Math = Love: November 2013

Saturday, November 30, 2013

Cuboctahedrons: A Perplexing Polyhedron Probability Problem

Due to the Thanksgiving holiday, we only had two days of school this week.  Needless to say, my students were not very excited about having to come to school on Monday and Tuesday.  Several of the schools around us took the entire week off, so that made the week seem even more torturous to my students.

My statistics students were especially restless.  We're in the middle of our unit on probability.  Monday, we looked at some probability word problems.  On Tuesday, I wanted to do something fun and interesting but still related to probability.  I did a quick google search of probability activities, and I ran across a net of a cuboctahedron.  Isn't that just a fun word?  Cuboctahedron.  Cuboctahedron.  Cuboctahedron.  It just makes me smile.

The activity instructed students to assemble their own cuboctahedron.  (The net is on page 4 of the linked PDF document.  I'm also intrigued by the probability activity on page 5 that involves acting out a Russian fable that predicts who will get married within the next year.)  Then, they were to toss the cuboctahedron 100 times and count how many times it landed on a square face and how many times it landed on a triangular face.

Assembled Cuboctahedron and Net Pattern
I let my students each pick a sheet of card stock from the cabinet, and I quickly ran off nets for them to cut out and assemble.  The cutting and gluing process was more time intensive than I realized.  This activity took the entire 50-minute period.  Since it was the day before Thanksgiving break, this was perfectly fine.  Most of my students ended up opting for tape because the net was so hard to put together.  I used glue, and it works fine if you have enough patience to let the glue dry a little between steps.

My Class' Finished Cuboctahedrons 
The cuboctahedron consists of 6 square faces and 8 triangular faces.  Students were asked to predict the probability that the cuboctahedron would land on each type of face BEFORE tossing it 100 times.

As a class, they decided that 6/14 of the faces were squares.  Therefore, the probability of landing on a square face was approximately 0.43.  8/14 of the faces were triangles.  Thus, the probability of landing on a triangular face was approximately 0.57.  '

Our Class Data 
In 299 trials, the cuboctahedron landed on a square face.  In 91 trials, the cuboctahedron landed on a triangular face.  So, the experimental probability of landing on a square face was approximately 0.77, and the experimental probability of landing on a triangular face was approximately 0.23.

My students were intrigued by this data.  I'm not sure what the authors' motivation was in writing this activity.  Were we supposed to get these surprising results?  We had a discussion of the difference between theoretical and experimental probability.  What is the reason behind this discrepancy?  Is it related to the differing areas of the faces?  Or, is it as one student suggested related to the way that the cuboctahedron lands?  It often hits on a corner, and when this happens it almost always favors the square faces for landing.

I liked this activity because it got my students thinking and talking about math on a day when they didn't feel like doing any math.  It's a rare thing when I give my students a problem I don't already know the answer to.  I need to do this more often!  Does anyone know more about this perplexing polyhedron probability problem?  

Friday, November 29, 2013

Radical Radicals

We are currently learning about radicals in Algebra 2.  I am loving this unit!  I'm building this unit off of a short unit I did on radicals with my Algebra 1 kiddos towards the end of last year.  After all, there's no need to reinvent the wheel...  We're still working on this unit, so there will be more pages posted sometime in the future.

Radical Functions Table of Contents
I'm doing something else radical with this unit on radicals.  I'm trying out SBG.  To be honest, I was disgusted with my Algebra 2 students' performance on the previous unit.  I have quite a few students who CANNOT factor a trinomial.  They bombed their last test.  But, they are still passing my class.  Therefore, they have no motivation to actually learn to factor.  I've always loved the idea of standards based grading, but I've always written it off before as too time consuming.

I take it all back.  Do you know what is too time consuming?  Spending almost an entire month on a unit and having students not master key concepts because your grading system tells them they don't have to.  Grades in my class have become almost meaningless.  I hate it.  When I look at a students' test score, I don't know what they know well and what they don't know at all.  I don't know if they completely mastered one topic and left another topic blank.  Or, maybe they made little mistakes on all of the topics.

It's gotten to the point where I hate to grade.  I let the pile of papers to grade sit there, growing, all week long.  Eventually, I bite the bullet and have a marathon grading session.  When I pass back papers and tests, very few students look to learn from their mistakes.  I hear other teachers talk about writing comments all over their students' papers.  I don't even do that.  Why?  I know they won't read them.

In a desperate measure, I threw this together in about an hour.  I wrote 6 learning goals for our unit on radicals.  I decided to grade on a scale from 0-4.  Is this the best scale?  I have no clue.  This was totally last minute.  Once a student achieves a 4 on a learning goal, they are exempt from questions that cover that learning goal on all future quizzes.

I had my students glue in a score tracking sheet on the page facing their table of contents for Unit 4.  Every day, I pass back the previous day's quiz.  Students update the sheet in their notebook.  Then, I review the questions students have questions on.  I like that this new method is forcing me to teach in layers, not lumps.

If a student doesn't get a 4, they immediately start looking for their mistake.  I have rarely been writing comments on the quizzes.  I just put a number.  It makes students think.  Where did I go wrong?  How can I avoid making this mistake on today's quiz?  They get mad at me when I give them a 3 for a tiny, tiny mistake.  But, by forcing them to retake the quiz, I am ensuring that they will never make that mistake again.

This system is a work in progress.  I'm liking it so far.  And, my students are liking it, too.  I still have a handful of students who aren't trying on the quizzes.  I'm not so sure what to do about that.  The only complaint from students is that now they can't get their name on the Star Student Bulletin Board for making an 85 or above on our unit tests since the quizzes are replacing the unit test.

I love looking through my grade book.  I can scroll down the columns and instantly know who needs work with each aspect of the chapter.  I'm still trying to figure out how to do homework with sbg.  And, I'm toying with trying out SBG with my Algebra 1 kiddos during our linear functions unit.  Decisions, decisions, decisions...

SBG Tracking Chart and Table of Contents
Before we could delve into simplifying radicals, I needed to refresh my students' memories regarding prime and composite numbers.  We color-coded a hundreds chart to keep in our notebooks for reference.

On the sides of the chart, we wrote definitions for prime and composite numbers.  Two of my students decided we should color our prime number definition the same color as our prime numbers on the chart and likewise for the composite numbers.  Color With A Purpose.  I like it!

Prime and Composite Numbers Chart
We used highlighters that I had ordered from Amazon.  They worked well for this activity.

I'm still in love with the birthday cake method for prime factorization.  I've written before about why I like this method better than factor trees.

The Birthday Cake Method for Finding Prime Factorization 
I typed out the steps for finding the prime factorization for my students to save time.  We also did examples together in our interactive notebooks.

Prime Factorization Birthday Cake Examples
Vocabulary is a very important part of this chapter.  When I started teaching algebra, I didn't know the terms index or radicand.  I'm sure my algebra teacher taught those words to me, but I never had to use them.  I will not let my students go down this same path.  We are constantly talking about the index and radicand!
Parts of a Radical - Index, Radicand, Radical Symbol
For practice, students had to determine the index and radicand of these radicals.  This further emphasizes the necessary vocab for this unit.
Parts of a Radical Examples
I also typed out the steps for simplifying radicals.  Last year, I had students write this out by hand, and it took WAY too long.
Steps for Simplifying Radicals
Then, we did examples of simplifying radicals in our notebooks.
Simplifying Radicals Examples
After simplifying radicals, we moved on to adding and subtracting like radicals.
Adding and Subtracting Like Radicals Notes
And, this was soon followed by multiplying radicals.
Multiplying Radicals Notes
We still have yet to cover dividing radicals, rationalizing the denominator, and converting between radical form and rational exponent form.

PDF Templates and Publisher Templates can be downloaded here.  

Thursday, November 28, 2013

One Variable Inequalities INB Pages (Algebra 1)

My Algebra 1 students just finished up a mini-unit on graphing and solving one variable inequalities.

As usual, every new unit starts with a table of contents.  Last year, I had students keep one table of contents at the beginning of their notebook.  I like the individual unit table of contents SO much better!  Almost all of my students keep them up to date!

Table of Contents
This foldable was inspired by a foldable I saw on Coffee Cups and Lesson Plans.  I started this unit by drawing the different inequality symbols on my Smart Board.  I wrote numbers above each symbol.  Then, I asked a simple question, "Who can tell me which number is above the symbol that means less than?"  Hands shot up in the air.  Some students gave me the correct number.  Others had confused the symbol for less than with the symbol for greater than.  Still others had confused it for the less than or equal to symbol.  One student argued that there was no way of knowing because there were no numbers for the alligator to eat. 

Graphing Inequalities Foldable - Outside

Oh, the alligator.  I remember learning about the alligator in elementary school.  The alligator is a really hungry animal.  He always chooses to eat the largest number of fish possible.  I even did a project for my ed psych class in college where I taught a kindergartner about inequality symbols using the alligator.  Now that I'm teaching algebra, I can't stand the alligator.  The alligator doesn't help my students translate symbols into words.  The alligator holds my students back.  So, I told them that they had to take everything they ever learned about the alligator and throw it out the window.  I thought there was going to be a revolt in my classroom.  One student proclaimed, "That's half of my life!  You can't expect me to just throw away half of my life because you don't like the alligator."       

When I look at an inequality symbol, I just know its name.  I guess I just memorized them somewhere along the way without realizing it.  Last year, one of my college algebra students was having a terrible time remembering which symbol meant less than and which symbol meant greater than.  One day, she had an epiphany.  If you make an L with your hand, it looks like the less than symbol.  Less than starts with L.  This was perfect!  So, making a backwards L with your hand creates the greater than symbol.  If you are familiar with sign language, you could also make an argument that there are similarities between a sign language G and the greater than symbol.  My student who discovered the hand trick was so proud of herself!  Her exact words were "This just made my day!"  And, she never had to ask me about what a symbol meant again.  
Since then, I've been teaching this hand trick to my students.  After a couple of days, they usually get to where they just know what they symbol is without thinking about Ls and backwards Ls.    

Graphing Inequalities Foldable - Inside

I had my students create the examples for each inequality symbol.  I would let one student pick the variable and another student pick the constant.  Some students were pretty mad at me because they didn't get a chance to pick a variable for the class.  We graphed each example and talked about the difference between open circles and closed circles.  My favorite discussion was on how to graph s is not equal to 3.  

I teach my students that the order you write an inequality in matters.  Does it really?  No.  However, on standardized tests, inequalities are always (almost always?) written with the variable first and constant last.  Of course, compound inequalities are the exception to this rule.  But, more about them later.  As soon as I passed out this half-sheet of paper, I had the full attention and curiosity of my students.  Why are there flip flops on my paper?  Don't you realize it's winter?  

I know, it's a little corny, but I told that "If you have to flip flop the sides of the inequality (to achieve the correct order), you must also flip flop the inequality symbol."  They affectionately called this the "Flip Flop Rule."  

Inequalities INB Page: Order Matters - The Flip Flop Rule

This next page was attempt to modify the page I created last year for graphing two-variable inequalities.  I don't like it quite as well, but I guess it did the job.

This probably isn't the best thing to admit, but I didn't teach my students why it was necessary to flip the inequality symbol when you multiply or divide by a negative last year.  I mean, I taught them to do it.  I just never explained why.  This year, one of my Algebra 2 students asked.  I wrote 2 < 4 on the Smart Board.  Then, I divided both sides by -1.  If we keep the inequality symbol the same, we get -2 < -4 which is a false statement.  Light bulbs went off.  It was a beautiful sight.  I am constantly becoming a better teacher.  I've heard before that experience is the best teacher.  I don't think I quite realized how true that is until I started teaching.  

This year, I wrote 5, a large space, and a 7 on the board.  Then, I asked, "Who can tell me what symbol should go between these two numbers?"  The class agreed that a less than symbol belonged in the middle.  Next, I asked for volunteers.  Tell me something we could do to both sides of this inequality.  Add 2.  So, we added 2 to both sides.  Guess what?  The inequality symbol is still true.  What else could we do to both sides of this inequality?  Subtract 7.  The inequality symbol is still correct.  Give me something else we could do.  Multiply by 2.  The inequality symbol is STILL correct.  By now, my students were convinced that the inequality symbol would always remain the same.  So, I issued them a challenge.  The first person to come up with an operation that would require us to change the inequality symbol would win a Tootsie Pop.  I bribe my students with A LOT of candy...

It took quite a while to come up with multiplying or dividing by a negative, but I like to think it was time extremely well spent.  I must have still been half-asleep when I typed up these notes because I called it The Golden Rule of Inequalities: Whenever you multiply or divide both sides of an inequality by a negative number, you must flip the inequality symbol.  In retrospect, the name makes no sense.  My kiddos didn't seem to notice, though.

Solving and Graphing Inequalities in One Variable
The last part of our unit focused on compound inequalities.  I used the topic of Christmas presents to motivate the difference between AND and OR Compound Inequalities.  After completing our bellwork, I sat down on my stool at the front of the classroom and announced, "Today we are going to talk about Christmas."  I called on a student and asked them what was on their wishlist for Santa.  There were snickers.  I guess high school students are too cool for Santa Claus...

The first time I did this, I lucked out.  The first student I called on said that he wanted a PS4 or an X-Box One for Christmas.  I became incredibly excited and started peppering him with questions.  What gaming system do you currently have?  If you get this new gaming system, will you have to get all new games or will your old games still work?  Finally, I asked him which system he was leaning towards.  "So, all you want for Christmas is an X-Box One?"  "You'd be happy if the only thing you got for Christmas was an X-Box One."  See what I'm building towards here?

Does the gaming system do you any good if you don't have any games to play?  So, I told him that what he really wanted was an X-Box One AND some games for Christmas.  Another student said my statement should actually say "I want an X-Box One, some games, AND some controllers for Christmas."  If you get the gaming system but no games, you will be disappointed.  If you get the games but no gaming system, you will be really disappointed.  The second time I did this during the day, I had to ask 3-4 students before I found someone that wanted a new gaming system for Christmas.  With my last class of the day, I asked almost every single person in the class before I found someone who mentioned video games.  That was a shocker!

Then, I wrote an OR statement on the board.  I want an X-Box One or a PS4 for Christmas.  We had a similar conversation.  If this was your wish, would you be happy if you found an X-Box One under the tree?  Would you be happy if you found a PS4 under the tree?  What if you found both under the tree?  With an AND statement, you had to get both of your wishes to be happy.  With an OR statement, you only have to get one of your wishes to be happy.  If you get both of them, you will just be ecstatic.  This was the perfect transition for looking at AND and OR inequalities.

Compound Inequalities Foldable - Outside

The inside of this foldable didn't photograph the best where I wrote in pencil on this dark purple paper.  Sorry about that!

Compound Inequalities Foldable - Inside

Files are uploaded here!  

Wednesday, November 27, 2013

Two Cubed Times Three: How a Math Teacher Celebrates Her Birthday

Monday, I was blessed with the opportunity to do a rare thing: celebrate my birthday on a school day!  My birthday almost always falls on Thanksgiving Break or the weekend.  So, I can count on one hand the number of times that I've been able to celebrate at school.  This was my first birthday celebration at school as a teacher, and I was pretty excited.

I had been talking up my birthday for weeks.  Well, I guess it was actually more of a lament.  I lamented the fact that my age was no longer going to be a prime number.  In fact, I was going to have to spend the next 1,826 days as a prime number.  There were two typical responses.  Response 1: What is a prime number?  This response was quickly followed by an "I'm so glad you asked!" and a short math lesson.  Alternately, Response 2: You love math way too much.  To this, I just smiled.  It's true.  I do love math.  And, I'm glad my students can see my passion for the subject through my words and actions.

The celebration started a little early on the Friday before my birthday.  I found sweet messages like this one written on my dry erase board.

A sweet birthday wish...

On the morning of my birthday, I changed the date and holiday to match the occasion.
My Mathematical Birthday Celebration

I was quickly informed that my monthly celebration was a tad too restrictive.  Apparently, other people are born in November, too.   After going to the teacher's lounge to make some copies for the day, I returned to find a lovely birthday surprise. 

Birthday Cake!

A birthday cake!  The giver was so sweet and apologetic: "I got you a birthday cake!  They left off the Miss and misspelled your name.  I'm so sorry, but I got you a cake!"

Here's a picture of me and my beautiful, thoughtful cake.  

A student brought me a birthday cake!

See the birthday hat?  I bought it myself.  :)  I went into Family Dollar on Sunday night, and I bought 3 bags of candy and a birthday hat.  The cashier asked, "Are you going to a party tonight?"  I just smiled and said, "Not quite."  The candy was for my students to eat on my birthday.  The birthday hat was for me to wear on my birthday.  I've stopped even trying to explain my random purchases to cashiers.  I've decided that weird looks from cashiers are just a part of being a teacher.  

Soon, I was greeted by two more students bearing another birthday cake!  My students sure know how to make someone feel special!  
Homemade Birthday Cake - Complete with a Math Problem for My Age

See the math problem that equals my age?  I have trained my students well!  :)  And, the decoration above the word "Happy" is a number line.  We've been graphing one variable inequalities, so this was especially fitting.  (Though, I didn't realize it was a number line.  I thought it was one of those markings that you would find on a football.  Of course, it wouldn't make any sense to put that on MY cake.  I didn't figure out it was a number line until the girl who drew it was bragging about the awesome number line she had drawn on my cake to her friends.  Oh, that's what that was.  There weren't any numbers on it, so I couldn't tell.  The longer dash in the middle of the number line was supposed to give it away.  Oops...)  

The second birthday cake also came with a card and a "Birthday Girl" ribbon.  I was pretty excited about this!  

My Birthday Girl Ribbon - I was so proud of this!

A Happy Birthday Card

Of course, the best way to celebrate your birthday is mathematically themed birthday bellwork.  My Algebra 1 students have been working on compound inequalities.  So, they found this problem when they entered my classroom:  Someone who doesn't know Ms. Hagan well enough to know that today is her 24th birthday guesses that she is less than 30 and greater than or equal to 22.  Write this statement as a single inequality, if possible.  Is it an "and" inequality or an "or" inequality?  Then, graph this inequality on a number line.

Algebra 1 - Birthday Bellwork - Compound Inequalities

This problem ended up leading to some great discussion!  Half the class was convinced it was an "or" inequality, and the other half was equally convinced it was an "and" inequality.  It was fun to see them realize that both "and" and "or" could be found in the problem. I love using the highlighters on the Smart Board to graph the intersection of two inequalities.    

We spent the rest of the hour reviewing inequalities for our test the next day.  One of my students decided to make fun of my birthday hat.  "Are you sixteen?  Because that's what sixteen year olds wear on their birthday."  I'm pretty proud of my response.  "Well, the package said "Ages 3 and Up" when I bought it.  Why don't we practice writing this as an inequality?"  

Inequalities to the Rescue!  

And, just so you know, my age is in the solution set for this inequality!  

My Algebra 2 students have been working with radicals.  We've been using the birthday cake method to find the prime factorization of the radicand.  So, their bellwork was to find the prime factorization of my age.  I told them there would be a contest for the most beautiful birthday cake.  My second period Algebra 2 class took this contest very seriously.  My fifth period Algebra 2 class couldn't have cared less.  

Algebra 2 Birthday Bellwork - Prime Factorization
Here are pictures of some of my favorite birthday cakes:

Prime Factorization Birthday Cake

Prime Factorization Birthday Cake

Prime Factorization Birthday Cake
One student didn't exactly follow the instructions.  They just drew me a birthday cake without the prime factorization.  I asked, "Where did the x squared come from?  That isn't a prime factorization."  "Oh, I just thought you wanted us to draw you a cake.  I decided your cake should be decorated with something mathematical."  I guess that works, too.

Another Birthday Cake

One of my 8th graders surprised me with a birthday cupcake!  

A Birthday Cupcake

Me and My Birthday Cupcake!
All in all, it was a fabulous day.  We had fun.  We ate cake.  We did math.  We did lots of math.

And, I can't forget to write about my birthday presents from my family!  They were appropriately themed.  I received an infinity necklace, an infinity scarf, and some awesome math games!  (There were non-mathy things, too.)  I'm so excited to try these out!

Mathematical Birthday Gifts

Things Teenagers Say...Volume Three

It's been over a month since my last edition of Things Teenagers Say.  You can check out the previous crazy things that I have overheard in my classroom in Volume One and Volume Two.



Me: Today we are going to be learning about polynomials!  Aren't you guys excited?
Student: Why would we be learning about party animals?  I'm confused...


Student: Doesn't the US cover all seven continents?
Me: No.  Let's take a look at this globe.
Student: This globe must be wrong.  My seventh grade geography teacher told us that the US was on all seven continents.

(I guess the globe my mom bought for my classroom did serve a purpose!)


Student:  The worksheet I just turned in has Icy Hot on it.  Sorry!

(Later the Same Day)

Another Student: I apologize that my homework I just put in the tray has tabouli on it.

(This is only something that would happen in Drumright.  Tabouli is like a way of life here.  It is served every single day in the cafeteria.  I'd never even heard of it before moving here.  I wasn't impressed the first time I tried it at, but it definitely grows on you!)


(During a Celebrity Age Guessing Game to Motivate Linear Regression in my Stats Class)
There is a photo of Clint Eastwood on the Smart Board.  Students have to guess the name of the celebrity and their age.

Student 1: Isn't that the guy from all those westerns?
Student 2: Yeah - that's John Wayne, right?


Student discussing me with another student: "Of course she doesn't wear makeup.  She's a vegetarian.  They're against stuff like that."


Student: I don't understand why my teachers always count my answers wrong when I put a line through my Ts.
Me: Do you know how to write an F in cursive?
Student: No.  Why?
Me: Well an F in cursive is a T with a line through it.  So, when you write a T with a line through it, your teachers think you are writing an F.
Student: That makes so much sense now.


Student:  What are you eating?
Me: An enchilada.
Student: Why are you eating an enchilada?
Me: Because I brought leftovers for lunch.  My mom made enchiladas when I went to visit my parents this weekend, and she sent me back some leftovers.
Student: Are your parents Hispanic?
Me: No.  Why do you ask that?
Student: Well, only Hispanic people eat enchiladas.
Me: Yeah, that's not quite a true statement.  Do I look like my parents are Hispanic?
Student: Yes. You look like you are part Hispanic and part Jewish.
Me: (Awkward Silence.  Yeah, I didn't have any words to respond to that.)


Student: Can we listen to some music today?
Me: Sure, who do you want to listen to?
Student: How about some Michelangelo?


(Upon entering my classroom on the first day that the desks went from groups of four to rows)

Student: Why is this set up like an actual classroom?  I'm confused.

Tuesday, November 26, 2013

A Tiny Change

Have you ever just made a tiny change to how you teach something and been shocked by the difference it made?

A couple of weeks ago, I was tutoring one of my Algebra 2 students after school.  We were multiplying polynomials, and he was getting the multiplication correct.  But, he was messing up when it came to combining the like terms.  After trying the same problem multiple times, I decided to write it out on the dry erase board as he told me each step.

Usually, I have students put squares or circles around like terms or squiggles or lines underneath them.  This usually does the trick.  This time, however, I decided that I would line up all of the x cubed terms, x squared terms, etc.  This way, when I went to combine like terms, I just had to add the coefficients instead of worrying about where all the like terms were.  The student seemed to like this method.  And, I was kinda impressed with myself for my last minute creativity.  I had already been at school for almost twelve hours at this point, so I was a little brain dead.

I didn't erase the board before going home, so this problem was still up when my Algebra 2 class came in the next morning.  A few of my students noticed it and mentioned how they really like this method of organizing your work.

I took a picture, but the problem had been up for multiple days by that time.  So, the answer has been partially erased.  Sorry about that!  Maybe everybody else has been teaching this way all this time...  Or, maybe this will be new to you as well.  If this post helps or inspires one person, it was worth it.

Method of Organizing Work when Multiplying Polynomials

Monday, November 25, 2013

Celebration of Mind Day (aka Hexaflexagon Day)

October 21st was Celebration of Mind Day.  Celebration of Mind Day is a holiday that honors Martin Gardner and his contribution to recreational mathematics.  This was the first day my students came back from Fall Break.  So, they had been out of school for five entire days.  I decided this would be the perfect opportunity to expand their horizons and show them a side of mathematics that they maybe hadn't seen before.

We talked about who Martin Gardner was.  He did not invent the hexaflexagon, but he did tell the world about it.  I first learned about hexaflexagons when I was in the tenth grade.  My sister was in the seventh grade at the time, and she got to make one in her math class.  I was a tad jealous, and I was also extremely frustrated.  I couldn't figure out how to make it work!  And, she made it look so easy.

Last year, I made my first hexaflexagon with my Algebra 1 students, and I instantly fell in love.  I could probably sit for hours and play with a hexaflexagon!

If you want to host your own hexaflexagon party, there are all the resources you could ever need here.  And, here is a link to PDF templates to make your own trihexaflexagon or hexahexaflexagon.  

We started our discussion of hexaflexagons by watching a Vi Hart video.

My students were amazed.  Of course, they instantly wanted to make one.  I printed off templates for a trihexaflexagon.  I have found that the best way to ensure success is to tell students to cut out their template and to double crease (once each way) each fold.  If students do this, they will be MUCH less frustrated.  And, their teacher is, therefore, much less frustrated!

After showing students how to assemble their hexaflexagons, I got the amazing opportunity to show students how to make their hexaflexagons work.  I always let students try to figure out how to work them on their own.  After all, they have seen a video of how it works.  But, most students need someone to show them where to pinch and how to open up the center of their hexaflexagon.  My favorite thing to do is to watch students' faces the first time they are able to open up the center of their hexaflexagon.  Their expression is PRICELESS.  I want to find a way to see that expression more on a day-to-day basis as we are learning algebra.

After creating their hexaflexagons, we watched the Hexaflexagon Safety Guide because it shows a lot of cool ways to decorate your hexaflexagon.  Plus, who wouldn't want to see a hexaflexagon made out of a tortilla?

The day after we made our hexaflexagons, one of my students came back to tell me that she had stayed up late watching all of Vi Hart's videos.  Her words: "Math is fun!"  

To those of you who will say that I wasted a day of instruction with my Algebra 1 and Algebra 2 students, I would beg to differ.  I wasn't hired just to teach math.  I was hired to change lives, to inspire students.  I have a group of Algebra 1 students from last year who come to visit me multiple times a week.  When they realized it was Hexaflexagon Day, they almost all told me that they still had their hexaflexagon from last year.  This is something they will remember for a long time.  I can't say the same about worksheet or a quiz.  

My Hexa-Hexaflexagon and My Tri-Hexaflexagon

One of my Statistics students is an amazing artist.  This was her creation.  I am inspired by it, and I just had to share!  Isn't this just gorgeous?  She puts my hexaflexagon to shame.

Sunday, November 24, 2013

Algebra 2 INB Pages - Exponential Functions, Exponent Rules, and Factoring

Unit 3 was a weird one for Algebra 2.  I wanted to cover exponential functions right after linear functions to see if teaching y=a+bx actually helped my students to graph y=a(b)^x.  I also needed to review exponent rules with my students before we could move onto quadratic functions, radical functions, or logarithmic functions.  And, I also needed to make sure that my students were solid on the distributive property, combining like terms, and factoring before we got any further into the semester.  So, I sort of combined all of these into a hodgepodge Unit 3.  

Unit 3 Table of Contents - I apologize for the fact that this was not up-to-date when I took the picture.  Oops...

Exponential Functions Frayer Model

Exponent Rules

Ms. Hagan's Book of Exponent Rules worked so well with my Algebra 1 students that I decided to try it out with my Algebra 2 students.  

Parts of a Power and Unwritten Exponents

Exponent Rules for Like Bases

Negative and Zero Exponents
You can read more details about how I taught exponent rules in this post.

Naming Polynomials Graphic Organizer

Combining Like Terms and the Distributive Property Foldable - Outside

Combining Like Terms and The Distributive Property Foldable - Inside

Introduction to Factoring Notes
If/when I ever do this again, I will type up the polynomials that are written in marker.  Students will have to cut them out and decide if they are the factored version or the distributed version.  Students will glue them in the appropriate places on the table and then come up with the corresponding version.

I used the same factoring graphic organizers with my Algebra 2 kiddos that I used with my Algebra 1 kids.  You can read more about my approach to factoring this year here.

Factoring Quadratic Trinomials when a=1

Factoring Quadratic Trinomials when a>1

Factoring Difference of Squares
We closed out the unit with an exponential growth and decay of skittles lab, but that will have to be another post for another day!

PDF Templates to Download:

Unit Table of Contents (PDF)

Blank Frayer Models (PDF)

Naming Polynomials Graphic Organizer (PDF)

Factoring Quadratic Trinomials (a=1) Graphic Organizer (PDF)

Factoring Quadratic Trinomials (a>1) Graphic Organizer (PDF)

Factoring Difference of Squares Graphic Organizer (PDF)