5.2.1: Changing Temperatures

Lesson

Let's add signed numbers.

Exercise \(\PageIndex{1}\): Which One Doesn't Belong: Arrows

Which pair of arrows doesn't belong?

1.

Figure \(\PageIndex{1}\): Number line. 21 evenly spaced tick marks. Scale negative 10 to 10, by 1s. Two arrows. One arrow points to the right from 0 to 3. One arrow points to the right from 3 to 7.

2.

Figure \(\PageIndex{2}\): Number line. 21 evenly spaced tick marks. Scale negative 10 to 10, by 1s. Two arrows. One arrow points to the right from 0 to 3. One arrow points to the left from 3 to negative 6.

3.

Figure \(\PageIndex{3}\): Number line. 21 evenly spaced tick marks. Scale negative 10 to 10, by 1s. Two arrows. One arrow points to the right from 0 to 3. One arrow points to the left from 3 to 0.

4.

Figure \(\PageIndex{4}\): Number line. 21 evenly spaced tick marks. Scale negative 10 to 10, by 1s. Two arrows pointing to the left, one from 0 to negative 4 and another from negative 4 to negative 9.

Exercise \(\PageIndex{2}\): Warmer and Colder

1. Complete the table and draw a number line diagram for each situation.

Table \(\PageIndex{1}\)

	start (\(^{\circ}\)C)	change (\(^{\circ}\)C)	final (\(^{\circ}\)C)	addition equation
a	\(+40\)	\(10\) degrees warmer	\(+50\)	\(40+10=50\)
b	\(+40\)	\(5\) degrees colder
c	\(+40\)	\(30\) degrees colder
d	\(+40\)	\(40\) degrees colder
e	\(+40\)	\(50\) degrees colder

a.

Figure \(\PageIndex{5}\)

b.

Figure \(\PageIndex{6}\)

c.

Figure \(\PageIndex{7}\)

d.

Figure \(\PageIndex{8}\)

e.

Figure \(\PageIndex{9}\)

2. Complete the table and draw a number line diagram for each situation.

Table \(\PageIndex{2}\)

	start (\(^{\circ}\)C)	change (\(^{\circ}\)C)	final (\(^{\circ}\)C)	addition equation
a	\(-20\)	\(30\) degrees warmer
b	\(-20\)	\(35\) degrees warmer
c	\(-20\)	\(15\) degrees warmer
d	\(-20\)	\(15\) degrees colder

a.

Figure \(\PageIndex{10}\)

b.

Figure \(\PageIndex{11}\)

c.

Figure \(\PageIndex{12}\)

d.

Figure \(\PageIndex{13}\)

Are you ready for more?

Figure \(\PageIndex{14}\): Number line. 2 tick marks. First tick mark is labeled a plus b. Second tick mark, to the right of the first tick mark, is labeled 0. Arrow a starts at zero and extends to the right. Arrow b is directly above arrow a, points to the left and begins where arrow a ends, and ends at the point labeled a plus b.

For the numbers \(a\) and \(b\) represented in the figure, which expression is equal to \(|a+b|\)?

\(|a|+|b|\qquad |a|-|b| \qquad |b|-|a|\)

Exercise \(\PageIndex{3}\): Winter Temperatures

One winter day, the temperature in Houston is \(8^{\circ}\) Celsius. Find the temperatures in these other cities. Explain or show your reasoning.

1. In Orlando, it is \(10^{\circ}\) warmer than it is in Houston.
2. In Salt Lake City, it is \(8^{\circ}\) colder than it is in Houston.
3. In Minneapolis, it is \(20^{\circ}\) colder than it is in Houston.
4. In Fairbanks, it is \(10^{\circ}\) colder than it is in Minneapolis.
5. Use the thermometer applet to verify your answers and explore your own scenarios.

Summary

If it is \(42^{\circ}\) outside and the temperature increases by \(7^{\circ}\), then we can add the initial temperature and the change in temperature to find the final temperature.

\(42+7=49\)

If the temperature decreases by \(7^{\circ}\), we can either subtract \(42-7\) to find the final temperature, or we can think of the change as \(-7^{\circ}\). Again, we can add to find the final temperature.

\(42+(-7)=35\)

In general, we can represent a change in temperature with a positive number if it increases and a negative number if it decreases. Then we can find the final temperature by adding the initial temperature and the change. If it is \(3^{\circ}\) and the temperature decreases by \(7^{\circ}\), then we can add to find the final temperature.

\(3+(-7)=-4\)

We can represent signed numbers with arrows on a number line. We can represent positive numbers with arrows that start at 0 and points to the right. For example, this arrow represents +10 because it is 10 units long and it points to the right.

Figure \(\PageIndex{15}\)

We can represent negative numbers with arrows that start at 0 and point to the left. For example, this arrow represents -4 because it is 4 units long and it points to the left.

Figure \(\PageIndex{16}\)

To represent addition, we put the arrows “tip to tail.” So this diagram represents \(3+5\):

Figure \(\PageIndex{17}\): A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the right, and ends at 3. A second arrow starts at 3, points to the right, and ends at 8. there is a solid dot indicated at 8.

And this represents \(3+(-5)\):

Figure \(\PageIndex{18}\): A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the right, and ends at three. A second arrow starts at 3, points to the left, and ends at negative 2. There is a solid dot indicated at negative

Practice

Exercise \(\PageIndex{4}\)

1. The temperature is \(-2^{\circ}\text{C}\). If the temperature rises by \(15^{\circ}\text{C}\), what is the new temperature?
2. At midnight the temperature is \(-6^{\circ}\text{C}\). At midday the temperature is \(9^{\circ}\text{C}\). By how much did the temperature rise?

Exercise \(\PageIndex{5}\)

Draw a diagram to represent each of these situations. Then write an addition expression that represents the final temperature.

1. The temperature was \(80^{\circ}\text{F}\) and then fell \(20^{\circ}\text{F}\).
2. The temperature was \(-13^{\circ}\text{F}\) and then rose \(9^{\circ}\text{F}\).
3. The temperature was \(-5^{\circ}\text{F}\) and then fell \(8^{\circ}\text{F}\).

Exercise \(\PageIndex{6}\)

Complete each statement with a number that makes the statement true.

1. _____ < \(7^{\circ}\text{C}\)
2. _____ < \(-3^{\circ}\text{C}\)
3. \(-0.8^{\circ}\text{C}\) < _____ < \(-0.1^{\circ}\text{C}\)
4. _____ > \(-2^{\circ}\text{C}\)

(From Unit 5.1.1)

Exercise \(\PageIndex{7}\)

Decide whether each table could represent a proportional relationship. If the relationship could be proportional, what would be the constant of proportionality?

1. The number of wheels on a group of buses.

   Table \(\PageIndex{3}\)

   number of buses	number of wheels	wheels per bus
   \(5\)	\(30\)
   \(8\)	\(48\)
   \(10\)	\(60\)
   \(15\)	\(90\)

2. The number of wheels on a train.

   Table \(\PageIndex{4}\)

   number of train cars	number of wheels	wheels per train car
   \(20\)	\(184\)
   \(30\)	\(264\)
   \(40\)	\(344\)
   \(50\)	\(424\)

(From Unit 2.3.1)

Exercise \(\PageIndex{8}\)

Noah was assigned to make 64 cookies for the bake sale. He made 125% of that number. 90% of the cookies he made were sold. How many of Noah's cookies were left after the bake sale?

(From Unit 4.2.2)