Jim Hogan Mathematics

A self paced opportunity to improve your mathematics and numeracy.

What the Math Teacher Never Told You!

One can be anything I choose one to be.

When combining or comparing things in Mathematics we do so using the same size bits.

An axiom is a self evident truth. No one ever told me these two fundamental ideas and I think I am the first and only person to ever state these as two axioms.

Explanations of the axioms
A self evident truth should not actually need explaining but I think these ones do need some clarity!

Axiom #1 comes into play with this problem.
I have half a small pie and half a big pie. How much do I have all together? 

Of course we do not have one pie! We actually have half a small pie and half a big pie, and, that is all we can say.

Small Pie problem

I can choose one to be a group of objects. I could choose a context such as a paddock of sheep, tools in a garden shed or something more alike such as members of a family. These can all be "one".

The question "Where is my unit or one" will arise many many times and will cause issues if you are not aware! When dividing we create a new "one" which is the same as the divisor. 10 divided by 2 is actually "how many groups of 2 are there in 10?" If we divide 2 by 1/2 we change the 2 (made up of 2 units or ones, into 4 halves, the half becoming our new unit.

Axiom #2 appears in this problem.

Add 1/2 + 1/3

Many children (and adults!) think the answer should be 2/5. It could be if you are talking about marks in a test where we get 1 out of 2 and 1 out of 3 words right in a spelling test. Overall we can say we got 2 out of 5. This is fine as long as we understand the context and we are not actually talking about fractions.

It can not be if we are talking about the fractions called "a half" and "a third". Fractions have a structure called numerator/denominator. The numerator is the "count" hence we can have 1 half, 2 halves, three halves, 4 halves and so on. These are represented as 1/2, 2/2, 3/3, 4/4 and so on.

A half (1/2) of one is bigger than a third (1/3) of the same one. The parts are not the same size. Look at the diagram.

Half and
              third problem

Using Axiom #1 and choosing my "one" or "unit" to be shaped like a box of this size, I visualise the 1/2 and the 1/3 and combining these is, or should be, clearly not possible as they are.

To solve this problem we transpose the fractions and rename them so they are made up of the same size bits. Here are a a couple of ways of doing this.


We use the 2 and the 3 to split the unit box into 2 parts horizontally and 3 parts vertically. This can always be done and is what call a very multiplicative thought. The unit box now has 6 parts, each of which being called a sixth or 1/6. The half is now seen, or transposed as 3/6ths and the 1/3 as 2 sixths.

Our problem of adding 1/2 + 1/3 now becomes 3/6 + 2/6 and as long as we know the numerator is the count we get 5/6ths as the answer.
None of this is easy and it should be carefully constructed for young minds or as adults we continue to have misunderstandings and resort to "panic", "I give in" or "hand throwing!" If you are having this sort of confusion it ius almost certain that this learning was not constructed carefully for you, once upon a time. Never think it is about you not being "math minded", only think "I do not understand it,YET!". There is a lot about having a positive mindset to learning mathematics on Prof Jo Boaler's website youcubed.org and on my webpage is a picture of her book.

Equivalent fractions

A more traditional approach to solving 1/2 + 1//3 is to rename using equivalent fractions as above. This is more pattern based or algebraic and we can also see perhaps that there are many ways of choosing the equivalent fractions. We could have chosen 6/12 and 4/12 and got 10/12 which is also correct. There are an infinite number of different ways but we usually choose the easiest.

I hope that helps clarify these axioms. We should return to them often when solving problems and in the following we must. You can enjoy a bit a struggle here!

(a) The Farmer Brown Problem
When Farmer Brown travels to town on his tractor at 20km/hr he arrives half an hour late. When farmer Brown travels to town at 30km/hr he arrives half and hour early.

- What speed should he travel at to arrive on time?
- How far is it to town?
- How long is the journey in hours?
- What colour is his tractor?
- What is his wife's name?

(b) What ratio is exactly half way between 1:1 and 1:2 ?

Struggle is good! You can email answers to me and ask questions.
Reflection is always good. So what did you learn? I liked the Mind Reader the best.

Remember always that anyone can learn anything if you put your mind to it.