Lungs diagram with internal details (Photo credit: Wikipedia)
Just a quick idea here… To make a lesson on calculating the volume of a sphere contextually meaningful how about using a balloon to calculate the capacity of your lungs? Get a pupil to take a deep breath and then fully exhale into a balloon. Then measure the radius of the balloon and calculate the volume. You’ll need too get balloons that form spherical shapes for it to work well.
An interesting chat about the accuracy of the results could follow. Is the surface tension of the balloon similar to that of your lungs? Is it an over estimate or under estimate and why? Etc…
As an extension, how about relating the volume of a sphere to that of a cylinder? Archimedes’ famous cylinder and circumscribed sphere discovery states the volume of a sphere circumscribed inside a cylinder is 2/3 the volume of the cylinder. Compare the formulae; can you prove it!?
Archimedes sphere and cylinder. The sphere has 2/3 the volume and surface area of the circumscribing cylinder. (Photo credit: Wikipedia)
Numberphile.com is a website your pupils simply must know about. It describes itself as “videos about numbers and stuff” which is a pretty good summary in my opinion. The videos are not tutorial skill-based offerings, but engaging adventures into interesting mathematical concepts and problems. I really like them as they show maths at its best: interesting problems and fascinating results, rather than contrived contextual links. That’s not to say there isn’t context, but where there is it is meaningful and adds to the intrigue.
These videos are the KS3 maths curriculum we’d all love to be teaching if standardised testing and judging teachers and students on exam-based success we not the order of the day. The passion and enthusiasm of the presenters combined with the engaging subject matter definitely stoke the flames of enquiry. Pupils I have shown the Numberphile videos to did react very positively to them; many wanting to know more about the maths behind them.
There are links in the videos to topics we do teach however, and I think there are plenty of opportunities to get these videos in as starters, plenaries or interesting homeworks. I commend them to you!
A colleague of mine, Claire Nealon wanted to adapt a resource I made a while ago of a triangle constructions tangram (see previous post). She wanted to make it more accessible for lower attaining students. Claire simplified the constructions so that the triangles are all right angled with side lengths that are Pythagorean Triples. This made the activity more accessible for her pupils who struggle reading and measuring lengths that are decimal numbers.
A colleague of mine recently stumbled across the website of Peter Bland, a maths tutor. It contains some excellent GCSE revision resources in the form on booklets of exam questions on particular topic. They are all available form his website here, but for your convenience I have also linked to his hosted resources by topic below:
This time Craig Barton and his guests from King’s College, London discuss falling standards in maths, advice on running discussion based lessons, and some festive maths ideas. In the attached Word document you will find links to all the resources, ideas and activities discussed, and a link to the Forum discussion as well.
This site adds to the growing number of websites providing interactive games and learning applets based on Adobe Flash. Other favourites of mine include FlashyMaths.co.uk and TeacherLed.com. Flash Maths is a worthy addition and well worth bookmarking. They have numerous applets that are well designed and add a little bit of a zing to your lessons. Get on over to Flash Maths to sample their offerings. Top tip… Shape Shoot is very good!
How do you give pupils a beautiful visual picture of why the surface area of a sphere is 4 pi r squared? A super lesson idea I heard recently involved an orange!
Give pupils an orange and a blank sheet of paper. They have to draw circles around the orange onto the paper. The idea is that the circles on the paper should have the same radius as the orange. They draw as many circles on the paper as possible.
Then get the pupils to peel the orange and arrange the peel so that it fills the circles they drew on the paper. Some careful ‘sculpting’ of the peel will be required to get it to fill the circles with no gaps or overlaps. Pupils should fully fill one circle with peel before moving onto the next circle and so on.
A beautiful image should arise whereby the peel completely fills four of the circles proving the surface area of the sphere is 4 pi r squared!
If you give this a go with your class please take some pictures and email them to us at email@example.com so we can feature them on this post!
Finally, time for an orange joke:
What did the little chic say when it’s mother laid an orange?
I want to give my current GCSE year 10 class more practice on identifying angles in parallel lines; alternate, corresponding and co-interior. I have put together the following worksheet that gets them to colour each type of angle a particular colour. Please feel free to use it with your own classes if you like it!
The TES Maths Panel are delighted to announce the arrival of The TES Maths Podcast! This is a brand new series hosted by Craig Barton, featuring recommended resources, discussions of best practice, conversations with leading educational professionals and more.
You can listen to the first episode by clicking here. TES are working on putting the series on iTunes shortly so they can be downloaded onto your iPhone, iPod, computer etc.
The other day, a friend of mine brought an excellent little podcast to my attention called The Math Factor. Suitable for pupils and teachers alike, the 10 minute episodes feature interesting problems and concepts within mathematics. Hilbert’s Infinity Hotel, Algebra on the Radio and Space Walkers; there is something here to engage any mathematically minded person. The show is pitched beautifully, making it accessible to people of all mathematical abilities. Highly recommended.
How often do your pupils calculate the answer to an angle fact question correctly, but then don’t get the mark for the ‘give a reason for your answer‘ question? Rather than ‘alternate angles are equal‘ they write any manner of things such as ‘it is a z shape and so the angles are the same‘. Since ‘z angles‘ is no longer accepted on the GCSE exam, they stand no chance of getting the mark, even though they do know the concept.
Pupils must know the appropriate angle fact language and use it. Repetition is the key and so I have created a set of flashcards that you can use with your classes on a regular basis as a quick starter to revise the angle facts. The idea is you print them out, laminate them and then shuffle them. You hold them up one at a time and the class have to say the angle fact using the correct language. You could project them on the board instead or even turn it into a mini-test. They are levelled from 3 to 8 and include types of angle, angle facts (basic and in parallel lines), the 3 rules of bearings and all the circle theorems so you can use whichever ones are appropriate for your class.
Disk florets of yellow chamomile (Anthemis tinctoria) with spirals indicating the arrangement drawn in. (Photo credit: Wikipedia)
I have known about the beauty of the pattern of seeds in the heads of sunflowers for a long time now, but still stare in wonder just as I did when I saw it the first time. You count the number of spirals in each direction and find that they are two consecutive numbers in the Fibonacci Sequence.
I recently heard that the pattern also includes the Golden Angle. This is when you divide up 360 degrees into two angles in the Golden Ratio. The smaller angle is the Golden Angle which is irrational and measures 137.508… degrees. The seed pattern is formed by seeds being ‘fired’ from the centre of the flower head outwards one at a time. The angle that each seed heads off from the centre when measured from the last one is the Golden Angle.
For a bit of fun I thought I’d write a computer simulation of this procedure to see if I could replicate the pattern. I then also changed the angle of consecutive seeds to plus and minus one degree from the Golden Angle to see how sensitive the pattern is to the angle. The results are surprising!
It seems as though the seed pattern is extremely sensitive to the angle that the seeds are released! Therefore we must conclude that sunflowers are great admirers of the beauty of mathematics! They are applied mathematicians at heart, putting their knowledge into a wonderful real world application!