 This is a video about using Grid Algebra in the classroom, and here I'm going to concentrate on making movements on the grid, maybe making movements for the first time. With Grid Algebra, if you go into the interactive Grid Algebra part of the software, you start off with a default grid of 6 rows. And I find it best to actually start with a smaller grid with movements. So I suggest you choose New Grid and choose Allow Negatives No, and just have one to two times tables with five columns. And then if you click on the button 1, 2, 3, 4, you'll get the numbers appearing in the grid. And here I'm looking, considering movement between the numbers. There's been a video earlier where I've talked about getting familiar with the structure of the grid, and this time we're looking at actually movement between numbers. So after start with a class, asking them if I were to, here we've got the one time stable, if I start at 2 and I go to 3, what am I doing to 2 to get to 3? And when I get that I add 1, then I click on 2 and drag it towards 3 and then let go. And that movement of expresses that 2 plus 1 is the operation that is done going from 2 to 3. So the operation is seen with, of course, the mathematical operation of adding 1, but with grid algebra there's also this moving to the right. It's seen as also moving 1 to the right. So there's a peel-back corner here, and that just tells you that there's something else in the cell as well. And if I click on that, the number 3 is still there, that hasn't gone away. And so clicking on the peel-back corner toggles between the expressions that are in any particular cell. And then likewise I can say consider starting at 5 and moving to 1. So I click on 5 and here I'm getting 1 less each time, and so that ends up being 5 take away 4. And again I've got the peel-back corner because the number 1 is still there. Then 5 plus 4 can also be picked up as well. If I were to go from 1 to 4, I would be adding 3. And likewise if I'm going from here 5 minus 4 up to the same cell, I would still be adding 3. So in this way expressions can be built up. And then this in turn can be taken for example to the cell with 2 in at the moment, and that would be subtracting 2. So I've got this expression and I'm subtracting 2. So Grid Algebra helps people to build up expressions and see them as objects in their own rights which can be manipulated not only as processes to be carried out. But if I were to carry out the process then 5 take away 4 plus 3 take away 2 will give me 2. Now there are different things in expressions in some of these cells. So for example I've got 2 plus 1 and 3 in this cell. If I click on the magnifier icon and then click take that over to this cell and click on it. Then a separate window comes up that shows me everything that's in that cell. And with another cell here for example I've got those two expressions I can put in a magnifier into that cell as well. It's colour coded to tell me what's in that cell. So with a class I work on movements horizontally first in the 2 times table. If I'm going from 4 to 6 in the 2 times table because it's the 2 times table I'm adding 2 each time. And likewise coming the other way if I would start at 8 I'd be subtracting 2 and then 4 and then 6. And then once again this can be picked up and dragged in its own right. And often pupils like to keep dragging really and start building up quite big expressions. Now obviously there comes a point when the software will decide that it can't go any further. But it can build up quite a lot of expressions. Now you can put in a magnifier into a cell it will show everything in that cell. So we've got several expressions in that cell we can't see them very well here. But if you click on the bottom right hand corner and drag then it can make the window bigger and we can see them so the whole class could see it if you're using an interactive whiteboard. So we've got a number of expressions in here and with grid algebra if they're in the same cell they're arithmetically the same. And if I wanted say for example 6 to be the subject of these equations here then I can just click on 6 and it will turn it so that 6 becomes the subject and 6 will appear on the top in the cell. So I'm just returning these back to their single digits. I tend to work on horizontal movements first before I consider any movements up or down. And as I'm working with horizontal movements gradually I will ask my pupils to write down in their books what they expect to see. So for example if I was starting with this number 6 and I was going to move it to where number 2 is here then I'd ask them maybe to write in their book what they're expecting to see when I make that movement. And then we will make the movement and see whether that's what they're expecting. Having worked with horizontal movement I then think about movements vertically. I tend to start by thinking about the one times table and thinking about what would you do to the one times table to turn it into the two times table. And somebody will invariably say that we'd multiply by 2. So I'd get hold of the 5, I'd drag it down and indeed with algebra shows that it is multiplied by 2. So here we can see the two times table as taking the one times table and multiplying it by 2. And then of course if I were to have made a movement such as the 2 plus 1 going to 3 this is an object which I can now also multiply by 2. And if I take that and come down then that's the way it is written. So sometimes with classes what I tend to do is I get them to articulate what they're the operation they're expecting to happen in this case multiplied by 2. I get them to say that first. So then when I make the movement they are already aware of the operation and they bring that to then the notation that they end up seeing. So they know it's going to be multiplied by 2 and apparently that's how it's written. So they're gradually getting used to more formal notation. And again I might for example make a bigger expression here and say I'm going to multiply this by 2 and again they might write down in their books what they might expect to see and then afterwards we make the movement and they see what they've written their book is what they see with the software. And of course although I'm doing the movements here quite often I'm inviting pupils to come up to the board and make the movements themselves. So I try to get them involved after I might begin by having control over the mouse and the movements first but then gradually I'm getting them to come up and make movements on the board. So having established something about moving down this multiplication I'll then maybe return to here to remind them that coming down is multiplying by 2 in this case. So what would going up be? And again I'll get them to articulate the operation first so it would be dividing by 2 then I would grab hold of 10 and move it up so that they are aware that it's going to be dividing by 2 and apparently that's how the computer is expressing it. And for some of them that might be an unusual way of expressing dividing it all depends but since we're heading towards algebra here for me what's important is they're getting used to formal notation. So likewise I may have made another journey here and then this is going to be divided by 2 I would tend to show them that first so they can see that the line has been expanded across 8 minus 6 and then perhaps begin to work on another expression here asking them perhaps to write down what this would look like when we move up dividing by 2. And again I get them to express the operation first so that they're aware of the operation and then see the notation afterwards. Now here's an expression and I'm going to take that and move it 1 to the right. So again I get them to express what happens when you move 1 to the right and after they've expressed that it's adding 1 I'll then move that and they will then look at and we might look at how it's ended up expressing the adding 1 in a particular significance is that you've got the line and then the addition sign is at the end of it and then I might get them to write down if I were to subtract 1 what would that look like then? So again they're beginning to practice writing down what they think it would look like before we then make the movements and then we might take this and think about we're going to multiply that by 2 I'm going to come down again what would that look like and so we come down and we see how what that ends up looking like so gradually we're building up different expressions in this way Okay so now I just want to change the grid into a bigger grid to show you some issues you need to be aware of with a larger grid and I'm going to put it in all the numbers again by pressing the button 1, 2, 3, 4 and so on here of course I'll be adding 1 as I normally do in the one-time stable with a three-time stable I'm adding 3 each time and the five-time stable adding 5 or adding 10, adding 15 and so on so what's important is that they're aware that whatever the time table is and the column on the left-hand side tells you that then that's what we're adding on each time as we move from one cell to the other or subtracting from one cell moving from one cell to the other coming down then if I start at the one-time stable and I want to go to the four-time stable then if I take a number and I come to the four-time stable I'll be multiplying by 2, 3 or 4 multiplying by 4 so that's similar to what we've had before and again if I have an expression here and I drag that down to the four-time stable then it would look like this an issue comes maybe when I'm thinking of the two-time stable if I were to take a number in a two-time stable and times it by 2 where do I end up? so if I were to take this 4 and times it by 2 where do I end up? and the answer is I end up here so one issue with a larger grid is that you have to think about which rows you're going to when you're multiplying by 2 or 3 and so on and if I were to start at 4 and I were to multiply it by 3 I will actually end up where the 12 is in the six-time stable now if I want to go from the four and I'm going to drag it down to where the 6 is below it well we're multiplying by not a whole number and I've decided with the software that I'm going to stay with whole numbers so in fact that is what's shown if you try to go and multiply by something that's not a whole number I've decided that we're going to stick with whole numbers it can get complicated enough anyway with whole numbers and so that's what happens if you try to multiply by something that's not a whole number and likewise with division if I'm taking something in the six-time stable like this 18 and if I divide by 2 I'm actually going to get to the 9 in the three-time stable so I'm not going to allow anything here but we would be dividing 2 to get to the three-time stable if I continued up I would be dividing by 3 here and dividing by 6 here so that's an issue that needs to be worked on if you've got a larger grid and that's why I recommend starting with just a grid with two rows because it takes away this extra complexity that's around but with some classes it would be absolutely appropriate to work on this so they've got the freedom of the whole grid to move and that's the end of this video there will be another video that we'll then go on and look at exploring further making journeys in the grid and giving tasks related so that for children which could help build up their awareness of orders of operations