 Hello, my name is Anne Haworth and today I want to share a simple mathematical starting point with you. In the classroom I have found it can be productive to start with something simple that everyone can get engaged with and then build on it to explore and develop some mathematical ideas. Maybe have simple starting points of your own that you use in this way. So to start off today's activity I invite you to think of a pair of numbers that add up to six. Now think of another pair and another. I wonder what your first thoughts were and what number pairs you thought of? What pairs of numbers do you think your students would suggest? How many pairs of numbers are there? I invite you to pause and think about that question and about what the responses may be in the classroom. Will your students see the infinite possibilities both with the numbers between 0 and 6 and with the numbers outside that range? If not, how can you help them to do so? Pause and think what questions you could ask. Perhaps you thought of some questions like these. If one of the numbers is three and a half what's the other one? If one of the numbers is 2.1 what's the other one? If one of the numbers is nine three-quarters what's the other one? If one of the numbers is a thousand what's the other one? If I give you a number can you always find another one so that the total is 6? If so how do you do it? What is the biggest number you can use? What is the smallest? Can you find pairs where both numbers are positive or both negative or one of each? To me it seems magical that you can go infinitely far in either direction getting some useful practice with directive numbers as you go and you can dig infinitely into the numbers between 0 and 6 and learn more about decimals and fractions. Who was it that said every mathematics lesson should include a glimpse of infinity? These activities provide the opportunity to practice working with numbers while scaling understanding of mathematical structures. I like practice to be embedded in real mathematics whenever possible. Pause and think where you would want to take this now in your classroom. I'm going to step into the spatial world by using the pairs of numbers as coordinates and plotting them on x y axes. Think how you would want to do this. Would you want each student plotting their own points? Would you want a class plot maybe on the interactive wired board? Or maybe you'd want to do it as a people mass activity. And how will you cope with the fact that any axes you draw will not accommodate all the number pairs students have thought of? So what do we get when we plot these points? It's another bit of mathematical magic. All the points line a straight line. You can connect them up with the ruler. Pause and think how you know that all the points line a straight line. Can you explain it to yourself? Can you convince others? This straight line has two important properties I'd like to emphasize. First of all, every point whose coordinates add up to 6 lies on this line. And the line consists of every possible such point, even the ones nobody had thought of. And some points people chose that won't fit on these particular axes are nevertheless part of that same line. The fact that we cannot mark them on these particular axes is a matter of practicality, not mathematics. They are all part of the line. The line that we physically draw is just a representation of the perfect mathematical line containing all these infinities of points. I think it's important to provide opportunities for students to explore and appreciate such ideas. I wonder if you agree? Where on the line are the pairs that were chosen earlier? Where are the pairs with both coordinates positive, for example, or one negative and one positive? And why are there none with both coordinates negative? What happens to the y-coordinates as the x-coordinates get bigger? Why? We can, of course, now use the line to solve the problem posed earlier, finding the second number in the pair given the first, or finding the first number given the second. A useful introduction to solving equations graphically, linked with the numerical work already done. An equation of this straight line? Well, of course, because the x-coordinate and the y-coordinate add up to 6, it's x plus y equals 6. The symbolic representation follows the words, and the same thing will happen with other straight line graphs. So now we have several different representations of this mathematics. The list of pairs of points, the line itself, and now this equation, which, of course, we could write in other forms, such as y equals 6 minus x, for different purposes. So perhaps now we could try to draw the graph of x plus y equals 9, say. What's the same and what's different compared with x plus y equals 6? What about x plus y equals negative 6? Or x plus y equals 0? What other lines are in this set? And can you plot such lines without having to list the points first? How? What other questions could you ask? That's as far as I'm going to take this activity with you today, but I hope you can see some ways forward. So to finish, I'd like to set you two tasks. First, the question I mentioned earlier, where could we take this work now? What would you want to do with it? Second, think about other simple starting points that you could use to lead to mathematical exploration and development in your classes. Have you got some that you use already? Or can you think of some new ones to try? And thank you all very much for watching.