Patterns — Class 3 Mathematics (Samacheer Kalvi)
TN State Board (Samacheer Kalvi) Class 3 Mathematics. Finding rules, extending sequences and spotting patterns in the tables.
1. Every sequence has a rule
A sequence is a list of numbers in order. The rule tells you how to get from one term to the next.
Look at 3, 7, 11, 15. Each term is 4 more than the one before, so the rule is add 4. The next term is 19.
Look at 100, 90, 80, 70. Each term is 10 less, so the rule is subtract 10. The next term is 60.
To find a rule, write the gap between each pair of terms. If every gap is the same, the rule is an addition or a subtraction.
2. When the gaps are not equal
Sometimes the gaps change, and that is a clue rather than a dead end.
Take 2, 4, 8, 16. The gaps are +2, +4, +8, which are not equal. So try multiplying instead: each term is twice the one before. The next term is 32.
Take 2, 4, 7, 11. The gaps are +2, +3, +4. They are not equal either, but they grow by one each time. So the next gap is +5 and the next term is 16.
Always check at least three gaps before deciding on a rule. Two terms can fit almost any rule, so a pattern guessed from a single gap is usually wrong.
3. Missing terms
To fill a gap in the middle, work out the rule from the terms you do have.
In 5, __, 15, 20 the terms rise by 5, so the missing one is 10.
In 50, 45, __, 35, __ the rule is subtract 5, so the missing terms are 40 and 30.
4. Odd and even
Even numbers can be split into two equal whole parts: 2, 4, 6, 8, 10. Odd numbers cannot: 1, 3, 5, 7, 9.
These combine in a fixed way, and the results never vary.
| Sum | Result | Example |
|---|---|---|
| odd + odd | even | 3 + 5 = 8 |
| even + even | even | 4 + 6 = 10 |
| odd + even | odd | 3 + 4 = 7 |
Multiplication has its own pattern. odd × odd stays odd, as in 3 × 5 = 15. But even × anything is even, because an even factor already supplies the pair.
5. Patterns hiding in the tables
The multiplication tables are full of patterns, and noticing them makes the tables easier to hold.
- The table of 5 always ends in 0 or 5.
- The table of 10 always ends in 0.
- In the table of 9 up to 90, the two digits always add to 9: 18 gives 1 + 8, 27 gives 2 + 7, 81 gives 8 + 1.
- The table of 2 is exactly the even numbers.
6. Growing shape patterns
Patterns are not only numbers. Build separate triangles from matchsticks: one triangle takes 3, two take 6, three take 9.
The number of sticks goes 3, 6, 9, 12, which is the table of 3. A shape pattern has turned into a number pattern, and that is what makes it possible to answer "how many for ten triangles" without building them.
7. Worked examples
Example 1. Find the next three terms: 5, 10, 15.
The gap is +5 each time, so the sequence continues 20, 25, 30.
Example 2. Find the next three terms: 3, 6, 12.
The gaps are +3 and +6, which are unequal, so try multiplying. Each term doubles, giving 24, 48, 96.
Example 3. Fill the missing terms: 1, 2, 4, __, 16, __.
Each term doubles, so the missing terms are 8 and 32.
Example 4. Is 7 + 9 odd or even? Answer without adding.
Both are odd, and odd + odd is always even. So the answer is even, and indeed 7 + 9 = 16.
8. Practice
- Find the next three terms: 80, 70, 60.
- Find the next three terms: 100, 95, 90.
- Find the missing terms: 4, 8, __, 16, 20.
- Is 6 + 8 odd or even? Say how you know without adding.
- Give the rule for 4, 8, 16, 32.
- In the table of 9, what do the digits of 63 add up to?
- How many matchsticks are needed for six separate triangles?
9. Answers
- 50, 40, 30.
- 85, 80, 75.
-
- Even, because even + even is always even.
- Multiply by 2 each time.
- 6 + 3 = 9.
- 6 × 3 = 18.
10. Summary
- Find a rule by writing the gaps between terms; equal gaps mean add or subtract.
- Unequal gaps mean try multiplying, or look for gaps that grow in their own pattern.
- Check three gaps before settling on a rule.
- odd + odd = even, even + even = even, odd + even = odd; even × anything = even.
- The table of 5 ends in 0 or 5, and the digits of the table of 9 add to 9.
- A growing shape pattern can always be turned into a number pattern.
