0 The #1 Thing I Won't Teach Next Year: Integer Operations

I realize it's still April.  Our last day of school is May 14th.  But, I'm already thinking about next year.  In fact, I'm thinking a lot about next year.  Things I want to do.  Things I don't want to do.  Things I need to do.  Things I should think about doing.

Here's the deal.  Oklahoma is changing standards next year.  Originally, we were a PARCC state.  Next year is supposed to be our first year of full implementation of Common Core.  Except, we decided to not call it Common Core.  We're calling them the Oklahoma Academic Standards (OAS).  I referred to them as Common Core the other day in the presence of our elementary principal, and I got yelled at.  I'm sorry.  Changing the name does not change what they are.

Then, Oklahoma decided that we don't have the necessary technology infastructure for PARCC testing.  So, we will write our own test that aligns with PARCC and CCSS/OAS.  All was fine and well.  Then, this month, the Oklahoma senate approved a repeal of Common Core.  Instead of teaching CCSS/OAS, Oklahoma is supposedly going to write their own standards that are more rigorous than Common Core.  As a teacher, this is frustrating.  I just want to know what I'm supposed to be teaching.  Will we teach/test CCSS for a few years while writing our own standards?  Or will we jump directly into new standards that haven't even been written yet?  I doubt we will go back to our old PASS standards, but it seems like anything can happen lately.  How can I prepare to provide the best education possible to my students when we keep changing our minds?!?

So, I tell you all this to say, I'm still not exactly sure what I'm teaching next year.  I'm teaching Algebra 1, Algebra 2, and Trig/Pre-Calculus.  But, exactly what those classes will consist of is still up in the air.  I may not have a for-sure set of standards for next year, but I do know this:

Next year, I will NOT spend class time teaching integer operations.

I teach in a school with a history of poor academic achievement.  For the past two years, I have spent at least a week in Algebra 1 on adding, subtracting, multiplying, and dividing integers.  And, I'm sick of it.  This is one of those concepts that is first introduced in 6th or 7th grade in Oklahoma.  I should not have to teach it in high school.  This year, I gave my Algebra 1 kiddos integer speed tests.  64 questions.  4 minutes.  I told them that they were going to take one every day until they passed.  After a week, I gave up.  Not a single student could even finish the test in 4 minutes.

I've got to raise the bar.  Every day in my class is valuable.  And, I could teach so much more if I can regain that week of review.  Because let's face it: reviewing integers is the biggest waste of time.  They've heard the explanation a million times before.  The ones who need the most help are the quickest ones to tune me out.  After all, a negative and a negative make a positive.  Somehow they missed out on the fact that it only works that way when you are multiplying or dividing.  -3 - 7 does not equal positive 10!

This year, I attempted to go back to the basics. We did exploratory activities with algebra tiles.  I saw a few light bulbs go off, but there was not enough light put off to justify the insane amount of time that was spent learning how to use the algebra tiles.  Last year, I made foldables and taught my students memorization tricks.  That didn't work any better.  

Over the course of the year, most of my students seem to (finally!) internalize the rules for integers.  Maybe it was my incessant singing of "Same signs add and keep.  Different signs subtract.  Keep the sign of the bigger number.  Then, you'll be exact" every time someone made a mistake with adding/subtracting integers this year that helped???

When I took Calculus I in college, we had two gateway exams that were in addition to our 3 regular exams.  One gateway exam was over differentiation, and the other gateway exam was over integration.  If I remember correctly, there were 20 questions, and you were required to get 18 or more correct to pass.  Each gateway exam was taken for the first time in class.  If you passed, you were done!  If you did not pass, you were allowed up to four additional chances to take and pass the exam.  These had to be done outside of class time, and they were scheduled with the professor or TA.  If you could not pass the gateway exam, your final grade would automatically be dropped one letter grade.

I want to take the strategy and apply it to integer operations.  I will give the first integer operations test in class.  The tests will be graded and passed back to the students.  Students who score below a 90% must re-test.  Re-tests must happen on their own time.  And, there will be a set date that students must demonstrate mastery of integer operations by.  Ideally, I want 100% of my students to demonstrate mastery of integer operations by the beginning of September.  We start school in mid-August.  This means I will have to get on this ASAP!  

Students who make less than a 90% on their integer operations test will be provided with a list of websites, online games, flash cards, etc. that they can use to practice with.  I will also be available to help them before school, after school, or at lunch.  Students who have not demonstrated integer mastery by the beginning of September will be required to log 1 hour of integer practice in my classroom each week until they pass.  This time can be spent before school, during lunch, or after school.  This is going to be absolutely 100% non-negotiable.  Students who do not log their hour of practice in my classroom will be given mandatory lunch detention until mastery is achieved.

This accomplishes multiple things.  It communicates to students that integer operations are an important concept for algebra success.  It communicates that this is something that students should have learned in middle school.  It also communicates that if they didn't learn it in middle school, then I am here to help them.  I care so much about their success that I am willing to inconvenience myself to help them master their integer operations.  I am identifying a weakness that I see in the students at my school as a whole.  And, instead of lamenting the weakness, I'm acknowledging it and providing practice opportunities and support.

I want students to take control of their own learning before that first gateway exam.  I will provide them study materials, practice tests, and links to online resources.  They can assess their own needs.  My students are capable of so much more than I give them credit for.  It's time I let them show me what they can achieve when they set their mind to it.

For my Algebra 2 students, I think their gateway exam needs to cover exponent rules.  I should not have to spend a week reteaching exponent rules to Algebra 2 students.

Have you ever done anything like this with your students?  Do you see any changes I need to make?  I'm very open to constructive feedback.  My teaching style is slowly morphing, and I think it's a good change!


P.S.  This blog post did not go where I planned for it go at all.  I started writing this post to tell you about an awesome new game I learned about at my last Math Teachers' Circle meeting.  I will be using it to emphasize integer operations and the order of operations.  That'll have to wait until tomorrow, though!                

Detail: The #1 Thing I Won't Teach Next Year: Integer Operations

0 Fabulous Function Machines

This is a lesson I taught way back in early January.  I've put off writing it because I want to make sure I do it justice.  Do you ever have a lesson that you are just insanely proud of?  This is that lesson for me.  (Feel free to link to your lesson or tell us about it in the comments!  Sharing is caring!) 

[Warning: This is probably the longest post I've ever written.  It's also got a ridiculous amount of photos and videos. I hope you will take the time to read through it and offer your feedback in the comments, though.  If you were coming to my class to observe me teach, this is the lesson I would want you to see.  I know there are tons of things I could do better that you can see that I can't.  I'm also very interested in hearing your ideas on how I can capture the spirit of this activity and use it to engage students throughout the year.]

Let me set the stage so you can see where this falls in my Algebra 1 curriculum.  At this point in the year, my Algebra 1 kiddos have reviewed the pre-algebra basics of the real number system, integer operations, and the order of operations.  We learned exponent rules and named polynomials.  We're experts (hopefully) at distributing and undistributing (factoring) expressions.  Equations have been written and solved.  Inequalities have been solved and graphed (in one variable.)  Students have learned the definitions of relations and functions.  We've also discussed domain and range and independent and dependent variables.  We can graph points on the coordinate plane.  It's now time to delve into function notation, expand on the idea of a function machine, and graph functions by making an input/output table.  We haven't discussed slope or looked at the difference between linear and non-linear functions.  Those will come in the next unit.

When we discuss how to differentiate between a relation that is a function and a relation that is not a function, I introduced the concept of a function machine.  Humans like functional machines.  We like to be able to predict what's going to happen.  When we put bread in our toasters, we expect toast.  If we put a toaster strudel in our toaster, we expected a warm, toasted toaster strudel.  If we put a bagel in our toaster, we expect a toasted bagel.  Because we know the function of a toaster (to toast things), we can predict the output based on the input.

What if this wasn't the case?  What if I put in a piece of bread, expecting toast, and a toasted poptart comes out?  What if I sometimes get toast and sometimes get a poptart?  Life would certainly be exciting, but nobody in their right mind would buy that toaster for anything more than novelty purposes.  If I'm eating breakfast and I like poptarts, I might not be too broken-hearted if I put in bread and got out a poptart.  But, what if I'm wanting to make a BLT on toasted bread?  I definitely don't want to eat my bacon, lettuce, and tomato sandwiched between two strawberry poptarts with lots of icing.  Yuck!  (And, for the record, my BLT would be made with these.  I've never actually eaten actual bacon.)   

After introducing the concept of a toaster as a function machine, I ran across an amazing comment on Sarah Rubin's Everybody is a Genius blog.  The comment was from Ms. Mac/Mrs. G.  And, she suggested showing students that a toaster allows several inputs with the same output.  But, it doesn't allow one input to have several outputs.  For example, if I put wheat bread in my toaster, I get out toast.  If I put rye bread in my toaster, I get out toast.  If I put pumpernickel bread in my toaster, I still get out toast.  (I've never actually eaten rye bread or pumpernickel bread.  I guess I need to get out more!)  But, if I put in a bagel, there is no way that I could ever get anything but a bagel out of the toaster.  There is zero possibility of getting scrambled eggs or a waffle.  Every input has exactly one output.  I love this addition, and I think it will really help my students.  I'm definitely adding this to my unit next year!  

I guess I made a pretty big deal out of my toaster analogy before Christmas.  After Christmas break, I asked the following question without really thinking about the whole toaster thing.

Me: "Guys, I brought some function machines from my house today.  Don't you have function machines at your house?"
Student: "Did you bring your toaster?"
Me: "No... I guess a toaster is a function machine, but I brought a different type of function machine today."

Now, remember the flip chutes I made this summer?  They are made out of saltine boxes.  Though, I'm pretty sure I was supposed to make them out of empty juice cartons.  They've been sitting in my house for months, just waiting to make their debut in this lesson. 

My Flip Chute Function Machines

I'm building up to the idea of function notation.  Last year, my Algebra 1 students had a terrible time wrapping their brains around function notation.  The whole idea behind this lesson is to try to avoid the frustration and confusion that seemed to persist last year with my students.  Why is the function notation necessary?  Because humans don't just like predictable things.  They like things with names.  Here's what I told my kids:

Me: "Humans love to name things.  We have an obsession with naming the things around us.  We name our children.  We name our pets.  We name our cards."
Student: "We name our guns." (That's news to me.  Of course, I don't actually own a gun.)
Another Student: "We name our shoes."  (Yeah, I don't do that either.)
A Third Student: "I give names to my favorite t-shirts."  (That is yet another thing I don't name.)
A Fourth Student: "I named my straightener."    

So, apparently, humans like to name things even more than I realized!  Mathematicians like to name things, too.  After all, they are humans.  Mathematicians are also fans of parsimony.  If they name something, they want to name it in the easiest and shortest way possible.  Therefore, usually the name of a function is notated by a single letter of the alphabet.  The word function starts with f, so a lot of times, mathematicians simply call a function f.   

I have to function machines, so I named one f and the other g.  To further emphasize that f and g are just the name of the function, I gave them more traditional names, too.  Francine.  And George.  After I gave them these names, I realized that these are both the names of characters on Arthur.  I loved to watch Arthur when I was growing up.  I can remember being so scared of going to the third grade.  I mean, what if my teacher ended up being as mean as Mr. Ratburn?  Okay, enough with the reminiscing.

Once I realized that I was channeling Arthur with my names, I made a comment about it to my students.

Me: "I guess I'm channeling Arthur today."
Student 1: "Who?"
Me: "You know!  Arthur, the aardvark."
Student 1: "What?"
Me: "Have you really never seen that show on television?"
Student 2: "Oh, I know what show you're talking about!  What did you say he was again?"
Me: "Arthur was an aardvark."
Student 2: "He was an aardvark?!?  I always thought he was a bear!"

I didn't realize that giving my functions names was going to lead to so many issues.  My 6th period class decided to rename George as Georgeio.  Another student came in at lunch to give me her opinion on the names.  "The function machines should be named George and Francisco.  Then, they can be gay, Italian lovers.  Italians make the best lovers.  I know this."  I didn't even respond to that comment.  I did not want any more details.   
      

And, let's not forget to mention the fact that Francine is taller than George.  One of my classes assumed that George and Francine were dating.  And, Francine is taller than George.  The girl is taller than the guy.  Oh, what outrage!

Next, it was time to take some notes over function notation.  These notes were heavily influenced by (read: stolen from) Everybody is a Genius.  We labeled each part of a function in function notation.  And, then we practiced evaluating functions using a function machine.

I'm going to be really honest.  After going over the parts of a function, my Algebra 1 kids were VERY confused.  But, after we worked through the two examples with the function machines at the bottom, light bulbs started going off.  "Oh, that's what that means!"      

Function Machine / Function Notation Booklet Foldable - Outside

Inside the foldable, students were given two new functions to evaluate for various inputs.  

Function Machine / Function Notation Booklet Foldable - Inside

I included some problems where students added functions and found the composition of functions.  Normally, I would have skipped over these because they are not tested in Algebra 1, but I'm slowly starting to work in increasing the rigor of my Algebra 1 class in preparation for Common Core.  I also included two problems at the bottom where students had to evaluate the function for a different variable.

On the right side of the inside of the page, I include 3 sample EOI questions for students to solve.  I want my students to be familiar with how the questions will be phrased on their end-of-instruction exams.  

This took probably 2/3 of our 50-minute class period.  For the last third of class, we started a hands-on activity using my shower curtain coordinate plane and the function machines.

Before class, I prepared five pairs of functions.  Each pair of functions f and g were written on a post-it note.  Since I have a limited domain and range on my shower curtain, I prepared cards for each ordered pair of the function that fit my restricted domain and range.   

Baggies of Function Cards

Here's a close-up of the cards in one bag.  The post-it note is a reminder for me.  It tells me what functions to write on the SMART Board.  One one side of the cards (cut out of card stock), students are told what to solve for.  For example, if a student picks up a g(2) card, they know that they are finding the output that results from putting 2 into g.  The resulting output is written on the other side of the card.  

Input/Output Cards for Function Machines

To give students an element of choice, I laid out all of the cards on the desk.  Students were instructed to pick a card up off the desk and return to their seat.  At their seats, they have calculators, dry erase boards, markers, and erasers.  The students were to evaluate the correct function for the given value of x.  

Students choose a card to evaluate.

If a student picked up the f(3) card, they would return to their seats and substitute 3 in for x in the function f(x) = 2x-3.  They should get an output value of 2(3)-3 = 6-3 = 3 if all their work was done correctly.  Once they have an answer, they must go back to the function machine and feed the problem to the function machine.  This is their input.  They feed the input into Francine because they were asked to plug the 3 into f. 

Putting the card into the function machine.

This is where things get cool.  Upon placing the f(3) card in the function machine, the function machine will flip the card over so that it comes out reading 3.  To the students, it is MAGIC!  

Function Machine Output

I would LOVE to set up a video camera to capture my students' faces the first time they put a card in the function machine.  Their expressions are priceless!  Of course, they soon realize that the answer is written on the back of the card.  And, all the function machine does is flip the card over.  Even after discovering how it works, they still seem to love it.  I might have seen one or two kids all day forgo feeding their cards into the function machine and just flip it over to check their answer instead.  

I like that this got my students checking their answer, and I hope that it reinforced the idea of inputs and outputs. If you've never seen a flip chute in action, I made a short video.  I've never really done anything like this before so I'm not sure it's that great, but I hope it will give you a better idea of the process.  Don't worry.  It's not that long of a video!  (If you're reading this via a RSS reader, you may need to click through to view the video file.)  

After checking their answer, students now had to graph their input and output as an ordered pair on the coordinate plane.  My giant coordinate plane was put in the floor for this activity.

Giant Coordinate Plane

To plot the points, I created points out of some giant foam stars that my sister had given me.  She had used part of them for an art project.

Ordered Pair Stars for Giant Coordinate Plane

To further emphasize function notation, I labeled the stars with generic ordered pairs.  (1, f(1))  or (5, g(5)).

I was hoping that students would make a connection between the fact that f(0) means the output of f when the input is 0.  I guess I didn't do the best job of explaining this on Day 1 because my students did really well until we got to the graphing part.

Instead of nice linear functions, our attempts usually ended up looking a little something like this.  

Graphing Mishap

After graphing each point, students picked up a new card and repeated the process. They placed the cards that had been graphed on top of the function machines to avoid further confusion.  Once all the ordered pairs had been found and graphed, we had a class discussion.  

Students discard cards after evaluating and graphing.

What stars look like they are in the wrong place to you?  A student would pick a star that they believed was incorrect.  And, as a class, we would evaluate the function for that value and determine the correct place for it to go.  Sometimes the star had been placed correctly.  Most of the time, however, students made the exact same mistake.  (3, f(3)) would be placed on (3, 3).  (-2, g(-2)) would be placed at (-2, -2).  Students were seeing the input of the function and were confusing that with the output.  

Here's an example of some mis-placed stars. 

Mis-placed Stars

Eventually, after some hard work, we concluded that all the stars were now in their right places.

Corrected Graph

Students were shocked to discover that the points formed two lines!  Will this always happen?  Students were convinced that sometimes they would make lines.  Other times they would make X's.  After graphing one pair of functions, time was up.

I felt like we had made some progress, but I knew we had so much more to go.

Enter Day 2.  We started off class by watching the Meat-A-Morphosis: An Introduction to Functions video from youtube.

If you haven't seen this video before, take seven and a half minutes and just do it!  I told my students that we were going to watch a movie, and they were instantly excited. I show lots of math songs and short youtube clips, but I tell my students that I don't believe in movies.  They think that we should be watching feature films before every single holiday.  Ummm....no.  I have way too many exciting mathematical things to share with them.  I don't have time for us to watch Finding Nemo.

They soon realize that we are watching another math video, so their excitement fades.  When the music starts, the complaints start up again.  "How old is this movie you're making us watch?!?"  "I don't know.  Just watch and see."  The girls ooh and aah over how cute the little chickens are.  The class collectively gasps when the (spoiler alert) chickens are chopped up to make chicken nuggets.  "Why are you showing us this?"  "Wait!  You're a vegetarian.  Why are you showing us a video about meat?"  "Guys!  I think she's trying to turn us into vegetarians!"  No, I'm trying to turn you into mathematicians.  At the end of the video, one particularly opinionated student said, "Some of the things I see in here scare me."         

At this point, I'm still frustrated over my students' struggle with function notation from the previous day.  How do I fix this?  I know!  We'll do an example problem, and I'll force them to use function notation an obnoxious number of times! 

Graphing Functions On The Coordinate Plane - INB Page

I gave the class a function to graph in function notation.  And, I also gave them their inputs, the integers from -5 to 5.  Students had to write the output in function notation, the output as a number, the ordered pair in function notation, and the ordered pair using only numbers.  We did the first two lines of the table.  Then, I gave students several minutes to work.  We would check their work, and they would continue on.  It was time-consuming, but I think it was a positive learning experience.

It was awesome to see students start to make connections.  Some realized they could very easily fill in the function notation columns.  Others noticed that the numerical output increased by two every time.  I walked around the room, pointing at student work, congratulating them on finding patterns.  We graphed the ordered pairs, and they formed a line!

With about fifteen to twenty minutes in class, we once again tried graphing functions using the function machines, cards, stars, and shower curtain coordinate plane.  When students looked confused, I reminded them of the function notation we had just used earlier that day in our notebook.  That seemed to help some.  But, I still had some very frustrated kiddos who could not figure out how to pick out which star to use or where to place it on the coordinate plane.

I was starting to write off this whole exercise as a failure.  Maybe I am just terrible at teaching function notation.  Maybe my kids are just doomed to never understand this.  I wasn't about to give up, though.  My students were going to understand function notation one way or another!

The next day, I decided we were going to spend our entire 50-minute period graphing functions.  If we continually graphed, checked our work, discussed discrepancies, and tried again, they were bound to have to get it!  I had students open their notebooks to our pages on function machines and function notation to use as a reference. 

Function Notation INB Pages

My students were rocking the process of evaluating the functions.  They were checking their answers using the function machines.  Life was good.  Well, life was good until they got to the stars.  I decided to just position myself by the stars and help students with the graphing process.  They were going to get this! 

After helping student after student find their correct star, I had an epiphany.  They already have the card in their hand that has the output written in function notation on one side and the output written in numerical notation on the other side.  Why don't I have them use this card to graph their ordered pairs?! 

My conversation with students went something like this.

Student: "I can't figure out which star I need!"
Me: "Well, what was your input?"
Student: "My input was 2, and my output was 1.  But, I don't see a star labeled (2,1)."
Me: "Do you remember how we had two different ways of writing the output in our notes?"
Student: "Yes."
Me: "Well, if your card says f(2), you need to find a star that also has f(2) in the ordered pair." 

Finding the Correct Star

Problem solved.  Find the star that matches the card.

This was great except for the fact that they still wanted to put the (2, f(2)) star on (2, 2).  Every.  Single.  Time.

It was time for a second epiphany.

Me: "So, your output was f(2)?"
Student: "Yes."
Me: "And, what was the numerical output that you got when you plugged 2 into the function?"
Student: "When I plugged in 2, I got 1."
Me: "So, f(2) is 1?"
Student: "Yes, I just said that."
Me: "So, you're saying that f(2) is equal to 1?"
Student: "Yes. f(2) equals 1."
Me: "If f(2) equals 1, then I can replace the f(2) with 1.  And, that will tell me where to graph the ordered pair."

Star With Ordered Pair In Function Notation

At this point, I made a show of flipping the card over several times.  I would show them that (2, f(2)) was the exact same as (2, 1).  After all, they had just told me that both sides of the card meant the same thing. 

Star With Ordered Pair in Numerical Notation

All of a sudden, my students were graphing points like nobody's business.  They didn't need my help to find their stars.  And, they certainly didn't need my help to graph their ordered pairs.  It was so fun to watch them match the cards with the stars and then lay the cards on top of the stars.  They would flip the card over just like I had shown them in order to graph their ordered pair.  Success! 

My students were able to successfully graph horizontal lines and quadratic functions using the function machines as well.  We ended up calling f(x) = 3 the Spam equation.  No matter what our input, 3 (spam) always comes out.  (If you're confused by this, you should really go back and watch the Meat-A-Morphosis video.  I promise; you'll be much less confused!)  

So, why do I love this lesson so much?

It's hands-on.  It got my kids out of their seats.  It was self-checking.  It was a safe environment to make mistakes.  When students point out that they think (3, f(3)) is in the wrong place, nobody knows who graphed that point unless the student announces that it was their star.  Each student worked at their own pace.  It was discussion-provoking.  Students were craving vocabulary words like slope and y-intercept that we had yet to learn.  They were asking questions, making predictions.  "I bet this one makes a line, too!"  After, we graphed a quadratic, one student wanted to know how he could tell from the equation if the graph would make a straight line or a parabola.  I also felt like I could differentiate this activity for my students.  If f(x) was a quadratic and g(x) was linear, I could push my advanced students toward solving the f(x) cards, and my students who needed more support could be further practicing linear functions with the g(x) cards.   

Basically, I want this to be what my classroom is like every single day. 

Want these files to use in your own classroom?  I've embedded them below as PDF and PUB (Microsoft Publisher) files.  If you have any trouble accessing these files, please send me an e-mail.  I will be happy to attach the files and send them your way!

Detail: Fabulous Function Machines

0 HOY VUX REDUX

My Algebra 1 students just recently finished learning to graph horizontal and vertical lines.  For this, I used, as before, the mnemonic device - HOY VUX.  

H - Horizontal Line

0 - Zero Slope

Y - Graph Crosses the Y-Axis

V - Vertical Line

U - Undefined Slope

X - Graph Crosses the X-Axis

HOY VUX has always reminded me of those word puzzles that stand for a common phrase.  Like last year, I gave some of these to my students to figure out before we jumped into the math part of the day's lesson.  

Mind Over Matter

Surprisingly, my students weren't as familiar with these as I expected.  They also weren't quite as competitive as I had anticipated.

That's okay.  I know how to fix that.

Jolly Ranchers

Jolly Rancher Bribery.  First person to raise their hand and give the correct answer gets a jolly rancher.

Word Puzzle

Just when I thought my students were about to get the hang of these puzzles, I put up this one.

None of my Algebra 1 classes were able to figure it out.

One of my student's answer to this one made me laugh.  The answer is "Excuse Me."  However, one of my students interpreted it as "Execute Me."  

After going through about 7 or so more puzzles, I put up the letters HOY VUX.  I pretended it was a puzzle, and I waited to see what my students would make of it.  

In one class, students started arguing about what language it was written in.  Apparently, it must be a German word!  After listening to their debate for a little while, I told my students that I needed to apologize.  This wasn't actually a puzzle at all.  This would be our lesson for the day.  

Last year's HOYVUX foldable worked fine, but I knew I wanted to change it up somehow this year.  First, I'll show you last year's notebook page.  Then, I'll show you the changes I made for this year.   

Last Year's HOY VUX Foldable - Outside

Last Year's HOY VUX Foldable - Inside

I was inspired by Jan's post about HOY VUX at Equation Freak.  I loved how she had arranged her interactive notebook page so that the HOY notes were written horizontally and the VUX notes were written vertically.  I decided to take her layout and turn it into a foldable.  

This may not have been my brightest idea.  

First, I had students cut out the templates for their foldables.

HOY VUX Template

  Next, I showed students which lines to cut on to form their flaps.

I highlighted the cut-lines for you above.  Here's the picture I drew for my students on the SMART Board.

It's not pretty, but it got the job done.

HOY VUX Foldable

We labeled the outside flaps HOY and VUX.  The corner rectangle contains notes on when to use HOY and when to use VUX.

HOY VUX Foldable - Outside

Remember when I said this wasn't my brightest idea?  Oh, my students were plenty impressed with the fact that HOY was written horizontally and VUX was written vertically.  It was fun to see students realize the connection and remark, "I see what you did there!"  

But, they were more impressed with what our foldable reminded them of.  Can you see it?  Maybe this will help.

In all three sections of Algebra 1, I had at least one student exclaim to the class that we had just made a gun-shaped foldable.  They boys were soon holding their foldables up to one another and pretending to shoot each other.  "Look!  You even gave us a trigger!"  Yeah.  That was definitely NOT what I intended.  

Here are the inside of the notes:

HOY - Graphing Horizontal Lines Notes (Inside of Foldable)

VUX - Graphing Vertical Lines Notes (Inside of Foldable)

We also created a small poof booklet to practice graphing horizontal and vertical lines with.

Graphing Horizontal and Vertical Lines Poof Booklet

Inside, there are 6 equations for students to practice graphing.  Students are asked to identify if the equation falls into the category of HOY or VUX.  Then, they have to say which axis the graph will cross and where it will cross that axis.

Inside of Poof Booklet

The file for this booklet has been uploaded below.  If you want more information on how to fold this style of booklet, check out this post.  The instructions are written for assembling a booklet to practice finding slope and intercepts, but the steps will be the same.  

Here's a look at the completed page.

Completed Notebook Page

Other than the fact that my students think the foldable looks like a gun, I'm really liking the changes that I made to this page.  

Files have been uploaded below.  If you have trouble downloading the files, please e-mail me.  I will be happy to attach the files you need and send them to you.  

Detail: HOY VUX REDUX

0 A Google Surprise

Do you ever google yourself?  I do.

I still remember the first time I ever heard about google.  I think I was in middle school.  I was riding in the car with my grandmother, and we were listening to NPR.  They were talking about a new phenomenon of googling someone before you met them.  The piece told the story of two people who had been set up on a blind date.  Both had googled each other before meeting in person, and it ended up coming out during their first date because they ended up revealing that they knew more about the person than they should have.

Seeing as Google has now been around longer than most of my students have been alive, they can't imagine life without google.  I remember life without google.  I remember it vividly.  On my first day of sixth grade, my science teacher gave us a worksheet that asked us questions about various inventions and their inventors.  After doing the ones I knew, I went to my parents for help.  My mother directed me to the set of encyclopedias she received as a graduation gift in the late 70's.  That worked great except for the fact that some of the inventions were not around in the late 70's.  So, do you know what we did?  We called the library, and a reference librarian kindly looked up the information for us.  I can't even imagine doing that nowadays.

By the time I started 7th grade, my parents decided that it would probably be a good idea to have a home computer.  Before that time, if I wanted to use a computer, I had to do it at my parents' work place.  It's crazy to think about how much things have changed in just my short life span.  When I tell my students that I was 16 when I got my first cell phone, they are amazed.  The iphone wasn't introduced until my senior year of high school.  And, they still can't quite comprehend how I can survive with a non-smart phone.  "You mean you use twitter from a computer?  I didn't know you could do that!"

So, anyway, back to my story about googling myself.  A couple of days ago, I googled my blog username just to see what popped up.  Most of the search results were my blog.  And, a few mentions of my blog on other blogs were there as well.  What I wasn't expecting was a youtube link.

I found a video describing how to do my Pre-Algebra Road Trip Project.  I thoroughly enjoyed watching the video and seeing how another teacher explained things.  If you're interested in the project, click here.

Or, alternatively, press play below.

  

Detail: A Google Surprise

0 Slope Name Art

I've seen this idea multiple places online, so I'm not exactly sure who to credit.  Most recently, I've seen it from Pam J. Wilson and Mrs. Hester.  It's a variation on two activities that I did last year.  So, my students have been working on slope.  We started out by discussing the four types of slope.  Slope Dude was a must for this!  Y'all are probably tired of hearing me go on and on about Slope Dude.  But, I just can't help it.  My students think the video is stupid, but I can guarantee you that they will NEVER forget Slope Dude's most infamous words.  One of the new rules I've made in my class is that students are only allowed to say the word "undefined" if they say it like they are falling off of a cliff!  "UNDEFIIIIIIIIIIINED!"

So, last year, we classified the slope of each line segment of each letter of the alphabet.  I enjoyed this activity, but this year I wanted students to create something to keep in their interactive notebooks.  This year, I have made it my goal for our interactive notebooks to be comprehensive.  Last year, there were a lot of gaps in our notebooks.

Last year, I also had students draw pictures and classify the slope of each line segment.  Again, it was a fun activity, but my students had barely anything in their notebooks regarding the four kinds of slope.

Slope Name Art

This year, I decided to have my students create what I deemed "Slope Name Art."  On the next blank page in their notebooks, I instructed students to write their name as large as possible.  The only caveat was that they could only use straight lines.  Any letters of the alphabet that would normally be written using curves would need to be modified.

I illustrated on the board with my own name.  Once students wrote their names, their next task was to classify the slope of each line segment.  I only classified the slope on the first two letters in my example.  My students were required to do this for every single letter.  They were given the chance to write their first name, last name, or both.        

Once they had classified the slope of each line segment, I asked them to color-code their name.  Choose one color and highlight all the line segments with a positive slope with that color.  Choose another color to highlight all the line segments that have a negative slope, etc.

In my notebook, I simply created a key that told which color represented which type of slope.

Slope Name Art

My students really enjoyed this activity.  It was almost a sort of competition between them.  "I only have one negative slope in my name."  "Almost all my letters are made up of zero slopes and undefined slopes."

Detail: Slope Name Art

0 Taking The Flyswatter Game Up A Notch

Remember my giant coordinate plane?  I am just so ridiculously proud of this thing.  I love it.

Giant Coordinate Plane

I'll be honest.  Not all of my kids thought it was as cool as I did.  A bunch of them had a "Ms. Hagan, really?" expression on their faces when I introduced it to the class.  But, I'm used to that expression by now.  There were also the students who stayed after class to graph points by standing on the coordinate plane because they wanted to have a turn even though we ran out of time.

Remember the fly swatter game?

Well, a couple of weeks ago, I decided to combine the two.  We played the fly swatter game on the giant coordinate plane.  Two students would come up to the coordinate plane.  They could stand anywhere around the coordinate plane that they wanted.  I would show them an ordered pair.  (I reused my flies that I had hung around the room for this.)  The first one to slap the correct place on the coordinate plane with their fly swatter won.

Coordinate Grid Fly Swatter Game

I had to make several rules to keep students from just slapping random places.  For example, only their first slap counted.  If they slapped the wrong place, they could not win the round.  The competition was intense.

This activity definitely worked better for my smaller classes.  I even turned this into a quiz for my smallest Algebra 1 class of 10 students.  Each student had to come up to the coordinate plane.  I asked them three questions similar to these: 1. Show me where (-3, 2) is on the coordinate plane.  2.  Show me where (0, 4) is on the coordinate plane.  3.  Choose a point in the fourth quadrant.  Show it to me, and tell me its ordered pair.

Detail: Taking The Flyswatter Game Up A Notch

0 Algebra 1 - Introduction to Relations and Functions

Unit 5 for my Algebra 1 kiddos is an introduction to relations and functions.  My goal with this unit is to review pre-algebra concepts such as ordered pairs and graphing points on the coordinate plane and emphasize vocabulary.  The EOI my students will take in the spring will have questions that specifically ask them to identify which relation is or is not a function, identify the domain or range of a given relation, and classify variables as dependent or independent.

After we get back from Christmas Break, we will finish up this unit with notes on function notation and determining the rule that a function follows.  I can't wait!  I'm soooo ready to get to Unit 6 - Rate of Change and Linear Functions.  I think I love to teach about slope and graphing more than anything else.  With Common Core, that will be shifting to eighth grade which makes me extremely sad.  When that happens, I guess I will have to find a new favorite unit to teach.

Here is our table of contents for this unit so far...

I already posted pictures and templates for the ordered pair foldable and coordinate plane foldable.

We made a frayer model to discuss an important vocab word: relation. 

Relation Frayer Model

Then, we discussed various ways to represent a relation.  I focused on the four types that my students will see on their end-of-year exam: ordered pairs, input/output table, coordinate plane, and mapping diagram.  My students were convinced that an input/output table was the same as a stem and leaf plot.  At least they have heard of a stem and leaf plot...

Ways to Represent a Relation Foldable - Outside

Ways to Represent a Relation Foldable - Inside

Once we had defined a relation, we could now define a specific type of relation: a function.  

Function INB Pages

Can you tell I love the Frayer Model for vocabulary?  We made one to summarize what we learned about functions.  I gave them the definition of a function and nothing else.  Then, I went through all of my released EOI test questions and copied and pasted all of the relations and/or functions into a Smart Board file.  I asked for a volunteer.  I showed them and the entire class a picture of a relation.  I instructed the students who had not volunteered to write down function or not a function.  Once everybody had written down their answer, the student who volunteered said their answer aloud.  Next, I had all of my students hold up their dry erase board to show the student.  After seeing the responses of their classmates, the student had to decide whether they should keep their original answer or change their answer.  Only after going through this process did I reveal whether it was a function or not.

This was a new way of doing things for me, and I kinda liked it.  I have students write their answers down and show them to me a lot.  But, I rarely have students show their boards to other students.  It reminds me of the "Poll the Audience" feature of Who Wants to Be a Millionaire.  I guess you could say that the person who has to voice their answer aloud is in the "Hot Seat."

Function Frayer Model

My students loved this.  I had lots and lots of students wanting to volunteer.  The conversations that ensued between students were AWESOME.  I also did something new this year.  I didn't teach my kids about the vertical line test to begin with.  I sort of let them discover it for themselves.  Of course, this worked better in some classes than others.

We ended this day's lesson by completing the function/not a function card sort that I love and have used a lot.  I used it last year with both my Algebra 1 and Algebra 2 classes.  And, I used it with both classes again this year.  This card sort is available from Math Tales From the Spring.

Function / Not a Function Card Sort

The next day, I had students create examples and non-examples of functions.  And, we summarized the vertical line test strategy that we had arrived during the previous lesson.  To give my students further practice, I assigned an error analysis task from Algebra's Friend.  My students are terrible at following directions.  Even though I told them that they were grading the answers given on the paper, I still had students who answered function or not a function instead of correct or incorrect.  My IEP students were especially bad about making this mistake.  I just don't know how to get them to slow down and read the instructions.

Domain and Range INB Pages

Next, we moved into our discussion of domain and range.  I used the DIXROY mnemonic device with my students.  I don't really know how many students even use this to help them remember the meaning of domain and range.  DIX: Domain, Inputs, X-Coordinates. ROY: Range, Outputs, Y-Coordinates.  I did see a student write the acronym on their index card to use on their semester test, so I guess at least one student has found it helpful.

Domain and Range (DIXROY) Notes

Domain and Range / Representations of a Relation Practice

This half-sheet was stolen, ahem, borrowed from Mrs. Hester.  She posted a picture of this organizer from her 8th grade interactive notebook, and I fell in love.  So, I typed up one at the last minute.  I uploaded the template for you below.  Everything isn't perfectly aligned and centered, but it was the best I could do with the limited time I had.  If I wasn't such a procrastinator, my notebooks would be better.

Independent and Dependent Variables INB Pages

Last year, some of my students really struggled with the difference between dependent and independent variables.  So, this year, I set out to teach this topic better.  I'm pretty sure I failed, but I'll share with you what I did anyway.  

Independent and Dependent Variables Notes

We took notes over the difference between independent and dependent variables.  I filled out the arrows above three different ways for the three different sections of Algebra 1 that I teach.  None of the ways led to universal understanding.  When I look at a scenario, I can easily tell you which variable depends on the other.  This is the dependent variable.  Thus, the other variable is independent.  Easy.  

Many of my kids cannot do this.  I will think I have come up with the perfect example that will make all things clear to all students.  Ms. Hagan's Outfit.  The Weather.  Which one is dependent?  Which one is independent?  They will tell me something like this: "The weather depends on Ms. Hagan's outfit."  This leads to me saying, "So, if I change my outfit, the weather will change?"  It sounds crazy.  But, they will answer yes to this and truly mean it.  I don't think my kids understand the word "depend."  Or, maybe they don't understand cause and effect.  

There were basically two groups of students in each section where I taught this lesson.  The students who thought this was the most obvious thing we had ever done in Algebra 1.  Why in the world would we spend an entire 50-minute period on this lesson?  Then, there were the students who missed every single question.  They would flip the dependent and independent variable.  Every single time.  I was almost tempted to tell them to just write the opposite of what they thought the answer should be.  Okay, that's terrible.  And, I would never actually do that.  But, why can't my students get this?  

I created a card sort that I thought would take 10 minutes tops.  We spent at least 35-40 minutes on it.  And, some students still didn't finish.  

Independent and Dependent Variable Card Sort

* I accidentally glued one pair on backwards.  The number of songs performed should be independent, and the duration of the concert should be dependent.  Oops.  I was in a hurry, and it shows!  

I did a quick google search for sets of independent and dependent variable scenarios.  I found such a set on www.ixl.com.  I copied and pasted these scenarios and edited them to remove the names.  I thought this activity would be too easy for my students if I left the names in the scenarios.  I think I could have left the names in, and my students STILL would have struggled.

I passed out a sheet containing the 20 statements.  I gave students these instructions:

1. Cut out the 20 rectangles.

2. Recycle your trash.

3. Pair up the statements.

4. Have Ms. Hagan check to make sure your statements are paired up correctly.

5. Classify each statement as dependent or independent.  Arrange these in two columns.  

6. Have Ms. Hagan check your classifications.

7. Glue these in your notebook on page 52 and 53.  Be sure to label the columns as dependent and independent.  

Students were okay with steps 1-2.  But, Step 3.  Oh my goodness.  "How am I supposed to know how to pair these up?"  "This is so hard."  "What's a potluck?"  "Is that where people get together and smoke pot together?"  "This is impossible."  "What does the word duration mean?"  "You mean pickles come from cucumbers?!?"  

I would go around and check my students' statements to make sure they were paired up correctly.  It was not out of the ordinary for students to only have 3/10 pairs made correctly.  They were pairing up statements that had NOTHING in common.  It was a nightmare.  

A few of my high-achieving students caught on early and were able to complete this activity with little assistance.  The rest of my students.  I just don't know.  This isn't supposed to be that hard.  They couldn't get them paired up let alone classify them as dependent or independent.  I was frazzled.  My students were frazzled.  

If you have any insight on how to teach independent and dependent variables, please leave a comment.  I'm begging you.      

I have uploaded the PDF templates that I created for my students to use below.  If you can't get them to load, please make sure that you have Flash/Shockwave installed.  If problems persist, feel free to send me an e-mail.  I will be happy to attach the files and send them to you!

Detail: Algebra 1 - Introduction to Relations and Functions