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).
Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.
Predict and Run 10 min · Investigate 10 min · Modify 10 min · Practise and Make 15 min
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.
Answering from what is known, not what could be worked out. Ask 'does the description say that?' before every yes.
| What you will see | A question to ask | The fix |
|---|---|---|
| A student answers yes to SSS for the RHS pair (by working out the third side). | Does the description give all three sides? | Answer from what is given; RHS has its own question. |
| A student answers yes at step B for the RHS pair. | Is the right angle between the hypotenuse and the 5 cm side? | The right angle is between the two shorter sides; RHS has its own question at E. |
| A student puts the similarity question first. | What would a congruent pair be called then? | Ask the most specific question (congruent) first. |
| A student's algorithm has no answer for some pairs. | Where does a 'no' at your last question go? | End with a catch-all: 'not enough to tell'. |
List the equal parts of the two triangles.
If all three pairs of sides are equal, say "congruent (SSS)" and stop.
If not, and two pairs of sides and the angle between them are equal, say "congruent (SAS)" and stop.
If not, and two pairs of angles are equal, check the sides next.
If a matching pair of sides is equal too, say "congruent (ASA or AAS)"; if not, say "similar".
Otherwise, say "not enough to tell".
One possible solution for each task. Other correct programs are fine.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
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.
For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.
You are a patient maths coach for a Year 8 student in Australia. Help me understand the ideas in ONE lesson only, from the lesson "Algorithms for congruence". The ideas: - Designing a congruence/similarity classification algorithm: I can Students design a step-by-step algorithm that correctly classifies two triangles as congruent, similar, or neither, using the appropriate conditions. - Evaluating and describing algorithms: I can Students trace through a given algorithm with test cases, identify errors or limitations, and describe in plain language how it works. What understanding looks like: - Designs an algorithm that correctly classifies triangles using at least two conditions. - Includes clear decision points with yes/no branches. - Tests the algorithm with at least two examples and corrects any errors. - Correctly traces an algorithm with given inputs. - Identifies at least one error or limitation in a given algorithm. - Describes the algorithm's logic in plain language. Mistakes students often make here: - A student answers yes to SSS for the RHS pair (by working out the third side). - A student answers yes at step B for the RHS pair. - A student puts the similarity question first. How to coach me (rules from the Waypoints Kit coach): - Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question. - Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit). - If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go. - Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back. - After you help, ask me one reflective question, such as how I know my answer is right or what I would check. - Praise my effort and method, not my ability. - You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so. - Use simple words. - Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Algorithms for congruence".
Why must an algorithm for congruence not answer 'congruent' when two sides and any angle are equal?
If the angle is not between the sides (SSA), two different triangles can fit, so they might not be congruent.
This covers AC9M8SP04.
Where it leads: algorithms in code (the Code pathways), and similar triangles in trigonometry (Year 10).