- Find and place positive and negative numbers on a number line
- Order rational numbers
- Interpret statements of inequality involving positive and negative numbers
- Use absolute value notation
- Under the Sea: Absolute Value and Ordering Rational Numbers printable
- Answer Key: Diving Into the Number System printable
- Standards Chart: Diving Into the Number System printable
- Whiteboard or chart paper and markers
- Make a class set of the Under the Sea: Absolute Value and Ordering Rational Numbers printable.
- Print a copy of the Answer Key: Diving Into the Number System printable for your use.
Step 1: Ask the class to consider the following problem: On a frosty winter day on the steppes of Mongolia, the temperature was 2°F at noon. Shortly after sunset, the temperature was 17°F below zero. Write the temperatures as 2°F and -17°F. Ask the class if the temperature was greater at noon or after sunset. Point out that since 2° is warmer than -17°, 2° is the greater temperature, even though 17 (without the negative sign) has a greater value than 2.
Step 2: Draw a number line on the board, with points at 2 and -17. Show how 2 is farther to the right on the number line than -17, so it is the greater of the two numbers. Indicate to the class that when comparing numbers on a number line, the number farther to the right is greater.
Step 3: Point out how number lines can also be drawn vertically, and that in certain situations, vertical number lines may be easier to use and more representative of the real-world situation they depict than a traditional horizontal number line. Situations where vertical number lines may be preferred include depicting concepts that literally go up and down, such as elevation and depth, as well as concepts that can be figuratively thought of as going up or down, such as temperature changes or money being “raised” toward a fundraising goal.
Step 4: Give the class another example with two negative numbers. Say the temperature in Mongolia at noon was -5°F and after sunset was -21°F. Which temperature was greater? Depict the example on a number line, showing how -5 is farther to the right on the line, so it is greater than -21.
Step 5: Mention that placing fractions and decimals on a number line works the same way as integers (whole numbers). For example, -2.7 degrees is to the left of -2.2 degrees on a number line, so -2.2 is the greater temperature.
Step 6: Ask the class if they know of real-world examples, in addition to temperature, where negative numbers might be compared to other negative numbers or positive numbers. Some examples might include:
a. A company with a loss of $20,000 (-20,000) had better financial results than a company that lost $2,000,000 (-2,000,000).
b. It’s better for a running back in football to lose a yard on a play (-1) than 6 yards (-6).
c. A spot on the ocean floor 200 meters below the surface of the ocean (-200) has a greater elevation than a spot 2,000 meters below sea level (-2,000).
d. To break a tie in soccer standings, a team that scored 30 goals and gave up 34 (goal differential of -4) has a higher ranking than a team that scored 40 goals but gave up 50 (goal differential of -10).
Step 7: Reiterate that when comparing two negative numbers, the number farther to the right on a horizontal number line (higher on a vertical number line) has the greater value.
Step 8: To introduce the concept of absolute value, draw a vertical number line on the board and pose the following scenario to the class: Dexter, the night guard at Acme Enterprises, sits at a guard station on the lobby floor of corporate headquarters at street level (label at 0 on the number line). While going through some paperwork, he gets a nasty paper cut and needs a bandage. He knows there is a first-aid kit in the human resources office on the third floor (label at 3 on the number line) and one in the parking garage office on the second floor of the underground parking garage (label at -2 on the number line). If Dexter wants the bandage quickly, which floor should he go to? Point out that the distance from -2 to 0 is 2 and the distance from 0 to 3 is 3, so Dexter should go to the underground parking garage office two floors below street level. Tell the class that this is an example of the concept of absolute value, which shows the distance of a number from zero.
Step 9: Introduce absolute value notation by pointing out that the distance from floor -2 to 0 is 2 floors. Write this as |-2| = 2 on the board. Say “The absolute value of negative 2 equals 2.” Point out that the distance from 0 to 3 is 3 and that |3| = 3. Say, “The absolute value of 3 is 3.” Reiterate that absolute value shows the distance of a number from zero.
Step 10: Write the following numbers on the board: 9, -5, 1, -7, -12. Group students in pairs and ask them to:
a. Determine the absolute value of each number. Answer: |-12| = 12, |-7| = 7, |-5| = 5, |1| = 1, |9| = 9
b. Draw a number line and plot the absolute values of the numbers on the line to order them from least to greatest. Answer: -12, -7, -5, 1, 9
Step 11: Checking for Understanding: Review answers as a class and respond to any questions. Make sure students understand that the value of -12 is less than the value of -7, but the absolute value of |7| is less than the value of |12|.
Step 12: Assign the Under the Sea: Absolute Value and Ordering Rational Numbers printable for classwork or homework.
Step 13: Checking for Understanding: Review the answers to the Under the Sea: Absolute Value and Ordering Rational Numbers printable, which are provided on page 1 of the Answer Key: Diving Into the Number System printable. Make sure students explain their mathematical thinking. Address any misconceptions that may arise.
- Grade 6: Absolute Value and Ordering Rational Numbers (CCSS 6.NS.5, 6.NS.6, and 6.NS.7)
- Grade 6–8: Constructing an Argument, Modeling, Using Appropriate Tools Strategically, and Attending to Precision (CCSS Practice MP3–6); NCTM Number and Operations
For more information, download the comprehensive Standards Chart: Diving Into the Number System printable.