Describing routes and relative positions | Year 8

01 / LEARN

Understand the idea.

Record the position after each instruction

For a route in space, record a new triple after each instruction. Positive $x$ adds to the first coordinate; negative $y$ subtracts from the second; movement along $z$ changes the third. Every move begins from the previous endpoint.

Distance travelled adds the positive length of every leg. Moving from $x = 1$ to $x = 5$ , then back to $x = 3$ , covers $4 + 2 = 6$ units. The final position changes by only 2 units, so position change and journey length differ.

Compare locations and plan a route

To plan an axis-aligned route from $A$ to $B$ , compare their $x$ entries, then their $y$ entries, then their $z$ entries. Increase or decrease each coordinate until it matches the destination. Changing the order can give another valid route.

If each coordinate changes directly to its target without reversing, add the magnitudes of the three coordinate differences. This gives the length of that axis-aligned route. A route with detours is longer, and a straight diagonal path is a different measurement.

Point B is labelled with coordinates 4, 1, 2 on a perspective diagram of x, y and z axes. A dashed gold route follows those coordinate directions in order. Numbered ticks represent one-unit steps.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Follow a route

Start at (1,2,0). Move 3 units in positive x, 2 units in positive z and 1 unit in negative y. Give the endpoint and route length.

Worked solution

1
Move 3 units in positive x.
$(1, 2, 0) \mapsto (1 + 3, 2, 0) = (4, 2, 0)$
An x-direction movement changes only the first coordinate.
2
Move 2 units in positive z.
$(4, 2, 0) \mapsto (4, 2, 0 + 2) = (4, 2, 2)$
The second move starts at the first endpoint and increases height.
3
Move 1 unit in negative y.
$(4, 2, 2) \mapsto (4, 2 - 1, 2) = (4, 1, 2)$
Negative y decreases the depth coordinate by 1.
4
Add the lengths of all three legs.
$d = 3 + 2 + 1 = 6 units$
All travelled lengths contribute positively, including the negative-y leg; the endpoint is $(4, 1, 2)$ .

Answer

Endpoint (4,1,2); route length 6 units.

EXAMPLE 02

Plan between two points

Give an axis-aligned route from A(1,4,2) to B(3,1,5), and its length.

Worked solution

1
Change x from 1 to 3.
$(1, 4, 2) \mapsto (3, 4, 2), 3 - 1 = 2$
This leg moves 2 units in positive x.
2
Change y from 4 to 1.
$(3, 4, 2) \mapsto (3, 1, 2), 4 - 1 = 3$
This leg moves 3 units in negative y.
3
Change z from 2 to 5.
$(3, 1, 2) \mapsto (3, 1, 5), 5 - 2 = 3$
This leg moves 3 units in positive z and reaches B.
4
Calculate this route's total length.
$d = 2 + 3 + 3 = 8 units$
The three axis-aligned leg lengths add; the negative direction of one leg does not subtract distance.

Answer

2 in positive x, 3 in negative y, 3 in positive z; 8 units.

03 / YOUR TURN

Try it yourself.

Check your understanding

Start at (2,1,3) and move 4 units in negative x. What is the endpoint?

Show answer & reasoning

(−2,1,3)

Moving 4 units in negative $x$ changes only the first entry: $2 - 4 = - 2$ . The endpoint is $(- 2, 1, 3)$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Start at $(0, 0, 0)$ . Move 2 in positive $x$ , 3 in positive $y$ and 1 in positive $z$ . Give the endpoint.
2 marks
Start at $(3, 2, 1)$ . Move 2 in negative $y$ . Give the endpoint.
2 marks
Start at $(1, 1, 1)$ . Move 4 in positive $x$ then 2 in negative $x$ . Give endpoint and distance.
2 marks
Describe the change from $A (2, 5, 1)$ to $B (2, 1, 4)$ .
2 marks
Find the length of an axis-aligned route from $(1, 2, 3)$ to $(4, 1, 5)$ without reversing direction.
2 marks
Mia says returning to the start means the total distance is zero. She walks 3 units out and 3 back. Correct her.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Start at $(0, 0, 0)$ . Move 2 in positive $x$ , 3 in positive $y$ and 1 in positive $z$ . Give the endpoint.
2 marks
Apply each change to its matching entry.
$(2, 3, 1)$
Start at $(3, 2, 1)$ . Move 2 in negative $y$ . Give the endpoint.
2 marks
$2 - 2 = 0$ in the $y$ -entry.
$(3, 0, 1)$
Start at $(1, 1, 1)$ . Move 4 in positive $x$ then 2 in negative $x$ . Give endpoint and distance.
2 marks
Final $x = 1 + 4 - 2 = 3$ . $Distance = 4 + 2 = 6$ .
$(3, 1, 1)$ ; 6 units.
Describe the change from $A (2, 5, 1)$ to $B (2, 1, 4)$ .
2 marks
Compare corresponding entries: $Δ x = 0$ , $Δ y = - 4$ , $Δ z = 3$ .
4 units in negative $y$ and 3 units in positive $z$ ; $x$ stays unchanged.
Find the length of an axis-aligned route from $(1, 2, 3)$ to $(4, 1, 5)$ without reversing direction.
2 marks
The leg lengths are 3, 1 and 2. Add: $3 + 1 + 2 = 6$ .
6 units.
Mia says returning to the start means the total distance is zero. She walks 3 units out and 3 back. Correct her.
2 marks
Both legs contribute distance: $3 + 3 = 6$ .
Her final position change is zero, but she travels 6 units.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Coordinate transformations review; identifying positions using three coordinates; interpreting 3D axes; describing routes and relative positions; comparing diagrams, views and digital representations. This is spatial location, not senior vector algebra.

Official curriculum source ↗

In this lesson

Understand the idea.

Record the position after each instruction

Compare locations and plan a route

See the method in action.

Follow a route

Move 3 units in positive x.

Move 2 units in positive z.

Move 1 unit in negative y.

Add the lengths of all three legs.

Plan between two points

Change x from 1 to 3.

Change y from 4 to 1.

Change z from 2 to 5.

Calculate this route's total length.

Try it yourself.

Check your understanding

Keep practising