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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