2+2 != Maths

April 22, 2012

Intro to Pythagoras

Filed under: classroom,teaching — Numbat @ 11:21
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I’ve been trying a lot of new things this year and I thought I would share my most recent experiment with you all.  It was time to introduce Pythagoras to my Year 8 (accelerated) class.  Here’s how I went about it.

Lessons 1 and 2

Firstly, I introduced the notion of doubling the square via the above video, which you can view on this page.  http://mathforlove.com/2012/02/pythagorean-theorem-part-1-video/

I stopped this video a few times to get student input. The first stoppage was at the 25 second mark, where I asked the students for ideas on how we can get a square twice the size of the first.  They were reluctant to offer suggestions, but at least none of them suggested a 2×2 square.  I was a little disappointed that I couldn’t get to discuss the error of that approach, but quietly happy that they were perhaps smart enough to know that wouldn’t work.

The second stoppage was at 1:10 (where asked) and I had the students look for ways to get half that square.  For this, I handed the students a piece of square paper (I have origami paper so I used that but any square paper would work).  One student was onto the solution straight away, but some of the others struggled a little till the solution was passed around.  I found this exercise quite useful and the students seemed to like it.

The third stoppage was around the 3:00 mark where I asked students to calculate the area of the square as shown above.  For this I handed out centimeter-grid paper which I created and printed from the following site.  http://www.math-aids.com/Graphing/Graph_Paper.html  I also set the students the task of calculating the area of squares for 1×3, 1×4, 1×5 and 3×4 diagonals.

I found this an excellent task.  The students really got involved and were genuinely engaged.  I was surprised at the number of students who had some difficulty drawing these squares on the grid paper, they didn’t understand the nature of the right-angle and had no process for ensuring that the diagonals were going to the right place. They could see that they didn’t get a square (they got parallelograms generally) but couldn’t figure out where they went wrong.

We built a table on the board of the side lengths and the areas and many of the students offered suggestions on patterns they thought were evident. I think next time I do this I will pick different side lengths so that one basic (but incorrect) pattern doesn’t appear. The students didn’t manage to come up with the pythagorean theorem themselves, but were easily led to the pattern and I sensed that they were happy to accept the pattern.  At this stage, we hadn’t even discussed triangles.

For the second part of this lesson I had the students construct their own right angled triangles (using the same grid paper from above to provide the right-angles) and measure the side lengths.  I then had them fill in the a, b, a2, b2, a2 + b2 + c2, c table and verify for themselves that it all worked (within the margin of error).  This part of the lesson didn’t go quite as well as I had hoped and I need to work on something here for next time.

The above took the better part of a double lesson and we resumed the next day.

Lesson 3

On the second day (third lesson) I had the students start with the above “puzzle”.  I like to start off with puzzles and the students jumped into this one with gusto.  I think the reason they took to it so well was that it was easy and there was little “thinking” involved.  The PDF file is available for download here : pythagorean triples PDF  The students got about 2/3 of the way through this file and started complaining about the repetition, and I suspect I need to cut the sheet down some for next time.

From here, I then proceeded to discuss Pythagoras Theorem and finding side lengths of triangles.

What I’ve found is that many students are more than happy to use the process from the above table to work out all their Pythagoras questions now.  They don’t want to set out the work in what I would call a “classic manner” (formula, substitution, manipulation, answer) but will happily write the numbers, square them and work out the missing values.

While I’m happy that these students actually understand the concepts involved, they’re complaining about the amount of time it takes to complete each question and I recognise that we need to do something about this at some point in time.

Overall, I’m happy with the way this intro to Pythagoras went. The students mostly enjoyed the activities and are more than comfortable with the “arithmetic” of Pythagoras. They are struggling with the “layout” and “rigor” I require but can do the Math which is more important at this stage I believe.

For next time, I think I need to work on the transition from the “calculate the area of the squares” to “measure the sides of the triangles”.  Perhaps I can get the students to create their own diagonals and work on those.  I also need to reduce the size of the “puzzle” I gave the students and work on the transition from that puzzle to the application to Pythagoras type questions.

I’m very happy that the students actually understand the concept of the squaring of sides of the triangles but I would be happier if the students were able to introduce a little more “rigor” into their working.  Perhaps that’s just a hold over from the old me?

Comments and Suggestions are welcome.



February 23, 2012

To the moon

Filed under: classroom,teaching — Numbat @ 20:06
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what to do in Maths class period 6 on a hot day when 1/3 of the class are off on excursion? I decided it was time to try something different.

The question is “how many times would I have to fold a piece of paper for it to reach the moon?”. Obviously, you have to assume it’s a pretty big piece of paper to begin with. I started with asking the students to estimate how thick a piece of paper was, or as one student corrected me, how “thin” was the piece of paper.

I then asked them to estimate the distance between the goal posts on the playing field outside the classroom (for the non Aussie’s amongst the readers – there is no standard length of an Aussie rules field).

I then asked them to estimate how many times they’d have to fold a piece of paper to fill the playing field. Then I let them loose. Most of them figured out you have to at least measure the playing field, and at least half the groups came up with a measurement that was reasonably accurate.

Only one group managed to make an estimate which was reasonably close – they calculated 10 times. The comment by the student was that it was a “surprisingly small” number. The actual number was closer to 17, but the lowest estimate by the students was 120. I will concede that 10 is surprisingly small against 120.

However, as a group they did not manage to convert that to the moon question. They all knew the rough distance to the moon (we’d used that number in a class a few weeks ago so they had a good idea), but they all still thought of a number in the thousands range.

Although I was disappointed in the quality of the estimates (even after the preparatory work) what I took away from this is that all the students engaged in the activity, and the discussions I overheard were very encouraging. “what’s the formula for ….”, “how do you work out ….”, “what should we do here…”

When I exposed the answer, a number of the students asked “how can that be”, and a couple of students attempted to explain it to them. I even heard one use the term “exponential”. I will count that part of the exercise as a success.

I was a bit rushed in this exercise as I wasn’t sure how long it would take and how long the students would need to measure the oval and make their calculations. I think I could prepare them a little better next time round. They were in no state after the exercise to debrief and that was a little disappointing. Next time I’d like to challenge the non-believers to prove me wrong.

Overall, however, I am very pleased with the outcome and I think many of the students saw a different side to Math than sitting in a classroom answering questions from a book. I’m hoping I can do many more of these sessions this year.


February 15, 2012

How Sorry Helped

Filed under: classroom — Numbat @ 18:10
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Yesterday I had a shocker in class. A problem arose that I was unprepared for. A problem which shouldn’t have happened. A problem caused by the kids being… well kids.

And I over reacted. Badly. I didn’t yell or scream (I tend not to do that) but I jumped on them hard. I issued ultimatums and outlined consequences which were, in hindsight, unreasonable. And the kids knew it. While most just bit their lips and took it, there were a few who tried to protest and I shut them down. I could tell by their body language that they were signing out.

The thing is that if these were adults, or even senior students, I think I would have been justified in my expectations. But these students are only Year 8 and my expectations were too high.

Last nite I spent considerable time thinking about what to do. Should I bunker down and ride it out – follow through on my unreasonable expectations and issue the consequences outlined? I knew I would have a number of students simply unable to comply with my timeline, was I prepared to make them suffer unreasonably? A few years ago there would have been no question, I’d be donning the protective gear and going full tilt.

This morning I walked into class and said “Sorry”. I admitted that I had over reacted and explained to them what they had done to upset me. We set new a deadline and put in place procedures to get us through to that deadline. I reiterated the consequences for failing to fulfill their part of the bargain and offered extra assistance for anyone who needed it. When I asked if this was acceptable they were enthusiastic in their response.

While it was certainly a hard thing to do, I will count it as a huge success. The students responded very well, including those who wouldn’t even talk to me yesterday. They worked, asked questions and were generally back on track.

I know that I won’t get away with the same thing again with this class for quite some time, but I feel like I’ve prevented a very dark few weeks. Hopefully I can build on that.


December 19, 2011

Our first attempt at SBG

Filed under: classroom,teaching — Numbat @ 16:01
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I make this post with some trepidation. Opening up our first attempts at converting to SBG is a little daunting. How will others react? Are we on the right track or have we completely gone bananas?

A few months ago I blogged about how I’d presented SBG to my Maths staff and how they’d started to think about and adopt the ideas in a small way. Since then we’ve been making some progress and have spent many hours going over our existing “work related” courses and coming up with a list of concepts.

We still have a ways to go but we have mostly finished the concepts list for our Year 7 course and would like comments from those of you who have been down this road previously. Please be gentle – this is our first attempt and I know that we have much to improve upon.

My own thoughts are that we have too many concepts, but I’m not really sure how to condense them any further. I think I’ll figure out some of that throughout next year but perhaps others can help.

So, without further ado, here is our list of Year 7 Concepts for 2012. Please make comments below.


September 17, 2011

Update: iPad Gaming in Math and Science

Filed under: classroom — Numbat @ 17:45
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Hi All,

a few weeks ago I was really excited about this lesson I’d prepared for my Year 7 ICT class.

As the early indicators showed, the kids were even more excited about the class than I anticipated, perhaps too excited.  During the experiments they were far more interested in the “procedure” than the “recording” and while many of them obtained some excellent data, getting that data out of them in a usable format was quite difficult.

Still, the whole thing was a huge success both with the students and for myself and I am already planning on improving it for next year.  I can also see how I can expand on this lesson for stats class in Maths by introducing an analysis and graphs of the results.

The first thing I need to do is to give the students more time. If there’s anything I’ve learned this year it’s that I am underestimating the amount of time students take to do things.  Even simple things like fill in a short survey take them a lot longer that I expect.

With that in mind, I offer the following data to anyone who’d like to analyse some real world, student collected data.  I asked the students to fill in a short survey I’d created before they did their own experiment so that I could collect some data for myself.  It was a rushed job, a last minute thought and I needed to do better.  Next year I will plan it and present it to the students better, perhaps even get them to assist with the preparation, so that the data might be more comprehensive.

If you use this data and find some conclusions please come back and share them with us here in the comments.


The data file is in excel format and has notes for each of the columns on a separate sheet.  The data file can be downloaded here iPad Experimental Data

August 18, 2011

iPad Gaming in Math and Science

Filed under: classroom — Numbat @ 17:23
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I cannot remember ever being as thrilled by both the prospects and the response to one of my lessons.

The aim was to get students to explore different settings to assist in extending their iPad battery life.  Rather than giving them a simple research assignment, I thought why not get them to actually prove or disprove these methods, let’s conduct a (semi) scientific experiment.

So I tasked them to design an experiment. They were required to research a number of different theories on conserving battery life and pick one to investigate.  They were required to write a hypothesis and a method to test their theories, to design a data collection method, to perform the experiment and then write up a conclusion.

My ultimate aim is to collect as much of this data that I can and do some statistical analysis on the results.  This would certainly be relevant for an older Math classroom although I doubt I’ll be getting these students to do this further analysis.  (These are Year 7 students, 12/13 years old).

The response has been phenomenal. I had a hunch that the students would respond positively, but I never envisaged the response I received. During the class discussion stage, virtually every student participated and we developed the framework for our experimental method collaboratively.  Even students who wouldn’t normally participate were right in it, and one particular student who normally participates with completely random and irrelevant remarks was on track and contributing thoughtful comments.

Two students in particular, who have hardly done any work all year, were almost the first to be ready.  They had done their research and planned their experiment in record time, and were even enthusiastic when I sent them back (repeatedly) to flesh out the process for their experiment.  I don’t recall ever having to write notes to parents asking them to “allow” their child to complete their homework, but I have written a dozen or so of those so far.

This started as a bit of a crazy idea that I thought I would run with, but I’m already seeing how I can expand and extend it for future years.  I can see how it could be useful in both Science and Math class, and the results are certainly relevant for my IT classroom.

If you give this a try in your class please report back on how it works out for you.


18 September:  there’s an update here which includes feedback and data from these sessions.

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