5 Maths Gems #5

Twitter is a different place now everyone's back at school. During the day it's really quiet because teachers are hard at work, but in the evening it comes alive again. Despite being the first week of term, the UK maths twitter chats (#mathscpdchat on Tuesdays and #mathschat on Wednesdays) were as busy as ever - as usual they gave me plenty of inspiration.

This week also saw the return of teacher blogs. I really enjoy reading reflections of what teachers have been trying in their lessons, what went well and what didn't. These blogs are a fantastic source of ideas.

This fifth weekly gems post is (unintentionally) brought to you by the letter P. It features patterns, puzzles, primes, projects, plenaries and posters.

1. Patterns and Puzzles

Fawn Nguyen's (@fawnpnguyen) excellent website visualpatterns.org has over 100 patterns. Students are asked to think about what step 43 looks like in each sequence. Fawn's website is a collaborative effort with contributions from dozens of teachers. Further contributions to the website are welcomed. The patterns can be copied and pasted into worksheets for maths lessons. This week @JemmaPDuck wrote a lovely blog post about a successful lesson in which her students were really engaged in exploring these sequences.

Sticking with visual maths, Denise Gaskins (@letsplaymath) shared a puzzle from the website 1001 Visual Puzzles. I explored this site and found that it's full of very unusual and original puzzles. I also learnt a little about topology - what a bizarre branch of mathematics. Here's an example of a topological puzzle (students would need to refer to a map of South America for this):

2. Primes

Primes are really interesting, aren't they? I've seen some great stuff on primes this week.

Richard Green (@edfromo) is a mathematician who posts regularly on Google+. He writes really interesting things about a wide range of mathematical topics. He often features animations which you'll know I'm a fan of if you read my post about animations and simulations. For example this is a very clear explanation of Mersenne primes and perfect numbers.

If you're thinking of doing something on primes with your students, @the_chalkface shared this excellent true or false activity which students will find interesting and stretching.

I also like this activity 'Further Factor Properties' from @TeachitMaths, mainly because it involves sexy primes and I'm a bit immature like that.

4. Projects

I've never thought of textbook redesign before, but it could be a really good mini-project at the end of a topic. On @JemmaPDuck's blog she says that she showed pupils a boring maths textbook and challenged them to come up with a different page on the topic they'd studied. The new page should be informative, well presented, clearly explained and contain challenging questions. I love this idea and it's perfect for peer assessment. It would also work well in a revision lesson. You could pick the best page for each topic and create a whole textbook designed by your students.

4. Plenaries
How do you end your lessons? Exit tickets? Diagnostic questions? Texts or tweets? Suffolk Maths has a helpful Plenary Review Grid with lots of good ideas.

Zoe Elder (@fullonlearning) shared this BBC article via @katiecpd '10 things we didn't know last week'. I learnt from this that Hello Kitty is not a cat! The article is worth reading every week and could be useful for tutor time. It would also make a nice plenary activity in maths lessons. Perhaps once a week students could brainstorm '10 things we didn't know last week' in terms of what they've learnt in maths. They may be surprised by how many new things they've learnt.

5. Posters
The maths periodic table display by @Just_Maths would be a fantastic addition to any maths classroom or corridor. By the sound of it, many maths teachers up and down the country have devoted a few hours of their time this week to making this awesome display. Here's a great example from Tim Jefferson (@DocendoTim).

Tom Riley (@riley_ed) shared this picture of a staircase to a maths department, which features primes, squares, cubes, times tables and formulae.

I love this post by Lois Lindemann (@MoreThanMaths) about her aims for the year, where she shared a fantastic idea - she asked students to draw portraits of their maths teachers. What a lovely way of celebrating their artistic talent.

Finally, I also spotted this excellent idea for a simple but effective message from Cristina Milos (@surreallyno). I had fun making a poster out of this message using postermywall.com.

While I was in a creative mood, I made some posters of @joboaler's positive norms messages. As I said last week, I don’t actually have my own classroom to decorate but hopefully someone will find them useful!

Resourceaholic.com update
I'm really pleased to see that my blog is being used by teachers to access great resources. This week I wrote a post about teaching Pythagoras' Theorem and added a topic specials page so you can quickly find similar posts. I also had a rant about student targets in which I questioned whether they are effective or even necessary. After Ed Southall (@edsouthall) shared these awesome wall decals, I updated my mathsy gifts page. I've also added a couple more homeworks to the Pret homeworks page.

How good are you at drawing lines of best fit? Test yourself here (thanks to @MARitchings) - fun for teachers and also a useful lesson resource.

Finally - look, Spiked Math has a similar idea for name learning to the one I mentioned in my last maths gems post.

Pythagoras' Theorem

Pythagoras' Theorem is one of my favourite topics to teach. In my experience, students quite like it too. At first they may be intimidated by the new and seemingly complex mathematics. It's probably the first time they've seen words such as theorem, proof and hypotenuse so no wonder they're apprehensive. But they'll quickly get the hang of it.

When A level students have to find the distance between two points or verify that a triangle is right-angled, their first thought should be Pythagoras. By Year 12, Pythagoras' Theorem should be a staple in their mathematical toolkit, along with other skills such as arithmetic, fractions, simplifying algebra, using index laws and factorising quadratics. In my opinion these things are the bread and butter of secondary school higher mathematics.

I'm going to touch briefly on proofs at the end of this post but my main focus is on interesting Pythagoras questions and resources for Key Stage 3 and 4. Before we start, I should issue my usual caveat - there's absolutely loads of excellent Pythagoras resources online and here I'm just showcasing a small selection.

Interesting Questions
It's not always necessary to force a question into a 'real world' context to make it engaging. Problems don't have to be contextual to be interesting - sometimes the very best questions are abstract but accessible.

In this example from illustrativemathematics.org, students have to work out whether the shaded triangle is right-angled.

This Spiderbox question, also from Illustrative Mathematics, appears at first glance to be a 3D Pythagoras question but actually the spider is walking on the outside of the box. Students make a net, draw the spider's path and work out its length.

Don Steward (yes, I'm still #stalkingdon) provides his usual excellent selection of Pythagoras questions. Here's an example called 'two diagonals' and another called 'kite areas'.

Helpful Worksheets

horacek.com.au

This 'Investigating the sides of right-angled triangles' worksheet from Teachit Maths is a straightforward activity in which students attempt to 'discover' Pythagoras' Theorem by drawing right-angled triangles and investigating the relationship between the lengths of their sides. Teachit Maths also provides a 'Pythagoras' Theorem - complete topic booklet' which covers the full topic from simple practice questions to problem solving, using surds and 3D Pythagoras.

These Pythagoras' Theorem Student Sheets (& notes) from Nuffield Foundation contain good problem solving questions, as does Pythagoras Problems from The Chalk Face.

I've written a Pythagoras Pret homework - see this page for more information about Pret homeworks.

Adding a Third Dimension
During my PGCE I had my students build cuboids out of pipe cleaners and work out how to find the length of the space diagonal by themselves. The pipe cleaners were a faff that I won't be repeating but figuring out how to find the length of a space diagonal is definitely something students can do themselves. Perhaps the easiest prop to use is your classroom. When I teach 3D Pythagoras I normally gesture at an imaginary space diagonal being drawn from the lower left corner to upper right corner of the room. In fact it would be better to attach some string or wool to the walls so you have an actual space diagonal going across the room, to help students visualise the right angled triangles.

There's two approaches to teaching 3D Pythagoras. One is to have the pupils derive the formula for the space diagonal (eg d² =  x² + y² + z²) and then memorise it. This may be useful later on, for example when they study vectors in C4, but it's not helpful when they come to 3D trigonometry. My preferred approach for 3D problems is that they identify and separately draw the two right angled triangles in question. I encourage them to keep their workings in surd form for accuracy.

Good 3D questions are available in these Trigonometry Worksheets (see exercise T7) by Frank Tapson (from the Cleave Books website).

Also, here's a nice 3D Pythagoras activity 'Cuboid Diagonal' from Don Steward.

Triples
Pythagorean Triples are fun to explore with your students. Before you do, take the opportunity to impress them with your apparent super-quick mental arithmetic ("the two sides are 5 and 12 you say? Well the hypotenuse must be 13").

Activity 3.4 in this set of activities from the Mathematics Enhancement Programme is a short and accessible investigation into Pythagorean Triples. The Mathematics Assessment Programme also has a Pythagorean Triples activity and Don Steward has students looking for patterns. There are many more substantial investigations available online.

Vocabulary

If your students have trouble remembering the word hypotenuse then a terrible joke like "What do you call a kettle of boiling water on top of Mount Everest?" might help them. This picture is taken from this amusing post from Math Jokes 4 Mathy Folks.

I also think it's worthwhile spending a few minutes discussing the meaning of the word theorem and how it differs from a theory. A theorem is a result that has been proven to be true, for example using logical arguments and previously established mathematical statements. Theory is a word used both in everyday life (eg 'the detective had a theory about who committed the murder') and in science (such as Einstein's theory of relativity). In everyday language it has a meaning similar to hypothesis. In science, a theory is a contemplative and rational type of abstract thinking. A theory simply attempts to explain something whereas a theorem is established mathematical fact. There's more on this here.

Pythagoras of Samos
Apparently Pythagoras was the first person to call the heavens a universe and the earth round. He also (allegedly) headed up a secret cult of Pythagoreans who weren't allowed to eat beans. Pythagoras said that "each number has its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly". The mysterious story of Pythagoras of Samos is really interesting and would make a good topic for research or display work.

Pythagorean Proofs
I won't talk too much about proofs because I'll be here all day. I must mention this brilliant water demonstration video though.



If you want your students to try some proof activities, this task 'Proofs of the Pythagorean Theorem' from Mathematics Assessment Project is about comparing three different proofs.

Finally, Don Steward's blog features Herve Lehning's posters of Pythagoras by means of dissections which are very pleasing to the eye.

I hope this has been helpful if you're planning Pythagoras lessons. Please comment if you have any ideas to add.

I've now put all my topic resource specials here so they are easier to browse.