Why should we develop our mental mathematics?
There are many mathematics calculations which we do so often that we should rather memorize these than having to calculate them when we need the answer. Once memorized, the answers to these calculations are immediately available to us from our memory while we are working on more complex problems.
If we have not memorized these basics mathematical facts, then this will slow us down while we are doing more advanced mathematics work. For example, while learning about factoring quadratic algebraic expressions (Grade 9-10) we are required to find two numbers which sum to one number and multiply to another number, and this is much easier if the basic knowledge of addition and multiplication of small numbers is already memorized without us having to use our calculator or to waste too much thinking about this.
These types of simple calculations occur in almost every mathematical problem and the faster you can use memory the more time you will have to work on the problems you are given.
Mental mathematics is fundamental to all mathematical skills, since we do mathematics in our heads and the more skilled we become at using our heads then the less we have to use the pencil and paper and calculators, both of which will slow us down in many cases.
I divide mental mathematics into two categories:
- DIRECT MEMORIZATION: Those calculations for which you draw the answer directly from your memory without having to think about it. For example, 100/25 which you should immediately recognize as 4 without having to think about this – it should be retrieved immediately from memory without computation. You MUST develop your memory for as many of these problems as possible and continue to develop these.
- BREAK DOWN OPERATIONS: These are calculations for which you can break down a calculation into elements for which you can use DIRECT MEMORIZATION. For example 400/25, which you can break down to 4 x 100/25 which becomes 4 x 4= 16. For this you will not memorize 400/25 but will have memorized 100/25=4 and 4×4=16 and you then apply these together.
You can read more here: Mental Mathematics 1: Why this is important.
Task 1: Memorize single digit addition where the sum is less than 10
Your first task is to memorize all of the additions of the digits 1-9 where the sum is less than 10. For example 3+4=7. All other operations with numbers will rely on this knowledge and you must be able to these immediately.
The addition of two single-digit numbers is the most fundamental of all arithmetic operations and this is divided into this task for numbers which add to less than 10, and the next task in which these add to more than 10.
As you move up the grades I expect you to be able to recall this information quicker. I may give you 1-2 minutes to do the tests while you are in the Fundamental Phase(Grades 1-3) but only 10 seconds for the FET Band (Grades 10-12).
Whereas some of these mental mathematics will seen trivial to you, my focus is on the speed at which you can do this, and can get the right answer every time.
Task 2: Memorize single digit addition where the sum is greater than 9
Here you need to memorize all additions of single digits in which the sum is at least 10, such as 5+5=10 and 7+8=15. For these there is always the carry of the ten and these all must be memorized since they will form the basis for all additions which you conduct. These will draw on the additions memorized in Task 1 above.
Task 3: Adding double-digit numbers
Now let us used the information from the previous two tasks to apply this to be able to calculate the sum of two double-digit numbers such as 13+45.
This can be broken down into two parts. Firstly calculating the sum of the units digits, here being 3+5=8, and second the sum of the tens digits, being 1+4=5. The answer is then 58.
Task 4: Number Bonds
Consider the number 8. How many ways can we divide this up into two parts which sum to 8?
We can have 1+7=8, and also 4+4=8, and many others.
This is the opposite function to addition, since in addition we add two numbers to get a new number, and in number bonds are taking a single number apart to find how it can be created from using two other numbers.
There are many other areas of mental mathematics which I would like you to master at this time. The include subtraction, multiplication, and division, divisibility rules, factoring, prime numbers, squares, square roots, cubes, cube roots, and then the application of these new mathematical facts into other areas of mathematics including HCF, LCM, common fractions, decimal fractions, estimation, algebra, geometry, analytical geometry, trigonometry, calculus, as well as all of the practical situations found in the Mathematical Literacy curriculum.
Dr Roger Layton