Every 1D motion problem starts with two choices you make before doing any math: a reference point (where is x = 0?) and a positive direction (which way does x increase?). Neither choice is dictated by the physics — they're conventions, the same way choosing which way is "up" on a map is a convention.
The walker below always moves the same physical way — from the left mark to the right mark. Flip which direction you're calling positive and watch the sign of the displacement flip, while the actual walk on the floor stays identical.
With a sign convention in hand, displacement's rate gets its own name: velocity.
This looks almost identical to average speed from Lesson 1-1-1, but the ingredients are different: velocity is built from displacement(signed), while speed is built from distance (always positive). That single difference is why velocity can be negative and speed cannot.
Taking rightward as positive, a skateboarder's position changes from x_i = 12 m to x_f = 4 m over 4 seconds. Find the average velocity.
This is the single most common error students carry out of this lesson: treating a negative sign on velocity as if it meant "slowing down" or "losing speed." It doesn't.
Whether an object is speeding up or slowing down depends on comparing velocity to acceleration — a topic for Unit 1's next lesson set (1-2-1 and 1-2-2). For now, keep "sign of velocity" and "speeding up or down" completely separate in your head. They are answering different questions.
Speed is the magnitude of velocity — the size of the number with the sign stripped off.
This is exactly the same relationship as distance and displacement from Lesson 1-1-1, just one level up: distance is built from magnitudes of individual movements, displacement keeps the signs. Speed strips the sign from velocity the same way |Δx| would strip it from a single displacement.
For a trip made of several legs, the total displacement is just the sum of each leg's signed value. Distance, by contrast, sums the magnitude of each leg, ignoring sign entirely — which is exactly why the two totals can end up so different.
Build a multi-leg trip out of signed movements. Distance adds every leg's size; displacement adds the signed values — so opposite-direction legs can cancel.
You now have the full signed-number toolkit for one-dimensional motion: position, displacement, velocity, and speed, all connected by consistent sign conventions. Lesson 1-2-1 will put velocity on its own vertical axis — a velocity-time graph — where the sign of the line's height (not its slope) tells you direction, and the slope becomes a brand-new quantity: acceleration.