## 2 September 2014

### 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'.

 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.

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 d=  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.

1. Great post, some lovely questions and resources here I've not seen before. On the subject of triples, you may find this interesting: http://wp.me/p2z9Lp-bP

And my very first ever (albeit not great) blog post was about Pythagoras in 3d! http://wp.me/p2z9Lp-9

1. Thanks Cav. Aah, your first post from 2012! I bet that feels like a long time ago. Love your triples post, I'll definitely use that method for finding triples. Also the proof by induction for FP1, great stuff.

2. Here’s a lesson I did when I visited a teacher with a year 8 class in a middle school in the summer term. She asked me if I could do a lesson which tied in some assorted revision and practice while leading the way towards Pythagoras – without spoiling the fun for their teachers in secondary school.

Phew! I asked for five minutes to think about it and this is what we did.

1…..I asked the children to construct a rectangle as perfect as they could make it. (Lots of revision of basic 2-D shapes and their properties, vocabulary, and of course accurate use of instruments.)

2…..I then asked them to draw a diagonal of their triangle and rub out one of the two triangles.

3…..I asked them to construct an equilateral triangle on each side of their triangle and to find the area of each of the three triangles. (So there was lots more practice in constructing triangles and using equipment, not to mention finding an assortment of methods to find the areas.)

4…..Finally, I asked them to post their three answers on the board and see what they could observe.

I was pretty pleased that I'd managed to come up with the whole lesson quickly from scratch. The word Pythagoras was never mentioned, but I rather hoped the lesson would have to come mind when they met one of the standard diagrams in a few months’ time. Of course, constructing semi-circles on the sides would have worked just as well, but I thought there’d be more mileage in using equilateral triangles.

1. Thanks for sharing this lesson Alan, it's fantastic. I love the use of accurate constructions, a skill that's always worth practising. I'm certainly going to borrow these ideas.

3. Two animations that might be of interest to readers of this post:

http://www.mathimation.co.uk/triangles/pythagoras-theorem-proof/

and http://burymathstutor.co.uk/Pythagoras.html

4. Interesting extensions of Pythagoras theorem.

Bet you didn't know this about Pythagoras.

1. Lovely, thank you!

5. Hi Jo,

Just popping by for some inspiration! So many great ideas for my year 9s!

A great activity I've done for Pythagoras is giving each half of the class a rope with 13 knots in at equal distances and then ask them to make me a right angled triangle.

It works as a great little race for students to start problem solving.

Dan

1. Thanks Dan, fantastic idea!

6. I'm not maths teacher (or a physics teacher) but a former engineer turned trade union official. I still love maths and physics though and during my attempt to get a decent understanding of special relativity I came across this very simple way of understanding special relativity and time dilatation using only Pythagoras.

There is, of course, more complex maths involved in relativity, but this use of only Pythagoras is brilliant and could be used with older students or 6th formers to get a conceptual understanding of an important physics concept. I wish someone had shown this to me when I was at school as I may have studied physics instead of going to Uni to chemistry and dropping out.

Is worth checking out

http://www.emc2-explained.info/The-Light-Clock/#.VgBs6ZcYNOI

7. Great post and ideas, thanks.
Been inspired to post a link to an interactive Pythagoras tool of our own:
http://www.mathelize.co.uk/pt.html