097i - Describing Position in Three Dimensions

Lesson 97 — Describing Position and Location in Three Dimensions

Strand: Space | Descriptor: AC9M8SP03 | Duration: 45 minutes

Equipment: isometric or grid paper; small classroom objects (box, ruler) for the warmup demonstration.

Learning Intentions

• To describe the position of an object in three dimensions using everyday and informal language.
• To describe position formally using a three-dimensional coordinate system, .

Success Criteria

I can:

1. Describe an object’s 3D position using informal methods: grid reference plus height, or compass direction plus elevation.
2. Explain why two dimensions (a single grid reference) are not enough to fully locate an object in space.
3. Set up and read a three-dimensional coordinate system with axes , and .
4. Plot and name the coordinates of points in 3D, including points on the axes and coordinate planes.

Warmup

(5 minutes — “find the object”, whole class)

Hold up a small box or ruler at a specific point in the room (e.g. balanced on top of a filing cabinet, off to one side).

1. If you could only say “it’s near the window”, would a classmate find it? Why not?
2. If you gave a grid reference for the room (like a seating plan, e.g. “column 3, row 2”), would that alone be enough to find the object?
3. What extra piece of information is missing?

Answers: 1. No — “near the window” is too vague; many objects could be near the window. 2. No — a grid reference only tells you a horizontal position on the floor; the object could be at floor level, desk height, or near the ceiling above that same grid square. 3. Height (how far up) is missing — position on a flat plan is only two dimensions, but the real room is three-dimensional.

Activities

Activity 1 — Explicit Instruction: from 2D to 3D Description (10 min)

I do: Recap that a 2D grid reference (like a map or seating plan) needs exactly two pieces of information: how far across, and how far up/down the page. Describe the classroom’s warmup object informally two ways:

Grid reference + height: “Column 3, row 2, at a height of about m above the floor.” Compass direction + elevation: “Roughly north-east of the door, at an elevation of about m.”

Both descriptions use three independent pieces of information — two to fix a horizontal position, and one for height. This is the essential idea of 3D position: three independent measurements are needed to locate a point in space, compared to two for a flat surface.

We do: Together describe the position of the classroom clock, using both informal methods (grid reference + height; compass + elevation).

You do: Choose three different objects in the room (e.g. a light fitting, a poster, a doorknob) and describe each one’s position using both informal methods. Compare with a partner — does your partner’s description let them correctly identify the object without looking?

Activity 2 — Explicit Instruction: Formal 3D Coordinates (13 min)

I do: Introduce the formal three-dimensional coordinate system. Just as locates a point on a flat page using two perpendicular axes, locates a point in space using three mutually perpendicular axes, all meeting at the origin .

• -axis: typically drawn running “forward” (out of the page, towards the viewer).
• -axis: typically drawn running “across” (left–right).
• -axis: typically drawn running “up” (vertical height).

Sketch the standard orientation on the board (drawn on isometric-style axes to suggest depth):

        z
        |
        |
        |________ y
       /
      /
     x

A point means: go units along , then units along , then units up along .

Worked example — plotting on isometric paper:

We do: Together plot and on isometric grid paper, tracing each step aloud.

You do: Plot the following points on your own isometric grid, labelling each clearly: , , , .

Discuss after plotting: What do you notice about the position of point ? (It lies exactly on the -axis, since its and coordinates are both zero.) What about ? (It lies in the “floor” — the -plane — since its -coordinate is zero, meaning no height.)

Activity 3 — Inquiry Task: Designing a 3D Coordinate Description for a Real Space (14 min)

Pairs. Choose a real or imagined multi-storey space — a school building, a block of storage shelves, or a multi-level car park.

Design a three-dimensional coordinate system for your chosen space. You must:

• Decide where the origin is located, and state it clearly (e.g. “the origin is the bottom-left corner of the ground floor, at the main entrance”).
• Decide what each axis measures and its unit (e.g. = metres east, = metres north, = metres above ground floor, or = floor number).
• Give the coordinates of four real locations within your space (e.g. “the library is at ” — meaning m east, m north, on the second floor).
• Explain one situation where your coordinate system would be genuinely useful in real life (e.g. for a delivery robot, an elevator control system, or an evacuation plan).

Socratic scaffolding:

Prompt	Purpose
Understand: what does a workable coordinate system actually need?	A fixed origin, three consistent axis directions, and consistent units.
What do you know about 2D coordinate systems that transfers here?	The origin is a fixed reference point; axes must be perpendicular and consistently labelled.
Devise a plan: where is a sensible, findable origin in your space?	A corner or entrance is easier for others to locate than an arbitrary interior point.
Carry it out	Assign coordinates to your four chosen locations, checking each against your origin and axis definitions.
Looking back	Could someone unfamiliar with your space find each location using only your coordinates and your stated origin/axis rules?

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Explain why a single grid reference (e.g. “B4”) is not enough to describe a location inside a multi-storey building.
2. Name the three coordinates needed to describe a point in 3D space, and what each one typically measures.
3. A point has coordinates . Which axis does it lie on?
4. Reasoning. A drone’s position is given as . What does the -coordinate of tell you about the drone?

Answers: 1. “B4” only fixes a horizontal grid square — it says nothing about which floor, so many different locations (one per floor) could share that reference. 2. , (horizontal position, two directions) and (height/vertical position). 3. The -axis (since and ). 4. The drone is at ground level (zero height) — it has not taken off, or has landed.

Common Misconceptions

Misconception	How to pre-empt it
Believing a 2D grid reference is sufficient to locate any real-world object.	Use the warmup’s “find the object” demonstration — height is invisible on a flat grid.
Treating the three axes as interchangeable, with no fixed meaning.	Insist that , , each have a defined direction and unit, stated explicitly at the start of any coordinate system.
Assuming can be placed anywhere without saying so.	Require the origin’s real-world location to always be stated in words alongside any 3D coordinate system.
Plotting by moving up first, then across.	Model the fixed order — , then , then — consistently in every worked example.
Believing a point with a coordinate of zero is “not really a point” or an error.	Discuss and explicitly — a zero coordinate is a valid, meaningful position (on an axis or a plane).

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A point lies on the -axis only. What must be true about its - and -coordinates?

Answer

Both must be zero — any point on the -axis has the form , since it has no displacement in the or directions.

E2 (AMC Junior style). A storage warehouse uses coordinates , each numbered from . There are aisles, shelves per aisle, and levels per shelf. How many distinct storage positions exist in total?

Answer

288 distinct positions.

E3 (Challenge). A point and a point are given. Describe, in words, the relationship between their positions, and state the length of the straight segment joining them.

Answer

and share the same - and -coordinates, so lies directly above . The segment is vertical, with length equal to the difference in -coordinates: units.

Homework

1. Describe the position of your bedroom light fitting at home using both informal methods from Activity 1 (grid reference + height; compass + elevation).
2. Plot the points , , on isometric or grid paper, showing your working for each.
3. State which axis or plane each point lies on: (a) (b) (c) (d) .
4. A library uses the coordinate system (metres east, metres north, floor number) with origin at the main entrance. The reference desk is m east, m north, on floor . Write its coordinates.
5. Reasoning. Explain why an elevator only needs to report a single number (the floor) to describe its vertical position, but a delivery drone needs a full -coordinate in metres.
6. Challenge. A cube has one corner at the origin and sides of length units, aligned with the axes. List the coordinates of all corners of the cube.

Answers: 3(a) the -axis (b) the -axis (c) the -plane (floor) (d) the -plane. 4. . 5. An elevator only ever stops at fixed, evenly-spaced floor levels, so a single integer floor number is sufficient and unambiguous; a drone can be at any continuous height, so it needs a precise, continuously-variable measurement in metres. 6. , , , , , , , .