Math 7 · 6.NS.C.5

Temperatures below zero and temperature change

Zero on a thermometer is not the coldest it can get. How do you name a temperature below it, compare two of them, and say how far the temperature moved?

You need

Adding and subtracting integersGrade 7 Reading a thermometerGrade 3

Where you are 4 in Temperature

Measurement12

Temperature4

Reading a thermometer Temperature scales: Celsius, Fahrenheit and Kelvin

Temperatures below zero and temperature change

Converting between Celsius and Fahrenheit

When you reach for it

A thermometer has a mark called zero, but zero is not the coldest anything can be. On the Celsius scale, zero is the temperature at which water freezes, and a winter night can be far colder. On the Fahrenheit scale, zero is a colder mark again, and a night can still be colder than that. The scale simply carries on downwards.

Temperatures under the zero are named with negative numbers. Seven degrees below zero on the Celsius scale is written -7^\circ\text{C} and read “negative seven degrees Celsius” or “seven degrees below zero”. Temperatures above zero are positive, and 0^\circ\text{C} itself is neither.

Reach for this skill whenever a reading lies below zero: to write it down, to decide which of two temperatures is colder, and to work out how much the temperature rose or fell, especially when it crossed zero on the way. A thermometer is a number line stood on end, so each of these questions is a question about integers, and adding and subtracting integers answers it.

The procedure

Reading a temperature below zero

Figure 1A thermometer is a number line stood on end. Higher is warmer, so -3^\circ\text{C} is warmer than -12^\circ\text{C}.

A temperature carries two pieces of information. Its sign says which side of zero it is on: positive above, negative below. Its size, the number without the sign, says how many degrees it is from zero. That size is the absolute value: |-12| = 12 because -12^\circ\text{C} is 12 degrees from zero, and |5| = 5.

So “twelve degrees below zero” means start at zero and go down 12 degrees, which is 0 - 12, that is -12. Reading a thermometer below zero works the same way: count the marks down from zero to the top of the liquid. Below zero the numbers grow larger in size as you go down, which is the opposite of what happens above zero, so it pays to find the zero first and count from there.

Comparing: higher is warmer

On a thermometer the numbers increase upwards, exactly as they increase to the right on a horizontal number line. So of two temperatures, the one higher on the scale is both the warmer temperature and the greater number. In the figure, -3 sits above -12, so -3^\circ\text{C} is the warmer of the two, and

-12 < -3

The sizes alone point the wrong way: 12 > 3, but -12^\circ\text{C} is 12 degrees below zero and -3^\circ\text{C} only 3, so -12 is further down the scale. Among temperatures below zero, the larger the size, the colder the temperature. Every positive temperature is warmer than every negative one, and zero sits between them.

Temperature change: later minus earlier

The change in temperature is the number that, added to the earlier temperature, gives the later one:

\text{earlier} + \text{change} = \text{later}

Finding the number that must be added to one number to reach another is exactly what subtraction does, so

\text{change} = \text{later} - \text{earlier}

The sign of the change says which way the temperature went. A positive change is a rise, a negative change is a fall, and a change of 0 means the two readings are the same.

Figure 2A move across zero is counted in two pieces, to zero and then past it. Subtracting later minus earlier gives the same count, with a sign that says which way.

On the left of the figure the temperature goes from -6^\circ\text{C} to 4^\circ\text{C}. Counting on the scale, it climbs 6 degrees to reach zero and then 4 more, 10 degrees in all. The subtraction gives the same answer, because subtracting a number is adding its opposite (subtracting integers):

4 - (-6) = 4 + 6 = 10

The +6 in that sum is the 6 degrees climbed below zero. Subtracting a negative temperature counts the degrees the temperature had to climb just to get back to zero.

On the right the temperature goes from 3^\circ\text{C} down to -8^\circ\text{C}: 3 degrees down to zero and 8 more, a fall of 11. Later minus earlier gives -8 - 3 = -11, the same 11 degrees with a minus sign that says “fall”.

Running the relation the other way

The equation \text{earlier} + \text{change} = \text{later} has three quantities, and knowing any two gives the third. To find a new temperature, add the change to the earlier one; a fall is added as a negative number. To find a starting temperature, undo the change by subtracting it from the later one:

\text{earlier} = \text{later} - \text{change}

Change and difference are not the same question

“By how much did the temperature change?” has a direction, so the order matters. From -1^\circ\text{C} to -6^\circ\text{C} the change is -6 - (-1) = -5, a fall of 5. From -6^\circ\text{C} to -1^\circ\text{C} it is -1 - (-6) = 5, a rise of 5. Swapping the order of a subtraction gives the opposite answer.

“How far apart are the two temperatures?” has no direction. It is the distance on the number line between them, found as the higher minus the lower, and it is never negative: -1^\circ\text{C} and -6^\circ\text{C} are 5 degrees apart whichever is named first. A day’s range, its high minus its low, is a difference of this kind.

Worked examples

Example 1

Writing temperatures as numbers and in words

(a) Write “eleven degrees below zero Celsius” as a number with its unit.

(b) Write -15^\circ\text{F} in words.

(c) One night the low is nine degrees above zero Celsius. The next night the low is the same number of degrees from zero, but below it. Write both lows.

Solution

(a) Eleven degrees below zero is zero take away eleven, 0 - 11 = -11, so the temperature is -11^\circ\text{C}.

(b) The minus sign means below zero and the size is 15, so this is “fifteen degrees below zero Fahrenheit”, or “negative fifteen degrees Fahrenheit”.

(c) The first low is 9^\circ\text{C}. The second is the same distance from zero on the other side, which is the opposite number, -(9) = -9, so -9^\circ\text{C}. The two lows are opposites, and both have absolute value 9.

Example 2

Ordering a week of overnight lows

The overnight lows on five nights were -4^\circ\text{C}, 2^\circ\text{C}, -11^\circ\text{C}, 0^\circ\text{C} and -7^\circ\text{C}.

(a) Put them in order from coldest to warmest.

(b) Write an inequality comparing the two coldest lows.

Solution

(a) Picture the five on the thermometer. The only positive one, 2, is highest. Then comes 0. The three negatives sit below zero, and the one with the largest size is lowest. So from coldest to warmest:

-11,\ -7,\ -4,\ 0,\ 2

(b) -11 is below -7 on the scale, so -11 < -7. Read aloud: negative eleven is less than negative seven, or -11^\circ\text{C} is colder than -7^\circ\text{C}.

(c) Distance from 0 is the absolute value, and the largest is |-11| = 11. The night of -11^\circ\text{C} was furthest from freezing. Notice that 2^\circ\text{C}, the warmest night, was only 2 degrees from freezing: closeness to zero and warmth are different things.

Example 3

A rise across zero

At 7 a.m. the temperature was -13^\circ\text{C}. By 2 p.m. it was 6^\circ\text{C}. What was the change?

Solution

The change is later minus earlier:

6 - (-13) = 6 + 13 = 19

The change is positive, so the temperature rose 19 degrees. On the thermometer, it climbed 13 degrees to reach zero and 6 more beyond it, and 13 + 6 = 19. Check: start at -13 and add the change, -13 + 19 = 6, which is the afternoon reading.

Example 4

A fall, and which way round to subtract

(a) The temperature went from -2^\circ\text{F} to -10^\circ\text{F}. What was the change?

(b) The next day it went from -10^\circ\text{F} back to -2^\circ\text{F}. What was that change?

Solution

(a) Later minus earlier, with subtracting a negative written as adding its opposite:

-10 - (-2) = -10 + 2 = -8

The change is -8: the temperature fell 8 degrees. Both readings are below zero, so no crossing is involved; it simply went 8 degrees further down the scale.

(b) Now the later reading is -2 and the earlier one -10:

-2 - (-10) = -2 + 10 = 8

A rise of 8. The two changes are opposites because the subtraction was done the other way round, and the two readings are 8 degrees apart either way.

Example 5

A night of changes

At 6 p.m. the temperature is -6^\circ\text{F}. It falls 9 degrees by midnight, then rises 17 degrees by noon the next day. Find the midnight and noon temperatures, and the change from 6 p.m. to noon.

Solution

A fall of 9 is a change of -9, added to the earlier temperature:

-6 + (-9) = -15

so at midnight it is -15^\circ\text{F}. The rise of 17 is then added to that:

-15 + 17 = 2

so at noon it is 2^\circ\text{F}.

The change from 6 p.m. to noon is later minus earlier, 2 - (-6) = 2 + 6 = 8, a rise of 8 degrees. It also equals the two changes added, -9 + 17 = 8. That is no accident: the noon reading is \bigl(-6 + (-9)\bigr) + 17, and by the associative property of addition this is -6 + \bigl(-9 + 17\bigr), so the overall change is the sum of the separate ones.

Example 6

Finding where the temperature started

After rising 12 degrees, the temperature is -5^\circ\text{C}. What was it before the rise?

Solution

The relation is \text{earlier} + 12 = -5. Undo the rise by subtracting it:

\text{earlier} = -5 - 12 = -5 + (-12) = -17

The temperature started at -17^\circ\text{C}. It makes sense: after a rise of 12 the reading is still below zero, so it must have started more than 12 degrees below zero. Check: -17 + 12 = -5.

Example 7

Ranges in a table of highs and lows

Day	High	Low
Monday	3^\circ\text{C}	-8^\circ\text{C}
Tuesday	-1^\circ\text{C}	-13^\circ\text{C}
Wednesday	5^\circ\text{C}	-2^\circ\text{C}
Thursday	-4^\circ\text{C}	-10^\circ\text{C}

(a) Find each day’s range, its high minus its low. Which day had the largest range?

(b) Which day had the warmest low, and which the coldest high?

Solution

(a) The range is how far apart the high and the low are, the higher minus the lower:

Day	Range
Monday	3 - (-8) = 11
Tuesday	-1 - (-13) = 12
Wednesday	5 - (-2) = 7
Thursday	-4 - (-10) = 6

Tuesday had the largest range, 12 degrees, even though both of its readings were below zero. The range depends only on how far apart the readings are, not on where they sit.

(b) The lows are -8, -13, -2 and -10. The warmest is the highest on the scale, -2, so Wednesday had the warmest low. The highs are 3, -1, 5 and -4, and the coldest is the lowest, -4, so Thursday had the coldest high.

-10 - (-8) = -10 + 8 = -2

Thursday’s low was 2 degrees colder than Monday’s.

Boundaries

Both temperatures must be on the same scale. The two scales put zero in different places, so “below zero” depends on the scale: -5^\circ\text{C} is 23^\circ\text{F}, above zero Fahrenheit. Subtracting a Celsius reading from a Fahrenheit one gives a meaningless number, and a change keeps the unit of its scale: a rise of 10 Celsius degrees is a rise of 18 Fahrenheit degrees. Moving between them is its own skill.

Differences mean something; ratios do not. Zero on these scales is a chosen mark, not “no warmth at all”, so 10^\circ\text{C} is not twice as warm as 5^\circ\text{C}, and -10^\circ\text{C} is not twice as cold as -5^\circ\text{C}. Saying one is 5 degrees warmer than the other is correct on any scale.

There is a floor. The integers go down forever; temperatures do not. Nothing can be colder than absolute zero, -273.15^\circ\text{C} or -459.67^\circ\text{F}, which is the zero of the Kelvin scale. A claim that the temperature fell 300 degrees from 20^\circ\text{C} is impossible, since 20 - 300 = -280 is below that floor.

A change describes the start and the end, not the journey. Two equal readings give a change of 0, but the temperature may have risen and fallen a long way in between.

Size is distance from zero, not warmth. Above zero a larger size is warmer; below zero a larger size is colder. To compare, look at position on the scale, never at size alone.

Where this goes next

Everything here works for any quantity measured from a chosen zero. Height above and below sea level, money held and money owed, and points scored for and against are all read, compared and subtracted in exactly this way, which is why temperature is often the first place the integers are met. Readings with decimals or fractions, such as -2.5^\circ\text{C}, are compared by the same picture, and ordering rational numbers does that in general.

The “how far apart” question becomes the distance between two points on a number line, written with absolute value as |a - b|, which gives the same answer whichever point is named first. And the fact that two scales put their zero in different places is the reason converting between Celsius and Fahrenheit needs both a stretch and a shift, while the Kelvin scale moves its zero down to the floor so that no temperature is ever negative.

Exercises

Reading and comparing

Write each as a number with its unit. (a) Twelve degrees below zero Celsius. (b) The opposite of 8^\circ\text{F}.
Which is colder, -9^\circ\text{C} or -14^\circ\text{C}? Write an inequality that compares them.
Order from coldest to warmest: 3^\circ\text{C}, -8^\circ\text{C}, -1^\circ\text{C}, -15^\circ\text{C}, 0^\circ\text{C}, -6^\circ\text{C}.
Order from warmest to coldest: -2.5^\circ\text{C}, -7^\circ\text{C}, 1.5^\circ\text{C}, -0.5^\circ\text{C}.
Write < or > between each pair. (a) -3 and -13 (b) -20 and 5 (c) 0 and -4
Of -6^\circ\text{C} and 4^\circ\text{C}, which is warmer, and which is further from zero?
A temperature, a whole number of degrees, is colder than -10^\circ\text{C} and warmer than -12^\circ\text{C}. What is it?

Temperature change

Find the change in temperature, and say whether it is a rise or a fall.

From -5^\circ\text{C} to 7^\circ\text{C}.
From 4^\circ\text{F} to -9^\circ\text{F}.
From -14^\circ\text{C} to -6^\circ\text{C}.
From -3^\circ\text{F} to -17^\circ\text{F}.
From 10^\circ\text{F} at sunset to -22^\circ\text{F} at dawn.

New and starting temperatures

The temperature is -7^\circ\text{C} and rises 15 degrees. What is it now?
The temperature is 2^\circ\text{F} and falls 11 degrees. What is it now?
After falling 9 degrees, the temperature is -13^\circ\text{C}. What was it before?
After rising 16 degrees, the temperature is 5^\circ\text{F}. What was it before?
Starting at -4^\circ\text{C}, the temperature falls 6 degrees, rises 3, then falls 8. What is the final temperature, and what is the overall change?

Problems

At the foot of a mountain the temperature is 7^\circ\text{C}, and at the summit it is -9^\circ\text{C}. What is the change on the way up? A storm then cools the summit 5 more degrees. What is the temperature there now?
One winter morning the lows in four towns were Ashford -12^\circ\text{C}, Bexley -5^\circ\text{C}, Carrow -19^\circ\text{C} and Dunmore 3^\circ\text{C}. (a) Which town was coldest? (b) How many degrees apart were the warmest and the coldest? (c) How many degrees colder was Carrow than Bexley?
On one day the high was -2^\circ\text{C} and the low -11^\circ\text{C}. On the next day the high was 4^\circ\text{C} and the low -6^\circ\text{C}. Which day had the larger range, and by how much?

Reasoning

Explain why -15^\circ\text{C} is colder than -5^\circ\text{C}, even though 15 is greater than 5.
Explain why the change from a temperature a to a temperature b and the change from b back to a are always opposite numbers.
Use the thermometer to explain why the change from -4^\circ\text{C} to 6^\circ\text{C} is 10 degrees, and show that 6 - (-4) gives the same number.
Is 10^\circ\text{C} twice as warm as 5^\circ\text{C}? The same two temperatures are 50^\circ\text{F} and 41^\circ\text{F}. Use that to explain your answer.
A thermometer reads -2^\circ\text{C}, and three hours later it reads -2^\circ\text{C} again. What is the change? Must the temperature have stayed the same the whole time?

Answers

(a) -12^\circ\text{C}, since twelve below zero is 0 - 12 = -12. (b) -8^\circ\text{F}, the same distance from zero on the other side.
-14^\circ\text{C} is colder, and -14 < -9. Fourteen degrees below zero is further down the scale than nine below.
-15, -8, -6, -1, 0, 3. The negatives come first, the one with the largest size lowest.
1.5, -0.5, -2.5, -7. The only positive one is warmest; then the negatives, nearest zero first.
(a) -3 > -13 (b) -20 < 5 (c) 0 > -4. In each pair the number higher on the scale is the greater.
4^\circ\text{C} is warmer. -6^\circ\text{C} is further from zero, since |-6| = 6 and |4| = 4.
-11^\circ\text{C}. It is the only integer between -12 and -10.
A rise of 12: 7 - (-5) = 7 + 5 = 12.
A fall of 13: -9 - 4 = -13.
A rise of 8: -6 - (-14) = -6 + 14 = 8.
A fall of 14: -17 - (-3) = -17 + 3 = -14.
A fall of 32: -22 - 10 = -32. That is 10 degrees down to zero and 22 more.
8^\circ\text{C}: -7 + 15 = 8.
-9^\circ\text{F}: 2 + (-11) = -9.
-4^\circ\text{C}. Earlier is later minus the change: -13 - (-9) = -13 + 9 = -4. Check: -4 - 9 = -13.
-11^\circ\text{F}: 5 - 16 = -11. Check: -11 + 16 = 5.
Final -15^\circ\text{C}, overall change -11. -4 + (-6) + 3 + (-8) = -15, and -15 - (-4) = -11, which is also -6 + 3 + (-8).
A fall of 16, since -9 - 7 = -16: 7 degrees down to zero and 9 more. After the storm, -9 + (-5) = -14, so -14^\circ\text{C}.
(a) Carrow, at -19^\circ\text{C}, the lowest on the scale. (b) 22 degrees apart: 3 - (-19) = 22. (c) 14 degrees colder: -5 - (-19) = -5 + 19 = 14.
The second day, by 1 degree. The ranges are -2 - (-11) = 9 and 4 - (-6) = 10.
On the thermometer, -15^\circ\text{C} is 15 degrees below zero and -5^\circ\text{C} only 5 below, so -15 sits lower on the scale, and lower is colder. Below zero, a larger size means further down, so -15 < -5.
The change from a to b is b - a, and from b to a it is a - b. Since a - b = -(b - a), the two are opposites: the temperature goes the same number of degrees, once up and once down.
From -4 the temperature climbs 4 degrees to reach zero, then 6 more to reach 6, so 4 + 6 = 10 degrees. Subtracting a negative is adding its opposite, so 6 - (-4) = 6 + 4 = 10, and the 4 in that sum is exactly the 4 degrees climbed below zero.
No. In Fahrenheit the same two temperatures are 50 and 41, and 50 is not twice 41. A ratio that changes when the scale changes says nothing about the weather; it comes from where each scale happens to put its zero. The difference is meaningful: 10^\circ\text{C} is 5 Celsius degrees warmer.
The change is -2 - (-2) = 0. That does not mean nothing happened: a change compares only the first and last readings, and the temperature may have risen and fallen in between.