Let’s generalise 1

OK, mathematicians. Are you ready for some sweaty brains? Good. What do you notice when you see this?

Uh-huh, you might notice the pink outline and there's some numbers. Can you have a look at what you notice about the numbers? In particular the mathematical noticing's?

OK, let's take a close look at the relationship between the numbers. That sounds like a really fancy word, but don't worry too much. Look, that just means, what do you see happening here between 23 and 24? 24 is one bigger than 23. What about with 9 and 10? Yeah, 10 is one bigger than 9 and what do you notice about the difference?

It's still the same. So that's really interesting to me. Let's have a look at a few more number sentences or equations and see what happens. OK, what can you see happening here? Yes, like you, I can see the same thing happening. So as one number gets bigger, the other number also gets bigger by the same amount. But the difference is still 14 and it looks like this happens for each pair of equations.

Yeah. Look. So here we have 23 - 9 is 14 and 24 - 10 is 14. 24 is one bigger than 23. 10 is one bigger than 9, but the difference is still the same, still 14. Let's look at this pair of numbers. What do you see? Yeah, the same pattern. 24. To get from 24 to 25, I add one. To get from 10 to 11, I add one, but the difference is still 14.

OK, what about this pair of numbers? Yeah, the same thing is happening. 25 is one less than 26. 11 is one less than 12, still 14. Or you could say it the other way. 26 is one more than 25. 12 is one more than 11, still 14.

This is pretty awesome. What if though, I don't look at just numbers in the counting sequence? What about 23 - 9 is 14 and 26 - 12 is 14? Let's have a look together.

Here is 23. And here is 9. And the distance between them or the space between them is 14. OK, here is 26. Here is 12. And the difference between them is. 14. Look. They are the same. So in this case 26 is 3 bigger than 23, 12 is 3 bigger than 9. But the difference is still the same. It's 14, so I think we're noticing something, that it doesn't matter how much you increase or decrease, so go up or go down. Or that you do to the numbers, the difference between them seems to stay the same.

So now I'm really curious. Does this always work? 'cause this would be an awesome strategy for us to know about. Let's have a look what happens with addition, ready?

Here's two number sentences or equations. What do you notice? Uh-huh, I see that too. Look. This time. Yeah, 23 is one more than 22. 9 is one less than 10. So 23 is one more. 9 is one less. But this sum is still 22.

Let's have a look at a few more number sentences or equations with addition. What can you see happening here? Ah, I see that too. Look. As each of these numbers goes down by one, these ones increased by one. But the sum stays the same.

So I think we need to revise what we were thinking before. 'cause it doesn't seem to work, this strategy doesn't seem to work, in the same way with addition, does it? So maybe what we need to say is that when, our working conjecture at the moment, is that when we're working with subtraction, it doesn't matter how much you increase or decrease the numbers by the difference between them seems to stay the same. So long as you do the same thing to both numbers, I think that could be a really important idea.

OK, so here's your challenge now, mathematicians. Here's our working conjecture, and what does happen with addition? This is what we'd like you to explore now so you can use any numbers you feel comfortable working with to investigate this idea. So you might work with numbers between one and 10, or you might work with numbers in, with fractional quantities. The choice is yours, but make sure you record your thinking using pictures as well as words and symbols and see if you can come up with a conjecture for what happens in addition.

OK mathematicians. Over to you. Enjoy investigating.