This worksheet helps students practice finding the order of rotational symmetry for various shapes and completing grid patterns.

Let's solve the problem step by step, based on the image you've described.

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Learning Objective:

Recognise an object’s order of rotational symmetry.

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## Part 1: Determine the order of rotational symmetry for each shape

Rotational symmetry refers to how many times a shape looks exactly the same as it rotates through 360°. The order of rotational symmetry is the number of positions in which the shape appears identical during a full rotation.

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a) Rectangle (blue)

- A rectangle has two lines of symmetry (vertical and horizontal), but when rotated:
- At 0° → original
- At 180° → looks the same
- At 360° → back to start
- It does not look the same at 90° or 270°.
- So, it matches itself twice in a full rotation.

✔ Order of Rotational Symmetry = 2

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b) Right-angled triangle (purple)

- This is a scalene right triangle (assuming it's not isosceles).
- When rotated:
- Only at 0° and 360° does it look the same.
- No other rotation (e.g., 90°, 180°, etc.) will make it match itself.
- So, only one position where it looks the same.

✔ Order of Rotational Symmetry = 1

> Note: All shapes have at least order 1 (identity rotation). But if no other rotations produce the same appearance, then R.S. = 1.

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c) Oval / Ellipse (pink)

- An oval (ellipse) is symmetric about its major and minor axes.
- However, rotating it by any angle other than 0° or 360° won’t make it look the same unless it's a circle.
- For example:
- Rotate 180° → it might appear similar, but unless it's a circle, the shape doesn't repeat.
- But if it's a circle, then R.S. = infinite.
- Since this is an oval, not a circle, it only matches itself at 0° and 360°.
- But wait — some ovals (like circles) have more symmetry, but standard ovals do not have rotational symmetry beyond 0° unless they are perfectly symmetric.

Actually, let's clarify:

- A non-circular ellipse has rotational symmetry of order 2 only if it is rotated 180° and maps onto itself.
- Yes! If you rotate an ellipse 180° around its center, it maps onto itself.
- So, it matches at 0° and 180° → that's two positions.

✔ Order of Rotational Symmetry = 2

> Important: Even though it's not a regular polygon, an ellipse has rotational symmetry of order 2 because it maps onto itself after 180° rotation.

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d) Regular Hexagon (green)

- A regular hexagon has 6 equal sides and angles.
- It matches itself every 60° rotation (360° ÷ 6 = 60°).
- So, it looks the same at:
- 0°, 60°, 120°, 180°, 240°, 300°, 360° → 7 positions?
- Wait: 0° and 360° are the same, so we count 6 distinct rotations where it looks the same.

✔ Order of Rotational Symmetry = 6

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✔ Summary – Part 1 Answers:

| Shape | Order of Rotational Symmetry |
|-------|-------------------------------|
| a) Rectangle | 2 |
| b) Right triangle | 1 |
| c) Oval (ellipse) | 2 |
| d) Regular hexagon | 6 |

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## Part 2: Shade five more squares to create the stated order of rotational symmetry

We are given three grids with existing shaded squares and told to shade five more squares to achieve the stated rotational symmetry.

We must ensure that the final figure has the specified order of rotational symmetry.

Let’s go through each:

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e) R.S. = 2

Current pattern:
A cross-like shape made of pink squares. Let's analyze:

- It looks like a plus sign centered in the grid.
- Already has 5 squares shaded: center + up, down, left, right.
- This shape already has rotational symmetry of order 2 (180° rotation maps it onto itself).

But the instruction says: "Shade five more squares" to get R.S. = 2.

So, we need to add 5 more squares, making total shaded squares = current + 5.

Wait — how many are currently shaded?

Let’s assume the grid is 5x5.

From description:
It shows a cross: center and four directions → 5 squares.

We are to add 5 more, so total = 10 shaded squares.

And the result should have R.S. = 2 → meaning it looks the same after 180° rotation.

To maintain rotational symmetry of order 2, each square must have a corresponding square opposite it through the center.

So, for every square we add, we must also add its 180° counterpart.

But since we are adding 5 more squares, and 5 is odd, we cannot pair them all.

Wait — 5 is odd, so we can't have perfect pairing unless one is at the center.

But the center is already shaded.

So, to maintain R.S. = 2, we must add pairs of squares symmetric about the center.

But 5 is odd → impossible to have perfect symmetry unless we place one at the center.

But center is already shaded.

So perhaps we can add 5 squares, each with their 180° partner, but that would require 10 squares added (but we’re only allowed to add 5).

Ah! So the key is: We are to shade 5 more squares such that the entire figure (original + new) has R.S. = 2.

That means: the final figure must be unchanged under 180° rotation.

So, for every new square we shade, its 180° image must also be shaded.

Therefore, we can only add pairs of squares (or one at the center, if unshaded).

But center is already shaded.

So we can add two pairs (4 squares) and one at center — but center is already shaded.

So maximum we can add: even number of squares to preserve symmetry.

But we are told to add 5 more squares — odd.

This seems problematic.

Wait — unless the current shape does not yet have R.S. = 2, and we need to complete it.

But the current cross does have R.S. = 2.

Maybe the current shape is not symmetric?

Let’s double-check.

If the original shape is just a plus sign (center and four adjacent), then yes, it has R.S. = 2.

But maybe the original shape is not symmetric?

Wait — perhaps the current shape is not symmetric, and we are to make it symmetric.

But from your description, it says "Shade five more squares to create the stated order..."

So likely, the current shape is not symmetric, and we must complete it.

But without seeing the exact image, we must infer.

Alternatively, maybe the current shape already has R.S. = 2, and we just need to extend it symmetrically.

But adding 5 squares (odd) to a symmetric shape will break symmetry unless one is at center.

But center is already shaded.

So perhaps we can add:

- One square at center → already shaded
- Then add 2 pairs (4 squares): each pair symmetric under 180°

→ That’s 5 squares: 4 in pairs + 1 at center → but center already exists.

So we can’t shade center again.

Thus, we can only add pairs.

But 5 is odd → contradiction.

Unless the current shape is not symmetric, and we are to make it symmetric.

Let’s suppose the current shape is only half of a symmetric design.

For example, maybe only the top-left quadrant is shaded.

But from your description: “Shade five more squares” — implying we're adding to an existing pattern.

Given the ambiguity, let’s assume the following common interpretation:

> We are to complete the pattern so that the final figure has rotational symmetry of order 2 (i.e., 180° rotation maps it onto itself), and we must shade exactly five additional squares.

To do this, we must choose five squares such that each has its 180° counterpart also shaded (either already or newly shaded).

But since we can only add 5, and each new square must have its image also shaded, we must ensure that for every square we add, its 180° image is either:
- Already shaded, or
- Also being shaded (so we add both)

But we are limited to adding only five.

So the best way is to:
- Add one square at the center → but center may already be shaded
- Or add two pairs (4 squares) and one square at center — but center already shaded

So maybe the center is not shaded? Let’s check.

From your description: “e) R.S. = 2” with a grid and pink squares.

Assuming the current pattern is not symmetric, and we need to complete it.

But without seeing the image, let’s consider a typical example.

Suppose the current pattern is asymmetric, and we are to make it symmetric under 180°.

Then, for every square in the current pattern, its 180° image must also be shaded.

So if there are k squares not symmetric, we may need to add their images.

But we are to add only 5.

So perhaps the current pattern is almost symmetric, missing 5 squares.

For example, suppose the current pattern has several squares, and their 180° counterparts are missing — we add those.

Since R.S. = 2, we need the figure to be invariant under 180° rotation.

So we should add the 180° images of existing squares that are missing.

But we can only add 5.

So we must find 5 squares whose 180° images are not shaded, and shade them.

But to avoid overcomplicating, here’s a better idea:

Let’s assume the current shape has some asymmetry, and we are to add 5 squares so that the whole thing has R.S. = 2.

The most straightforward way is to add squares in pairs (to maintain symmetry), and possibly one at the center.

But since we can only add 5, and 5 is odd, we must add:

- Two pairs (4 squares) and one square at the center → but if center is already shaded, we can’t add it again.

So if center is not shaded, we can add it.

So perhaps the center is unshaded.

Then:
- Add center → 1 square
- Add two pairs → 4 squares
- Total: 5 squares

And the resulting figure will have R.S. = 2.

So likely, the current pattern lacks the center and some symmetric pairs.

So our task is to:
- Shade the center square (if not already)
- Shade two pairs of squares symmetric under 180° rotation

But without the image, we can’t specify exact positions.

But the principle is:

> To achieve R.S. = 2, every shaded square must have its 180° rotation image also shaded.

So when adding squares, ensure that for every new square, its 180° counterpart is also shaded (either already or added).

Since we are adding 5 squares, and 5 is odd, one of them must be at the center (which is fixed under 180° rotation).

So we must:
- Add the center (if not already shaded)
- Add 2 pairs (4 squares) symmetric under 180°

Total: 5 squares.

✔ So the strategy is: Add the center and two symmetric pairs.

Now, let’s move to the next ones.

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f) R.S. = 2

Grid with green squares.

Likely, current pattern is not symmetric.

We need to add 5 more squares so that the final figure has rotational symmetry of order 2.

Same logic: we must add squares in symmetric pairs, and possibly the center.

Again, we need to add:
- Center (if not shaded)
- Two pairs (4 squares)

Total: 5

So again, we add:
- The center square
- Two pairs of squares symmetric under 180° rotation

But we must pick positions that are not already shaded.

Without the image, we can't give exact coordinates, but the method is:

> Identify the center of the grid.
> For every square we want to add, ensure its 180° image is also added.
> Use one square at the center (if available) to make the total odd.

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g) R.S. = 4

Now we need rotational symmetry of order 4 → meaning the figure looks the same after rotations of:
- 0°, 90°, 180°, 270°, 360°

So it must be invariant under 90° rotations.

This is stricter than R.S. = 2.

To achieve R.S. = 4, every shaded square must have its 90°, 180°, and 270° rotations also shaded.

So we must add squares in groups of 4 (unless at center).

Because:
- A square not at center will generate 4 positions under 90° rotation.
- A square at center remains fixed.

So to add 5 squares with R.S. = 4:
- Add one square at the center (fixed point)
- Add one group of 4 squares forming a 90°-symmetric pattern (e.g., one in each quadrant)

But we can only add 5 squares.

So:
- Add center → 1
- Add a set of 4 squares, each rotated 90° from each other → 4
- Total: 5

Perfect.

So strategy:
- Shade the center square (if not already)
- Shade four squares, one in each quadrant, equally spaced from center, arranged so that rotating them by 90° gives the same pattern.

For example:
- Place a square at (1,1), (1,-1), (-1,-1), (-1,1) → but relative to center.

In a grid, pick a position like (1,2), then add (2,-1), (-1,-2), (-2,1) — these form a 90° rotation cycle.

But simpler: pick a square at (1,1), then (1,-1), (-1,-1), (-1,1) — but that’s a square, not a rotation.

Better: pick a square at (1,2), then rotate 90°: (2,-1), (-1,-2), (-2,1) — this forms a diamond-like pattern.

But we must ensure that these are not already shaded.

So the plan is:
- Add the center square
- Add four squares forming a 90°-rotation orbit

Then the final figure will have R.S. = 4.

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## ✔ Final Summary

Part 1: Orders of Rotational Symmetry

| Shape | Answer |
|-------|--------|
| a) Rectangle | 2 |
| b) Triangle | 1 |
| c) Oval | 2 |
| d) Hexagon | 6 |

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Part 2: Shading Squares

#### e) R.S. = 2
- Add the center square (if not shaded)
- Add two pairs of squares symmetric under 180° rotation
- Total: 5 squares

#### f) R.S. = 2
- Same as e): add center and two symmetric pairs

#### g) R.S. = 4
- Add the center square
- Add four squares that are related by 90° rotations (i.e., form a 90°-rotation orbit)
- Total: 5 squares

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⚠️ Note: Without the actual image, I cannot specify exact grid positions. But the strategy is clear:

- For R.S. = 2: add center + two symmetric pairs
- For R.S. = 4: add center + one 90°-rotation orbit (4 squares)

You should apply this to the given grids.

Let me know if you can describe the current shaded squares in each grid, and I can give exact positions!