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Common maths mistakes (and how to fix them!)

Here are some common maths mistakes which I have encountered many times during my maths tutorials.


1. Negative numbers!  This can be a minefield but here is one of the "best":


If x = -4 what is the value of x2 ?  

Many students say -16 and show me their calculator which confirms -42 = -16.

The correct answer is 16 (positive).  To do this correctly on the calculator you should enter (-4)2 = 16.  The calculator says -42 = -16 because it is squaring 4 first and then making it negative.


Remember that maths operations are not necessarily applied in the order as written from left to right, you must use the priority system of BIDMAS.


This is quite a common issue when using the quadratic formula (you know the one!).


2. Squaring brackets


(a + b)2 = a2 + b2  is incorrect.  

You should do it this way: (a + b)2 = (a + b)(a + b) = a2 + b2 + 2ab


3. Dealing with time in hours and minutes.


For example: a car travels at a speed 80 km/h, how long will a journey of 112km take (give your answer in hours and minutes)?

Most students will correctly calculate the time = 112 / 80 = 1.4 hours, but now comes the tricky bit!

Remember 0.1 (one tenth) of an hour is 6 minutes so 0.4 hrs = 4x6 = 24 min.  The answer is then 1 hr and 24 min.


4. Significant figures (SF)


Eg1 express 4.05629 to 3 SF.  It is commonly thought that "0" does not count as a SF however it does when in between other digits so the correct answer is 4.06.  


Eg2 express 28,304 to 1 SF.  Correct answer is 30,000.  In this case the zeros after the 3 are needed to get the decimal place in the correct position.  


5. Unit conversions for area and volume


We know that 1m = 100cm but how many cm2 are there in 1m

Imagine a square measuring 1m by 1m, this is an area of 1m2.  But in terms of centimetres the square is 100cm by 100cm giving an area of 10,000cm2

So 1m2 = 10,000cm2.


In a similar way, by thinking about a 1 metre cube you will discover that 1m3 = 1,000,000cm3.


6. Repeated percentage changes


Eg the population of a town increases by 20% in year 1 then increases by 14% in year 2. What is the overall percentage increase over the two years?


Note that you can't just add the two percentages together! 


This is best tackled using multipliers: a 20% increase is equivalent to multiplying by 1.2 and the multiplier for a 14% increase is 1.14.  Therefore the combined effect is 1.2 x 1.14 = 1.368.  This corresponds to an increase of 36.8%.


7. Factorise don't cancel!


This one applies more to A level maths than GCSE. In solving an equation it is sometimes tempting to divide both sides by a variable since this will simplify the equation. However you run the risk of losing solutions. Here's a simple example:


Solve the equation  x2 - x = 0

                                     x2 = x  (adding x to both sides)

                                      x = 1  (dividing both sides by x)


This is a valid solution but there is another solution which we have 'lost'.

It should be done by factorising like this:


x2 - x = 0

x(x-1) = 0

x = 0 or x = 1.


This situation often arises when solving trigonometric equations, for instance:


sinxtanx = sinx

sinxtanx - sinx = 0

sinx(tanx - 1) = 0

So sinx = 0 or tanx = 1 leading to the solutions x = 0o, 45o, 180o or 225o

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