Patterns for CTET: number and shape patterns and the road to algebra

Give a child the beads red, blue, blue, red, blue, blue and ask what comes next. Most will say red. Now ask what the 20th bead will be. That question is much harder, because the child can no longer just look at the last bead: they have to find the unit that repeats and reason about it. That shift, from continuing a pattern to describing its rule, is the start of algebra, and it is why patterns appear in the CTET Paper I syllabus and in pedagogy questions for both papers.

This guide covers the kinds of patterns, the number patterns worth knowing, worked examples with every answer checked, the link to algebra and the teaching approach CTET expects.

Kinds of patterns

Repeating patterns come first in the primary years because children can build them with beads, blocks, claps and actions. Growing patterns follow, and they lead directly into number patterns and algebra.

Number patterns worth knowing

Patterns that surprise children

• Square numbers as sums of odd numbers: 1 = 1², 1 + 3 = 4 = 2², 1 + 3 + 5 = 9 = 3², 1 + 3 + 5 + 7 = 16 = 4². The sum of the first n odd numbers is n².
• Ones multiplied: 11 × 11 = 121, 111 × 111 = 12,321, 1,111 × 1,111 = 12,34,321.
• The 9 times table: 09, 18, 27, 36, 45, 54, 63, 72, 81, 90. The tens digit goes up by one as the ones digit goes down by one, and the digits of each of these add to 9.
• Even and odd: even + even = even, odd + odd = even, even + odd = odd.

Worked examples

1. The 20th bead. The pattern is red, blue, blue, repeating. The core has 3 beads. 20 = 6 × 3 + 2, so after six full cores the 20th bead is the 2nd bead of the core: blue. The 25th: 25 = 8 × 3 + 1, so red.

2. Matchstick squares. A row of 1 square needs 4 matchsticks, 2 squares need 7, 3 squares need 10. Each new square adds 3 sticks, so n squares need 3n + 1. For 10 squares: 3 × 10 + 1 = 31.

3. A calendar block. Take any 3 × 3 block of dates on a calendar, for example 3, 4, 5 / 10, 11, 12 / 17, 18, 19. The sum is 12 + 33 + 54 = 99, which is 9 × 11, nine times the centre date. It works for every such block, because the dates are balanced around the centre: each date above is 7 less than the one below it, and each date to the left is 1 less than the one to its right.

4. A number trick. Think of a number, double it, add 6, halve the result, then subtract the number you started with. The answer is always 3. Why? If the number is n: (2n + 6) ÷ 2 − n = n + 3 − n = 3. This is a perfect bridge from pattern to algebra.

5. A magic square. The numbers 1 to 9 can be arranged so that every row, column and diagonal adds to 15:

Top row: 2, 7, 6
Middle row: 9, 5, 1
Bottom row: 4, 3, 8

Rows: 2 + 7 + 6, 9 + 5 + 1 and 4 + 3 + 8 are all 15. Columns: 2 + 9 + 4, 7 + 5 + 3 and 6 + 1 + 8 are all 15. Diagonals: 2 + 5 + 8 and 6 + 5 + 4 are both 15. The magic sum is 15 because 1 + 2 + … + 9 = 45 is shared equally among 3 rows.

Why patterns matter: the road to algebra

Finding a pattern and saying its rule is generalising, one of the central processes in mathematics. The progression looks like this:

1. Copy a pattern.
2. Extend it: what comes next?
3. Find the core or rule and describe it in words: "add 3 each time".
4. Predict a far term: what is the 20th?
5. Write the rule with a letter: 3n + 1.

Steps 1 and 2 are Paper I territory; steps 4 and 5 are where Class 6 algebra begins. NCERT's upper-primary books have long introduced algebra by this route, starting from matchstick patterns. See algebra.

Patterns also build number sense. A child who notices that 9 × 7 is 10 × 7 minus 7 is using a pattern to calculate. And shape patterns in rangoli, kolam, border prints and tiled floors connect school maths to the child's home and community.

How children go wrong

The fourth row is worth remembering. A short sequence can often be continued in more than one way, so a good question gives enough terms, or asks the child to explain the rule used.

Teaching patterns through activity

• Bodies and sounds: clap, clap, stamp; stand, sit, sit. Patterns do not need paper.
• Beads, blocks and bottle caps for building, extending and breaking patterns into cores.
• Hundred square: colour the multiples of 2, then 5, then 9, and describe the pictures that appear.
• Calendars and multiplication tables: hunt for patterns and explain why they happen.
• Create and swap: children make their own pattern, and a partner must find the rule.
• Matchsticks and dot patterns in upper primary, moving towards rules with letters.
• Rangoli, kolam and border designs from children's homes, linked to symmetry. See geometry and shapes.

Practice set

1. What comes next: 3, 6, 12, 24, ?
2. What comes next: 1, 3, 6, 10, 15, ?
3. What comes next: 100, 91, 82, 73, ?
4. What comes next: 1, 1, 2, 3, 5, 8, ?
5. In the repeating pattern A, B, C, C, A, B, C, C, … what is the 30th letter?
6. One triangle needs 3 matchsticks, two triangles in a row need 5, three need 7. How many for 8 triangles?
7. Find 1 + 3 + 5 + … + 19 without adding term by term.
8. A teacher asks children to make their own bead patterns and explain the rule to a partner. The main purpose is to develop: (a) fine motor skills (b) the ability to generalise (c) colour recognition (d) obedience

Answers

1. Doubling: 48.
2. Add 6: 21.
3. Subtract 9: 64.
4. 5 + 8 = 13.
5. The core has 4 letters. 30 = 7 × 4 + 2, so the 2nd letter of the core: B.
6. 2n + 1 = 2 × 8 + 1 = 17.
7. These are the first 10 odd numbers, so the sum is 10² = 100.
8. (b). Creating and explaining a rule is generalisation, the root of algebraic thinking.

What to do next

• Write the number-pattern table from memory with the rule for each.
• Make three repeating patterns and find the 20th, 25th and 50th term of each by calculation.
• Find the matchstick rule for a row of pentagons (start: 1 pentagon, 5 sticks).
• Continue with algebra, then revise data handling.