Stage 4 Geometry Worksheet for Age 12 | Portal 2 Mathematics and Design

Instructions

1. You are a geometry designer working on an Aperture-style test chamber. Your mission is to understand shapes, space, measurement, and movement.
2. Choose the questions that interest you most first. You may draw on paper, use graph paper, build with blocks, or use a digital geometry tool.
3. Show your working when you calculate. If you get stuck, use the hints before checking the answer key.
4. For reflection questions, there may be more than one good answer. Aim for thoughtful explanations rather than perfect wording.
5. You may pause, skip a question, or return to it later.

Mission Goals

By the end of this worksheet, you should be able to:

• Explain what geometry is and why it is useful.
• Recognise geometry in the world around you.
• Use rules about angles, shapes, transformations, scale, area, and volume.
• Apply geometry to design or solve a practical problem.
• Reflect on what you learned and how you could improve.

1. What Is Geometry?

Geometry is the study of:

• shapes and their properties
• size, position, distance, and space
• how shapes can move, turn, flip, or change size

A. Complete the sentence.

Geometry is useful because it helps people ____

B. Circle the best answer.

1. Which object is mainly a 3D shape?

   a. A line segment
   b. A cube
   c. An angle
   d. A point

2. Which measurement tells us the distance around a shape?

   a. Area
   b. Volume
   c. Perimeter
   d. Height

3. Which transformation slides a shape without turning or flipping it?

   a. Translation
   b. Reflection
   c. Rotation
   d. Enlargement

C. Match each word to its meaning.

Answers: 1. 2. 3. 4. 5. _____

2. Where Do I See Geometry?

Look around your home, classroom, workplace, playground, or a game environment. Record examples of geometry you notice.

Reflection prompt: Choose one example from your table. How would geometry help someone design, build, repair, or improve it?

3. The Rules of the Test Chamber

Use these rules as your geometry toolkit.

• Angles on a straight line add to 180°.
• Angles around a point add to 360°.
• The angles in a triangle add to 180°.
• The angles in a quadrilateral add to 360°.
• Area of a rectangle = length × width.
• Area of a triangle = 1/2 × base × perpendicular height.
• Volume of a rectangular prism = length × width × height.
• A translation slides a shape.
• A reflection flips a shape across a mirror line.
• A rotation turns a shape around a centre.
• An enlargement changes size using a scale factor.

A. Find the missing angles. Show your calculation.

1. A triangle has angles of 45° and 65°. Find the third angle.

   Calculation: __

   Answer: __°

2. A quadrilateral has angles of 90°, 90°, and 110°. Find the fourth angle.

   Calculation: __

   Answer: __°

3. Three angles meet at a point. They are 120°, 85°, and x°. Find x.

   Calculation: __

   Answer: __°

B. Area and volume in a test chamber

A rectangular chamber is 8 m long and 5 m wide.

1. What is its area?

   Calculation: __

   Answer: __ m²

The chamber is 3 m high.

2. What is its volume?

   Calculation: __

   Answer: __ m³

3. What is the perimeter of the floor?

   Calculation: __

   Answer: __ m

4. Plot, Move, and Solve

On squared paper, plot the four points below and join them in order.

• A = (1, 1)
• B = (5, 1)
• C = (5, 3)
• D = (1, 3)

This shape represents a rectangular test chamber.

A. Describe the original shape.

1. What is its length? __ units
2. What is its width? __ units
3. What is its area? __ square units
4. What is its perimeter? __ units

B. Activate a translation.

Move every point 3 units right and 2 units down.

New coordinates:

• A' = (, )
• B' = (, )
• C' = (, )
• D' = (, )

What stays the same after a translation? Choose all that apply.

• [ ] Side lengths
• [ ] Angle sizes
• [ ] Area
• [ ] Position
• [ ] Shape

C. Activate a reflection.

Reflect the original points in the y-axis.

New coordinates:

• A' = (, )
• B' = (, )
• C' = (, )
• D' = (, )

What happens to the x-coordinate during a reflection in the y-axis?

5. Use Geometry to Design Something

Choose one design mission. You may draw it, build it, or describe it with measurements.

Mission A: Design a test chamber

Create a rectangular or composite floor plan. Include at least three of the following:

• two different shapes
• one pair of parallel lines
• one right angle
• one angle that is not 90°
• labelled lengths
• a calculated area or perimeter

Mission B: Build a safe ramp

Design a ramp for a small robot. Decide its length, height, and width. Explain why its measurements would make it useful and safe.

Mission C: Create a portal route

Draw a route on a coordinate grid. Include at least one translation, reflection, or rotation. Label the starting point and finishing point.

Mission D: Improve a real object

Choose a room, box, playground, desk, garden, or game level. Explain how you could use geometry to make it more efficient, attractive, stable, or accessible.

My chosen mission: _____

Sketch, plan, or description:

Measurements and calculations:

How geometry helped me solve the problem:

Optional Challenge: The Companion Cube Crate

A square crate has a side length of 4 units.

1. Find its perimeter. __ units
2. Find its area. __ square units
3. The crate is enlarged by a scale factor of 2. What is the new side length? __ units
4. What is the new area? __ square units
5. Did the area double, become four times larger, or become eight times larger? Explain why.

6. What Did I Learn?

Complete the reflection. You can write, draw, use bullet points, or talk through your answers with a partner or teacher.

One geometry idea I understand better now is:

A rule or formula I used was:

A part I found easy or enjoyable was:

A part I found confusing or would like to practise is:

I could use this geometry somewhere else, such as:

My next small step is:

Answer Key

1A. Possible answer

Geometry helps people describe, measure, design, build, compare, and solve problems involving shapes and space.

1B. Multiple choice

1. b. A cube
2. c. Perimeter
3. a. Translation

1C. Matching

1. A
2. C
3. B
4. D
5. E

2. Geometry sighting log

Answers will vary. Accept examples such as windows, floor tiles, sports courts, stairs, wheels, buildings, packaging, maps, screens, or game environments. A strong answer explains how geometry could help with designing, measuring, building, or improving the example.

3A. Missing angles

1. Triangle: 180° − 45° − 65° = 70°
2. Quadrilateral: 360° − 90° − 90° − 110° = 70°
3. Angles around a point: 360° − 120° − 85° = 155°

3B. Area and volume

1. Area = 8 × 5 = 40 m²
2. Volume = 8 × 5 × 3 = 120 m³
3. Perimeter = 2 × 8 + 2 × 5 = 26 m

4A. Original rectangle

1. Length = 4 units
2. Width = 2 units
3. Area = 4 × 2 = 8 square units
4. Perimeter = 2 × 4 + 2 × 2 = 12 units

4B. Translation 3 units right and 2 units down

• A' = (4, −1)
• B' = (8, −1)
• C' = (8, 1)
• D' = (4, 1)

The side lengths, angle sizes, area, and shape stay the same. The position changes.

4C. Reflection in the y-axis

• A' = (−1, 1)
• B' = (−5, 1)
• C' = (−5, 3)
• D' = (−1, 3)

During a reflection in the y-axis, the x-coordinate changes sign while the y-coordinate stays the same.

5. Design mission

Answers will vary. A successful design should include clear labels, sensible measurements, at least one accurate calculation, and an explanation of how geometry was used. Check that area, perimeter, volume, angles, or transformations have been calculated using the correct rule.

Optional Challenge

1. Perimeter = 4 × 4 = 16 units
2. Area = 4 × 4 = 16 square units
3. New side length = 4 × 2 = 8 units
4. New area = 8 × 8 = 64 square units
5. The area becomes four times larger because both the length and width are multiplied by 2, so 2 × 2 = 4.

6. Reflection

Answers will vary. Look for a specific geometry idea, a correctly named rule or formula, an honest description of progress, and a realistic next step.