Lesson 1

The purpose of this task is to introduce students to temperatures measured in degrees Celsius. Many students have an intuitive understanding of temperature ranges in degrees Fahrenheit that are typical of the city or town in which they live, but many are unfamiliar with the Celsius scale.

Arrange students in groups of 2. Tell students that they will look at an image, and their job is to think of at least one thing they notice and at least one thing they wonder. Display the image for all to see. Ask students to give a signal when they have noticed or wondered about something. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After each response, ask the class if they agree or disagree and to explain alternative ways of thinking, referring back to the images each time. Explain to students that temperatures are usually measured in either degrees Fahrenheit, which is what they are probably most familiar with, and degrees Celsius, which may be new for them. Tell them that many other countries measure temperature in degrees Celsius and that scientists use this temperature scale. One thing that is special about the Celsius scale is that water freezes at 0 degrees and boils at 100 degrees (at sea level).

The purpose of this task is to use the previously introduced context of temperature 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).

Notice the arguments students make to decide whether Elena or Jada are correct in question 2. Some students may defend Elena because they see the liquid is above -2 and conclude that the temperature is -2.5 degrees. Other students will defend Jada by noting the temperature is halfway between -1 and -2 degrees, concluding that it must be -1.5 degrees.

Allow students 5–6 minutes quiet work time followed by whole-class discussion.

Some students may have difficulty identifying the non-integer temperatures on the thermometers. This difficulty arises when students are unable to identify the scale on a number line. This may be significantly more challenging on the negative side of the number line as students are accustomed to the numbers increasing in magnitude on the positive side as you go up. This issue is addressed in task item number 2. It may be helpful to draw attention to the tick mark between 1 and 2 and its label. This previews the idea of opposites addressed in the next activity.

The purpose of the discussion is to use temperature to explore the concept of negative numbers and introduce the vocabulary of rational numbers. Select students to share their reasoning as to whether they agreed with Jada or Elena in question 2. 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, and so it must be -1.5 degrees.

Tell students that rational numbers are like fractions except they can also be negative. So rational numbers are all fractions and their opposites. The term “RATIOnal number” comes from the fact that ratios and fractions are closely related ideas. Display some examples of rational numbers like 4, -3.8, \(\text-\frac{4}{3}\), and \(\frac12\) for all to see. Ask students whether they agree 4 and 3.8 are fractions. Tell them these might not look like fractions, but they actually are fractions because they can be written as \(\frac{16}{4}\) and \(\frac{38}{10}\). All rational numbers can be plotted as points on the number line and can be positive, zero, or negative just like temperature.

The purpose of this activity is for students to continue interpreting signed numbers in context and to begin to compare their relative location. A vertical number line shows the heights above sea level or depths below sea level of various animals. The number line is labeled in 5 meter increments, so students have to interpolate the height or depth for some of the animals. Next, they are given the height or depth of other animals that are not pictured and asked to compare these to the animals shown.

As students work, monitor for whether they are expressing relative distances in words, for example “3 meters below,” or if they are expressing the same idea with notation, as in -3 meters. Both are acceptable; these ideas are connected in the discussion that follows (MP2). Also monitor for students who notice that there are two possible answers for the last question.

Display the image for all to see. Tell students to measure the height or depth of each animal's eyes, to the nearest meter. Remind students that we choose sea level to be our zero level, in the same way that we chose a zero level for temperature.

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 main point for students to get out of this activity is that we can represent distance above and below sea level using signed numbers. The depths of the shark, fish, and octopus can be expressed as approximately -3 m, -6 m, and -7.5 m respectively, because they are below sea level.

Signed numbers can also be used to represent the relative vertical position of different pairs of animals. Have selected students share their responses and reasoning for how the heights of the albatross, seabird, and clownfish compare to the dolphin, seagull, and octopus. Record and display their verbal descriptions using signed numbers. For example, if a student says the albatross is 7 meters below the seagull, write "-7".

Finally, ask whether students noticed the ambiguity in the last question (about the height of the new dolphin). Ask such a student to explain why there are two possible answers to the last question.

The purpose of this task is to present a second, natural context for negative numbers and to start comparing positive and negative numbers in preparation for ordering them. Monitor for students who make connections between elevation and temperature or come up with strategies for deciding which points are lower or higher than other points. Students may use the structure of a vertical number line in order to compare the relative location of each elevation (MP7).

Some students may have difficulty comparing negative elevations. For example, when students are asked to find a higher elevation than Coachella, CA, they may think that -35 feet is a higher elevation than -22 feet because 35 > 22. Encourage students to create a vertical number line and plot elevations before comparing them. Alternatively, provide them with a pre-made number line to use.

The important concept is that elevation measures how far below or above sea level something is. Positive elevation tells us that something is above sea level, whereas negative elevation tells us that something is below sea level. In the same way, positive numbers are greater than zero and negative numbers are less than zero. Zero is neither greater than or less than zero; therefore, it is neither positive or negative. Invite selected students to share their thinking about how they compared different elevations and any similarities they may have noticed between elevation and temperature.