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.



April 14, 2012

What do you mean I don’t need a formula?

Filed under: standards based grading,teaching — Numbat @ 14:16
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I have had a very hard time introducing concepts based assessment (aka standards based grading) to my Year 11 General Maths class this year. I have always tried to teach concepts and so assessing this way was a very good fit. I’ve also tried to emulate some of Dan Meyer’s “be less helpful” ideas. To say it’s been a tough task would be an understatement.

My students are all invested in “formula, formula, formula” and copying worked examples from “cheat sheets” onto test papers – these tasks they can do, but ask them to actually think about something, anything, and they run for the hills.

Imagine my pure joy, then, when during marking of my most recent concepts test I came across the following.  Here is a student who is complaining about having no notes, no formula and yet they were able to think the problem through and get the correct answer (albeit with the wrong units)

If only more of my students would actually engage their brains and take a chance to apply some of their knowledge to their work.

Contrast the above to this second student who has clearly found the right formula and put the numbers into the correct spot, but can’t use a calculator properly or recognise that the answer the calculator has given them is totally wrong.  This is a student with very little understanding of the question or of the concept.  A “formula” driven student who has probably experienced some success in the past but is totally out of sorts when asked something outside their comfort zone.

Clearly the first student has a better understanding of the concept and the processes involved, and I would like to think that if a few more students in this class could make the same effort then they would start to see Maths in a different light.

Unfortunately, many of my students would rather expend their efforts attempting to get me to “teach them real Maths”.


P.S.  I must admit to having a chuckle over this student’s comment about the photocopier. 🙂

March 31, 2012

Maths and my Lathe License

Filed under: standards based grading,teaching — Numbat @ 19:57
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When I was in Year 9 or 10 (I can’t remember which) I studied a year of Fitting and Machining. This was the “big leagues” at my school where we were ushered into a room with many grand looking machines and the smell of machine oil permeated everything.

Before we were allowed to even go near the lathes – the ultimate tool in that particular shed – we were all required to get our lathe license. This consisted of being able to name all 108 parts of the lathe as identified by our teacher. Our teacher handed out mimeographed sheets similar to the one shown here, although luckily the ones we received had each part numbered with all 108 names listed in order as well.

The “test” consisted of a sheet with a diagram of a lathe, with all 108 parts numbered, and we were required to write in the name of the part in the numbered lines below. A few of us got together and set about “memorising” the 108 names, in order. We managed to devise a pattern, almost a song, consisting of those 108 names. It wasn’t easy, but after quite a bit of “study”, we managed to memorise all 108 names. Come test time, we regurgitated those names, in order, in writing, and “passed” with flying colors.

Not a single one of us could identify any part of the lathe by sight or by function. All we had managed to do was to learn the words of the “lathe song”, very much like the words to our favorite songs of the time or of the national anthem. If our teacher had walked up to a lathe, pointed to a part and asked us to name it we would have been exposed as the frauds we were, but he didn’t, we had our lathe licenses and were allowed to use the machines.

The parallel between my lathe license and the way that many students are taught or learn Maths has become more and more evident to me this past year. Many students learn a “mantra” for each question. They follow a sequence of steps from start to finish never really understanding what they are doing. They reach a conclusion which they neither understand nor could repeat if the question were presented to them in a slightly different manner.

On twitter recently there was a bit of discussion about how students say they like Math, but what they really like is the fact that they can memorise a procedure and regurgitate that without having to think. I am beginning to realise that many of my students have not been asked to think in Math class too often.

This year in my concept tests I have disallowed worked examples. I am getting huge resistance from my Year 11 students in particular, many of whom have been bought up on being allowed to bring their notebooks into their tests. At parent teacher interviews the other evening I had one particular student announce to both myself and her parents that of course she knew the work and had learned it, but could only “do it” if she had a worked example in front of her. She was quite serious in demanding that I allow her to bring worked examples into tests. Luckily for me, I was able to convince her parents (and hopefully, eventually the student herself) that all she had managed to learn was how to translate a worked example into a test question by substituting the test values. In essence, she has learned her “lathe song” and will quite happily repeat it when requested.

This year it is my intention to actively prevent students from learning “lathe songs”. Hopefully I can get them to actually learn the concept and in doing so get them to engage their brains a little more frequently. It’s hard, time consuming and the students don’t appreciate the efforts yet. Hopefully one day they will.

What about others? Are your students learning “lathe songs” or are they learning Maths?

March 14, 2012

Pi Day

Filed under: teaching — Numbat @ 20:46
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this year was the first time I’ve partaken in Pi Day activities. I enjoyed it, and so did my students, even if they now think I’m even stranger than before. Many of them don’t understand how I can be passionate about numbers and Math but hopefully some of that will rub off on them.

For today’s activity I had my Year 7 students calculate Pi. We measured the circumference and diameter of a dozen different circular objects and then calculated out the Pi ratio.

I was happy enough with the results, and the students were definitely engaged in the activity. What continues to amaze me, however, is how little my students understand the actual process of measuring. We used dressmaking (or tailor’s) tapes and quite a few of the students had issues. Which end of the tape to measure, what the numbers actually mean, how to read the measurement off the tape. I will put that down to failure on my part – assuming that something so basic an action would be understood by all.

I was very happy with the way they went about the task, and many of them chose to use a spreadsheet to store the results rather than pen and paper – yet another example of giving the students choice providing a surprising result.

I still need to get much better at my follow-up and debriefing from these types of activities but the more I do the more I am becoming far more confident. I still worry the students won’t engage and will go haywire but that worry is reducing with every success. I suppose the inevitable fall is just around the corner but I shall face that when it arrives.


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.


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.


October 30, 2011

Division as sharing helps with fractions.

Filed under: teaching — Numbat @ 15:50
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Hi All,

Thanks to a number of twitter references of late I have managed to really solidify my thinking on the concept of division.  I had never really given it much thought as division came a little too easily to my mind, however there are two very subtly different concepts of division.

The first, which is commonly referred to as “chunking”, involves breaking an amount up into groups of a certain size.  So, for example, 10 ÷ 5 becomes “break 10 into groups of 5”.  This is what I have found to be the more prevalent way of thinking about division amongst high school students.

The second method doesn’t appear to have a common name, and so I am going to call it the “sharing” method.  This involves sharing the amount between a given number of groups.  So, the same sum, 10 ÷ 5 becomes “share 10 items amongst 5 groups”.

The difference is subtle, and best represented graphically, as follows.

1) 10 ÷ 5 : Thought as 10 divided into groups of 5. How many groups of 5?

2) 10 ÷ 5 : Thought as 10 shared between 5 groups. How many does each group receive?

When teaching division as the opposite of multiplication, it becomes easy for students to concentrate on the “chunking” idea of division.  10 ÷ 5 = 2 because 2 x 5 = 10, or 2 groups of 5 equals 10. As stated, I have found this to be the more prevalent way of thinking amongst high school students.  However, I think the “sharing” method is probably the more “natural” way to think of division.

If I were to give 6 sweets to a group of three children, they would very easily calculate that they would each receive two sweets.  6 ÷ 3 = 2.  However, if I handed the same group of three an unknown quantity of sweets, they would still be capable of dividing them equally.

I very much doubt that would add up the sweets then divide by three. No, they would simply dole them out one at a time to each and end up with an equal share.

The two concepts of division become very important when it comes time to introduce fractions to students.  When presented with the fraction ¼, students who’s primary concept of division is chunking find the image of ¼ difficult to understand.  How can I split 1 into groups of 4?  It just doesn’t compute.

However, students who see division as “sharing” find it easier to compute that fraction.  “Share 1 pie among 4 students” is easier – simply cut the pie into 4 equal parts and dole them out.  This is what they’ve been doing as kids all along.


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

September 9, 2011

My first problem

Filed under: problem solving — Numbat @ 11:01
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Hi All,

I awoke a little “mathy” today.The first order of business most days for me is to check the blogs and twitter feeds that I follow.  It kickstarts the brain quite nicely.

I had to watch the following video twice.

The first time I watched, just as he got into the 2nd example (26 seconds into the video), my brain suddenly kicked into gear and screamed “No way!”.  As my early morning brain grappled with what I was seeing and James continued on with his explanation, it took a few minutes for me to realise the “proof” was staring me right in the face.

I immediately wondered if my students would see through these examples as quickly (or slowly as the case may be) as I did.  So there is the basis of my problem.  The first problem I have attempted to “create” I might add.  Please feel free to let me know what you think and make suggestions for improvement in the comments.

The problem:

Watch the first 45 seconds of the above video, then ask your students to either prove or disprove the method. You could make it a little harder (or perhaps easier) if you ask them to only use the 4 examples shown.


Thanks go to @jamestanton for the video and @ddmeyer for the inspiration.

August 22, 2011

Maths must be beautiful

Filed under: Musings — Numbat @ 20:06

Just wandering around the twitter-verse tonite and came across these in two separate tweets.

Coincidence?  Me thinks not!

“The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. … There is no permanent place in the world for ugly mathematics.

– G. H. Hardy

And then this..

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