Lesson 5

The purpose of this warm-up is to use opposites to get students to think about distance from 0. Problem 3 also reminds students that the opposite of a negative number is positive.

Notice students who choose 0 or a negative number for \(a\) and how they reason about \(\text-a\).

Arrange students in groups of 2. Give students 5 minutes of quiet think time, then 2 minutes of partner discussion. Follow with whole-class discussion.

For problem 3, students might assume that \(\text-a\) is always a negative number. Ask these students to start with a negative number and find its opposite. For example, starting with \(a = \text-3\), we can find its opposite, \(\text-(\text-3)\), to be equal to 3.

The main idea of discussion is that opposites have the same distance to 0 (i.e., same absolute value) and that the opposite of a negative number is positive. Ask students to discuss their reasoning with a partner. In a whole-class discussion, ask a student who chose \(a\) to be positive to share their reasoning about how to plot \(\text-a\) and whether they agreed with Noah in problem 3. Then, select previously identified students who chose \(a\) to be negative to share their thinking. If not mentioned by students, emphasize both symbolic and geometric statements of the fact that the opposite of a negative number is positive. For example, if \(a=\text-3\), write \(\text-(\text-3) = 3\) and show that 3 is the opposite of -3 on the number line because they are the same distance to 0. If time allows, select a student who chose \(a\) to be 0 and compare to cases where \(a\) is negative or positive. The number 0 is its own opposite because no other number is 0 units away from 0. Sequencing the discussion to look at positive, negative, and 0 values of \(a\) helps students to visualize and generalize the concept of opposites for rational numbers.

Students distinguish between absolute value and order in the context of elevation. Students express their ideas carefully using symbols, verbally, and using a vertical number line. Placing possible elevations on the number line serves as a transition to thinking about solutions to inequalities. Look for students who choose positive and negative elevations for Han and Lin to compare in the discussion.

Arrange students in groups of 4. Distribute one set of sticky notes to each group, where each note contains one name: Clare, Andre, Han, Lin, and Priya. Display the image for all to see throughout the activity.

Ask students to read the instructions for the task and the description of each person's elevation. Give them a few minutes to use their sticky notes, as a group, to decide where each person (except Priya) could be located.

Place Clare’s sticky note on the number line according to the completed first row of the table. Explain the completed first row of the table to students as it pertains to Clare’s description. Use precise language when explaining the symbols in the table:

• One possible elevation for Clare is 150 feet because 150 is greater than -100, and it is also farther from sea level.
• 150 is greater than -100.
• The absolute value of 150 is greater than the absolute value of -100.

Ask groups to complete the rest of the table for the other people (except Priya), and then answer the question about Priya. Note that it is possible to come up with different, correct responses that fit the descriptions. Give students 10 minutes to work followed by whole-class discussion.

The purpose of the discussion is to let students practice using proper vocabulary to express ideas that distinguish order from absolute value with positive and negative numbers. Select previously identified students to share different elevations for Han and for Lin that show both positive and negative possibilities. Encourage students to explain why the elevation they chose satisfies the description in the problem. As students speak, record their statements using \(<,>,=\) and \(|\boldcdot |\). Allow students to rearrange sticky notes on the vertical number line display. If time allows, use the sticky notes to show the range of possible solutions for each character; this will help to further prepare students for the concept of graphing solutions of an inequality on the number line.

In this info gap activity, students use comparisons of order and absolute value of rational numbers to determine the location of unknown points on the number line. In doing so students reinforce their understanding that a number and its absolute value are different properties. Students will also begin to understand that the distance between two numbers, while being positive, could be in either direction between the numbers. This concept is expanded on further when students study arithmetic with rational numbers in grade 7.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

Arrange students in groups of 2. In each group, distribute the first problem card to one student and a data card to the other student. After debriefing on the first problem, distribute the cards for the second problem, in which students switch roles.

Students may struggle to make sense of the abstract information they are given if they don't choose to draw a number line. Rather than specifically instructing them to use this strategy, consider asking them a question like “How could you keep track of the information you've learned about the points so far?”

Select students with different strategies to share their approaches. Invite them to share which of the clues they thought were more helpful and which were least helpful. Ask students to explain how drawing a number line helped them and how they decided on the appropriate order for the unknown numbers.

The goal of this activity is for students to practice comparing rational numbers.

Notice students who compare fractions to decimals, fractions to integers, or who compare absolute values to negative numbers.

Arrange students in groups of 2. Give students 10 minutes to work before whole-class discussion.

The goal of discussion is to allow students to use precise language when comparing rational numbers and absolute values verbally. Select previously identified students to share their responses that compare fractions to decimals, fractions to integers, or absolute values to negative numbers. Display their responses using absolute value and \(>, <, =\) symbols for all to see. Ask students to indicate whether they agree that each response is true, and ask students to share their reasoning about whether they agree or disagree.