A favourite starter
Sunday 17 January 2016
This is a quick post just to share one of my favourite little starting activities and a couple of related thoughts. These things are just little gems to be collected and treasured. This one was first shared with me by John Mason, author of 'Thinking mathematically' and more. I had the same experience then that most of my students do. If you have'nt seen it then please have a go before you read on.
- Think of a number between 2 and 3
- Think of another number between 2 and 3 that does not involve the digit 5
- Now another that doesn't use the digit 5, but does use the digit 7
- And finally, think of a number between 2 and 3 that doesn't use the digit 5, does use the digit 7, doesn't use the digit 9 and is a close to two and a half as possible.
Spoiler alert! Try not to read any further until you have had a go at this yourself, otherwise, you kind of miss the experience........
This is a terrific little video from 'Mathsnacks' which I have shown on numerous occassions because it is both a bit of fun and resonates with lots of issues in terms of how we see numbers. It will also take up a bit of page space before I discuss the solution to the above problem. There is a subtle hint in the video too!
How did you do?
So now to discuss the little starter. In looking for a solution to the last part, depending on the class, most students will iterate until they get to the conclusion that the number they want is 2. 48 (followed by an infinite number of 8s) and then a 7. If a student offers 2.487, then the next says 2.4887, then 2.48887 and so on as we realise that we can always just get a little bit closer to two and a half by adding another 8.
What then follows is a great discussion about whether or not this number that we are trying to describe actually exists which is often very fruitful and entertaining. It is a nice model for demonstrating how quickly we can get in to uncertain territory within our number system.
Depending on the time you have available and the unpredictability of classes you might then show the number rights video. This hits on some really important points. Because we so often use fractions as proportions, we are often confused by their identity as actual numbers and recogninsing which sense a fraction is being used in is really important. Of course, the astute will notice that despite the committed actvism in the video, irrational and complex numbers have been sidelined - again (big sigh of resignation)
On rare occasions a student in my class has put 2 and 2 together (sorry, couldn't resist) and you can watch their expression go from possibility to delight as they realise they have trumped everyone. Then there is a collective self kicking exercise when they share their answer of ... (click the hidden box icon to reveal)
OK, so maybe you got there already, but I didn't and I am amazed and delighted with how often this goes exactly this way in my classroom.
So many issues wrapped up in all of this and thats why I enjoy it. Now off to further update the numberline on my classroom wall as I commit to being less 'numberist' with every passing year.