Coordinates in 3D

A third axis, z, gives a point in space three coordinates (x, y, z). Real systems do the same with other names: a drone, a building, a plane's latitude, longitude and altitude, and a 3D printer.

Two drones are both above the same spot in a park. How can a controller tell where each one is, so they don't crash? Two numbers are not enough.

On a flat grid, a point has two coordinates, $(x, y)$ . To describe a point in space, add a third axis, $z$ , at right angles to both. How do three numbers fix a point?

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

How high?

The $z$ -axis points up. A point is $(x, y, z)$ :

$x$ : how far along the x-axis
$y$ : how far along the y-axis
$z$ : how far up

The origin is $(0, 0, 0)$ . Points with $z = 0$ are on the floor (the x-y plane). The order is always $x$ , then $y$ , then $z$ .

Real systems use the same idea with other names:

System	Three numbers
a drone	metres east, metres north, metres up
a building	metres across, metres back, metres up (about 3 m a floor)
a plane	latitude, longitude, altitude

Each real system has its own fixed order. Latitude (how far north or south) is always written before longitude (how far east or west), then altitude.

A point is at (2, 5, 3). How far up is it?235

TryTry it on the board and compare the result with your prediction.

Reach Q

The blue axis is x, the orange is y and the purple points up (z). The dashed path shows how to get from the origin to P.

Try it yourselfYour edits are saved in this browser.

Move P onto the ring Q at (3, 2, 4), one axis at a time. Which slider moved P up, as you predicted?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Reading points

Here is the routine.

Worked exampleBefore each step, say what you think comes next.

$\text{5 along } x$ The first number.
$\text{0 along } y$ The second number: no distance is 0.
$\text{2 up}$ The third number, z.
$(5, 0, 2)$ Write them in order, in brackets.

Read Kim's working. Which number is the height?

P is 4 along the x-axis, 1 along the y-axis and 6 up. What are its coordinates?(4, 6, 1)(6, 4, 1)(1, 4, 6)(4, 1, 6)

Kim records a drone's position. Click the line where she goes wrong.

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 longitudeThe altitudeThe latitudeNone of them

ModifyChange the problem on purpose and see what happens.

Boxes, axes and floors

Need a hint?

The box's volume is x × y × z.

Try it yourselfYour edits are saved in this browser.

Move P so the box from the origin to P has a volume of 24 cubic units.

Use the board. This ticks when you get there.

Put P on the z-axis, 5 units up.

Use the board. This ticks when you get there.

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?

Where are all the points with z = 0?On the floor: the x-y planeNowhereAt the origin onlyOn the z-axis

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

A corner of a box

Then write the coordinates of all eight corners of the box, and check two of them on the board.

Try it yourselfYour edits are saved in this browser.

Need a hint?

The box goes from the origin to P, so P is the far top corner.

Another hint

Its edges are x, y and z long.

Make a box 4 units along x, 1 along y and 3 high.

Use the board. This ticks when you get there.

For teachers

Students plot and read points (x, y, z) on 3D axes, and use 3D systems for drones, buildings, planes and 3D printers (AC9M8SP03, nodes 1 and 2). About 40 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

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.

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

Predict: ask which number tells the height.
Run: ask what each slider changes, and what stays the same.
Investigate: ask Kim where (3, 5, 2) actually is.
Modify: ask why a box from the origin has edges x, y and z.
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

A student writes the height first. Ask: What order did we agree: along, along, up? Fix: Always (x, y, z): the height is last.
A student reads a point from the picture by where it seems to sit. Ask: Follow the dashed path: how far along each axis? Fix: Read the path, not the position on the screen.
A student thinks floor 4 is 15 m up (counting the ground floor). Ask: How many floors above the ground floor is floor 4? Fix: Ground floor is 0, so floor 4 is 4 × 3 m.

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.

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).

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.

Now you can

Plot and read a point (x, y, z) in 3D.
Say what each coordinate means in a real system, such as a drone's position.
Describe positions with latitude, longitude and altitude.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.You reached the endBack to Angles and shapes

Source: content/maths/angles/3d-coordinates/lesson.md

How high?

Reach Q

Reading points

Boxes, axes and floors

3D positions

A corner of a box

Students will be able to

Before class

Running the lesson

The hard part (on purpose)

When students get stuck

Stretch and support

Where it leads

What each step is for