1. Investigational Work Following through and seeing the steps and paths that MAY be taken

2. The Problem as seen on Web

3. Remember to Choose four consecutive EVEN numbers; Multiply the two outer number Multiply the two middle numbers Find the difference. Allow time for the thoughts and ideas to develop to try and answer WHY? So we have a go!

4. You might get … “I tried a few and then did a kind of algebra, so I wrote 2n, 2n+2, 2n+4, 2n+6 Multiplying : 2n x (2n+6) and (2n+2)x(2n+4).... [various calculations to end up with] This left me with 8.”

5. 2 4 6 8 24 16 Or they may show it spatially … Click to show the 2,4,6,8, which has to be two wide, when counting in twos. Click to show the 4 x 6 in yellow Click to show the 2 x 8 in blue Click [three times slowly] to show the (4 x 6) minus the (2 x 8, bent over) and so we see the 8 that remain !

6. There may be a lot more going on to explore these two very different ways of working (thinking/learning), but this may be enough. BUT asking “I wonder what would happen if … “ will most likely lead to other opportunities. SO, “I wonder what would happen if instead of using 2 as the ‘step’ size we tried other sizes?” OR, “I wonder what would happen if instead of using four lots of numbers we had three, five, six, seven etc.?” Either or both of these questions can be explored, the next slide shows the results that follow and allow further investigation into what can be seen.

7. Size of Step click We could also go up to, say, … twenty one numbers in the line click and what about taking bigger steps in the line? click 8 So if we have not just four numbers in a line, stepping up in 1s and 2’s see what we get … Number in a Line

8. It’s also a good idea, when you have a lot of numbers in a sequence or a table, to have a look at the Digital Roots of those numbers. And now … The next slide gives an opportunity to do this, and it’s worthwhile to allow lots of time for further exploration of these new numbers.

9. Numbers in a Line Size of Step to see their Digital Roots. Look and explore … then Click Click again to see the top line extended Numbers in a Line Difference Digital Root So let’s go for the Digital Roots

10. Exploration may focus on the reflective pattern seen in the Digital Roots. click This may be exciting in itself and so stop here. BUT you could ask “What is so special about the numbers and their reflective partner?”

11. A further investigation can start just now! Let’s have a look at the numbers either side of the middle 9. We’ll have to have a look at where they came from. click The next slide looks at pairing off those numbers and then looks at the difference between the differences that were found earlier on. click Number of numbers Difference Digital Root

12. 3131 4242 5454 6666 7979 8 12 9 16 10 20 11 25 12 30 13 36 14 42 15 49 16 56 17 64 18 72 19 81 20 90 21 100 22 110 23 121 24 132 25 144 26 156 27 169 28 182 29 196 30 210 31 225 32 240 33 256 34 272 35 289 3131 4242 5454 6666 7979 8 12 9 16 10 20 11 25 12 30 13 36 14 42 15 49 16 56 17 64 18 72 19 81 20 90 21 100 22 110 23 121 24 132 25 144 26 156 27 169 28 182 29 196 30 210 31 225 32 240 33 256 34 272 35 289 Remember the top number is how many numbers in the line and the lower number is the difference between the two multiplications. 18 ---36--- ------54------ ---------72--------- ------------90------------ ---------------108--------------- ------------------126------------------ ---------------------144--------------------- ------------------------162------------------------ ---------------------------180--------------------------- ------------------------------198------------------------------ ---------------------------------216--------------------------------- ------------------------------------234------------------------------------ ---------------------------------------252--------------------------------------- ------------------------------------------270------------------------------------------ ---------------------------------------------288--------------------------------------------- Click to see the differences between the pairs So here are the same ones. Now click to see them paired Investigating these relationships only came about because of looking at the Digital Roots

13. Click to see the difference of the differences spatially Here is the yellow showing 18 numbers in line with a difference of 90 Here is the brown showing 20 numbers in line with a difference of 72 Click to show 90 - 72 Click to show the difference of 18 separated Click again and Click again

14. Where have we got to? 1/ We completed the question, allowing for different kinds of answers. 3/ We’ve changed the numbers into digital roots, which gave new ideas that could be investigated further. 2/ We extended the rules to look at new arrays of numbers, that could be investigated. Notice how the numerical can lead to the spatial and visa versa. Each investigation may give rise to number patterns that the pupils have come across before. Going back to the original challenge and then changing something allows more work to develop and comparisons can be made between the sets of results

15. So one more suggestion before I sign off. Working effectively with pupils doing an investigation I believe that the teacher has to resist taking the pupils down a route that they have seen. I hope that this presentation will just open a door for some to see that so very much can be explored from a simple starting point.

16. Some pupils may be prepared to go further with asking the question, “I wonder what would happen if …” So, let’s take as an example, considerably changing the initial rule to … “Multiply the first and middle numbers [ lower middle in the case of an even number] Multiply the last and middle numbers [ upper middle in the case of an even number] ” (e.g. 1, 2, 3, 4 so multiply 1 x 2 and 3 x 4 we get 12 - 2 = 10 ) (e.g. 1, 2, 3, 4, 5 so multiply 1 x 3 and 3 x 5 we get 15 - 3 = 12 ) EVEN number of numbers ODD number of numbers

17. Working with this new rule … … we can get a new table for stepping up in 1s and 2s Again, an array of numbers to explore, as was done with the original rule. The next slide looks at just one of these investigations.

18. 1 x 2 click 3 x 4 click 12 - 2 click and click again for the next step move those extra ones-click Looking at this spatially Using 1,2,3,4 as an example So we suddenly find that we have a triangular number formed. There are of course many questions that can now be pursued further. What about …?