Chapter Notes: Patterns & Symmetry

What is Pattern?

A pattern is a design or arrangement that repeats regularly and predictably. Patterns can be made by repeating shapes, lines, colours or numbers. We can also make patterns by drawing a shape and its mirror image placed next to each other.

What is Symmetry?

Symmetry means that an object can be divided into two identical halves that match exactly in size and shape. The straight line that divides an object into two identical halves is called the line of symmetry. Objects that can be divided this way are called symmetrical figures.

Look carefully at the pictures below and the dotted lines. If you fold each picture along the dotted line, one half will exactly cover the other half. This shows that the picture has a line of symmetry.

If an object can be folded so that both halves fit on top of each other, it is symmetrical. The line along which it is folded is the line of symmetry. For example, the line passing through the middle of a flower may be its line of symmetry.

Not all objects are symmetrical. Objects that do not divide into two identical halves are called non-symmetrical or asymmetrical objects.

Reflection - Like a Mirror Image

When you look in a mirror, you see your reflection. The mirror shows a reversed copy of the original object. If one half of a shape is the mirror image of the other half, we say the shape has reflectional symmetry or mirror symmetry.

Many letters and numbers also have mirror images. The mirror image shows how the shape looks when reflected across the line of symmetry.

Symmetry in Some Geometrical Shapes

1. A square has four lines of symmetry.
2. A rectangle has two lines of symmetry.
3. An isosceles triangle has one line of symmetry.
4. An equilateral triangle has three lines of symmetry.

Symmetry in Letters and Numbers

Some capital letters and numbers have one or more lines of symmetry. Others do not have any line of symmetry. Learning which letters and numbers are symmetrical helps us recognise mirror images.

However, there are some letters and numbers which are not symmetrical.

Patterns

Patterns are shapes, designs or sequences of numbers that repeat predictably. Patterns help us to make guesses about what comes next and to see regularity in shapes and numbers.

Patterns in Addition

Patterns of the sum of three consecutive numbers
1 + 2 + 3 = 6, 6 is a multiple of 3 and 3 × 2 = 6.
2 + 3 + 4 = 9, 9 is a multiple of 3 and 3 × 3 = 9.
3 + 4 + 5 = 12, 12 is a multiple of 3 and 3 × 4 = 12.
4 + 5 + 6 = 15, 15 is a multiple of 3 and 3 × 5 = 15.
5 + 6 + 7 = 18, 18 is a multiple of 3 and 3 × 6 = 18.

Rule: The sum of three consecutive numbers is always a multiple of 3. The sum equals three times the middle number.

Patterns in Multiplication

I. Multiplication of a number ending in 5 by itself

When a number ending in 5 is multiplied by itself, the product always ends in 25. The other digits come from multiplying the part left of 5 by its next higher number.

Rule: Every such product ends in 25. The digits before 25 are found by multiplying the number formed by the digits to the left of 5 with the next higher whole number.

II. Multiplying a number made of all 1s by itself

We have,
1 × 1 = 1
11 × 11 = 121
111 × 111 = 12321
1111 × 1111 = 1234321

Rule: The product builds up with digits increasing to the middle and then decreasing; the middle digits often show the sum of the digits of the factor.

III. Multiplying pairs like 19 and 21

19 × 21 = 399 = 400 - 1 = 20 × 20 - 1
29 × 31 = 899 = 900 - 1 = 30 × 30 - 1
39 × 41 = 1599 = 1600 - 1 = 40 × 40 - 1
Observing the above pattern, we can write
49 × 51 = 50 × 50 - 1 = 2500 - 1 = 2499    
59 × 61 = 60 × 60 - 1 = 3600 - 1 = 3599

IV. To multiply a number by 11

There are quick methods to multiply by 11 that show a pattern in digits. See the pictures below for examples and practice.

Patterns in Division

Observe the following patterns to see how the quotient changes when the dividend or divisor changes.

These patterns show that, with the divisor fixed, increasing the dividend increases the quotient and decreasing the dividend decreases the quotient.

Now keep the dividend the same and change the divisor:

As the divisor increases (keeping the dividend same), the quotient becomes smaller. As the divisor decreases, the quotient becomes larger.
Find the missing number in the sequence: 3125, 625, 125, __, 5, 1
The rule is: each number is divided by 5 to get the next number. So the missing number is 25.

Geometrical Patterns Based on Symmetry

Many decorative and traditional patterns use symmetry to look balanced and beautiful. These patterns often repeat shapes and colours so that halves match when folded along a central line.

Rangoli patterns are drawn during festivals. Some rangoli designs are symmetrical - one half is the mirror image of the other half.

The pattern above is symmetrical. If folded in half along the middle, the lines and colours in each half match.

Quilt patterns are another example. Some quilt blocks are symmetrical, and others are not. If a pattern is not symmetrical, folding it in half will not make the two halves match exactly.

The quilt pattern shown here is not symmetrical; when folded, some lines and colours do not match.

How to check symmetry: Fold the figure along a straight line (or imagine folding), or place a mirror on the line you think is a line of symmetry. If both halves match exactly, the shape has that line of symmetry.