Real house prices bend with size: the first bedrooms add a lot, huge homes add less per foot. A single neuron can only draw a straight line, so it can't follow that bend — which is exactly why we add hidden neurons. Here's a real network (size → 2 hidden neurons → price). Train it, watch the curve snap onto the houses, then step through one backward pass.
This is the network's average miss, in dollars, after each training step. It starts high — the untrained curve is way off — then drops fast as the curve finds its shape, then fine-tunes. Press Train in Card 1 to watch it fall.
Pick one house, then step through. This uses the weights as they are now — train first for clean gradients, or walk through untrained to see messy ones.
You've watched backprop compute the gradients. Now be one: grab a knob in the trained network and turn it. Watch the price curve bend and the error move — the gradient is simply the slope of the bowl underneath, pointing downhill.
That slope under the ball is the gradient backprop computed for this weight. Drag the knob to the bottom of the bowl and the curve snaps onto the houses. Training just rolls every knob to the bottom of its own bowl — all at once.
The intuition you wanted, on something real. Each weight is a knob, and backprop works out one number for each: its gradient — which way to turn this knob to shrink the miss, and how hard. That's exactly the slope you turned by hand in Card 4, found for every knob at once.
To get it, backprop starts at the miss and walks backward, multiplying the pieces it passes along each path (the chain rule). An output weight's gradient is the miss × the hidden value it carried. To reach the weights further left, it first works out how hard each hidden neuron is pushing the miss — the miss passed back through that neuron's weight and scaled by its own sigmoid slope — then multiplies by the size that fed in.