Fun with Symmetry · Learner pages

Lesson 1 · Test a mirror line

Explain line symmetry using a fold or reflection.

Name or initials: ____________   Date: ____________

A line of symmetry divides a shape into matching mirror halves.

1. Look at the whole shape. Start with a rectangle whose length is greater than its width.
2. Try a fold. Fold along the vertical line through its centre.
3. Match the points. Each point on one side has a matching point the same distance from the line on the other side.
4. Check the full shape. Every edge and point must match. This vertical centre line is a line of symmetry.

Worked example: A non-square rectangle has a vertical and a horizontal line of symmetry. Fold along either centre line: matching edges meet. A slanting diagonal is not a symmetry line.

1. A line of symmetry makes two halves that…

   1. fit exactly when folded · 2. only have the same colour · 3. always become circles

2. For a non-square rectangle, is a diagonal a line of symmetry?

   1. Yes, always · 2. No · 3. Only if it is coloured

3. A matching point across a mirror line is…

   1. always above it · 2. always closer to it · 3. equally far from the line

Discuss: Cut a paper rectangle with unequal adjacent sides. Test its two centre folds and one diagonal fold.

Lesson 2 · Complete a reflected pattern

Place a reflected point on the same row and at equal distance from a vertical mirror.

Name or initials: ____________   Date: ____________

For a vertical mirror, a point moves across the line while staying at the same height.

1. Find the original point. Point A is two grid spaces left of the vertical mirror.
2. Measure to the mirror. Count horizontally from A to the mirror: two spaces.
3. Copy the distance. Count two spaces to the right on the same row and place the image point.
4. Check a matching pair. The mirror lies halfway between the two points. Each is two spaces from it.

Worked example: A point is 3 grid spaces to the right of a vertical mirror. Its image is 3 spaces to the left on the same row. The two points are 6 spaces apart.

1. A point is 2 spaces left of a vertical mirror. Its image is…

   1. 2 spaces above · 2. 2 spaces right, same row · 3. 1 space right

2. A point lies directly on the mirror line. What happens to it?

   1. It disappears · 2. It moves one space · 3. It stays at the same position

3. A point is 4 spaces from a vertical mirror. How far apart are it and its image?

   1. 8 spaces · 2. 4 spaces · 3. 2 spaces

Discuss: Draw a vertical line on squared paper. Place three points on the left and reflect them one at a time.

Lesson 3 · Count symmetry lines

Find all symmetry lines of familiar regular and non-regular shapes.

Name or initials: ____________   Date: ____________

Test each proposed line by matching the entire shape, and count each distinct line once.

1. Try the vertical centre line. A square matches across its vertical centre line. That is one symmetry line.
2. Try the horizontal centre line. Its horizontal centre line also works. There are now two.
3. Test one diagonal. A square has equal sides, so this diagonal fold also matches. This is the third line.
4. Test the other diagonal. The other diagonal works too. These four are all the symmetry lines of a square.

Worked example: An equilateral triangle has 3 symmetry lines, each joining a vertex to the midpoint of the opposite side. A non-square rectangle has 2, and a square has 4.

1. How many lines of symmetry does a square have?

   1. 2 · 2. 3 · 3. 4

2. How many lines of symmetry does a non-square rectangle have?

   1. 2 · 2. 4 · 3. 0

3. How many lines of symmetry does an equilateral triangle have?

   1. 1 · 2. 3 · 3. 4

Discuss: Draw a square, a non-square rectangle and an equilateral triangle. Use folds to explain each count.

Final check · Set 1

Try one set after the lessons. Use the other set after reviewing any errors.

1. A symmetry line requires…

   exactly matching mirror halves / equal colour only / any two equal-area pieces

2. Which is a symmetry line of a non-square rectangle?

   Either diagonal / A centre line parallel to a side / Any line through a corner

3. A point is 3 spaces left of a vertical mirror. Place its image…

   3 spaces up / on the mirror / 3 spaces right, same row

4. Two reflected points are each 2 spaces from the mirror. Their separation is…

   4 spaces / 2 spaces / 1 space

5. A square has how many symmetry lines?

   2 / 4 / 6

6. An equilateral triangle has how many symmetry lines?

   0 / 2 / 3

Final check · Set 2

Try one set after the lessons. Use the other set after reviewing any errors.

1. A line cuts a shape into equal areas. Must it be a symmetry line?

   Yes / No / Only for coloured shapes

2. A diagonal of a non-square rectangle is…

   always a symmetry line / its only symmetry line / not a symmetry line

3. A point 5 spaces right of a vertical mirror reflects to…

   5 spaces left, same row / 5 spaces up / 10 spaces right

4. A point on the mirror maps to…

   a neighbouring row / itself / a missing point

5. A non-square rectangle has how many symmetry lines?

   1 / 4 / 2

6. Which shape has 4 symmetry lines?

   A square / An equilateral triangle / A non-square rectangle