue card and all the green cards they used to make the winning expression.  A new blue card is then put down and green cards are added until there are 9 green cards again.

(5) Once all the green cards are gone, the game is ended and the player with the most green and blue cards wins.

Examples:

Here are several examples of how the players in the set-up above could reach the target expression. 

Standards and Pre-Algebra

My husband moved us out to NY so that he could get a physics PhD (I know, I couldn't bring him over to the much more beautiful and elegant world of math.)  He has an Iranian classmate that we've started hanging out with.  The other night he invited us over for dinner with his roommates and friends all of whom are Iranian  and all of whom are either studying physics, mechanical engineering or computer science.  Because most of them have TA-ships and are teaching undergraduate courses, when they found out I was a math teacher they all turned to me and asked a ton of questions along the vein of "why don't American undergrads know any math?!  What DO they learn in high school?"  My husband's friend had been struggling with his undergrads in a physics lab because they couldn't make a simple algebraic substitution (I can't remember what the problem was, but something like if a=b/c and d=2a, then d=2(b/c).  Of course instead of a, b, c, and d they had maybe q with subscripts.)  I asked him if maybe the subscripts had confused them, and he said he went back to simple a, b, c and d variables and they were still stumped.  It took him 2 hours to explain this substitution to these students.  He said they had no sense of variable at all.  They could solve equations by rote, and they had bits and pieces of algebraic techniques, but no logical understanding of what algebra is and why they need to know it for physics.  The other Iranian PhD students chimed in with their own anecdotes of students who have come to college to study the hard sciences with very little mathematical aptitude.  They spent a while discussing how the Iranian education system is much more rigorous compared to what we have in the US.

This is not a low ranked college.  The students who come to Stony Brook University should know their algebra, especially those who want to study the hard sciences and math because it has very competitive science and math departments.  And New York has the Regents.  How can students who passed the grueling Algebra 1, Geometry and Algebra 2/trig Regents exams not know simple substitutions (and not be able to grasp them even when a physics TA comes over and personally explains the process for over an hour?)  With such a small sample and only anecdotes from overworked TAs who aren't trained to teach math  this is not a fair base from which to judge the New York high school math curriculum, but I'm feeling a little judgy at the moment especially after wrestling with the New York math standards and the regents for the first time this year.

Pre-algebra was a sacred class for me at my old school because it creates the base the rest of students' algebra understandings must rest on.  For this reason I went really slowly and carefully in my pre-algebra class and made sure students were really understanding the jump from concrete to abstract mathematics.  I strongly believe that pre-algebra should spend as much time as possible on cementing the ideas of what variables are, how to write expressions, and how equations and formulas are linked to variables and expressions.  These are DIFFICULT ideas.  Students need time to process them.  They need the freedom to explore them in their own ways.  They need to see how variables aren't just unknown numbers- that they're so much richer and more flexible than number- that's why they're so useful in algebra.  Students should spend time observing patterns in variables (specifically, combining like terms and the exponent rules are a great way to do this) and how we can generalize number patterns using variables in simple and elegant ways.  I believe this is what pre-algebra is for.  It's NOT for statistics!  It's NOT for quadratics and FOIL.  It's NOT for re-drilling fractions, decimals, ratios and percents again for the 50th time.  The New York (and Oregon for that matter) standards cram so much into each school year that students don't cement their knowledge or have time to make meaningful connections.  This means that each topic appears in the math standards for at least four years in a row because students have to constantly review stuff they should have learned last year but only "covered" because there wasn't time to go into it in depth.  (i.e. adding and subtracting fractions appears from 5th-9th grades.)   Each topic gets "covered" each year but not taught each year.  So quick students have to relearn the same content year after year, while students who struggle never properly learn it at all.

I know this argument doesn't necessarily have traction.  Students need to review no matter how deeply you taught the material the year before, but I do know that I spent a month on developing variable sense and then another month showing students the usefulness of variables and expressions in writing out general number patterns placing specific emphasis on exponent rules and geometric patterns at my previous school and when the students needed the exponent rules again in algebra 1, we only needed a half-hour review and ALL my students were fluid with using them in very complex situations.  I'm getting algebra 2, pre-calc, and calc students now who don't understand their exponent rules and their eyes glaze over every time I try to show them the logic behind the rules because to them, they're just a random assortment of letters to be memorized when needed and forgotten the rest of the time.  You can't learn differentiation in calc without being able to turn roots and rationals into exponential expressions instead.  This inability to understand that this one seemingly random technique (exponent rules) is rooted deeply in mathematical logic and needs to be understood logically because it is a foundational piece of the structure of algebra I believe is a symptom of the standards push for breadth over depth.  Students have memorized math techniques as a history student memorizes dates.  They may sort of have a sense of order, but no sense of significance.

Variable sense is important and deserves time.  If given time in pre-algebra, students will be much more successful in their higher math classes.  It does not deserve a week a year spread over 4 years.  So to answer the question posed by our Iranian friends on what is wrong with American education, I think it's the standards.  And more specifically, that no one seems to know what should be shoved into pre-algebra so they make it a hodgepodge of random techniques they think will be useful for algebra 1 rather than spending that year to really develop variable sense.  And I am a part of the problem too because I'm correlating my lesson plans to NY state standards so that my students will be able to pass the Regents.  I'm scared of going off in the direction I feel is right because it doesn't cover the "standards" I'm supposed to cover.  I think pre-algebra is the problem and I wish I could go shake the people who put "determine if a relation is a function" and "describe and identify transformations in the plane, using proper function notation (rotations, reflections, translations, and dilations)" on the PRE-ALGEBRA standards.  There's a reason we have a whole year of highschool geometry and two years of algebra.  Give them time to get used to the idea of variable BEFORE rushing them into function translations!

Don't Break the Quotient Rule or Our Friendly Giant will Stick You Between His Couch Cushions that Never get Vacuumed!

I've been meaning to put these up for a while.  My pre-algebra students finished their exponents brochures (description here) and they're kind of awesome so I thought I would post them.  Unfortunately, they did not do as well on the test following the brochures as I would have liked (80% average which is ok, but I wanted better).  I think they performed poorly because even though the brochures helped them cement the basic rules, it didn't help them review how to tackle the different types of problems they might see.  I was trying to avoid doing a drill and kill review worksheet by doing a project instead, but since the test had drill and kill problems, the only way to really prepare them for the test was with a worksheet.  So we're going to spend another few days reviewing exponents and we'll have a retake in a week.
I already started the review.  With our first extra day on exponents, I made a fake test for them that I had "taken" by compiling all the mistakes they made on their first test.  I showed them how I grade tests and then asked them to grade "my" test without an answer key.  I tried to make it a little bit fun in that I said I would award a prize to the student who's final grade was closest to the grade I would have given this test.  Every single student graded the test to within two or three points of the grade I would have given it.  They also found all the mistakes and discussed how frustrating it was when I didn't show my work or circle my answer.  It actually turned into a pretty fun activity because they got to scold me and they were having great discussions about which errors constituted arithmetic errors (which is only -1/2 a point) and which errors constituted understanding errors (which is -1 point).  So clearly they know the material well enough to recognize good work from bad work.  I just need to get them to recognize their own good work and bad work.  When I was wondering aloud about how the same student who got a 65% on his own test could have identified and corrected 100% of my mistakes the very next day, one of our Japanese teachers mentioned that it's easier to understand a language that you're studying than it is to speak it.  So I need to give them more speaking practice?  Maybe I've been focusing too much on error correction.  I decided to make one of my teaching goals this year to help kids learn to recognize their own errors and I guess I went a little overboard.  I have a nice, long drill and kill practice test lined up for them next.
Anyway, without further ado, here are my students' "exponent planet" brochures.

Front of brochure

Inside of brochure

Here's another one 

Front

Inside (The pictures are adorable)

And just one more because I can't resist the math puns this student used

Don't Break the Product Rule or you'll be Slathered in Butter and Grilled!

My Pre-algebra class is reaching the end of their exponent unit.  I blogged about this unit before, but it was one of my first posts so I'm sorry about the roughness (I was also a little scribd happy because I was excited by how easy it was to throw documents right into my post.  It's magical!) I still feel like this unit is really important because really internalizing the exponent rules makes for a much smoother transition into algebra 2 and beyond, but after reading around a lot on other people's blogs, I'm not sure that stomping along through all the rules in order is the best way to teach them.  I also am aware that in the real world students will never have to simplify these ridiculous exponent problems. Though I still think that understanding these rules is necessary in creating a foundation for high school math and is a good way to introduce the logical system of math, I'm a little uncomfortable with how hard it is to tie them to the real world.  I ended the unit with exponential growth and decay and scientific notation which use the exponent rules in context, but I still wish I had a more concrete way to make these rules relevant to students.

I've been trying to think of a way to review what we learned throughout the whole unit.  Last year, I had the students just do a poster project where they had to neatly and creatively demonstrate all the rules, but I think that was just a desperate attempt on my part to have them review the material without adding a whole bunch more prep work on my part because I was swamped.  This year I created this review activity for them:

Exponent Unit 'Going on Vacation' Review Project After reading around so many blogs and seeing what the larger teaching community is doing, I've realized that even the activities I'm most proud of are lacking the real world context that has been stressed by so many other bloggers.  I've stressed teaching the logic of mathematics to my students because that is what is beautiful about math to me, but my students probably need more context driven activities and examples.  The problem is that my education in mathematics has been entirely traditional (i.e. contextless) and I don't think about applications and I don't interpret the "real world" through math.  I don't know how to see math in the world around me.  Yet I guess, just as we tend to repeat our parent's mistakes, it's so easy to teach the way I was taught and to focus strictly on logic.  I've realized that this is a grave deficiency in my teaching that I need to learn to correct, but changing the way I think about mathematics is going take a lot of time.  At least I'm pretty good at making fun and silly math assignments, even if they're not tied to context.  Two months ago when I came up with the idea for this project I was pretty proud of it.  Now I realize that it doesn't give students any deeper insights into math.  It will be silly and engaging I think, but I need to do better.

Exponent Rules Unit

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.