We see so many patterns around us in our daily life. We know that a pattern is an arrangement of objects, colors, or numbers placed in a certain order. Some patterns neither grow nor reduce but only repeat. Such patterns are known as repeating patterns. A pattern has a group of units that follow a rule while repeating or changing. Some examples of patterns are given below:
Whole numbers can be represented by line segments, triangles, squares, rectangles, etc.
1. Every whole number greater than 1 can be arranged in a line as shown below:
2. Some whole numbers can be represented by triangles. Such numbers are called triangular numbers.
3. Some whole numbers can be represented by squares. These numbers are also known as perfect squares.
4. Some whole numbers can be represented as rectangles.
3 Consecutive Numbers |
5 Consecutive Numbers |
1 + 2 + 3 = 6 2 + 3 + 4 = 9 3 + 4 + 5 = 12 4 + 5 + 6 = 15 5 + 6 + 7 = 18 |
1 + 2 + 3 + 4 + 5 = 15 2 + 3 + 4 + 5 + 6 = 20 3 + 4 + 5 + 6 + 7 = 25 4 + 5 + 6 + 7 + 8 = 30 5 + 6 + 7 + 8 + 9 = 35 6 + 7 + 8 + 9 + 10 = 40 |
Sums are the multiples of 3 |
Sums are the multiples of 5 |
Let us have a quick review of what we have learnt earlier about patterns.
I. Complete the series by drawing the next figure:
Answer:
A pattern can also be created with numbers. The set of numbers which follow a common rule form a pattern. For example, the sequence 2, 4, 6, 8, …… can be extended by using the rule of even numbers. Patterns with numbers can also be created using mathematical operations like addition, subtraction, multiplication and division.
For example:
1. Write the next 3 terms of the pattern 11, 15, 19, 23, ……..
The first term is 11. The common difference is 4. The next 3 terms are 27, 31, 35.
The mathematical calculations can be simplified by observing certain patterns.
1. Addition of 9, 99, 999, 9999, etc. to a whole number
For Example:
115 + 9 115 + 99 115 + 999 115 + 9999 115 + 99999 |
= 115 + 10 - 1 = 115 + 100 - 1 = 115 + 1000 - 1 = 115 + 10000 - 1 = 115 + 100000 - 1 |
= 125 - 1 = 215 - 1 = 1115 - 1 = 10115 - 1 = 100115 - 1 |
= 124 = 214 = 1114 = 10114 = 100114 |
2. Subtraction of 9, 99, 999, etc. from a whole number.
For Example:
2345 - 9 2345 - 99 2345 - 999 |
= 2345 - (10 - 1) = 2345 - (100 - 1) = 2345 - (1000 - 1) |
= 2345 - 10 + 1 = 2345 - 100 + 1 = 2345 - 1000 + 1 |
= 2335 + 1 = 2336 = 2245 + 1 = 2246 = 1345 + 1 = 1346 |
3. Multiplication of a whole number by 9, 99, 999, etc.
For Example:
125 × 9 125 × 99 125 × 999 |
= 125 (10 - 1) = 125 (100 - 1) = 125 (1000 - 1) |
= 1250 - 125 = 12500 - 125 = 125000 - 125 |
= 1125 = 12375 = 124875 |
Here are some patterns. Follow them and extend them:
(i)
1 × 9 + 2 = 11
12 × 9 + 3 = 111
123 × 9 + 4 = 1111
1234 × 9 + 5 = 11111
12345 × 9 + 6 = _______
123456 × 9 + 7 = _______
(ii)
1 × 8 + 1 = 9
12 × 8 + 2 = 98
123 × 8 + 3 = 987
1234 × 8 + 4 = 9876
12345 × 8 + 5 = _______
123456 × 8 + 6 = _______
(iii)
111 ÷ 3 = 37
222 ÷ 6 = 37
333 ÷ 9 = 37
444 ÷ 12 = 37
555 ÷ 15 = _______
666 ÷ 18 = _______
(iv)
9 + 1 = 10
90 + 10 = 100
900 + 100 = 1000
9000 + 1000 = _______
90000 + 10000 = _______
900000 + 100000 = _______
Follow the pattern of the following products and extend them further:
(i)
5 × 5 = 25
55 × 5 = 275
555 × 5 = 2775
5555 × 5 = 27775
_ _ _ _ _ _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ _ _ _ _ _
55555555 × 5 = 277777775
_ _ _ _ _ _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ _ _ _ _ _
(ii)
404 × 404 = 163216
505 × 505 = 255025
606 × 606 = 367236
707 × 707 = 499849
808 × 808 = 652864
_ _ _ _ _ _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ _ _ _ _ _
(iii) Study the following pattern and write the next two steps.
1 × 1 = 1
11 × 11 = 121
111 × 111 = 12321
1111 × 1111 = 1234321
____ × _____ = ________
_____ × _____ = _________
The next two steps will be
11111 × 11111 = 123454321
111111 × 111111 = 12345654321
Worksheet on Patterns in Whole Numbers:
In patterns in numbers here are some unsolved questions for the students to understand and complete the questions on patterns.
I. Complete the given series:
(i) 1, 3, 5, 7, …….., …….., …….., ……..
(ii) 5, 10, 15, 20, …….., …….., …….., ……..
(iii) 10, 20, 30, 40, …….., …….., …….., ……..
(iv) 3, 5, 8, 12, 17, …….., …….., …….., ……..
(v) 10, 20, 35, 55, 80, …….., …….., …….., ……..
(vi) 2, 2, 3, 3, 3, 4, 4, 4, 4, …….., …….., …….., ……..
(vii) 1, 3, 6, 8, 11, 13, 16, …….., …….., …….., ……..
(viii) 35, 45, …….., 65, 75, …….., 95
(ix) 100, 90, 80, …….., 60, 50 ……..
(x) 1, 3, 5, 7, 9, …….., …….., …….., ……..
Answer:
I. (i) 9, 11, 13, 15
(ii) 25, 30, 35, 40
(iii) 50, 60, 70, 80
(iv) 23, 30, 38, 47
(v) 110, 145, 185, 230
(vi) 5, 5, 5, 5, 5
(vii) 18, 21, 23, 26
(viii) 55, 85
(ix) 70, 40
(x) 11, 13, 15, 17
II. Complete the series:
(i) 1, 4, 7, 10, 13, 16, …….., …….., …….., ……..
(ii) 12, 17, 22, 27, 32, …….., …….., …….., ……..
(iii) 6, 16, 26, 36, 46, …….., …….., …….., ……..
(iv) 10, 30, 50, 70, …….., …….., …….., ……..
(v) 38, 35, 32, 29, …….., …….., …….., ……..
(vi) 54, 50, 46, 42, …….., …….., …….., ……..
(vii) 53, 49, 45, 41, …….., …….., …….., ……..
(viii) 23, 30, 37, 44, …….., …….., …….., ……..
Answer:
II. (i) 19, 22, 25, 28
(ii) 137, 42, 47, 52
(iii) 56, 66, 76, 86
(iv) 90, 110, 130, 150
(v) 26, 23, 20, 17
(vi) 38, 34, 30, 26
(vii) 37, 33, 29, 25
(viii) 51, 58, 65, 72
III. Complete the following series:
(i) 7, 14, 21, 28, …….., …….., …….., ……..
(ii) 25, …….., 75, 100, …….., …….., …….., ……..
(iii) 10, 100, …….., 10000, 100000, ……..
(iv) 2, 10, …….., 250, 1250
(v) 500000, 50000, …….., 500, …….., 5
(vi) 64, 32, 16, …….., 4, …….., 1
(vii) 5, 10, 20, …….., 80, …….., 320
(viii) 8, 16, 24, 32, …….., …….., …….., ……..
(ix) 45, 54, 63, …….., …….., …….., ……..
(x) 2, 6, 18, 54, ……..
Answers:
III. (i) 7, 14, 21, 28, …….., …….., …….., ……..
(ii) 25, …….., 75, 100, …….., …….., …….., ……..
(iii) 10, 100, …….., 10000, 100000, ……..
(iv) 2, 10, …….., 250, 1250
(v) 500000, 50000, …….., 500, …….., 5
(vi) 64, 32, 16, …….., 4, …….., 1
(vii) 5, 10, 20, …….., 80, …….., 320
(viii) 8, 16, 24, 32, …….., …….., …….., ……..
(ix) 45, 54, 63, …….., …….., …….., ……..
(x) 2, 6, 18, 54, ……..
IV. Using the shorter method simplify the following.
(i) 226 + 99
(ii) 2157 + 9999
(iii) 1239 + 99999
(iv) 6712 - 999
(v) 42785 - 9999
(vi) 1235 × 999
Answer:
IV. (i) 226 + 99
= 226 + 100 - 1
= 326 - 1
= 325
(ii) 2157 + 9,999
= 2,157 + 10,000 - 1
= 12,157 - 1
= 12,156
(iii) 1239 + 99999
= 1239 + 100000 - 1
= 101239 - 1
= 101238
(iv) 6712 - 999
= 6712 - (1000 - 1)
= 6712 - 1000 + 1
= 5712 + 1
= 5713
(v) 42785 - 9999
= 42785 - (10,000 - 1)
= 42785 - 10,000 + 1
= 32785 + 1
= 32786
(vi) 1235 × 999
= 1235 × (1000 - 1)
= 123500 - 1235
= 1233765
2. Observe the pattern and fill in the blanks.
1 × 9 + 1 = 10
12 × 9 + 2 = 110
123 × 9 + 3 = 1110
1234 × 9 + 4 = _____
12345 × 9 + 5 = ______
Answer:
2.
1234 × 9 + 4 = 11110
12345 × 9 + 5 = 111110
3. Study the following pattern and write the next two steps.
1 × 8 + 1 = 9
12 × 8 + 2 = 98
123 × 8 + 3 = 987
1234 × 8 + 4 = 9876
12345 × 8 + 5 = 98765
______ × _ + _ = ______
_______ × __ + __ = _______
Answer:
3.
123456 × 8 + 6 = 987654
1234567 × 8 + 7 = 9876543
Related Concept
● Patterns and Mental Mathematics
● Counting Numbers in Proper Pattern
● Puzzle
● Patterns
4th Grade Math Activities
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