# Ramblings from the Classroom – Week 12

Thought I would stay on the theme of using mental models – using this new phrase I appear to picked up somewhere!

As we have moved through different types of equations we got to an equation with unknowns on both sides. Before I started modelling on the room I had a discussion with the group looking at what was the same and what was different. Once we had highlighted that an x term was on both sides I then created the model below:

We then started discussing what it meant when we were solving an equation. This eventually developed the idea that we wanted to finish with a solution where x = something. We then focussed on the green x blocks and one student suggested we match up with a -x red block. So I placed one of the left hand side and one of the right so it remained equal.

The student all immediately said they wanted to get rid of the 1x on the right hand side. So we added another -x block to each side.

What is fascinating with this approach is they see the equation in the middle of the screen get simpler but you can see them facially experience a sense of cognitive overload because the visual model looks more complex. We then had a discussion focussed on the right hand side of the equation and how the x terms had disappeared from the equation and but visually they were still present. We then returned to the idea of a ‘zero pair’ but this time with the x term. This was then strongly reinforced when I then dropped the -x terms on top of the x terms and they cancelled out and vanished.

We then followed the steps covered in the previous blog to arrive at our solution.

I kept explaining to the group that I was new to this style of Maths and then we had a interesting discussion about what division it when look at the final two pictures above. With students describing the idea of dividing by 3 as placing into 3 equally sized groups.

As this was being modelled on the board I wrote the steps for the equation on the whiteboard to the right of the projector so students could see all the steps together. To reinforce I then showed another example but with the steps already completed which allowed me to focus on the discussion about the process and I annotated the written example.

I had not put the final step down because I wanted to remind them of the process when the answer will not be an integer and emphasised this when modelling.

This was the fourth lesson and whilst the modelling and discussion took slightly longer than direct modelling students were then naturally able to work then for a longer period of time and started to look at solutions to equations like:

5 + 3x = 7 – 4x

In previous year students would have needed more guidance but do to their increased visual understanding a lot more gave this equation a go without needing additional support.

Have a good week everybody.

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