MULTIPLICATION TABLES - REMEMBERING MADE EASY BY SPECIAL TIPS, EASY TO FOLLOW
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Please study the introductory part of Multiplication where we discussed the multiplication as repeated addition with examples and exercises.
There, we also discussed the need to remember the values of the products of one single digit number with another single digit number.
Learning these products is much like vocabulary in a language. Till you master the vocabulary, the language looks like a foreign language. Once you master, the language becomes your mother tongue.
Our aim here is to make the single-digits' products as easy as your mother tongue.
We try to establish the connections between any two single-digit factors and the resulting product, until the connection becomes intuitive.
A multiplication table ("times table") is a grid where rows and columns are headed by the numbers to multiply, and the entry in each cell is the product of the column and row headings. The heading for the first row and first column contains the symbol "x" which is multiplication operator.
So, for example, 6 x 8 = 48 by looking up where 6 and 8 intersect.
The traditional rote learning of multiplication was based on memorization of columns in the table, in a form like
1 x 8 = 8
2 x 8 = 16
3 x 8 = 24
4 x 8 = 32
5 x 8 = 40
6 x 8 = 48
7 x 8 = 56
8 x 8 = 64
9 x 8 = 72
10 x 8 = 80
Learning the contents of the (10x10) table is much easier than it superficially seems to be.
Here are some TIPS which are helpful in remebering the contents of the Table.
Tip 1 of Multiplication Tables :
Half of the table is a mirror image of the other or Order of Multiplying Does Not Matter.
The entries in the Multiplication Table above and below the Principal Diagonal Elements are the same.
Because of the symmetry of the Multiplication Table 45 entries are in fact duplicates. (45 + 10 principal diagonal elemements = 55 entries left).
For Example, If you remember 4 x 7 = 28, then, you also know the value of 7 x 4. 4 x 7 = 28 = 7 x 4
If you remember 3 x 8 = 24, then, you also know the value of 8 x 3. 3 x 8 = 24 = 8 x 3
Because of Tip1, it is enough if you remember half of the Table which is shown below.
x
1
2
3
4
5
6
7
8
9
10
1
1
2
3
4
5
6
7
8
9
10
2
4
6
8
10
12
14
16
18
20
3
9
12
15
18
21
24
27
30
4
16
20
24
28
32
36
40
5
25
30
35
40
45
50
6
36
42
48
54
60
7
49
56
63
70
8
64
72
80
9
81
90
10
100
Tip 2 of Multiplication Tables :
1 times table and 10 times table are easy to remember.
1 times any number = that number. 10 times any number = that number followed by zero.
For Example, 1 x 5 = 5; 1 x 7 = 7 etc. 10 x 5 = 50; 10 x 7 = 70 etc.
So, no effort is required in remembering 1 times table and 10 times table.
This reduces the entries of the Table to remember to the following.
x
1
2
3
4
5
6
7
8
9
10
1
2
4
6
8
10
12
14
16
18
3
9
12
15
18
21
24
27
4
16
20
24
28
32
36
5
25
30
35
40
45
6
36
42
48
54
7
49
56
63
8
64
72
9
81
10
Tip 3 of Multiplication Tables :
We know, the number multiplied by itself is called the Square of the Number.
The product of the Numbers which differ by 2 is equal to the square of the in between number minus one.
To apply this Tip, you need to remember the Squares of the Numbers.
I strongly advise you to remember the squares of numbers which are given below.
1 x 1 = 1
2 x 2 = 4
3 x 3 = 9
4 x 4 = 16
5 x 5 = 25
6 x 6 = 36
7 x 7 = 49
8 x 8 = 64
9 x 9 = 81
10 x 10 = 100
Note that these are nothing but the elements of the principal diagonal.
Now, to apply the above Tip, consider 7 x 9. 7 and 9 differ by 2. So their product = 7 x 9 = square of in between number (which is 8) minus 1. = 8 x 8 - 1 = 64 - 1 = 63.
Let us see another example. consider 4 x 6. 4 and 6 differ by 2. So their product = 4 x 6 = square of in between number (which is 5) minus 1. = 5 x 5 - 1 = 25 - 1 = 24.
With this Tip, the elements 2 x 4, 3 x 5, 4 x 6, 5 x 7, 6 x 8, 7 x 9 can be remembered and hence the table reduces to
x
1
2
3
4
5
6
7
8
9
10
1
2
4
6
10
12
14
16
18
3
9
12
18
21
24
27
4
16
20
28
32
36
5
25
30
40
45
6
36
42
54
7
49
56
8
64
72
9
81
10
Tip 4 of Multiplication Tables :
The product of the Numbers which differ by 4 is equal to the square of the middle number minus 4.
Now, to apply the above Tip, consider 4 x 8. 4 and 8 differ by 4. So their product = 4 x 8 = square of the middle number (which is 6) minus 4. = 6 x 6 - 4 = 36 - 4 = 32.
Let us see another example. consider 3 x 7. 3 and 7 differ by 4. So their product = 3 x 7 = square of the middle number (which is 5) minus 4. = 5 x 5 - 4 = 25 - 4 = 21.
With this Tip, the elements 2 x 6, 3 x 7, 4 x 8, 5 x 9 can be remembered and hence the table reduces to
x
1
2
3
4
5
6
7
8
9
10
1
2
4
6
10
14
16
18
3
9
12
18
24
27
4
16
20
28
36
5
25
30
40
6
36
42
54
7
49
56
8
64
72
9
81
10
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5 times even number = half of the even number followed by 0. 5 times odd number = half of the (odd number - 1) followed by 5.
For example, 5 x 6 = half of 6 followed by 0 = 30 5 x 8 = half of 8 followed by 0 = 40
5 x 7 = half of (7 - 1) followed by 5 = half of 6 followed by 5 = 35 5 x 9 = half of (9 - 1) followed by 5 = half of 8 followed by 5 = 45
With this Tip, the table reduces to
x
1
2
3
4
5
6
7
8
9
10
1
2
4
6
14
16
18
3
9
12
18
24
27
4
16
28
36
5
6
36
42
54
7
49
56
8
64
72
9
81
10
Tip 6 of Multiplication Tables :
Remembering Nine Times Table made easy :
To remember 9 times table, we follow an easy method which is explained below.
Put your hands in front of you with your palms facing you. Your fingers represent the numbers one to ten in an order.
1 = left thumb 2 = left index finger 3 = left middle finger 4 = left ring finger 5 = left little finger 6 = right little finger 7 = right ring finger 8 = right middle finger 9 = right index finger 10 = right thumb
To find 4 x 9 : Close your fourth finger (left ring finger). The number of fingers to the left of the closed finger (3) is the left digit and to the right of the closed finger (6) is the right digit.
So the answer is 36.
To find 6 x 9 : Close your sixth finger (right little finger). The number of fingers to the left of the closed finger (5) is the left digit and to the right of the closed finger (4) is the right digit.
So the answer is 54.
To find 7 x 9 : Close your seventh finger (right ring finger). The number of fingers to the left of the closed finger (6) is the left digit and to the right of the closed finger (3) is the right digit.
So the answer is 63.
To find 9 x 9 : Close your ninth finger (right index finger). The number of fingers to the left of the closed finger (8) is the left digit and to the right of the closed finger (1) is the right digit.
So the answer is 81.
With this method you can find any value from 1 x 9 to 9 x 9.
Another method to write 9 times table as a whole is, For the values from 1 x 9 to 10 x 9, increase the left digit from 0 to 9 and decrease the right digit from 9 to 0.
This is clear from the following vertical presentation.
1 x 9 = 09
2 x 9 = 18
3 x 9 = 27
4 x 9 = 36
5 x 9 = 45
6 x 9 = 54
7 x 9 = 63
8 x 9 = 72
9 x 9 = 81
10 x 9 = 90
onetwo equals three times four i. e. 12 = 3 x 4. fivesix equals seven times eight i. e. 56 = 7 x 8.
See the order of the numbers 1, 2, 3, 4, 5, 6, 7, 8.
With this Tip, we can remember 3 x 4 and 7 x 8and hence the table reduces to
x
1
2
3
4
5
6
7
8
9
10
1
2
4
6
14
16
3
9
18
24
4
16
28
5
6
36
42
7
49
8
64
9
10
Tip 8 of Multiplication Tables :
Remembering Two times table :
2 times any number = Adding the number to itself.
Adding the single digit number to itself can be done with the help of fingers, as explained in
Addition.
Here the addition is only once and hence can be done easily.
You may use this tip till you are comfortable in remembering the 2 times table.
With this Tip, the table reduces to
x
1
2
3
4
5
6
7
8
9
10
1
2
3
9
18
24
4
16
28
5
6
36
42
7
49
8
64
9
10
With these 8 Tips, the 100 elements in the Table are reduced to a mere 9 elements.
Tip 9 of Multiplication Tables :
Remembering Six times table :
6 times any number = 5 x the number + the number.
Since, we have already seen the Tip (Tip 5) to remember the 5 times table, we may use that to find 5 x the number and add the number to get 6 times the number.
For example, 6 x 6 = 6 x 5 + 6 = half of 6 followed by 0 + 6 = 30 + 6 = 36 6 x 8 = 8 x 5 + 8 = half of 8 followed by 0 + 8 = 40 + 8 = 48
6 x 7 = 7 x 5 + 7 = half of (7 - 1) followed by 5 + 7 = half of 6 followed by 5 + 7 = 35 + 7 = 42 6 x 9 = 9 x 5 + 9 = half of (9 - 1) followed by 5 + 9= half of 8 followed by 5 + 9 = 45 + 9 = 54
With this Tip, the table reduces to
x
1
2
3
4
5
6
7
8
9
10
1
2
3
9
24
4
16
28
5
6
7
49
8
64
9
10
With these 9 Tips, the 100 elements in the Table are reduced to a mere 6 elements.
For example, for a number to be divisible by 3 the sum of the digits is to be divisible by 3.
Suppose you write 3 x 8 = 24. You may check that 2 + 4 = 6 is divisible by 3.
For a number to be divisible by 7, the difference of two times the last digit and the remaining number is to be divisible by 7.
Suppose you write 4 x 7 = 28. You may check that 2 x 8 - 2 = 16 - 2 = 14 is divisible by 7.
Note that all these Tips are only to help to remember the MultiplicationTables. If you can directly remember the Multiplication Tables, these tips are not necessary.
With Tips or without Tips, you have to remember the Multiplication Tables upto 9 x 9, accurately.
Put the required time and energy to remember these Multiplication Tables (1 x 1 to 9 x 9), without which you can not proceed further in calculations and in math.
With the help of the above Multiplication table, you can find the product of any two numbers.
If you still have difficulty, in remembering the Tables (1 x 1 to 9 x 9), Try the
Video game
mentioned at the top of this page.
Some students, who have good memory and good capacity for attention may remember the Multiplication Tables upto 20 x 20.
If you can also do the same, here is the Multiplication Table upto 20 times20.
Note that memorising the Multiplication tables beyond 9 x 9 is optional.
With the knowledge of Multiplication Tables upto 9 x 9 and the knowledge of
Multiplication, you can find the other entries in the following table.
For example, we have found the values of 4 x 12, 6 x 20, 2 x 30, 4 x 15. 4 x 11 and 5 x 13 in Examples 4 to 9 of
Multiplication. Multiplication Table upto 20 times20.
x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
3
3
6
9
12
15
18
21
24
27
30
33
36
39
42
45
48
51
54
57
60
4
4
8
12
16
20
24
28
32
36
40
44
48
52
56
60
64
68
72
76
80
5
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
6
6
12
18
24
30
36
42
48
54
60
66
72
78
84
90
96
102
108
114
120
7
7
14
21
28
35
42
49
56
63
70
77
84
91
98
105
112
119
126
133
140
8
8
16
24
32
40
48
56
64
72
80
88
96
104
112
120
128
136
144
152
160
9
9
18
27
36
45
54
63
72
81
90
99
108
117
126
135
144
153
162
171
180
10
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
11
11
22
33
44
55
66
77
88
99
110
121
132
143
154
165
176
187
198
209
220
12
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
204
216
228
240
13
13
26
39
52
65
78
91
104
117
130
143
156
169
182
195
208
221
234
247
260
14
14
28
42
56
70
84
98
112
126
140
154
168
182
196
210
224
238
252
266
280
15
15
30
45
60
75
90
105
120
135
150
165
180
195
210
225
240
255
270
285
300
16
16
32
48
64
80
96
112
128
144
160
176
192
208
224
240
256
272
288
304
320
17
17
34
51
68
85
102
119
136
153
170
187
204
221
238
255
272
289
306
323
340
18
18
36
54
72
90
108
126
144
162
180
198
216
234
252
270
288
306
324
342
360
19
19
38
57
76
95
114
133
152
171
190
209
228
247
266
285
304
323
342
361
380
20
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
360
380
400
We will see one more Tip here.
Tip 11 of Multiplication Tables :
Rememebering 11 times Table
11 times single digit = digit twice. 11 times two digit number = sum of the digits between the digits, if the sum of digits is single digit.. If the sum of the digits is double digit, carry over 1 to the left.
For Example, 11 x 6 = 66; 11 xx 8 = 88; 11 x 3 = 33; 11 x 12 = 1 (1 + 2) 2 = 132; 11 x 18 = 1 (1 + 8) 8 = 198; 11 x 19 = 1( + 1) 0 9 = 209; 11 x 89 = 8( + 1) 7 9 = 979;
In 11 x 12, 1 + 2 = 3 is put beteen 1 and 2 to get 132. ∴ 11 x 12 = 132.
In 11 x 19, 1 + 9 =10. So 1 is carried over to the left and 0 is put between the digits. The carried over 1 added to the left 1 gives 2. ∴ 11 x 19 = 209.
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