Understand the idea.
Make every decision precise
An algorithm is a finite sequence of precise instructions. A shape-sorting rule should test given or verified properties, such as side count, equal lengths and right angles. “Looks square” is unreliable because orientation and sketch accuracy can mislead.
Follow the supplied table in order and stop at its first output. An AND condition requires both properties. The square test therefore checks both four equal sides and four right angles before a broader rectangle or rhombus test can stop the process.
| Step | Condition | Output or next step |
|---|---|---|
| 1 | Not exactly four sides | Not a quadrilateral; stop |
| 2 | Four right angles AND four equal sides | Square; stop |
| 3 | Four right angles | Rectangle; stop |
| 4 | Four equal sides | Rhombus; stop |
| 5 | None of the earlier conditions | Other quadrilateral; stop |
Test the boundaries, then improve the rules
Test ordinary cases and boundary cases: a square, a non-square rectangle, a non-square rhombus, another quadrilateral and a triangle. The triangle should stop at the side-count test before any four-sided classification is considered.
A counterexample shows why a proposed rule fails. Four equal sides alone do not guarantee a square: a rhombus without right angles passes that test. Add the right-angle condition and place this more specific test before the broader family tests.
See the method in action.
Trace the decisions
Use the sorting table for a shape with four sides, four equal side lengths and four right angles.
Worked solution
- 1
Trace the first condition.
The shape has exactly four sides, so the “not a quadrilateral” output does not apply.
- 2
Evaluate the square condition.
The shape has both four right angles and four equal sides, so both parts of the AND condition are true.
- 3
Output Square and stop.
Step 2 succeeds; the ordered algorithm does not continue to the broader rectangle or rhombus rules.
Square.
Repair an early stop
A program first asks “Does the shape have four right angles?” and outputs Rectangle if yes. Why does this prevent it giving the most specific name for a square?
Worked solution
- 1
Trace the original first test for a square.
A square has four right angles, so the program immediately outputs Rectangle and stops.
- 2
Compare that output with the requested precision.
Rectangle is a correct family name, but Square is the more specific classification for this input.
- 3
Insert the more specific test first.
First require four right angles AND four equal sides; if both hold, output Square and stop. Otherwise continue to the rectangle test.
- 4
Check both square and non-square rectangle cases.
A square now stops at Square. A rectangle with unequal adjacent sides fails the square test and then correctly stops at Rectangle.
Check the square condition before the rectangle condition.
Try it yourself.
Check your understanding
The table receives a triangle with two equal sides. What is its output?
Show answer & reasoning
Not a quadrilateral.
A triangle has three sides, so it satisfies step 1: not exactly four sides. The output is “Not a quadrilateral”, and the program stops before testing its equal sides.
Keep practising
6 worksheet questions · 14 marks
Read the questions online
- Use the sorting table for a four-sided shape with four right angles but not four equal sides. State the output.2 marks
An ordered algorithm for four-sided shapes. Stop at the first output. Step Condition Output or next step 1 Not exactly four sides Not a quadrilateral; stop 2 Four right angles AND four equal sides Square; stop 3 Four right angles Rectangle; stop 4 Four equal sides Rhombus; stop 5 None of the earlier conditions Other quadrilateral; stop - Use the sorting table for a four-sided shape with four equal sides but no right angles. State the output.2 marks
An ordered algorithm for four-sided shapes. Stop at the first output. Step Condition Output or next step 1 Not exactly four sides Not a quadrilateral; stop 2 Four right angles AND four equal sides Square; stop 3 Four right angles Rectangle; stop 4 Four equal sides Rhombus; stop 5 None of the earlier conditions Other quadrilateral; stop - The rule “If a shape has four equal sides, output Square” is incorrect. Give a counterexample and repair the condition.3 marks
- A five-sided polygon is given to the table. At which step does it stop, and what is the output?2 marks
An ordered algorithm for four-sided shapes. Stop at the first output. Step Condition Output or next step 1 Not exactly four sides Not a quadrilateral; stop 2 Four right angles AND four equal sides Square; stop 3 Four right angles Rectangle; stop 4 Four equal sides Rhombus; stop 5 None of the earlier conditions Other quadrilateral; stop - Write an ordered side-based algorithm that gives the most specific name Equilateral, Isosceles or Scalene for a triangle.3 marks
- A four-sided shape has no right angles and not all sides equal. The table outputs Other quadrilateral. Does this prove it has no parallel sides? Explain.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Use the sorting table for a four-sided shape with four right angles but not four equal sides. State the output.2 marks
An ordered algorithm for four-sided shapes. Stop at the first output. Step Condition Output or next step 1 Not exactly four sides Not a quadrilateral; stop 2 Four right angles AND four equal sides Square; stop 3 Four right angles Rectangle; stop 4 Four equal sides Rhombus; stop 5 None of the earlier conditions Other quadrilateral; stop Step 2 fails because the sides are not all equal.
Step 3 succeeds because all four angles are right.
Rectangle. - Use the sorting table for a four-sided shape with four equal sides but no right angles. State the output.2 marks
An ordered algorithm for four-sided shapes. Stop at the first output. Step Condition Output or next step 1 Not exactly four sides Not a quadrilateral; stop 2 Four right angles AND four equal sides Square; stop 3 Four right angles Rectangle; stop 4 Four equal sides Rhombus; stop 5 None of the earlier conditions Other quadrilateral; stop Steps 2 and 3 fail because the angles are not all right.
Step 4 succeeds because all four sides are equal.
Rhombus. - The rule “If a shape has four equal sides, output Square” is incorrect. Give a counterexample and repair the condition.3 marks
A rhombus can have four equal sides without any right angles.
Adding the right-angle condition distinguishes a square.
A non-square rhombus is a counterexample. Require four equal sides AND four right angles. - A five-sided polygon is given to the table. At which step does it stop, and what is the output?2 marks
An ordered algorithm for four-sided shapes. Stop at the first output. Step Condition Output or next step 1 Not exactly four sides Not a quadrilateral; stop 2 Four right angles AND four equal sides Square; stop 3 Four right angles Rectangle; stop 4 Four equal sides Rhombus; stop 5 None of the earlier conditions Other quadrilateral; stop Five is not exactly four.
The first condition is true, so no later step is used.
Step 1; Not a quadrilateral. - Write an ordered side-based algorithm that gives the most specific name Equilateral, Isosceles or Scalene for a triangle.3 marks
An equilateral triangle also has at least two equal sides.
Therefore the all-three-equal condition must be tested before the broader at-least-two condition.
First test all three sides equal: Equilateral. Otherwise test at least two equal sides: Isosceles. Otherwise: Scalene. - A four-sided shape has no right angles and not all sides equal. The table outputs Other quadrilateral. Does this prove it has no parallel sides? Explain.2 marks
Other quadrilateral means only that earlier categories were not selected.
For example, a non-rectangular parallelogram with unequal adjacent sides reaches that output.
No. The table has not tested parallel sides.
Curriculum reference & lesson scope
Australian Curriculum v9 / SA Mathematics
Year 7 angle relationships, polygon properties and classification algorithms. Chapter grouping follows the approved Ascent outline.
Official curriculum source ↗