Grade 3

Reading a thermometer

The liquid in a thermometer stops at a mark with no number beside it. What temperature is it, and how much warmer is it than this morning?

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Measuring tools and reading their scalesGrade 2

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Temperature scales: Celsius, Fahrenheit and KelvinMath 6 Temperatures below zero and temperature changeMath 7

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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 is a narrow glass tube with a little bulb of colored liquid at the bottom. When the air or water around it gets warmer, the liquid swells and climbs the tube. When it gets colder, the liquid shrinks and drops. Beside the tube is a scale of marks, like a ruler stood on end. The temperature is the number on the scale where the top of the liquid stops.

The trouble is that most marks have no number. A tube has room for a mark every degree or every 2 degrees, but not room to write a number beside each one, so only some marks are labeled. To read the thermometer you have to work out what the unlabeled marks stand for. That is the skill from Measuring tools and reading their scales (find what one space is worth, then count spaces from a labeled mark), used on an upright scale.

Two more things make a thermometer its own job. First, there are two common temperature scales, degrees Fahrenheit, written ^\circ\text{F}, and degrees Celsius, written ^\circ\text{C}. Many thermometers carry both, one on each side of the tube. Second, the question is often not just “how hot?” but “how much warmer than before?” or “how much did it cool?” That is a difference between two readings, found by counting up or by subtracting.

Reach for this skill whenever a temperature has to be read off a scale, written down with its unit, or compared with another temperature on the same scale.

The procedure

The lesson runs over two days. Part 1 is reading the scale. Part 2 is finding how much warmer or colder.

Part 1: reading the scale

Find the top of the liquid. The reading is where the liquid stops, not where the tube stops and not where the bulb is. Look straight across at the top of the liquid, level with it, and see which mark it lines up with.

Find what one space is worth. Pick two labeled marks next to each other, say 40 and 50. Count the spaces between them, not the marks. Five spaces between 40 and 50 share the 10 degrees equally, so each space is worth 10 \div 5 = 2 degrees. The spaces are equal because the tube is the same width all the way up, so each degree of warming pushes the liquid up the same distance.

Counting spaces rather than marks matters. Between 40 and 50 there are 4 unlabeled marks but 5 spaces, because the last space ends on the 50 itself.

Count from the nearest labeled mark. Start at a labeled mark below the liquid and count up one space at a time, adding what each space is worth. Or start at the labeled mark above the liquid and count down. Both must give the same reading, which makes a good check.

Write the unit. A reading is a number and a scale together: 54^\circ\text{F}, said “fifty-four degrees Fahrenheit”. The small circle means degrees. The bare number 54 is not a temperature, because the same number on the other scale is a very different temperature.

Two scales on one tube. A thermometer with both scales has one column of liquid, so the top of the liquid gives one reading on the Fahrenheit side and another on the Celsius side. Read each side on its own, with its own labeled marks and its own space value. The two numbers are different but they describe the same warmth, the way 1 yard and 3 feet describe the same length. A Fahrenheit degree is a smaller step than a Celsius degree, so the Fahrenheit number is the larger of the two at any warm temperature you will meet here.

Part 2: how much warmer or colder

To compare two temperatures, both must be read on the same scale. Then the higher reading is the warmer one, because the liquid climbs as things warm.

“How much warmer” asks for the distance between the two readings on the scale. There are two ways to find it, and they always agree.

Count up. Start at the colder reading and jump up to the warmer one in friendly steps: first to the next ten, then by tens, then the last few. Add the jumps. This is counting up to subtract on an open number line stood on end, and it is how you would count marks on the thermometer itself.

Subtract. Warmer reading minus colder reading. Counting up from the smaller number to the larger one finds the number that makes the sum come out right, which is exactly what subtraction finds, so the two ways must match.

“How much colder” is the same distance, looked at from the other end. If the soup cooled from one reading to another, the drop is the larger reading minus the smaller one. The answer is a number of degrees on that scale: “23 degrees Fahrenheit warmer”.

Worked examples

Example 1

Part 1: the bird bath

Ratana has a thermometer floating in her bird bath. It is shown below. What temperature is the water?

Figure 1The thermometer in Ratana’s bird bath. Only every fifth mark has a number.

Solution

First, what is one space worth? Take the labeled marks 40 and 50. Between them there are 4 small marks, which cut the gap into 5 spaces. Five equal spaces share 10 degrees, so

10 \div 5 = 2

and each space is worth 2 degrees. The marks go 40, 42, 44, 46, 48, 50.

The top of the liquid is 2 marks above the 50. Counting up from 50 by 2s: 52, 54. So the water is 54^\circ\text{F}.

Check from above. The next labeled mark up is 60, and the liquid is 3 marks below it. Counting down by 2s: 58, 56, 54. The same reading, 54^\circ\text{F}.

Example 2

Part 1: soup at the soup kitchen

Sakchai checks a big pot of soup at the soup kitchen with the thermometer below. The top of the liquid stops at a mark with no number. What temperature is the soup?

Figure 2Sakchai’s soup thermometer. The liquid stops at a mark with no number.

Solution

This scale is in degrees Celsius, and the numbers go up by 20: 0, 20, 40, 60, 80, 100. Between 60 and 80 there are 3 small marks, which make 4 spaces. Four equal spaces share 20 degrees, so each space is worth

20 \div 4 = 5

degrees. The marks between 60 and 80 stand for 65, 70 and 75.

The liquid stops at the third mark above 60. Counting up by 5s: 65, 70, 75. The soup is 75^\circ\text{C}.

Check from above: the liquid is 1 mark below 80, and 80 - 5 = 75. The same reading.

Notice that the space value is not the same on every thermometer. On the bird bath it was 2; here it is 5. It has to be worked out fresh each time, from the labels.

Example 3

Part 1: one tube, two scales

At noon Paarma reads the thermometer in her attic. It has a Fahrenheit scale on the left and a Celsius scale on the right. What does each side say?

Figure 3Paarma’s attic thermometer. One tube, two scales, two different numbers for the same height.

Solution

Read each side on its own.

The left side, Fahrenheit, has numbers every 10 and 4 small marks between, so 5 spaces share 10 degrees and each space is worth 10 \div 5 = 2 degrees. The liquid stops 3 marks above the 80: 82, 84, 86. The left side reads 86^\circ\text{F}.

The right side, Celsius, has numbers every 5 and 4 small marks between, so 5 spaces share 5 degrees and each space is worth 5 \div 5 = 1 degree. The liquid stops right at the 30 label. The right side reads 30^\circ\text{C}.

So the attic is 86^\circ\text{F}, which is the same as 30^\circ\text{C}. Both numbers are true; they count in different-sized degrees. Look at the picture: a Celsius space, which is 1 degree, is nearly as tall as a Fahrenheit space, which is 2 degrees. A Celsius degree is the bigger step, so the same height takes fewer of them.

Example 4

Part 1: between two marks

Later in the day the liquid in Ratana’s bird bath thermometer stops exactly halfway between the 62 mark and the 64 mark. What is the temperature?

Solution

The marks go up by 2, so there is no mark for 63. But the spaces are equal, and each space covers 2 degrees, so halfway along a space is 1 degree past the mark below it:

62 + 1 = 63

The water is 63^\circ\text{F}. When the liquid stops between marks and not exactly halfway, the honest answer is “between 62^\circ\text{F} and 64^\circ\text{F}, closer to 62”. A reading can only be as exact as the marks allow.

Example 5

Part 2: the attic warms up

At 8 o’clock in the morning Paarma’s attic is 68^\circ\text{F}. By 3 o’clock in the afternoon it is 91^\circ\text{F}. How much warmer is the attic in the afternoon?

Solution

Both readings are Fahrenheit, so they can be compared. The warmer one is 91^\circ\text{F}.

Count up from 68 to 91 in friendly jumps:

Figure 4Counting up from 68 to 91 in friendly jumps. The jumps add to 23. The line is not drawn to scale.

From 68 up to the next ten, 70, is a jump of 2. From 70 up to 90 is two tens, a jump of 20. From 90 to 91 is a jump of 1. Add the jumps:

2 + 20 + 1 = 23

The attic is 23 degrees Fahrenheit warmer in the afternoon.

Check by subtracting the colder reading from the warmer one. In 91 - 68 the ones need regrouping: trade a ten for 10 ones, making 8 tens and 11 ones. Then 11 - 8 = 3 ones and 8 - 6 = 2 tens, so

91 - 68 = 23

The same answer.

Example 6

Part 2: the soup cools

Sakchai ladles soup that is 82^\circ\text{C} into a bowl. By the time it reaches the table it is 64^\circ\text{C}. How much did the soup cool?

Solution

The question asks how much colder, and both readings are Celsius. The distance between 64 and 82 on the scale is the same whichever end you start from, so count up from the colder reading:

64 \to 70 \text{ is } 6, \qquad 70 \to 80 \text{ is } 10, \qquad 80 \to 82 \text{ is } 2

6 + 10 + 2 = 18

The soup cooled by 18 degrees Celsius. Subtraction agrees: 82 - 64 = 18.

Example 7

Part 2: where will the liquid stop?

At sunrise Ratana’s bird bath is 47^\circ\text{F}. By noon it has warmed by 18 degrees. What does the thermometer read at noon, and where on the bird bath scale (a mark every 2 degrees) does the liquid stop?

Solution

Warming by 18 degrees means the reading goes up by 18. Jump up from 47: 3 to reach 50, then 15 more, since 3 + 15 = 18:

47 + 3 = 50, \qquad 50 + 15 = 65

At noon the water is 65^\circ\text{F}.

The marks go up by 2 from 60: 60, 62, 64, 66. There is no mark for 65. It is 1 degree past 64, and a space is 2 degrees, so the liquid stops halfway between the 64 mark and the 66 mark.

Boundaries

A reading is only as exact as the marks. When the liquid stops on a mark, the reading is that mark’s number. When it stops between two marks, you can say which two, and halfway gives the number in between, but there is no way to read finer than the marks allow.

The space value comes from the labels on that thermometer. It is the gap between two labeled marks divided by the number of spaces between them. It changes from one thermometer to the next, and on a two-scale thermometer it can change from one side to the other.

Compare and subtract only on one scale. 30 is less than 86, yet 30^\circ\text{C} and 86^\circ\text{F} are the same temperature. Two readings can be compared, or subtracted, only when they are on the same scale. A difference keeps its scale too: a rise of 23 degrees Fahrenheit is not a rise of 23 degrees Celsius.

This lesson stays above zero. Every reading here is 0 or more, and every difference is within 100. Thermometers go below zero, and those readings need negative numbers, the subject of Temperatures below zero and temperature change.

Where this goes next

The two scales were simply read here, side by side. Temperature scales: Celsius, Fahrenheit and Kelvin explains where their numbers come from: why water freezes at 0^\circ\text{C} and boils at 100^\circ\text{C}, and what the Fahrenheit numbers are for the same two moments. Temperatures below zero and temperature change carries the scale down past zero, where a cold night is a negative number and a change can cross zero. Much later, Converting between Celsius and Fahrenheit gives a rule that turns any reading on one scale into the reading on the other, the rule that the two-scale thermometer in Example 3 has built into its marks.

Exercises

Part 1: reading the scale

Exercises 1 to 3 use the three thermometers below.

Figure 5Thermometers A, B and C for Exercises 1 to 3.

Read thermometer A. Write the temperature with its unit.
Read thermometer B in the same figure.
Read thermometer C in the same figure.
A thermometer has labeled marks at 30 and 40, with 9 unlabeled marks between them. What is one space worth?
A thermometer has labeled marks at 0 and 20, with 3 unlabeled marks between them. What is one space worth?
Ratana’s bird bath thermometer has a mark every 2 degrees and a number every 10. The liquid stops 3 marks above the 70. What is the temperature in degrees Fahrenheit?
A thermometer at the soup kitchen sink has a mark every 5 degrees Celsius and a number every 20. The liquid stops 1 mark below the 60. What is the temperature?
On the Celsius side of Paarma’s attic thermometer (a mark every degree, a number every 5), the liquid stops 2 marks above the 25. What does the Celsius side read?
On a Fahrenheit scale with a mark every 2 degrees, the liquid stops exactly halfway between the 44 mark and the 46 mark. What is the temperature?
A Fahrenheit scale has a number at every ten and a mark every 2 degrees. Which of these temperatures land exactly on a mark: 38^\circ\text{F}, 41^\circ\text{F}, 50^\circ\text{F}, 67^\circ\text{F}?

Part 2: how much warmer or colder

The bird bath is 46^\circ\text{F} at 7 in the morning and 63^\circ\text{F} at 1 in the afternoon. How much warmer is it in the afternoon?
A pot of soup is 88^\circ\text{C} when the burner goes off and 69^\circ\text{C} an hour later. How much did it cool?
Paarma’s attic is 59^\circ\text{F} before dawn and 84^\circ\text{F} in the afternoon. How much warmer is the afternoon?
The Celsius side of the attic thermometer reads 17^\circ\text{C} in the morning and 31^\circ\text{C} in the afternoon. How much warmer is the afternoon, in degrees Celsius?
The bird bath is 49^\circ\text{F} and warms by 15 degrees. What is the new reading? On the bird bath scale (a mark every 2 degrees, a number every 10), where does the liquid stop?
Soup at the soup kitchen cools by 26 degrees and is then 57^\circ\text{C}. What was it before it cooled?
The attic reads 71^\circ\text{F} in the morning, 94^\circ\text{F} in the afternoon and 80^\circ\text{F} in the evening. How much did it warm from morning to afternoon? How much did it cool from afternoon to evening? Is the evening warmer or colder than the morning, and by how much?
One day the bird bath warmed from 52^\circ\text{F} to 70^\circ\text{F}. The same day the attic warmed from 66^\circ\text{F} to 83^\circ\text{F}. Which warmed more, and by how much more?
Two pots at the soup kitchen have Celsius thermometers with a mark every 5 degrees and a number every 20. In the first pot the liquid stops 2 marks above the 80; in the second, 2 marks below the 60. Read both. How much hotter is the first pot?
On Ratana’s bird bath thermometer (a mark every 2 degrees, a number every 10), the liquid stops 4 marks above the 50 in the morning and 2 marks above the 60 at noon. Read both, then find how much warmer the water is at noon. How many marks did the liquid climb?

Reasoning

Why is a temperature written with its unit, as in 81^\circ\text{F}, and never as the bare number 81?
In the figure for Exercise 1, thermometer B shows a smaller number than thermometer A. Does that mean B is colder? Use the two-scale thermometer in Example 3 to decide.
Explain why counting up from the colder reading to the warmer one always gives the same answer as subtracting the colder reading from the warmer one.
On a scale with a mark every 2 degrees, numbered at 40, 50, 60 and so on, can the liquid ever stop exactly on a mark that stands for 49? Where would the liquid be at 49^\circ\text{F}?

Answers

66^\circ\text{F}. The numbers go up by 10 with 5 spaces between, so each space is 10 \div 5 = 2 degrees. The liquid is 3 marks above 60: 62, 64, 66.
37^\circ\text{C}. The numbers go up by 10 with 10 spaces between, so each space is 1 degree. The liquid is 7 marks above 30, so 30 + 7 = 37. From above: 40 - 3 = 37.
85^\circ\text{F}. The numbers go up by 20 with 4 spaces between, so each space is 20 \div 4 = 5 degrees. The liquid is 1 mark above 80, so 80 + 5 = 85.
1 degree. Nine marks make 10 spaces, and 10 spaces share 40 - 30 = 10 degrees: 10 \div 10 = 1.
5 degrees. Three marks make 4 spaces sharing 20 - 0 = 20 degrees: 20 \div 4 = 5.
76^\circ\text{F}. Up from 70 by 2s: 72, 74, 76.
55^\circ\text{C}. One mark below 60 is 60 - 5 = 55.
27^\circ\text{C}. Each mark is 1 degree, so 25 + 2 = 27.
45^\circ\text{F}. A space covers 2 degrees, so halfway is 1 degree past the lower mark: 44 + 1 = 45.
38^\circ\text{F} and 50^\circ\text{F}. Counting by 2s from a labeled ten lands only on even numbers, so 38 and 50 have marks and 41 and 67 fall halfway between two marks.
17 degrees Fahrenheit warmer. Count up: 46 \to 50 is 4, 50 \to 60 is 10, 60 \to 63 is 3, and 4 + 10 + 3 = 17. Or 63 - 46 = 17.
It cooled by 19 degrees Celsius. Count up from 69: 1 to 70, then 18 to 88, and 1 + 18 = 19. Or 88 - 69 = 19.
25 degrees Fahrenheit warmer. 59 \to 60 is 1, 60 \to 80 is 20, 80 \to 84 is 4, and 1 + 20 + 4 = 25. Or 84 - 59 = 25.
14 degrees Celsius warmer. 17 \to 20 is 3, 20 \to 31 is 11, and 3 + 11 = 14. Or 31 - 17 = 14.
64^\circ\text{F}, since 49 + 15 = 64. The marks above 60 are 62, 64, so the liquid stops on the second mark above the 60.
83^\circ\text{C}. Cooling took 26 degrees away, so add them back: 57 + 26 = 83. Check: 83 - 26 = 57.
It warmed 23 degrees (94 - 71 = 23) and cooled 14 degrees (94 - 80 = 14). The evening is warmer than the morning by 9 degrees Fahrenheit, since 80 - 71 = 9. Also 23 - 14 = 9.
The bird bath warmed 70 - 52 = 18 degrees and the attic 83 - 66 = 17 degrees. The bird bath warmed more, by 18 - 17 = 1 degree. Both are Fahrenheit, so they can be compared.
Each space is 20 \div 4 = 5 degrees. The first pot is 80 + 5 + 5 = 90^\circ\text{C}; the second is 60 - 5 - 5 = 50^\circ\text{C}. The first is 90 - 50 = 40 degrees Celsius hotter.
Morning: 50 + 4 \times 2 = 58^\circ\text{F}. Noon: 60 + 2 \times 2 = 64^\circ\text{F}. The water is 64 - 58 = 6 degrees warmer. The liquid climbed 6 \div 2 = 3 marks, from 58 to 60, 62, 64.
A number alone does not say which scale it is on, and 81 degrees Fahrenheit is a warm day while 81 degrees Celsius is close to the soup in a pot. The unit tells which scale, so it is part of the reading.
No. A is in degrees Fahrenheit and B in degrees Celsius, so their numbers cannot be compared directly. On the two-scale thermometer in Example 3, the 35^\circ\text{C} mark sits higher than the 90^\circ\text{F} mark, so 35^\circ\text{C} is warmer than 90^\circ\text{F}. Then 37^\circ\text{C} is warmer still, and 66^\circ\text{F} is lower than 90^\circ\text{F}. So B shows the warmer temperature, even with the smaller number.
Counting up finds how many degrees must be added to the colder reading to reach the warmer one. Subtracting the colder from the warmer finds the number that, added to the colder, gives the warmer: that is what subtraction means. Both find the same missing addend, the length of the piece of scale between the two readings, so they must agree.
No. Counting by 2s from 40 gives 42, 44, 46, 48, 50: only even numbers get a mark, and 49 is odd. At 49^\circ\text{F} the liquid stops halfway between the 48 mark and the 50 mark.