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Year 9AC9M9SP03

Algorithms from Constructions and Theorems

Design, test and refine algorithms involving a sequence of steps and decisions based on geometric constructions and theorems; discuss and evaluate refinements.

An algorithm turns a construction into steps anyone can follow. Testing it on an awkward case is what refines it.

Builds onYr 9 · Enlargement and What It Preserves

Method

Steps and decisions, in a fixed order
Same input, same output, every time.
It must terminate
A procedure that can loop forever is not an algorithm.
Test the edge cases
The awkward case is what exposes a missing step.
Refine, then say what you changed and why
The evaluation is part of the task.

Worked example

e.g. Write an algorithm to test whether two triangles are congruent.\text{Write an algorithm to test whether two triangles are congruent.}
  1. Start with the cheapest test to check.Are all three sides equal? If yes, SSS.\text{Are all three sides equal? If yes, } SSS.
  2. If not, move to the next test.Two sides and the included angle? If yes, SAS.\text{Two sides and the included angle? If yes, } SAS.
  3. Continue through the remaining tests.ASA, then RHSASA, \text{ then } RHS
  4. Add the exit if nothing matches.Otherwise, not congruent.\text{Otherwise, not congruent.}
  5. Test it on an awkward case: two sides and a non-included angle.SSA — must not return congruentSSA \text{ — must not return congruent}
  6. Refine. State the angle must be included, and say why.SSA can give two different triangles\text{SSA can give two different triangles}

Practice

Write the steps, then break your own algorithm before trusting it.

1
Why must an algorithm terminate?\text{Why must an algorithm terminate?}
AnswerOtherwise it never gives an answer.\text{Otherwise it never gives an answer.}
2
Which test should come first, and why?\text{Which test should come first, and why?}
AnswerSSS — three side lengths are easiest to compare.SSS \text{ — three side lengths are easiest to compare.}
3
Why does SSA fail as a test?\text{Why does SSA fail as a test?}
AnswerTwo different triangles can fit the same data.\text{Two different triangles can fit the same data.}
4
Write an algorithm to bisect an angle.\text{Write an algorithm to bisect an angle.}
AnswerArc from the vertex, arc from each crossing point, join the vertex to where they meet.\text{Arc from the vertex, arc from each crossing point, join the vertex to where they meet.}
5
How do you know an algorithm is reliable?\text{How do you know an algorithm is reliable?}
AnswerTest it on the hardest case, not the easiest.\text{Test it on the hardest case, not the easiest.}
6
What must a refinement report include?\text{What must a refinement report include?}
AnswerWhat changed, and why.\text{What changed, and why.}
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