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Patterns in Mathematics - Patterns in Shapes

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Patterns in shapes involve sequences of geometric figures that follow a specific rule of growth or repetition.

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Matchstick patterns are a common way to understand shape patterns. We use a variable, usually nn, to represent the number of shapes formed.

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If one shape requires kk matchsticks, then nn such separate shapes will require k×nk \times n matchsticks.

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In patterns where shapes are joined (sharing a common side), the rule changes. For example, for nn squares joined in a row, the rule is 3n+13n + 1.

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Identifying patterns helps in translating visual arrangements into algebraic expressions.

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The variable nn in the rule can take any natural number value like 1,2,3,…1, 2, 3, \dots to find the number of matchsticks for that specific position in the sequence.

📐Formulae

Number of matchsticks for n separate triangles=3n\text{Number of matchsticks for } n \text{ separate triangles} = 3n

Number of matchsticks for n separate squares=4n\text{Number of matchsticks for } n \text{ separate squares} = 4n

Number of matchsticks for n separate hexagons=6n\text{Number of matchsticks for } n \text{ separate hexagons} = 6n

General rule for n joined squares sharing one side=3n+1\text{General rule for } n \text{ joined squares sharing one side} = 3n + 1

General rule for n joined triangles sharing one side=2n+1\text{General rule for } n \text{ joined triangles sharing one side} = 2n + 1

💡Examples

Problem 1:

Observe the pattern of the letter 'L' made using matchsticks. One 'L' uses 2 sticks, two 'L's use 4 sticks. Find the rule for the number of matchsticks required to make nn such 'L's.

Solution:

2n2n

Explanation:

To make one 'L', we need 2 matchsticks. To make two 'L's, we need 2×2=42 \times 2 = 4 sticks. To make three 'L's, we need 2×3=62 \times 3 = 6 sticks. Following this pattern, for nn number of 'L's, the matchsticks required will be 2×n2 \times n, which is written as 2n2n.

Problem 2:

A pattern is formed by joining squares side-by-side. 1 square needs 4 sticks, 2 squares need 7 sticks, and 3 squares need 10 sticks. Find the general rule for nn squares.

Solution:

3n+13n + 1

Explanation:

If we look at the sequence: For n=1n=1, sticks =4= 4. For n=2n=2, sticks =7= 7. For n=3n=3, sticks =10= 10. Each additional square adds 3 sticks because they share one side. We can write 44 as 3(1)+13(1)+1, 77 as 3(2)+13(2)+1, and 1010 as 3(3)+13(3)+1. Thus, the rule for nn squares is 3n+13n + 1.

Problem 3:

Find the number of matchsticks required to make a pattern of 15 hexagons if they are all kept separate and each hexagon requires 6 matchsticks.

Solution:

9090

Explanation:

The rule for nn separate hexagons is 6n6n. Here, n=15n = 15. Substituting the value: 15×690\begin{array}{r} 15 \\ \times 6 \\ \hline 90 \end{array} So, 90 matchsticks are required.