Learning to start small in Cabri3D

I was interested in AbsentCat’s Active Geometry post, talking of a “geometry processor” doing wonderful things, but learning to use it sounded too much to cope with in the endless pressure of a school day. I work with young teenagers who should be in early Key Stage Four, but, because their previous education has been disrupted by events beyond their control or a teacher’s capacity to imagine, are in most cases struggling to master KS2 or even KS1. How could this active geometry business help them or me? But, during a staff development workshop, I was shown the ready made examples accompanying a copy of the three dimensional version, Cabri3D, mentioned by Philip Yorke. One of them addressed a topic which I was due to tackle with my youngsters: the nets of a solid. When I expressed interest in that example, I was offered a short loan of the laptop on which the workshop demonstration had been run.

Very uncertainly, I rigged up the machine and waited for my class. And they loved it. They are very videogame savvy, and related to the direct manipulation of an onscreen object in a way they had never related to paper or cardboard equivalents. The software allows them to pick up a single section of the flattened net, swing it along a guide path, and have the attached panels follow it. The cube assembles itself. Then a face can be swung back, restoring the 3D solid to flat net. As many times as they wish. With the ice broken, they were then amenable to physical exploration in a way that they had never been before.

Since then, I have read Chandra’s account of her Beanbag Thrower - another example of starting small, with a simple and manageable aspect, not being overawed by the greater power available. I have learned a lot from the experience, and learned a lot too from her account. Next time, I shall be bolder - integrating the software into a lesson plan, as she has, rather than just starting with it. I have borrowed the demonstration machine again, and next week I shall be using the 2D version, Cabri II Plus, for a more ambitious project around bicycle wheels. I intend to watch the demonstration videos, have signed up for another workshop on geometric software in CDT, and plan to spend the summer reinventing myself - bring it on.

[contributed by BobTheBumbler]

Throwing beanbags in Mathematica 6

This is the first in a series, illustrating how Mathematica 6 makes it possible to package complex ideas in a way which is suitable for primary education use. You do not need any mathematical knowledge to use the ideas and materials here. Nor do you need any knowledge of the Mathematica 6 program, although you will need a copy of the software.The equation for motion of a projectile on the horizontal plane is not normally encountered until level three in either mathematics or physics. The intuitive idea of parabolic motion dependent on velocity and angle of launch, however, is easily grasped much earlier. In this case. we experimented with class of 8/9 year olds in a British school, as part of an integrated sports theme.

We started by having one pupil toss a beanbag to another, from one side of a classroom whiteboard to the other, while the rest of the class watched and commented. The shape of the path followed was sketched by each child.

This was then repeated several times, the teacher marking one point on the trajectory each time until the shape emerged. The class discussed the similarities and difference between the shape on the board and their own sketches.

Next we moved outside to the playground, where sheets of flipchart paper had been taped to the side of a school building, from ground level to a height of 2 metres, over a distance of five metres. The beanbag throwing was repeated, but this time the children did the marking in of trajectory sample points themselves. In groups of five or six, they spaced themselves between thrower and catcher and each marked the point at which the beanbag passed in front of her or him; after each throw, a new group replaced the last until each pupil had made one mark. Once again, we discussed the shape of the curve.

The throwers were now asked to throw the beanbag higher, but still to try and drop it into the catcher’s hands. The changing shape of the parabola was watched, and discussed, by teacher and class.

The catcher was then removed, and each child took a turn at throwing. They were encourage to vary the strength of their throw, but to aim always just above head height. The effect of throw strength on range was one they already understood, but the changing yet conserved shape of the trajectory as new to them.

Having established the idea of curved trajectory, affected by both angle and strength of throw, the action moved back indoors where the pupils worked in threes and fours. In the centre of the room was laptop with a copy of Mathematica 6, showing a graphic demonstrator (prepared beforehand) which was projected onto an electronic whiteboard.

The demonstrator, which we had called The Beanbag Thrower, allowed the pupils to experiment very rapidly with different combinations of strength and angle of throw, by adjusting “slider” controls. (Double click the illustration on the right to see it full size.)

We decided not to issue specific prompt material at first (though we had them ready, just in case), preferring to see what would spontaneously emerge from free experimentation. WE gave each child a sheet with two columns labeled “what did you do?” and “what happened”, then left them to devise their own strategies. This optimistic but risky approach was vindicated when, after less than minute, the first group announced that “setting the angle half way always sends the beanbag furthest, whatever strength you throw at”. The rest of the class immediately switched to testing this hypothesis, and quickly confirmed it.

The next surprise was when another group correctly interpreted the meaning of negative y values, where the curve drops below the start point. “That’s where nobody catches the beanbag” said one girl, “and so it falls lower than your hand was when you threw it”. Her friend extended this to “And then it goes on falling, lower than your feet, like if the ground wasn’t there”, to which a boy added “Like if you were on top of a building, and your throw took it over the edge”.

Clicking a “+” symbol at the end of a slider opens an input box and a small set of CD player type controls. The box lets you type in an exact number for height or angle, so the pupils were able to check that maximum throw distance really did occur at exactly half way along the angle range, not just approximately.

The CD type controls bar lets you run variable through its full range automatically in various ways. When they discovered this. the pupils’ reaction was to set both height and angle running simultaneously, as fast as possible, in opposite directions - which, while fun, didn’t reveal very much. After a while, though, they realised the value of setting one variable to a fixed value and letting the other roam repeatedly up and down its range in slow motion. Watching the height, the range, and the value of the changing variable, they came to several useful insights. When they realised that you don’t have to maximise the range but can find an infinite number of ways to deliver the beanbag to a particular point, they became very excited: this related directly to football, netball, cricket, tennis, badminton, and darts.

They were also intrigued by the realisation that, for any given distance and strength of throw, there are two possible trajectories. There was much animated discussion over the relative advantages and drawbacks of high versus flat trajectories, eventually leading to understanding that the flat option usually give a competitive advantage.

We, too, were excited by the outcomes of the experiment. The class teacher felt that her pupils had learned lessons in sports, ICT and applicability of mathematics, not to mention transferable skills in group work, co-operation and the efficient conduct of a scientific enquiry. I felt that I had a new range of useful possibilities to explore in wedding maths and science to wider curriculum through IT.

You can download our Mathematica demonstrator model here (see bottom of this post), to experiment with it yourself in a copy of Mathematica. Unfortunately it will not run in the free Mathematica Player, as such graphics are only usable in a special file format can only be produced by the publishers, Wolfram Research . As soon as I have time (during the Spring break in a couple of weeks, perhaps) I hope to work through the process of submitting it for publication as a demonstration for the player.

There are, if you want to try them out, already some projectile models on the Wolfram Demonstrations Project site, but they are more complicated and not (in my view) suitable for primary school use. The exception is “Dart Practice”, which I plan to try with older (age 10/12) pupils.

• The free Mathematica Player can be downloaded from http://www.wolfram.com/products/player/
• The Dart Practice model (usable in the free player) is at http://demonstrations.wolfram.com/DartPractice/
• The Wolfram Demonstrations Project site is at http://demonstrations.wolfram.com/
• You can look at all of the available projectile models here.

You may also like to look at the Kinetic Books Virtual Experiments which were reviewed by AbsentCat last month.

The record sheet and Mathmatica notebook which we used are in a zip file which you can find here. If you are interested in the works behind the demonstrator, and brave enough to tackle the level three mathematics involved, there is also full explanation sheet in the same zip file. If you are unable to use the zip file, send an email to the editorial address for these education pages (you can find in on the “contributors” tab) with “Beanbag materials request” in the subject line.

[contributed by Chandra]

Mathematica 6 opens primary possibilities

Over the last two weeks, I’ve been working with fellow contributor AbsentCat to trial in primary education the use of artefacts from Wolfram Research Inc’s new release six of Mathematica.

I have investigated computer mathematics packages in the past but they never really seemed to be a viable option in real primary schools where the teacher is a generalist with level two mathematics (I am only formally qualified to level two myself), while both the cost and complexity of the software are high.

The availability of a free player, easily packaged graphic entities, and the demonstrations project changes all that - or, at least, I believe that it does, based on this first taste.

I will be back, with concrete tested examples, in a few days.

[contributed by Chandra]

Sun, moon and stones

“Can we go on a trip?”

It’s the perennial cry of every class, and educationally it pays dividends, but mathematics students tend to lose out to more field based subjects like Biology, Geography, Environmental Studies and English literature.

Where can a maths class go, that is both useful and enjoyable?

Today I spent the afternoon and evening with a mathematics lecturer Ivor McGillivray, a film team, a group of students studying maths at GCSE level and A-level, two laptops and an electronic theodolite, at Stonehenge - the megalithic ruin on Salisbury Plain in south western England. The focus was the history of sun and moon in mathematics, and practical activity was central.

It was an eye opening trip, inspirational and remarkably successful in pedagogic terms.

It’s not my story, so I won’t go into detail now, but I hope to persuade Ivor to write an article for us.

Photographs at left (click them for a larger view) show, from top…

• Calculating the height of a standing stone.

• Cross checking measurements for accuracy.

• Setting up the theodolite.

• Relaxing between tasks.

[Contributed by Felix Grant]