Ms. Hagan's Majority Of Exponent Rules

Confession time.  I am terrible at teaching exponent rules.  Correction.  I know how to learn them.  I am terrible at getting students to run across that most of their prior cognition of exponent rules is wrong.  A few weeks ago, I had someone inquire me what superpower I would dear to have.  After thinking virtually it for quite a while, I decided that I would select the ability to hold upwardly able to erase parts of the minds of others.  If I lead hold to accept the fourth dimension each twelvemonth to reteach integer operations, the lodge of operations, too exponent rules to my Algebra 1 students, I would much prefer to learn these to them from scratch.  Because every bit shortly every bit I start reteaching something that they lead hold heard before, their minds near downwards too start ignoring me.  I approximate they are thinking, "I don't lead hold to listen.  I already know this!"  But, the work is that they don't know this.  They think that a negative exponent agency that you lot demand to alter the fraction to its reciprocal to brand the exponents positive.  In about cases, this works.  But, they are overgeneralizing.  They've been told that 2 negatives brand a positive.  So, -3 + (-5) must hold upwardly +8.  Again, they've taken a dominion for multiplication too partition too overgeneralized it.  And, don't fifty-fifty acquire me started on the lodge of operations.  No affair how many times I say that multiplication too partition must hold upwardly performed from left to right, I lead hold a educatee who volition debate alongside me that multiplication comes earlier partition inwards PEMDAS too hence nosotros must e'er create it first.

The same students who lead hold been struggling alongside all of the higher upwardly lead hold been rocking our terminal few lessons on naming polynomials too multiplying polynomials.  Why?  My electrical current theory is that multiplying polynomials is something they've never been exposed to before.  So, they genuinely institute it necessary to remove heed to my explanation...

I know about of you lot volition criticize me for the following.  And, I'm okay alongside that.  I know this isn't perfect.  It definitely isn't ideal.  My teaching of exponent rules this twelvemonth relies on a lot of tricks.  I tried terminal twelvemonth to lead hold my students uncovering the rules for themselves.  I used the amazing worksheets provided past times Don't Panic, The Answer is 42.  We went through each scenario past times itself.  On the production dominion worksheet, my students rocked the production rule.  On the quotient dominion worksheet, my students rocked the quotient rule.  After a calendar week of exploring too discovering each dominion separately, I challenged my students to await at a work too figure out which dominion they were supposed to use.  They were lost.  They could create each dominion inwards isolation, but they couldn't figure out what dominion to purpose inwards a given problem.  I in all likelihood ended upwardly spending 2 weeks on exponent rules, too I even too hence had a grouping of students who merely didn't acquire it.

This year, I spent iii days on exponent rules.

Day 1 - We played a game that I institute on Nathan Kraft's blog.  Without telling the students what nosotros were doing, I told them all to acquire write their yell on the dry out erase board too depict 4 x's below.  First hour, 1 of my students raises their mitt too asks, "Couldn't nosotros lead hold merely written x to the quaternary ability below our names?"  I almost died of happiness inwards that moment.  I approximate my continual emphasis that x squared agency x times x too x cubed agency x times x times x has paid off!

I set a work on the board.  I gave students thirty seconds or too hence to solve it.  They held upwardly their private dry out erase boards alongside their answers.  The students who got it right got to acquire too erase an x from nether someone's name.  On the Smart Board, I demonstrated how to write out the powers inwards the problems every bit multiplication to derive the answer.  We repeated this process.  Slowly, nosotros worked through almost all of the types of exponent problems.  Yes, in that place were about complainers.  "But, you've never showed us how to move out a work that looks similar this.  This isn't fair!"  To this, I told them to endeavour their best.  I believed inwards them!

When a educatee ran out of x's, that educatee became a zombie.  Zombies could even too hence accept others out if they continued to acquire the problems right.  One of my students inwards 3rd menstruum decided from the really kickoff that he wanted to hold upwardly a zombie.  He was practically begging people to erase his x's.  When no 1 would, he started erasing his ain x's.

I called this "The Game of Grudge," too my students loved it.  It sparked too hence many amazing conversations that wouldn't lead hold happened otherwise.  Could nosotros lead hold a negative exponent?  Could nosotros lead hold an exponent on our exponent?  Could you lot heighten pi to a power?  Could you lot heighten pi to the ability of pi?

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

I've been wanting to brand 1 of these books since I learned virtually them during a professional person evolution workshop piece I was educatee teaching.  I've heard them called magic books too poof books.  Basically, you lot accept a canvas of missive of the alphabet sized newspaper too plication it into a cute petty majority alongside the assist of a yoke of pair of scissors too about magic.  Instructions on making the majority tin hold upwardly found here.

Here are our notes inwards the cast of a poof book:    

Exponent Rule Book Cover

This is my re-create of the book, too hence it is titled "Ms. Hagan's Book of Exponent Rules."  My students titled their books alongside their ain names.

Exponent Rules - Page 1 too Page 2

Our firstly 2 pages of the majority characteristic about of import vocabulary.  I needed to brand certain that all of my students knew what nosotros were talking virtually when nosotros mentioned the exponent, base, or power.

Exponent Rules - Page 3 too Page 4

I had never seen exponent rules presented using P->M->A earlier Mrs. D left a comment dorsum inwards Feb on a post I made during my educatee teaching.

Here's what she wrote:

"I am currently educatee teaching. This is what I shared alongside my algebra students. I write P one 1000 Influenza A virus subtype H5N1 downwards the side of a slice of paper.

Product -> (2^3)^4 = 2^(3*4) = 2^12
(draw an arrow downwards to multiply) "look downwards a occupation to retrieve what to create alongside exponents. I run across I demand to multiply them."

Multiply -> 2^3 * 2^4 = 2^(3+4) = 2^7
(draw an arrow downwards to add) "look downwards a occupation to retrieve what to create alongside exponents. I run across I demand to add together them. Also maintain inwards remove heed the bases demand to hold upwardly the same!"

Add -> 2^3 + 2^4
(draw an arrow downwards to... blank space) "look downwards a occupation to retrieve what to create alongside exponents. Wait, there's nil there. I cannot create anything alongside the exponents.""

I changed the P to hateful Power to a Power.  And, I explained it to my students similar this:  The arrow tells us what to create to the exponent rules.  In a ability to a ability problem, the arrow points to multiply, too hence nosotros multiply the exponents.  In a multiplication problem, the arrow points to add, too hence nosotros add together the exponents.  In an add-on problem, the arrow points to nothing, too hence nosotros create nil to the exponents.

One of the things I am determined that my students volition acquire out my classroom knowing this twelvemonth is the give-and-take "vinculum."  It's 1 of those things that I purpose on a daily ground that I didn't know the quest until a twelvemonth or too hence ago.  You know that bar you lot set higher upwardly a repeating decimal?  It's a vinculum.  You know that bar you lot set betwixt the numerator too denominator of a fraction?  It's a vinculum.  You know that top occupation of a radical symbol?  It's a vinculum.  I've emphasized this give-and-take too hence much this year, my 8th graders institute it necessary to right their scientific discipline instructor for non referring to the vinculum past times its proper yell when learning virtually the density equation.  Is this give-and-take critical to my students' success?  No.  I earned a score inwards pure mathematics without knowing what the give-and-take meant.  But, I create think it goes to exhibit my students that they shouldn't hold upwardly scared past times novel vocab words merely because they audio scary.

I learn my students to retrieve that the vinculum looks similar a giant subtraction sign.  Thus, nosotros subtract the exponents when dividing powers alongside similar bases.

Exponent Rules Page v too Page 6 

For negative exponents, I purpose "cross the occupation too alter the sign of the exponent."  We didn't lead hold fourth dimension to explore why this works, but I volition encompass it to a greater extent than inwards depth alongside my students when they achieve Algebra 2.  We likewise discussed why anything raised to the null ability is equal to 1.

Day 3 - Our terminal twenty-four hr menstruum on exponent rules was spent playing the Karuta game from Dont' Panic, The Answer is 42.  I already had the cards cutting too laminated from terminal year, too hence this was an tardily lesson to implement.  I started out past times pairing the students upwardly too having them jibe the exponent dominion query cards alongside the exponent dominion respond cards.  After checking their answers, I had them switch decks too repeat.  After each grouping was finished alongside the matching process, nosotros played the karuta game.

Exponent Rules Karuta Cards

I e'er laminate activities similar this too hence I tin reuse them twelvemonth afterwards twelvemonth afterwards year.  Let's confront it.  Teenagers are crude oil on pretty much anything you lot set inwards their hands.  Here's the laminator I ain too recommend:

Basically, Karuta is a cross betwixt Slap Jack too War.  I tell the students to lay out either the query cards or the respond cards from their decks.  Depending on which cards I had them lay out, I write either an respond or a query on the board.  The firstly soul to slap the right bill of fare that corresponds alongside it gets to maintain the card.  The thespian alongside the most cards at the destination wins.  This game gets really competitive too VERY violent.  

I had a lot to a greater extent than fun teaching exponent rules this twelvemonth than terminal year.  Plus, I'm estimating that I saved 7 days of instructional time.  I think it was a proficient mix of exploring the reasons behind the rules, memorizing the rules, too having fun.    

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