design and create algorithms involving a sequence of steps and decisions that will sort and classify sets of shapes according to their attributes, and describe how the algorithms work.
We are learning to design precise spatial algorithms that use ordered steps, decisions, repetition and stopping conditions.
A spatial algorithm is a finite set of precise instructions for constructing, transforming or navigating a geometric object. Its input and coordinate system must be stated, and a translation needs a vector, a reflection needs a mirror line, and a rotation needs a centre, angle and direction.
Iteration repeats a defined block of steps, but the algorithm must say whether each new transformation acts on the original object or the latest image. A counter or geometric condition controls repetition, and a stopping condition prevents an endless loop or an unwanted duplicate image.
Decision branches make the next action depend on a test such as whether a point lies inside a boundary. A reliable algorithm gives an action for every possible result, can be traced manually on a small input, and produces the same output whenever the same input and conditions are used.
Success criteria
I can state the input, transformation details and order of steps in a reproducible spatial algorithm.
I can trace iterations and decision branches to predict an algorithm's geometric output.
I can identify ambiguity or a faulty stopping rule and repair the algorithm.
Key vocabulary
algorithm
An algorithm is a finite, ordered set of instructions that produces a reproducible result.
iteration
An iteration is one repetition of a specified group of steps.
decision branch
A decision branch selects the next step according to whether a stated condition is true or false.
stopping condition
A stopping condition is the test that ends a repeated process at the required output.
Visual models and representations
Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.
1InputRecord the original triangle and set image count to 1.
2DecisionIf image count equals 6, output the design and stop.
3TransformOtherwise rotate the latest triangle 60 degrees anticlockwise about the origin.
4RepeatRecord the image, add 1 to the count and return to the decision.
A four-step sixfold-pattern algorithm records the original triangle, tests whether six images exist, rotates the latest image 60 degrees…
Ambiguous instructionPrecise repairMove the shape over.Translate every vertex by the vector (4, −2).Repeat until it looks complete.Repeat until exactly 8 images have been recorded.If it leaves the grid, do something.If any vertex leaves the grid, stop; otherwise record the image.
A comparison table replaces ambiguous spatial instructions with precise translation vectors, image counts and complete boundary decisions.
4 worked numerical & application examples
Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.
Example 1
Writing a finite algorithm for a sixfold rotation design.
Input one triangle, record it as the first image, and set image_count to 1.
Test whether image_count equals 6; if it does, output the design and stop.
If image_count is less than 6, rotate the latest image 60 degrees anticlockwise about the origin, record the new image and add 1 to image_count.
Return to the decision, producing images at 0, 60, 120, 180, 240 and 300 degrees without adding a duplicate at 360 degrees.
Final answer: The algorithm records exactly six equally spaced images and stops when image_count reaches 6.
Check: Tracing the counter gives five rotations after the original, and one further rotation would reproduce the original position, so the stopping condition is correct.
Example 2
Tracing and debugging a repeated translation.
Start with triangle A(0, 0), B(2, 0), C(1, 2), record it, and set triangle_count to 1.
While triangle_count is less than 4, translate every vertex of the latest image by (3, 0), record the image and add 1 to triangle_count.
The next A-vertices are (3, 0), (6, 0) and (9, 0), with the B- and C-vertices changing by the same vector each time.
Stop at triangle_count equals 4, because using 'repeat until the row looks long enough' would not give a reproducible output.
Final answer: The output contains four congruent triangles whose A-vertices are (0, 0), (3, 0), (6, 0) and (9, 0).
Check: Every corresponding coordinate differs by the fixed vector (3, 0), and the counter ends at exactly four recorded triangles.
Example 3
Application problem 1
Problem: An algorithm says: rotate a shape 90° clockwise about the origin, repeat 4 times, then stop. Explain the final position and identify the condition that guarantees the algorithm terminates.
Plan: Represent the information first, then calculate, interpret and independently check the result.
Work: Four 90° rotations total 360°, so the shape returns to its starting position. The fixed repeat count of 4 is the termination condition.
Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Final answer: Four 90° rotations total 360°, so the shape returns to its starting position. The fixed repeat count of 4 is the termination condition.
Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Example 4
Application problem 2
Problem: Design an algorithm that creates a row of four congruent triangles from A(0, 0), B(2, 0), C(1, 2) using translation by (3, 0), and include an explicit decision and stopping rule.
Plan: Count the original as the first triangle and test the count before adding each new image.
Work: Record the original and set count to 1; if count equals 4, output and stop, otherwise translate every vertex of the latest triangle by (3, 0), record it, add 1 to count and return to the decision, producing A-vertices at (0, 0), (3, 0), (6, 0) and (9, 0).
Interpret: The algorithm combines a fixed transformation with a counter-controlled loop that outputs exactly four triangles.
Final answer: Record the original and set count to 1; if count equals 4, output and stop, otherwise translate every vertex of the latest triangle by (3, 0), record it, add 1 to count and return to the decision, producing A-vertices at (0, 0), (3, 0), (6, 0) and (9, 0).
Check: The algorithm combines a fixed transformation with a counter-controlled loop that outputs exactly four triangles.
Common misconceptions
Common mistake: The instruction 'rotate the triangle' is precise enough for a computer or another person to follow.
Correction: A rotation must specify the centre, angle, direction and which image is rotated.
Common mistake: A repeated geometric process will stop when the design looks finished.
Correction: A reproducible algorithm needs a measurable stopping condition such as an image count.
Common mistake: Testing only the first iteration proves that an algorithm works for all later iterations.
Correction: Later values can trigger a different branch or reveal a faulty loop, so boundary and stopping cases must also be traced.
10 important problems to solve
Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.
1. State the missing information in the instruction 'Rotate point P by 90 degrees.'
Check answer
Answer: The instruction must state the centre of rotation and whether the turn is clockwise or anticlockwise.
Hint: Ask where the point turns around and in which direction it moves.
Why: An angle alone does not uniquely define a rotation.
2. A loop records the original square as image 1 and adds one translated image each time until image_count equals 5; how many translations are performed?
Check answer
Answer: The loop performs 4 translations.
Hint: Count the additional images needed after image 1.
Why: The original is already counted, so four new images are needed to reach a total of five.
3. Explain the purpose of a stopping condition in a spatial algorithm.
Check answer
Answer: A stopping condition ends repetition when the required geometric output has been produced.
Hint: Consider what would happen if a loop had no instruction telling it when to end.
Why: A measurable stopping rule makes the algorithm finite and reproducible.
4. Start at point (1, 2) and repeat the translation (3, −1) twice; what are the coordinates after each iteration?
Check answer
Answer: After the first iteration the point is (4, 1), and after the second iteration it is (7, 0).
Hint: Use the first image, not the original point, as the input to the second iteration.
Why: Each iteration adds the same vector to the latest image coordinate.
5. Repair the instruction 'Move every vertex to the right and down' so it defines one exact coordinate transformation.
Check answer
Answer: One valid repair is 'Translate every vertex by the vector (4, −2), so add 4 to x and subtract 2 from y.'
Hint: Choose and state an exact horizontal change and an exact vertical change.
Why: A numerical vector replaces vague direction words with a reproducible movement.
6. An algorithm tests whether any transformed vertex has x greater than 10; write complete actions for both outcomes so the design remains inside the grid.
Check answer
Answer: If any vertex has x greater than 10, discard that image and stop; otherwise, record the image and continue to the next iteration.
Hint: Write one instruction for when the boundary is crossed and one for when it is not.
Why: A complete decision specifies distinct actions for both the true and false results.
7. An algorithm reflects the latest image in the y-axis repeatedly and is meant to make two distinct images; identify an appropriate stopping condition.
Check answer
Answer: Record the original, reflect it once in the y-axis, record the image, and stop when image_count equals 2.
Hint: Trace the effect of applying the same reflection twice.
Why: A second reflection would return to the original position and create a duplicate.
8. Trace this algorithm: begin with P(1, 0), set count to 0, and while count is less than 3 rotate the latest point 90 degrees anticlockwise about the origin and add 1 to count; list every new coordinate.
Check answer
Answer: The three new coordinates are (0, 1), (−1, 0) and (0, −1), after which count equals 3 and the loop stops.
Hint: Record the coordinate and counter value after each rotation before testing the loop again.
Why: Repeated 90-degree anticlockwise rotations apply (x, y) to (−y, x) to the latest point three times.
9. Design an algorithm that creates a row of four congruent triangles from A(0, 0), B(2, 0), C(1, 2) using translation by (3, 0), and include an explicit decision and stopping rule.
Check answer
Answer: Record the original and set count to 1; if count equals 4, output and stop, otherwise translate every vertex of the latest triangle by (3, 0), record it, add 1 to count and return to the decision, producing A-vertices at (0, 0), (3, 0), (6, 0) and (9, 0).
Hint: Count the original as the first triangle and test the count before adding each new image.
Why: The algorithm combines a fixed transformation with a counter-controlled loop that outputs exactly four triangles.
10. An algorithm says: rotate a shape 90° clockwise about the origin, repeat 4 times, then stop. Explain the final position and identify the condition that guarantees the algorithm terminates.
Check answer
Answer: Four 90° rotations total 360°, so the shape returns to its starting position. The fixed repeat count of 4 is the termination condition.
Hint: Represent the information first, then calculate, interpret and independently check the result.
Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Curriculum coverage and elaborations
Content description: design and create algorithms involving a sequence of steps and decisions that will sort and classify sets of shapes according to their attributes, and describe how the algorithms work.
International curriculum mapping
The Australian Curriculum code above is exact. Victorian Year 7, NSW Stage 4, US Grade 7, England Key Stage 3, New Zealand Level 4 and comparable international curricula contain broadly related learning, but code-to-code equivalence varies by jurisdiction.
🎥 Optional Video Lesson
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Math Antics — Compare quadrilateral properties for classification. The video’s US term “trapezoid” corresponds to “trapezium” in Australian usage.
As you watch: Can a shape belong to more than one category?
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Curriculum equivalents: Victoria, NSW and international
Curriculum equivalents for Design and create algorithms involving a sequence of steps and...
Mapped skill: design and create algorithms involving a sequence of steps and decisions that will sort and classify sets of shapes according to their attributes, and describe how the algorithms work
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Topic reference: AC9M7SP04 — Spatial Algorithms and Decision Processes