Portal 2-Inspired Geometry Worksheet for 12-Year-Olds | Maths

Instructions

1. You are the designer of a Portal-style test chamber. Your mission is to investigate how shapes move and change position.
2. Work at your own pace. You may complete the sections in order, choose a different starting point, or return to a question later.
3. You may draw on squared paper, use a ruler, or make a coordinate grid. Label your work clearly.
4. In this worksheet, a transformation is a movement or change made to a shape. The main transformations are:
   • Translation: slides a shape without turning or flipping it.
   • Reflection: flips a shape across a mirror line.
   • Rotation: turns a shape around a fixed point.
5. Show your thinking when you can. There may be more than one sensible design for the open-ended tasks.

Get Curious

What is this? Why do people use it?

A test chamber uses portals to move objects from one place to another. In geometry, we can describe these movements exactly using coordinates and transformations.

1. Circle the best answer.

   A transformation is:

   • a) A measurement of how heavy an object is
   • b) A movement or change made to a shape
   • c) The number of sides in a shape
   • d) The distance around a shape

2. Match each transformation to its description. Write the correct letter beside each number.

   1. Translation: _____
   2. Reflection: _____
   3. Rotation: _____
   • A. A shape is flipped over a line.
   • B. A shape is turned around a fixed point.
   • C. A shape slides to a new position.

3. A square is moved 4 spaces to the right and 2 spaces up. Has its size changed? Explain your answer.

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4. Choose one:

   The transformation I would most like to test in a Portal chamber is:

   • Translation
   • Reflection
   • Rotation

   Why? _____

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Look Around

Where do I see this? How does it connect to my world?

Transformations appear in maps, video games, animations, logos, building designs, floor plans, and sports tactics.

1. Find or imagine four examples of shapes moving or changing in real life. One example has been completed.

2. Look at these situations. Write translation, reflection, or rotation.

   a) A logo is flipped to make its mirror image: ____

   b) A door swings around its hinges: ____

   c) A chess piece is moved three squares forward: ____

   d) A photograph is turned upside down: ____

3. Imagine a Portal test chamber on a coordinate grid. Why would coordinates be useful for placing a portal or moving a cube accurately?

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4. Draw a small object, such as a cube, arrow, or robot, in one position. Draw a translated copy beside it. Add arrows to show the movement.

   Drawing space:

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Find the Pattern

What is the big idea? What rules make it work?

On a coordinate grid, the horizontal position is x and the vertical position is y. A point is written as (x, y).

Example: The rule (x, y) → (x + 3, y - 2) moves every point 3 units right and 2 units down.

1. Complete the missing coordinates using the rule (x, y) → (x + 3, y - 2).

   a) (1, 5) → (____, ____)

   b) (-2, 4) → (____, ____)

   c) (6, 0) → (____, ____)

2. Complete the rules.

   a) Moving 5 units right: (x, y) → (x + ____, y)

   b) Moving 4 units down: (x, y) → (x, y - ____)

   c) Reflecting in the y-axis: (x, y) → (____, ____)

   Hint: A reflection in the y-axis changes the sign of x but keeps y the same.

3. The points of a triangle are A(1, 1), B(4, 1), and C(1, 3). The triangle is translated 2 units right and 3 units up.

   Find the new coordinates.

   A' = (, )

   B' = (, )

   C' = (, )

4. What stays the same when a shape is translated? Circle all that apply.

   • The shape's size
   • The shape's side lengths
   • The shape's angle sizes
   • The shape's position

5. Explain the difference between a translation and a reflection in your own words.

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Have a Go

How can I use this idea? What can I make, test or solve?

Mission 1: Move the Companion Cube

A triangular floor panel has vertices P(2, 2), Q(6, 2), and R(2, 5).

1. Translate the panel 3 units left and 1 unit down. Find the new coordinates.

   P' = (, )

   Q' = (, )

   R' = (, )

2. What is the length of side PQ? What is the length of side PR?

   PQ = __ units

   PR = __ units

3. Has the perimeter changed after the translation? Explain.

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Mission 2: Activate the Reflection Portal

Reflect the point T(5, -2) in the y-axis.

1. The reflected point is T' = (____, ____).

2. Reflect the point U(-3, 4) in the y-axis.

   U' = (____, ____)

3. Describe what happened to the x-coordinate and the y-coordinate.

   x-coordinate: _____

   y-coordinate: _____

Mission 3: Design a Test Chamber

Create a shape using at least three points on a coordinate grid. Then choose one transformation and apply it.

• Original points: __
• Transformation rule: __
• New points: ___

Draw both shapes on a coordinate grid. Add arrows, a mirror line, or a centre point to show what happened.

Drawing space:

Mission 4: Find the Best Route

A robot starts at A(1, 1) and needs to reach B(7, 4) using only horizontal and vertical moves.

1. Give one possible route using movements such as “6 right, 3 up.”

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2. How many total units does your route travel?

   __ units

3. Can you find a different route with the same total distance? Explain.

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Think Deeper

What did I learn? How can I improve or use this somewhere else?

1. Complete the sentences.

   A translation is __

   A reflection is ___

   A rotation is _____

2. Which rule feels easiest for you to use? Why?

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3. Which part would you like to practise more? Choose one and explain how you could practise it.

   • Reading coordinates
   • Translating shapes
   • Reflecting shapes
   • Rotating shapes
   • Explaining my reasoning

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4. Imagine you are designing a real game, map, building, or app. Where could transformations help you?

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Optional Challenge: The Rotating Portal

A point V(2, 1) is rotated 90° anticlockwise around the origin. For this rotation, use the rule (x, y) → (-y, x).

1. Find the new coordinates of V.

   V' = (____, ____)

2. Use the same rule to rotate W(4, -3).

   W' = (____, ____)

3. Explain one way you could check that your answers are sensible.

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Answer Key

Get Curious

1. b) A movement or change made to a shape

2. 1. C 2. A 3. B

3. No. A translation changes the position but keeps the size, shape, side lengths, and angle sizes the same.

4. Any well-explained choice is acceptable.

Look Around

1. Answers will vary. Suitable transformations must match the examples given.

2. a) Reflection b) Rotation c) Translation d) Rotation

3. Coordinates give exact positions, so a portal or cube can be placed and moved accurately.

4. Answers will vary. The second object should be the same shape and size, shifted without turning or flipping.

Find the Pattern

1. a) (4, 3) b) (1, 2) c) (9, -2)

2. a) 5 b) 4 c) (-x, y)

3. A' = (3, 4) B' = (6, 4) C' = (3, 6)

4. The shape's size, the shape's side lengths, and the shape's angle sizes stay the same. The position changes.

5. A translation slides a shape without flipping or turning it. A reflection flips a shape across a mirror line.

Have a Go

Mission 1

1. The rule is (x, y) → (x - 3, y - 1).

   P' = (-1, 1)

   Q' = (3, 1)

   R' = (-1, 4)

2. PQ = 4 units and PR = 3 units.

3. No. A translation does not change the side lengths or angles, so the perimeter remains the same.

Mission 2

1. T' = (-5, -2)

2. U' = (3, 4)

3. The x-coordinate changes sign. The y-coordinate stays the same.

Mission 3

Answers will vary. The new points must follow the stated transformation rule.

Mission 4

1. One possible route is 6 units right, then 3 units up.

2. The total distance is 9 units.

3. Yes. For example, moving up first and then right still travels 3 + 6 = 9 units.

Think Deeper

1. Sample answers:

   • A translation is a slide.
   • A reflection is a flip across a line.
   • A rotation is a turn around a fixed point.

2. Answers will vary but should include a reason.

3. Answers will vary but should name a useful practice method.

4. Answers will vary. Suitable examples include moving characters in a game, flipping a logo, rotating a map, or designing a repeating pattern.

Optional Challenge

1. Using (x, y) → (-y, x), V(2, 1) → V'(-1, 2).

2. W(4, -3) → W'(3, 4).

3. A sensible check is to plot the original and new points, check that the distance from the origin stays the same, or confirm that the point has turned 90° anticlockwise.