Year 9 · 80 minutes · Optimisation
Why can trying harder move a model further from its goal?
Gradient descent updates a parameter by subtracting learning rate times the loss gradient. For this one-parameter quadratic, loss is (w−3)² and gradient is 2(w−3). Small steps converge slowly; suitable steps converge quickly; overly large steps oscillate or diverge. The simple surface makes each update inspectable. Real neural losses can be non-convex, noisy and much harder to optimise.
Work through w=0 with learning rates 0.1, 0.5 and 1.1. Prepare a graph of the quadratic. Explain that an oversize parameter triggers an explicit numerical-stop state rather than silently clipping the result.
Evaluate a quadratic and follow a repeated numeric update. Useful earlier investigations: y8-neuron Use pairs for investigation, with operator/reviewer swaps after each comparison. Keep individual predictions, journals and a short oral defence so group work does not hide understanding.
Australian Curriculum Version 9 · Digital Technologies: AC9TDI10P06, AC9TDI10P08. Selected aspects only. This activity contributes evidence; it does not cover the full descriptor or achievement standard. A programming descriptor is not claimed for merely moving controls. ACARA AI curriculum connection · V9 Technologies These are planning connections, not ACARA endorsement or exhaustive descriptor alignment.
Locate w=0 and reason about the direction of decreasing loss.
Ask: “Why subtract the gradient here?”
Listen for: “The negative gradient points toward lower local loss.”
Calculate updates for a chosen rate before running.
Ask: “Will the next point cross the minimum?”
Listen for: “That depends on the size of the update.”
Run individual updates, inspect arrows and loss history, then reset and compare equal-step runs at different rates.
Ask: “Is a larger movement the same as more learning?”
Listen for: “No, it can increase the loss.”
Set learning rate 1 on the quadratic and observe repeated equal-distance jumps.
Ask: “Why does the loss fail to decrease?”
Listen for: “Each step overshoots by the same distance.”
Choose a fixed or decaying rate, state a stopping criterion and compare with the baseline from the same start.
Ask: “What evidence tells you to stop rather than take another step?”
Listen for: “A small gradient or sufficiently small loss change.”
Submit two calculated updates and stable/unstable traces.
Ask: “Which part of this result depends on the simple quadratic?”
Listen for: “The exact safe learning-rate range and guaranteed shape.”
Bigger steps always learn faster.
Calculate two updates from w=0 at a chosen learning rate before stepping the optimiser.
At learning rate 1, this quadratic oscillates without improvement; above 1, the distance from the minimum grows.
Specify a stable update schedule and stopping rule, then compare equal-budget traces from the same initial parameter.
Check arithmetic for two updates and interpretation of increasing loss. Ask learners to separate optimisation success from generalisation.
Use the quadratic and whole-number starting parameters; provide gradient values before asking for differentiation.
Add a second parameter with a steeper curvature and investigate why one learning rate can be unstable along one axis.
An optimiser notebook page with numeric updates, traces and a stopping policy.
All loss functions are synthetic. Divergent computations are halted explicitly and no external training job is launched.
Optimise many quadratic surfaces with different curvature on MPS/CUDA, recording stability regions and synchronised timing rather than assuming GPU speedups.
| Criterion | Beginning | Secure | Extending |
|---|---|---|---|
| Update mechanics | Moves toward a minimum by intuition | Calculates gradient and update | Predicts oscillation and divergence |
| Experiment design | Compares unequal starts or budgets | Uses matched starts and steps | Justifies stopping and limits transfer to complex losses |
Australian Curriculum Version 9 · Digital Technologies
References: AC9TDI10P06, AC9TDI10P08. Read the current source (checked 2026-09-07).
Evidence to assess: An optimiser notebook page with numeric updates, traces and a stopping policy.
Selected aspects only. This activity contributes evidence; it does not cover the full descriptor or achievement standard. A programming descriptor is not claimed for merely moving controls. Moderate the supplied illustrative responses against your school unit and current achievement standard.
These are planning estimates to test with your class. A short session develops one supported claim; it does not compress the whole senior project.
| Stage | 45 minute focus | 60 minute investigation |
|---|---|---|
| Readiness and prediction | 0–5 | 0–5 |
| Trace the supplied example | 5–13 | 5–15 |
| Author and run cases | 13–25 | 15–35 |
| Counterexample and redesign | 25–35 | 35–45 |
| Explain and discuss | 35–42 | 45–55 |
| Export and handover | 42–45 | 55–60 |
For a longer project, use three 50-minute sessions. Session 1 (0–50): readiness, model, hypothesis and initial cases. Export a project and record the next test. Session 2 (50–100): reopen, check settings, author counterexamples and revise the design. Export the changed project and identify unresolved evidence. Session 3 (100–150): independent peer test, final artefact, individual explanation and moderation. If using two 60-minute sessions, stop at minute 60 after saving the first comparison; use 60–120 for redesign, independent test and defence.
Entry check: Evaluate a quadratic and follow a repeated numeric update. Ask the learner to demonstrate it before choosing the level of support.
Preparation: allow about 15 minutes to run the starter, print the cards and check a project can be reopened. This estimate has not yet been measured in a classroom pilot.
Read the entry question aloud, model one row, and label the units. Offer the case table as a large-print sheet. Keep mathematical derivations optional until the learner can explain the comparison.
For one device, use a projector: one pair predicts, one operates, and the class records on paper. Swap roles after the first comparison. For individual access, support keyboard controls and a written table equivalent to each visual. Learners may explain orally or with an annotated diagram. Never require personal data, a recorded voice, or a photograph.
Mixed readiness: if the entry check is difficult, use the linked prerequisite and the first two case cards; retain the same central question. If secure, ask the learner to design an unseen test and state which explanation it could disprove.
For each rate, update w ← w − rate × 2(w−target). Retain every step and loss.
Starting parameters: Target = 3
3 cases calculated from your supplied inputs. Compare the evidence with your prediction.
| start | steps | final | loss |
|---|---|---|---|
| -2 | 4 | 0.952 | 4.194304 |
| 0 | 4 | -3.2208 | 38.698353 |
| 8 | 3 | 3 | 0 |
These are authored examples, not work collected from children. Assess reasoning using the lesson rubric, not whether the first prediction was correct.
Beginning: “It worked because the result looks right.” This identifies no exact case, control or measurement. Ask the learner to point to one row and say what happened.
Developing: “In the first case I recorded start: -2; steps: 4; final: 0.952; loss: 4.194304.” This cites evidence, but does not yet explain how the result follows from the rule. Ask the learner to trace the relevant step.
Secure: “For the first supplied case, start: -2; steps: 4; final: 0.952; loss: 4.194304. I can trace it using this mechanism: For each rate, update w ← w − rate × 2(w−target). Retain every step and loss. My result supports a claim about these supplied cases. It does not establish that the same result holds outside them.” Look for an accurate trace, the actual settings and a bounded claim; accept equivalent oral or visual evidence.
Extending: The learner constructs and reruns a new case, reports whether the first explanation survives, and defends a revised design. Use this concrete challenge: Specify a learning-rate schedule and compare repeated updates from two starting points. Require the original and changed evidence and this boundary: This convex quadratic is a controlled teaching case, not a neural training benchmark.
Moderation: first assess independently against each lesson criterion. Compare the exact trace or artefact that led to your judgement. Resolve differences using evidence, not polished language. Keep each learner's individual explanation even when the artefact was produced in a group.