Unit 5, Lesson 1
Interpreting Negative Numbers
Grade 7
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The purpose of this Warm-up is to remind students about negative numbers using a familiar context. The context of a thermometer also reminds students of vertical number lines.
Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2. Introduce the context image, and explain that degrees Celsius is a way of measuring temperature, like degrees Fahrenheit. If necessary, tell students that 0 degrees Celsius is the freezing point of water and 0 degrees Fahrenheit is the freezing point of brine, a type of saltwater. Use Co-Craft Questions to orient students to the context and elicit possible mathematical questions.
Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before comparing questions with a partner.
Four vertical thermometers measured in degrees Celsius. There are 16 evenly spaced tick marks and starting from the bottom of the thermometer, negative 5 is on the first tick mark, zero on the sixth, 5 on the eleventh, and 10 on the sixteenth. The first thermometer is shaded starting from the bottom of the thermometer to the tenth tickmark. The second thermometer is shaded starting from the bottom of the thermometer to the third tickmark. The third thermometer is shaded starting from the bottom of the thermometer to between the eleventh and twelfth tickmark. The fourth thermometer is shaded starting from the bottom of the thermometer to between the fourth and fifth tickmark.
Invite several partners to share one question with the class, and record responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “positive,” “negative,” and “difference.”
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In this activity, students return to the context of a thermometer to examine rational numbers that are not integers. Students compare and interpret the signed numbers, including a temperature that is not pictured, to make sense of them in context (MP2).
Give students 2–3 minutes of quiet work time, and follow with a whole-class discussion.
What temperature is shown on each thermometer?
Four vertical thermometers measured in degrees Celsius. There are 16 evenly spaced tick marks and starting from the bottom of the thermometer, negative 5 is on the first tick mark, zero on the sixth, 5 on the eleventh, and 10 on the sixteenth. The first thermometer is shaded starting from the bottom of the thermometer to the tenth tickmark. The second thermometer is shaded starting from the bottom of the thermometer to the third tickmark. The third thermometer is shaded starting from the bottom of the thermometer to between the eleventh and twelfth tickmark. The fourth thermometer is shaded starting from the bottom of the thermometer to between the fourth and fifth tickmark.
The purpose of this activity is to remind students of how to compare two values using inequality notation. Begin by inviting students to share their responses for the temperature shown on each thermometer.
Then discuss the last question, asking students to explain their reasoning until they come to an agreement that
MLR2: Collect and Display
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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.
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 0 level, in the same way that we chose a 0 level for temperature.
Here is a picture of some sea animals. The number line on the left shows the vertical position, in meters, of each animal above or below sea level.
A graph. Vertical axis is labeled vertical position, in meters. 5 tick marks. Scale from negative 15 to 15, by 5s. Horizontal axis has no label or scale. Seagull is at 10. Dolphin is at 3. Penguin is at 0. Shark is at negative 3. Fish is at negative 7. Octopus is at negative 10.
The jumping dolphin?
The flying seagull?
The octopus?
The jumping dolphin?
The flying seagull?
The octopus?
The jumping dolphin?
The flying seagull?
The octopus?
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.)
We can use positive numbers and negative numbers to represent temperature and elevation.
When numbers represent temperatures, positive numbers indicate temperatures that are warmer than zero and negative numbers indicate temperatures that are colder than zero. This thermometer shows a temperature of -1 degree Celsius, which we write
A thermometer positioned horizontally and with the bottom of the thermometer on the left, is labeled “degrees Celsius.” There are 4 evenly spaced large tick marks. Starting on the left, the large tick marks are labeled negative 5, 0, 5, and 10. Between each large tick mark, there are 4 evenly spaced small tick marks. The thermometer is shaded, starting from the left, up to negative 1.
When numbers represent elevations, positive numbers indicate positions above sea level and negative numbers indicate positions below sea level.
We can see the order of signed numbers on a number line.
A number is always less than a number to its right. So
We use absolute value to describe how far a number is from 0. The numbers 15 and -15 are both 15 units from 0, so
Help us improve by sharing suggestions or reporting issues.
If some students struggle to estimate temperatures that are between two markings, consider asking:
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.