Lesson 6

The purpose of this Number Talk is to elicit strategies and understandings students have about the distance from 0 on the number line. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to think about distance from 0 for various rational numbers. While four problems are given, it may not be possible to share every strategy. Consider gathering only two or three different strategies per problem.

Reveal one problem at a time. Give students 30 seconds of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all previous problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their reasoning for each problem. Record and display their explanations for all to see. To involve more students in the conversation, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Did anyone have the same answer but would explain it differently?”
• “Did anyone reason about the problem in a different way?”
• “Does anyone want to add on to _____’s reasoning?”
• “Do you agree or disagree? Why?

The purpose of this task is to help students understand the absolute value of a number as its distance from 0 on the number line. The context is not realistic, but helps students visualize relationships on the number line in a more concrete way. Students have used the concept of absolute value informally in previous lessons, but this is where the term is formally introduced and used precisely (MP6).

Students may confuse absolute value with opposites, thinking that absolute value changes the sign of the number. Remind students that absolute value represents the distance of the number from 0 without worrying about the sign or direction. Use some concrete examples: in a board game, 3 moves forward and 3 moves backward both involve 3 jumps but you land in different places; traveling 5 miles east or 5 miles west would put the same number of miles on your car’s odometer, but you end up in different places depending on the direction; earning $10 and spending $10 both involve the amount $10 but in one case you gain it and in the other you lose it. Absolute value is just the amount involved, not including the sign.

Define the absolute value of a number as its distance from 0. Ask students to contrast \(|\text-8|\) and \(|8|\) to come to the conclusion that they have the same value but represent the distance between two distinct points and zero. Also ask for situations where they think the absolute value might be useful (example: your car’s odometer tracks the miles you drove, but if you make a round trip—the same distance in two opposite directions—the difference between where you started and where you ended is zero).

To help clear up misconceptions related to opposites and absolute values, ask:

• “What is the difference between a number’s opposite and a number’s absolute value?” (Opposite is another number on the number line whose distance from zero is the same; absolute value is a number that describes that distance.)
• “Does finding a number’s absolute value always mean changing the sign?” (No, absolute value represents the distance from zero. If the number is positive, the number and its absolute value are the same. If the number is negative, the distance is represented by the number without its negative sign.)
• “If \(n\) is any number that can be positive or negative, what is the sign of the absolute value of \(n\)?”

The purpose of this task is for students to develop their understanding of the \(|x|\) notation in familiar contexts. They should build their understanding that \(|x|\) represents the distance from zero to \(x\) and that \(|\text-x|\) and \(|x|\) are equal.

Allow students 10 minutes quiet work time followed by whole-class discussion.

To summarize students’ work, consider displaying the following diagram and these four expressions. Give students a minute to study the diagram and match each letter on the diagram to an appropriate expression.

• -4 feet
• \(|{15}|\) feet
• \(|{\text -4}|\) feet
• 15 feet