Lesson 2

This warm-up prompts students to compare four expressions that will prime them for writing inequality statements involving signed numbers in later activities. It encourages students to explain their reasoning, hold mathematical conversations, and gives you the opportunity to hear how they use terminology and talk about characteristics of the inequalities in comparison to one another. To allow all students to access the activity, each inequality has one obvious reason it does not belong. During the discussion, listen for important ideas and terminology that will be helpful in upcoming work of the unit. 

At this time, students might explain the direction of the inequality symbol in terms of the size of the numbers. This explanation is acceptable for students to give during this warm-up, but the next activity introduces a more correct concept of ordering that includes negative numbers.

Arrange students in groups of 2–4. Display the questions for all to see. Ask students to indicate when they have noticed one question that does not belong and can explain why. Give students 1 minute of quiet think time and then time to share their thinking with their group. In their groups, tell each student to share their reasoning why a particular question does not belong and together find at least one reason each question doesn’t belong.

Ask each group to share one reason why a particular image does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as “greater than” or “less than.” Also, press students on unsubstantiated claims.

The purpose of the task is for students to compare signed numbers in a real-world context and then use inequality signs accurately with negative numbers (MP2). The context should help students understand “less than” or “greater than” language. Students evaluate and critique another's reasoning (MP3).

1. Use a straightedge or a ruler to draw a horizontal line. Mark the middle point of the line and label it 0.
2. To the right of 0, draw tick marks that are 1 centimeter apart. Label the tick marks 1, 2, 3 . . . 10. This represents the positive side of your number line.
3. Fold your paper so that a vertical crease goes through 0 and the two sides of the number line match up perfectly.
4. Use the fold to help you trace the tick marks that you already drew onto the opposite side of the number line. Unfold and label the tick marks -1, -2, -3 . . . -10. This represents the negative side of your number line.

Allow students 5–6 minutes quiet work time followed by whole-class discussion.

Some students may have difficulty comparing numbers on the negative side of the number line. Have students plot the numbers on a number line in order to sequence them from least to greatest. Make it evident that temperatures become warmer (i.e., greater) as they move from left to right on the number line.

Ask students to explain how they could tell which of the two negative numbers was greater in problem 2. After gathering 1 or 2 responses, display the following image for all to see:

Explain to students that as we go to the right on the number line, we can think of the temperature as getting hotter, and as we go to the left, we can think of the temperature as getting colder. Numbers don’t always describe temperature, though. So we use the word “greater” to describe a number that is farther to the right, and “less” to describe numbers that are farther to the left. For example, we would write \(6 > \text-50\) and say “6 is greater than -50 because it is farther to the right on the number line.” Equivalently, we could write \(\text-50 < 6\) and say “-50 is less than 6 because -50 is farther to the left on the number line.”

Ask students to make up their own list of negative numbers and plot on a number line. Then ask them to write several inequality statements with their numbers. As time allows, have students share one inequality statement with the class to see if all agree that it is true. If time allows, have students swap their list with a partner to plot and check their partner’s number lines and inequality statements.

The purpose of this task is for students to understand that for a given number, numbers to the left are always less than the number, and numbers to the right are always greater than the number. The precise use of the term “absolute value” is not expected at this time.

Some students may have difficulty comparing numbers on the negative side of the number line. Have students plot the numbers on a number line from least to greatest. It may be helpful to provide an example that students can use as a visual aid while they are working independently.

Some students may have difficulty comparing fractions in question 4. Remind them that comparing fractions is easier using a common denominator. Suggest that they subdivide the interval from 0 to 1 into fourths or eighths.

The key takeaway from the discussion is that now that we have numbers on both sides of 0, distance from 0 isn’t enough to compare two numbers. Instead, we call numbers farther to the left on the number line “less” and numbers farther to the right “greater.” Display the following number line for all to see:

One inequality at a time, ask students to indicate whether they think each of the following is true or false:

• \(\frac54 > \text{-}\frac {3}{2}\) (true)
• \(\frac54\) is farther from 0 than \(\text{-}\frac {3}{2}\) (false)
• \(\text{-}\frac {3}{2} < \text{-}\frac {3}{4}\) (true)
• \(\text{-}\frac {3}{2}\) is farther from 0 than \(\text{-}\frac {3}{4}\) (true)

Invite students to share their reasoning. Here are some sentence frames that might be helpful:

• “__ is greater (less) than __ because __.”
• “__ is farther from 0 than __ because __.”