Position in three dimensions and time zones
Locating a point with three coordinates, and solving duration and clock problems across time zones
About three lessons of 45 to 60 minutes
Where, and when?
A warehouse robot needs more than 'across' and 'along' to find a parcel; it needs a third number for which shelf level to reach. A delivery drone needs to know its height above the ground, not just its spot on the map below. Adding a third coordinate, z, turns a flat map into a full 3D position.
Meanwhile, a video call with a cousin overseas, or a multiplayer game played with a friend on another continent, runs into a different kind of 'where': when. Every place on Earth uses its own local clock, and those clocks are offset from each other by whole (and sometimes half) hours. This unit covers both: pinning down a position with three coordinates, and converting a time correctly from one city's clock to another's.
- Warehouse robot position(aisle, row, shelf level) is a real (x, y, z) coordinate
- Delivery drone altitudeheight above the ground is just the z-coordinate
- A video call across time zonesthe same moment, but a different number on each clock
- 24-hour time avoids AM/PM mix-ups22:00 is unambiguous; 10 o'clock is not
What students will be able to do
Students will describe the position of a point using three coordinates (x, y, z), find a new position after a point moves by given steps in each direction, read a single coordinate from a 3D position, convert a time in one city to the time in another using a stated time-zone offset, and solve duration problems that cross midnight or a time-zone boundary.
- I can describe the position of a point using an (x, y, z) coordinate.
- I can find a new position after a point moves by given steps in each direction.
- I can read off a single coordinate, such as height, from a 3D position.
- I can convert a time in one city to the time in another city using a stated offset.
- I can solve a time-zone problem that wraps past midnight into the next or previous day.
Standards this unit teaches
- AC9M8SP03Australian Curriculum v9 (ACARA)Position in three dimensions
Describe the position and location of objects in 3 dimensions in different ways, including using a three-dimensional coordinate system, with the use of dynamic geometric software and other digital tools.
- AC9M8M04Australian Curriculum v9 (ACARA)Duration and time zones
Solve problems involving duration, including using 12- and 24-hour time across multiple time zones.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Year 7 transformations on the Cartesian plane worksheetthe 2D (x, y) groundwork this unit's third coordinate extends
- Grade 3 time, mass & volume teaching unitthe elapsed-time reasoning this unit's clock arithmetic builds on
- Coordinates in the glossarya refresher on (x, y) position before the third coordinate is added
Words to teach and display
- Coordinate
- a number that fixes a position along one axis
- x, y, z axes
- three directions at right angles to each other, used together to fix a position in 3D space
- 24-hour time
- a way of writing time from 00:00 to 23:59 with no AM or PM needed
- Time zone
- a region that agrees to use the same local clock, offset by a fixed number of hours from other regions
- Offset
- how many hours ahead of or behind another time zone a given time zone is
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. A third coordinate: describing position in three dimensions
ConcreteA 2D position needs two numbers, (x, y): across and along. A 3D position needs a third, (x, y, z), for up or down, such as a warehouse robot's shelf level or a drone's height above the ground. The order always stays x, then y, then z, exactly as (x, y) always stays x then y.
Moving in 3D works exactly like moving in 2D, just with one extra number: add the change in each direction to the matching coordinate, separately. A move can change all three coordinates at once, or just one of them.
A warehouse robot is at position (4, 2, 3): 4 metres along the aisle, 2 metres across, on shelf level 3. It receives the instruction to move (+2, -1, +1). Find its new position.
- Add the x-change: 4 + 2 = 6.
- Add the y-change: 2 + (-1) = 1.
- Add the z-change: 3 + 1 = 4.
Answer: The robot's new position is (6, 1, 4).
- Which coordinate tells you the robot's shelf level?
- If only the z-coordinate changes, has the robot moved sideways, or up and down?
2. Reading a single coordinate from a 3D position
PictorialOften you only need one piece of information from a position, such as just the height. Practise picking out exactly the x, y or z coordinate that answers the question, and ignoring the other two.
A drone hovers at position (15, 8, 12), measured in metres from the launch pad, where z is height above the ground. What is the drone's height?
- The height is described by the third coordinate, z.
- Read off z from the position: z = 12.
Answer: The drone is 12 metres above the ground.
- Which coordinate would tell you how far sideways the drone has travelled from the launch pad?
- Could two drones share the same height but be at different (x, y) positions?
3. Converting a time between two time zones
PictorialCities in different time zones are offset from each other by a number of hours. 'Ahead of' means add the offset; 'behind' means subtract it. Using 24-hour time avoids any AM/PM confusion while you do the arithmetic.
It is 14:00 in City A. City B is 8 hours ahead of City A. What time is it in City B?
- 'Ahead of' means add the offset: 14 + 8 = 22.
- 22 is within 0 to 23, so no day change is needed.
Answer: 22:00 in City B, the same day.
- If City B were 8 hours behind City A instead, what would you do differently?
- Why does using 24-hour time make this arithmetic safer than using AM/PM?
4. Crossing midnight: wrapping the result into the next or previous day
AbstractSometimes adding or subtracting the offset pushes the result past 23:00 or below 00:00. When that happens, add or subtract 24 to bring the answer back into the 0-23 range, and that adjustment is also the signal that the day has changed.
A result of 24 or more means the time has rolled into the NEXT day, so subtract 24. A result below 0 means the time has rolled into the PREVIOUS day, so add 24.
It is 20:00 in City A. City B is 9 hours ahead. What time is it in City B, and is it the same day?
- Add the offset: 20 + 9 = 29.
- 29 is 24 or more, so subtract 24: 29 - 24 = 5.
- Because the result wrapped past 24, it is the next day.
Answer: 05:00 in City B, the next day.
- What would tell you the day had gone BACKWARDS instead of forwards?
- Would a result of exactly 24:00 mean the same day or the next day?
Common misconceptions and how to address them
MisconceptionReading the three coordinates of a 3D position in any order, e.g. reading (4, 2, 3) as (2, 4, 3).
Why it happens: Without a fixed, practised order it is easy to mix up which number measures which direction.
How to address it: Fix a consistent order, x then y then z, and say what each one measures every single time a position is read, the same discipline as reading (x, y) in 2D.
MisconceptionBelieving a 3D move can only ever change one coordinate at a time.
Why it happens: Working with 2D moves that often change just one coordinate makes a simultaneous 3-way change feel unusual.
How to address it: Add each coordinate's change separately, all in the same step, exactly like translating a point in 2D by (dx, dy) but with a third number added alongside.
MisconceptionAdding the time-zone offset without first checking whether the city is ahead or behind, so subtracting when it should add or the reverse.
Why it happens: Once a number (the offset) is in view, it is tempting to just add it without re-reading the direction word.
How to address it: Underline 'ahead' or 'behind' before touching any numbers. Ahead always adds, behind always subtracts, with no exceptions.
MisconceptionLeaving a converted time outside the 0-23 range, e.g. reporting an answer as '26:00'.
Why it happens: The addition or subtraction was done correctly, but the final wrapping step was skipped.
How to address it: Any result of 24 or more, or below 0, must be adjusted by 24 before it is a valid answer; that adjustment also tells you whether the day moved forward or back.
MisconceptionAssuming that crossing into a different time zone changes the actual moment in time, not just its clock label.
Why it happens: Different numbers on the clock make it feel like time itself is different in each place.
How to address it: The same instant is happening everywhere at once; only the local clock's LABEL for that instant changes between time zones, the same event just gets a different number depending on where you read the clock.
Guided practice (with answers)
1. A crane's hook is at position (5, 10, 2). It moves by (-2, 0, +4). Find its new position.
Answer: (3, 10, 6), because 5 - 2 = 3, 10 + 0 = 10, and 2 + 4 = 6.
2. A submarine is at position (30, 12, -18), where z is height relative to sea level (negative means below). How far below sea level is it?
Answer: 18 metres below sea level, because z = -18.
3. A submarine is at position (12, -7, 25). What is its y-coordinate?
Answer: -7.
4. It is 09:00 in City A. City B is 3 hours ahead. What time is it in City B?
Answer: 12:00, because 9 + 3 = 12.
5. It is 23:00 in City A. City B is 4 hours ahead. What time is it in City B, and is it the same day?
Answer: 03:00, the next day, because 23 + 4 = 27, and 27 - 24 = 3.
6. It is 02:00 in City A. City B is 6 hours behind. What time is it in City B, and is it the same day?
Answer: 20:00, the previous day, because 2 - 6 = -4, and -4 + 24 = 20.
Independent practice worksheets
Set the matching ChalkBee worksheet for independent work. The answer keys are computed in code, so they are never wrong.
Differentiation
- Introduce a 3D move by changing just one coordinate at a time before combining all three.
- Start with 'ahead of' (addition) time-zone problems only, adding 'behind' (subtraction) once that is secure.
- Give a printed 0-23 number line so students can count along it by hand before trusting the arithmetic alone.
- Keep every time-zone problem within the 0-23 range at first, introducing the wraparound case as its own explicit lesson.
- Introduce negative z-coordinates for depth below ground or below sea level.
- Pose a combined problem: a call starts at a given time, lasts a stated duration, AND crosses a time zone.
- Research real time zones and note that some, such as India and parts of Australia, use a half-hour offset rather than a whole hour.
- Ask students to design a delivery-drone flight plan as a sequence of 3D moves, then state its final position.
Assessment: exit ticket
A short exit ticket covering a 3D position move and two time-zone conversions.
1. A point is at position (7, -3, 5). It moves by (+1, +3, -5). Find its new position.
Answer: (8, 0, 0), because 7 + 1 = 8, -3 + 3 = 0, and 5 - 5 = 0.
2. It is 11:00 in City A. City B is 6 hours behind. What time is it in City B?
Answer: 05:00, because 11 - 6 = 5.
3. It is 21:00 in City A. City B is 7 hours ahead. What time is it in City B, and is it the same day?
Answer: 04:00, the next day, because 21 + 7 = 28, and 28 - 24 = 4.
Teacher notes and timings
- Rough timing: Lesson 1 3D position and moves (sections 1 and 2), Lesson 2 time-zone conversion without wraparound (section 3), Lesson 3 wraparound problems and the exit ticket (section 4 and assessment).
- AC9M8SP03 and AC9M8M04 previously had no dedicated Year 8 lesson.
- AC9M8SP03 explicitly names 'dynamic geometric software and other digital tools'; a free 3D grapher (such as GeoGebra 3D, or a Minecraft-style coordinate world students already know) makes an excellent live demonstration for section 1.
- City A and City B are used deliberately instead of naming real cities, since real time-zone offsets shift with daylight saving; if naming real cities, check the current offset first.