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Year 4 Maths 2D Shapes & Symmetry

Create symmetrical patterns

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Theory

Creating symmetrical patterns is a Year 4 maths symmetry skill: completing a design by reflecting it across a line of symmetry. This page explains how to shade the mirror image on a grid across a vertical or horizontal line, why a square on the line is its own image, and how a reflection is different from a slide or a turn. In Western Australia this is taught in Year 5. Curriculum: VC2M4SP04 / AC9M4SP03.

A pattern is symmetrical when it has a line of symmetry: a line you could fold along so one half lands exactly on the other. The two halves are mirror images.

To build or finish a symmetrical pattern you reflect (flip) the shape across the line. A reflected square sits the same distance from the line, on the other side, in the same row for a vertical line or the same column for a horizontal line. A reflection keeps the same size and shape and only flips the direction.

A square sitting on the line is its own mirror image, so it needs no partner. A pattern can have a vertical line of symmetry, a horizontal line, or both at once. Reflecting is not the same as a slide (translation) or a turn (rotation) — only a reflection makes a mirror image.

A symmetrical pattern with a vertical line of symmetry A six by four grid. Navy squares on the left of a red dashed vertical mirror line are matched by red squares the same distance to the right, so the two halves are mirror images.
Each navy square on the left has a red mirror square the same distance across the dashed line.
Completing a symmetrical pattern across a horizontal line Before: a four by four grid with four navy squares shaded above a red dashed horizontal mirror line. After: the same grid with the four navy squares reflected to four red squares below the line, completing the pattern. Before After
Before and after: reflect the top squares to finish the pattern below the line.

Three moves can change where a shape is. All three keep the same size and shape.

MoveWhat happensMakes a mirror image?
Reflection (flip)The shape flips over a lineYes
Translation (slide)The shape slides to a new place, facing the same wayNo
Rotation (turn)The shape turns about a pointNo
\[ \text{total shaded} = 2 \times \text{one side} + \text{squares on the line} \]
Two lines of symmetry. With an upright and a horizontal line, every square you shade in one quarter gives a square in each of the four quarters.

How to complete a symmetrical pattern

  1. Find the line. Look for the line of symmetry and check whether it is upright (vertical) or across (horizontal).
  2. Measure each square. For every shaded square, count how many squares it is from the line.
  3. Reflect it across. Shade the matching square the same distance on the other side, in the same row for a vertical line or the same column for a horizontal line.
  4. Check the fold. Imagine folding along the line — every shaded square should land on a shaded square.

How to count the squares in a finished pattern

  1. Count one side. Count the shaded squares on one side of the line.
  2. Double it. The other side has the same number.
  3. Add the line squares. Squares sitting on the line are counted once only.
Example 1 — Squares on the line
The mirror line runs down the middle column. On the left there are \(4\) navy squares, and \(2\) squares sit on the line. How many squares are shaded in the finished pattern?
A pattern with squares on the mirror line A seven by four grid with a red dashed upright line through the middle column. Four navy squares on the left are matched by four red squares on the right. Two grey-blue squares sit on the line itself.
Solution

Each of the \(4\) left squares needs a mirror square on the right: \(4\) more.

A square on the line is its own mirror image, so the \(2\) line squares need no partner.

Total: \(4 + 2 + 4 = 10\) shaded squares.

Example 2 — Reflect across a horizontal line
The top half of a pattern is shaded. How many squares must be shaded below the dashed line to finish it?
The top half of a pattern above a horizontal mirror line A five by six grid with a red dashed line across the middle. Six navy squares are shaded above the line. The squares below the line are empty.
Solution

Each top square is reflected straight down, the same number of rows from the line, in the same column.

There are \(6\) shaded squares above the line, so shade \(6\) squares below it.

Example 3 — Find the mirror column
Columns are numbered \(1\) to \(8\). The mirror line is between columns \(4\) and \(5\). A square is shaded in column \(3\). Which column holds its mirror square?
Solution

Column \(3\) is the second column to the left of the line (columns \(4\), then \(3\)).

The mirror square is the second column to the right of the line (columns \(5\), then \(6\)).

Its mirror square is in column \(6\).

Example 4 — Two lines of symmetry
A pattern needs both an upright and a horizontal line of symmetry. You shade \(2\) squares in the top-left quarter. How many squares are shaded when the pattern is finished?
A pattern with two lines of symmetry A six by six grid with a red dashed upright line and a red dashed line across. Two navy squares in the top-left quarter are reflected into the other three quarters as red squares, eight shaded squares in all.
Solution

Each square is reflected across the upright line, then both are reflected across the horizontal line, so every square gives \(4\) squares (one in each quarter).

\(2\) squares \(\times\) \(4\) quarters \(= 8\) shaded squares.

Example 5 — A border pattern
A border is made by repeating a unit. Each unit has \(3\) shaded squares, and the border has \(4\) units in a row. How many shaded squares are there?
Solution

Each unit is a slide (translation) of the first one, so every unit has \(3\) shaded squares.

\(4 \times 3 = 12\) shaded squares.

Example 6 — Name the move
Name each move: reflection (flip), translation (slide) or rotation (turn).
(i) A picture of a boat moves \(4\) squares to the right and still faces the same way.
(ii) An arrow that pointed right now points down, after turning about its tail.
Solution

(i) The boat only moved along without turning or flipping: a translation (slide).

(ii) The arrow turned about a point: a rotation (turn).

Common pitfalls

Reflect, do not slide. Watch out for copying a square straight across. A mirror square is flipped over the line, so the square nearest the line matches the square nearest the line on the other side.
Keep the distance equal. A square \(2\) columns from the line reflects to a square \(2\) columns from the line on the other side, not \(1\) or \(3\).
A reflection keeps the size. Watch out for making the mirror image bigger or smaller. A reflection only flips the direction, so both halves match exactly.

Frequently asked questions

What is a line of symmetry?

A line of symmetry is a line you could fold along so one half of the pattern lands exactly on the other half. The two halves are mirror images.

How do you complete a symmetrical pattern?

Reflect each shaded square across the line of symmetry. Its mirror square sits the same distance from the line on the other side, in the same row for a vertical line or the same column for a horizontal line.

What is a reflection in maths?

A reflection is a flip of a shape over a line. It makes a mirror image that is the same size and shape, with only the direction reversed.

What is the difference between a reflection, a translation and a rotation?

A reflection is a flip over a line, a translation is a slide to a new place, and a rotation is a turn around a point. Only a reflection makes a mirror image.

Can a pattern have more than one line of symmetry?

Yes. A pattern can have a vertical line, a horizontal line, or both at once. Shading one square then forces its mirror squares across every line of symmetry.

Does a reflection make a shape bigger?

No. A reflection keeps the same size and shape. It only flips the direction, so the mirror image matches the original exactly.

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