Class 6 Mathematics

Chapter 1 Solutions — Patterns in Mathematics

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Solutions

Overview

Step-by-step NCERT solutions for Patterns in Mathematics (Chapter 1, NCERT Class 6 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Patterns in Mathematics textbook chapter.

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What these solutions cover

All 17 questions in Patterns in Mathematics are solved in the PDF. Here's what's inside, exercise by exercise:

Patterns in Numbers

Rewrite each sequence of Table 1 in your notebook, along with the next three numbers in each sequence! After each sequence, write in your own words what is the rule for forming the numbers in the sequence. Table 1 sequences: 1, 1, 1, 1, 1, 1, 1, ... (All 1's) 1, 2, 3, 4, 5, 6, 7, ... (Counting numbers) 1, 3, 5, 7, 9, 11, 13, ... (Odd numbers) 2, 4, 6, 8, 10, 12, 14, ... (Even numbers) 1, 3, 6…

Visualising Number Sequences

Why are 1, 3, 6, 10, 15, … called triangular numbers? Why are 1, 4, 9, 16, 25, … called square numbers or squares? Why are 1, 8, 27, 64, 125, … called cubes?
What would you call the following sequence of numbers? 1, 7, 19, 37, ... (The dots are arranged in hexagonal patterns as shown in the figures.) What is the next number in the sequence?

Relations among Number Sequences

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?
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 + ... + 99 + 100 + 99 + ... + 3 + 2 + 1?
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?
Which sequence do you get when you start to add the counting numbers up? Can you give a smaller pictorial explanation?
What happens when you add up pairs of consecutive triangular numbers? That is, take 1 + 3, 3 + 6, 6 + 10, 10 + 15, … Which sequence do you get? Why? Can you explain it with a picture?
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, … ? Now add 1 to each of these numbers — what numbers do you get? Why does this happen?
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?
What happens when you start to add up hexagonal numbers, i.e., take 1, 1 + 7, 1 + 7 + 19, 1 + 7 + 19 + 37, … ? Which sequence do you get? Can you explain it using a picture of a cube?

Patterns in Shapes

Can you recognise the pattern in each of the sequences in Table 3?

Relation to Number Sequences

Count the number of sides in each shape in the sequence of Regular Polygons. Which number sequence do you get? What about the number of corners in each shape in the sequence of Regular Polygons? Do you get the same number sequence? Can you explain why this happens?
Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?
How many little squares are there in each shape of the sequence of Stacked Squares? Which number sequence does this give? Can you explain why?
How many little triangles are there in each shape of the sequence of Stacked Triangles? Which number sequence does this give? Can you explain why? (Hint: In each shape in the sequence, how many triangles are there in each row?)
To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment '—' by a 'speed bump'. As one does this more and more times, the changes become tinier and tinier with very very small line segments. How many total line segments are there in each shape of the Koch Snowflake? What is the corresponding number sequence? (The answer is 3, 12, 48, ..., i.e., 3…

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