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What Makes A Good Resource - Rearranging Formulae

This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 08 September 2009 by ncetm_administrator
Updated on 03 September 2010 by ncetm_administrator

What Makes a Good Resource

Rearranging Formulae

Resource description:
This is a set of cards organised in four colour coded collections (red, peach, orange and green). Each set has a “master” card (the one with a border around it) and students are asked to start with the red collection and sort them into equivalent formulae – i.e. formulae that can be transformed into any other one in the same set.

Download the resource (Excel)
Teacher comment:
I wanted an activity which would give my students a wide range of formulae that they could compare and contrast and so encourage them to make decisions about equivalent structures. They already have feel for the link between certain equivalent number equations like 3 × 2 = 6; 6 ÷ 2 = 3 and 6 ÷ 2 = 3 and I wanted them to generalise from this to the idea of transformation of formulae.I was interested in testing out the idea that the rules for transformation of formulae could be drawn out of their existing awareness of number relationships rather than (as I have done in the past) teach it as a separate, bolt-on skill.
What I did:
I began the lesson by writing the following number equation on the board:

3 × 2 = 6

… and asked “How else could we write this statement?”

This generated things like 6 = 3 × 2, 6 ÷ 3 = 2, etc.

I then wrote:
a × b = c

and asked the same question “How else could we write this statement?”

Students seemed to very quickly see the common structure and offered c = a × b, a ÷ c = b and a ÷ b = c.

I then handed out the set of 4 colour coded cards, asked them, in pairs, to begin with the red set and to sort them into those that were equivalent to the “master” card and those that weren’t.

The range of four differentiated card sets allowed a good degree of differentiation and all pupils were able to gain success. The more able pupils attempted all four activities and had quite enough to occupy them for the lesson.

I was also pleased with the way the colour coding allowed me to see, at a glance, how far each group had progressed and this enables me to circulate around the class and intervene productively.

Here is a flavour of some of the interesting discussions that occurred:

From the red cards:
It reinforced such fundamental concepts as

and if 


(unless, as one insightful student pointed out, M = V)

From the yellow cards:
We had a discussion about the fact that if

and so if        

From the peach cards:
The concept of multiplying every term by a denominator (a common source of error with this class) was discussed.

From the green cards:
We were able to think about adding two fractions with different denominators, e.g. in:

I then got a measure of how well they felt they had done with a show of hands. No mention had been made up to this point of the significance of the colour coding and the students were asked to put their hands up if they were confident with their answers for the red, peach, yellow and green. All students were happy with the red cards but only a few were confident on the green.

I then handed out a set of exercises where the questions were set at the corresponding levels to the colour coded cards. The students then chose the level of question to attempt, appropriate to their level of understanding.
The students worked on these for the rest of the lesson.
At the end of the lesson the rearrangement of A = πr2  and C=2πr were discussed to show a simple use of the skill of rearranging formulae.


I really feel that this approach of starting with my students’ intuitive feel for connections between numbers as a way of them constructing the rules for transforming formulae worked really well. I am definitely going to try this approach again.

Secondary Resources
What makes a good resources - using the materials

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