In this task, students are introduced to certain contexts for which it would be difficult to answer mathematical questions if restricted to only positive numbers. The need for negative numbers leads to the natural representation of them on the number line. 

This task is not about operations with signed numbers, but rather why we extend our number system beyond positive numbers. Students reason abstractly and quantitatively when they represent the change in temperature on a number line (MP2).

In the digital version of the activity, students use an applet to represent changes in temperature. The applet allows students to quickly represent an increase or decrease in temperature.

The purpose of this discussion is to introduce a negative number as a number that is less than zero, in contrast to a positive number as a number greater than zero. Some students will have a pre-existing understanding of positive and negative numbers. 

Begin by inviting several students to share the different ways they described the final temperature in the second situation (4 degrees below zero, -4 degrees Celsius). Tell students that this is an example of a negative number because it describes a number less than zero, and we use the - symbol to show that.

Ask students how they would describe a number that is greater than zero (a positive number). Explain that a + symbol is used to indicate a positive number, though it is not always written out. For example, +7 and 7 both represent positive 7. Negative 7 is represented as -7.

The purpose of this task is to build understanding of the negative side of the number line, both by reading values and assigning values to equally spaced divisions. Non-integer negative numbers are also used. Students reason abstractly and quantitatively as they interpret positive and negative numbers in context (MP2).

The purpose of the discussion is for students to share their strategies for making sense of negative values on the number line. Begin by inviting students to share their reasoning as to whether they agreed with Jada or Elena in the last question. If not mentioned by students, connect this question to the Warm-up by pointing out that the temperature is halfway between -1 and -2 on the number line, so it must be -1.5 degrees.

Then display this number line for all to see.

Discuss the following questions:

• “What could be the value of point ? Explain your reasoning.” (A value between 3 and 3.5 is reasonable. The point is between 3 and 4 but looks closer to 3.)

• “What could be the value of point ? Explain your reasoning.” (A value between -3 and -3.5 is reasonable. The point is between -3 and -4 but looks closer to -3.)

• “Do you think the value of point is closer to -0.75 or -1.25? Why?” (Point is closer to -0.75 because it is located between 0 and -1.)

• “What do you notice about the location of negative values on a vertical number line?” (Negative numbers are at the bottom. The negative numbers are like a mirror of the positive numbers.)

In this activity, students continue to interpret signed numbers in context and begin to compare their relative locations on a vertical number line in order to make sense of signed number (MP1). The number line is labeled in 5-meter increments, so students have to interpolate the height or depth for some of the animals. Students are also given the height or depth of other animals that are not pictured and asked to compare these to the height or depth of the animals shown.

Monitor for students who recognize that there are two possible answers to the last question depending on whether the new dolphin is 3 meters above or below the original dolphin. 

If students measure to the top or bottom of the animal, remind them that we are using the eyes of the animal to measure their height or depth.

Some students may struggle to visualize where the albatross, seagull, and clownfish are on the graph. Consider having them draw or place a marker where the new animal is located while comparing it to the other animals in the picture.

The purpose of this discussion is for students to share their reasoning for the height or depth of each new animal. A key idea of this discussion is that distances above and below sea level can be represented using signed numbers. The depths of the shark, fish, and octopus can be expressed as negative numbers because they are below sea level, while the heights of the remaining animals can be expressed as positive numbers because they are above sea level.

Begin by inviting students to share their responses and reasoning to the fourth question about the relative position of the clownfish. Record and display their verbal descriptions using signed numbers. For example, if a student says the clownfish is 5 meters below the dolphin, write “-5”.

Finally, ask students to share their responses and reasoning to the last question about the height of the new dolphin. Invite a previously selected student to explain why there are two possible answers to the last question. (The new dolphin could be 3 meters above or below the original dolphin.)

This activity presents a second, natural context for negative numbers, and students start comparing positive and negative numbers in preparation for ordering them in a following activity. Students may use the structure of a vertical number line in order to compare the relative location of each elevation (MP7).

In the digital version of the activity, students use an applet to represent the elevations of the highest points on land and lowest points in the ocean on a vertical number line. The applet allows students to drag points to a vertical number line to mark different mountains or trenches and quickly check their answers.

The goal of this discussion is for students to compare negative elevations. Begin by displaying a blank vertical number line for all to see. 

Ask students where the elevation of New Orleans, LA could be on this number line, and plot and label the point for all to see. Continue adding points to the number line for the 5 elevations given for the new city not on the list (-11 feet, 3 feet, -4 feet, -9 feet, 0 feet). Discuss the following questions:

• “What does it mean when a point is above 0 on a vertical number line?” (The number is positive, and the elevation represented by that point is above sea level.)

• “What does it mean when a point is below 0 on a vertical number line?” (The number is negative, and the elevation represented by that point is below sea level.)

• “Is -11 feet higher or lower in elevation than -8 feet?” (-11 is lower in elevation because it is lower than -8 on the vertical number line.)

• “Is -4 feet higher or lower in elevation than -8 feet?” (-4 is higher in elevation because it is higher than -8 on the vertical number line.)