Grid Algebra is a free piece of software developed by Dave Hewitt (who leads the PGCE course at Loughborough) available at gridalgebra.com. This year I used it with two alpha groups, but it is useful across the age and attainment range. Previously, I used it with higher attaining groups.
Its greatest strengths are for developing familiarity and fluency with formal mathematical notation and for working with expressions without the need to evaluate them. But this is perhaps underselling it! Have a watch of the first video at https://gridalgebra.com/intro/overview to get an understanding of the software.
There are a number of resources and student tasks available from the menu on the left of the main screen (or click on the three lines). The aim of this blog post is to suggest a possible route into the software and what students might do with it.
1-2 lessons – Getting to know the grid
Have a read of the guidance at https://gridalgebra.com/resources/number , “Getting to know the grid”. This suggests some good whole class modelling and questioning to familiarise students with the grid. Follow this up with either tasks 15 or 16 (if you want something iPad based) and/or the “What number goes here” worksheet (same link as above) – 2 rows was mostly right for the alpha group. To demo the grid with the ability to scroll you need to use your computer – the iPad won’t allow you to scroll (but most of the rest of the time it works well).
1-3 lessons – Moving round the grid
This is where Grid Algebra is really powerful. I started with an empty grid with two rows and then inserted a number (e.g. 7) into the middle of the top row. I asked what number would come next and discussed that it is 8 because it is 7+1. There is a nice wow moment when you drag the 7 to show 7+1. Note that you can also insert the number 8 and use the magnifying glass so that it says 7+1=8. We continued with more whole class questioning and modelling, sometimes dragging from the 7 and sometimes dragging from an expression to make e.g. 7+1-2. You may need to delete some expressions or use the three dots in the bottom right of each cell to get the grid to look as you want it. Before moving onto the second row, I arranged mine to read: 7-2, 7-1, 7, 7+1 and 7+2.
The question about what happens on the next row naturally crops up. Starting with the number under 7, we discussed that it was 14 because it is 2x7. A misconception to be aware of is that the number to the left of 14 might be 15 – remind students that it is going up in twos for the two times table and 16 from double (7+1). After modelling moving the 7 down, I tried to avoid asking students what they thought it would say for moving 7+1 down since this is a notation thing, and can’t really be inferred from prior knowledge (in Dave Hewitt’s language this is arbitrary rather than necessary). We looked at 2(7+1) and discussed the omitted multiplication sign.
At this stage the students had a play with dragging numbers round the grid themselves – it could be from a filled grid or an empty grid with just one number. A discussion about dividing and the use of the fraction bar will naturally follow.
We built towards task 12 – make the expression, numbers – small grid. It has a fairly short, and unchangeable, time limit so it can be a bit stressful. So first, we did some similar questions altogether. They set up a blank grid with 1 or 2 rows on their iPads and inserted a number of my choice in a specific cell (e.g. put an 11 in the right most box). Then, they had to create a specific expression by dragging (e.g. 11-3+2). I found it best to keep the same grid to create 5 or so expressions, one at a time, before clearing the grid and inserting a new number. Initially, I wrote the expression and read it out before progressing onto mixing up which one came first, how long a gap I left and finally only writing it. We built up the complexity and speed until they seem ready to have a go at task 12. I chose not to progress onto larger grids (task 11) in order to keep things simple and keep confidence up.
1 lesson – Using letters
At this point it was a short leap to do the same as we had just done but starting with a letter, and moving that around the grid rather than just a number. Tasks 13 and 14 are analogous to tasks 11 and 12. Students found this step very natural.
We also used task 9, find the journey. It is easier in that there is no time limit but harder because you don’t get the feedback from moving the letter round the grid.
1-2 lesson – Substitution
In my lesson sequence I left grid algebra for a week whilst we did some work with function machines (although I did make some links). We returned to look at substitution. Dave Hewitt (in the substitution notes here) suggests dragging a letter around an empty grid to create a few different expressions. He then drags a number onto the original letter and uses the magnifying glass on that cell so that it says, for example, x = 6. One can then take the 6 on the same journey (the three dots in the bottom of each cell allows one to switch between the different expressions in the cell). “Don’t say x, say 6” and then point to the x in different expressions.
I did this and then the students individually worked on task 19. I allowed them to use a calculator in order to focus on reading and understanding the expressions rather than getting hung up on the arithmetic. I was astonished at how well they worked on this task, successfully substituting into expressions with 3 or 4 operations including brackets and long fraction bars. In this task the grid is visible for students for the first 4 questions and then it disappears and there are often two letters to substitute in.
Future work
Given the success of grid algebra with the groups I intend to make more use of it. They were not fluent with the concept of inverse operations (when we were looking at number machines) so I think that task 10 will be helpful. This will lead nicely into solving equations (tasks 18).