There are many ways to solve a maths problem. As long as the person solving the problem sticks to the rules of mathematics, then the problem can usually be solved in a variety of ways. There is no single path to the correct solution. This is one of the attributes of maths that makes it EASY. This is also central to the philosophy of Singapore Maths.
By way of example, if we take the multiplication 7 x 9, one way of arriving at the correct answer is to learn the times tables. Learning these is boring, that is why at Puppet Maths we use the Sands-Daniels Musical Times Tables. These uniquely, in my experience, set the times tables to well known tunes that you can hum, and they are without the vast quantities of extraneous verbiage that have nothing to do with multiplying numbers that so many musical times tables are afflicted with. The music triggers the memory of the lyrics and so removes the fear of getting the words wrong thereby making the times tables fun to learn, and fun to recite. We advise that having learnt the songs that the pupil recite them in their head when they require to recall the product of two digits.
However, if one hasn’t been lucky enough to have learnt your times tables using the Sands-Daniels songs, one can still arrive at the correct answer to the calculation by other means. Another way would be to put 7 dots on the page nine times over and then count them up. Alternatively one might write down 7, add 7 to it to get two lots of 7, then add 7 again, and again, and again, until one had added 7 nine times. Both these method would produce the answer.
Another method would be to notice that 9 is almost 10. One could multiply 7 by ten to get 70, and then reason that since we did not want ten lots of 7 only nine lots, so we could then subtract one seven to get our answer.
But there are more approaches… one could look for a pattern in the 9 times table. Whenever 9 is the multiplicand, the product of 9 and some multiplier is such that its 10s digit is one less than the multiplier, and the tens digit and the units digit of the product add up to nine… so in the case of 7 x 9, seven is the multiplier, so the tens digit of the product will be one less, i.e. 6, and the units digit of the product will be whatever added to 6 makes nine, i.e. 3. The product is 63.
Alternatively, one might look for a pattern in the 7 times table. The seven times table follows the pattern on a mobile phone number pad. For this you ignore the 0 button, and use just the other nine buttons. Starting with 7 at the bottom left hand corner, that is one seven. To find two sevens move up the key board. The rule is every time you move up the keypad then you add 10 and the units is given by the number on the key pad. So for two sevens you move one up the keypad, which gives you a ten for moving up the keypad and a four for the units, as that is the number on the keypad button – result 14. For three 7s you move up the keypad again, that adds a ten for moving up, we now have two tens, and the units are given by the number on the button which is now 1 – three sevens are 21. For four sevens we go back down to the bottom key of the middle row of the keypad (that is the button marked 8). Because we have not gone upwards with this move we don’t add another 10, so our tens digit is still 2, but now our units digit, given by the button, is an 8, - four sevens are 28! Five sevens… we move up the keypad so we add a ten giving us 3 tens now and the number on the keypad button is a 5, five sevens are 35, and so on. Doing this we discover that nine sevens are 63.
At Puppet Maths we teach that as long as the pupil sticks to the small number of rules of maths (there aren’t many… and unlike language where there are irregular verbs, there are no irregularities in maths, it always obeys the rules) then they should find the answer by hook or by crook.