Dynamic Symmetry

Years 7, 8 and 9 have been working on some dynamic symmetry problems. These are exemplified by the videos below. The task is quite simply described but less simply achieved..... 'Can you make the following animations using dynamic geometry? ( See the following links for more details... Kaleidoscope, Animated Questions for a discussion on these ideas and 'One Question Lessons' for more thoughts on this type of lesson.)

The activity ticks a lot of boxes, 

1. The task is easily understood, 
2. Students really want to have a go because they want to make these really cool animations,
3. As they work and try things out, they can see straight away if they have achieved what they set out to do. As such, they are getting instant regular feedback on what they have done. they then refine each time trying to get closer to the stated aim. 
4. In doing the above, students have to reason with each other and articulate mathematical definitions of what they think is going on, of what properties give rise to what behaviours. these are the sort of conversations that make me as a maths teacher delighted and they are very productive for students!

The end results are very pleasing indeed, both in terms of what students produce, but perhaps more importantly in terms of what exploration and discovery they have done together.

For the ambitious, we can then move on to this combination of rotation and reflection!

Older pupils, perhaps in year 9 or above might get interested in this great problem!! Watch carefully to see if you can figure out how this was made! there are lots of surprises in this one. See Dr Who for more details!

Students might go on to making some of the fabulous designs on show with this task - Olympic Logos

Our provision for technology opens lots of doors for us here in Toulouse! Firstly the ease with which we can have a go at these types of tasks and secondly, our choice of media for students to work in. Below are some of the videos that students created whilst doing this task!....

Minigolf

This is a lesson about a minigolf hole, a ball and some bounces off the wall, adapted from a great activity shared by Fawn Nguyen on her blog.

We'd looked before at how to find he quickest route between two points that also touches a line: reflect one of the points first.

Luckily, plenty of children remembered this when we tried out some tricky shots in a game of minigolf.

First we had a go at creating some courses with an obstacle between the ball and the hole.

We did this using Geogebra. Within a rectangle we put a ball and a hole, and an obstacle between the two.

Have a look at it in GeoGebra Tube.

Then we looked at how the Reflect Object about Line tool could be used to find the reflected point. From there it was mostly straightforward to work out the correct shot. If you can see the GeoGebra image above, you can have a go yourself.

There were one or two cases where people had put the ball in a very tricky place. This meant two bounces were needed:

Then, we had a go at doing this on paper, with different tools. This time we used the ruler to work out where the reflection of the ball should be, and the protractor to check the angle the ball hit the wall and the angle the ball left the wall:

In most cases we found that the two angles were the same! 

The idea of a game of minigolf seemed to work well as a practical context for talking about angles and reflections, and also for using geometry tools, online and physical. So much so that some students were all for a visit to the local minigolf "to study angles"!

Where is the middle?

Finding an easy Geogebra challenge for Year 5s that's just right is itself a challenge, but this one went well.

Where is the centre of a regular pentagon?

We know how to make these, because we've had a go at making stars (another interesting and more open-ended challenge).

Some of the children remembered how to find the mid-point between two points (we'd done this finding the parallelograms inside any quadrilateral).

So they found the mid-point between A and B. Here it is, F. Then they found the midpoint between F and D and "I've found it!"

There it is - G.

Except that it isn't.

We checked and, doing the same thing on the four other sides, we got four other "centres":

So, it's got us close, but not close enough. After a bit more experimenting, some of the class did it with lines: We were now pretty confident that we'd found the centre! To finish off, I asked the children to play a little, and see what they could create from that. Later on, some of them had a bit of spare time to colour in their creations on Paint.

As if by magic, a right angle appeared...

Year 5 looked at a certain kind of triangle. It's one inside a circle where the long side of the triangle is the diameter of the circle. The opposite corner is on the circle too.

At first we looked at these triangles on a big scale. The school pitch, like every football pitch, has a very convenient circle, with a straight line right through its middle. If that line is the long side of a triangle, what angle would the opposite corner have?



Next we used a pair of compasses and a ruler to recreate the situation:

We've also started using GeoGebra  in Year 5. It's great for showing how the angle is always 90°:

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com I didn't take it on to this with Year 5, but there is a fairly simple proof that this will be always the case...