First PrinciplesStart anywhere. Prove it, then move on.

Rung 6

The coordinate plane

Where a function stops being a table and becomes a shape.

Best attempted after 4. Function as a black box and 5. Balancing equations. Nothing stops you trying this now — the gate will tell you if you were right.

The gate

Given a line in any one of three forms — a graph, an equation, or a table of points — produce the other two. Do it for a line with a negative slope and for one with a slope between 0 and 1, and state what the slope means in words.

Nobody checks this but you. Do it honestly and the rungs above hold; do it loosely and they will not, somewhere further up where the cause is much harder to find.

Two number lines crossed at right angles, and suddenly every pair of numbers has a place to stand. Put every input-output pair from rung 4 in its place and the function stops being a list and becomes a picture.

For a linear function, that picture is a straight line, and its steepness has a name: slope. Slope is a rate. It answers "if the input goes up by one, what happens to the output?"

Why this is on the ladder

Slope is the derivative in its first, easiest disguise. Rung 20 will ask what slope means for a curve that is not straight, and the answer will be "the slope of the straight line that hugs it here". Everything about gradient descent — rung 41, the mechanism by which every model on earth is trained — is the claim that if you know the slope, you know which way is downhill.

Get comfortable now, where the line is straight and you can see it.

Do this

For each of these, produce the other two representations:

  1. The equation y = -2x + 5
  2. A table containing (0, 1), (2, 2), (4, 3)
  3. A line drawn through (1, 4) and (3, 0)

For each, say in plain words what the slope means: for every one step right, the line falls two. Not "the slope is -2" — what it means.

Then answer without drawing: which is steeper, slope 0.5 or slope -3? Say why.

Where people get stuck

The sign and the size get confused. A slope of -3 is steeper than a slope of 0.5, though it is a smaller number. Steepness is the size; direction is the sign. Later, at rung 33, the gradient will carry exactly this pair of facts — how fast, and which way — and the habit of reading them separately starts here.

Fractional slopes also trip people who have only met whole ones. A slope of 1/2 is a real slope and a perfectly ordinary line.

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