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Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Keep all previous problems and work displayed throughout the talk.
For each pair of expressions, decide mentally which one has a value that is closer to 0.
To involve more students in the conversation, consider asking:
In this task, students are introduced to the absolute value of a number as its distance from zero on the number line. The context is not realistic, but helps students visualize relationships on the number line in a more concrete way.
In the digital version of the activity, students use an applet to visualize a bug jumping on a number line. The applet allows students to pick a starting point for the bug, choose the direction it jumps, and then check where it lands.
If students don't have individual access, displaying the applet for all to see would be helpful during the Launch and Activity Synthesis.
Give students 7–8 minutes of quiet work time, and follow with a whole-class discussion.
A bug is jumping around on a number line.
If the bug starts at 1 and jumps 4 units to the right, where does it end up? How far away from 0 is this?
If the bug starts at 1 and jumps 4 units to the left, where does it end up? How far away from 0 is this?
If the bug starts at 0 and jumps 3 units away, where might it land?
If the bug jumps 7 units and lands at 0, where could it have started?
The absolute value of a number is its distance from 0. The bug is currently to the left of 0 and the absolute value of its location is 4. Where on the number line is it?
If the bug is to the left of 0 and the absolute value of its location is 5, where on the number line is it?
If the bug is to the right of 0 and the absolute value of its location is 2.5, where on the number line is it?
We use the notation to say "the absolute value of -2," which means "the distance of -2 from 0 on the number line."
What does mean, and what is its value?
The purpose of this discussion is to formally introduce the term absolute value. The absolute value of a number is its distance from zero. Begin by asking students to contrast and . (They are both finding the distance from a number to zero. They both have the same value, but represent the distance between two different points and zero.) If needed, draw a number line, and demonstrate how -8 and 8 are both 8 units away from 0.
Here are some additional questions for discussion:
Conclude the discussion by asking students to think of situations that use the concept of absolute value. (A car’s odometer tracks the miles driven regardless of the direction. A swimmer swimming laps in a pool will end up in the same location, but absolute value can be used to measure the total distance swam. People can get on or off a city bus. Absolute value measures the total number of riders.)
In this activity, students continue to develop their understanding that represents the distance from zero to and that and are equal.
Tell students that a part of the city of New Orleans is 6 feet below sea level. Explain that we can use “-6 feet” to describe its elevation, and “ feet” to describe its vertical distance from sea level.
Remind students that we write “” to describe a temperature that is 5 degrees Celsius below freezing and “” for a temperature that is 5 degrees above freezing.
Give students 5–6 minutes of quiet work time, and follow with a whole-class discussion.
The goal of this discussion is for students to explain how absolute value relates to distance and elevation in a situation. Arrange students in groups of 2, and display the image and expressions for all to see.
Give students 1 minute to study the image and match each letter with one of the expressions. (A: 15 feet, B: feet, C: feet, D: -4 feet) Have students discuss their matches with a partner before inviting students to share their responses and reasoning. Record and display student responses on the image for all to see.
We can compare two numbers by looking at their positions on the number line: The number farther to the right is greater. The number farther to the left is less.
Sometimes we want to compare which one is closer to or farther from 0. For example, we may want to know how far away the temperature is from the freezing point of , regardless of whether it is above or below freezing.
The absolute value of a number tells us its distance from 0.
For example, the absolute value of -4 is 4, because -4 is 4 units to the left of 0. The absolute value of 4 is also 4, because 4 is 4 units to the right of 0. Opposites always have the same absolute value because they are both the same distance from 0.
The distance from 0 to itself is 0, so the absolute value of 0 is 0. Zero is the only number whose distance to 0 is 0. For all other absolute values, there are always two numbers—one positive and one negative—that have that distance from 0.
To say, “the absolute value of 4,” we write "."
To say, “the absolute value of -8 is 8,” we write "."
If students confuse absolute value with opposites and think that finding the absolute value of a number changes the sign of the number, consider sharing these examples and explaining how they contrast opposites and absolute value:
Match each number with a description of a situation.
Which temperature is colder: or ?
Which temperature is closer to the freezing point: or ?
Which temperature has a smaller absolute value? Explain how you know.
Match each number with a description of an elevation situation.
The elevation of a city has a difference from sea level of 10 feet. Name the two elevations that the city could have.