Thanks to +Jerry Smith for bringing this to my attention. Lovely!
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.
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!
I looked into the feasibility of using iPads in the maths classroom last year and concluded that the time wasn’t quite right. The ‘planets were moving into alignment’, with the direction of technological development leading towards the iPad being a brilliant learning tool in the maths classroom, but they weren’t there yet.
The lack of Flash compatibility was a big issue as sites like MyMaths and Manga High use it. Also, QuickGraph is an excellent free graph plotting app but I couldn’t find an app that also handles dynamic geometry. If only Geogebra had an app for the iPad…
There is a Kickstarter project to develop the Geogebra app for the iPad. They need to raise $10 000 to develop it which will then be a free download from the App Store.
You can read more about the project on the superb Math and Multimedia blog here.
Start with an equilateral triangle, a pencil, a ruler and a die. Label one vertex of the triangle “1 and 2″, the next “3 and 4″ and the final one “5 and 6″. Mark a dot on one of the vertices of the equilateral triangle. Now roll the die. This tells you which corner of the triangle to move towards. Mark your next dot half way between your last dot and the corner of the triangle your die identified. Then roll the die again and mark your next dot half way between your last dot and the corner of the triangle your die identified. Repeat this until you discover the beautiful result!
You may like to write a spreadsheet to do this for you rather than constructing it by hand. I wrote a spreadsheet that did 10 000 trials and look at the result:
You get the famous fractal, Sierpinski’s Triangle!
If you extend the idea into three-dimensions, adding a fourth vertex directly above the centre of the equilateral triangle, the points form a Sierpinski Tetrahedron!
- the length of the outspread arms is equal to the height of a man
- from the hairline to the bottom of the chin is one-tenth of the height of a man
- from below the chin to the top of the head is one-eighth of the height of a man
- from above the chest to the top of the head is one-sixth of the height of a man
- from above the chest to the hairline is one-seventh of the height of a man
- the maximum width of the shoulders is a quarter of the height of a man
- the distance from the elbow to the tip of the hand is a quarter of the height of a man
- the distance from the elbow to the armpit is one-eighth of the height of a man
- the length of the hand is one-tenth of the height of a man
- the foot is one-seventh of the height of a man
- from below the foot to below the knee is a quarter of the height of a man
- the distances from the below the chin to the nose and the eyebrows and the hairline are equal to the ears and to one-third of the face
Get your class practising fractions of an amount by asking them to see if they agree with Leonardo’s claims. Get the tape measures out and get investigating!
As a fun summer activity with the kids we decided to construct a geodesic dome! We got the instructions from cutoutfoldup.com at this link. Our first attempt was a failure as the newspaper struts were too weak and flexible and the suggested joint method of using masking tape just didn’t hold the struts in place under sustained load.
We realised that the cutoutfoldup.com design came from desertdomes.com, a superb website that gives you full designs for any size domes you want to build. We took another design from this website and changed to using bamboo skewers and a glue gun for the joints. The result was a geodesic dome of 0.85m radius that made a perfect ‘tent’ for the kids to play in… Here is a picture of the finished dome:
I have seen lovely demonstrations using coloured water and perspex containers that the volume of a pyramid is one third the volume of a prism with the same base area and height. There are lots of videos on YouTube showing this; here’s one:
However, here is an alternative activity for showing the same result from the excellent cutoutfoldup.com website:
The activity gives you a net of a pyramid. You build three of them and they fit together to form a cube. I did this activity with my children this week and here are a couple of pictures of the results:
Buffon’s Needle is a wonderful probability experiment you can do with a class that has a most surprising result. Out of a seemingly ordinary, unspectacular experiment involving dropping a pencil between a pair of parallel lines, the relative frequency is related to pi! Here is a superb website that explains the experiment and gives a derivation of why the results tend towards pi.
I tried Buffon’s Needle with a class this week and they loved it. We did over 1500 trials and got 2.90 to 2 d.p., a slightly disappointing result but after showing them the Java applet simulation on the website, they were convinced. Being a top set they were able to follow an explanation of why the result is linked to pi, although they did have to take my word for the integration of 1/2 sin theta between zero and pi!
Buffon’s Needle is a great lesson if you want to show the magical, mysterious side to maths where seemingly unrelated topics such as random pencil drops and pi are shown to be intertwined!
Countdown is one of my favourite ways to start a lesson. The wonderful NRich Maths website have created a brilliant set of interactive resources based on the countdown game including:
You may not get onto the last three with secondary school pupils but they would be great for A level study. Besides the traditional game, Countdown Fractions is particularly challenging and engaging
CGP have really raised their game recently. Traditionally known for their excellent revision guides, regular readers of Great Maths Teaching Ideas will know how highly I rate their relatively new GCSE MathsTutor product. This DVD-ROM disc that retails for just £3, contains video lessons and worked exam questions for every topic on the maths GCSE. View a demo of MathsTutor here. They also produce a VLE/network version that you can upload to your learning platform school-wide for just £250 a year. The videos are engaging, informative and contain regular bouts of cheesy humour. The pedagogy is good as the vides are made by maths teachers. The kids love them and say they really help them with their learning.
I have been reflecting on my teaching recently and have realised that I need to be giving pupils more time to consolidate their learning. Once pupils have ‘remembered’ new knowledge and shown the ability to apply it I often press on, conscious of getting through the whole scheme of work on time. Homework should provide time for consolidation but I want to offer that opportunity in an environment where pupils can get help if they need it. I am reading Eric Jenner’s brilliant book Super Teaching at the moment that discusses the way in which our brains learn and the necessity of processing time. Giving your subconscious brain time to process and organise new information is vital to the learning process and you have to make time for it or you won’t retain the learning. I’ve decided to do once-a-week consolidation lessons where pupils focus on practising lots of questions on topics that they have learned previously in a supportive environment. Rather than teaching something once and hoping they remember it in 2 years’ time I hope these regular consolidation lessons will allow my pupils to keep more plates spinning at once so their learning gets better embedded and they have less to resurrect when it comes to revision time. They should also find it easier to make links between topics which is so vital to learning. As Jenner says, it’s the connections that count in learning. You have to provide them with as many things to anchor new learning to as possible. My classes will still get high-paced dynamic lessons focussed on learning with lots of collaborative work and discussion, but they’ll also do some good old-fashioned consolidation. I hope by putting lessons aside for this I can strike a better balance that ensures my pupils get time for each part of the whole learning process.
If you are interested in giving your pupils more structured consolidation time you will need a collection of questions to give them. Step forward CGP and their new series Mathematics for GCSE & IGCSE. In house, CGP call these the no nonsense textbooks. They are just that. There are no glossy photographs and no dumbing down of technical maths language. Each section contains a worked example and then dozens of questions in the exam style that get progressively harder. Answers are included at the back. The books come in three flavours: Higher Level, Foundation Level and Foundation- The Basics. View them on the CGP site here. The books retail to the public at £19.99 each but they will sell them to schools for just £10 each! Amazing value for the thousands of questions in each book.
Am I taking a step backwards here by advocating such an old-fashioned style of textbook? I don’t believe so. I am a fan of collaborative working and discussion in a maths classroom. I believe in AfL and engaging learners with ICT and situated learning experiences. However, in a true student-centred classroom it is the pupils who have to put the work in to learn. I work hard for them by providing an environment conducive to learning but they need to match that effort themselves. If used in an old-fashioned sit and work in silence, teacher sits at front of room lesson where pupils who were struggling are not identified, these textbooks would not be effective. No textbook would be. On the contrary, used in a consolidation lesson that incorporates instant feedback through self-marking, a brain, book, buddy, boss sequence of who to ask when you are stuck, collaborative working and support, these textbooks could form the spine of a true student-centred learning environment. It’s what you do with them that counts!
At just £10 each these are top value and would see a student through a whole GCSE course for both class work and home work questions. Well done CGP. Mixing the best of the old with the new at a cracking price.
Disclosure: The author of this post received free sample copies of the textbooks for review.
A few weeks ago I was approached by the TES Secondary Maths Advisor, Craig Barton to join the new TES Secondary Maths Panel. The vision of the Panel is to raise awareness of the treasure trove of teaching resources that have been uploaded to the TES resources site and to act with editorial control to bring the best of them to your attention as easily as possible. Panel members will also write articles relating to maths teaching in the TES Magazine. I accepted the invite along with 12 other teachers and the TES Maths Panel was formed. You can see the profiles of the Panel members at this page.
The first project of the Panel was to create a series of Topic Specials for the secondary maths area on the TES Resources site. By TES’ own admission, the search functionality on the site has room for improvement and thus it is not always as easy as it should be to find the highest-rated, most-relevant resource to your search. The Topic Specials are the result of the panel sifting through the resources and choosing the 10 best on the whole of TES for each topic. Click here to view the Topic Specials. The Topic Specials feature some really excellent teaching and learning resources and will save you lots of time searching. We have divided up the whole secondary maths curriculum and written a Topic Special each. My own contribution was the Pythagoras and Trigonometry Topic Special. You will need to sign up for a free TES account to download the resources.
In addition to the Topic Specials you will see other resource collections on the Secondary Maths Resource Collections Page. These include GCSE revision resources, revision videos, MEP resources, National Strategies resources and much more. I’d strongly encourage you to take a look and bookmark the page.
An alternative route to the Secondary Maths Resource Collections Page is to go to the main TES website (www.tes.co.uk), then click resources, then secondary, then maths, then TES maths resource collections.
A+ Click is a new website that provides a vast collection of puzzles and challenges to develop logical reasoning skills in your pupils. You can search by topic or year group, making finding puzzles at your class’ ability easy. Like NRich Maths in many ways, A+ Click also keeps a record of how many answers your pupils got right. Ideal for computer room lessons. Craig Barton has made one of his Webwizz videos that explains all the details:
I’ll certainly be using these regularly for starters with classes who I want to show some more initiative in solving problems.
Further to my previous post where I launched the Revision Grids series of resources, I am pleased to announce that the Level 5 Revision Grids are now ready for you to download! I hope you and your pupils find them engaging and useful for learning.
If you like them please do pass them on to your teaching friends and share them through your social networks.
Watch this space for future Revision Grids at other levels. To be notified of new resources you can subscribe to Great Maths Teaching Ideas by email, Like our Facebook Page, and follow us on Twitter, Google+ and Linked In.
If you like the resources on Great Maths Teaching Ideas you’ll love reading 100 Things Awesome Teachers Do, available in the Amazon Kindle Store, the Apple iBookstore and many other online retailers.