Coordinates in 3D

Classroom resources

Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.

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• Worksheet Download Copy address for Prepare Print the student copy; the Teacher edition has the answers.

Students will be able to

• I can plot and read a point (x, y, z) in 3D.
• I can say what each coordinate means in a real system, such as a drone's position.
• I can describe positions with latitude, longitude and altitude.

Timing

Predict and Run 8 min · Investigate 10 min · Modify 12 min · Practise and Make 10 min

Before class

Do the Run board yourself first. Students need the Cartesian plane (Year 7 transformations) and views of solids.

The one idea: one more axis, at right angles to the other two, gives one more number; the order is always x, y, z. The common mistake is putting the height first or in the order it was read (Kim's mistake).

The board draws the axes in isometric projection, so a point can look like two different points; the dashed path and the label settle it. A corner of the room makes a good real set of axes: two floor edges and the vertical edge.

Wider: plan a 3D printer's path for a small box, or describe a position in a game such as Minecraft (which puts y up: a different convention, which is fine once it is stated).

Running the lesson

1. Predict: ask which number tells the height.
2. Run: ask what each slider changes, and what stays the same.
3. Investigate: ask Kim where (3, 5, 2) actually is.
4. Modify: ask why a box from the origin has edges x, y and z.
5. Make: ask for one corner on the floor and one on the z-axis.

The hard part (on purpose)

Seeing depth on a flat picture. Ask students to trace the dashed path with a finger: along x, along y, then up.

When students get stuck

Answers to the checks

1. how high?: A point is at (2, 5, 3). How far up is it?
   3. The third number, z, is the height. The point is 2 along x and 5 along y.
2. reach Q: Move P onto the ring Q at (3, 2, 4), one axis at a time. Which slider moved P up, as you predicted?
   
   x = 3 along, y = 2 along, z = 4 up. The dashed lines show the three moves.
3. Reading a point: P is 4 along the x-axis, 1 along the y-axis and 6 up. What are its coordinates?
   (4, 1, 6)
   Always x, then y, then z: (4, 1, 6).
4. Find the mistake: Kim records a drone's position. Click the line where she goes wrong.
   line 2
   Line 2: the order is always (x, y, z), so the drone is at (5, 2, 3). Anyone reading (3, 5, 2) puts it 3 m east and 2 m up.
5. Latitude, longitude and altitude: A plane's position is given as latitude 34° S, longitude 151° E and altitude 10 000 m. Which number plays the part of z?
   The altitude
   Latitude and longitude place it over the ground; altitude says how high, like z.
6. A box of 24: Move P so the box from the origin to P has a volume of 24 cubic units.
   
   Such as P (2, 3, 4) or (4, 3, 2): its edges are x, y and z long, and 2 × 3 × 4 = 24.
7. On the z-axis: Put P on the z-axis, 5 units up.
   
   (0, 0, 5): no distance along x or y, so it is on the z-axis.
8. Floors: In an apartment building, the ground floor is 0 m up and each floor is 3 m higher than the one below. A flat is on floor 4. What is its z-coordinate, in metres?
   12
   4 floors × 3 m = 12 m up.
9. z = 0: Where are all the points with z = 0?
   On the floor: the x-y plane
   No height means the point is on the flat x-y plane.
10. A plane climbs: A plane is at latitude 33.9° S, longitude 151.2° E and altitude 9000 m. It climbs straight up 1000 m. Where is it now?
    33.9° S, 151.2° E, 10 000 m
    Only the altitude changes: 9000 + 1000 = 10 000 m. Latitude and longitude still place it over the same spot (near Sydney).
11. Flying up: A drone at (6, 4, 10) flies straight up 5 m. What is its new z-coordinate?
    15
    Up changes only z: 10 + 5 = 15. x and y stay 6 and 4.
12. A 3D printer: A 3D printer's nozzle is at (20, 35, 2) mm. It prints the next layer 0.2 mm higher. Where is it?
    (20, 35, 2.2)
    A new layer is higher, so only z changes: 2 + 0.2 = 2.2 mm.
13. Review: views: Review: a pile of cubes is seen from above, from the front and from the side. Which view shows how tall the pile is?
    The front view (or the side view)
    From above you see the floor (x and y); from the front or side you see the heights (z).
14. a corner of a box: Make a box 4 units along x, 1 along y and 3 high.
    
    P = (4, 1, 3), the corner furthest from the origin. Its volume is 12 cubic units.

Solutions

One possible solution for each task. Other correct programs are fine.

boxes, axes and floors › A box of 24

Open task: any program that does what the step asks.

boxes, axes and floors › On the z-axis

Open task: any program that does what the step asks.

boxes, axes and floors › Floors

Open task: any program that does what the step asks.

boxes, axes and floors › z = 0

Open task: any program that does what the step asks.

a corner of a box

Open task: any program that does what the step asks.

Stretch and support

Stretch: Find the distance from the origin to (3, 4, 12) by using Pythagoras twice.

Support: Use the corner of a room or a box: point along each edge in turn while saying the coordinate.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 8 student in Australia. Help me understand the ideas in ONE lesson only, from the lesson "Coordinates in 3D".

The ideas:
- Extending the coordinate system to three dimensions: I can Students extend the 2D Cartesian plane to three dimensions by adding a third axis (z), and plot and read coordinates in the first octant.
- Describing positions in real-world 3D systems: I can Students describe positions in practical 3D reference systems including latitude/longitude/altitude and multi-storey coordinate systems.

What understanding looks like:
- Plots a point given 3D coordinates in the first octant.
- Reads the coordinates of a plotted point.
- Identifies which axis extends the 2D plane to 3D.
- Correctly describes a position using a practical 3D reference system.
- Interprets a 3D coordinate in a real-world context.
- Connects the three coordinates to three independent dimensions of location.

Mistakes students often make here:
- A student writes the height first.
- A student reads a point from the picture by where it seems to sit.
- A student thinks floor 4 is 15 m up (counting the ground floor).

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Coordinates in 3D".

Exit ticket

A drone is 8 m east, 3 m north and 20 m up from its base, with x east, y north and z up. Write its position. Then it flies 5 m north: where is it now?

(8, 3, 20), then (8, 8, 20).

What each step is for

• how high?: Committing to which number gives the height shows whether they read the coordinates in order.
• reach Q: Moving P along each axis in turn shows that each coordinate is one independent direction.
• reading points: Write (x, y, z) in that order whatever the context; putting the height first is the slip (Kim); latitude, longitude and altitude are a 3D system too.
• boxes, axes and floors: A box from the origin to P has edges x, y and z; a point on an axis has two zeros; a building's floors give z in a real system.
• 3D positions: Reading and writing 3D positions in several contexts (a plane, a drone, a 3D printer), mixed; the views review links to solids.
• a corner of a box: Placing P to make a given box joins coordinates, edges and the box's corners.

Where it leads

This covers AC9M8SP03.

Where it leads: 3D modelling and games in Digital Technologies, vectors in senior mathematics, and Pythagoras in 3D (Year 10).