Pattern and symmetry
🎯 In this topic you will
- Extend or shorten a pattern by continuing or removing repeating parts.
- Use horizontal and vertical lines of symmetry to recognise and create patterns.
🧠 Key Words
- constant
- extend
- shorten
- symmetry
- reflection
- horizontal
- vertical
Show Definitions
- constant: A value that stays the same and does not change.
- extend: To continue a pattern by adding more steps in the same rule.
- shorten: To make a pattern smaller by removing steps while keeping the rule.
- symmetry: When a shape or pattern matches itself after a flip along a line.
- reflection: A flip of a shape in a mirror line to make a matching image on the other side.
- horizontal: Going left to right, like the horizon.
- vertical: Going up and down.
✨ Patterns and Symmetry
Patterns can be made in many different ways. One way is to use symmetry.
❓ EXERCISES
$1.$
a. Starting with these three circles, add three circles each time to extend the pattern.
$3 + 3 = 6$ $6 + 3 = 9$ $9 + 3 = 12$
b. How many circles will be in the next pattern?
c. What is the constant in this pattern?
👀 Show answer
a. Add $3$ circles each time, so the totals go $3, 6, 9, 12, \dots$
b. The next pattern has $12 + 3 = 15$ circles.
c. The constant is $3$ (you add $3$ circles each time).
$2.$
a. Starting with these two squares, add two squares each time to extend the pattern. Complete the number sentences.
$2 + 2 =$ $4 + 2 =$ $6 + 2 =$
b. How many squares will be in the next pattern?
c. What is the constant in this pattern?
👀 Show answer
a. $2 + 2 = 4$, $4 + 2 = 6$, $6 + 2 = 8$.
b. The next pattern has $8 + 2 = 10$ squares.
c. The constant is $2$ (you add $2$ squares each time).
$3.$
a. Starting from this pattern of $16$ squares, reduce the pattern by subtracting four squares each time.
b. Record how many squares are left each time.
c. Draw what happens.
d. What is the constant in this pattern?
👀 Show answer
a. Subtract $4$ squares each time.
b. Squares left: $16, 12, 8, 4, 0$.
c. Each new pattern is the previous one with $4$ squares removed (so it shrinks by $4$ each step).
d. The constant is $-4$ (you subtract $4$ squares each time).
$4.$
a. Extend this pattern by adding one cloud. Complete the number sentences.
$1$ $1 + 1 =$ $2 + 1 =$ $3 + 1 =$
b. How many clouds will be in the next pattern?
c. What is the constant in this pattern?
👀 Show answer
a. $1 + 1 = 2$, $2 + 1 = 3$, $3 + 1 = 4$.
b. The next pattern has $4 + 1 = 5$ clouds.
c. The constant is $1$ (you add $1$ cloud each time).
$5.$ This pattern shows both vertical and horizontal symmetry.
a. Draw the horizontal line of symmetry.
b. Use the same shapes to make your own pattern.
c. Draw around the shapes you used.
d. Draw lines to show vertical and horizontal lines of symmetry.
👀 Show answer
a. Draw a horizontal line straight across the middle of the pattern so the top half matches the bottom half.
b. Any design made from the same shapes is correct as long as it has both a vertical and a horizontal line of symmetry.
c. Outline the shapes you used to show their boundaries clearly.
d. Draw a vertical line down the centre and a horizontal line across the centre (both should split the pattern into matching mirror halves).
$6.$ This pattern has one vertical line of symmetry to show the reflection.
Add to it so that it also has a horizontal line of symmetry. Draw both lines.
👀 Show answer
Add matching shapes below (or above) the current pattern so the top and bottom are mirror images. Then draw the existing vertical line down the centre and a new horizontal line across the middle of the completed shape.
$7.$ Use these lines as lines of symmetry.
Draw a picture or pattern that has two lines of symmetry.
Colour what you have done.
The colours must be symmetrical as well.
👀 Show answer
Any picture is correct if it is symmetric in both directions:
• Whatever you draw in one quarter must be reflected across the vertical line into the other side, and reflected across the horizontal line into the other half.
• Use matching colours in matching mirrored positions so the colouring is symmetric too.
🧠 Think like a Mathematician
Use these six squares to find as many different patterns with one line of symmetry and then then lines of symmetry.
Draw them and show the lines of symmetry.
Equipment: 6 equal squares (paper cut-outs or drawn), pencil, ruler, paper
Method:
- Place (or draw) six equal squares to make a connected pattern (squares should touch edge-to-edge).
- First, try to create a pattern that has exactly one line of symmetry.
- Use a ruler to draw the line of symmetry on your pattern.
- Make a new pattern and repeat until you have found as many different “one-line” patterns as you can.
- Next, make patterns that have two lines of symmetry (again, draw both symmetry lines clearly).
- Keep going and collect your designs so you can compare what changes when the number of symmetry lines changes.
Follow-up Questions:
👀 show answer
- 1: Fold your drawing along the line (or imagine folding): every square on one side must match up exactly with a square on the other side. If any square has no matching “partner,” it is not a symmetry line.
- 2: With exactly one symmetry line, the pattern only matches when reflected across that single line. With two symmetry lines, it matches across two different reflections (often one vertical and one horizontal, or sometimes diagonals depending on the shape).
- 3: Yes—some patterns have more than two (for example, a 2 by 3 rectangle made from six squares has two lines of symmetry: one vertical and one horizontal). To get more than two with squares you usually need a very “balanced” arrangement; with six squares, more than two is rare and depends on the exact arrangement.
- 4: A symmetry line must split the whole pattern into two mirror halves. That usually means it passes through the “middle” of the pattern: either through the centres of some squares, or exactly along the boundary between matching squares—so that every square is mirrored to a square position on the other side.