Let me be clear here; it makes maths a whole lot quicker to do if the result of adding or multiplying together any two numbers from nought to ten does pop straight into your head. If these do not pop into your head then it is a bit like taking part in an obstacle race where, when the gun goes off, you have to first build the obstacles before having to get over them. When you do know the times tables instead of being obstacles to keep reconstructing they become the bricks to use in getting over future obstacles. However simple addition and the times tables are the only bits you should expect to pop into your head.
In the mathematics of numbers the obstacles we meet do become increasingly difficult and are made of more and more pieces. Fitting the pieces together at any stage can be daunting but becomes easier when you realise that each new obstacle is built not afresh but from previously constructed ones which have now turned into useful stuff.
The first piece is the number One and the method of construction is counting. Using One and counting an awful lot of useful stuff can be built.
Counting is simple. Most people can count without thinking about it. Which is a problem when doing maths because then counting needs thinking about. A first thought is we have to be clear what is being counted in terms of type and location.
2. How many cars in the picture?
3. How many racing cars on the transporter?
4. How many racing cars in the picture?
5. How many vehicles in the picture?
To be really clear the answers should not just be a number. So the answer to question 1 is
There are four cars on the transporter.
We can combine objects when counting when they have something in common.
In Picture 1 there are:
|4 saloon cars||4 cars||4 vehicles|
|4 racing cars||4 cars||4 vehicles|
|1 transporter||1 vehicle|
|Totals||8 cars||9 vehicles|
In the picture below how many yachts are there? Each yacht has a crew of two. How many crew, in total, on these yachts?
In this case totaling yachts and crew does not make sense as yachts and crew have no common type even though they have location in common.
When things have nothing in common all you can do is list them. This is the first secret of algebra.