TACHS: Temperature changes: move along the number line
Track rises and falls through zero, find the size of a change, work back to a starting temperature, and use a steady hourly change.
How do you keep track of temperatures that go above and below zero?
Short answer
Start with the signed temperature, add each rise and subtract each fall, one at a time. For the size of a change, subtract later minus earlier; to work backward, undo the changes in reverse order.
Know first: Adding and subtracting negative numbers, Reading a number line
Step 1 of 6
A rise moves right, a fall moves left
Picture temperature on a number line. A rise moves right, which is adding. A fall moves left, which is subtracting. When you cross zero, keep counting: from 4 degrees, a fall of 12 goes 4 steps down to 0 and 8 more below, landing at −8.
Keep the sign of the starting temperature. Most wrong answers drop a negative sign somewhere.
| The question says | Do this |
|---|---|
| It rises 5° | Add 5 |
| It falls 11° | Subtract 11 |
| How many degrees did it fall? | Later − earlier. A negative result is a fall of that many degrees |
| What was it before these changes? | Undo in reverse order: undo a fall by adding, a rise by subtracting |
| It changes steadily from 6 a.m. to noon | Total change ÷ number of hours = change per hour |
Write each step on its own line so the signs stay visible.
Step 2 of 6
One change at a time
Worked example: A day of rises and falls
At 6 a.m. it is −5°C. By noon the temperature rises 9°C. By evening it falls 12°C. Overnight it rises 2°C. What is the final temperature?
- A−28
- B−6
- C4
- D6
- 1
Start
−5
- 2
Rise 9
−5 + 9 = 4
- 3
Fall 12
4 − 12 = −8, crossing zero
- 4
Rise 2
−8 + 2 = −6
- 5
Check by totaling the changes
+9 − 12 + 2 = −1, and −5 + (−1) = −6.
- 6
See why the other choices are there
−28 treats every change as a fall. 4 drops the sign of the start: 5 + 9 − 12 + 2. 6 drops the sign at the end.
Answer
The final temperature is −6°C.
How far did it fall?
It is 3°C at 6 p.m. and −7°C at midnight. How many degrees did the temperature fall?
- 1
Later minus earlier: −7 − 3.
- 2
−7 − 3 = −4.
What went wrong
Subtracting 3 from −7 moves 3 more steps left, to −10. Getting −4 adds the 3 instead.
- 3
It fell 4 degrees.
The fix
−7 − 3 = −10, so the temperature fell 10 degrees. On the number line: 3 steps down to 0, then 7 more, for 10 in all.
Step 3 of 6
Run it backward, or spread it out by the hour
To find a starting temperature, undo the changes from last to first. If a temperature rose 6° and then fell 9° to reach −4°C, undo the fall first: −4 + 9 = 5. Then undo the rise: 5 − 6 = −1. Check forward: −1 + 6 − 9 = −4.
For a steady change, find the total change and divide by the hours. From −8°C at 6 a.m. to 4°C at noon is a rise of 4 − (−8) = 12 degrees in 6 hours, or 2 degrees per hour. At 8 a.m. it is −8 + 2 × 2 = −4°C.
Step 4 of 6
Watch every sign
Check yourself · Question 1
The temperature is −7°C. It rises 10°C, falls 5°C, then rises 1°C. What is the final temperature?
Answer: B
−7 + 10 = 3. Then 3 − 5 = −2. Then −2 + 1 = −1.
Check yourself · Question 2
After falling 5°C and then rising 8°C, the temperature is −3°C. What was the temperature before these changes?
Answer: B
Undo the last change first. Undo the rise: −3 − 8 = −11. Undo the fall: −11 + 5 = −6. Check: −6 − 5 + 8 = −3.
Trap answer: 0°C
Running the changes forward
- Why it looks right
- The question lists the changes in order, so it feels natural to apply them in that order to the number given.
- Why it's wrong
- −3°C is where the temperature ended, not where it began. Applying the changes again moves further away from the start.
- The right answer
- −6°C: undo the rise, then undo the fall.
Check yourself · Question 3
A freezer reads 5°C at 1 p.m. and −13°C at 7 p.m. Its temperature falls steadily by the same amount each hour. What does it read at 3 p.m.?
Answer: C
Total change: −13 − 5 = −18 over 6 hours, so −3 degrees per hour. From 1 p.m. to 3 p.m. is 2 hours: 5 + 2 × (−3) = −1.
Common mistake
"It went from 3°C to −7°C, so it fell −10 degrees."
- Why it's tempting
- Later minus earlier gives −10, and the minus sign seems like part of the answer.
- Do this instead
- The sign tells you the direction. The question "how many degrees did it fall" asks for a size, which is positive: it fell 10 degrees.
Step 5 of 6
What to remember
Remember
- Add rises and subtract falls, one change at a time, keeping the starting sign.
- The size of a change is later − earlier, written as a positive number of degrees risen or fallen.
- To work backward, undo the changes from last to first; for a steady change, divide the total change by the hours.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, which change takes the temperature furthest below zero, and which wrong choice comes from treating every change as a fall?