# The All I Can Throwers – Sessions with Den and Jenna. #1 – Consecutive Numbers

I think most of us are pretty uncomfortable with labels like “slow learners”, “less able” etc. And terms like “gifted”, “quick”, “high ability” aren’t much better. Apart from anything else, many are highly pejorative and they all suggest that everyone has a fixed level of learning.

All the same, I do from time to time meet children who do have the experience and background that lets me know they can take all I can throw at them, and recently I had the chance to work with a couple of them for a few weeks. The challenge was not to accelerate them through the syllabus but to give them the opportunity to explore some ideas at greater depth than is normally possible.

So what did I do with my All I Can Throwers? In looking for ideas I had many of the same criteria I’d use for any group, but basically I wanted themes that were accessible, intriguing, offered scope for asking questions, and lots of things to find out.

My first almost chose itself. I was asked to go to Cambridge a while back to talk to NRICH about my favourite activities and we both agreed that number 1 on our list would be the Consecutive Numbers question.

• I can write 12 as the sum of three consecutive numbers; 12 = 3 + 4 + 5
• Another example: 9 = 4 + 5
• Another: 14 = 2 + 3 + 4 + 5
• (and some numbers can be made in more than one way: 21 = 10 + 11, and also 21 = 5 + 6 + 7).

So the question is whether all numbers can be made in this way.

In truth, it’s the perfect enquiry for almost anyone. It’s immediately accessible using the simplest arithmetic, and offers scope to explore in your own way. There’s lots of scope for formulating questions, making observations, and reasoning and generalising. Perhaps best of all, it gives up its secrets gradually. It won’t take long to make the first observation, which explains half of all numbers, but others may take a little longer. There are further generalisations on the way, each adding a little more to the understanding.

Some of these discoveries will be made by any pupil who tackles the question, but there’s a final gorgeous climax in store for those who, like my pupils Den and Jenna, are able to dig deep and make generalisations. It had been a great way to spend my first hour with them, and with just five minutes to go they realised, and were able to explain, just which numbers cannot be made as the sum of consecutive numbers. But you don’t have to have a Den or Jenna in your group – it’s a great topic to explore (and makes for a fine display for the first parents’ evening of the year).

.

.