One of the great things about my role(s) this year is that I have had the opportunity to meet some fantastic Maths teachers and educationalists and last week I hosted Mike Ollerton for two separate events. Mike has made many significant contributions to Mathematics education over the years and he has kindly permitted me to write about the ideas he shared with us last week.
This is a simple activity that feels quite fun and personal but could lead to some rich discussions. Mike’s description of it is here:
After going round the class, asking several children for their BDVs, there are many questions which might present themselves. Can you ask children to work out someone else’s birthday given their BDV? Mike suggests lots more questions:
- Which BDVs only have one birth date?
- What are the minimum and the maximum BDVs in a class?
- Which BDVs have the most dates?
- What is the smallest BDV which cannot be made?
- What is the largest unique BDV?
- Which dates are square BDVs?
- Which dates are triangular BDVs?
- In a group of people who has the average BDV?
What other problems can you devise based upon BDVs?
Here’s a little idea for a team activity that could get quite competitive and hopefully “fun”. I haven’t tried it yet, and it might be a while before I use this. When I do, I’ll try to update this post with any tweaks depending on how it runs.
It’s based a on this really simple website Is This Prime created by@christianp which I saw on Jo Morgan’s MathsGems.
It presents you with numbers and you click YES (i.e. it is Prime) or NO. It’s not an app so it can be used on a laptop / desktop although it works really well on browser on a tablet. I’ll be doing this in a computer room as a group activity.
I reckon this could work with classes from Year 5 to Year 8, but most pupils in the class will need to have a reasonably good grasp on their times tables or it could be frustrating. It provides consolidation of times tables and primes but I think the real objective here is actually to use this as a lead in to various data and averages topics. I always try to teach KS3 Statistics using data that the students have created themselves as they are far more engaged and care about what the data is telling them. This not only provides that meaningful data, but does so in a way which consolidates some fundamental number facts at the same time
I plan to use Google Sheets to collect the data which we will then analyse in a later lesson. Google Docs in general is great for this type of collaboration. I have created a template for a group.
Each group has a separate sheet that they fill in as they go. Just duplicate the sheets for as many groups as you have, making sure that each group is working on their own sheet before you start.
Talking of groups, here are my general rules for planning any group activity:
- I chose the groups. I have nothing against pupils working in friendship groups but I know who to avoid putting together and the process of self-selecting can be painful for some.
- Everyone has a role. Some pupils will see group work as a chance to sit back and some will naturally dominate. Assigning specific roles reduces this.
- Everyone contributes equally. By rotating the roles I will try as far as possible to make sure that everyone ends up doing the same activities by the end of the session.
To get some excitement going, I’ll keep a running commentary on the highest score. I also plan to write up the “Errors” that the Error Recorders give me as we go. I want to make sure we have some time at the end for reflection on how it went, i.e.
- Did you work together as a team? How did you support each other?
- What was a good strategy for a high score? (When I play it, I rarely use all 60 seconds as I am trying to go too quickly and so I am often tempted to guess ones I don’t know)
- As a team what did you do to make sure your scores were improving (Write down the errors on a big piece of paper? – I didn’t say you couldn’t!)
I would definitely leave the data analysis part until the following lesson. There is lots you can do with this and it could form the basis of a series of lessons on Averages, Data representation including Box plots. We can start with the question, “Who was the winning team?”, which in itself is open to interpretation.
Datapointed: I like to start this one at the beginning of the lesson, let them watch up to about 20, then teach the lesson leaving it running in the background. Then come back at various points to see how far it has got.
Primitives: This one is more interactive and can be used when teaching Prime Factor Decomposition.
Chalkface: And finally a blog post with lots of interesting ideas and this nice graphic.
A nice couple of demonstrations of what makes a proof, i.e. just because you’ve got lots of examples doesn’t mean you’ve proven it.
Mathematicians of the 18th century proved that numbers 31; 331; 3331; 33331; 333331; 3333331; 33333331 are all primes. This was no mean feat without a calculator. It was a big tempation to think that all numbers of such kind are primes. But, the next number is not a prime:
333333331 = 17 x 19607843
Another classic example is the question of how many areas you get when you cut a circle with chords formed by joining points on the circumference.
You might think you’ve spotted a pattern of doubling each time (or 2^n). And indeed the next one is 16. But the one after that is 31.
The formula is not quite so straightforward and involves combinations:
In its expanded form it looks even more crazy!
Wolfram has more details on this problem here.
And some nice discussion of the problem here.
I like to use this as a sort of a crescendo to teaching prime factor decomposition which is itself a very satisfying experience.
Although it sometimes feels a bit procedural it’s a nice way of:
- Practicing times tables
- Getting to know your primes
- Appreciating the commutativity of multiplication.
Anyway, here’s the trick (everyone needs a calculator in front of them)
- Chose any three digit number. Write it down somewhere.
- Type your number into your calculator and divide by 7.
- Hands up if you got an integer answer. Opportunity here for a nice discussion that we might expect 1 in 7 hands to be up at this point.
- Press clear and divide the same number by 11. Repeat again with 13. Right, now they’ve appreciated that not many numbers are divisible exactly by 7, 11 and 13. Time to blow their minds…
- This time type your 3 digit number into your calculator twice so you have a 6-digit number. e.g.
- Divide by 7. Hands up if you have a whole number. Wow, everyone. Now don’t press clear, but divide by 11. And then 13. Wow. Gets you exactly back to your original 3-digit number.
How much you chose to explain this will depend on the ability of the class, but the points are:
- Whatever 3-digit number you chose, the 6-digit number is 1001 times the 3-digit.
- 1001=7x11x13. Weird but true. And this is why it works.
If your students seem to like it, I always ask them to try it out on their family when they get home. I love the idea that I just might have created a discussion about maths around the dinner table – you never know…