L2 · Slope as rate of change in context

Learning goal

“I can compute the slope between two points of a real-world linear function and interpret it as a RATE OF CHANGE WITH UNITS - including negative rates - and I can compare two functions by their rates.”

Slope — the steepness number of a line: how much the output changes when the input grows by 1.

Rate of change — slope wearing its context: the change in the output quantity per one unit of the input quantity, e.g. 30 litres per minute.

Rise — the change in the output between two points (change in output). It can be negative.

Run — the change in the input between two points (change in input). Slope = rise / run.

Worked example

A tank is being filled. At 0 minutes it holds 40 L; at 5 minutes it holds 190 L. Step 1: Rise = change in output = 190 - 40 = 150 L. Run = change in input = 5 - 0 = 5 min. Step 2: Slope = rise / run = 150 / 5 = 30. Step 3: ATTACH THE UNITS: the slope is 30 litres per minute. That sentence is the answer a bare "30" is not: it says what grows (litres), by how much (30), per what (each minute). In function form the filling is V(t) = 30t + 40.

Check yourself — explain it

A slope computed as change in distance / change in time has units...

Paired problem — example still visible

A reader's page count: at 2 hours they are on page 30; at 6 hours, page 110. Find the reading rate. -> Rise = 110 - 30 = 80 pages; run = 6 - 2 = 4 h; rate = 80 / 4 = 20 pages per hour.

Faded completion — last step is yours

"Points (0, 15) and (3, 60): slope = (60 - 15) / ___ = ___" -> 3; 15

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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