Rich Mathematical Problems in Astronomy

Mass and Escape Velocity (KEY)

A Math Teacher's Journal

I finally have some time to sit down and write a reflection on my first day (days, since I’m on an A/B schedule) of school. I really think this was my best start to a school year in the 9 years that I have been teaching.

Because our school is trying to create a 9th grade center without actually building a separate building, we have grouped almost all of the freshmen teachers in one end of the building. I volunteered to teach the 9th graders who were taking Geometry and Pre-AP Geometry (i.e. honors). As I mentioned in an earlier post, I really want to orient the class around problem-solving, using the Exeter problems as a base.

After doing some thinking, I really agreed with the people who had written about not using the first days of school for review. I decided I didn’t want to spend the first day of class reading the class procedures and policies to them — I wanted to do math! Therefore, here’s how my day went:

- The classroom is set up with students in groups of 4.
- I randomly distributed the kids by having them draw colored craft sticks that had numbers from 1 to 8 as they entered the room — that determined their group number.
- I used a Google form to collect student information. They could either use their phones, or school iPads to enter the information. I also had a slip of paper for them to write their names so I could make a seating chart.
- If it was a large class, I used a spinner to pick a color at each group. The person who had that colored stick had to stand up and introduce himself and the other people in his group.
- I told them that I was not going to read the procedures to them because I wanted them to get in the habit of reading instructions for themselves. Their homework was to read the procedures, sign the acknowledgement form, and get their parents/guardians to sign off as well. I also handed out instructions on how to sign up for a Khan Academy account, and told them that was the other part of their homework. Of the approximately 150 students I have, only about six or so don’t have internet access, so I think this will work.
- Here’s where the fun part started. I showed them the picture of the people laying the penny floor and asked what they noticed/wondered. Most of them wondered things along the expected lines: how long did it take, how many pennies, etc. Then, I handed each group a baggie containing four 6″ rulers and six pennies and a short worksheet that had them generate a rectangular floor to cover in pennies.
- It was hard for me, but I managed to be much less helpful than I normally have been.
- Several students were done very quickly, but it usually ended up that they had their units confused (ex. they had a 250 ft
^{2}floor that would be covered by 262 pennies). - Some of the sharper students immediately thought of dividing the area of the floor by the area of the penny, and they were a little indignant when I told them they were wrong. This was hard for me to balance how much to lead them, but for the most part, I think I did okay.
- One really sharp girl noticed that the penny was 3/4″ wide and calculated the area of a
**square**that was 3/4″ wide. She then divided the area of the floor by the area of her square. Her only mistake was getting her units mixed up at the beginning, but after she fixed that, she got it. - Once a group had settled on a number using 256 pennies (16×16) per square foot, I would use the six pennies to show them how their floor would be tiled and have them compare it to the picture. They then realized that it wasn’t quite as simple as they had thought. One girl actually used google to see how many pennies were in a square foot.
- At the Open House we had on Wednesday, one of the girls actually told me that she really liked it when I had lined the pennies up the way they had figured the area and then shifted the pennies over the way it would be tiled. She basically said that it blew her mind in a good way.
- When we had about thirty minutes left in class, I then handed out an excerpt from a book that really changed my life when I was in high school,
*Have Space Suit, Will Travel*. They were able to finish with about five minutes left, so we talked about some of their answers.

All of my classes (to varying degrees) were engaged in math on the first day of class! I also noticed that I did not lose my voice the way I usually do, which tells me that I didn’t have to do as much talking as usual. My principal came by to observe on both days — the first day, he came in while they were working on the penny problem; the next day, he came while they were reading the excerpt. I think he liked what I was doing.

The rest of the week was actually fairly smooth as well. We decided to have freshmen orientation during each freshman’s math class, so on Wednesday and Thursday, I took my classes to the auditorium for most of the period. Friday, we were finally able to actually start on our first objective, which was algebraic proofs.

This coming Tuesday will be the best measure of how my plans are going to work: students are supposed to take notes from my powerpoint and do some Khan Academy problems before they come to class. In class, we will work some of the Exeter problems on slope in groups, and then I will give them a quiz over both algebraic proofs and slope. I’m a little nervous that the problems are too hard, but since I have never taught honors kids, I need to see how far I can push them. We’ll see how it goes.

Graphing Whiteboard: I have been *obsessed* with this idea all summer. My new room only has 20′ of board space, and I get tired of having to draw axes by hand. This board is the leftover piece from the two tileboards that I had cut down into 10 32″x24″ group whiteboards. I then used spray adhesive to glue down chart paper, and then came the tricky part: to finish it off, I adhered clear plastic vinyl and smoothed out *most* of the air bubbles and stretches.

Sierpinski Triangles: Every year I have my students make Sierpinski triangles as a project. I usually keep the nicest ones of these, but I never really had a great place or idea to display them. Then I had a brainstorm: a Sierpinski triangle of Sierpinski triangles! This is on my window wall. As I get more triangles, I can add to it.

Penny Floor Tile: I want to have students doing math on their first day of class, and I thought it would be cool to show them this:

I will ask the “What do you notice? What do you wonder?” questions, and hopefully someone will wonder either how many pennies it takes or how much it costs. I will then pass out pennies and rulers to each group and give each group a room to tile. Their job is to figure out how many pennies it will take. I plan to let them struggle with this for a bit. For the solution, I made a 12″x12″ tile of pennies:

I have put together a group of posters to get printed, so I thought I’d share the ones I have made. Most of these PDFs are set to print out 11×17.

I actually found this quote on schoolfailblog.org, but I wanted it formatted landscape instead of portrait:

In honor of Max’s great presentation at TMC13 (and @druinok’s favorite font), I made:

If you want to get lost in great mathematically-related quotes, check out Furman University Mathematical Quotations Server. Some of the good ones I’ve found:

and

I get very tired of students who want to exert the least amount of effort possible. It occurred to me that just about any time a student asked me “Can I just …”, the student wanted to take an easy way out. Thus:

To Make

~~SEEC proposal~~~~Group whiteboards~~~~Graph whiteboard~~- Viking head wallhanging
- Circle design wallhanging
- Make posters
~~“What do you notice…what do you wonder?”~~~~“Let’s get better and do it on purpose!”~~~~Scott Kim’s “Mathematics” inversion~~~~Scott Kim’s “Alphabet” inversion~~~~To Understand the Universe…~~~~“Everything should be made as simple as possible, but not simpler.” – Albert Einstein~~~~“Whereas at the outset geometry is reported to have concerned herself with the measurement of muddy land, she now handles celestial as well as terrestrial problems: she has extended her domain to the furthest bounds of space.” – W.B. Frankland~~- “No human investigation can be called real science if it cannot be demonstrated mathematically.” – Leonardo da Vinci
- “The knowledge of which geometry aims is the knowledge of the eternal.” – Plato
~~Learning Objective posters~~~~SBG grading scale explanation~~~~Print posters~~~~Practice wall~~~~Group table labels~~- Group question stems
~~Penny Floor tile~~

Curriculum Materials

~~Parent letter and signature sheet~~~~Rules and procedures~~~~Finish google student info forms~~- Set up blog/livebinder (decide)
~~Penny Floor Tile WS~~- Unit 1 Test and Review

While I have always been a tech-savvy person who likes new devices/technologies, I also have a serious dislike of, and prejudice against, mobile phones. Part of this stems from my general dislike of talking on the phone; part of this dislike goes back to when I was a network administrator and my company needed to be able to get ahold of me 24/7. My students are always shocked when I tell them that, while I do have a cell phone, it’s usually locked up in my car or my scooter and always kept turned off.

Even as cell phones became smart phones, I felt no desire to join the crowd (in addition to being an introvert, I’m also an iconoclast). As more and more people seemed to walk around with phones held to their ears or staring down at their hands, I just could not see the attraction.

I started to do a little bit of rethinking during a staff development when one of my fellow teachers just took a picture of the powerpoint slide with his phone instead of writing everything down. I also found it useful when my friends could look up some info or find an address, but again, I had no desire to spend all of that money (plus the monthly charges) for some phone.

Then, I had my “Eureka!” moment. I’m not sure what started this thought process, but it finally occurred to me that a smartphone wasn’t really a phone at all–it was a handheld computer that took pictures, gave directions, played music, and oh, by the way, also could be used as a phone. When I started thinking of it as a mini-computer, I suddenly realized how handy it would be! The next hurdle was the price. Since I didn’t even know if I would find it useful, I didn’t want to be tied to a contract for two years. Fortunately, I found a no-contract plan that I can live with (Virgin Mobile).

One of my favorite book series is the “In Death” series of futuristic detective novels by J.D. Robb (Nora Roberts). In them, everybody carries around a PPC (portable personal computer). I joke with my friend S that I am not carrying around a smartphone, but a PPC, so that’s what we call it.

Here’s why I am writing this essay: I was pretty sure that I would find my new ~~smartphone~~ handheld computer to be useful and maybe even fun. What I had not realized is that it would make certain things that I had previously done on my desktop or laptop so much easier to do. I enjoy Twitter (@sandramiller_tx), but I had fallen off on keeping up with it because I wouldn’t always think of it when I was on my computer, and so many messages would accumulate that it just seemed futile to try to keep up. On my PPC, I can easily scroll up the list and keep up with what’s going on. This made an incredible difference as I went into #TMC13. What some have called the “back-channel” conversations were almost as important as the presentations, and I would not previously have been able to watch or participate in them.

Tieing in with what a couple of other people have written about introverts and extroverts, as I said, I’m definitely an introvert. I joke with my students that I’m anti-social, but I’m not that far along the scale (my father definitely is, however). I agree with Greg’s post that online writing and collaboration is easier for the introvert because there is a distance and an implied lack of obligation (on both sides). I love to lurk on both Twitter and Facebook, but I rarely post on either. What my “handheld computer” has allowed me to do is to access those conversations more easily, which makes me feel more a part of what’s going on, even though I rarely share anything.

In the month or so that I’ve had my new phone, I’ve only used it as a phone three times. I’ve been somewhat shocked that I have texted S as much as I have. I can see the attraction–if I have something short I want to tell her about, but don’t want to bother her with a phone call (the introvert again). More than that, though, this new electronic device allowed me to participate more fully than I would otherwise have been able to do in a truly incredible professional learning event. (And the Amtrak app kept me from going crazy waiting on trains.)

I just thought of this as I was reading some recaps of TMC13: During “My Favorites”, Chris Lusto offered the idea of having students work out the definition of a circle.

What just struck me was one of the activities challenged the students to come up with an example that fit a classmate’s definition but was **not** a circle. COUNTEREXAMPLE!! Counterexamples (especially **geometric** counterexamples) are so tricky for students to get. Here’s a built-in example of why they’re a big deal. Love!