Lesson 7-3: Understand Absolute Value of Rational Numbers

Warm-Up 1

What is the absolute value of −6?
|−6| = ?

Vocabulary: Absolute value is the distance from zero on a number line. Distance is always positive or zero. The symbols | | mean absolute value.

✓Correct! |−6| = 6. The number −6 is 6 units from zero on the number line.

✗Not quite. Absolute value means distance from zero. −6 is 6 units from zero, so |−6| = 6.

Warm-Up 2

True or false: Absolute value is always positive or zero.

Think about it: Distance can never be negative. Since absolute value measures distance from zero, can it ever be a negative number?

✓Correct! Absolute value is distance from zero, and distance is never negative. It is always zero or positive.

✗Actually, this is true. Since absolute value measures distance from zero, it can never be negative.

Practice 1

Find the absolute value of each number.

Remember: Absolute value = distance from zero. Remove any negative sign. If the number is already positive or zero, it stays the same.

a. |−3.5| =

b. |7| =

c. |0| =

Sentence frame: "|___| = ___ because ___ is ___ units from zero on the number line."

✓All correct! |−3.5| = 3.5, |7| = 7, and |0| = 0. Each answer is the distance from zero.

✗Some answers need another look. Remember: absolute value is always the distance from zero (drop the negative sign). |−3.5| = 3.5, |7| = 7, |0| = 0.

Practice 2

Which number has the greater absolute value: −8 or 5?

Strategy: Find the absolute value of each number first.
|−8| = ? and |5| = ?
Then compare: which distance from zero is larger?

Explain your reasoning:

Sentence frame: "|−8| = ___ and |5| = ___. Since ___ > ___, −8 has the greater absolute value."

✓Correct! |−8| = 8 and |5| = 5. Since 8 > 5, −8 has the greater absolute value even though −8 is less than 5 on the number line.

✗Find each absolute value: |−8| = 8 and |5| = 5. Since 8 > 5, the answer is B. Remember: absolute value is about distance from zero, not which number is greater.

Practice 3

Drag the expressions into order from least to greatest absolute value.

Steps: First, find the absolute value of each expression. Then arrange from smallest to largest.
|0| = 0, |−1| = 1, |2| = 2, |−3| = 3, |4| = 4, |−5.5| = 5.5

|2|

|−3|

|0|

|−5.5|

|4|

|−1|

Least → Greatest

Sentence frame: "I know |0| is the smallest because 0 is the closest to zero. |−5.5| is the largest because 5.5 is the farthest from zero."

✓All correct! The order is: |0| = 0, |−1| = 1, |2| = 2, |−3| = 3, |4| = 4, |−5.5| = 5.5.

✗Not quite. Find each absolute value first, then sort: 0, 1, 2, 3, 4, 5.5. The correct order is |0|, |−1|, |2|, |−3|, |4|, |−5.5|.

Practice 4

True or false: "|−4| > |3| means −4 > 3."

Be careful! Comparing absolute values is NOT the same as comparing the numbers themselves.
|−4| = 4 and |3| = 3, so 4 > 3 is true.
But is −4 > 3? Where is −4 on the number line compared to 3?

Explain your reasoning:

Sentence frame: "|−4| = ___ and |3| = ___, so |−4| > |3| is ___. But −4 > 3 is ___ because −4 is to the ___ of 3 on the number line."

✓Correct! |−4| > |3| is true (4 > 3), but −4 is NOT greater than 3. Comparing absolute values is different from comparing the numbers.

✗This is false. While |−4| > |3| (because 4 > 3), that does not mean −4 > 3. In fact, −4 < 3. Absolute value comparison does not equal number comparison.

Practice 5

A submarine is at −200 feet (below sea level). A bird is at 150 feet (above sea level). Which is farther from sea level?

Sea level = 0. Use absolute value to find each distance from sea level:
Submarine: |−200| = ?
Bird: |150| = ?
Which distance is greater?

Explain using absolute value:

Sentence frame: "The submarine is at ___ ft, so its distance from sea level is |___| = ___ ft. The bird is at ___ ft, so its distance is |___| = ___ ft. Since ___ > ___, the ___ is farther from sea level."

✓Correct! |−200| = 200 and |150| = 150. Since 200 > 150, the submarine is farther from sea level even though its elevation is a negative number.

✗Use absolute value to find distance from sea level: |−200| = 200 and |150| = 150. Since 200 > 150, the submarine is farther. The answer is B.

Challenge

Two hikers are exploring a canyon. Hiker A is at −350 feet (below sea level). Hiker B is at 275 feet (above sea level). Compare their distances from sea level using absolute value. Who is farther from sea level, and by how much? Explain your reasoning.

Plan your response:
1. Find each hiker's distance from sea level using absolute value.
2. Compare the distances.
3. Find the difference: subtract the smaller from the larger.
|−350| = ? and |275| = ?

Your response:

Sentence frame: "Hiker A is at ___ ft, so their distance from sea level is |___| = ___ ft. Hiker B is at ___ ft, so their distance is |___| = ___ ft. Hiker ___ is farther from sea level by ___ − ___ = ___ ft because ___."

🟠 Level 0 — Extra Support

Warm-Up

Practice

Challenge