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To a maths educator, the concepts of rearranging equations seem well defined and methodical: do the same to both sides of the equation and apply inverse operations to remove various entities until the subject of the equation is isolated. However, by examining students’ work and listening to students thoughts on the subject, it became apparent that some are not applying these concepts and principles of rearranging equations. Instead, they are 'moving' entities across equations in a random fashion (see some examples in the figure). When asked about their method of ‘moving’ and ‘bringing across’, the students are often resolute: "isn't this what you are supposed to do?" As maths educators, we need to ask ourselves, how do students come to think that this is the correct method for rearranging equations?

There appears to be a significant variation in the language that maths educators use around rearranging equations. Indeed many of us habitually use the words 'moving', 'shifting', 'bringing across'. Could it be that this terminology, when used in class, and heard over and over again by the students, seeds the wrong idea of the principles of transposition? It seems that it does, and especially so with the students who have not yet grasped the concept of transposition fully.

This means we need to 'watch our language'. The table below shows how the same transposition problem can be worked through with and without the use of unhelpful language. We need to actively try to avoid words such as ‘move’, ‘shift’, ‘bring over’ and similar but instead use ‘get rid of’, ‘remove’, ‘have … on the other side of equation’, ‘add … to both sides of the equation’, ‘divide both sides of the equation by …’. The bottom line is: we do not ‘move’ anything. We do the same thing to both sides of an equation in order to ‘get rid of’ things which are in the way. In addition to using appropriate language ourselves, we also need to step in and correct a student who is using misleading language in class to prevent the spread of unhelpful language amongst their peers.

Scenario 1: unhelpful language


Scenario 2 : language that is not misleading

"I move 5 to the left side and it changes the sign".


"We bring the 2 across the equals sign and divide by it".




"I want to have 5 on the left side of the equation. What should I do? Subtract 5 from both sides".

"Now I want to get rid of 2. How do I remove multiplication by 2? Divide both sides by 2".





Join us in adopting the ‘good language of transposition’. It is only for the benefit of all students, their professional careers and our future as a numerate society!

If you have any further ideas on how to foster the correct concepts when teaching students how to rearrange equations then we would love to hear from you.