Fun with infinity

I’m lucky enough to teach two top set year 8 classes (year 7 last year) who are at least semi-geeky about their maths. Last summer I did a lesson with them on the nature and craziness of infinity, with the able assistance of the amazing Vi Hart (or at least her Youtube vids), and Jordan Ellenberg’s book: How Not to Be Wrong. They loved it, so I’ve reproduced some of the resources below.

Vi Hart videos:

The infinite series paradox (1):

What is the sum of: 1 + 2 + 4 + 8 + 16 + …?

To answer this, first multiply the sum by 2: 2(1 + 2 + 4 + 8 + 16 + …) = 2 + 4 + 8 + 16 + …

Now: 2(1 + 2 + 4 + 8 + 16 + …) – 1(1 + 2 + 4 + 8 + 16 + …) = 2 + 4 + 8 + 16 + … – 1 – 2 – 4 – 8 – 16 – … = -1

But the LHS simplifies to: 1 + 2 + 4 + 8 + 16 + … = -1

Thus we seem to have proved that the infinite sum of ever-increasing terms is -1. Huh.

The infinite series paradox (2):

What is the value of the infinite sum 1 – 1 + 1 – 1 + 1 – 1 + …?

Answer 1: (1 – 1) + (1 – 1) + (1 – 1) + … = 0 + 0 + 0 + … = 0

So far so good. But hang on…

Answer 2: 1 – (1 – 1) – (1 – 1) – (1 – 1) … = 1 – 0 – 0 – 0 … = 1. Huh.

Answer 3: Suppose T is the value of our mystery sum: T = 1 – 1 + 1 – 1 + 1 – 1 + …

Take the negative of both sides: – T = – 1 + 1 – 1 + 1 …

But this is just what you’d get if you take 1 away from T i.e. T – 1.

So T – 1 = -T, which is only satisfied when T = 1/2. Eek.

I find answer 3 both most counterintuitive and most satisfying. Brilliantly, some of my year 7s came up with this solution themselves.

 

Relevance to the classroom:

This stuff isn’t going to be on any curriculum, and one observer in my classroom was heard afterwards to mutter ‘he just let them doodle!’ But for some of our most gifted mathematicians, we have to squeeze in real maths around the staid content of the GCSE curriculum – it might just inspire them to carry on with the subject to A-level and beyond. Plus everyone loves to have their brains messed with.

Abraham Wald and the bullet holes

So I’m constantly on the hunt for ways to relate maths to real-life, and to find hooks for topics. Over the summer I read Jordan Ellenberg’s book ‘How Not to Be Wrong’ and the following story is lifted straight from that book. It’s a great intro to a probability topic.

The premise
You’re in the air force during World War 2. Your job is to stop your planes getting shot down by enemy fighters, by armouring them. But armour makes a plane heavier, less manoeuvrable and uses more fuel. So you have to compromise between weight and protection from bullets.

The data
You are presented with the table below, which gives average bullet holes on planes returning from missions:

Section of plane Bullet holes per square metre
Engine 1.11
Fuselage 1.73
Fuel system 1.55
Rest of the plane 1.8

The question:

Where is the best place to put extra armour?

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Summer Institute highlight: An amazing day on number

I’m posting a series on the highlights as I saw them from the Teach First Summer Institute at Warwick. One of these was the first day we devoted to studying how to teach number. Diana Spurr and Marcus Shepheard covered a wonderfully rich variety of content in impressive depth – it was a really valuable day. Highlights include:

More highlights to come in a few days…

The wonderful world of the empty number line

Screen Shot 2013-08-11 at 09.36.44

How interesting can an empty number line be? Not very, I assumed. I’d read before about teachers raving about the concept. I’d always thought they were a little funny in the head. I am now officially a convert.

It turns out the empty number line can be used in a whole range of ways. Most simply, it can be a great mental tool for students’ addition and subtraction, by using multiples of 10, counting on etc.

For example, 12+13 becomes:

Screen Shot 2013-08-11 at 09.36.51

25-12 becomes:

Screen Shot 2013-08-11 at 09.36.56

So far so good, but it was what Diana and Marcus did next that was so exciting…using empty number lines to visually represent linear equations. (I’m aware there are many other ways to represent linear equations – this just feels like a particularly powerful one)

For example, taking the equation 3x+8=23:

Screen Shot 2013-08-11 at 09.37.02

So:

Screen Shot 2013-08-11 at 09.37.11

so x=5

 

You can even use the empty number line to explore relationships between fractions, decimals and percentages:

Screen Shot 2013-08-11 at 09.37.18

What could A be?

Summer Institute highlight: IWBs and technology in the classroom

This is part of a series blogging the highlights (as I saw them) from the Teach First Summer Institute 2013.

Hannah Tuffnell and Joe Ambrose hosted a great session on IWBs and technology in the classroom. It was brilliant partly because of the content but also partly because, on the penultimate day of the Summer Institute, in a boiling hot room, at 7:30 they kept 20 or so exhausted participants engaged and learning for an hour.

The new technologies they promoted were:

  • Prezi: Most of us knew this one – a more engaging version of powerpoint. It’s time-consuming, though, so best for the occasional lesson e.g. introducing a new topic.
  • Powtoon: Use to easily create animations up to 5 minutes long – students can make them too.
  • Padlet: An online noticeboard where students can write on a wall (you can moderate comments!). Could be great for homework.
  • Edmodo: A safer version of facebook. You can use it to set up assignments, polls, resources etc.
    • Useful for homework setting
    • Marks multiple choice answers
    • Students can download an app that lets them do homework on their phone.
  • Google Forms: (find via Google Drive) Create a questionnaire to give to students. Useful for feedback on you as a teacher.
  • Poll everywhere: allows students to text into polls in real-time, and can embed within powerpoint.

In terms of IWBs themselves, some specific tricks:

  • Revealer tool: pull down a box to reveal specific parts of the board
  • Infinite cloner tool allows you to create duplicates of e.g. a coin
  • Layering can be used to allow some things into a box and not others
  • Using two colours can allow you to magically ‘reveal’ an answer
  • It is possible to lock things into place.
  • Shape recognition and text recognition tools are available
  • The magic pen allows you to:
    • Write in text that slowly fades
    • Circle (everything else goes dark)
    • Zoom in by drawing a square
  • You can create a random name generator with a hat (this allows you to differentiate by having different names in different areas of the hat)

SMART exchange/SMART world is the source of more information and training.

These IWB ideas are great, but have made me realise what I really need is a bit of time with the board to play around with creating and using things. I can already see how I could use these tools to sort shapes, or to reveal the correct answer to factorising problems.

Tips from behaviour for learning session

We had a behaviour for learning session as part of our Teach First training last week. The points that came out of it that struck a chord with me were:

  • When taking the register, be human! Ask how each person is/something about them and listen to the answer.
  • For a tricky class/to assert my authority early on, consider telling students to line up at the back and tell them individually where to sit.
  • Get students to repeat instructions back to you to ensure they’ve heard, listened and understood.
  • You can set/maintain expectations when meeting students at the door (e.g. uniform, chewing gum) which reduces the amount you have to talk about things as a whole class.
  • If I take any classes with a reputation, I can make clear that as I’ve come from another school the class gets a fresh start with me, and that I’m really looking forward to teaching them.
  • Consider writing extra minutes on the board for late students.
  • Be very clear and specific with your instructions – often poor behaviour comes from students not understanding what they’re supposed to do.
  • Put classroom rules on the inside cover of students’ books.
  • At the end of the lesson, give students jobs, and act quite authoritatively to counteract the end-of-lesson excitement.