Task taken from

Worksheet from Grid Algebra written by Dave Hewitt https://gridalgebra.com/resources/equations.

If you have not used Grid Algebra before, you may find the introduction videos published by the ATM useful https://www.youtube.com/watch?v=LaH3S_0igak

The Task

For this task you need to use the Grid Algebra programme (https://gridalgebra.com) set up with multiplication tables 1 to 6, with 5 columns and negatives allowed. You may also want to pre-prepare the grids used in the task.

Put a letter in the top row and make a journey similar to the one below

Image of a grid from the software showing the steps from x to two times three x plus nine

Put a number in the final cell via either the ‘Number Chooser’ or the ‘Expression Calculator’. In my example, I will put in LaTeX: 54 .

Click on ‘Magnifier’ and then the final cell in the journey to show the equation. In my case it will say $LaTeX: 54=2\left(3x+9\right)$ .

Say that you are going to take the number (54 in my case) back along the inverse journey to where the letter is. Ask students where you are going to go first, and what will the new expression look like when you make that movement. Get one of them to write it on the board. In my case it will be $LaTeX: \frac{54}{2}$ . Make this movement to show that expression in the cell, and compare what it looks like compared with what the student wrote on the board. For example, if they had written $LaTeX: 54\div2$ then it does not look the same as $LaTeX: \frac{54}{2}$ and encourage them next time to write it as the software will show it. Then make the rest of the movements back to the letter cell, each time getting students to write down what they think the expression will look like before making each step. Be fussy about such things as brackets, the length of a division line, and also where the LaTeX: + or LaTeX: - signs appear following a division (i.e. at the same level as the division sign).

Placing the 54 into the cell and showing the equation using the magnifier establishes the equality relationship between 54 and $LaTeX: 2\left(3x+9\right)$ . Using the language of “in my case” emphasises the use of 54 as a specific example of the general relationships that are represented by the journeys. This generality is further emphasised by talking about the journey in terms of “taking the number” rather than “taking the 54”.

This task focuses on reversing the journey taken to create an expression, in this case $LaTeX: 2\left(3x+9\right)$ . The first step in reversing this journey can be described as “going up three rows”, "dividing by two", or “going to the LaTeX: 3x+9 ". The first of these focuses on the journey, the second focuses on the operation, and the third focuses on the destination. By asking what “the new expression will look like”, it is the result of this journey that is emphasised, as well as the operation the journey represents. “Going up three rows” can disguise what this journey represents mathematically, as there’s no clear connection between this movement and the operation of dividing by 2 that it represents.

This idea of reversing the journey connects to the inverse of an operation. The idea of a route or a journey emphasises that this process of 'undoing' and 'doing' the inverse operations has an order. Here, paying attention to how you are using the word inverse, as 'the inverse' (a noun) is an abbreviation for a compound phrase where inverse is an adjective, i.e. the inverse operation or the inverse journey etc..

Asking learners to write the resulting expression on the board makes the connection between the notation and the journey. This distinction could be between writing $LaTeX: 54\div2$ and writing 54 over 2, or it could be reflected in how the learner speaks this expression, “divided by 2” or “over 2”. As the reverse journey continues, further connections between the operations represented by the steps of the journey and the notation representing the destinations of these steps can be made. The task emphasises the precision of this notation, and it is this precision that connects the notation to the operations it represents.

Being fussy about the brackets, the length of a division line, and also where the LaTeX: + or LaTeX: - signs appear following a division (i.e. at the same level as the division sign) also draws attention to how the notation reflects the order of operations, and again the precision of this notation.

Now click the ‘Open Route Panel’ button and mark the route going from the number back to the letter.

In the Route Panel, click on the Button for listing the contents of a cell icon, and a new window will appear showing all the expressions along that route, from LaTeX: 54 to $LaTeX: \frac{\frac{54}{2}-9}{3}$ (in my case).

In the coloured bar at the very top of this new window, click on the $LaTeX: \uparrow\downarrow$ sign and you will find that the expressions involving the letter appear with those involving the number. It shows, for each stage, what expressions would be in each cell if both the number and letter at each end of the route were dragged along the route.

Screen capture of the grid with the expression window showing each step of the route

Once the students are clear about what appears in this window, click and drag the three dots at the bottom right of the window so that only the first line is seen.

The destination equation

Ask the students what the next line will say. Get them to write it down and one student to write on the board what it will look like. Then drag the three dots down so that the second line is shown. Compare what this looks like with what the student wrote on the board. Get all the students to look carefully at what they had written and see whether it looks the same or not. Again, be fussy about the notation and get the students to be fussy as well.

Repeat for the remaining lines.

This stage focuses on how undoing each operation affects the notation. By treating the number (in this case 54) on the left-hand side of the equation as an object, rather than carrying out the arithmetic at this stage, students experience how applying the same inverse operation to both sides, in this case dividing by 2, affects the notation. Undoing the multiplication by 2 on the right-hand side means dividing the left-hand side by 2.

Some students may have written something like $LaTeX: \frac{54}{2}=\frac{2\left(3x+9\right)}{2}$ or some students may have written $LaTeX: 54\div2$ . Comparing these with what the equation looks like in the expression window can help students to be fussy about the notation, and can emphasise the connection between the original operation and the inverse operation.

Then get the students to carry out the arithmetic to find the single number which the letter must equal. Check that with this value you do get the number in the end cell of the route.

Leaving the arithmetic until the end and using it to check the journey can help to focus on the operations and the connection between these operations and the notation.

Click on 'Clear Grid' and choose 'Everything'.

Repeat the above one or two more times, with a new letter with a different journey each time.

When you think it is the right time, reduce the support by:

Hiding the grid (click on 'Hide Grid') after getting the route window open.
Using a pre-preared and saved grid, which has the grid hidden and only has the route window in view, showing the equations for a route you had previously made. You can save it when just the first line is shown, as below. Then you can gradually reveal the rest line by line, with the students writing down first what each line will look like

Once the students are comfortable reading, writing and saying the notation, removing the grid, using either of the ways suggested above, can shift the focus from doing and undoing as a route or journey and the accompanying operations and notation, to solely the operations on algebraic expressions and the representation of these using notation. This fading of the visual scaffolding needs to be accompanied by a fading of the linguistic scaffolding, where descriptions of movements around the grid are replaced with descriptions of the operations these movements represent.

A key aspect of this activity is that students become familiar with moving something from one side of the equation to the other. This is leading towards them also moving a letter from one side to the other as well, as this will be useful when dealing with equations with an unknown on both sides of the equals sign.

Give the students some equations for them to solve step by step.

Consider including some equations which involve a second letter, such as $LaTeX: \frac{2x+6}{p}-10=14$ , saying that you will let them know what LaTeX: p is later.

Saying that you will let them know what LaTeX: p is later makes the idea of a letter representing an unknown explicit, as well as implying that the work will be focused on manipulating this unknown, rather than the number that it represents. In the earlier activities, the letter also represented an unknown, but an unknown that is eventually found. Here, there is no need to find out what number is being represented by LaTeX: p .

Extensions/Simplifications

Journeys can be long or short! Sometimes students enjoy really long ones once they have sense of what they are doing.

The grid can be changed to one with less rows.

Towards the end, involve equations with other letters involved, saying that you will let them know the value of those letters later.

Reflection tool

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