A blog for sharing my math interests on the web, to post new materials for elementary, secondary and teacher ed, and vent mathematical steam when needed.
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What makes for these mathematical mini-obsessions?
Whenever I put something up on 101qs.com, I make sure that I pose questions on
at least 10 posts from other people. (I also never skip. [So will I ever have questioned everything? #anyqs] Though for some it can be hard to find mathematical questions. [Orange.]) On a recent trip I saw Alex Shum's cool revolving door picture.
It reminded me of the problems where you're trying to take a sofa around a corner. (Which reminded me of my very first favorite screensaver, inspired by Douglas Adams, of trying to fit a sofa down a staircase.) Whatever the reason, I immediately wanted to make a GeoGebra sketch.
I made a first pass. That got me wondering how big are those openings? In particular, would a revolving door ever have and opening straight through like the one in the diagram to the left? (Seems to defeat the purpose of a revolving door.)
What are the standards for revolving doors? Thanks to Google and the International Revolving Door Company, I now know more than I ever knew I wanted to know.
In particular:
Data!
So much to wonder about these. I love the idea of describing circles as circumscribing a rectangle and it makes perfect sense for construction. But now I want to know about this data. What kind of function is it? Does it make it so the opening is always more narrow the one sector of the door? Is the angle of the opening from the center constant? How do you choose between a 3 wing and 4 wing?
First pass on the data was pretty curious.
Good for making the sketch, but weird. Why would the three wing doors have narrower openings?
I made a weird little function to give the door openings, 2*radius (0.475 + 0.21 (doors - 3)), so that the ratio is .475 for 3 wings, .685 for four wings.
I liked the opportunity for modeling that this turned into, and it's also a good problem to show where modeling either supports calculation or is more efficient than calculations. Constructing the model in GeoGebra required data fitting, and some algebra to find the appropriate boundaries for the geometric objects.
And it helped me understand why 3 wing doors have narrower opening than 4 wing doors.
"So does this bore the heck out of you?" a student asked me.
The problem is that this was after two days of doing the Barbie Bungee Jump activity. The fabulous Barbie Bungee Jump. (Cf. Julie and Fawn) I was assisting a very nice and competent substitute teacher.
This is a good school with good students and good teachers. What's going on, or not going on? My first brief observations:
Students were given all the steps to follow. Being told to do a, b and c and then doing a, b and c is not engaging.
There was no hook. How much of a hook depends on the lesson. This one could have used the video, a discussion about bungee jumping, etc. Going straight into 'here's what you do' gives no chance for wondering. Even if it's what the students want or are asking for.
There was no expectation that this was worth their time or could be interesting. There is always time to start, but this might also be about developing a culture of inquiry. Students need to learn that this is what math is, and this is what math class is like or could be like.
Recount: Day 1: Students were given a worksheet with a table, told how to assemble the rubber bands and washers and to collect data for 1 to 6 rubber bands. Then graph all of their data and freehand a line of best fit. This is the beginning of a functions unit that will end with linear functions. Then they were asked to make a prediction for how many rubber bands they would need for the drop. We didn't have the actual heights, so they predicted for 3 m. Mostly, their prediction method was pick a number that was bigger than 6. 20 seemed nice to them, though some went with 18, since 6 rubber bands was close to a meter. Two groups found the average increase per rubber band.
We weren't clear about how to do the drop in the stairwell. We didn't have a set (or maybe even one) tape measure for long distances. Two teachers wound up determining 3 drop spots and measuring the distances, between 3 and 3.5 m.
Day 2: (After a snow day and a PD day.)
The students coming back were not much more enthralled than they were Day 1. I shared how this was the start of a functions unit, the math idea of having a rule to go from input to output. I tried to phrase the question as given the input of how many rubber bands, could they predict how far it would drop. (Not much traction, as there were already instructions on the screen.) The substitute gave each group their drop height. The two groups that had figured out the averages used this to make quite specific predictions, and one group made the complete table that this would generate. At the last second they cut 2 rubberbands off of their total, to allow for the length of the disk and acceleration. They were worried that it would be traveling faster at the bottom and that would make it stretch more. Two other groups adjusted their number a bit, but without reasoning that they could share.
We proceeded to the stairwell and groups took turns making their single drop. 2 hit the floor, including one of the more mathemaical groups, 2 got about 70 cm away, 1 was more than 1 m, and the group that had made the table got to within 10 cm. Went back to the classroom, shared the results and had the winning group describe their efforts while few listened. I talked with the mathy group that hit the floor about what went wrong. Basically they felt math failed them. Double checking their work I saw the problem was that they were computing for the wrong drop height! Their calculation would have put them quite close.
So What?
The students were pretty happy. Better than a typical math class, playing with rubber bands, leaving the classroom. The sub was okay with it, as students were mostly in control and made it through all of the steps. I felt like we missed an opportunity.
So what would you change about the lesson? What would add/create/inspire intellectual need in what is a (potentially) great activity?
Post Script:
Excellent discussion! I just want a few of the shared links to be more visible here. But many people put great thinking below so don't skimp on the comment reading.
I got to give a whiz-bang 60 minute (with an option for 30 extra minutes) intro to GeoGebra at the New Tech network conference this week. 50 plus tech-savvy teachers... so it was good. I am always worried that people expect me to tell them about GeoGebra for an hour, when purpose is to get them started using it on the spot, in ways that make sense of their potential use. (Note that if you are in driving distance, I am more than happy to come do this at your school. No GeoGebra lectures, however.)
Purposes. So what are the ways that people make use of it? Oh, let me count them:
World's best graphing calculator. (A little weak on statistics and CAS, but that's improving quickly.) For you and your students. For algebra, calculus, or geometry.
Mathematical image editor. For uses in reports, papers, handouts or assessments.
Demonstration tool. Project a great visualization on your screen to show to or discuss with students.
Focused mathematical activity for students.
Open-ended inquiry tool. Pose a question and let students investigate.
Requirements. The AMAZING thing about this tool is that with version 4.0, all of these are accessible to teachers in that 60-90 minute start up.
Open the program, start typing equations on the input bar.
Needs some quick familiarity with the tool bar to make your image, then File > Export > Graphics View As A Picture.
GeoGebraTube. If you have not looked at this, you are missing out. 14,000 sketches and counting; free accounts, search, likes, tagging and you can collect them in teacher mode or show collections in student friendly mode. This is why you need minimal expertise to start using the program deeply. If you can run YouTube and you are a teacher, you can do this.
See #3.
Students today are geared for this kind of tool. You give them access, they'll figure things out about it that I don't know.
Really, any training beyond that first 90 min. is about if you want to become proficient in number 5, or if you want to be designing your own activities. Some teachers are doing that anyway by the end of an hour, most by the end of a half day. Once you start using it, there's a big danger of being sucked in by the possibilities of what you can make. The power of dynamic examples is as much greater than static electronic images as static electronic images were than hand drawn. (My opinion. No research. Actually yes research, but they would never quantify so crazily.)
After my session, I got to go to Geoff Krall's (@emergentmath) session on formative assessment. He was using the MARS MAP (Mathematics Assessment Resource Service - Mathematics Assessment Project) materials. In particular, he used the Ferris Wheel lesson to get us collaborating and specific in discussion.
As we discussed, it really got me thinking about how I would use the task early on. It would make a good project or assessment, I think, but what about as an inquiry? The basic problem was to make a symbolic model [find a, b and c for a+b*cos(ct)] for the height of a car on a specific Ferris wheel. Then there was a card sort which got students comparing context, equation and graphs.
I've given many explorations before that got students experimenting with parameters to see the effect on graphs, but I love the idea of tying it to a context. That doubles up on the intuition they can apply - physical and visual. If the students had access to that, they might be able to do enough trials to start to generalize. Even without much trigonometry understanding, it's a nice context for graph transformations. For me, these kind of thoughts now lead to GeoGebra. I made a quick sketch, with the Ferris wheel in a 2nd graphics window, and was delighted to find that even the 2nd window worked on GeoGebraTube.
But since then I thought it would be worthwhile to develop a bit more. Both to familiarize myself with using the 2nd graphics window and to make the single model into a reusable activity. I knew I wanted to have either a customizable or random Ferris wheel, some animation of the situation and a way for the students to enter the equation.
That bore some thought: sliders, input boxes for parameters or an input box for the function? Sliders are best for seeing continuously linked examples, but can make a problem like this too easy! The input boxes for the parameters helped support the idea of structure, require some thinking before making a new guess, and don't require as much typing as entering the whole function. Plus you can isolate one parameter and just adjust that. That might be a positive or negative. It feels like a support for learners early in this, by encouraging them to focus on one parameter at a time.
The trick to working on two graphics views is the advanced tab of object properties. You can use any tools from the main window. Just select the tool and then use it in the 2nd graphics window. The objects you make there show up in the algebra view. But when you edit things, or create them in the input bar, they migrate or appear in the first graphics window. The solution is in the object properties, advanced tab; just check the box you need. Note that you can have something appear in both ... there just has to be a cool use of that.
I don't think there's anything else too tricky about the sketch. I used the Function[ , , ] to get the modeled equation to move with the tracing point, the ZoomIn[1] command on the button to clear traces, and the UpdateConstruction[] command to reset the Ferris wheel dimensions. (I had slick graphics window dimensions based on the Ferris wheel, but then the ZoomIn[1] command doesn't work. Ultimately I thought it was better to see the Ferris wheel changing sizes anyway.)
The sketch is on GeoGebraTube: Teacher page for download or Student worksheet for in browser use. You have to click in the main window to get the animation button to show.
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