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Year 8 · Neural networks
What can one artificial neuron actually compute?
Go to the investigation ↓A four-case neuron specification with a demonstrated architecture limit.
What can one weighted signal separate?
This artificial neuron multiplies each input by a weight, adds a bias and applies a sigmoid. It computes a number; it is not a miniature person. The weights determine how inputs influence the output. A single linear boundary can represent an AND-like decision but cannot separate XOR’s diagonal classes. A high activation is not automatically a calibrated probability.
Before this lesson: Substitute values into a weighted sum. Useful earlier investigations: The playground planner.
This artificial neuron multiplies each input by a weight, adds a bias and applies a sigmoid. It computes a number; it is not a miniature person. The weights determine how inputs influence the output. A single linear boundary can represent an AND-like decision but cannot separate XOR’s diagonal classes. A high activation is not automatically a calibrated probability.
Substitute values into a weighted sum. Useful earlier investigations: The playground planner.
Write z = w1x1 + w2x2 + b and sigmoid(z) = 1/(1+exp(−z)). Prepare AND and XOR truth tables. Explain that learners adjust parameters directly; automatic training is a later investigation.
ACARA V9 AI curriculum connection ↗ · Technologies ↗
A complete teaching sequence · 75 minutes
Calculate a weighted sum with inputs 1 and 0, then inspect the activated output.
Ask: “Which contribution disappears when its input is zero?”
Listen for: “That input’s weight times zero.”
Predict the output direction when w1 increases with x1=1 and all else fixed.
Ask: “Would that same change matter when x1=0?”
Listen for: “No, its contribution would remain zero.”
Adjust weights and bias while linked edge contributions, sum and output update. Test all four input pairs, not just the currently visible one.
Ask: “Why did changing the bias affect every case?”
Listen for: “It is added regardless of the inputs.”
Switch the target table to XOR and try to separate its diagonal positives with one boundary.
Ask: “Can one straight boundary put both opposite corners on one side?”
Listen for: “Not without including a negative corner.”
Return to AND, find parameters that classify all four cases, then document why this architecture is insufficient for XOR.
Ask: “What change is needed beyond more tuning?”
Listen for: “A different representation or a hidden layer.”
Submit weights, four sums and the architecture limitation.
Ask: “Does a 0.95 activation prove 95% accuracy?”
Listen for: “No; calibration requires evidence across cases.”
A neural unit thinks like a person.
XOR cannot be separated by a single linear boundary, regardless of tuning these weights.
Build an AND-like gate, verify four cases and propose a two-stage representation for XOR on paper.
Ask learners to calculate an unseen weighted sum and explain the XOR geometry. Do not require a formal proof, but reject a claim that more random tuning will necessarily solve it.
| Criterion | Beginning | Secure | Extending |
|---|---|---|---|
| Mechanism | Describes a thinking node | Calculates contributions and activation | Predicts zero-input and bias effects |
| Architecture critique | Tests only one input | Verifies all four AND cases | Explains XOR’s representational limitation |
Begin with step activation and integer sums, then introduce sigmoid. Provide a formula sheet.
Construct two hidden units for XOR and trace their outputs into a final unit.
Inputs are synthetic binary signals. No brain-like or human-understanding claims should be inferred from the diagram.
Shared device? Turn remembering off. A project file lets you continue on another device.
Your browser is the laboratory
Increase one weight with its input fixed at 1, then at 0. Predict both output changes before testing.
Inputs start at 1 and 0. Weights are 2 and 2; bias is −3. The neuron sums weighted inputs and bias, applies sigmoid and predicts 1 if the sum is at least zero. Test all four binary inputs.
The experiment opens after your prediction.
There is no penalty for being surprised.
Apple Silicon · PyTorch MPS
Use real MPS/CUDA matrix operations to evaluate thousands of weight pairs against AND and XOR truth tables. Change bias and compare the best and mean number correct over the weight grid.
Students extend the experiment in teams, documenting parameters, outputs and limitations.
The bundle contains lesson-specific working code, a configuration file, a reactive notebook, a deterministic CPU check and hardware setup instructions. Acceleration is reported only after a tensor operation and result read-back succeed.
Download Mac Studio investigation ↓unzip y8-neuron-mac-pathway.zip -d y8-neuron-mac cd y8-neuron-mac bash setup-mac.sh source .venv/bin/activate python experiment.py --device mps --output results marimo edit notebook.py
CPU and available-device execution status is recorded in the downloaded README and validation report. DGX Spark execution requires that hardware; static validation alone does not establish GPU compatibility or performance. The browser lesson remains fully available without this extension.
NVIDIA DGX Spark · PyTorch CUDA
Use real MPS/CUDA matrix operations to evaluate thousands of weight pairs against AND and XOR truth tables. Change bias and compare the best and mean number correct over the weight grid.
Students extend the experiment in teams, documenting parameters, outputs and limitations.
The bundle contains lesson-specific working code, a configuration file, a reactive notebook, a deterministic CPU check and hardware setup instructions. Acceleration is reported only after a tensor operation and result read-back succeed.
Download DGX Spark investigation ↓unzip y8-neuron-dgx-pathway.zip -d y8-neuron-dgx cd y8-neuron-dgx bash run-dgx.sh # Open the localhost notebook URL printed by the container.
CPU and available-device execution status is recorded in the downloaded README and validation report. DGX Spark execution requires that hardware; static validation alone does not establish GPU compatibility or performance. The browser lesson remains fully available without this extension.
A four-case neuron specification with a demonstrated architecture limit.
Download the editable handout →