The Importance of Change

I was at a car parts store the other day, and I saw a sun shade for your windshield that advertised this:

I had to double-take.

A change of 44F is the same as a change of 7C?  Now, I'm not as savvy at converting from Celsius to Fahrenheit as Kate Nowak, but even I knew this was some faulty converting by a marketing department.

No.  I take that back.  I admit, they did the conversion fairly well.  44F is indeed (approximately) equal to 7C.  But a change of 44F and a change of 7C?  That's a little different.

Couldn't they just have tried an example?  One example.  That's all I ask.  Like, take two numbers whose change is 7, say 0C and 7C.  Covert those to Fahrenheit (32F and 45F, respectively), and you'd see right away--that's not a 44 degree change.

Check for the reasonableness of your answer, as I tell my students.

So, what is a change of 7C equal to in Fahrenheit?  Let's do a little algebra!

We know to get from Celsius to Fahrenheit we can use:  

So, let's take two temperatures, in Celsius and call them C1 and C2.  Then their change, in Fahrenheit, can be calculated through the following:

In other words, the change in Fahrenheit, is equal to 9/5 times the change in Celsius.[1]  Which means that a change of 7C is only equal to a change of 12.6F, a far cry from 44F, in my opinion.  And this verifies the consistency of our example that we tried earlier.  Which is good since this is a linear function we're talking about.  No change in slope here.

So, which is it?  Does this sun shade keep our vehicles up to 7C (12.6F) cooler, or 44F cooler?  Maybe we just get to choose.


[1]  This can be changed from an algebra problem to a calculus problem rather nicely.  I think this would be a lovely introduction to using differentials as approximations for actual change (or in this case, since the function is linear, the differential will be equal to the true  change).

Step 1:  Give the kids the function for F with respect C.
Step 2:  Have them calculate dF when dC=7. [dF=F'*dC=12.6]
Step 3:  Show them the  picture at the top.
Step 4:  Watch them pee their pants (they're math nerds too, right?).

Function Composition

I love composing functions.  I always have.  It's one of the topics in algebra I get most excited about.  I don't really know why.  It's cool that functions have this operation that the real numbers don't have.  It's cool that this operation shows up repeatedly in calculus.  But, ultimately, I just think it's really fun.

Every time I go to teach function compositions, I think I finally have it nailed.  I use all the right colors to differentiate between the inside function and outside function; I start out with an application problem to motivate the use; and I'm pretty sure my enthusiasm during this lesson is off the charts.

Still, something was missing.

In the past, I've started with composing at a general expression.  We would start by finding f(g(x)), for example.  I guess I did it that way because that's how the textbooks always did it.

This year, I started with composing at a number.  We started by finding f(g(3)), for example.  I'm not sure why I never thought of this before.  I feel like a total idiot that it took me this long, but it is what it is.  I did 2-3 examples of evaluating at a real number, and the kids took off from there.  As I walked around the room, checking their work, it was clear they nailed it (at least for the day, right?).

Now onto the general case.  But first, a bit of a digression.  I showed them this slide:

We worked these together.  The magic bullet?  f(a+1).  In the past, I would skip from f(a) to f(x+1).  This is where I lost the kids.  x+1 was too much too fast because the original function is in terms of x.  But adding that one little line made all the difference in the world.  From here, we could find f(g(x)) where g(x)=x+1.  No problemo.  Here is my notebook file.

Now if they can just remember that (x+1)^2 isn't x^2+1 I will be a very happy lady.

Kate Nowak wrote this in one of her posts that I've always remembered:

I just find it stunning that you can plan out a lesson 95% correctly and it will miss most of your kids. And you can change one little thing - add three rows to a table - and now all the kids basically get [it]...and tell you why they don't get why this is such a big deal. I feel a little like I have super powers.

Math History: We're all learning here

Like every other math teacher I know, I love me some math history.  Mathematicians are just some of the craziest people to read about and study, I'm sure of it.  I'm also sure that putting historical context into what you're studying helps you remember it more.

In Pre-Calc this year we've had a new mathematician for each unit we've covered.  The mathematician correlates to what we're studying (sometimes I do a better of job of picking the mathematician than other times).  If students write a paragraph about said mathematician and then write a fact on the white board, they get bonus points for that unit's test.  The catch is that they have to write something new on the board.  So, the longer they wait to do the assignment, the harder it gets, because the dude's birth and death dates go pretty quick.

One unexpected outcome of this has actually been my Algebra II classes' response to this board.  They'll see something they think is interesting and it starts a conservation about our spotlight mathematician.  [Read: kids are excited about math!]  I've also heard them say things like, "Mrs. Peterson!  Wasn't that a super smart answer?! Maybe my picture and name should be on that white board!"  To which I usually reply something snarky.  But, the point is that they're finding mathematicians cool, and that makes my heart happy.

Here are the mathematicians we've done so far, in chronological order.  I think the kids are getting the hang of it.

Archimedes

Pythagoras

de Moivre

I still have a long ways to go in incorporating math history; there's much more that can be done here.  But, I'm learning.  And so are the kids.

Dear Mom, I know something you don't know

This is something I did in the very beginning of the year that I meant to blog about and just never did.

At our school, students are introduced to the graphing calculator in Algebra II, which means I've gotten to show kids lots of awesome techniques to problem-solving that they haven't been exposed to yet.

One of our first tasks of the year was solving systems of linear equations by graphing, specifically using the graphing calculator (TI-84).  I asked the kids if they knew when the graphing calculator was invented and when it became standard use in high school mathematics courses.  I asked if their parents would know how to use such a device.

The students assured me their parents would have no idea how to use a graphing calculator, so I asked them to write their parents a letter describing how to solve a system using the TI-84.  I was somewhat surprised--every student (I have about 100 young algebraists) sat quietly and wrote.  And wrote.  And wrote.

This was my favorite:

Yo momma, it’s yo boy ________ and I’m telling you how to solve for y.  Add x on both sides then you will end up with y=x+0.  On the other problem you are going to subtract 8x on both sides and then divide by 4 on both sides.  You should end up with y=-2x-6.  Then you want to put this in your graphing calculator and press graph.  BOOM!  There’s the graph.  Then will want to press 2^nd then calc, then press enter three times and you should have your coordinates.

What being a dance teacher taught me

Before I was a math teacher, while I was still in undergrad, I was a dance teacher.  It seems silly writing this now, because at the time I was going to school to become a math teacher, but I didn't take the job to learn a thing about teaching.  I took the job because I wanted to keep dancing.  That's all.  It was very selfish.

Looking back now, I realize that I actually did learn a thing or two about teaching that can carried to the math classroom.

A lovely demonstration doesn't do the students squat if they don't get to practice.  I would have gotten fired from my job if I showed up every day and put on a mini-production for my students, but never taught them how to do any of the moves I just demonstrated.  I know that probably seems like a "duh" to most middle school and high school teachers, but that's honestly how I spent my first couple years as a teacher (it was at the college level, but still).  I would come to class, having prepared every word I was going to say, and deliver what I felt was a pretty darn good lecture.  And, then...I would cross my fingers and hope my students did the homework.  Sorry to all my students those years.  Truly.

A good teacher gets everyone involved.  All the time.  A ballet student is not going to learn a thing if she says, "I think I'll just sit in the back and take notes on this one."  Ok, I didn't have any dance students want to take notes, but I did have students who were very reluctant to try new jumps or new turns, or new shoes for crying out loud.  I learned that in those cases, it's important to break down every single move until you can figure out where the block in the body is.  It was also important to point out what the student was doing right.

Mirrors.  Mirrors are essential to a good dance lesson.  Why?  Because dancers have to learn what their bodies look like when they move in different ways.  Eventually, a good dancer who has practiced in front of mirrors long enough can just feel what the body looks like.  She knows when she royally screwed up a pirouette and she knows when she nailed a grande jete.  This is what we try to foster in our math students, too.  We want students who can feel when something isn't going right.  When their answer isn't reasonable, when their proof is just getting messier and messier, when the calculator's window needs to be adjusted.  These things don't happen, though, unless students get mirrors with which to evaluate their work on regular basis.  Metacognition and all that jazz.

I tried to find some kind of picture of me as a dancer as proof that this is a real thing.  But, I guess my mom must have most of that stuff.  The only thing I could find on my laptop was this video.  If I remember correctly, this was my last semester of college.  I was practicing at the studio I taught at for a dance I was going to perform at my school's Chapel later that week.

A Day in the Life: 11.14.12

6:30  Phone alarm goes off and cats know it's time for breakfast.  Text hubby Good Morning and that I can't wait for him to get back from a business trip tonight.

6:31  By now cats are furious.  Feed them.  Shower.  Wake up.  Dry hair.  Get ready for the day.  Check ingredients for a cake I know I need to bake later.

7:40  Drive to school.

7:48  Arrive at school and wonder why the faculty lot is full already (7:50 is our official start time, but still...). Oh, yeah, Faculty Meeting today.  That's why I opt for the afternoon meeting.  Walk up to the third floor with one of the science teachers.  I love my floor of all math and science teachers.  I arrive at my door, and a student is waiting for me, ready to ask questions about a Pre-Calc quiz.  I tell him we're going to take one more day to review.  I start arranging my desks in five groups.  Recruit aforementioned student to help.  Turn  on SMARTBoard, TV, and computer.  Check emails.  Log onto school attendance/grading system.  Quickly enter the 100 or so scores from yesterday's Algebra II test that I graded util 10:30 last night.  Co-worker comes in and greets me with Starbucks.  This is going to be a good day.

8:15  First bell rings.  Students enter, surprised by the arrangement of desks.  "Are we not having a quiz today?  Is it going to be a group quiz?!"  I give them instructions to take a seat anywhere for now but not unpack.

8:20  Second bell rings.  I explain that we're taking one more day to review the material on the upcoming quiz, and why what we're studying now is valuable and I want them to know it well.  I also explain that the students will lead the review today.  I rearrange them a bit, making sure there's at least one math superstar in each of the five groups.  I assign each group a team leader (the superstar).  Each group has a problem on every desk (Group 1 has a different problem from Group 2, etc).  I explain how today's review will go:  each group will work on the exercise on their desks.  If they have any questions, they can ask their team leader.  After seven minutes, the team leader will choose a new leader.  That leader will stay behind to explain the problem to the next group. Every one else rotates on to the next set of desks.  We'll repeat this five more times.  Because this is a small class (24 students) everyone will get a chance to lead at least once.

8:40  The students have rotated a couple times now and have mastered the format of the review.  I learned this technique from a colleague and make a mental note to thank her.  I hear great discussions going on.  Students teaching students!  A teacher's dream.  It's unnatural for me to just stand by my desk and not help, but, today, they really don't need me.

9:10  We finish the last round and I ask the students to rearrange my desks to their normal positions.  I ask for feedback on the review.  Students say it was very helpful and that they're much more optimistic about the quiz now.  We'll see how it goes tomorrow.  I remind students to check the homework folder for any graded homework they need to pick up.

9:15  Bell rings and I wish my students a wonderful morning.  I open my Notebook slides for today's Algebra II lesson.  As students walk in, I hand back tests, congratulating most of them, as this class did much better on this assessment than on the last one (both were on quadratics).

9:20  Bell rings again and I welcome the students.  There were instructions on the board to grab a worksheet on the way in.  Most have done so.  I let them know that if they are not happy with their test grade, they are required to come see me three times before Thanksgiving (with their test on them), and then I will give them access to a make-up version of the test.  The intercom announces the voice of a principal, asking teachers to turn their televisions to Channel 78 for today's school news.  My TV is already on this channel, so I just turn up the volume.  As students watch, I take attendance.

9:25  Today's lesson is a bit of review material that I want to make sure the kids are solid on before we enter the next chapter on polynomial functions.  I start with the extra credit writing assignment I gave them on yesterday's test.  The assignment was to explain what the calculator is graphing when you type in y=x^2+7x+10 and to give at least one point they knew was on the graph for certain and to explain their logic.  No one gave a good answer to this.  No, not one of out my eight dozen Algebra II students.  So, we'll try again today.  I ask someone--anyone!--to give me a point s/he KNOWS is on this graph, without looking at the calculator. *Cue crickets chirping*  Eventually we get to (0,10).  I ask for another point.  Then another.  Could we do another?  On the back of the worksheet, I ask them to find three more points they know will be on this graph.  We talk some more.  Then I ask them to answer the original question again.  What does the graph of y=x^2+7x+10 mean??  I call on some students to read their answers.  I hear good phrases like "infinitely many ordered pairs" and I'm satiated for the time being.  But we are not done with this story.

9:40  Have students turn to the front of the worksheet again.  Explain the reasoning for today's lesson (review so we get to the really good stuff).  My go-to strategy with Algebra II is to do a problem with them and then have them try some on their own while I walk around taking questions.  I'm not a homework-giver in Algebra II (though I am in Pre-Calc), so it's important that they get the practice time they need in class.  We work on adding and subtracting polynomials.

9:55  Switch to multiplying monomials with like bases.  Ask students to turn to the back and summarize their findings in complete sentence(s).  Call on students to share their answers.  Explain how "adding the exponents together" doesn't qualify as a complete sentence.

10:00  Try some slightly more difficult exercises to practice the shortcut they remembered/discovered.

10:10  Give students a couple review problems on graphing linear inequalities.  (Yeah, it's random, but I'm a firm believer in review.)

10:15  Bell rings.  Tell students where to turn in worksheets, remind them about the opportunity to take a make-up test, and wish them a great morning.  Next class (Algebra II) starts to enter.  I hand their tests back as well.

10:20  Bell rings again.  I welcome the class back.  This hour goes very similar to the one before.

11:12  Dismiss students who are enrolled in a tech course off campus and need to catch the bus.

11:15  Recruit the students who are finished with the worksheet to rearrange my room for next hour's Pre-Calc class.

11:20  Bell rings.  Wish students a good day.

11:25  Bell rings again.  This class runs similar to my first hour.  Students are thrilled they have one more day to study.  I explain how the review will go and they have at it.  I intentionally gave the hardest problem to the group closest to my desk, so I could help those groups throughout the hour if need be.  But, they don't need me (hoorah!).

12:18  I ask for formula sheets back as well as feedback.  This class liked the review as well.

12:20  Bell rings.  No time for students to rearrange my 36 desks, so I'll have to do that after remediation time.  A few students stay to ask questions.  This is their lunchtime, but it's my time to remediate any students that need it (I assign those who have D's and F's).  One student also comes from 2nd hour Algebra II to gain access to that make-up test.

12:50  I dismiss students who are there for remediation (I assigned 4 Algebra II students, but most were gone today).  They go eat lunch.  One stays behind to tell me the reason she's been missing so much class is because she's 9 weeks pregnant.  I try to focus on the joy (a new human being!) and not the hardship I know she'll have to endure for the next several years.

12:52  I start to rearrange the desks.  Turns out it's a lot easier to mess them up than get them back together.  Another student from my 2nd hour class comes in.  He says he wants to talk about the back part of the test...but doesn't have the test with him.  I bring out another copy and we discuss how to use the discriminant to determine the number and types of solutions a quadratic equation will have.

1:05  I wonder if I have enough time to grab lunch today.  Most days I just wait until my plan (2:30), but I think I can grab something to eat today.  I head to the math lounge to heat up my lunch.

1:10 Take my lunch to my colleague's room across the hall from my room.  This is my first interaction with an adult all day.

1:23  Head back to my room.  Thirteen full minutes for lunch--that's actually a good day.  Once I get to my door, students are waiting for me.

1:25  First bell rings.  I check to see if there are any emails I need to respond to asap.  I try to stay on top of emails as they come in throughout the day.  Sometimes that's easier said than done.

1:30  Second bell rings.  I welcome students while I continue to pass back their tests.  Class runs pretty similar to my last Algebra II classes, with the exception that this class wants to see one of the test problems worked.  It seems like they're just buying time, so I work the problem somewhat quickly and move on to the lesson.

2:25  Bell rings and I wish students a wonderful day.

2:27  Collapse in my chair.  Have I sat down at all today?  Have I gone to the restroom yet?  I should do that.

2:28  Use the restroom.  This is my planning period, but I oftentimes have one or two students in my classroom during this time for one reason or the other.  Today, a student from 5th hour is staying to gain access to the make-up test.  I ask her to correct her first test on a clean sheet of paper, showing all work.  When she finishes, we talk about her mistakes.  I feel confident enough to give her a make-up tomorrow.  While she works, I work, too, planning for Friday's Pre-Calculus lesson (Law of Sines) and tomorrow's Algebra II lesson (multiplying polynomials).  We're going to try lattice multiplication tomorrow and see how it goes.

3:25  Bell rings.  School's out, but I feel nowhere close to being ready to leave work.  I put my planning aside for a few minutes, meet another math teacher in her room, and we get ready to attend the afternoon faculty meeting (whoever thought of offering two meetings on the same day was a GENIUS).

3:30  Faculty meeting.

4:10  I head back to my room.  A student I don't recognize meets me at the door.  "Are you the math Mrs. Peterson?!"  "I am!"  "I need help in Pre-Calc!  Can you tutor me?"  I make a quick judgment call.  She's not my student, but I'm not going to refuse helping her.  I tell her she's welcome to come to my remediation sessions anytime.

4:12  Back to planning.

4:30  A friend calls and asks if I want to have dinner with her (she knows my husband won't be home til late).  I say sushi sounds amazing.  I have to stay a bit longer, but I promise to give her a call when I get ready to leave.

4:32  More planning.

5:30  I make all the copies I need for tomorrow's Pre-Calc quiz.

5:32  I turn off my TV and computer and lock up for the day.  I'm quite impressed with my early departure time, especially with the faculty meeting.

5:45  Arrive at friend's house.  We decide to pick up Target sushi because I have to bake some cakes (and hence pick up some supplies) for an International Dinner my husband's work is hosting tomorrow.  I decide on Philadelphia Rolls.  Yum.

6:00  Check out at Target and drive to my apartment.

6:10  Arrive at my apartment.  The cats want more food.  I feed them before I change out of work clothes.

6:15  My friend and I scroll through Netflix, deciding on a movie to watch.  We settle with First Wives Club.

6:18  Movie and sushi.  If hubby has to be away, this is the way to spend it.

7:40  Movie finishes and I take my friend back to her house.

8:00  Back to my place.  I need to start those cakes for tomorrow.  I decide I'll bake the cakes tonight and finish the glaze tomorrow before we leave.  The cake I'm baking is called Silvia Torta, named after the queen of Sweden.  I want to bake three of them, so I triple the recipe, converting from metric to English as I go along (good thing I'm decent at math).

8:30  Realize this is not going to be enough batter for three cakes.  I pour the batter into two pans.

8:35  Start another batch for the third pan (1.5 times the recipe, I remind myself).

8:50  Finish the third batch.  Pour.  Put all three pans in the oven.  Set timer for 10 minutes.

8:52  Check personal email.

9:00  Timer goes off for cakes.  I rotate them and set the time for another 10 minutes.

9:02  Math Twitterblogosphere time!  I check Twitter and read some of my favorite math blogs.

9:10  Timer goes off again.  Cakes are almost done I rotate them once more and set the timer for a final 5 minutes before going back to the internet.

9:15  Cakes are done.  Apartment smells tasty.  Glad my husband will come home to this smell tonight, because it's not often I have food made any more.

9:30  Start to write my Day in the Life post.  It's getting too long.  Oh well.

9:47  Get a text from husband that he just landed in Tulsa.  Yes!

10:05  He enters the apartment with a "Hi, Love!  You've been baking?"  My heart is happy to hear his voice again.  We talk and catch each other up on the day before heading to bed.

10:30  Ablutions.

10:37  Sleep.

When I had to defend my profession twice in one hour

This Saturday I was at wedding, and I started talking to one of my friend's brothers, whom I've known for several years.  This brother is a senior engineering major and is currently tutoring two students in Algebra II.  He said something to the effect of, "I don't think the teacher took the traditional route to become a teacher, and you can tell, she just doesn't really know what she's doing!"

"How can you tell?" I asked, trying to mask my anger at his accusation.

"Well, one of the kids will come home with stuff she's written on the board, and it's just totally wrong!"

"And how do you know that the student didn't copy it incorrectly?"

Mmmmhmm.  That's what I thought.

My grief:  I hate how many times people assume we went into teaching because we couldn't do anything else.  This is a kid who hasn't even graduated college yet, and he feels he has more math knowledge than someone who breathes it day in and day out.  Listen up.  We went to school, too.  And most of us were freaking awesome at it.  That's why we became teachers--to stay in school and share what we love with other students.

Fast forward a few minutes, and my husband and I are leaving the reception early to catch up with a couple from our church, both of whom are youth pastors.  When they asked me how my job is going the man proceeded to tell me:

"You just have to make it interesting for the kids, you know?  Relate it to their lives.  Like video games and stuff."[1]

My grief:  Why do people assume that we don't try to make mathematics interesting?  Do they really think I sit at my desk pondering, "My, now how boring could I possibly make this?"  I'm also sick of hearing, 'Relate it to their lives!'  Good God, I'm trying, but these kids can't add or multiply rational numbers, and by the end of the year they're supposed to recognize conic sections, sequences and series, and logarithmic functions.  If it's so easy, you try it.  Please.  Oh, and while you're at it, go ahead and throw in some video games.  Because that's not a disaster waiting to happen.

Here's the thing:  I'm not trying to be down on my profession.  (Although it was a rough day at school, so I should probably wait to post this.)  But, it's just very clear to me that teaching is something people think they know so much about (because they've all been in the presence of multiple teachers), but unless you are a teacher or you live with one, you really have no idea what it's like.

That's why I'm very enthusiastic about A Day in the Life of  Math Educator.  I plan to write mine sometime this week.  Consider what you just suffered through the prologue.



[1]  Do I tell you how to do your job?

Quadratic RAFTs

I felt like my students were not yet ready to test on quadratics again (all methods of solving), so on Friday we took a day to recap what they've learned so far by writing RAFTs.

When I first heard about RAFTs, I was pretty excited, but I wasn't sure how juniors and seniors would respond.  To be honest, we do some pretty cheese-ball stuff in my classes, and I think this qualifies as such.  But, cheese-ball can be hilarious.  Here's evidence of the hilarity:

Role: Discriminant

Audience: America

Format: Campaign Ad[1]

Topic: The usefulness of the discriminant

Dear people of America,

They drew this on their paper,
but I'm too lazy to scan it

Are you tired of solving quadratic equations and wondering what the answer should be?  You waste minute upon upon trying to figure out when to stop solving.  Not with the discriminant.  With the discriminant you can instantly know what to look for while solving.  The problem under the square rot becomes hardly a problem at all if you vote to keep discriminant around.  Do yourself a favor, and check 'Yes" for "Vision D"!

Sincerely,

Board of Discriminants  

*****

Role:  i

Audience: Negativity

Format: Letter

Topic: How i and the negative numbers work together

Dear Negativity,

Your square root is always bringing us together.  At first we had a problem because you were always being fake, but then I came around and made being a real a possibility.  I know that sometimes your square root makes you feel imaginary but I'm always there to rescue you when he does that.  Many, many years ago you were a problem to everyone and no one knew how to fix you.  When I came along things changed and your negativity no longer was a problem.  I love you and your square root.

Sincerely,

i

*****

Role:  i, The Illusionist

Audience: Potential magic show-goers

Format: Ad

Topic: The coolness of i

Hi, my name is "i."  Some call me "Imaginary," and some call me "Illusion."  If you come out to this amazing show you won't regret it!  There are many fascinating things about me that I would like to show you!  Depending on when you catch me at the show, depends on my reality.  Let's just say there is a certain pattern to me.  Sometimes I am just imaginary when I feel like being myself, but I can also be in the form of -1, -i, and 1.  Do you think you can figure me out?  Come to the show and you will see!  Or will you...?

*****

I let my students work in "groups" of ones, twos, or threes.  They brainstormed on whiteboards, and then wrote their final product on a clean sheet of paper.  The activity took about thirty minutes.  A few of my students had written RAFTs before, but most of them had not.  So there was a lot of "I don't get what you want us to do."  And there were a few kids who just sat there for the first few minutes, which I'm pretty ok with.

I definitely had to encourage some students more than I did others.  But, reading the final drafts was both fun and enlightening for me.  It's clear that there are some topics that the students really understand, and some that we really need to discuss further.  This is what I love about writing in math class:  it's incredibly revealing, is it not?

[1] Sounds more like an infomercial to me, but to each his own.

Warm Up for i

Sometimes we have to relish in the little things, right?

This is a warm up I gave to my Algebra II students, just a couple days after they had first been introduced to i:

While I do like the warm up, what I'm really quite proud of is how I implemented/graded it.  When students felt like they had finished the warm up, I had them let me know.  I checked their work quickly.  If I liked what I saw, they were given the day's assignment (and a 100% for the warm up).  If not, they were given some verbal questions from myself, such as...

"You say i is imaginary, but you also say it's equal to -1?  Are you saying it's impossible (not real) to lose a dollar (-1)?"

"i is equal to the square root of 1?  But the square root of 1 is...?  Oh, so we need two symbols for the multiplicative identity now?"

"i is equal to i?  Try again.  This time tell me something."

Yes, I was harsh and sarcastic.  But this is an important concept.

Eventually, everyone had true sentences on his/her paper (which means everyone who came to class got a 100).

Each day I've been doing an "EOI Preview" as a warm up and I've been taking the highest 3-4 grades for the week.  This warm up gave everyone a chance to get an excellent grade in for the week, and I didn't let students move on until they could articulate the truth.

I know, I know...I really need to switch to Standards-Based Grading.  Sigh...

They're going to be prepared for calc...so help me God

I've written before about how I feel like a concept we think our students get that they really don't get is the composition of certain functions, specifically trig functions and log functions.  I made a vow to myself to emphasize compositions with my Pre-Calc students a lot this year, so that when they do get to calculus, the Chain Rule and u-substitutions will be two of their best friends, as opposed to worst enemies.

I was reminded of this vow when I asked a student to read an exercise from the book out loud.  The exercise started like this:



And this is how she read it:

"Sin" [as in a transgression, not a trigonometric function] "times pi over two minus x."

I wanted to say, "When have you EVER heard anyone say it like that, girl?"  But, I remained calm.  I ignored the mispronunciation (we have bigger fish to fry here), and focused on the "times" part.

It seems like every time I have this conversation ("It's not 'f times x,' it's 'f of x,' guys."), I feel like the kids are just nodding to get me to shut up.  I can't blame them.  I did the same thing in grad school [way] more than once.  As long as I make the prof think I understand what he's saying, all will be well.

But, inputs.  They're kinda a big deal.  What worries me is that it seems like students often view inputs as some kind of multiplication as opposed to actual arguments, which makes sense as the notation is very similar (parenthesis for both).

I continued to notice this was a problem as we were verifying trig identities.  I don't know if the kids just got so into the proofs that they forgot a few fundamental things...like what sine and cosine are...or what was going through their heads exactly.  But, let me tell you, I saw crap like following slide all. the. time.  So, I made them figure it out:

I would not tell them what was wrong, but I did mention it was subtle.  When they finally started figuring it out, we talked about why we need all those theta's!  Our dear trig functions are meaningless without them!

It's a small step, but if it gets them to remember that these trig functions must have an angle at which they're to be evaluated, even if that angle is arbitrary, well, then, that's a good thing.

Is mathematics invented or discovered?

We've been solving some quadratic equations in Algebra II currently, and I've had an ulterior motive this whole unit.

Quadratic equations seem to lend themselves particularly well to math history lessons (or "math commercials" as my principal calls them--love that).  For example, when we talked about the Square Root Principle, I gave a mini-lesson on Christoph Rudolf (names that rhyme are the best, aren't they?) and the introduction of the square root symbol and how it's supposed to resemble a lowercase r, etc.

I asked them innocently here, "So, do you think mathematics is invented or discovered?"

If they said, "invented," I said something like, "So, the square root of two didn't exist until Rudolf came up with a name for it?"

If they said, "discovered," I said something like, "So, you're just going to ignore the contributions people like Rudolf made to mathematics?"

I let them hash it out a little, playing devil's advocate all the way.  And then I ended with, "Well, interesting conversation, guys," and proceeded to my next slide.  Which, inevitably had the effect of "WAIT!  Aren't you going to tell us?"

"No."

Which then had the effect of, "I'm going to Google it!"

"Go for it."[1]

The next day, Day 2, we continued with the square root principle, but now we tried to solve equations like x^2=-1.  I let them try to convince each other that there is no real solution to this equation (though their multiplication skills are still lacking, so...sigh).  Here's where we talked about imaginary numbers and a mini-lesson on Euler ensued.  I told them Euler couldn't stand not having an answer to this problem, as it--along with other problems like it--had been appearing in mathematics for nearly two thousand years.  So, Euler made his own solution, and called one of the solutions i.

"Now do you think mathematics is invented or discovered?"  We took a poll[2]:

3rd Hour

5th Hour

At the end of class, I had them write a letter to me defending their answer.  They were instructed to choose only one (invented or discovered).  Here are two really great letters, one from each point of view:

Dear Mrs. Peterson,
Mathematics was discovered, because just because a human didn't know the answer to something doesn't mean it doesn't exist.  Before the Pythagorean Theorem was invented, a right triangle still had an area.  Humans simply put words/letters/numbers and theorems to help us find the answer, and explain math, but the problem they solve, and answers they find, were always there.  Some species of animals haven't been discovered yet, but when someone finds them, they didn't invent the animal, they discovered it.

Dear Mrs. Peterson,
I believe mathematics is invented.  I believe this because invented means to create or design something that has not existed before, or make up an idea, name, story, etc.  You have to create a name for mathematics to exist.  One apple is not one apple unless you give a name to the number or quantity of the apple.

The next day, something happened that I think will go down as one of my favorite teaching moments of all time.  A student, who has said from Day 1 that she's not good at math, came up to me before class and looked at me with her precious, sincere, huge brown eyes:

"Mrs. Peterson?  I really need to ask you something."

"Go for it.  What's up?"

"Can you PLEASE tell me--is mathematics invented or discovered?  I can't stop thinking about it."

Cue burst of emotion and huge cheesy grin on my face.  Why was this so wonderful?  Because she just experienced what makes mathematics so addictive:  the deep longing to solve or to prove, and the pleasure that follows the accomplishment.

Luckily (or maybe unluckily) for her, on this day, Day 3 of our discussion, I wrote a letter to my classes, defending my point of view.  Now, I didn't give myself the same restrictions I gave them, and I typed this up the night before (I know, bad Rebecka), so it's pretty rough around the edges (hey, that's the great thing about teaching--now I have a whole year to make it better).  But, here's what I wrote:
Discovered or Invented

What was so, so cool about this whole discussion is that it really appealed to most of my students.  They were hooked.  They kept asking about it.  They wouldn't let it go.

What more could I ask for?

So, what do you say:  Is mathematics invented or discovered?  I'd really like to know your input...so I can make my letter better for next year.


[1]  When I checked in on these students, they seemed more confused than when they started.  Let's hear it for UnGoogleable Problems!

[2] Don't let the total numbers fool you.  Only 1/2-2/3 of my classes participated (don't want anyone thinking I have a class of 17!).  Not sure if the rest didn't want to commit to a single answer, if they didn't have access to a phone, or if I just didn't quite hook 'em...

Trig Art Projects 2012

At our school, our Pre-Calc/Trig students create trig art projects every fall.  I know it's a pretty popular thing to do, but I gotta say, I'm pretty sure I got some of the most amazing trig art ever.

The math criteria:

• Use at least 3 trigonometric functions
• Each function must have at least 2 periods
• There must be at least 2 shifts (vertical or horizontal)

Once students figure out their functions, they graph each function on a transparency or a sheet of wax paper.  This acts as their blueprint.  From there, they copy their blueprint onto a quarter of a poster board and add their artistic genius.  Finally, they fill out a rubric, telling me things about each function, such as its domain and range (they must state the domain and range that they used for THEIR project; i.e., "all real numbers" does not cut it), amplitude, and any shifting.

Students working with Desmos in the Math Lab

I took a day to let them play on Desmos.  This was fantastic because they were able to see immediately what  their transformations did to each function, without having to graph each function fifty times by hand.  Also, I showed them how to restrict the domain of their functions, which taught them a teeny bit of programming.  I feel that using math to understand technology is one of the greatest things I can teach my students.  So a day in the Math Lab to play with trig functions was well worth it, in my opinion.


I let my students vote on the Most Creative Award.  Here's their winner from 1st hour:

And from 4th hour:

Each winner got a gift card to Target in the approximate amount of $3pi ($9.42).  And, no, when I asked for that amount, the cashier's face was not nearly as disconcerted as I was hoping.

But the kids laughed.  And that's what matters.

Teepees for Factoring

One thing that's been very shocking to me as I've made the transition from college instructor to high school teacher is the lack of number sense that many of my [Algebra II] students possess. Estimation skills, knowledge of times tables 0-12, and the recognition of a negative sign are so sorely lacking.  On a daily basis, I get told that a negative plus a negative is a positive, because, "two negatives make a positive, Mrs. Peterson."  Students also regularly explain to me that when we multiply a negative times a positive, the product will take the sign of the larger factor.

I haven't figured out how to break these bad habits.  But I'm working on it.

Needless to say, the thought of teaching factoring was a bit daunting.  How can I teach them to un-distribute, if they can't distribute correctly in the first place?

And then a couple colleagues of mine introduced me to the x-method, or what I call the teepee method.  This may be old news to many, but I had never seen it before, and I found that it was just the bit of organization some of my students needed in order to factor trinomials.

So, let's say we want to factor x^2-9x+20.  We create the following teepee:

Then we find two numbers that multiply to be the top number and add to be the bottom number:

And, viola!  Then we can factor the original trinomial: (x-4)(x-5).

This by no means solves all my problems.  How can we find those numbers in the first place if we don't know how to multiply?  However, it is a nice little organizer for those students who are visual learners.

I can't take any of the credit for this visual organizer.  I'm just passing along what I learned from my wonderful department.  But, in the words of LeVar Burton, "Don't take my word for it."  Here's what some of my students wrote when I asked them to choose their favorite form of factoring from the ones we had discussed so far (GCF, difference of squares, and trinomial factorization) and tell me why...

“Trinomials are my favorite because I like to make lil x’s and then put the numbers inside that would make the others true.”

“My favorite factoring exercise is trinomial factoring because it really makes you think.  The x’s really help too.”

“The trinomial factoring is my favorite because it’s easy to use the teepee.”

Review Game: TRIG BRAIN POWER!

To prepare for an upcoming test last week, my Pre-Calc/Trig students and I had a great time playing TRIG BRAIN POWER! (because I can't think of a better name, and yes, it must be in all caps).

It's quite simple.  On the SMART Board, my "home page" was a unit circle with six adorable colored brains sitting on the positive x-axis:

I split students up into six teams, one team for each colored brain.  I would show a review exercise, and when every single person on the team had a written response, the team raised their hands, which signaled me to come check their answer.  The first team with the correct answer got to move three spaces on the unit circle, the second team got to move two, and the third team got to move one.  After each exercise, I had the winning team explain their strategy, and if there were no questions we moved on to the next exercise.  The team that rotated through the greatest angle at the end of the hour won (we had to add 2pi rads a couple times!).

I essentially adapted a game called BrainSavvy that I found on SMART Exchange.  If you want the Notebook file for this particular edition of TRIG BRAIN POWER!, just shoot me an email.

I liked this because it involved minimal prep, the students had to work with each other, they worked hard, and they had fun.

Function Transformations/Domain and Range: Day 2 (and a bit more)

Day 2:  Domain and Range of Parent Functions

I started the year with domain and range of a finite set of points, because, it's an easy concept and I wanted my students to know--Algebra II is totally conquerable.

Fast forward a month, and I think they might just ready to handle a continuous case.  We broke it up sloooooowly and built towards finding the domain and range of the four functions they found the day before.

Domain and Range

Obstacle #1:  Closed circle v. Open Circle
I really enjoyed this part of the lesson because the students were totally in to the closed circle v. open circle, which is awesome...but not before we jumped over a few hurdles.  When I asked them what an open circle denotes in mathematics, a few people proudly reported, "Parenthesis!"

No, sweethearts, an open circle does not mean parenthesis, last time I checked the dictionary.

So, we got to talk about my favorite branch of all--analysis.[1]  How could we write all the numbers between -5 and 4, but not including -5 (see fourth slide above)?  Of course, [-4.9, 4] was suggested.  But then poor -4.99 (just to mention one) gets left out!  Any number they suggested that was close to -5, but bigger, I could always find a number that was even closer (thanks, density!).

Hmmm...

And then a kid suggested something that was absolutely brilliant.

Incorrect, but brilliant nonetheless.

"Could we use -4.999...?"

Cue look of How far do I dare take this subject with these kids?

I wasn't quite sure.  And still am not so sure.  I decided to write this on the board:

-4.999... = -5

"This is what we know, and can prove, mathematically:  that -4.9 repeating is equal to -5.  So, sadly, writing [-4.999..., 4] does not help our quest because that's the same as [-5, 4], which is what we were trying to avoid in the first place."  BRILLIANT thought though, loquacious kid in the front row.

So, new notation is all that we could come up with in order to fix this dilemma.  Parenthesis.  As was wildly suggested before.

Now if only they could remember a few of the deeper ideas as opposed to just "open circle=parenthesis."

Obstacle #2:  Domain of Deceiving Functions
Here's what I mean by a deceiving function.  I took y=x^3 and showed a graph like this:

Most students were convinced that the graph would never pass x=-3 on the left, and x=3 on the right; hence the domain must be something like [-3,3].  And who can blame them?  They haven't developed a good sense for what the graph of a function is yet.

But that's ok.  Because we have Desmos.

So, to abettercalculator.com we went to graph the function, along with the line x=-3:

The kids' case was looking good.  Until we started scrolling down:

And down...

Aw, isn't that a lovely linearization...

And changing the y-axis view even more...

Gah!

And this convinced many (though not all)...the domain is indeed all real numbers.

Days 3-5 were spent focusing on vertical and horizontal translations.  We tried vertical stretching/shrinking, too, but I started to lose several of them, so I decided parent functions, domain/range, and translations were plenty for now.  We can come back for the rest later.  That's the beauty of 180 days, as opposed to the 48 that I'm used to.

Before the unit test, we played Kate Nowak's Speed Dating Game, but I adjusted it for these topics.  You can find the game cards I created here.

[1]  I claim analysis/advanced calculus as my emphasis in grad school...mostly because that's what I took the most classes in and that's what I took my written comprehensive exams in.  In any case, while I can't say I've read that many texts on analysis, I can say, that Understanding Analysis by Stephen Abbott is, by far, the best text I've ever seen on introductory analysis.  All calculus teachers should be required to read it.  Truly!  It's the best.  Get the whole thing for free here.  Go.  Read.  ENJOY.  He's a master teacher.

Function Transformations/Domain and Range: Day 1

As I've said before, I'm all for the motto of "The person doing the work is the person doing the learning," and I fully believe in making the students do the work in class.  However, I do think something gets overlooked a lot with this motto:  if I want my kids to be doing the work in class, I usually have to prepare a crapload outside class.  I'm willing to do this (most days).  But, I don't think that gets stated enough.

Being new to this age group (and never having attended a public high school myself), the trick for me is to anticipate the students' every move:  to come up with activities and lessons that are challenging enough to keep 36 students at 36 different levels engaged for an hour, but that are not too difficult so students just give up (or call your name so many times that by the end you're dizzier than a Turkish whirling dervish).

Some days these lessons flow out a lot faster than other days.

This was not a lesson that came quickly.  My insomnia from grad school is back and running, so this is a lesson that got started around 5 in the morning on a Saturday (I repeat, 5 in the morning, on a Saturday), and got finished sometime in the late morning.  For all that work, it has a lot of flaws.  But it has some good aspects, too.

So, here's Day 1 of Parent Functions, Domain/Range, and Transformations

Day 1:  Introduce Parent Functions

I really believe one of the most important skills I can teach my students is to read, comprehend, and subsequently follow directions.  There are so many cool things I've learned in my life, and so many more cool things I hope to learn.  But I couldn't have learned most of those cool things if I hadn't taken the time to read, comprehend, and apply my knowledge.  With that in mind, we had a big-time literacy day in Algebra II.

Students were given these xy tables, graph paper, and the directions below.  Not a whole lot else.  At first, they were livid.

"I don't get it!"

"Read the directions."

"What do you want us to do?!"

"Read the directions."

"You haven't taught us this!"

"Read the directions."

"UGH!"

"Read the directions."

It was an exhausting day, I'm not going to lie.  But they eventually caught on, and I learned that some of them are great at reading, comprehending, and applying, and some are not.  Here are the directions.  Many thanks to @Fouss for the subtitle. ;)

Parent Function Directions

Not everyone got to Part II, which was actually nice as it allowed for differentiated instruction.  I posted the best graphs from Part II in the front of the classroom so that students have these four parent graphs in front of them at all times for now.

What I liked

• Students READ.
• Students did the work.
• Students focused on a small amount of information:  four rather important graphs.

What I didn't like/Questions I still have

• Do they really understand that the graph of an equation is the representation of every single solution of that equation?  I feel like I say that a lot, but that doesn't mean anyone actually understands what I'm saying.
• It's not super exciting.  I know these graphs have a lot more interesting aspects to them then just "Draw an xy-chart and plot the points," and I feel like maybe I stripped them of a lot of their intrigue.
• I let the students pick their own groups.  I still don't know if that was good or not.  The complainers tend to be friends with each other.
• I only printed one set of directions for each group because I'm more than a little frugal with my copies.  Also, I wanted them to work together.  However, I think the students would have benefited from everyone having his/her own set of directions.

I will post more on the unit soon!  Hopefully!

Systems of Linear Equations Activity: 3 Cases

So, this isn't anything super fancy, but it worked quite well with my Algebra II kiddos (without much prep on my part, which doesn't happen often), so I wanted to archive the idea and hopefully get some feedback/ways to improve it.

I gave four lines in slope-intercept form and had my students get out a clean sheet of paper, fold it twice to create four quadrants, and write one of the lines at the top of each quadrant.

Then they were to write four categories (in each and every quadrant--oh my!):

• Given line (Y1)
• No solution line (Y2)
• Infinitely many solutions line (Y3)
• One solution line (Y4)

The given line (Y1) is the line I gave them.

For Y2:  we talked about what would need to be true about the second line in order for it to never touch the given line.  The kids were pretty quick to tell me that the lines would have to be parallel, and for that to be the case, the lines would need to have the same slope (and different y-intercepts, btw, cherubs).  A-ha!  We do remember some things from Algebra I!  So, we decided on a line that was parallel to Y1 and wrote it in the category of "No solution line."

For Y3:  we discussed what would need to be true about a line in order for it touch the given line at each and every point on that line.  Well...it's gotta be the same line!  Write that in the category of "Infinitely many solutions line."

For Y4:  the typical case, but I love that this activity made them think a little deeper about this case.  "So...what has to be true for a line to touch the given line once and ONLY once?"  Pause.  Pause.  Pause.  

Still, small voice:  "Different slopes?"

Oooo...

"So, a line with ANY slope other than that of the given line will intersect with the given line somewhere?"

Pause.  Pause.  Pause.

Unanimously:  "YEAH!"

Wohoo!  So, we made up a line with different slope and wrote it in the category of "One solution line."[1]

Graphing calculator time...

For the given line of y=2x+1, our y= screen may have looked something like this:

We changed the features for Y1 and Y3 so we could distinguish between the lines and actually see the calculator graph the given line again for the special case of infinitely many solutions.  I had them sketch these lines at the bottom in addition to stating the point of intersection for Y1 and Y4 (they could use their calculator).

I did one of these exercises with them and then had them do the same thing for the remaining three given lines on their own/with their partner.

About half-way through the period, I had them turn their papers over.  Using the same quadrants and the same lines they created, we solved each of the three cases algebraically.  That's twelve systems they solved in half a lesson.  The goal was to get them to see that algebraically a false statement is related graphically to two lines that never intersect (no solution); that a true statement is related to two lines that always intersect (infinitely many solutions); and that a conditional statement is related to two lines that intersect once (one solution).

Again, it didn't take lots of prep and I think it really brought together the geometry with the algebra.  I hope you're proud, Descartes.

[1]  Lots of students would just change the slope of the given line but keep the y-intercept.  Then, when they solved the system, they noticed that x always turned out to be zero.  "Mrs. Peterson!  I keep getting x=0!  What's going on?"  "What did you keep the same?"  "The y-inter...oooo..."  Light bulb.  One kid was so excited about this I truly thought he was going to pee his pants.  It's the little things in life.

Math Teachers at Play #54

Welcome to the fifty-fourth edition of Math Teachers at Play!  We have a great roundup of articles this month...

Literacy            

• Yan Kow Cheong at Singapore Math gives some ideas for journal writing in math class.
• Terrance Banks at So I Teach Math and Coach? gives a great literacy strategy that encourages students to be the teacher.
• Bowman Dickson at Bowman in Arabia shares how to start class with whiteboarding (reading a graph is a literacy component, if you ask me!).
• For more graph reading, I posted two AP Calculus problems I think are great.

54:  The number of countries in Africa

Instruction

• Guillermo Bautista at Mathematics and Multimedia explains common errors in MDAS operations.
• Dan MacKinnon at mathrecreation shows beautiful patterns that arise in the multiplication tables that we were all told to memorize in grade school.
• Glenn gives a pretty snazzy visual representation of quadratic equations that have complex solutions.
• Jeff Brenneman  at Trust Me-I'm a Math Teacher shares a project idea in which students identify the key issues to winning the 2012 Presidential election.

54:  The number of playing cards
(with 2 jokers)

Gamification

• Bon Crowder at Math is not a Four Letter Word shows how to get children thinking about graphing using a Lite-Brite.
• Lilac at Learners in Bloom shares some great ideas for introducing inequalities and pattern-recognition to your preschooler (whoa)!
• Mrs. M at The Daily Cupcake shares eleven awesome activities for kindergarten students learning numbers 0-5.
• @reminoodle at The MathSmith describes a group project she gives every trimester for students to create their own review games. Be sure to check out the impressive pictures!
• Christy at Just Another Step to Take gives five ways to create homemade board games for practicing math facts.
• Jennifer Smith-Sloane at Live. Teach. Create. shares an awesome foldable for converting fractions, decimals, and percents.

54:  The number of colored squares
on a Rubik's Cube.

Great Advice and Insight

• Denise at Let's Play Math! gives tips on how to teach math to your children, even if you experience math anxiety.
• Peter Glynn at Nature Math has started a discussion on why some kids hate math.
• Colleen Young at Mathematics, Learning, and Web 2.0 gives some great back-to-school advice.
• Jimmy Pai at The Pai Intersect discusses both his interests in and his concerns about flipping a classroom.
• Cindy Johnson at The Conic Card Lady's New Adventure shares fantastic advice to new high school teachers.

+54:  The international phone code for Argentina

The next installment of MTaP will be held at Mathematical Palette!  Submit your responses here.

Also, be sure to check out the latest Carnival of Mathematics and Mathematics and Multimedia Carnival.