Mistakes mean we're getting better, right?

We tried a modified version of Bowman’s Mistake Game in PreCalc this week to review for an upcoming game.  I split the class into six groups and gave each group a problem to work.  Then I gave them these:

Directions

·As a group, work your given problem correctly.  Then, have me check your answer.

·Once you have a correct answer, work the problem incorrectly, hiding your mistake as cleverly as possible.  Your "mistake" must be a true pitfall of the given problem (i.e., what kinds of conceptual errors would students likely make?).  Your error cannot be a simple arithmetic or algebraic mistake.

·When you're happy with your lie, put it on a whiteboard (no need to write out the original question).

·When every group is done, you will find the errors on the other whiteboards and vote on the group with the sneakiest mistake.  Winners get candy. :)

After everyone had looked through and analyzed each group's whiteboard, I brought the boards to the front and had a student from each group summarize the mistake one more time.  

They taped the original question face-up and the mistake face-down

Then, students voted on the best error.  We had previously discussed that the errors needed to be conceptual, big-picture mistakes.  Something that would tell me, “Uh, this kid doesn’t really know what’s going on here…”  Not something like forgetting to distribute a negative or simplifying incorrectly. 

Before the kids left, I had them give me one mistake they promised not to make, write it on a post-it note, and stick it to my door on their way out.

I plan on leaving these up as they enter the door tomorrow so they can be reminded of those promises right before they start the test.

_____________

Aside, and probably more important...

As usual, my first run at this activity wasn't perfect.  There's a lot that needs to be changed.  It's easy for me to get discouraged when an activity doesn't go exactly as I had planned.  But I've been thinking lately (dangerous, I know):  

(1) My class activities have to start somewhere; they can't just magically be perfect...isn't that what we tell our kids:  you have to practice and have patience if you want to become really good at something?  I guess the same goes with becoming good at making the students do the work.  Learning how to scaffold; learning how to ask engaging questions; learning when to step in and when to stay out.  This takes a lot of practice.  No matter how much preparation I put into a lesson or activity, I have to practice delivering it, too...and that can't be done without kids in the room.

(2)  My students have to be taught how to talk about math.  It's a language.  Providing places for them to talk about what they're learning is great...but I can't expect that the conversations will just magically happen.  If the conversations aren't flowing quite as well as I'd like, it's a-ok.  It probably means we're doing good stuff here, actually.  Because we're practicing something they're not particularly good at...yet.

Asymptote Bingo

We're starting rational functions next week in Algebra II, but I knew my kids weren't super strong with asymptotes yet.  Truth be told, we still have a hard time recognizing that vertical lines will be an x=____; horizontal will be a y=____.

So, we played some Bingo on Friday, and I'm hoping for a solid start on rational functions tomorrow.

We play Bingo a good deal in Algebra II, so the kids are pretty familiar with my set-up by now.  Here's how we do it:

1. On a personal whiteboard, draw four horizontal lines and four vertical lines, creating a 5x5 game board.  Mark the middle box as your FREE SPACE!
2. Fill in  your board with these equations.  You may fill them in however you'd like, but you must use all 24 boxes--so keep track as you go!

Now we're ready to play!

I showed a graph of a function and then told the kids to cross out the correct asymptote.  The first several functions were strictly exponential and log functions--which we've already studied.  Anytime someone got a Bingo, s/he got a piece of candy (thank you, dear parents and guardians).  We played all hour, which gave nearly every kid a chance to get at least one piece of candy.  I kept track of the equations we had used and had the students call off their equations so I could check their answers.

After awhile the students started asking, "When are we going to get to the ones with two or three equations?  I need one for a Bingo!"

Mwahahahahaaaaa!  They were asking to learn about rational functions, and they didn't even know.

Before they knew it, they were analyzing the asymptotes of rational functions without any real instruction from me.  Just good progressions from what they already knew to what they needed to learn.

And now I feel at least a little better about their knowledge of asymptotes.

If you want the Notebook Bingo file, click here.

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.

Math Dominos

We're down to the last three weeks of classes and then it's finals week!  At our school, all College Algebra students take the same final, and if they don't pass the final, they don't pass the class.

No pressure.

Actually, in all honesty, we have about a 97% college-wide pass rate, but it's still nerve-wracking giving a final that you, as an instructor, have never seen.  So, today we started reviewing.  With dominos.

We started with the 6/6 domino in the middle of the floor with the rest of the dominos spread out around it.  From there, each team was given a set of four problems.

Let's say Group 1's answer to their first question is 2.  Then they are to find the 6/2 domino and place it in their team's slot.  On to the next question, whose solution, let's say, is also 2.  Then they find the 2/2 domino and place it next to their first domino.  The first team to finish their row of four dominos wins.

I'm not gonna lie, this game took a bit of prep work.  I started out by creating the following domino creation (the pack I bought came with that nifty plastic octagon--perfect!).  I did this to ensure that no domino would need to be used more than once.

Each "ray" represents a team's problems.  So, from here I had to create problems that had the correct solutions.  For example, looking at the leftmost ray, I had to create four problems that had solutions of 5,1,2, and 5, in that order. (I decided to just do four problems instead of five.  So ignore the last domino in each ray.)  Obviously, having only seven numbers to work with isn't so fun.  To combat this, if a solution was 12.34 and I wanted the team to pick a domino with a 2, I wrote:  "1DOMINO.34."  Not super elegant, but it made sense to my students, which is what matters.  It also allowed me to use the same problem for more than one team.

When a team was finished, I had them add their dominos together to see if their sum matched the sum I had in my notes (kuddos to my husband for this suggestion!).

What I liked:

Five teams with four problems each

• It got students talking.
• It got students thinking about the final.
• They asked to play again...? 
• There were "checks" built-in:  Did you get an answer other than 0,1,2,3,4,5, or 6?  Has your desired domino already been played (is this an error on your part or another team's part)?

What I didn't like:

• I had one or two superstar students in each class that basically did all four problems for their team.
• It took a bit of time to come up with the right problems.

Exponential Regressions: M&Ms

M+Ms by HazeyNut

I think this idea originally came from Virginia Tech, but I may be wrong.  In any case, here's how we did exponential regressions in College Algebra this semester.

Each student gets a "Fun Size" bag of M&Ms.  Students divide into teams of 3-4.  Each team gets a napkin that they're asked to unfold completely.  The teams spill out their M&Ms on their napkins, making sure all candy pieces are lying flat on the napkin.  Now for the math...

1. Count the total number of M&Ms on the napkin (this will correspond to x=0, where x is the number of "shakes").
2. Fold the napkin over the M&Ms and shake, shake, shake so that the candies get mixed up well.  When done, make sure all pieces are lying flat.  Take away any M&Ms that don't have the M facing up.  Eat them.  Now count how many M&Ms are left (this will correspond to x=1).
3. Fold up the napkin, shake, remove M&Ms that don't have the M facing up, eat them, and count the leftovers (x=2).
4. Repeat Step 3 until M&Ms are gone.

After each turn, I asked all teams to tell me how many M&Ms were left, and I inputted their results into an Excel spreadsheet, which was being projected on the board.  I created a template so that with each number I inputted, a scatterplot began to form.  [Download the template.]  After five turns, one group's data (whose M&Ms cooperated quite nicely) looked like this:

After the M&Ms were gone, I asked each team to find an exponential regression using their graphing calculators that fit their particular data (they could look up at the Excel spreadsheet, where the data had been recorded for them).  In a perfect world, their regressions would look something like y=a(0.5)^x, where a is the number of M&Ms they started with.  Of course, the number of M&Ms doesn't diminish perfectly to half its previous size every time, so we got results that looked more like this (again, this was a rather good trial):

But the imperfection is good.  For one, that's life.  For two, it makes it a little less obvious as to what's going on and creates a nice starting point for some discussion.

After the students gave me the regression equations, I plotted the regressions on Excel (which you can do easily in just a couple clicks).  I asked which team looked like they had the best regression and then we compared r^2 values to see if they matched the students' intuition.

I really wanted to use Skittles for this project so I could call it "Skittles:  Taste the Exponential Regression."  Alas, M&Ms were half the price of Skittles and my frugality won over.  Maybe next semester.

Polynomial and Rational Function Review Day

Review days are a continual dilemma for me.  After each unit is finished and before each test day, I schedule a Review/Catch Up Day.  I'm not sold on this format, but I'm not opposed to it either.  Due to the fact that we live in Oklahoma, I definitely have to have a few built-in catch up days since we close our schools and colleges at the slightest sign of snow/"cold" weather.

I've been using these built-in catch up days for review, which seems natural enough.  The downside is that I often have very low attendance on these days. So, today I tried something new.  I offered something that nearly every one of my students will come to class for:  extra credit. 

You would think I would have had this revelation a while ago...alas.   

Math Mama wrote about a game of "Risk Your Beginning Algebra Skills."  I really liked the idea of the students having to wager a certain amount for each given question:  they have to be honest with themselves and decide how well they think they know something before seeing the answer.  So, I adapted Math Mama's idea and made a game of "Polynomial and Rational Function Jeopardy."  Before I showed them the answer slide, I would ask which students felt confident about their answer and then I had one of these students share with the class.  They were usually right on the money.  

I chose to do this in a slideshow format as opposed to a worksheet format because I know my students well enough to know that half of them would be working ahead and hence not contributing to the overall discussion (all my students are concurrent high school juniors and seniors).  Even if the overall discussion is not always where I would like it to be, some discussion still beats no discussion.

Overall, I think this review day was a success.  Students participated and hopefully got a good sense of what they needed to study for the upcoming test.

Here's the wager sheet I gave my students in Word format and PDF format.

Polynomial and Rational Jeopardy

View more PowerPoint from Rebecka Peterson

End Behavior Activities

We're currently studying polynomial functions in College Algebra.  Here are a couple activities I've done with the students to discuss end behavior.

Activity I:  The End Behavior Game
Not only do the students get to practice the Leading Term Test, but the teacher gets to enjoy a new variety of dance.

End behavior game

View more PowerPoint from Rebecka Peterson

Activity II:  The Polynomial Train
This is actually a twist on what I really did, but I think I will do it this way next time.

I'd start with a constant function (f(x)=1 in the example below) graphed using Desmo's calculator and ask a student to add a term to the function so that it would ________ to the left and ________ to the right.  The next student would be asked to add another term in order to change the function's tails to a new given end behavior.  After a few students, the function might look something like this:

So the first student was asked to add a term in order to change the end behavior so that it rises to the left and falls to the right.  The second student was then asked to add another term so that it rises to the left and to the right, etc.

I like using this calculator in class because the students can see how the graph changes as we change the output.

A final note...

One of my students showed me these hand motions and words (read from left to right) to go along with the end behaviors of polynomial functions.  He remembered it from years ago, so it must have stuck!  Maybe you've seen it before, but it was new to me.  I can just see a classroom of students taking a test and moving their arms in the air as they try to answer a question...

"Odd function"

"Even function"

"Nega..."

"...tively"