Two Freezers and a Data Logger: Signed Numbers in the Lab
Southside Middle School's science lab has two mini-freezers that the seventh grade uses for an insulation investigation. Team A plugs in Freezer A, whose interior temperature is 8¼ °C at the start. The team records that the temperature falls at a steady rate of 2¼ °C per hour and wants to know when the interior will reach −3 °C. Team B's data logger prints one reading per hour for Freezer B: 6:00 a.m. −3.25 °C; 7:00 a.m. −8 °C; 8:00 a.m. −8 °C; 9:00 a.m. −2.5 °C. A teammate looks at the first and last readings and says, 'It cooled by 0.25 °C each hour, because 3.25 − 2.5 = 0.75 and there were three hours.' Meanwhile two students are arguing about a different probe that read −7/8 °C, which one of them insists is colder than a reading of −0.9 °C taken on the same thermometer. The lab report is due at the end of the period.
1. How long after being plugged in does Freezer A reach −3 °C? Show the signed change and the division that produces the time.
Read the worked answer
The temperature change is final minus starting: −3 − 8¼ = −11¼ °C, that is, −11.25 °C.
Time = change ÷ rate = −11.25 ÷ (−2.25) = 5 hours.
Check by counting down in steps of 2.25: 8.25 → 6.00 → 3.75 → 1.50 → −0.75 → −3.00, which takes 5 steps.
The division sanity-checks itself: dividing a negative change by a negative rate gives a positive time, which is what a duration must be. Freezer A reaches −3 °C five hours after it is plugged in.
2. Find Freezer B's average hourly temperature change from the logger. Then explain what is wrong with the teammate's statement that it cooled 0.25 °C each hour.
Read the worked answer
Total change = final − initial = −2.5 − (−3.25) = +0.75 °C.
Average rate = +0.75 ÷ 3 = +0.25 °C per hour. The positive sign means Freezer B warmed on average over those three hours, not that it cooled.
The teammate subtracted 3.25 − 2.5, which gives only the size of the difference (0.75) and then attached a direction by guesswork. Order matters: final minus initial, signs kept.
The average also hides the pattern: the freezer dropped 4.75 °C in the first hour (−8 − (−3.25)), held steady for an hour, then rose 5.5 °C (−2.5 − (−8)). A single average rate describes the overall trend, not a steady process.
3. One student claims −7/8 °C is colder than −0.9 °C. Decide who is right and give a method that settles such arguments reliably.
Read the worked answer
−7/8 = −0.875. On a horizontal number line, −0.9 lies to the left of −0.875, and further left means smaller, so −0.9 < −7/8. The reading of −0.9 °C is the colder one.
A reliable method: write both numbers as decimals with the same number of places, −0.900 and −0.875, then apply the rule that for negative numbers the value with the larger digits is the smaller number.
The student making the claim is likely comparing 8 with 9, or 7/8 with 9/10, as if the numbers were positive - which reverses the order once a minus sign is in front.