Pattern Around Us — NCERT Solutions

Given: The multiplication table of 2: $2, 4, 6, 8, 10, 12, 14, 16, 18, 20, \ldots$

Concept: Any number multiplied by 2 gives a result that can be divided into pairs with nothing left over.

$$2 \times 1 = 2 \;(\text{even}), \quad 2 \times 2 = 4 \;(\text{even}), \quad 2 \times 3 = 6 \;(\text{even}), \ldots$$

Since every number in the 2-times table is obtained by multiplying by 2, each result can always be arranged in complete pairs.

Answer: Yes, all numbers in the times-2 table are even. This is because multiplying any number by 2 always gives a number that can be perfectly divided into pairs.

Given: Numbers shown using crayon arrangements (figures not fully visible, but standard numbers used in this activity are 1 through 8 or similar).

Concept:

• Odd number → crayons cannot be paired completely; one crayon is left alone → circle it.
• Even number → all crayons can be paired completely → put a square around it.

Standard answers for numbers 1 to 8:

$$\underbrace{1}_{\text{circle (odd)}}, \quad \underbrace{2}_{\text{square (even)}}, \quad \underbrace{3}_{\text{circle (odd)}}, \quad \underbrace{4}_{\text{square (even)}}$$
$$\underbrace{5}_{\text{circle (odd)}}, \quad \underbrace{6}_{\text{square (even)}}, \quad \underbrace{7}_{\text{circle (odd)}}, \quad \underbrace{8}_{\text{square (even)}}$$

Rule to remember:

• Numbers ending in $0, 2, 4, 6, 8$ → Even → put a square.
• Numbers ending in $1, 3, 5, 7, 9$ → Odd → circle them.

Given: Page numbers of a textbook.

Concept: Page numbers follow a strict alternating even-odd pattern.

Observation:

• The right-hand (recto) pages always have odd page numbers: 1, 3, 5, 7, …
• The left-hand (verso) pages always have even page numbers: 2, 4, 6, 8, …

This is the pattern Shirley has noticed!

Answer: In any book, right-side pages are odd-numbered and left-side pages are even-numbered. Even and odd page numbers alternate throughout the book.

Mark squares (□) on even-numbered pages and circles (○) on odd-numbered pages in your textbook.

Given: Numbers — 30, 46, 78, 67, 300, 154, 415, 99.

Concept: A number is even if its units digit (last digit) is 0, 2, 4, 6, or 8. A number is odd if its units digit is 1, 3, 5, 7, or 9.

$$\begin{array}{|c|c|c|}\hline \textbf{Number} & \textbf{Units Digit} & \textbf{Even / Odd} \\ \hline 30 & 0 & \text{Even} \\ 46 & 6 & \text{Even} \\ 78 & 8 & \text{Even} \\ 67 & 7 & \text{Odd} \\ 300 & 0 & \text{Even} \\ 154 & 4 & \text{Even} \\ 415 & 5 & \text{Odd} \\ 99 & 9 & \text{Odd} \\ \hline \end{array}$$

Odd numbers: 67, 415, 99

Even numbers: 30, 46, 78, 300, 154

Given: Digits — 1 and 6 (no repetition allowed).

Step 1: Form 2-digit numbers using 1 and 6 without repeating any digit.
$$16 \quad \text{and} \quad 61$$

Step 2: Identify each as even or odd.

• $16$: units digit is $6$ → Even
• $61$: units digit is $1$ → Odd

Answer: The two numbers are 16 (even) and 61 (odd).

Given: Choose any two digits to form even 2-digit numbers.

Concept: A number is even if its units digit is 0, 2, 4, 6, or 8.

So, to make an even 2-digit number, the units (ones) place must have an even digit.

Example using digits 3 and 4:

• $34$ → units digit 4 → Even ✓
• $43$ → units digit 3 → Odd ✗

Example using digits 2 and 7:

• $72$ → units digit 2 → Even ✓
• $27$ → units digit 7 → Odd ✗

Answer: To make an even number, always place an even digit (0, 2, 4, 6, or 8) in the units place. For example, using digits 3 and 8: 38 and 83 — here 38 is even and 83 is odd. Choose digits so that the units place has an even digit.

Given: Numbers from 1 to 100.

Concept: Even and odd numbers alternate.

Even numbers between 1 and 100: $2, 4, 6, 8, \ldots, 100$
$$\text{Count} = \frac{100}{2} = 50$$

Odd numbers between 1 and 100: $1, 3, 5, 7, \ldots, 99$
$$\text{Count} = \frac{100}{2} = 50$$

Answer: There are equal numbers of even and odd numbers between 1 and 100 — exactly 50 each.

Given: Shirley's observation — numbers before and after an odd number are even.

Concept: Even and odd numbers alternate on the number line.

Checking Shirley's observation (odd number):

• Before and after $5$ (odd): $4$ (even) and $6$ (even) ✓
• Before and after $11$ (odd): $10$ (even) and $12$ (even) ✓

Checking Shiv's wonder (even number):

• Before and after $6$ (even): $5$ (odd) and $7$ (odd) ✓
• Before and after $14$ (even): $13$ (odd) and $15$ (odd) ✓
• Before and after $100$ (even): $99$ (odd) and $101$ (odd) ✓

Answer: Yes, Shiv is correct! Both the numbers before and after any even number are always odd. This is because even and odd numbers alternate — even, odd, even, odd — so an even number is always surrounded by two odd numbers, and an odd number is always surrounded by two even numbers.

Given: Any 10 consecutive numbers.

Let us choose: $11, 12, 13, 14, 15, 16, 17, 18, 19, 20$

$$\begin{array}{|c|c|}\hline \textbf{Number} & \textbf{Even / Odd} \\ \hline 11 & \text{Odd} \\ 12 & \text{Even} \\ 13 & \text{Odd} \\ 14 & \text{Even} \\ 15 & \text{Odd} \\ 16 & \text{Even} \\ 17 & \text{Odd} \\ 18 & \text{Even} \\ 19 & \text{Odd} \\ 20 & \text{Even} \\ \hline \end{array}$$

What we notice:

1. Even and odd numbers alternate — they keep switching one after the other.
2. In any 10 consecutive numbers, there are always exactly 5 even and 5 odd numbers.
3. If the first number is odd, the pattern is: Odd, Even, Odd, Even, … and vice versa.

Answer: In any set of consecutive numbers, even and odd numbers always alternate. There are equal numbers of even and odd numbers in any group of 10 consecutive numbers.