These questions turn temperatures, heights, depths and balances into positions on a number line. You may need a final position, a signed change, a distance, an equal change per period or an unknown starting value.
Build the signed model first
Begin by naming the zero reference: freezing point for the temperature scale, sea level or the surface for height, and no money held or owed for a balance. Translate the starting position with its sign, then translate the movement. A rise or deposit moves right; a fall, payment or loss moves left. Starting at -6 and rising 8 crosses zero and ends at 2. Starting at 7 and paying 11 crosses the other way and ends at -4.
| What is asked | Method |
|---|---|
| Final position | Start at the signed value and apply the movement in its stated direction. |
| Signed change | Calculate end minus start. Keep the sign because it records rise or fall. |
| Distance or difference in height | Count the gap between the two positions. The result is nonnegative. |
| Equal change per period | Divide the signed total change by the number of equal periods. |
| Unknown start | Undo the stated movement from the ending value. |
For distance, first ask whether the positions lie on the same side of zero. Positions at -12 and -7 are 5 apart because their depths overlap. Positions at -3 and 8 are 11 apart because the route includes 3 units to zero and 8 beyond it. This differs from signed change: moving from 8 to -3 has change -11, although the distance travelled is 11. Repeated equal losses use multiplication, while sharing a total loss equally uses division.
Context traps to avoid
- Treating a rise as another fall moves in the wrong direction. Sketch the start and move right for a rise, even if the final value remains negative.
- Reporting only a depth or shortfall magnitude loses the side of zero. A signed height below the reference and a debt both need a minus sign.
- Subtracting magnitudes across zero makes the gap too small. When the positions lie on opposite sides, add their distances from zero.
- Keeping the minus sign of the starting debt after a deposit crosses zero treats the remaining money as debt. Use enough of the deposit to reach zero, then place the remainder on the positive side.
- Applying a stated fall again when recovering a start moves farther from the unknown. Reverse the movement: undo a fall by adding it back.
Try it
At dawn the temperature is 4 degrees Celsius. By midnight it has fallen by 9 degrees. Enter the midnight temperature in degrees Celsius.
Over 7 hours a balance changes by a total of -56 units, by the same amount in every hour. What is the change per hour?
- 8
- -8
- -49
- -63
One wreck lies at a height of -220 metres and another at a height of -85 metres. By how many metres do the two heights differ? Enter the difference in metres.