 This is a video that's about using the software Grid Algebra in the Mathematics Classroom. The software is available from the Association of Teachers of Mathematics. And it follows on from a video about making journeys on the grid. This video, though, concerns introducing letters and using letters. So I'm going to click on the interactive Grid Algebra. And just to remind everyone that it's based on the multiplication tables, the grid, the one times table, two times table, three times table, four times table, and so on. And a key feature of Grid Algebra is that you can make movements between these numbers. So going from two to three, we're adding one. And going from four to two, we'll be subtracting two. So addition to the right, subtraction to the left, multiplication is coming down. And division is coming up. And so expressions can be built through movements on the grid. More detail about that is in a different video. So now I'm going to clear the grid. And previously, when talking about journeys, we had placed a number somewhere on the grid and made a journey with that number. And then rubbed out the middle expressions. And the task given to students was to find out what is the journey, what journey was made to try to recreate the journey from 13 to this final expression. So where did that journey go so that we can recreate that expression? And that helped them learn order of operations. And before I'm about to introduce letters, I do this activity again. So here with students may be coming up to the interactive whiteboard and moving this in order to recreate that expression. And I would do that a few times with different start numbers in different places and having different expressions and so on at the end of the journey. I would do this a few times because what's interesting here is that the attention is on the operations, it's on the movements. The attention is not on what number we started with. 13 here is pretty irrelevant. What matters is that whatever that number was, it was 2 was added to it and then that was multiplied by 6 and then subtracted. 18 was subtracted and then it was divided by 2. So the 13 becomes not the focus of attention. The focus of attention are the operations that take place. And that gives an opportunity for a letter to be slipped in instead of the number 13. So having done this activity a few times of recreating the expression, I then maybe say that I'm bored with numbers. I'm going to choose a letter instead. Which letter should I have? And I choose a letter and then I do movement straight away so that there may be a reaction to the letter initially. But I very quickly start doing something to the letter and say what have I done to R? Asking them to say divided by 4. And now what have I done? Adding 3. And now what have I done? And so this is something that they're actually familiar with because they've done this activity a number of times. So that I'm trying to keep the attention on the operations so that they're not dwelling on the fact that I've decided to put a letter to start with rather than a number. So keeping the attention onto the operations is quite important. And then I might again rub out the middle expressions and I'll simply come up and recreate the journey to end up with this expression again. So we're actually reproducing a task that they have already very familiar with. Only this time it happens to be a letter that starts the journey rather than a number. And since that doesn't make any difference to the actual task then I generally find that after a little maybe surprise at the idea of a letter being put in, really it's never mentioned again. It's accepted really quite easily. So we might do similar tasks with letters as we did with numbers and that may include, if I close it down for a moment and get back to the original page, it might include computer generated tasks as well. So there's a make expression task that involves letters and here again I choose there's always different levels to start with. So here I've got an expression on the left hand side and I've got to make that expression within the time that I'm given. So here I'm dividing by 3 and then I'm going to add 6 and then I'm going to multiply by 3 and then I'm going to add 6 and I've done that within the time and so I will get a point and then I've got a new expression to make and so on. So that may be a task that they've done before but with numbers and this time it's a task that involves letters instead. And likewise there's another task of finding the journey that involves letters as well where here I have to click on where I think the journey is. So I click starting here then I think I'm going to add 18. I'm in the 6 times table so adding 18 will be moving 3 along and then I'm dividing by 3 which will get me to the 2 times table and then take away 2 will get me there. And when I think I finished I click on I finished and it will tell me whether it's right or not and I've got a new task. So those tasks will be done with letters now. If I return to the interactive grid damage per grid and get some letters then I'll choose a letter. And place it somewhere. Once the letter's on the screen I can draw a root from that letter or indeed a number in this case it's a letter. So I click on the root button and click on starting a root and then I click to where I'd like my root to go and then click back on the button once I finish my root. And if I click on the start here the letter X is there and I can drag X on the root. And the task for these students would be to write down what would the expression look like at the end if I were to take it along this root. So I'm starting with the letter X I'm going to take it along this root and what will it look like the final expression once I've done that. So this is something they might do on paper and then someone might come up and either do that root by dragging it or there's another option. There's a button here where you can actually get a sequence of what would happen to that letter X if it goes along that root. So first it's divided by two and then there's four added and so on and so on and this is the final expression we would end up in the cell with this marker seven on it. And here I'd get quite fussy about how people have written things. So with the end of that division line I'm expecting the subtraction sign to be right at the end of it. So zip, ping, take away three and likewise is zip, click, click for add four. So I'll be quite fussy about the positioning of this because my aim here is for students to become, begin to get quite fluent with writing and reading for more notation. There are also some handouts that are available in the resources section that relates to this. If I were to go on say order of operations then in the handout section there are journeys and here a sheet comes up with a number of these journeys printed out and that could be given to students for them to write down what the expressions would be if we start with a letter N. At the beginning what would the expression look like when you get to the end. And so you could also print out the screen dumps of those screens and print them out for students to write down the expressions. Generally I find that writing expressions needs to be worked on even when they're beginning to be familiar with reading for more notation. Usually the writing lags a little bit behind the reading for more notation. So if I go back to the main menu and interactive query algebra and have some letters available. This time I could, let me think, put a letter somewhere and then I might put another letter somewhere else. I've got two letters on the screen. And the task for students is to think about if I made a journey from one letter to the other, for example this one, then if I put a magnifier in here then it tells me what S is in terms of P. So I've made a journey with P to get to S and it tells me what S is in terms of P. And so students may be given the task of trying to express what would S be in terms of P or what might P be in terms of S. In fact you could have a whole load of letters on the screen. So let's go and collect a number of letters. And we can put them in different places and we can put them in different places. And then students might be asked to write all these letters in terms of X, for example. So if I made a journey from X to each of these letters what would the letters be in terms of X? And then you might say okay I'd like them all in terms of W please. And this could be done either with the screen shown on an interactive whiteboard or indeed it could be a sheet that could be given out to students. And they could be checked by actually making those journeys on the grid to see whether they're right or not. Sometimes it might be useful for students to have copies of the blank grid which is available in the resources section of the software for them to write on tasks like this and other tasks. Another idea with letters is the idea of making codes. I'm going to load a grid that I have prepared earlier. And this is a grid that has obviously filled up with different letters. One thing to note with this grid is that there are actually 30 cells here. There are 26 letters in the alphabet of course. So some have got a replication of the same letter. So we've got an E here and we've got an E there. Now there's an issue about E appearing in both in two places. It would have to mean that the number here is the same as the number there within the grid. So you can't just put two copies of the same letter anyway. More about that later. But with this grid of letters I do a little code here by giving someone an expression like p plus 4 and they have to work out what letter that is. So if you do a journey that is p plus 4 you actually get to the letter i. So p plus 4 is a code for the letter i. And what I have is a sheet with a number of expressions written in it where students have to work out what the letter is. So in this case p plus 4 ended up at the letter i. So i would be the first letter. Then we've got z minus 15. So if I go back to grid algebra, here's z minus 15. I mean the five times table. So 5, 10, 15. That will be the letter s. So the next letter in that code is s. So s would go here. And likewise each of these will represent a single letter. And then when all the letters are written here it will make a sentence. And here we've got another code with much more complicated expressions. Often students really like some complexity and quite engage in the idea of trying to work out where they would end up given each of these expressions and which letter they would be in that cell. So I picture students having a copy of this grid that they can use and work with along with the codes. And of course then another activity is that they can create their own codes for somebody else to decode. Okay, now I'm going to clear this grid and I'm going to take a letter and make a journey. And in this case we've got the letter k being taken on the journey. K at the moment could be any value. The icon up here has got no numbers inside it, this grid. And that indicates that the grid is not defined at the moment. So k is a variable. It could be any number along as it's in the five times table. But however I could choose a particular value for k. So let's choose the number 40. In which case if I put a magnifier in here it will tell me that k and 40 in the same cell. So k is 40, 40 is k. And so if k is 40 then we ended up with this expression. What number should be in here then if k is 40? So I get students to not say k but say 40. And so I point to this and they say 40. Then divide by 5 plus 3, multiply by 4, take away 8. And so we carry out those operations and work out what number should go here. And so in this case I believe it's 36. So I'm going to drag 36 into that cell. And it is accepted so it must be right. And so we found out that this expression then must equal 36 if k is equal to 40. And so we're beginning to do substitution tasks. So I'm going to close those down. I'm going to rub out the numbers again. And when I rub out the numbers then this icon will not have any numbers in it again because we're back to an undefined grid. And so sometimes I stay with the same expression for a little while putting in different numbers for k so they get a sense that k is a variable. It could be different numbers. It doesn't have to be a single one. And so again if k this time is 55 then what number should be in this expression? So if I were to put a letter in and make a journey around the grid supposing I do this journey. So the expression's reasonably complex. But if I know that r for example is 45 then in fact because I know it's in the three times table and that expression is just one cell to the right then in fact that will be the next number in the three times table. So in fact I don't have to do much work with this expression at all to know that it's 48. But if I put a magnifier in both of these cells I can hide the grid and in this case actually I do have to do the work. The grid isn't there to help me. So sometimes that's a nice feature and I'd have to take work out that it's 45 divided by 3 and 3 and so on in order to work out this 48. I still grab 48 and put it in that magnifier and if it's correct it will appear there. And there are tasks here that relate to substitution and I choose my level of difficulty again and here I'm told that d is equal to 8 and I have to drag in the number that relates to this expression down here. So I've got 6 times 6 is 36 so it looks like it's 30. And so I drag in the number 30 in there and I get a score and then I'm on to the next challenge. And for the first 10 questions the grid is available and sometimes is a support but from the 11th question onwards the grid is hidden and so we're just left with the value of the letter along with the expression and someone will have to work directly on the expression. And obviously there are paper tasks that could be given to students to do substitution tasks and in fact in the resources section there is a sheet there with some substitution questions. Lastly I'm just going to talk about the possibility of having more than one of the same letter. So if I were to choose a letter and place it here then at the moment the grid is not defined but it's possible for me to place another copy of this letter somewhere else and not all cells will work but if I place it in this cell for example you will see the icon up here has numbers in it that means that this grid is now defined. So it is now quite unique there is a unique number that M must be if this number in this cell is going to be the same as the number in that cell so if the number of the yellow cell is the same as the number in the blue cell then there is a unique value for M and in fact we can begin to give a little clue by making this M join with that M and this is a little clue about helping me maybe think about what that number must be for M and if I manage to work it out then I can always get what number I think it is and drag it in in this case 8's been accepted so indeed it is 8 if I were to drag 8 over here it would also be accepted and if I were to click on the numbers we can actually see the whole surrounding grid with those two 8's in those positions so there can be something quite interesting some interesting challenges about finding places where you can put more than one copy of the same letter and then try to work out what value that letter must be and indeed there is a computer generated task that's based on that idea which is called make them equal okay that's going to be the end of this video about grid algebra and it's available from the Association of Teachers and Mathematics of their website www.atm.org.uk