Visualising Times Tables — Patterns in whole numbers — Everything you never wanted to know about teaching your child basic maths

Why is an understanding of prime numbers and prime factors a critically important key to learning and understanding times tables and multiplication? Because the understanding of primes and prime factors turn your times tables into sets of related facts, instead of being merely a long series of isolated facts… More about this later. But to begin with, let’s start here:

Just to give you a flavour of what’s coming later in the article you will see how the units digits go through cyclical sequences, as shown in the follow graphic.

But let’s start slowly …

Before Times Tables … Learning to Add and Take

(1) Understand the concept of tens and units.
Use a number grid that starts at 0 (zero) up to 50 or so that wraps the next row every time the units digit gets back to zero.
To add on tens, go down rows (one row for each ten you are adding).
To add on units, go across.
Similarly, subtraction is going up rows, and back columns.
Get clear how many 10s and units in any given number

(2) Use numicon to discover and learn all the “number bonds” ie. pairs of whole numbers that can be used to make any given number from ten down to two. Why? Because adding by counting on is really slow. Once these number pairs are known, 9+8 can quickly be turned into 10+7, ie. 17 and 7+6 can quickly be turned into 10+3, ie. 13 etc. Practice which pairs cross the tens boundary and which don’t.
Once this is all clear and practiced, repeatedly adding on (say) 6 is no longer difficult.

To get going with young children

Stick numbers onto lego blocks with masking tape.

Then you can “count in twos”, “count in threes”, “count in fours”, “count in fives” etc.

You could use a different colour block for odd numbers and even numbers. Getting the little one to write the numbers on the masking tape was a particular popular part of the activity.

Count in twos.

Count in threes.

Count in fours and fives.

While you do the above, you can have her take note that multiplication is commutative (without using that word).

Three lots of four is the same as four lots of three. More on this below when we start using the Numicon.

Make sure she understands tens and units

Alongside writing out the numbers onto masking tape above, create a grid of counting numbers that wraps after every ten, and get your little one to fill it in. Check she can say each number, write each number and tell you how many tens and units there are in each number. (There’s more about using grids like this for addition and subtraction further below.)

Go down a row to add on tens. Go across a column to add on units.

EDIT: It may be better to start at 0 than at 1 — i.e. have 0 to 9 in the first line, 10 to 19 in the second line and so on. This way every row has the same number of tens in it, and every column has the same number of units in it.

Something that is obvious but easy to miss

Something that is obvious, and useful, but easy to miss about the conventional “times table” table is that to obtain the number that goes into any given square in the table, you just count the total number of squares contained within the rectangle that that square is the corner of.

Cut out a corner from a piece of card as shown in the photo:

You can use this mask to cordon off the cells in the grid that you are wanting to count for any given result cell. The counted number of cells goes into the cell that is in the bottom-right corner of the cut-out section, as indicated by the arrows drawn onto the card.

Move the masking card around over the grid, and fill in the counted number of cells or circles into the cell that is in the bottom-right corner of the cut-out section.

By using the above method, you can end up with a multiplication table without actually having done any multiplication !

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Multiplication tables

Multiplication tables like the one below are useful for reference, but perhaps less useful for understanding* or visualising what is going on in the progression of the increments of the table.

* Aside: That having been said, notice that the times table grid is symmetrical about the diagonal. Which is why when it comes to learning your times tables, you perhaps only need to learn any given times table up as far as its square, ie. the 2s up to 4, the 3s up to 9, the 4s up to 16, and so on (as I discuss in a section way down later in this article).

That is why presenting multiplication tables in the following forms may be useful in the initial understanding and presentation of what is actually being learnt.

Below are the two, three, four and five times tables (2 times table, 3 times table, 4 times table, 5 times table):

Students can see there are patterns created on the grid by colouring the squares belonging to every 3rd number (for the 3 times table), every 4th number (for the 4 times table), and every 5th number (for the 5 times table).

For the 3 times table, notice that the pattern repeats itself after the number produced in the units column, by the repeating increment, returns to 0 (it does this at 30). So after 30 we have 33 (which maps onto 3), 36 (which maps onto 6), 39 (which maps onto 9) and 42 (which maps onto 12) and so on.

Similarly for the 2 times table (the repeat happens after 10) — 12 maps onto 2, 14 maps onto 4 and so on. And also the 4 times table (the repeat happens after 20) — 24 maps onto 4, 28 maps onto 8 and so on. [More about the patterns created in the units column later in this article. These are fun!]

https://docs.google.com/spreadsheets/d/e/2PACX-1vR4_Nq1zvQC8_mfugZl-_5G9eDVupo1t2-NTfwXAlCdYelgihd1lzMvde7KLe3V7KeFzmwQBbKQo-ZZ/pubhtml

Now if we put the 3 or 4 times tables on a grid with row length of 3 or 4 repectively, naturally enough the column of answers goes straight down.

3 times table represented as a rectangular grid. Add 3 to any number by moving to the cell one row below.

4 times table represented as a rectangular grid. Add 4 to any number by moving to the cell one row below.

Interestingly, the columns of answers also go straight down if we put the 3, 4 or 6 times tables on a grid with rows going up in 12s.

Putting the 3, 4 (and 6) times tables onto a grid with rows of 12 instead of rows of 10 has the effect of lining the columns back up (because 12 is exactly divisible by 3,4 and 6).

Here are the six, seven, eight and nine times tables presented in this same form in a grid with rows of 10 (6 times table, 7 times table, 8 times table, 9 times table).

https://docs.google.com/spreadsheets/d/e/2PACX-1vR4_Nq1zvQC8_mfugZl-_5G9eDVupo1t2-NTfwXAlCdYelgihd1lzMvde7KLe3V7KeFzmwQBbKQo-ZZ/pubhtml

6 times table represented as a rectangular grid. Add 6 to any number by moving to the cell one row below.

7 times table represented as a rectangular grid. Add 7 to any number by moving to the cell one row below.

And here’s what it looks like when you ask your apprentice to do it in her homework book:

Here’s another way of figuring out times tables without actually having to do anything that someone who doesn’t know their times tables can’t already do… All you have to do is fill in the grid with your counting numbers. When you get to the end of a row, the next counting number just goes into the first cell on the next row, and so it goes on. Once you have filled in the grid, you “magically” discover you have just written out your times table for the given number in the last column of each of the tables.

Make the grid 10 numbers wide to enable addition and subtraction of two-digit numbers. Eg. on the grid below put your finger on 34. To add eleven move across one cell to the right and down one row.

A grid like those shown above, but one which is 10 numbers wide can be used to practice simple 2 digit addition and subtraction. To add your units you just count to the right, subtract count to the left. To add your tens, count down in rows, subtract count up in rows.

Putting 6 times tables onto a grid with rows of 12 instead of rows of 10 has the effect of lining the columns back up (because 12 is exactly divisible by 6). It does not make a lot of difference with the 7 times table though.

Here are the ten, eleven, twelve and thirteen times tables (10 times table, 11 times table, 12 times table, 13 times table):

https://docs.google.com/spreadsheets/d/e/2PACX-1vR4_Nq1zvQC8_mfugZl-_5G9eDVupo1t2-NTfwXAlCdYelgihd1lzMvde7KLe3V7KeFzmwQBbKQo-ZZ/pubhtml

https://docs.google.com/spreadsheets/d/e/2PACX-1vR4_Nq1zvQC8_mfugZl-_5G9eDVupo1t2-NTfwXAlCdYelgihd1lzMvde7KLe3V7KeFzmwQBbKQo-ZZ/pubhtml

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Adding on 9 is the same as adding on 10 and taking off 1 etc. (“number bonds”)

Remember those “number bonds” ? My Emily comes home from school singing : “One and nine (clap, clap), two and eight (clap, clap), three and seven, four and six, five and five! (clap, clap)” … If you want to sing the song there is a couple of rests or claps after nine and eight, and then the last three pairs kinda come straight after each other with two final claps on the end. But no matter.

Adding on 9 is the same as adding on 10 and taking off 1. Consequently the digit in the units column of the 9 times table is the same as the digit in the units column of the 1 times table when counted down from 10(x1).

[Because when you go backwards down through the times table numbers you take off (in this case) 1 each time, which as far as the unit digit is concerned is the same operation as adding 9.]

The same is true of the 8 times table and the 2 times table. Adding on 8 is the same as adding on 10 and taking away 2, so consequently the last digit in the 8 times table is same as the last digit in the 2 times table when it’s written backwards.

And so on… The same is true of the 7 times table and the 3 times table. Adding on 7 is the same as adding on 10 and taking off 3.

And the 6 times table and the 4 times table.

Showing the patterns in the units column graphically — the “number wheel”

You can see these patterns in the units column of times tables (described in the previous section) more visually if you get yourself a decagon (as shown in the graphic below).

I made the following graphic taking an image from Wikipedia and butchering it… I figure it’s fine to print it out if you want to. That’s what I did for my Emily.

https://en.wikipedia.org/wiki/Decagon

Now get your Emily or equivalent thereof to write out a times table underneath the picture when printed onto A4. Then tell them to start with their pencil on zero and go to that last digit (units) each number in the times table in turn with their pencil. You’ll get something like the following:

Below is the 7 times table … 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 … ie. 0,7,4,1,8,5,2,9,6,3,0.

Or if you start from zero and follow the lines in the oposite direction you get the 3 times table … 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30… ie. 0,3,6,9,2,5,8,1,4,7,0.

The 7 times table or the 3 times table, depending on which direction you go around the circle. [Start from zero.]

Below is one we made earlier… This is the how the drawing turns out for the 4 times table and 6 times table. 4,8,12,16,20,24,28,32,36,40.

This is what the 4 times table and the 6 times table look like, when the units column is put onto a decagon. Notice times tables of even numbers have no odd numbers in them.

Here is the pattern for the 8 times table (which is also the pattern for the 2 times table) … It is a pentagon!

Here is the pattern for the 8 times table (which is also the pattern for the 2 times table). It is a pentagon!

[And 1 and 9 is similarly just a decagon.]

Summary of the unit digits patterns of nine out of the ten possible times table unit digits patterns. If you start from zero, there are only 6 possible patterns, because 1 is the same as 9 (backwards), 2 is the same as 8 (backwards), 3 is the same as 7 (backwards), and 4 is the same as 6 (backwards). The 5 times table has nothing to differentiate forwards or backwards. And the 10 times table, of course, always ends with 0 zero as the units digit.

https://thort.space/523707003/for/arrangement/824/view/0.355563,0.373389,-0.856829/up/-0.057924,0.923776,0.378526/zoom/354.7614

What rectangles can be made out of a given number?

Although young children probably aren’t up for a discussion of prime numbers, one thing I discovered Emily is happy to do is find whether or not any given number can be made into a rectangle (or square) which is wider that one unit wide.

We tried doing this with a bunch of things, but the Numicon little cylinders has so far been the thing we found that works best.

What rectangles can be made out of 12 ?

Once she has found a rectangle that can be made with the number (that is other than one wide) she can draw a picture of it on squared paper (or not), and write next to picture how many long it is by how many wide with an “x” in between.

What rectangles can be made out of 18 ?

The 4 times table is just the 2 times table multiplied by 2 (ie. with every other number missed out)

Or to say the same thing in a slight different way, to get the 4 times table you just miss out every other number in the 2 times table.

So you can learn the 2 times table by missing out every other counting number (missing out the odd numbers) ie. every other number is divisible by 2.

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24 …
becomes
2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48 …

Once I have learnt the 2 times table by missing out every other counting number, I can now learn the 4 times table by missing out every other number in the 2 times table.

2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48 …
becomes
4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80 …

The 8 times table is just the 4 times table multiplied by 2 (ie. with every other number missed out)

Once I have learnt the 4 times table by missing out every other number in the 2 times table, I can now learn the 8 times table by missing out every other number in the 4 times table.

4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80 …
becomes
8,16,24,32,40,48,56,64,72,80,88,96,104,112,120,128,136,144,152,160 …

The 6 times table is the 3 times table multipled by 2 (ie. with every other number missed out)

Or to say the same thing in a slight different way, to get the 6 times table you just miss out every other number in the 3 times table (ie. miss out all the odd numbers).

3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60 …
becomes
6,12,18,24,30,36,42,48,54,60,66,72,78,84,90,96,102,108,114,120 …

Aside: By the way, notice these numbers 51 and 57 in the 3 times table. They look like they might be prime numbers, but they’re not! Here are the first bunch of prime numbers:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97, 101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181, 191,193,197,199 …

If we take a sum like 14 x 3 (which equals 42) we can halve the first number and double the second number, so:
14 x 3 = 7 x 6 = 42
similarly
12 x 3 = 6 x 6 = 36
and
10 x 3 = 5 x 6 = 30
and
8 x 3 = 4 x 6 = 24
and
4 x 3 = 2 x 6 = 12

Deepening your understanding with Primes

If you go back up four sections, you see we started having a conversation along the lines of “What rectangles can you make out of this number of Numicon cylinders?”

Numbers that can be made into rectangles that have a width and length that is at least 2 units, can be divided exactly by the number of units on each of the sides of such a rectangle. We say these numbers are “exactly divisible by” these two whole numbers.

Numbers that can’t be made into rectangles in this way are called “prime” numbers. The length in units of the sides of any such rectangles may themselves be makable into rectangles in the same way.

By repeating this process until the sides of any such rectangles are all primes, you have determined the “prime factors” of a number.

If your apprentice is able to get her head round prime factors then you can run through all the counting numbers from 1 upwards, and identify each number either as prime or as having its one and only set of prime factors:

1 — prime
2 — prime
3 — prime
4 = 2x2
5 — prime
6 = 2x3
7 — prime
8 = 2x2x2
9 = 3x3
10 = 2x5
11 — prime
12 = 2x2x3 = 4x3, 2x6
13 — prime
14 = 2x7
15 = 3x5
16 = 2x2x2x2
17 — prime
18 = 2x3x3 = 6x3, 9x2
19 — prime
20 = 2x2x5 = 4x5, 2x10
21 = 3x7
22 = 2x11
23 — prime
24 = 2x2x2x3 = 8x3, 6x4, 12x2
25 = 5x5
26 = 2x13
27 = 3x3x3
28 = 2x2x7 = 4x7, 14x2
29 — prime
30 = 2x3x5 = 6x5, 10x3, 15x2
31 — prime
32 = 2x2x2x2x2
33 = 3x11
etc. etc.

Once you have created a list like this, let’s say for all the whole numbers up to 100, you can combine the prime factors in all their various combinations and thereby understand why with a number such as 24(2x2x2x3), all of 8x3=24 and 6x4=24 and 12x2=24 … because you can multiply the prime factors together in any order.

Other key things to deepen understanding

Why is every number that is in a times table always in at least one other times table (unless it is a square number)?
Answer: Because AxB = BxA (This is called by commutative.)
Eg. 6x7 = 7x6
This show’s up nicely on the Times Table Grid by the grid being completely symmetrical along the diagonal with the square numbers on. Every number of one side of the square numbers diagonal is also in the mirror location on the other side of the diagonal.

What times tables is AxB in?
Answer: The A times table and the B times table; it is the Bth number in the A times table and it’s the Ath number in the B times table.
Eg. What times tables is 6x7 in?
Answer: The 6 times table and the 7 times table; it’s the 7th number in the 6 times table and the 6th number in the 7 times table.

Remembering times tables part 1 — using fingers with Times Tables

Let’s start by considering the times table grid in more detail. One thing to notice is that for times tables of even numbers, the units digit of the second five numbers is the same as the units digit for the first five numbers. This is because the units digit cycles and returns to zero after 5 numbers in even times tables (and b.t.w. all times table unit digits cycle and return to zero after 10 numbers — obviously !) This is why for times tables of even numbers we get the pentagonal patterns when plotting the units digit on the decagon as we showed up above.

I’ve added a visual hint in respect of this cycling of the units digit by using the blue, green and red arrows in the following graphic. I also put pink circles behind the numbers in the grid where the units digit is cycling back to zero after 5 numbers.

By the way, the sequences of numbers in the times table grid are obviously the same going across as they are going down — more about this in the next section.

For times tables of even numbers, the units digit of the second five numbers in the sequence is the same as the units digit for the first five numbers.

So now let’s map the fingers of our left hand onto the first five numbers and the fingers of the right hand on the second hand.

In the following pictures it would have been better really if we had drawn round our hands face up, but anyway. Hopefully you don’t make that mistake when you do it.

With the even times tables, 2 4 6 and 8, the units digits on the second hand are the same as the units digits on the first hand.

Get your little one to draw around their hands and put one of the times table numbers on each finger up to ten. Then practice counting up and and down the times tables, opening or closing each finger as they say the number. Make sure you do it the same way every time you practice, so that a specific finger becomes associated with a specific number in any given times table. Point out anchoring points and check points: eg. the little finger (or thumb — depending on which way round you face your palm) on the first hand always ends in a 5 or a 0 (because it is always 5 times the number of the current times table), and the last finger going up always ends in a 0 (because it is 10 times the number of current times table), and the first finger is always just the number of the times table, and on even times tables the first finger of the second hand always has a units digit the same as the number of the current table (eg. 4 and 24, 6 and 36, 8 and 48).

More pictures and maybe videos coming soon! :-)

Remembering times tables part 2 — you only actually need to learn any times table as far as its square !

For example you only need to learn the 2 times table as far as 2x2.
And you only need to learn the 3 times table as far as 3x3.
And you only need to learn the 4 times table as far as 4x4.

In case this is not obvious (and it wasn’t obvious to me for a long time, but now I cringe to admit it!) let me explain.

As we discussed in a section above AxB = BxA, eg. 6x7=7x6. This is obviously true if you just look at the rectangle on the numicon board. To see that it’s the same, just turn your AxB rectangle through 90 degrees, and there you have the same number of numicon pegs as BxA. You’ll notice if you fill out the times table grid above that the table is symmetrical about the shaded diagonal where the square numbers are (1x1, 2x2, 3x3, 4x4, 5x5, 6x6, 7x7, 8x8, 9x9 and 10x10), and this is because of the same observation that AxB = BxA.

So let’s say I learn each of my times tables up as far as the square; 2times as far as 2x2, 3times as far as 3x3, 4times as far as 4x4 and so on up 10times as far as 10x10 or even 12times up to 12x12.

To recap, I learnt my 3 times table up as far as 3x3. But now someone asks me what 4x3 is, and you might think I’m in trouble, cos only learnt 3,6,9; but 4x3 as well as being the 3 times table is also in the 4 times table … 4,8,12. The answer is 12. It is always quicker to go up to a given multiple if you count up in the larger number. For example if I want to know what 9x3 is, it’s quicker if I can say to myself: 9,18,27, than if I have to say: 3,6,9,12,15,18,21,24,27.

So now learning the 9 times table, the 8 times table, the 5 times table, and the 10 times table are all very easy (5s and 10s for obvious reasons, 8s and 9s see next section), and I only need to learn the 3s up to 9, the 4s up to 16, the 6s up to 36 and the 7s up to 49 and I’m done. And if I use the two-handed finger method to learn the 3s, 4s, 6s, and 7s (I’m going to fill this in the section above) it’s all over.

Given that the sequence for the 3s is just 3,6,9 (3 numbers)
and the sequence for the 4s is just 4,8,12,16 (4 numbers)
and the sequence for the 6s is just 6,12,18,24,30,36 (6 numbers)
and the sequence for the 7s is just 7,14,21,28,35,42,49 (7 numbers)
you may discover you have pretty much learnt all your times tables without having to have learnt very much of anything at all — a total of about 20 numbers, many of which by the time you are at the age when you’re expected to know times tables, you already know.

All of which having been said, it probably is still a good idea to the learn all the number sequences of each of the times tables at least up to 10, given the earlier discussions above and elsewhere about the importance of ratio in basic maths, and the discussion below about equivalent fraction sequences which gives a key lever into understanding ratio.

So I’m not saying it’s not useful to learn the sequences beyond the square numbers, just that you don’t need to learn these sequences if all you need to do is be able to multiple any pair of digits together.

Remembering times tables part 3 — the 9 times table and the 8 times table are very easy to learn (so learn them and you are almost done!)

There’s lots of videos like this online — here’s just a couple. I’m not going to bother making another one. But make sure you know this trick because it makes the 9s very easy to remember.

I’m going to add more to this section in soon as I get a spare moment.

Remembering times tables part 4— square numbers as anchoring points

I think for some reason the square numbers are easier to remember than other multiples. 4,9,16,25,36,49,64,81,100,121,144,169,196,225,256. So these can be used as anchoring points.

What is 7 8s? … Well 8 8s are 64, and 8 less than that is 56.

What is 6 8s? … Well 7 8s is 56 (we just figured out) so 6 8s must be 48.

What is 6 7s? … Well 7 7s is 49, and 7 less than that is 42.

[Also powers of 2: 2,4,8,16,32,64,128,256 may be useful anchoring points.]

Remembering times tables part 5— using the easier tables (eg. 5 times) as anchoring points

What is 6 x 8 ?
It’s 48, says Emily.
How did you work that out?
Well, says Emily, I worked out that 5 x 8 is 40 (because I can count up in 5s) and then I added on another 8.

This is a good example of how easier times tables can be used as anchoring points. You can do the same thing with 6 x 4 and 6 x 6.

5 x 4 is … 5,10,15,20 … so 6 x 4 must be 20 plus 4 … ie. 24

5 x 6 is … 5,10,15,20,25,30 … so 6 x 6 must be 30 plus 6 … ie. 36

Once we have started to get good at these anchoring points, we can put them into sequences…

What is 6 x 8? … Well 5 x 8 is 40, so 6 x 8 must be 48.
Ok, so what is 6 x 7? … Well 8 x 6 is 48, so 7 x 6 must be 6 less than that … 42.
Ok, so what is 7 x 7? … Well 6 x 7 is 42, so 7 x 7 must be 7 more than that … 49.

Remembering times tables part 6 — someone else’s take on this

In the discussion above I have presented some techniques I have found for engaging children in an understanding of decimal whole numbers, and particularly in understanding the way prime numbers and multiples and timestables are structured. However when it comes to “just being able to remember your times tables”, some other techniques might be useful. Below is a useful video I found in respect of this, and a diagram I made to accompany the video.

Division

Practice translating: “20 divided by 5” into “How many 5s in 20 ?” etc.

Then how many 5s in 21, 22, ie. 4r1, 4r2.

Now supposing we wanted to share 21 cakes between 5 people, what could we do with the last cake? … Fractions.

Equivalent Fraction Series

Using cakes student can see visually that

1/2 = 2/4 = 3/6 = 4/8 = 5/10 … etc.

1/3 = 2/6 = 3/9 = 4/12 = 5/15 … etc.

2/3 = 4/6 = 6/9 = 8/12 = 10/15 … etc.

1/4 = 2/8 = 3/12= 4/16 = 5/20 … etc.

Generalise from this to equivalent fractions.

More Key Stage 1 materials from me here:

Phonics Reading Systems

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