# Maths I Should Understand Better

As a maths teacher, I can comfortably do a GCSE maths paper getting close to full marks (odd mistake perhaps) without really trying. This probably comes as no surprise. However, there are some parts of the maths that I actually don’t fully understand or don’t know why they work. Here are some of the ones I’d like a better understanding of:

1) When dividing fractions, why does flipping the second one over and multiplying instead work? I have a vague grasp of those being two ‘inverse’ type operations but is that all there is to it? Is there a nice visual representation of why that works?

2) Using a quadratic curve and plotting a linear graph to solve a second quadratic. I don’t fully get why finding the ‘difference’ between the two quadratics and plotting that line means that the intersection of the line with the original quadratic gives the solutions.

3) In Statistics, for the standard deviation (Chi-squared for that matter) why do we square the terms? I understand the need to remove the negative signs but wouldn’t modulus do it?

4) Do trapeziums have to have exactly one pair of parallel lines? Why shouldn’t a rectangle count? Why does this shape appear to have some ambiguity in its definition? Who would be the person/group of people to have a definitive answer?

5) Perhaps not GCSE as such but is there any reason why 0 can’t be even? Or positive?

Am I alone in wondering these things (and think it’s important to wonder)?

Are there parts of GCSE maths that you teach methods for but don’t really get yourself? Do share – you’re amongst friends here!

What do you understand by ‘even’?How about ‘numbers in the two times table’ (or similar)?Well, 0 x 2 = 0 so 0 is in the two times table.So, 0 *is* an even number.

Formula for a trapezium also works when applied to a rectangle!I also feel obliged to explain to my students why we flip the second fraction. Here’s the logic of my explanation:4 divide by what gives me 8? Students derive that dividing by a half actually has the same effect as doubling. How many quarters are there in two? Eight! So dividing by a quarter has the same effect as multiplying by 4…..So dividing by 1/4 is the same as x 4 …which we can rewrite as 4/1Get a student to check another reciprocal 1/x with x/1.typically three divided by one third.I then ask the students to tell me what they think the rule is……THEN…. does it work for fractions with numerator >1? if so 2/3 div by 2/3 should give me 1!Hope this helps.

All rectangles are trapezia but not all trapezia are rectangles! German school maths books have diagrams explaining the logical relationships between the various quadrilaterals. Zero is the absence of number therefore it does not have the normal number attributes.

The standard deviation measures the distance from a constant sequence, in a certain sense. The formula for standard deviation involves the square root of a sum of squares, and it resembles the distance formula in higher dimensions.

For solving curves, you’re turning it into simultaneous equations. If you have (say):y = x^2 – 4; andy = 3x – 6Eliminate y to get x^2 – 4 = 3x – 6x^2 – 3x + 2 = 0(x-2)(x-1) = 0… which gives you the x-coordinates of intersection.

Also for fractions, I think it’s to do with distributive laws of inverses, a plausible term I just made up. Just like a – (b-c) = a – b + c: a ?? (b ?? c) = a ?? b * c = a * (c ?? b), because the order doesn’t matter.

I like this post. It??s good to know that other teachers struggle with these things as well. 2) Some reasoning behind Question 27 ?? 1/2 – is the same as how many half??s go into 7. so 7 ?? 1/2 = 14 (same as 7 x 2). 1/2 is the recipocal of 26 ?? 1/3 same as 6 x 3 for the same reason. 1/3 is the reciprocal of 34 ?? 2/3 – how many 2/3 go into 4? 4?? 2/3 = 6 which is the same as 4 x 3/2). 2/3 is the reciprocal of 3/2.1/2 ?? 1/4 = 2 same as 1/2 x 4 = 2a/b ?? c/d can be written as an entire fraction (a/b)/(c/d). Multiply both top and bottom of this fraction by d and get (ad/b)/c which can be written ad/b x 1/c = ad/bc = a/b x d/c.

Sorry, I meant to say reasoning behind Q1 in the above post.3) With Q3 I used to wonder the same thing. It is a perfectly good measure of spread doing it this way – don??t quote me on this but I think it is called the mean deviation. I guess it??s just a case of now that the product moment correlation coefficient and normal distribution are reliant on the sd then it??s now become the most common measure of spread. I??d love to hear a good reason why it is used more widely then the mean deviation. 4) I always thought that this is to do with the fact that there a number of different properties to each shape. For example, a diagonal has 2 sets of parallel sides but it also has equal length diagonals. Not all trapezia have equal length diagonals except the isosceles trapizum. For the same reason a square is different to a rectangle because its diagonals cross at 90 degrees.I guess it??s like a huge venn diagram with certain properties of some shapes branching into properties of others. 5) I guess zero definately can??t be odd because by definition an odd number is any number of the form 2k+1 where k = 0,1,2,3,… It surely must be more to do how you define these things. I have similar difficulty with whether 0 is a natural number. In my first ever lecture at uni a professor mentioned this point briefly. I remember that he said that the natural numbers are the counting numbers and 0 is a counting number because if you asked someone to count the number of elephants in the room then they would have to say 0 – but then to counter you haven??t actually counted anything so is it a counting number???Q2 I??ve never heard of doing that before! Have to go to a meeting now so will have a think about it later.I think more teachers should do what you??ve done. I may do the same on my blog for the next post.

Your (5) is just bizarre, and it’s getting international fame. Where did you get the idea that zero wasn’t even? I’d love a reference or link…Zero is unarguably even. There’s room for a bit of argument about whether it should be called positive. If you don’t like the explanation that it’s an integer of form 2n not of form 2n+1, you might like the rules that adding or subtracting even numbers gives you an even number, but adding an even to an odd gives you an odd, etc. It’s needlessly complicated to say "an even number or zero". All the places "or zero" comes in are with the evens. It’s only slightly tongue-in-cheek to suggest that zero has no special property in contexts of evenness because really even and odd is about clock arithmetic, and on a clock there is no zero.Zero isn’t counted as positive because in fundamental uses (when taking differences, say) it would have just as good a claim to be negative, and for this we really need to have three different cases. Compared with A, B is either greater, less than or equal; the difference is either positive, negative or zero. Here it really doesn’t fit into one side or the other. Zero does sit better with the positive numbers when you’re multiplying, in that a (positive or zero) times a (positive or zero) is a (positive or zero). But even here zero is a special case (anything times zero is zero) and it works better to mention it separately. Just say "non-negative" when you have to.

(3) To expand on daveinstpaul’s point, you can define the mean of data X1….Xn as that M which minimises (X1-M)^2 + (X2-M)^2 + … + (Xn-M). Call that expression the variance-from-M and when it’s minimised, M is the mean and variance-from-M is the variance.If you minimise |X1-M| + |X2 – M| + … + |Xn-M|, you’ll find this M to be the median. (Or, if the median is not on one of the data points, the minimum will be achieved at all points of an interval whose midpoint is usually called the median). I’d call this expression the Total Error.So variance lives intimately with mean, and total error with median. You need more mathematical structure in your data to measure mean than median — you can take the median note of a tune, but not its mean — and for completely unstructured data, like a fruitbowl, you are stuck with mode. The mode is that M which minimises the total count of "is-different-from-M".

(3) This example of ‘where to wait for an elevator?’ gives quite an intuitive explanation of what values the mean and median minimise.http://www.johndcook.com/blog/2010/11/29/where-to-wait-for-an-elevator/

Hi Dave,I wrote a blog article about explaining why when you divide fractions ‘turning the second one upside down and multiplying’ works.http://www.greatmathsteachingideas.com/2010/12/30/dividing-fractions-the-awkward-question/Would love to know what you think.

There is this video on flipping the second fraction http://www.youtube.com/watch?v=OKUVvAaCv_0