Some thoughts on solving equations at KS3 and KS4 level.
a) Tell the students that the equals sign is kind of like the fence between your garden and your neighbours garden. Whatever you get in your garden (a trampoline, swimming pool, zip line etc.), the Smith´s on the other side get exactly the same thing for their garden. It´s the same when you´re solving equations, whatever you do to one side of an equation, you have to balance it by doing the same to the other side (This also works for fractions and ratios).
b) When solving equations in year 9,10,11, I think it´s important to differentiate between adding/ subtracting to both sides of an equation and multiplying/dividing to both sides of an equation. When you add or subtract, you’re simply doing the same thing to both sides. When you multiply or divide, of course you’re doing the same thing to both sides but students get confused if they’re not taught to either put brackets around all the terms before they divide or multiply or if you do not emphasise the fact that you must multiply or divide every single term in the equation.
For Example, one of my year 9 students tried to solve the following equation.
2x + 3/x = 17 (multiply both sides by x)
2x + 3 = 17x
In this case, the student obvioulsy hasn’t taken the 2x into account when they multiplied through by x.
I had to emphasise that when adding/subtracting, you do it to both sides and multiplying/dividing you do it to every term.
(This applet is a good way to start year 7 students off with solving equations)