NCERT Solutions Class 6 Maths Chapter 1 Patterns in Mathematics Exercise 1.4

Exercise 1.4 focuses on understanding the connections between different number sequences. Students learn that triangular numbers can be formed by adding consecutive counting numbers, while certain combinations of sequences can result in square numbers. By examining these relationships, students develop a clearer understanding of how mathematical patterns are interconnected and how one sequence can be generated from another.

At ALLEN, we understand the importance of building these foundational skills, and our expertly crafted NCERT Solutions are tailored to help students grasp every nuance of pattern recognition. ALLEN's solutions provide clear, step-by-step guidance, enabling students to creatively extend and complete patterns while applying sound mathematical logic, ensuring a robust understanding of geometric and symmetrical principles.

1.0Download NCERT Solutions of Class 6 Maths Chapter 1 Patterns in Mathematics Exercise 1.4: Free PDF

Download step-by-step NCERT Solutions Class 6 Maths Chapter 1 Patterns in Mathematics Exercise 1.4 and boost your understanding of patterns in visual and geometric forms. Access the free PDF now and practice with accurate answers and explanations.

Key Concepts of Exercise 1.4

• Recognizing patterns in figures, shapes, and designs
• Recognizing patterns of repetition in visual sequences
• Completing symmetrical designs and artistic patterns
• Recognizing rotational symmetry and mirror symmetry of patterns 
• Encouraging creative thought while applying the logic of mathematics.

2.0NCERT Exercise Solutions Class 6 Chapter 1- Patterns in Mathematics : All Exercises 

3.0NCERT Class 6 Chapter 1 Patterns in Mathematics Exercise 1.4 : Detailed Solutions

1.4 Relations Among Number Sequences

1. Can you find a similar pictorial explanation for why adding counting numbers up and down, i.e., $1,1+2+1,1+2+3+2+1$,..., gives square numbers? Sol. When you add counting numbers up and down, like $1,1+2+1,1+2+3+2+1$ etc., you are essentially forming symmetrical shapes that resemble squares. For example:

1 = 1 (1 square dot) $1+2+1=4$ (forms a $2\times 2$ square) $1+2+3+2+1=9$ (forms a $3\times 3$ square) Each time, the numbers symmetrically increase and then decrease, giving a perfect square pattern.

2. By imagining a large version of your picture, or drawing it partially, as needed, can you see what will be the value of $1+2+3+\dots +99+100+99+\dots +3+2+1$ ? Sol. 1=1 $1+2+1=4$ $1+2+3+2+1=9$ $1+2+3+4+3+2+1=16$ $1+2+3+4+5+4+3+2+1=25$ $1+2+3+4+5+6+5+4+3+2+1=36$ $1+2+3+\dots .+99+100+99+\dots +3+2+1=10000$

3. Which sequence do you get when you start to add the All 1's sequence up? What sequence do you get when you add the All 1's sequence up and down? Sol. When we add all 1's sequence up we get the counting numbers, as $1=1$, $1+1=2$, $1+1+1=3$, $1+1+1+1=4$, When we add all 1's sequence up and down, we get counting numbers depend upon number of times 1 occurs.

4. Which sequence do you get when you start to add the Counting numbers up? Can you give a smaller pictorial explanation? Sol. If you add the counting numbers ( $1,2,3,4,\dots$ ), you get triangular numbers: $1=1$ $1+2=3$ $1+2+3=6$ $1+2+3+4=10$ This forms the triangular number sequence. Pictorial representation:

5. What happens when you add up pairs of consecutive triangular numbers? That is, take $1+3,3+6,6+10,10+15\dots$ ? Which sequence do you get? Why? Can you explain it with a picture? Sol.

6. What happens when you start to add up powers of 2 starting with 1 , i.e., take $1,1+2,1+2+4$, $1+2+4+8,\dots$ ? Now add 1 to each of these numbers -what numbers do you get? Why does this happen? Sol. When we start to add powers of 2: 1 $1+2=3$ $1+2+4=7$ $1+2+4+8=15$ When we add 1 to each of these numbers: $1+1=2$ $3+1=4$ $7+1=8$ $15+1=16$ We get a number sequence of powers of 2 again: $2,4,8,16,\dots$

7. What happens when you multiply the triangular numbers by 6 and add 1 ? Which sequence do you get? Can you explain it with a picture? Sol. Multiplying the triangular numbers by 6 and adding 1 gives: $1\times 6+1=7$ $3\times 6+1=19$ $6\times 6+1=37$ $10\times 6+1=61$ This forms the hexagonal number sequence: $7,19,37,61\dots$ Pictorial representation:

8. What happens when you start to add up hexagonal numbers, i.e., take $1,1+7,1+7+19$, $1+7+19+37,\dots$ ? Which sequence do you get? Can you explain it using a picture of a cube?

We get the cube of consecutive numbers. i.e., $1^3,2^3,3^3,4^3$,

9. Find your patterns or relations in and among the sequences in Table 1. Can you explain why they happen with a picture or otherwise? Sol. $3,6,9,12,15,18\dots \dots \dots .$. (consecutive multiples of 3 ). $10,15,20,25$........ (first number is 10 . Then increase of 5 in each term)

4.0Key Features and Benefits Class 6 Maths Chapter 1 Patterns in Mathematics: Exercise 1.4

• Offers the opportunity for visual thinking and spatial intelligence.
• Allows the student to become interested and enjoy geometry through creative avenues.
• Value in enhancing logical and reasoning skills and pattern recognition.
• Accepted basic understanding of symmetry in patterns, and recognizing symmetry related to a problem based design.
• Provides concrete, student proven solutions to allow student based discovery.