The timestable taking advantage of symmetries

 

The timestable taking advantage of symmetries

Enrique Alonso Vendrell

(As published into Epsilon Magazine number 101: Page 157. ISSN: 2340-714X)

Abstract	In this paper, the author develops a novel method of presenting the timestables to the student, with the following advantages: it is easy to learn, as it requires minimal use of memory, and is quick to write. Therefore, it can be useful to all students, but especially to students with memory difficulties.
Keywords	Innovation, Didactics, Mathematics, Multiplication tables, Primary, Study techniques, Memorization.

3. Instruction for the table to be written

The first step is to write a row and a column with the numbers 1 to 9, with number 1 being the coincident. Then, the units of the products of those numbers are written down, but only from 2 to 5, as shown in Figure 1. This is the only thing that needs to be memorised, and it is quite simple: the last column is made up by alternating digits 0 and 5 and the first row is formed counting by 2. This leaves only three digits to be memorised: 9, 2 and 6.

Figure 1. Unit of the products

The next step involves making a mirrored zone to columns 2,3 and 4, keeping column 5 in the middle, but writing the difference to 10, as shown in Figure 2. Therefore, the new zone will be made up of the numbers resulting from subtracting the previous corresponding values to 10.

 Figure 2. Difference to 10

To complete the rest of the table, we now have to write the simetric zone to the zone we already have, including the numbers 1 to 9 of the first row, as shown on the figure 3. This time no mathematical operation is needed.

 Figure 3. Specular image

Now, for the last step as shown in figure 4, we have to put the 12 marks in those digits that are greater than the digit that its above. That’s it, the first 6 digits on the diagonal, almost all the eights (not it the last column), and the 6 that its below the 3, and the 5 that its below the 0.

Figure 4. The timestable complete

4. Use of the table

The use of this table is very easy. The first factor needs to be lower than the second one. Otherwise, the commutative property should be used, switching the order of the factors.

In order to get the unit, we need to look for the digit that is on the row and column that indicates the product. For instance, as shown in Figure 5, to get the unit of 6x7 we look for the digit that is on the 6^th row and 7^th column which, in this case, is 2.

In order to get the tens, we have to count the number of digits in the column which are not marked in the table, starting from the unit digit previously found (included) to the number on the first row (not included). For instance, the tens of 6x7 is 4 because of the 4 digits, 2,5,1 and 4, that are not marked, as shown with the starts in Figure 5.

 Figure 5. Example of use: 6x7=42