Basic facts concept map

• Multiplication is joining together several sets of the same size.
• Repeated addition is a fundamental model for multiplication,
  e.g., 3 × 6 = (2 × 6) + 6 = [(1 × 6) + 6] + 6 = 6 + 6+ 6
• Multiplication includes the facts to 10 × 10.
• Students can also be exploring division strategies, and learning division basic facts at the same time as learning about multiplication.
• We will use the convention that 4 × 7 is the same as 4 lots of 7. This means that the 4-times table is column 4 of Tables 7 and 8.

The following principles apply to multiplication facts:

• Multiplying by 0 or by 1 is very easy. These are so self-evident that we probably won’t need students directly need to memorise them. However, it is useful for students to be aware of them and know how to use them.
• Some times tables are easier to learn than others
• Multiplication is commutative, so some basic facts can be derived from known ones e.g. 5 × 8 = 8 × 5

A most likely order to memorise the tables is:

1. 1- and 10-times table because adding on 1 is very easy, as is adding on 10 if the students understand place value.
   2-times table because adding on 2 is very simple
   5-times table which is just the intermediate jumps of the 10-times table.
2. 3-times and 4-times tables. Adding on small amounts is easier than large amounts.
3. 6- to 9-times tables.These can be approached by repeated addition, but it is increasingly difficult to do this using addition alone without exploiting derived facts or patterns. It is useful for students to derive each table using repeated addition.

A) 1-, 10-, 2-, and 5- times table
1-times table:    Purple
This is just the counting numbers; 1, 2, 3, 4, …, 9.
10-times table:  Blue
The sequence 10, 20, 30, 40,  is jumps (repeated addition) of 10 rather than 1.
2-times table:   Blue
It can easily be obtained from repeated addition of 2.
It is just the sequence of doubles. Students often know this from their addition knowledge (plus commutativity). e.g. 4 × 2 = 2 × 4 = 4 + 4 = 8
5-times table:    Blue
This follows from the 10-times table, just filling in the jumps half way between the successive 10s.

Table 7 - 1, 2, 5, and 10 times tables        

Second Number

Points

• This gives 40 basic facts (the striped regions). The 2-times table is the second column.
• Commutativity gives another 24 basic facts. (unstriped coloured region), e.g. 5 × 7 = 7 × 5 = 35

B) 4-, and 3- times table
4-times table: Green
This can be by doubling a known times table (the 2× table).  Almost all the doubles are already known or are easy, except 8 × 4 = 2 × 16 = 32 and 9 × 4 = 2 × 18 = 36
It can also easily be obtained from repeated addition of 4.

3-times table: Green 
This can easily be obtained from repeated addition of 3. It can also be derived from the 2-times table  e.g. 6 × 3 = 6 × 2 + 6 = 18 This is equivalent to working across the rows of Table 7.

Table 8 - 3- and 4-times tables    

Points

• This gives another 12 new basic facts: 3 × 3, 4 × 3, 6 × 3, …, 9 × 3; and 3 × 4, 4 × 4, 6 × 4, …, 9 × 4
• Commutativity gives another 8 basic facts.

C) 6- to the 9- times tables
There are only 16 basic facts to complete the 6- to the 9- times tables. These are the ones in Orange or Yellow . Of these facts, the six Yellow basic facts can be obtained using commutativity, i.e. 7 × 8 = 8 × 7. Strategies that underpin basic multiplication facts are as follows:

Patterns in multiplication
Exploring patterns in the times tables is another useful approach that both gives interest to, and aids the memorisation of the times tables. Every table has a repeating pattern of some sort, even the least accessible of all, the 7-times table.  Intriguing links can be seen between the last digits in the times table. See Table 9, which shows how the last digit in each pair of times tables is the exact reverse of each other. It looks as if this is a feature of having a base–10 number system, and this then explains the commonly known pattern of the 9-times table.

Table 9 - Patterns in the ones digit of the times tables

The nine-times table pattern
This pattern is very well known, and is found in many disguises.
The ones digit starts at 9 and goes down by 1 (9, 8, 7, …)
The tens digit starts at 0 and goes up by 1 (0, 1, 2, …),i.e. the digit in the tens place is one less than the number of times 9 is being multiplied.
The digit in the ones place plus the digit in the 10’s place = 9
6 × 9   = 54 because 5 is 1 less than 6, and 5 + 4 = 9
1 × 9   =    9
2 × 9   =  18
3 × 9   =  27
4 × 9   =  36 etc.

Repeating patterns in multiplication
In many of the tables, the digits cycle in useful ways. The ones digits have various “cycle lengths” before they repeat. Once the ones digit completes or begins a new cycle, it is a signal that something may also happen to the tens digit.

Cycle length 1:  10-times table
The ones digit is always 0, the tens digit goes up by one

Cycle length 2:   5-times table: 
Tens digit goes:     0, 1, 1, 2, 2,    3, 3, 4, 4, 5
Ones digit goes:     5, 0, 5, 0, 5,    0, 5, 0, 5, 0

Cycle length 5: The 2-, 4-, 6- and 8- times tables
These repeat the ones digits in a cycle of 5. This means that if you know the table up to 5, the table up to 10 is easy. It is especially memorable for the 2- and the 8-times table.

2-times table:

Tens digit goes:    0, 0, 0, 0, 1,      1, 1, 1, 1, 2
Ones digit goes:   2, 4, 6, 8, 0,      2, 4, 6, 8, 0 

8-times table:     

Tens digit goes:    0, 1, 2, 3, 4,      4, 5, 6, 7, 8
Ones digit goes:   8, 6, 4, 2, 0,      8, 6, 4, 2, 0

The 4- and the 6- times tables also have repeating patterns in the ones digit based on the even digits.
For the 4-times table the ones digit cycle is:  4, 8, 2, 6, 0
For the 6-times table the ones digit cycle is:  6, 2, 8, 4, 0, which is just the reverse of the 4- times table.

Cycle length 10: 3-, 7-, and 9-times table (plus the 1-times table)

9-times table:     

Tens digit goes:    0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Ones digit goes:    9, 8, 7, 6, 5, 4, 3, 2, 1, 0

There is, however, a pattern for the 3- and the previously intractable 7-times table.

3-times table:

Tens digit goes:    0, 0, 0,    1, 1, 1,     2, 2, 2,      3
Ones digit goes:   3, 6, 9,    2, 5, 8,     1, 4, 7,      0

7-times table:

Tens digit goes:    0, 1, 2,    2, 3, 4,     4, 5, 6,      7
Ones digit goes:   7, 4, 1,    8, 5, 2,     9, 6, 3,      0

Just look, in both cases the tens digit does something interesting at the end of each sub-cycle of 3! The pattern in the ones digit is the exact reverse in the 7-times tables as it was for the 3-times table.

Extending the multiplication basic facts
The extended 10-times table
Any number multiplied by ten is just that number with a 0 on the end. This is because our number system has a base of 10.

Example:
26 × 10 = 260, or equivalently 340 is 34 lots of 10.
We can then extend the basic facts as follows:
60 × 4 = 240 because it equals (6 × 4) × 10 = 24 × 10 = 240

The hyper-extended 10-times table
Students should know the effect of multiplying different powers of ten together. Again this is just the result of our base-10 number system. The first few of these are summarised in Table 10.

Example:
100 × 10 000 = 1 000 000
This is because 100 = 10 × 10 and 10 000 = 10 × 10 × 10 × 10
So 100 × 10 000 = 10 × 10 × 10 × 10 × 10 × 10 = 1 000 000 = 106
(Which is a 6-dimensional hyper cube!)

We can then extend the basic facts as follows:
60  × 400 = 24 000, because it equals (6 × 4) × 10 × 100 = 24 × 1000= 24 000
4000  × 50 = 240, because it equals (4 × 5) × 1000 × 10 = 20 × 10 000 = 200 000

Table 10 - The hyper-extended ten-times table

Doubling and halving
This is particularly useful once the hard-to-learn facts are known (i.e. where both numbers are between 6 and 9). This leads to the following:

• 12-times table (from 1 to 5)
  6 × 2 =   12  = 12 × 1,
  6 × 4 =   24  = 12 × 2,
  6 × 6 =   36  = 12 × 3,
  6 × 8 =   48  = 12 × 4,
  6 × 10 = 60  = 12 × 5
    
• The 14-, 16-, 18-, and 20-times tables can similarly be derived, e.g. 9 × 8 = 72  = 18 × 4.

Other useful multiplication tables
11-times table and the 11.1-times table
The ones digit and the tens digit are the same. So the sequence goes:

1 × 11 = 11,         1 × 11.1 = 11.1,
2 × 11 = 22,         2 × 11.1 = 22.2,
3 × 11 = 33,         3 × 11.1 = 33.3,
…,
9 × 11 =  99,         9 × 11.1 =  99.9 = 100
and 10 × 11 = 110 (using the extended 10-times table)

25-times table
All that we need to know is the following:

1 × 25 = 25     (which is obvious)
2 × 25 = 50    (which many people seen to know as a doubles fact)
3 × 25 = 75    (which is just 25 more than 50 and is often known)
4 × 25 = 100  (which is often known, or is double 50)

Then we get the repeating pattern:

25,  50,    75, 100
125, 150, 175, 200,
225, 250, 275, 300, …  where each hundred is 4 lots of 25.

The 5-times table can help us because 25 × 2 = 50 and
The 4-times table can also help because 25 × 4 = 100,

125-times table
All that we need to know is the following, with the bolded ones the only new ones once the 25-times table is known:

1 × 125 = 125 (which is obvious)
2 × 125 = 250
3 × 125 = 375
4 × 125 = 500  (which is double 250 from the 25-times table)
5 × 125 = 625
6 × 125 = 750  (which is triple 250 from the 25-times table)
7 × 125 = 875 
8 × 125 = 1000 (which is quadruple 250 from the 25-times table)

15-times table
This is related to the 3-times table because 15 × 2 = 30 and this can be seen in following patterns:

15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, …

35-times table
This is related to the 7-times table because 35 × 2 = 70 and this can be seen in following patterns:

35, 70, 105, 140, 175, 210, 245, 280, 315, 350, 385, 420 …

45-times table
This is related to the 9-times table because 45 × 2 = 90 and this can be seen in following patterns:

45, 90, 135, 180, 225, 270, 315, 360, 405, 450, 495, 540 …
This is particularly useful for angles in Geometry.