An algorithm can decide whether two triangles are congruent, similar or neither by asking about their equal parts in a careful order: the congruence tests first, then the similarity conditions, and 'not enough to tell' when nothing fits. You test it on one pair of every kind, find what it misses, and fix it.
A design program has to decide whether two parts it is given will fit each other exactly, fit after scaling, or not fit at all. What questions should it ask, and in what order?
An algorithm can decide whether two triangles are congruent, similar or neither. It looks at the parts that are known to be equal (or in the same ratio) and asks yes/no questions about them.
Three things make it work:
Order: test congruence before similarity, because congruent triangles are similar too.
Every case: each test (SSS, SAS, ASA or AAS, RHS, and AA, SSS and SAS for similarity) has a path, and "not enough to tell" catches the rest.
Testing: run it on one pair of every kind, including SSA, which must not be called congruent.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Angles only
TryTry it on the board and compare the result with your prediction.
Six pairs
Answer "yes" only when the description tells you the parts are equal. Open The whole key to see the algorithm as numbered steps.
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Follow the algorithm for all six pairs of triangles. Answer each question from what the description says is equal. How does it handle the twice-as-big pair, compared with your prediction?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Build and check it
Why must the congruence questions come before "two pairs of angles equal"?
Read Raj's algorithm. Which step lets the SSA pair through?
Put the steps of a short algorithm in order. It leaves out RHS and the other similarity tests; the Run algorithm has them.
Pieces (click to add)
Your program
Raj tests his algorithm on triangles with sides 6 cm and 8 cm and a 30° angle opposite the 6 cm side, and it says congruent. Click the line that is wrong.
ModifyChange the problem on purpose and see what happens.
Find a gap and fix it
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This algorithm has a new step G. Run it on all seven pairs. Does every pair, including the new one, reach the right answer?
Use the board. This ticks when you get there.
In the improved algorithm, how many questions does it ask before it says 'Neither, or not enough to tell'?
PractiseA few on your own, to check you have it.
Tracing algorithms
Review: in parallelogram ABCD, angle A = 75°. What is angle B, in degrees?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Your own algorithm
Write an algorithm that sorts pairs of triangles into congruent, similar and not enough to tell. Draw it as a flowchart or write numbered steps. Swap with a partner: trace their algorithm on your test pairs, and describe in words how it works.
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Need a hint?
Start from the tests you know: SSS, SAS, ASA or AAS, RHS.
Another hint
Write one test pair of triangles for each answer, including SSA.
For teachers
Students follow, order, fix and design algorithms that decide whether two triangles are congruent, similar or neither, testing them on one pair of every kind (AC9M8SP04, nodes 1 and 2). About 45 minutes.
I can follow an algorithm for two triangles and say which test it used.
I can find a pair of triangles an algorithm gets wrong, and fix it.
I can design and test my own algorithm, and describe how it works.
Before class
Do the Run board yourself first. Students need the congruence tests, similar triangles and the shape-classifying algorithms from Year 7.
The one idea: order matters (congruent before similar, because congruent triangles are similar too), every case needs a path, and the algorithm is only trusted once it has been tested on one pair of each kind. The common mistake is an algorithm that accepts any angle with two sides (Raj's SSA mistake).
The board is the same key used for classifying shapes; each specimen is a pair of triangles. Answer "yes" only when the description says the parts are equal. The improved key adds step G for SAS similarity, a gap students find in Modify.
Wider: compare two correct algorithms for the fewest questions; write the algorithm in Python with if and elif.
Running the lesson
Predict: ask what equal angles alone tell you.
Run: ask which question separates congruent from similar.
Investigate: ask Raj which angle the SSA pair has.
Modify: ask what test case would have found the gap.
Make: ask a partner to trace your algorithm on an SSA pair.
The hard part (on purpose)
Answering from what is known, not what could be worked out. Ask 'does the description say that?' before every yes.
When students get stuck
A student answers yes to SSS for the RHS pair (by working out the third side). Ask: Does the description give all three sides? Fix: Answer from what is given; RHS has its own question.
A student answers yes at step B for the RHS pair. Ask: Is the right angle between the hypotenuse and the 5 cm side? Fix: The right angle is between the two shorter sides; RHS has its own question at E.
A student puts the similarity question first. Ask: What would a congruent pair be called then? Fix: Ask the most specific question (congruent) first.
A student's algorithm has no answer for some pairs. Ask: Where does a 'no' at your last question go? Fix: End with a catch-all: 'not enough to tell'.
Stretch and support
Stretch: Write the algorithm in Python with if, elif and else, and test it on your six pairs. What is the fewest number of questions that separates all the answers?
Support: Give the questions on cards and the six test pairs on cards; students order the questions, then run each pair through.
Where it leads
This covers AC9M8SP04.
Where it leads: algorithms in code (the Code pathways), and similar triangles in trigonometry (Year 10).
What each step is for
angles only: Committing to what an angles-only algorithm says about a big and a small triangle shows whether they see that equal angles give similar, not congruent.
six pairs: Running the full algorithm on one pair of each kind shows every path, and that SSA ends at 'not enough to tell'.
build and check it: Ordering the steps shows why congruence comes first; an algorithm that accepts any angle lets SSA through (Raj); tracing a pair checks a path.
find a gap and fix it: A pair the first algorithm wrongly sends to 'not enough to tell' (SAS similarity) shows a gap; adding a step and re-testing is the fix.
tracing algorithms: Tracing pairs to their answers and describing the logic, mixed; SSA called congruent is the planted slip; the quadrilateral review keeps properties fresh.
your own algorithm: Designing and testing their own algorithm shows whether they order the tests, cover every case and check it.
Now you can
Follow an algorithm for two triangles and say which test it used.
Find a pair of triangles an algorithm gets wrong, and fix it.
Design and test my own algorithm, and describe how it works.