Patterns — Class 5 Mathematics (Samacheer Kalvi)
TN State Board (Samacheer Kalvi) Class 5 Mathematics, Unit 3. Patterns in shapes and patterns in numbers. Taught in July.
1. What a pattern is
Patterns are regular and repeated arrangements of colours, shapes, designs or lines on a surface.
Patterns are everywhere — in a kolam at the doorstep, in the border of a saree, in floor tiles, in the petals of a flower. Once you find the rule, you can continue the pattern for ever.
In a shape pattern the rule might be about the shape itself, its colour, its size, or the direction it faces. Read the first few terms carefully before deciding.
2. Square numbers
To square a number, multiply it by itself.
| Calculation | Square number |
|---|---|
| 1 × 1 | 1 |
| 2 × 2 | 4 |
| 3 × 3 | 9 |
| 4 × 4 | 16 |
| 5 × 5 | 25 |
| 6 × 6 | 36 |
These are called square numbers for a reason you can see. Take that many dots, or bindis, or seeds, and you can arrange them into a perfect square with no gaps and none left over.
Joshua tried to make a square with 12 bindis. He could not — there were gaps. So 12 is not a square number, because no whole number multiplied by itself gives 12.
A square number is always positive.
3. Triangular numbers
Now arrange dots in a triangle instead.
| Triangle | Dots added | Total |
|---|---|---|
| 1st | 1 | 1 |
| 2nd | 2 | 1 + 2 = 3 |
| 3rd | 3 | 1 + 2 + 3 = 6 |
| 4th | 4 | 1 + 2 + 3 + 4 = 10 |
| 5th | 5 | 1 + 2 + 3 + 4 + 5 = 15 |
So the triangular numbers are 1, 3, 6, 10, 15, 21, ...
To find the next one, add a new row of dots at the bottom, one longer than the last row, and count everything.
4. Where the two families meet
Here is something worth noticing.
Add two consecutive triangular numbers and you always get a square number.
- 1 + 3 = 4
- 3 + 6 = 9
- 6 + 10 = 16
- 10 + 15 = 25
Every answer is a square. It makes sense once you picture it: two triangles of dots, one upside down, fit together exactly into a square.
5. Square numbers and odd numbers
There is a second surprise, this time involving the odd numbers 1, 3, 5, 7, ...
Add up the odd numbers from 1, and the running total is always a square number.
| Sum of odd numbers | Total | As a square |
|---|---|---|
| 1 | 1 | 1 × 1 |
| 1 + 3 | 4 | 2 × 2 |
| 1 + 3 + 5 | 9 | 3 × 3 |
| 1 + 3 + 5 + 7 | 16 | 4 × 4 |
| 1 + 3 + 5 + 7 + 9 | 25 | 5 × 5 |
| 1 + 3 + 5 + 7 + 9 + 11 | 36 | 6 × 6 |
Count how many odd numbers you added, and that count is the number you square. Six odd numbers give 6 × 6 = 36.
So the difference between one square number and the next is always an odd number: 4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7.
6. Patterns in shapes and angles
Angles have names according to their size, and knowing the names lets you describe a pattern of shapes precisely.
| Angle | Measure |
|---|---|
| Zero angle | 0° |
| Acute angle | less than 90° |
| Right angle | exactly 90° |
| Obtuse angle | between 90° and 180° |
| Straight angle | 180° |
The three angles of an equilateral triangle are equal to one another, and together they make a straight angle. So each of them is 180° ÷ 3 = 60°.
That one fact unlocks other shapes. Six equilateral triangles fit exactly around a point inside a regular hexagon. The angle at the centre is a full turn of 360°, and each corner of the hexagon is made of two 60° angles, so it measures 120°.
7. Rotating angles
Turn a shape about a fixed point and it sweeps out an angle. A quarter turn is 90°, a half turn is 180°, three quarters is 270°, and a full turn of 360° brings the shape back exactly where it began.
Rotating one shape again and again about a centre builds a pattern. Place a triangle, turn it 60°, place it again, and keep going. After six placements the triangle has come full circle, because 6 × 60° = 360°, and the six copies together form a rosette.
This is how the rangoli patterns drawn at doorways are built, and why they always look balanced: every copy is the same shape, set at an equal turn from the last.
8. Word patterns
A pattern need not be made of numbers or shapes. A sequence of words can follow a rule just as strictly.
- BUS, BUST, BURST: one letter is inserted each time.
- AZ, BY, CX, DW: the first letter moves forward through the alphabet while the second moves backward.
- MADAM, LEVEL, ROTOR: each word reads the same in both directions.
To find the rule, compare one term with the next and ask what single change turns the first into the second. Then test that the same change also carries the second into the third. A rule that explains only one step is a coincidence, not a pattern.
9. Worked examples
Example 1. What is the square of 7?
Solution: 7 × 7 = 49.
Example 2. 64 is the square of which number?
Solution: 8 × 8 = 64, so 8.
Example 3. Is 24 a square number?
Solution: No. 4 × 4 = 16 and 5 × 5 = 25, so no whole number squared gives 24.
Example 4. Write the first six triangular numbers.
Solution: 1, 3, 6, 10, 15, 21.
Example 5. Find 1 + 3 + 5 + 7 + 9 + 11 + 13.
Solution: There are 7 odd numbers, so the total is 7 × 7 = 49.
Example 6. Add the consecutive triangular numbers 10 and 15. What kind of number is the answer?
Solution: 10 + 15 = 25, which is a square number, 5 × 5.
Example 7. What is the next triangular number after 15?
Solution: Add a row of 6 dots: 15 + 6 = 21.
10. Practice
- What is the square of 9?
- Circle the square numbers: 4, 6, 9, 12, 16, 20, 25.
- Is 30 a square number? Explain.
- Write the first five triangular numbers.
- What is the next triangular number after 21?
- Find 1 + 3 + 5 + 7 + 9 without adding them one by one.
- Add the consecutive triangular numbers 6 and 10. What do you get?
- What is the difference between 36 and 25, and what kind of number is it?
- What is the measure of each angle of an equilateral triangle?
- Through what angle must a shape be turned if a full turn is made in 8 equal steps?
- Write the next term: AZ, BY, CX, ____.
11. Answers
- 9 × 9 = 81.
- 4, 9, 16 and 25.
- No. 5 × 5 = 25 and 6 × 6 = 36, so no whole number squared gives 30.
- 1, 3, 6, 10, 15.
- Add a row of 7 dots: 21 + 7 = 28.
- Five odd numbers, so the total is 5 × 5 = 25.
- 16, which is a square number, 4 × 4.
- 36 − 25 = 11, an odd number. The gap between consecutive squares is always odd.
- 180° ÷ 3 = 60°.
- 360° ÷ 8 = 45°.
- DW. The first letter moves forward one place and the second moves back one place.
12. Summary
- A pattern is a regular, repeated arrangement of colours, shapes, designs or lines.
- A square number is a number multiplied by itself: 1, 4, 9, 16, 25, 36.
- Square numbers can be laid out as a perfect square of dots; 12 cannot, so it is not one.
- Triangular numbers come from stacking rows of dots: 1, 3, 6, 10, 15, 21.
- Each new triangular number adds a row one dot longer than the last.
- Two consecutive triangular numbers always add to a square number.
- The sum of the first few odd numbers is always a square, and the count of them is the number squared.
- The gap between consecutive square numbers is always an odd number.
