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					                                 The Decimal Square
                          A fresh look at Multiplication Tables

 Allan lawson describes how he devised a system of dynamic vector patterns for
                    use in teaching the multiplication tables.


The apparent universal ability for young children to manipulate small groups of dot
patterns in their mindʼs eye, in solving addition and subtraction problems led me directly
to a new challenge; how to find a method for solving the basic multiplication facts using
dot patterns.

I realized this challenge might require a completely different approach compared to that
of the Math Square which I developed earlier for teaching the addition facts. As I
discovered, the solution to the challenge did in fact involve the Math Square – but not
before some mathematical developments had been applied first.

Two stages were required - the first of these developments involved the transformation
of the Math Square into a more precise numeral system. The second stage involved
transforming this new numeral system into an arithmetic system to display the
multiplication tables.

The First Stage - a new numeral system

Math Square numerals are formed by groups of dots in various patterns. These patterns
are systematically built in a 3 by 3 square format, representing the numbers 1 to 10.
Typical dot patterns for 1 to 10 would look like this:-




[the purple dot at the centre of the ten dot pattern represents two dots, or counters,
placed one above the other]
The first stage in achieving my goal of solving the multiplications using patterns was to
transform these groups of dots into an ordinal numeral system.

Instead of recognizing each individual pattern as a whole group of dots (what I term
“group collection”) - for example the figure below is recognized as 8:




- If we label each counter in the Math Square with a designated fixed number from 0 to
9, we only need to know the position of any single counter to know its designated
number




For example, we now know that the bottom left-hand counter is 7, even if none of the
counters are actually labeled with numbers. We know it is 7 because of its position
within the framework of the square format.

Now if we reduce the circular counters to numbered points on a framework of 3
horizontal and 3 vertical lines, we would have a number framework - which I call the
Decimal Square:-
This configuration was an important step in my overall goal.

The Arabic numerals labeling each intersection of the Decimal Square become
redundant, as each intersection has its fixed numerical value from 0 to 9 (called
“number positions”). They are shown in the diagram above for indication purposes only.

So the decimal square is a positional numeral system. To demonstrate this, a series of
squares could be arranged horizontally, each square in ascending powers of 10, to
represent any number.

For example 53, 370 would be represented as:




The Decimal Square numeral system can now be adapted and form a solution to our
goal of creating a system of algorithms to solve multiplications using patterns - this
being the second stage of development referred to earlier.

The Second Stage - Vector-based Multiplication Tables

Finding patterns in multiplication tables is not new; what we are aiming to do, however,
is to discover a new kind of pattern – one formed by movement from one number
position to another in a positional numeral system. This is where the Decimal Square
comes to the rescue.




If we re-introduce Arabic numerals into the Decimal Square and label each number
position with its number, we could then explore the patterns of lines* formed by
connecting each number position in turn as we proceed through the multiplication
tables. In the process, we will be developing the Decimal Square into an arithmetic
system for solving our multiplication pattern question.

[* the connecting lines being referred to as “vectors” - a line that indicates connectivity
and direction]

The Decimal Square, labeled with its fixed unit number positions, looks like this:




The Decimal Square




Now, taking as an example the 3 times table. Starting from the 3 position on the square
(1×3), vectors are drawn in green on the square, joining each unit number position as
we move through the table from 1x 3 to 11 x 3:
We discover a simple pattern of parallel green lines formed by the vectors; 3 pointing
south, and one diagonal pointing North East, where the pattern repeats itself from the
11×3 position. And we also see a simple numerical progression where we write the tens
values (in red) in their relevant unit positions.

Let us examine the vectors formed in another multiplication table, the 7 times table:




The vector pattern is the same as that in the 3 times table, but in reverse direction.
Starting from the 7 unit position on the square (1×7), the vectors point north, following
the same route as to the 3 times table. The tens results (in red) written next to their
associated unit positions, again, reveal a simple numerical progression.

The 9 times table follows the same vectors, rotated 90 degrees anti-clockwise:
Shown for the sake of mathematical completeness, the 1 times table follows the same
course as the 9 times table, but in reverse:




[Note: I have not joined all the diagonal links with vectors, because it would be
unnecessary and confusing. The aim is to simplify arithmetic functions not complicate
them where it is not necessary.]

These multiplication tables (and here the use of the word “table” is perhaps more
appropriate than ever it was before) reveal the vector patterns * for the 1, 3, 7 and 9
tables.

* [I use this word in its plural form. In reality there is only a single pattern with its vectors
pointing in the same direction, for all 4 multiplication tables; it is simply rotated through
90 or 180 degrees.]

To complete the series of odd-numbered times tables, we should include the 5 times
table. But this is harder to show as a vector pattern using the Decimal Square, as zero
and 5 share the same number position at the centre. But if we separate the two
positions and place them on a single line “framework”, like this:




We can now reveal the simplest pattern of all the multiplication tables:
The single line framework is, of course, the same line throughout the table, but I have
shown it as separate lines to avoid confusion over the tens results.

To complete our investigations, we can use the same method to discover the vector
patterns for the even-numbered times tables; 2, 4, 6 and 8.
So the 8 times table is the reverse of the 2 times table; and the 6 times table is the
reverse of the 4 times table. All share the same pattern, but rotated through either 90 or
180 degrees.

Teachers using this system of Decimal Square Vector Times Tables, can show pupils
that we can continue following the same vector patterns in any table far beyond the
eleventh time (11 x n). Pupils can be set a challenge; what is 12 x , 13 x, 14 x the table
being discussed.

Naturally, it is impossible for a short article such as this to provide a suggested
framework for teaching the system in the curriculum. It can only be a brief presentation.

And I realize this is not a method for showing pupils the nature of multiplication - for
example, as a number fact resulting from a series of additions of the same number.
Instead, this system provides pupils with a method for obtaining the results of the times
tables without the use of ʻcounting onʼ or rote memory; rather it relies on spatial, visual
memory and the ability to visualize a planned sequence of events in 2 dimensional
space using points, vectors and numerals. It also reveals a beautiful, unified symmetry
that is the essence of mathematics.

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[About the author: Allan Lawson, 62, is a retired English lawyer now living in Canada,
who has had a lifelong interest in numeral theory and the teaching of mathematics]

				
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posted:3/27/2010
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