Height Problem from Graphing Stories

Height Image from Graphing Stories

This week in my Precal class, we were working on increasing, decreasing, and constant functions. I showed them a Graphing Story. These problems are perfect for analyzing graphs of functions. I chose the one that asks the viewer to graph height vs. time based on a lady swinging on a rope in Costa Rica.

If you haven’t already, first watch it for yourself here.

I’ve done some of the other graphing stories before and I love them. We did this one, discussed various aspects of the graph, and eventually hit on increasing, decreasing and constant intervals of the function. Its a great activity, but an interesting debate arose regarding the given answer.

You notice that the answer (i.e. graph) has a relatively large increase during 7-10 seconds.

Height Graph

When I initially displayed the solution to my students, they debated whether this interval was accurate. They found it intriguing that, according to the solution, she swung roughly 32 meters higher than her original height on the platform. Some of them said yes, it was possible for her to swing 44 meters in the air. I mean, looking at the video her upswing does look VERY high – much higher than the platform she started jumped from. Some students said no, that the camera angle was playing tricks on us. It was such an awesome discussion, and to be honest, I told them I wasn’t sure about who was right, but I’d do my best to find out.

After school I spoke to a physics teacher at my school, Shane Coleman (who these students also have), about the problem to get his scientific opinion. He informed me that it is physically impossible for the lady to have swung as high as 44 meters, unless she was pushed hard, had extra momentum when she left the platform, or pumped her legs on the swing, none of which happened. He told me for the footage in the clip, her kinetic energy would not have exceeded her potential energy. In other words, she wouldn’t have been able to swing higher than her original height; it would have been less and less with every swing. I probably should have known this, because its sort of common sense, but whatever. So then I brought all this back to my Precalc kids and it made for a pretty good discussion while we ironed out the actual solution. Later that day, the students probed even more while in Shane’s class. Awesome.

Now here’s my question: have my students, Shane, and I missed something here or is the solution inaccurate? If so, my follow up question is what assumption(s) could we make about the swing that would make the given solution valid?

On a slightly different note, isn’t it also interesting that her height is 0 at two different instances, yet she never reaches the ground?


Friday Letters

This summer I got a dynamite idea from Rebecka Perterson at TMC14. The premise is that students have the option of writing you a letter on Fridays instead of doing the Warm Up or bell work activity. Then, no matter how short or how long their letter, you write them back and give it to them on Monday.

When I first informed the students about Friday letters, I could see a refreshing look on most of their faces. But things got even more interesting when told them I would respond to every letter I received. Without one letter being written, I could tell I had already made a lasting impression.

Friday Letter Box

So here’s my box. It was literally a 5 minute job. Literally. It’s a small lidless cardboard box with a file folder taped to the top wrapped in blue paper. In other words, it’s the saddest mailbox ever. The kids got a pretty good kick out of it. One of the kids said “it’s the thought that counts!”

Last week was the first Friday the students had the opportunity to write me. I provided the kids with 1/4 sheets of paper and post it notes for those that chose to write. I received 17 letters, which I was stoked about. For the most part they were light hearted, things like “I really like your class so far” to “What’s your favorite color?” to “The first Varsity basketball game is December 3, can you come?” I thoroughly enjoyed reading and responding to them all. I suspect the letters will get more intriguing as the year goes on.

Friday letters opens up a private line of communication with my students that I’ve never had before. I’d like to think that my communication was always pretty good, but Friday letters adds a new dimension. It allows students to communicate to me their questions, feelings, and thoughts in a way that is accessible, non-judgmental, and meaningful. They feel that their voice is heard while getting individual attention from me. Plus, I really get to know my kids, which I love…they’re all so unique. This will really afford me the opportunity to learn who they are as people and tap into their strengths.

How will this ultimately effect my classroom this year? Will students just eventually stop writing and be “over” it? We’ll see. I’d love to hear similar things that other teachers have done to open up this type of communication with their kids.


Classroom Economy

Dollar Bill

I have been meaning to write about a rewards/management system that I use in my classroom for the last couple of years. It’s math money. I have heard of other teachers that use fictitious money in their classroom, but mine has interesting and meaningful twist: students pay taxes and file tax returns for all the money they earn. I cannot remember who I got the idea from, but I remember it was an elementary or middle school teacher. If I can find who it was, I will post the link.

A few years ago, I had one of my (now) former students create some $1, $5, and $10 math money “bills” that I use for currency. He created it in Photoshop (they’re somewhat elaborate) and gave me the files. The little I actually know about Photoshop allows me to update them from year to year.

The students earn money for lots of things. It’s all about positive reinforcement. I especially prize participating in class and collaboration – and will usually pay students for these things. Attendance, homework, hard work, Student of the Week, exam scores, etc….these are also things that will earn a student some dinero. We also have various paid positions that students must be elected to (attendance taker, runner, etc.), each of which receive a weekly salary.

Students spend their money during auctions every two weeks. I auction off candy, dollar store items, homework passes, positive calls home, among other things. We even have 50/50 raffles every now and then. The possibilities here are endless, plus its where all the fun is! It’s crazy, students love to compete to be the highest bidder and spend their money.

As students earn their money, I keep a simple tally of how much they earn. Other than Student of the Week payments, all income is taxed. Students can choose to withhold income as they earn it – they simply give it back to me (i.e. the government). Each class elects a treasurer that will take my tallies and enter them into our class database (Google spreadsheet), which sums all taxed and untaxed money each student earns. At the end each marking period I take this spreadsheet and mail merge it into individual W-2’s for each student, which I print and give to them.

I have created a tax form that students complete on tax day. We literally spend an entire day doing taxes. They ask questions, get confused, and eventually learn (like most people who do their own taxes). Students itemize their deductions and compute their taxes using our tax rate table. I used to have adjustable tax rates, but it got too complicated. Deductions include high exam scores, donating to a class pencil fund, seeing me for extra help after school, and other “good” deeds. After they submit their income tax forms, the treasurer examines them to ensure there are no discrepancies. Some students will receive a tax refund while most others will owe taxes.

The whole process really does mock the tax-filing process. I like to think that after students take part in this process six times every year, they have a pretty good understanding of how the IRS does its thing. It’s also an effective way to promote positive behavior in my class. Using money in my classroom has afforded me loads of flexibility in terms of classroom management and allowed me to mimic the real world in countless ways. Plus, its just plain fun.


The First Two Days

So the first two days of school year 2014-15 have now come and gone. Here’s what I did.

Day 1: I really try and get away from the whole class expectations, syllabus, parent information thing on the first day. I have never really liked it because I feel it doesn’t set the right tone for the classroom. First impressions are huge for students, so besides taking attendance and having whole group get-to-know you discussions, I decide to do some sort of fun competition on the first day. It usually is indirectly tied to math, but directly tied to problem solving. It helps set the tone for the year.

This year I did an activity that I experienced this past summer at a workshop. I don’t know the name, but I’ll describe it. If someone knows the name and has a link to it, please post it in a comment. I moved all my desks to the sides if the classroom and set up the chairs in two circles. Each circle had about 10-15 students. I gave one person in each circle a little ball and had them say their name and then throw it to a non-adjacent person in the circle. That person would then say their name and throw it to a different non-adjacent person. This continues until everyone has caught the ball. I point out that each student needs to remember who he/she threw the ball to. And herein lies the challenge: to repeat the same sequence as fast as possible while I time them. The only rule is that the sequence of how the ball travels must remain the same.

The hope is that each group finds an efficient way of getting the ball through the sequence of people. With a little bit of scaffolding, most groups throughout the day found a pretty good way of completing the task.

Day 2: This is when I hit the students with most of the expectations for my classroom. I discussed the grading policy, overall expectations, seating charts, and the like. I also introduced them to a couple of things that I absolutely love.

1. Math money. My students and I run a pretty robust classroom economy. I plan on posting about this eventually, but it includes students earning money and spending it on various things that I present to them. The kicker is that it also includes the student’s paying and filing taxes on their income. We also elect a class treasurer and the whole nine. The kids have a ball with this.

2. Friday letters. I got this idea from Rebecka at TMC14 and I am stoked about doing it this year. I’ll post about it in the coming weeks.

3. The impossible shot. This is just fun. I found this online this summer and thought it would be fun to implement in my classroom. I have a line on the floor on one side the room and I taped a box to the opposite wall close to the ceiling. One student per week will have a chance to make it. I’ve added a pretty sweet award to any student that can sink the shot, too. Talk about motivation! I allowed one student to take the shot and they were instantly hooked. (I also had several of my colleagues trying to make by the end of the day!)

Next year, I hope to be a little more “mathy” with first couple of days. Here’s a couple of ideas that I have found that I would like to try:

  1. #Mathis Tweet Strips
  2. Something with Tangrams or IQ Circles
  3. Lesson on fun facts about me and a “test” afterwards (from John Mahlstedt) (still not mathy, but really cool)
  4. There are others that I can’t find at the moment, but if I do, I will post them here…

Looking forward to an awesome year.