The idea, step by step
Make every decision precise
An algorithm is a finite list of instructions that can be followed consistently. A shape-sorting algorithm uses observable or given properties, such as the number of sides, equal lengths and right angles. Avoid instructions like “looks like a square”, because a rotated drawing may mislead.
Read an AND condition carefully: both statements must be true. In the ordered table, stop when a condition gives an output. This ensures a square receives the most specific name before the broader rectangle rule can catch it.
| 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 special cases. The table should identify a square, a non-square rectangle, a non-square rhombus and a general quadrilateral. Also test a triangle: it should stop at step 1 instead of falling through to a four-sided category.
A counterexample exposes a faulty rule. “Four equal sides means square” fails for a slanted rhombus. Repair it by requiring four right angles as well. You can express the same algorithm as a flowchart or implement its decisions in a simple program; the mathematical tests must stay the same.
A common trap
Any order of correct shape definitions gives the same most specific classification.
Instead: With an early stop, order matters. A square satisfies the rectangle rule too, so the square test must come first if the most specific name is required.
See the thinking
Trace the decisions
Use the sorting table for a shape with four sides, four equal side lengths and four right angles.
- Step 1 does not apply because there are four sides.
- Both conditions in step 2 are true.
- Output Square and stop; do not continue to the broader rectangle or rhombus checks.
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?
- A square has four right angles, so it stops at Rectangle.
- The output is a true family name but not the most specific name requested.
- Test four right angles AND four equal sides first; if both hold output Square. Then test the general rectangle condition.
Check your understanding
The table receives a triangle with two equal sides. What is its output?
Show answer
Not a quadrilateral.
It has three sides, so step 1 applies. The algorithm stops before testing equal sides or angles.
Your turn
6 questions · 14 marks · Work at your own pace.
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 ↗