Q I help adults with maths for use in everyday life. They find it difficult to add columns of figures quickly in their head. I have told them it is OK to use a calculator as long as they have a way to estimate that the answer is correct. Can you suggest a quick way to add columns of figures?
A I found this method in a secondhand bookshop many years ago: Jakow Trachtenberg's System of Speed Arithmetic, reprinted in 1962. This is a real-life illustration of maths being used to overcome adversity.
Trachtenberg was born in 1888 in Odessa, Russia. A brilliant engineer, he conceived his system while in a Nazi concentration camp for seven years.
This was his way of keeping his sanity.
He did not have access to books, paper, pen or pencil. He worked with large numbers and calculated in his head. He scribbled his theories on bits of wrapping paper and old envelopes, writing only the processes he was sure would work. The work was entrusted to a fellow prisoner when he thought he was to be executed, but he was transferred to a labour camp and escaped in 1945.
After the war, Trachtenberg taught disadvantaged children and in 1950 he founded the Mathematical Institute in Zurich, teaching children and adults.
The method I explain here is great for adding columns of figures.
Set up the sum as you would for column addition, allowing three rows at the bottom for the different stages of the answer. Fill in under the bottom right-hand column with a zero as this identifies the 10s. If decimals are used, set the place value (this will become clear as we work through the addition).
As you add up the numbers, you make a tick by row where a lump of 10 is reached (I prefer to write the ticks on the paper rather than keeping them in my head). Carry over the remainder to build towards the next 10.
In the example I have shown I begin from the bottom of the column (it doesn't matter which end you start, but you do need to start on the right hand column). 9 + 7 = 16 - a lump of 10 + 6 - so I put a tick by the 7 and carry the 6 in my head. Adding the "carry 6" + 2 + 3 = 11 - another lump of 10 + 1 - so I put a tick by the 3. Then I add "carry 1" + 5 + 6 = 12 - a lump of 10 + "carry 2". We have run out of figures in the column so we write this remainder under the zero we wrote earlier.
Counting the ticks, we have three. This 3 is entered to the left of the zero (the column added to 32 and you can now see why the zero was entered).
The second column is now added in the same way as the first. 1 + 9 = 10, so a tick is made by the '1'. There is nothing to carry so the next two numbers are added and so on.
This carries on until all the columns have been added in this way. Then a total of the ticks and remainders is made by adding the two rows of figures that have been recorded: in this case the answer is 4082.
The nice thing about this is that you don't have to remember big numbers in your head and it is easy to check for mistakes. Even so, it is always good to check the solution by approximation.
I used to run maths fun weekends and sometimes used methods from the book, which seemed to build confidence. I checked on the internet and the book is still available. There is also a website which is dedicated to the method and has a fuller version of Trachtenberg's life-story.
Q I am a support teacher in a primary school. We have been looking at triangle numbers. The pupil I work with asked me what they are used for.
A One use of triangle numbers is in pharmacies, for counting pills. They have a tray in the shape of an equilateral triangle, with the row totals marked somewhere on the tray (1 row = 1, 2 rows = 3, 3 rows = 6 and so on). The pills are put on the tray, the rows are counted and the total number of pills is read from the table (at the top of the triangle in the photo below). The diagram shows 5 rows of pills, a total of 15 + 2 (from an incomplete row), giving 17 pills.
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