Patterns — Class 4 Mathematics (Samacheer Kalvi)
TN State Board (Samacheer Kalvi) Class 4 Mathematics, Unit 3. Patterns in shapes and patterns in numbers. Taught in July.
1. Patterns in shapes
Hold a kaleidoscope to your eye. It is a tube with loose pieces of coloured glass or paper at one end and mirrors inside. Rotate the tube and the picture changes — yet it is never random. The mirrors reflect each piece several times, so whatever falls into the tube comes back as a balanced, repeating design.
That is the idea behind every shape pattern: something repeats under a rule. Once you find the rule, you can say what comes next without being told.
A spirograph is another toy that does this. A toothed wheel rolled inside a ring traces loops that come back on themselves, making a pattern from nothing but a rule about turning.
2. Multiples of 9 and their digit sum
Now for the most useful pattern in this unit.
Multiply any number by 9 and add up the digits of the answer. Keep adding until one digit is left. You will always land on 9.
| Product | Digit sum | Reduce again |
|---|---|---|
| 84 × 9 = 756 | 7 + 5 + 6 = 18 | 1 + 8 = 9 |
| 43 × 9 = 387 | 3 + 8 + 7 = 18 | 1 + 8 = 9 |
| 123 × 9 = 1107 | 1 + 1 + 0 + 7 = 9 | 9 |
| 9 × 9 = 81 | 8 + 1 = 9 | 9 |
| 81 × 9 = 729 | 7 + 2 + 9 = 18 | 1 + 8 = 9 |
This gives a test that works both ways: if the digits of a number add up to 9, that number is a multiple of 9.
3. Casting out nines
Because of that rule, you can check whether a big number is divisible by 9 without dividing at all. Add its digits — and any digit or group of digits that already adds to 9 can simply be thrown away, or cast out.
Is 46908 a multiple of 9? Cast out the 9. What remains is 4 + 6 + 0 + 8 = 18, and 1 + 8 = 9. So yes, 46908 is a multiple of 9.
The same trick checks your arithmetic. Cast out nines on both sides of a sum; if the two sides do not match, the answer is wrong somewhere.
Check 4355 + 5369 = 9724. Left: (4+3+5+5) = 17 → 8, and (5+3+6+9) = 23 → 5. Together 8 + 5 = 13 → 4. Right: 9 + 7 + 2 + 4 = 22 → 4. Both give 4, so the addition passes the check.
4. The reversal pattern
Take any two-digit number, reverse its digits, and subtract the smaller from the larger. The difference is always a multiple of 9.
| Number | Reversed | Difference | Digit sum |
|---|---|---|---|
| 92 | 29 | 92 − 29 = 63 | 6 + 3 = 9 |
| 38 | 83 | 83 − 38 = 45 | 4 + 5 = 9 |
| 71 | 17 | 71 − 17 = 54 | 5 + 4 = 9 |
5. Multiplying by 10 and 100
Multiplying by 10 puts one zero on the end. Multiplying by 100 puts two.
- 57 × 10 = 570
- 57 × 100 = 5700
- 9 × 400 = 3600
- 8 × 700 = 5600
For the last two, multiply the leading digits and then attach the zeros: 9 × 4 = 36, and 400 carries two zeros, so 9 × 400 = 3600.
6. Magic squares
A magic square is a square of numbers in which every row, every column and both diagonals add to the same total.
| 2 | 9 | 4 |
|---|---|---|
| 7 | 5 | 3 |
| 6 | 1 | 8 |
Check a row: 2 + 9 + 4 = 15. A column: 2 + 7 + 6 = 15. A diagonal: 2 + 5 + 8 = 15. Every line gives 15.
Here is a second one whose magic total is 30:
| 13 | 6 | 11 |
|---|---|---|
| 8 | 10 | 12 |
| 9 | 14 | 7 |
7. Number sequences
A sequence has a rule too — usually "add the same amount each time".
- 90, 180, 270, 360, 450, 540 (add 90; these are the multiples of 9 grown ten times)
- 125, 150, 175, 200, 225, 250 (add 25)
- 100, 400, 700, 1000, 1300, 1600 (add 300)
And one beautiful staircase built on 9:
- 9 × 6 = 54
- 9 × 66 = 594
- 9 × 666 = 5994
- 9 × 6666 = 59994
Each new 6 pushes another 9 into the middle of the answer.
8. Worked examples
Example 1. Is 24689 a multiple of 9?
Solution: 2 + 4 + 6 + 8 + 9 = 29, and 2 + 9 = 11, and 1 + 1 = 2. The result is 2, not 9, so it is not a multiple of 9.
Example 2. Is 23769 a multiple of 9?
Solution: Cast out the 9. Then 2 + 3 + 7 + 6 = 18, and 1 + 8 = 9. So yes.
Example 3. Complete: 90, 180, 270, ___, ___, ___.
Solution: The rule is add 90. The next terms are 360, 450, 540.
Example 4. Find 47 × 100.
Solution: Attach two zeros: 4700.
Example 5. A magic square reads 2 9 4 in the top row, 7 __ 3 in the middle row and 6 1 8 in the bottom row. Its magic total is 15. Find the missing centre number.
Solution: The middle row must also add to 15, so 7 + __ + 3 = 15. Since 7 + 3 = 10, the centre is 15 − 10 = 5.
Example 6. Complete the letter-and-number pattern: A9, B18, C27, D36, ___, ___.
Solution: Letters run in order and the numbers are multiples of 9, so E45, F54.
9. Practice
- Circle the multiples of 9: 25, 27, 35, 36, 45, 46, 54, 55.
- Use casting out nines to test whether 13476 is a multiple of 9.
- Complete: 125, 150, 175, ___, ___, ___.
- Find 6 × 800.
- Reverse 62 and subtract the smaller from the larger. Is the difference a multiple of 9?
- What is 9 × 66666?
- In the magic square 2 9 4 / 7 5 3 / 6 1 8, what do both diagonals add to?
- Complete: 100, 400, 700, ___, ___.
10. Answers
- 27, 36, 45, 54.
- 1 + 3 + 4 + 7 + 6 = 21, and 2 + 1 = 3. Not 9, so 13476 is not a multiple of 9.
- 200, 225, 250.
- 6 × 8 = 48, then attach two zeros: 4800.
- 62 − 26 = 36, and 3 + 6 = 9. Yes, it is a multiple of 9.
-
- 2 + 5 + 8 = 15 and 4 + 5 + 6 = 15. Both give 15.
- 1000, 1300.
11. Summary
- A pattern is anything that repeats under a rule; a kaleidoscope makes patterns by reflection.
- The digits of any multiple of 9 add up to 9 when reduced to a single digit.
- Casting out nines tests divisibility and checks addition, subtraction and multiplication.
- Reversing a two-digit number and subtracting always leaves a multiple of 9.
- Multiplying by 10 adds one zero; by 100, two zeros.
- In a magic square every row, column and diagonal share the same total — 15 for the 2 9 4 square.
- Sequences usually grow by adding a fixed amount; find that amount first.
