 This is a video about using the software Grid Algebra in the Mathematics classroom. The software is available from the Association of Teachers and Mathematics. And this video concentrates on inverse operations and helping students learn about solving equations. And it follows on from a video that's about introducing and using letters. Grid Algebra works particularly well with a whole class using an interactive whiteboard, although projecting it is also fine. There are also times when students work individually or in pairs in a computer room using Grid Algebra. I tend to work by offering challenges to the class rather than demonstrating something myself. So much of what follows are examples of challenges where students come up to the board and work with the software rather than doing something myself. I may start off by controlling the mouse initially, but I will soon ask students to come up to the board and collectively work on the challenges set. So although at times I am aware I use the word I when talking about what might be done, I am really thinking about the challenges students might be asked to work on. I'm going to start the software by going into the interactive Grid Algebra part of the software. And just as a quick reminder, Grid Algebra is based on a set of multiplication tables where I can pick up a number and move it to other numbers in the Grid. Addition is to the right so that when I'm in the four times table, I'll be adding four, then eight and then twelve and so on. Subtraction is to the left, multiplication is down, and division is up. And then these expressions can also be taken on journeys as well. And again this can be multiplied and so on to create bigger expressions. So I'm going to clear the Grid. I can have a number box or indeed a letter box where I can pick up a number and put it in the Grid and that number can be taken on a journey. Likewise a letter can be selected and put in the Grid and that can be taken on a journey. And what I'm going to do now is to clear all the numbers within the Grid but I'll keep the ones involving letters. And I've taken D on this journey around the Grid. And what I'm going to do is to delete all the expressions apart from the final one. And with a class I might then say, oh no, I've forgotten. Where was that letter D? Where was the letter D? We've ended up here with this expression in this cell. How can I use this expression to work out where that letter D is? How can I get back to the letter D? And there'll be some students that may be able to think about this and realize that they're going to need to subtract 18 and then divide by 6 and so on. But I've found that there'll be a number of other students in the class who are really not quite sure what to do. But I still do this in order to set up what the task is. The task is to try and work out, give an expression involving a letter, how I could get back to where the letter is. So having set that up, I then, as the main thing that we're going to focus on, I then clear the Grid and take another letter and go on a journey with that letter. But this time I'm going to offer support of making, marking that root. So I've clicked on the root button and there then becomes a new root button that appears underneath it. Clicking on that, I can click on where I've been in that journey. I click back on the button again so that I'm not continuing marking a longer root. And then this root indicates the journey that's been taken. If I click on any cell, then what's beneath that root marker is shown. And now I've got the visual dynamic of the root that was taken to get to this final expression. I'm going to hide the root again by clicking on that button. And I'm going to rub out everything but the final expression. So we've got a similar problem now where I've got an expression and I want to work out where's the letter m. But this time I'm going to click back on the root to show the root so that we know that in fact m was here and was taken on that journey to get to this expression. But that can help them visually think about what they have to do with this expression to get back to the letter m. So it's going back on the inverse journey round the grid. So at the same time I pay attention to the operations that are in the expression. So the first thing I'm going to do is I'm going to take 4 and then I'm taking 8. And we're noticing that the last thing we've done is to add 8 on the expression and now I'm subtracting 8. Students by this time I would have worked on order operations so by this time they would have a certain confidence with order operations within an expression. Now ideally it would be nice if it didn't say add 8, subtract 8. Really they cancel one other out and it would be nice if they did just cancel one other out. So I can make that happen by in the toolbar there's an inverse button. If I were to click that so that it's down then when I'm making a journey the journey builds up unless I'm making a journey which undoes the very last operation within the expression and then it will cancel it out. And so with this expression then again I would talk with a class and we would end up thinking about what we have to undo now. The last thing we did was multiply by 4 so we're going to divide by 4. And so we're going to divide by 4 and then that is cancelled out. And then it says subtract 3 so if I add 1, 2, 3 that will cancel that out and then it says dividing by 2 so if I multiply by 2 it will cancel that out. So the appearance of the root can help students know what we have to do to get back to letter M but talking about the actual expression allows us to work on what that means in terms of mathematical operations. So the actual visual support of a root on the grid supports their learning of inverse operations within the expression. Now I'm going to clear that. I would do several different examples each time I would deliberately go and choose a different letter. And perhaps the next example that I might have sometimes doing quite complicated expressions motivates students. I don't think once I get the idea of this actually you could have quite complex roots that take place. So I'm going to take the root away again. I'm going to rub out again everything but the final expression. And so again I might be asking well given this expression here how do we get back to the letter W. And I might click on the root button and then click again so it just appears briefly and that might just help students work out what they need to do. In this case subtract 6 first and of course there's feedback here. If they get it right then the expression does actually simplify. And then what's next and again I can flash the root on briefly if I need to. So we're dividing by 2 and so on. So the support of the root gradually is going to be faded so they can begin to do this without the root actually being on the screen. So the idea is to gradually reduce the support. Having the root shown to start with thing may be having another example where the root can be flashed on if it's needed. And then having another example where the root isn't shown at all then maybe doing a few of those examples. Then to reduce the support even more I would before the lesson starts actually create an example and rub out everything but the final expression. And then I would go up to file and I would save that grid so that the grid is saved on the computer and then in the lesson I would load up that grid so that it appears just like this and the students haven't been able to see that expression built up and so that reduces the support further to then try and work out where the letter Y is. In fact if you exit here and go to the start up page for grid algebra if you click on the resources and then under topics if you click on inverse then there are some prepared grids of which we have some saying find the letter so if I were to choose perhaps one of these open it up. Then here's one prepared earlier where there's an expression obtained from taking letter P on a journey and we have to try and find out where is the letter P within the grid. There are also some paper tasks that can be created where this is one here where there's a series of copies of the grid with just two rows each where there's an expression that's been created and students have to work out where that letter would be within the grid. And I've produced these just by doing screen dumps from grid algebra and putting them into a word file or they could be put into an Excel file whichever you prefer. These ones just involve the first two rows and here are some that I created where it involves all six rows and much more complex journeys. I've put in a magnifier into the cells so that there's a window that shows the expression a little bit larger and again the task is for students to work out where the letter is. Again I tend to find that big expressions often motivate students and once they get the idea of what they're doing then that's not necessarily any more difficult than simpler expressions. Another possibility with grid algebra within the resources section is there's a grid that says where are the letters. If I choose one of those and open it up here I've got a cell with the letter Z in it and there have been a number of other letters on the screen where they've been taken on a journey and arrived into this same cell and if I click on the peel back corner indeed I can scroll through all the expressions that have ended up in this cell. And the task is to find out where on the grid are each of these letters, the letter A, letter B, CD and so on and so on and so on. And this can be done as a whole class activity or indeed pairs of students could work on this on a computer. One feature is that in the toolbar there is a hide show button. If I were to click on that it actually reveals where each of these letters are. And in fact if I click on them then those letters become highlighted again and if I click on the hide show button again those that still haven't been clicked on or disappear and these remain. So that's a feature that some can sometimes be used. I will now return back to the interactive grid algebra. Having worked on students being able to find inverse journeys then there are two potential ways to go at this point. I'm going to talk about both ways. Each of these ways could be done immediately after students have got the idea of making inverse journeys. I'm going to think to start with about putting one letter somewhere and another letter somewhere else and then making a journey from one letter to the other letter. So I think I'll make this journey and then having made the journey from one letter to the other I'd like to take the other letter back on the inverse journey back to the letter A in this case. So I'm going to put a magnifier into this cell and that shows me everything that's in that cell. I've got the expression that I had by dragging A to this cell and I've also got the letter B that's in that cell as well. So again there's an option of creating a root. I could actually draw out the root that was taken, finish drawing it, and then take the letter B back along this root to end up in this cell. I'll put another magnifier in that cell as well. So that's the inverse journey done to get B back to where A was. Or I can be offering that challenge but without the root shown. But still there are the expressions there to guide me along the way. Or the next time gradually I'm reducing the support and another way to reduce the support is to rub out the middle expressions. And so now this is what I've ended up with. I want to take B back on the inverse journey of this back to the letter A. And so then I have to pay more attention to this expression to think about what I need to do. So I might divide by four first and then I've got to subtract two. And then each time I've got to work out from looking at the expression what it is I need to do next. And then to actually reduce the support even more having got these two magnifiers I can click on a button that actually hides the grid itself. And so now I've just got these two windows. And now I haven't got the support of the grid whatsoever. I've got to look at this expression and work out what I'm going to do to letter B to get back to where A is. To do this click on a button that's an expression calculator and put that into the window with A because this is where I want the expression to be. When there are letters in the grid then the letters become available within the expression calculator. So I'm going to start with B and take it back in a journey so it will end up in this cell. So I'm going to do B and then I'm going to divide by four and the expression calculator looks after the notation. I just have to think about what operations I have to do. Subtract two. Then I'm going to this is dividing by six so I'm going to multiply by six. Then subtract twelve and lastly divide by three. And then when I press enter if it's correct it will appear in this window. And if it's not I'll get a little icon that will tell me that it's not correct. So I'm going to press enter and it's correct in this case so it appears. So gradually I'm reducing support. There is a task related to this within the resources section. Sorry it's not the resources section it's one of the tasks here. And this is called multiple inverse journeys. And here there's been a journey made from the letter T to P. And in fact here's T and there's P. There have been some journeys made from T to get to P. And here are different journeys that have been made, different routes that have been taken. And over here are the inverse journeys going back to T that they've been mixed up. So here one is highlighted and I have to decide which of these is the inverse journey of this particular journey. And so in that case I think I'm going to decide it's that one. And then the next one is highlighted and I have to decide which one of these it is. And then again decide the next one and then the last one. And then I get my next task again. So this is something that students can do on their own or in pairs in a computer room. Going back to the interactive grid algebra. If I were to again choose two letters and make a journey from one letter to the other. And I'm going to put in magnifiers into each of the cells. And I'm arranging them one underneath the other. Then I'm going to click on the peel back corner so that I get the letter H. And you'll find that this will swap rounds showing on the left hand side whatever expression is on the top within that cell. I'm going to take H back along the journey. So when I take H along to here and let go that will appear now within this window as well. And I can look at and ask students then to look at what has happened from here to there. What has changed? What's changed on the right hand side? What's changed on the left hand side? And why has that happened? And then I can take that again going back a bit more on the journey and look again at what's happened here between this expression and that expression. How's the left hand side changes? How's the right hand side changed? And then returning gradually back to the letter B. And here that allows students to begin to see a step by step scenario of changing the subject of the equation. So these are some possibilities where I'm working from moving from one letter to the other. Another alternative that could come either after this or instead of this is where I have a letter. Again I'm going to take it on a journey but this time it's not going to another letter. Instead I'm going to choose to make this a particular number. At the moment the icon on the left hand side has got no numbers in it that indicates that this grid at the moment is undefined. Letter G could be any number. It has to be in the three times table but it could be any number in the three times table. So I'm going to choose to put a number in for this expression here and I'm going to choose, I don't know, I think I'll make it 43. So I'll put in a magnifier here to see that we've got an equation now where this expression is equal to 43. And the question is what number should be in the cell with just G then? What must be the number G now? And so just as before we're taking an inverse journey back to G but this time it's the number 43 we're taking back to G rather than a different letter back to G. So again I might want to put in a root that helps people see visually what they're doing and then gradually withdraw that support just as we've done where there were two letters involved. But taking the number 43 back along the inverse journey back to G will mean that I end up then having an equation where this will essentially tell me what number that letter G must be. So 43 take away 2 which is 41 and then I'm going to multiply that by 6. Well I've given myself a bit of mental arithmetic to do here but that's 246 and then I'm going to take away 12, 234 and then I'm going to divide by 2 is 117. So G should be 117 so I can go to the number box and get 117 and drag it into that cell and if it's right it will stay there and if it's not it will give me an error. Now interestingly enough actually the arithmetic is more difficult than the algebra in many ways which is sometimes an interesting phenomenon that sometimes it's actually easier doing it with the algebra. You don't actually have to do any hard mental arithmetic. But anyway I attempt to choose an example that gave me what I had to do a reasonable amount of mental arithmetic but it may well be that you choose an example more carefully where you don't have to do such complex mental arithmetic. Again just as before we can reduce supports where we can have come right down to the grid being hidden where I have to work directly with this equation to work out what I would do with a number 43. Again I put the expression calculator into that window and type out what I would do with a number 43. So I would subtract 2, I would then multiply by 6, take away 12 and then I would divide by 2. And then again the algebra is more straightforward than the arithmetic that's needed to be done but having done the arithmetic we can now just drag 117 into that window to see that G is 117 and I can click on the letter G if I want to make that the subject of the equation. The idea here is that we're withdrawing support so that students in the end are working directly with an equation of this kind and having to work with that equation without the support of the grid. And this will help them when they're working with questions that might have the support of the grid. So in this case I've created some questions that have used the print screen feature where I've pasted copies of grid algebra within a word file. And here we've got an equation set up but there's also the image of the grid of how that equation was created and that can be useful for students to work out what to do to find out what the letter E is. And so I've got examples of that here or questions that are just straight when they're working with questions that are quite traditional questions to solve equations. So within the resources section if I click on solving equations then one of the handouts is solving equations. It's just a sheet of equations to solve and you can create sheets like this yourself where students by this stage maybe have used the support of the grid and the imagery to begin to help them be able to work out what they have to do with inverse operations to work out the value of the letters. And having got to this stage then really grid algebra in many ways has done its job and then students are working with these questions, these situations on paper. So that is the end of this video about working towards solving equations. The software grid algebra is available from the Association of Teachers and Mathematics from the website www.atm.org.uk