Lesson 7

Comparing Numbers and Distance from Zero

Lesson Narrative

In this lesson, students use precise language to distinguish between order and absolute value of rational numbers (MP6). It is a common mistake for students to mix up “greater” or “less” with absolute value. A confused student might say that -18 is greater than 4 because they see 18 as being the “bigger” number. What this student means to express is \(|\text-18| > 4\). The absolute value of -18 is greater than 4 because -18 is more than 4 units away from 0. In the “Submarine” activity, students visualize possible elevations of characters with sticky notes on a vertical number line. The freedom to move a sticky note within a specified range anticipates the concept of a solution to an inequality in the next section.

Learning Goals

Teacher Facing

  • Critique comparisons (expressed using words or symbols) of rational numbers and their absolute values.
  • Generate values that meet given conditions for their relative position and absolute value, and justify the comparisons (using words and symbols).
  • Recognize that the value of $\text-a$ can be positive or negative, depending on the value of $a$.

Student Facing

Let’s use absolute value and negative numbers to think about elevation.

Required Preparation

For every 4 students, create a set of 5 sticky notes that read Clare, Andre, Han, Lin, and Priya. These are for the launch of the “Submarine” activity.

Learning Targets

Student Facing

  • I can explain what absolute value means in situations involving elevation.
  • I can use absolute values to describe elevations.
  • I can use inequalities to compare rational numbers and the absolute values of rational numbers.

CCSS Standards

Glossary Entries

  • absolute value

    The absolute value of a number is its distance from 0 on the number line.

    Horizontal number line, tick marks every 1 unit from -7 to 7. Above, there is a horizontal segment from -7 to 0 labeled 7, and a horizontal segment from 0 to 5 labeled 5. 

    The absolute value of -7 is 7, because it is 7 units away from 0. The absolute value of 5 is 5, because it is 5 units away from 0.

  • negative number

    A negative number is a number that is less than zero. On a horizontal number line, negative numbers are usually shown to the left of 0.

    number line with arrow that extends from 0 to -7
  • opposite

    Two numbers are opposites if they are the same distance from 0 and on different sides of the number line.

    For example, 4 is the opposite of -4, and -4 is the opposite of 4. They are both the same distance from 0. One is negative, and the other is positive.

    Number line that extends from -5 to 5, with points at -4 and 4.
  • positive number

    A positive number is a number that is greater than zero. On a horizontal number line, positive numbers are usually shown to the right of 0.

    Number line with green arrow extending from 0 to 7
  • rational number

    A rational number is a fraction or the opposite of a fraction.

    For example, 8 and -8 are rational numbers because they can be written as \(\frac81\) and \(\text-\frac81\).

    Also, 0.75 and -0.75 are rational numbers because they can be written as \(\frac{75}{100}\) and \(\text-\frac{75}{100}\).

  • sign

    The sign of any number other than 0 is either positive or negative.

    For example, the sign of 6 is positive. The sign of -6 is negative. Zero does not have a sign, because it is not positive or negative.