Up for adding, down for subtracting
Adding moves up the scale and subtracting moves down, whichever side of zero you start on. A move can carry you across zero on the way. The easiest way to work one out is to split the move at zero.
Taking away more than you have
5 − 9: start at 5 and move down 9. The first 5 steps reach 0, and there are 4 more steps to go, so the move ends 4 below zero. So 5 − 9 = −4.
Splitting the move at zero is splitting the 9 into 5 + 4: 5 steps to reach zero, and 4 beyond it.
From 5°, a fall of 9 passes zero after 5 steps and stops 4 below it, at −4°.
Walk the marker 9 steps to the left from 5. After 5 steps it reaches zero, and after 4 more it stops at −4.
Two falls stack
−3 + (−4): adding a negative number moves down the scale. Start 3 below zero and drop 4 more, and you are 7 below zero. So −3 + (−4) = −7.
Adding −4 gives the same answer as subtracting 4: −3 + (−4) = −3 − 4 = −7. Both moves point down, so they add up to a fall of 7.
From −3°, a further fall of 4 lands on −7°.
Climbing through zero
−3 + 5: start 3 below zero and climb 5. The first 3 steps reach 0, and the other 2 go above it. So −3 + 5 = 2.
Bigger climbs work the same way. −4 + 7 is 4 steps up to zero and 3 more beyond it, so −4 + 7 = 3.
From −3°, a rise of 5 passes zero after 3 steps and stops at 2°.
A day in the shed
At dawn a shed is at −4 °C. By noon it has warmed by 9 °C: −4 + 9 = 5, so it is 5 °C. By evening it has cooled by 8 °C: 5 − 8 = −3, so it is −3 °C.
Each change is a move along the scale. A rise is an addition and a fall is a subtraction, wherever the temperature starts.
The usual mistakes
Dropping the minus sign. 5 − 9 is not 4. Taking away more than you have must finish below zero, so 5 − 9 = −4.
Subtracting two falls from each other. −3 + (−4) is not −1 or 1. Both moves point down, so they stack: −3 + (−4) = −7.
Keeping a minus sign out of habit. −3 + 5 is not −2. A climb of 5 from −3 carries you above zero, to 2.