Lesson 6
Changing Temperatures
Let's add signed numbers.
6.1: Which One Doesn’t Belong: Arrows
Which pair of arrows doesn't belong?
6.2: Warmer and Colder
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Complete the table and draw a number line diagram for each situation.
start (\(^\circ\text{C}\)) change (\(^\circ\text{C}\)) final (\(^\circ \text{C}\)) 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 -
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Complete the table and draw a number line diagram for each situation.
start (\(^\circ\text{C}\)) change (\(^\circ\text{C}\)) final (\(^\circ\text{C}\)) addition equation a -20 30 degrees warmer b -20 35 degrees warmer c -20 15 degrees warmer d -20 15 degrees colder -
![Number line.](https://cms-im.s3.amazonaws.com/yHm1dxAgmbyWk7Bt1nsYfrh5?response-content-disposition=inline%3B%20filename%3D%227-7.5.B2.Revised.Image.03b.png%22%3B%20filename%2A%3DUTF-8%27%277-7.5.B2.Revised.Image.03b.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T162816Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=443bb07eea318b19579532c3d9123ee514b683aae45a8b0d425751da069c6d88)
For the numbers \(a\) and \(b\) represented in the figure, which expression is equal to \(|a+b|\)?
\(|a|+|b|\)
\(|a|-|b|\)
\(|b|-|a|\)
6.3: 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.
- In Orlando, it is \(10^\circ\) warmer than it is in Houston.
- In Salt Lake City, it is \(8^\circ\) colder than it is in Houston.
- In Minneapolis, it is \(20^\circ\) colder than it is in Houston.
- In Fairbanks, it is \(10^\circ\) colder than it is in Minneapolis.
- 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 \(\text-7^\circ\). Again, we can add to find the final temperature.
\(42 + (\text-7) = 35\)
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.
\(3+ (\text-7) = \text-4\)
We can represent signed numbers with arrows on a number line. We can represent positive numbers with arrows that start at 0 and point to the right. For example, this arrow represents +10 because it is 10 units long and it points to the right.
![A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the left, and ends at negative 4.There is a solid dot indicated at 4.](https://cms-im.s3.amazonaws.com/kWiPFEiEBLs5EqzjzzBXV4JV?response-content-disposition=inline%3B%20filename%3D%227-7.5.B.pdf-56.png%22%3B%20filename%2A%3DUTF-8%27%277-7.5.B.pdf-56.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T162816Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e99dc25fa10dc85b1450053befa76afb9608dc461a82a2a7d5b9096ea9e88036)
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.
![A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the left, and ends at negative 4.There is a solid dot indicated at 4.](https://cms-im.s3.amazonaws.com/2rwrpmtbxTua5yWsmiCi4nry?response-content-disposition=inline%3B%20filename%3D%227-7.5.B.pdf-53.png%22%3B%20filename%2A%3DUTF-8%27%277-7.5.B.pdf-53.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T162816Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=1c984dc3935d81dd06efd38d65dbb79d466c802f6201b725e46cc74e7b9b52e7)
To represent addition, we put the arrows “tip to tail.” So this diagram represents \(3+5\):
![A number line.](https://cms-im.s3.amazonaws.com/Uhg65HY5RGjUNmbNmYjGKt3F?response-content-disposition=inline%3B%20filename%3D%227-7.5.B.pdf-54.png%22%3B%20filename%2A%3DUTF-8%27%277-7.5.B.pdf-54.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T162816Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f310483ac8fb3790928ed846bf7f5ad660ef670338f8df88df6d2d1321a873fc)
And this represents \(3 + (\text-5)\):
![A number line.](https://cms-im.s3.amazonaws.com/VS2hkJGUGk86PuaK1rcqhPPH?response-content-disposition=inline%3B%20filename%3D%227-7.5.B.pdf-55.png%22%3B%20filename%2A%3DUTF-8%27%277-7.5.B.pdf-55.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T162816Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=70f6875efd755ea52740a771f47816c3452e1e44b34b2bf3fbd58cdbce9d8ca1)