Multiplication tables remain a nightmare for schoolchildren to see worse for students. Emmanuel Macron knows very well this cold and terrible fear of being questioned on it, and this still at present … .

Teacher I tried to find ways to help my students better memorize these tables and create quick computational automatisms. Generally there was no concern about memorizing tables from 1 to 5, it was tables 6, 7, 8 and 9 that posed more problems.

I discovered, one day, this old technique, simple to use. But then why, as students, none of our math teachers tried to pass it on to us? …. I remain perplexed by this finding. Calculating with your fingers is the basis of the calculation… At the premise of mathematics, it was the pebbles that were used. The word calculation comes from the latincalculus (“pebble”).

This multiplication process is indicated in the khulasat alhissab of the author Baha’ Al-din (1547-1622). But it is found, earlier written by Master Nicolas Chuquet in the Triparty in the science of numbers (1484), as well as the principle of multiplication laid still taught today in schools.

It has the advantage of holding only the products of the first whole numbers by heart up to $$5-times 5$$ and easily deducing the products from the other whole numbers up to $$9-times 9$$. In other words, this method will only be used if the two factors in the product are strictly greater than 5.

Example: Either the product $$8\times 9$$ to be determined.

Note that $$8 = 5+\color{red}{3}$$ and $$9 = 5+\color{red}{4}$$

Three fingers are raised with one hand and 4 with the other, the other fingers being low $$5-\color{red}{3}=\color{blue}{2}$$ for one $$5-\color{red}{4}=\color{blue}{1}$$ for the other.

The amount of fingers raised is $$3+4=7$$ gives the figure of the tens of the product $$8\times 9$$ .

The product of the numbers corresponding to the fingers lowered $$2 \times 1 =2$$ gives the number of the ones.

There are $$8\times 9 = \color{red}{7}\color{blue}{2}$$

## The detailed method in video:

To go further

With students of 3rd grade, after having them discover this method, it is quite wise to ask them to demonstrate it rigorously using the literal calculation … (Challenge posed in group work, class atmosphere guaranteed!)

Demonstration:  $$a$$ and $$b$$  two natural whole numbers such as $$5<a<10$$ , $$5<b<10$$.

$$a\times b = (5+c)(5+d)$$ so $$c$$ ,$$d$$ are two natural numbers such as $$5<a<10$$ , $$5<b<10$$.

$$= \color{green}{5\times 5 }+ \color{red}{5c+5d}+cd$$

$$=\color{green}{25}+ \color{red}{10c+10d -5c-5d}+cd$$

$$=25+\color{brown}{ 10c+10d} -5c-5d+cd$$

$$=\color{brown}{10(c+d)}+\color{blue}{25-5c-5d+cd}$$

$$=10(c+d)+\color{blue}{(5-c)(5-d)}$$

Thus: $$10(c+d)$$  represents the number of fingers raised multiplied by 10.

$$(5-c)(5-d)$$: the product of the numbers represented by the closed fingers

After this method acquired, it is important to work on the rapidity and automatism in mental calculation using such a high level athlete who works, every day, the same gestures to perform in his discipline …

A free application (and I hope that will remain so for a while) developed by Christophe Auclair has the gift of challenging students alone or in duo. This application combines gamification and mental calculation. Students see it as a real relaxation in class and like to challenge themselves in a variety of ways.

To download and use without moderation in class, as at home$$\sqrt{2}$$