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Most children meet "find the next number" puzzles long before Class 6. Chapter 1 of the new NCERT maths book asks for something more: once you have found a pattern, explain why it happens. That shift, from guessing to explaining, is what this chapter is really about. This guide covers what is in it, works through its key patterns with full answers, points out where students go wrong, and gives a simple routine for home.
What does Chapter 1 of Ganita Prakash cover?
Patterns in Mathematics is Chapter 1 of *Ganita Prakash*, NCERT's mathematics textbook for Grade 6, first published in August 2024 (the NCERT copy is marked "Reprint 2026-27"). The book's opening pages say this chapter was composed and contributed by Manjul Bhargava, a professor at Princeton University and co-chairperson of NCERT's National Syllabus and Teaching Learning Material Committee, who in 2014 became the first person of Indian origin to receive the Fields Medal.
It opens with a definition worth remembering: "Mathematics is, in large part, the search for patterns, and for the explanations as to why those patterns exist." Then it covers:
Number sequences: all 1s, counting numbers, odd and even numbers, triangular numbers, squares, cubes, Virahānka numbers, and powers of 2 and 3.
Pictures of sequences: dots arranged in triangles, squares and cubes.
Relations between sequences, such as adding odd numbers to get squares.
Shape sequences: regular polygons, complete graphs, stacked squares and triangles, and the Koch snowflake.
Why do odd numbers add up to square numbers?
The chapter shows that 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, and so on. Then it asks why. Picture a square of dots. Take the top-left dot (1). Add an L-shape of 3 dots round it to make a 2 by 2 square. Add an L of 5 to make 3 by 3. Each new L adds the next odd number, and each L turns one square into the next. Because you can always add one more L, the pattern goes on forever.
Worked examples
Example 1: sum of the first 10 odd numbers. By the picture, the first 10 odd numbers make a 10 by 10 square. So 1 + 3 + 5 + ... + 19 = 10 × 10 = 100. Check the last term: the 10th odd number is 2 × 10 − 1 = 19.
Example 2: the first 100 odd numbers. Same picture, 100 by 100: the sum is 100 × 100 = 10,000.
Example 3: up and down. 1 + 2 + 3 + 2 + 1 = 9 = 3 × 3. In general, adding up to a number and back down gives that number squared. So 1 + 2 + ... + 99 + 100 + 99 + ... + 2 + 1 = 100 × 100 = 10,000.
Example 4: hexagonal numbers. The chapter gives 1, 7, 19, 37. The gaps are 6, 12, 18, so the next gap is 24 and the next number is 37 + 24 = 61. The chapter also asks what happens when you multiply triangular numbers by 6 and add 1: 6 × 1 + 1 = 7, 6 × 3 + 1 = 19, 6 × 6 + 1 = 37, 6 × 10 + 1 = 61. The hexagonal numbers appear again.
Example 5: Virahānka numbers. In 1, 2, 3, 5, 8, 13, 21, each number is the sum of the two before it. The next three are 13 + 21 = 34, 21 + 34 = 55 and 34 + 55 = 89.
Example 6: complete graphs. A complete graph joins every pair of points with a line. With 2, 3, 4, 5 and 6 points you get 1, 3, 6, 10 and 15 lines: the triangular numbers again.
Example 7: powers of 2 plus one. 1 = 1, 1 + 2 = 3, 1 + 2 + 4 = 7, 1 + 2 + 4 + 8 = 15. Add 1 to each and you get 2, 4, 8, 16: the next power of 2 each time.
Where do students go wrong with patterns?
The most common mistake is deciding a rule too early. The sequence 1, 2, 4 could continue as 8 (doubling) or as 7 (add 1, then 2, then 3). Both are patterns. Without a rule or an explanation, three terms do not settle it. Other slips:
Writing the next term but not the rule. The chapter asks you to write the rule in your own words for each sequence.
Counting corners and sides separately for polygons and expecting different answers. In any closed shape made of straight sides, they are equal.
Mixing up squares and cubes: 1, 4, 9, 16 are squares; 1, 8, 27, 64 are cubes.
Assuming a pattern holds forever because it held five times. Draw the picture that shows why.
Why does pattern work matter for later maths?
It is more than a warm-up. In a long-running study of 517 children from low-income families in the United States, Rittle-Johnson and colleagues (2017) found that patterning knowledge in preschool, along with counting and a sense of quantity, predicted maths achievement in fifth grade, and patterning was still an important predictor at the end of first grade. Spotting and explaining patterns is part of the foundation for the algebra that comes later.
For parents: a useful question to ask is "How do you know it will keep going?" Do not supply the rule; ask for a drawing.
A weekly routine for this chapter
Day 1: Write each sequence from the chapter's Table 1 with the next three terms and the rule in words.
Day 2: Draw dot pictures for triangular numbers and squares. Show that 36 is both.
Day 3: Redo Examples 1 to 3 without looking.
Day 4: Invent your own sequence, give a friend four terms, and see whether they find your rule or a different one.
Day 5: Draw the first three shapes of the Koch snowflake and count the segments: 3, 12, 48.
The next chapters build on this: see our guides to number play in Class 6 and lines and angles in Class 6. If you want to explore a pattern beyond the book, Fermi's AI tutor can work out what you already know and build a plan for the session, and it checks your understanding as it goes.
Tonight, draw the L-shapes for a 5 by 5 square and check that 1 + 3 + 5 + 7 + 9 = 25.
