The Multiplication Table - Online Math Tutor Juan Castaneda, MS

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»Multiplication Table«

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The Multiplication Table

The Multiplication Table shown below is the 12 by 12 square array of numerical results of multiplication problems involving two integer factors, each between 1 and 12.
The multiplication table is basically formed by putting together all times tables for numbers from 1 to 12.

Multiplication Table
123456789101112
1 123456789101112
2 24681012141618202224
3 369121518212427303336
4 4812162024283236404448
5 51015202530354045505560
6 61218243036424854606672
7 71421283542495663707784
8 81624324048566472808896
9 918273645546372819099108
10 102030405060708090100110120
11 112233445566778899110121132
12 1224364860728496108120132144

Patterns within the Multiplication Table

Knowing the multiplication table by heart helps a lot in doing mental arithmetic calculations. This in turn allows you to quickly verify or reject individual numbers being proposed as solutions for abstract algebraic equations.

Many students struggle memorizing multiplication facts, and such a struggle can be a very big handicap when it comes to exams, homework, and general problem solving in any math course.

Having all multiplication facts from the times tables together in a single place and noticing the several existing patterns among these numbers is a powerful aid to memory in this respect because it provides a bigger context where multiplication facts are not isolated anymore. In this way they become more meaningful, making them easier to learn and to remember.

Symmetry

The multiplication table is Symmetric. This means the numbers down a given column are the same as those along the corresponding row.
This property is based on multiplication's commutative property. Since 8x4 = 32, and 4x8 = 32, then the number 32 shows up in two spots in the multiplication table: at the intersection of the 8th row with the 4th column, and also at the intersection of 4th row with the 8th column.

Every number in the multiplication table equals the number located directly across the main diagonal, and at the same distance from it.
If we split the multiplication table in half going down its main diagonal, where the square numbers are located, then the triangular region on the lower left is a mirror image of the triangular region on the upper right.

Corresponding columns and rows are equal
123456789101112
1 8
2 16
3 24
4 32
5 40
6 48
7 56
8 81624324048566472808896
9 72
10 80
11 88
12 96

Patterns shown by the easiest-to-remember times tables

The easiest times tables to memorize are those for numbers 1, 10, 11, and 5.

The times table for number 1 is just the sequence of natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
This is because the number 1 is the Multiplicative Identity.
That means any number times 1 equals itself. In algebraic notation: α × 1 = 1 × α = α

The next times table that is very easy to memorize is that of the number ten.
The reason for this is the result of multiplying any number times ten gets written with the exact same digits of the original number plus a zero at the end.
Examples: 2 x 10 = 20; 35 x 10 = 350; 70 x 10 = 700; 12345 x 10 = 123450, and so on.

Multiplying a single digit times 11 produces a two-digit number with the same digit repeated twice.
From 2 x 11 = 22 all the way to 9 x 11 = 99 those are very easy multiplication results to remember.

The number 5 is half of 10, so multipliying any number times 5 gives us half the result of multiplying it times 10.
It is easy to write down the 5 times table because we can do it adding by fives, and the resuls always end in 5 or in 0, they alternate.
The result of multiplying an odd number times five ends in 5, while the result of multiplying an even number times five ends in 0.

Writing down all these results in their corresponding columns and rows in the multiplication table, we can partially fill it as shown below.

Partially Filled Multiplication Table - Step 1
123456789101112
1 123456789101112
2 2102022
3 3153033
4 4204044
5 51015202530354045505560
6 6306066
7 7357077
8 8408088
9 9459099
10 102030405060708090100110120
11 112233445566778899110
12 1260120

A nice pattern in the 9 times table

After the times tables for numbers 1, 10, 11, and 5, the next ones easiest to memorize are those for numbers 2, 3, and 9.
The 2 and 3 times tables are easy to work out and to remember because 2 and 3 are small numbers.
The times table for number 9 is easy because it follows a very simple pattern.

The times table for number 2 is the sequence of even natural numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, ...
The result of multiplying a number times two is the same as adding that number to itself.
In algebraic notation: α × 2 = 2 × α = α + α

For multiplication times three the algebraic identity is: α × 3 = 3 × α = α + α + α

In my experience, it is very rare for students beyond 5th grade to struggle with the 2 times table or the 3 times table.

When it comes to memorizing the first ten multiples of 9, from 9x1 to 9x10, it is very useful to note their digits follow this simple pattern:

Digit Pattern for the First Ten Multiples of 9
MultiplicationProductProduct's
Tens Digit
Product's
Units Digit
Sum of
Product's Digits
9 x 1= 9090 + 9 = 9
9 x 2= 18181 + 8 = 9
9 x 3= 27272 + 7 = 9
9 x 4= 36363 + 6 = 9
9 x 5= 45454 + 5 = 9
9 x 6= 54545 + 4 = 9
9 x 7= 63636 + 3 = 9
9 x 8= 72727 + 2 = 9
9 x 9= 81818 + 1 = 9
9 x 10= 90909 + 0 = 9

The product's tens digit goes up by one each time, from 0 to 9, at the same time as its unit digit goes down by one, from 9 to 0.

This pattern very much goes along with a cool way you can use your fingers as a calculator to multiply digit values times nine.

Say you want to find out how much 9x4 is. What you do is you hold up both your hands in front of you, with your palms facing you. Then you number your fingers from left to right, your left thumb being number one, and your right thumb being number ten.

Then you count on your fingers up to four, which turns out to be your left hand's ring finger, and you bend it down. Then you can see three fingers up to its left, and six fingers up to its right. You put those two numbers together, 3 and 6, and you have the result: 9 x 4 = 36.

Now, filling in the corresponding rows and columns for the 2, 3, and 9 times tables into the multiplication table, it takes us to the following stage, where we have already filled in 108 spots out of a total of 144 spots in the whole multiplication table.

So, having taken care of the seven easiest times tables to remember, now we are able to focus our memorization efforts in the remaining 36 spots, having those 108 known spots as a background context to support our memory for the remaining part of the task.

Partially Filled Multiplication Table - Step 2
123456789101112
1 123456789101112
2 24681012141618202224
3 369121518212427303336
4 481220364044
5 51015202530354045505560
6 6121830546066
7 7142135637077
8 8162440728088
9 918273645546372819099108
10 102030405060708090100110120
11 112233445566778899110
12 12243660108120

Multiplication times 4 and times 8

Both 4 and 8 are powers of 2. The number 4 is the square of 2, and the number 8 is the cube of 2.

4 = 22 = 2 × 2, and 8 = 23 = 2 × 2 × 2

These properties give us a simple way to multiply times 4 and times 8 by using repeated addition.
For example, to calculate 6x4 and 6x8 we can go like this:

First, 6 + 6 = 12, second 12 + 12 = 24, and third 24 + 24 = 48

Therefore, with the above three additions we have found that 6 x 4 = 24, and also that 6 x 8 = 48, just by adding equal numbers each time.

Memorizing the multiplication table may take a lot of practice and repetition for some students but it is well worth it. It is time well invested, if only because it makes working with fractions and percentages a whole lot easier when you do not have to struggle with all these basic multiplication facts.

In my long tutoring experience, I have noticed that each student has his or her own history of memorizing the times tables, in many cases involving different degrees of struggle or difficulty, and different lengths of time.

Some multiplication facts that are easy to learn for some people are hard for other people.
This may or may not be your case, because there is a lot of variability there but for many students, the hardest multiplication item to fully memorize was 7x8.
Here is a simple, nice mnemonic involving the sequence 1, 2, 3, 4, 5, 6, 7, 8 that has helped some students as a memory anchor to remember the result of 7x8:

12 = 3 x 4, and 56 = 7 x 8

The more patterns you notice in the multiplication table, the more comprehensive, and cohesive the context your mind has to host all those multiplication facts in a way that they do not seem or feel isolated and meaningless any more but instead they are connected and shored up by several relationships among them.

Square Numbers Along the Main Diagonal

Square numbers in the multiplication table are located along the main diagonal because they are the result of multiplying a number times itself.

There is an important pattern we notice when moving perpendicularly away from the main diagonal.
Moving either up and to the right of a square number, or down and to its left, we find its predecesor, the number that is just one unit less than that square number.

In the table below square numbers are highlighted in pink, while their predecessors are highlighted in green.

By the way, it is very useful to memorize the square numbers as a sequence, because there are plenty of properties and algebraic formulas and identities where squares show up, making square numbers ubiquitous in homework and exam problems.
Square numbers are all over the place in math, so it is very convenient to know them well.

Square Numbers and their Predecessors
123456789101112
1 13
2 48
3 3915
4 81624
5 152535
6 243648
7 354963
8 486480
9 638199
10 80100120
11 99121
12 120144

To explain how the above pattern helps you with multiplication I will use an example.
Let's suppose you did not know that 7x9 = 63 but you knew 8x8 = 64.

If you notice that 7 and 9 are exactly two units apart from each other, and that 8 is right in between them, then you can automatically, without any further thinking or worry, apply your knowledge of the square 8x8 = 64, subtract one, and be 100%, absolutely sure that 7x9 = 63.

Similarly, just because 12 x 12 = 144, then you immediately know that 11 x 13 is going to be 143.
Why? Because 11 comes before 12, 13 comes after 12, and 144 - 1 = 143. Done.

This is no coincidence. The fact that 4x6 = 24 and 5x5 = 25 and all other cases like that highlighted in the table above, are particular examples of the following algebraic identity:

(n - 1)(n + 1) = n2 - 1

In words, the product of the predecessor of a number times the successor of that number, equals the predecessor of the square of said number.

Differences Along Lines Perpendicular to the Main Diagonal

This is another interesting pattern inside the multiplication table. It is really an extension of the previous one. When we keep moving perpendicularly away from the main diagonal, more than just one spot, we find the numbers decrease but not by the same amount all the time. Instead, they decrease more and more rapidly, as their successive differences keep increasing by 2 all the time.

When we start at a square number, like 36 for example, the next number is its predecessor 35, their difference being only one. However, the next difference is 3, the next difference is 5, the next one is 7, and so on and so forth.

When we start at a number like 72, that is not a square but a number times the next number (72 = 8 x 9), and we go out from there, the numbers also decrease but in this case the first difference is 2, the second difference is 4, the third one is 6, and so on and so forth.

Constantly Increasing Differences
123456789101112
1 11
2 20
3 27
4 32
5 3560
6 3666
7 3570
8 3272
9 2772
10 2070
11 1166
12 60

Let's take a closer look at these constantly increasing differences that show up between products as one factor decreases by one at the same time as the other factor increases by one:

(In the second set of products and differences in the next table I added the products 4x13 and 3x14 that would be there if we expanded the multiplication table to include 14x14)

Differences Increase Faster by 2 Each Time
Starting from 36Starting from 72
Product Product
6 x 6= 36Difference 8 x 9= 72Difference
5 x 7= 35and36 - 35=1 7 x 10= 70and72 - 70=2
4 x 8= 32and35 - 32=3 6 x 11= 66and70 - 66=4
3 x 9= 27and32 - 27=5 5 x 12= 60and66 - 60=6
2 x 10= 20and27 - 20=7 4 x 13= 52and60 - 52=8
1 x 11= 11and20 - 11=9 3 x 14= 42and52 - 42=10

This pattern of constantly increasing differences is a telling sign of quadratic growth, because each square number is the sum of consecutive odd numbers up to a certain point.

When you graph a series of contiguous vertical bars whose heights grow in such a way that the difference between the differences of consecutive heights stays constant, you get a parabola-shaped figure.

The XY graph of square numbers { (x, y) | y = x2 } is also a parabola.
So the 3D graph of the multiplication table has many parabolas in it.

The square numbers along the main diagonal of the multiplication table follow a parabola that goes up from the origin (0, 0).
At the same time, each square number in this central, upwards parabola is the vertex of a perpendicular, downward parabola that descends towards the coordinate axes.

Tracing the Loci (places) of Constant Products with Hyperbolas

The next image is kind of self-explanatory in the sense that color lines drawn inside the multiplication table here clearly pass through spots with the same numerical value for the product, although coming from different multiplication problems.

There is a simple 3D interpretation of the multiplication table where the two factors of each multiplication are taken as the 'x' and 'y' coordinates of a point (x, y, z) in space, while z, the height of the representative point above the XY-plane, is considered to be the resulting product: z = xy

In this 3D interpretation, all multiplications with the same result are represented by points located at the same height above the XY-plane, and the curved lines that join them, being made of points at the same height, are horizontal curves.

These curves turn out to be hyperbolas, since in the equation xy = constant, both variables are in an inverse relationship. Many of the numerical patterns found inside the multiplication table are translated into geometry as straight lines or curved lines in the 3D model { (x, y, z) | z = xy } of the multiplication table, a ruled, quadratic surface called a hyperbolic paraboloid, with a saddle shape.

• The constant increments along the rows and columns of the multiplication table result in straight lines embedded in the surface, its "rules," each parallel to the XZ-plane, or to the YZ-plane.
• The constanlty increasing differences along the main diagonal of the multiplication table, result in an upward parabola embedded in the surface.
• The constantly increasing differences along lines perpendicular to the main diagonal, result in downward parabolas embedded in the surface, with their vertices right on the upward parabola that comes from the main diagonal.
• The spots in the multiplication table where the values of the products are the same, result in horizontal hyperbolas embedded in the 3D model surface. These hyperbolas are the surface's level curves

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