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 about it, and this still currently…** .

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 **calculus** comes from the Latin **calculus** (“pebble”).

This process of multiplication is indicated in the ** khulasat alhissab** of the author Baha’ Al-din

**(1547-1622).**But we find it, earlier written by master Nicolas Chuquet in the

**as well as the principle of multiplication posed still taught today in schools.**

*Triparty in the science of numbers*(1484),It has the advantage of retaining by heart only the products of the first integers up to \( 5times 5 \) and easily deduce the products of the other integers up to \( 9times 9 \) . In other words, this method will only be used if the two factors in the product are strictly greater than 5.

**Example: Let be the product \( 8times 9 \) to be determined.**

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

We raise 3 doitgs with one hand and 4 with the other, the other fingers being lowered, i.e. \( 5-color{red}{3}=color{blue}{2} \) for one and \( 5-color{red}{4}=color{blue}{1} \) for the other.

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

The product of the numbers corresponding to the lowered fingers is \(2 times \) 1 = 2 gives the digit of the units.

We find \( 8times 9 = color{red}{7}color{blue}{2}\)

**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!)

Proof: let \( a and b be two natural \) \( \) integers such that \( 5 , <a <10 \) \( 5 <b <10 \) .

\( atimes b = (5+c)(5+d) \) so c , d are two natural numbers such that \( \) \( \) \( 0 and <c <5 \) \( 0<d<5\)

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

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

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

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

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

**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 acquired method, it is important to work **on rapidity and automatism** in mental calculation using like 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 good time) 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, like at home!