What the Math Teacher Never Told You!
UNWRITTEN AXIOM #1
One can be anything I choose one to be.
UNWRITTEN AXIOM #2
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!
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.
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
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
It can not be if we are talking about the fractions
"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.
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
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
on my webpage is a picture of her book.
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
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
- 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
Reflection is always good. So what did you learn? I liked the
Mind Reader the best.
always that anyone can learn anything if you put
your mind to it.