Stage 4 Geometry Worksheet for 12-Year-Olds | Portal 2 Math Activities

Instructions

1. You are a test subject helping Chell design safe test chambers using geometry.
2. Work through the chambers in order. The questions begin with coordinates and angles, then become more challenging.
3. Show calculations whenever you can. A calculator may be used after you estimate the answer.
4. For each reflection prompt, write one or two complete sentences.
5. Optional challenge questions are marked Challenge.

Learning goals:

• Plot and describe points on a coordinate plane.
• Identify and calculate angles.
• Use symmetry and transformations.
• Calculate perimeter, area, and volume.
• Explain how geometry is useful in a real design problem.

Chamber 1: Coordinate Portal Calibration

A test chamber is drawn on a coordinate grid. The horizontal axis is the x-axis and the vertical axis is the y-axis.

Helpful reminder: A coordinate is written as (x, y). Move across first, then move up or down.

1. Plot these points on graph paper or a digital coordinate grid:

   • Blue Portal: A(2, 3)
   • Orange Portal: B(-4, 3)
   • Cube: C(-4, -2)
   • Button: D(2, -2)

2. Join the points in order: A → B → C → D → A.

3. What shape have you created?

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4. What is the length of side AB? Explain how you know.

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5. What is the length of side BC?

   ---

6. Find the perimeter of the shape.

   Working: ___

   Answer: ____

7. Find the area of the shape.

   Working: ___

   Answer: ____

Reflection: Which coordinate helped you identify the shape most quickly? Why?

Chamber 2: Laser Angle Lock

A laser beam travels in a straight line. The angles in a chamber must be set correctly before the door opens.

1. Classify each angle as acute, right, obtuse, or straight.

   • 35°: ____
   • 90°: ____
   • 128°: ___
   • 180°: ___

2. Two angles lie on a straight line. One angle is 67°. Find the other angle.

   Working: ___

   Answer: ____

3. Two angles meet at a right angle. One angle is 24°. Find the other angle.

   Working: ___

   Answer: ____

4. A triangular test chamber has angles of 48° and 72°. Find the third angle.

   Hint: The angles inside a triangle add to 180°.

   Working: ___

   Answer: ____

5. A robot turns 90° clockwise, then 45° clockwise. What is the total turn?

   Answer: ____

Reflection: Where might engineers use angle measurements outside a geometry lesson?

Chamber 3: Symmetry and Transformations

A portal frame has the vertices P(1, 1), Q(4, 1), R(4, 3), and S(1, 3).

1. Reflect point P(1, 1) across the x-axis. Write the new coordinate.

   Answer: P'(__, __)

2. Reflect point Q(4, 1) across the y-axis. Write the new coordinate.

   Answer: Q'(__, __)

3. Translate point R(4, 3) three units left and two units down.

   Answer: R'(__, __)

4. The portal frame is a rectangle. How many lines of symmetry does it have?

   Answer: ____

5. A test chamber floor pattern is turned 90° clockwise around its centre. Is this a reflection, rotation, or translation?

   Answer: ____

6. Draw a shape with exactly one line of symmetry. Mark its line of symmetry with a dashed line.

   Drawing space:

   ---

   ---

Reflection: Which transformation would be most useful for copying a chamber design quickly? Explain.

Chamber 4: Weighted Storage Cube

A storage cube is a rectangular prism with a length of 6 cm, a width of 4 cm, and a height of 3 cm.

1. Find the volume of the storage cube.

   Formula: volume = length × width × height

   Working: ___

   Answer: ____

2. Find the area of the top face.

   Working: ___

   Answer: ____

3. Find the total surface area of the rectangular prism.

   Hint: There are three pairs of equal faces.

   Working: ___

   Answer: ____

4. Each cubic centimetre of the cube can hold 0.5 units of gel. How many units of gel can the cube hold?

   Working: ___

   Answer: ____

5. A larger storage container has dimensions 12 cm × 8 cm × 6 cm. How many times the volume of the first cube is it?

   Working: ___

   Answer: ____

Chamber 5: Design the Exit Route

You must design a rectangular escape chamber on a coordinate grid.

1. Choose four coordinates that make a rectangle with a length of at least 5 units and a width of at least 3 units.

   Vertex 1: ____

   Vertex 2: ____

   Vertex 3: ____

   Vertex 4: ____

2. Draw and label your rectangle on a coordinate grid.

3. Calculate its perimeter.

   Working: ___

   Answer: ____

4. Calculate its area.

   Working: ___

   Answer: ____

5. Add a circular portal inside your rectangle. Estimate or measure its radius.

   Radius: ____ units

6. Explain one design choice that makes your chamber easier or safer to navigate.

Final Test: Geometry Thinking

1. A rectangular room is 9 m long and 5 m wide. A strip of wall around the room must be painted. What is the perimeter of the room?

   Answer: ____

2. The same room has a square testing platform with side length 3 m. What area of floor remains outside the platform?

   Working: ___

   Answer: ____

3. A robot moves from (−2, 1) to (−2, 6), then to (4, 6). What is the total distance travelled?

   Working: ___

   Answer: ____

4. A triangular warning sign has angles 55°, 55°, and 70°. What type of triangle is it based on its angles?

   Answer: ____

5. Challenge: A rectangular chamber has an area of 48 m². Its length is 10 m. What is its width? Is the answer a whole number?

   Working: ___

   Answer: ____

Learning Reflection

Complete these sentences.

1. One geometry skill I used confidently was:

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2. One calculation I had to check carefully was:

   ---

3. A mistake I corrected or could correct is:

   ---

4. Geometry could help me in real life when I need to:

   ---

5. My next geometry mission is:

   ---

Answer Key

Chamber 1

1. The points form a rectangle.
2. AB = 6 units because the x-coordinates change from 2 to −4.
3. BC = 5 units because the y-coordinates change from 3 to −2.
4. Perimeter = 6 + 5 + 6 + 5 = 22 units.
5. Area = 6 × 5 = 30 square units.
6. Reflection answers will vary. Strong answers mention coordinates, horizontal or vertical sides, or opposite sides being equal.

Chamber 2

1. 35° is acute; 90° is right; 128° is obtuse; 180° is straight.
2. 180° − 67° = 113°.
3. 90° − 24° = 66°.
4. 180° − 48° − 72° = 60°.
5. 90° + 45° = 135° clockwise.
6. Reflection answers will vary. Suitable examples include construction, maps, robotics, architecture, sports, and navigation.

Chamber 3

1. Reflection of P(1, 1) across the x-axis: P'(1, −1).
2. Reflection of Q(4, 1) across the y-axis: Q'(−4, 1).
3. Translation of R(4, 3) three left and two down: R'(1, 1).
4. A non-square rectangle has 2 lines of symmetry.
5. A 90° turn is a rotation.
6. Drawing answers will vary. The shape must have exactly one line of symmetry.

Chamber 4

1. Volume = 6 × 4 × 3 = 72 cm³.
2. Top face area = 6 × 4 = 24 cm².
3. Surface area = 2(6 × 4) + 2(6 × 3) + 2(4 × 3) = 48 + 36 + 24 = 108 cm².
4. Gel capacity = 72 × 0.5 = 36 units.
5. The larger container has volume 12 × 8 × 6 = 576 cm³. Since 576 ÷ 72 = 8, it is 8 times the volume.

Chamber 5

1. Answers will vary. Any four coordinates forming a rectangle with the required dimensions are acceptable.
2. The drawing should show a labelled rectangle on a coordinate grid.
3. Perimeter depends on the learner's rectangle: 2 × length + 2 × width.
4. Area depends on the learner's rectangle: length × width.
5. The radius estimate depends on the learner's drawing.
6. Suitable design choices include a wider route, right-angle corners, clear labels, symmetry, or enough space around the portal.

Final Test

1. Perimeter = 2(9 + 5) = 28 m.
2. Room area = 9 × 5 = 45 m². Platform area = 3 × 3 = 9 m². Remaining area = 45 − 9 = 36 m².
3. First movement: 5 units. Second movement: 6 units. Total distance = 11 units.
4. It is an acute triangle because all three angles are less than 90°. It is also isosceles because two angles are equal.
5. Width = area ÷ length = 48 ÷ 10 = 4.8 m. It is not a whole number.

Reflection Guidance

Answers will vary. Look for specific references to coordinates, angles, transformations, perimeter, area, volume, checking work, or a real-world application.