Sequences of steps and decisions | Year 9

01 / LEARN

Understand the idea.

Make each step testable

A geometric algorithm specifies a finite procedure from stated inputs to a required output. Name the initial points, tools and order of operations. Each instruction should identify a visible action, such as joining two named arc crossings.

When an action depends on a condition, state the decision explicitly. For a segment of positive length $L$ , test whether the compass radius satisfies $r > \frac{L}{2}$ . If it fails, choose a suitable larger radius before drawing the arcs.

Order and repetition matter

Trace a single point through a sequence before transforming a whole figure. Translation followed by enlargement can produce a different point from enlargement followed by translation. The order belongs to the algorithm, along with any scale factor and centre.

For a repeated construction, specify the action, number of repetitions and stopping point. To construct a regular hexagon in a circle, step the radius around the circumference six times, then join consecutive marked points. Without stopping instructions, the procedure is incomplete.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Trace an ordered pair of moves

Start at P(1,2), translate by (2,1), then enlarge by factor 2 from the origin.

Worked solution

1
Apply the translation first.
$(1, 2) + (2, 1) = (3, 3)$
Translation adds the given displacement to the starting point.
2
Enlarge the translated point.
$2 (3, 3) = (6, 6)$
The next instruction doubles both coordinates from the origin.
3
Trace the reversed order for comparison.
$\begin{aligned} 2 (1, 2) & = (2, 4) \\ (2, 4) + (2, 1) & = (4, 5) \end{aligned}$
If enlargement comes first, the later translation is not itself doubled.
4
Report the output for the specified order.
$(6, 6) \neq (4, 5)$
The requested translation-then-enlargement procedure ends at $(6, 6)$ , demonstrating that this order matters.

Answer

(6,6).

EXAMPLE 02

A construction decision

Give the radius decision for a bisector of a segment of length L.

Worked solution

1
State the input and required test.
$L > 0, r > \frac{L}{2}$
A perpendicular-bisector construction needs a positive segment length and two arc crossings.
2
Handle a failed radius choice.
$r \leq \frac{L}{2} \Rightarrow choose a new r > \frac{L}{2}$
Smaller circles do not meet; at equality they only touch once.
3
Name the endpoints and draw the accepted arcs.
$A U = B U = A V = B V = r$
Call the endpoints $A$ and $B$ . Use the same accepted radius at both, naming the two crossings $U$ and $V$ .
4
Join the crossings and stop.
$\begin{array}{c} M = A B \cap U V \\ U V ⟂ A B, A M = M B \end{array}$
The equal-distance crossings determine the perpendicular bisector, which is the required output.

Answer

Use the same r>L/2 at both endpoints.

03 / YOUR TURN

Try it yourself.

Check your understanding

Translate (2,−1) by (1,3), then double its coordinates. Find the output.

Show answer & reasoning

(6,4)

The translation gives $(2, - 1) + (1, 3) = (3, 2)$ . Doubling this intermediate point gives $2 (3, 2) = (6, 4)$ .

Keep practising

6 worksheet questions · 15 marks

Open worksheet Worked solutions

Read the questions online

Start at $(- 1, 2)$ , translate by $(3, - 1)$ , then enlarge by factor 2 from the origin.
2 marks
From $(0, 0)$ , repeat “move 2 units right” four times. Find the final point.
2 marks
Complete the decision: for segment length $L$ , increase radius $r$ if ___.
2 marks
Enlarge $P (1, 1)$ by factor 3 from the origin, then translate by $(2, 0)$ . Find the image.
2 marks
Describe a finite compass algorithm to mark a regular hexagon on a circle.
4 marks
An algorithm says “if four sides are equal, the quadrilateral is a square”. Give a counterexample and repair the decision.
3 marks

Read the worked solutions

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

Start at $(- 1, 2)$ , translate by $(3, - 1)$ , then enlarge by factor 2 from the origin.
2 marks
Translation gives $(2, 1)$ .
Enlargement gives $(4, 2)$ .
$(4, 2)$
From $(0, 0)$ , repeat “move 2 units right” four times. Find the final point.
2 marks
Four moves give horizontal change $4 \times 2 = 8$ , with no vertical change.
$(8, 0)$
Complete the decision: for segment length $L$ , increase radius $r$ if ___.
2 marks
The radius must be greater than half the segment to produce two arc crossings.
$r \leq \frac{L}{2}$
Enlarge $P (1, 1)$ by factor 3 from the origin, then translate by $(2, 0)$ . Find the image.
2 marks
Enlargement gives $(3, 3)$ .
Translation gives $(5, 3)$ .
$(5, 3)$
Describe a finite compass algorithm to mark a regular hexagon on a circle.
4 marks
Keep the compass width equal to the radius.
Six equal chords close the hexagon.
Stop after six sides.
Choose a starting point, step the circle radius around the circumference six times, then join neighbouring vertices.
An algorithm says “if four sides are equal, the quadrilateral is a square”. Give a counterexample and repair the decision.
3 marks
Equal sides alone do not force right angles.
A quadrilateral with all sides equal and a right angle is a square.
A non-square rhombus; also require a right angle.

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Curriculum reference & lesson scope

Australian Curriculum v9 / South Australian Curriculum

Angle reasoning consolidation; enlargements using coordinates/digital geometry; invariant and changing properties; geometric constructions; sequences of steps and decisions; testing/refining construction algorithms; explaining geometric reasoning.

Official curriculum source ↗

In this lesson

Understand the idea.

Make each step testable

Order and repetition matter

See the method in action.

Trace an ordered pair of moves

Apply the translation first.

Enlarge the translated point.

Trace the reversed order for comparison.

Report the output for the specified order.

A construction decision

State the input and required test.

Handle a failed radius choice.

Name the endpoints and draw the accepted arcs.

Join the crossings and stop.

Try it yourself.

Check your understanding

Keep practising