Learning Objectives

By the end of this lesson, you should be able to:

1. Compare and order integers using a number line and the symbols <, >, and =.
2. Find absolute values and opposites and apply integer ideas to real-life situations.

God is Present in Every Season

Integers help us describe situations above and below a starting point: temperatures above and below zero, money saved or owed, and places above or below sea level. Life also has different seasons. Whether circumstances feel positive or difficult, God sees us, understands us, and gives us wisdom to respond faithfully.

Memory verse: “Teach us to number our days, that we may gain a heart of wisdom.” — Psalm 90:12

1. What Are Integers?

Integers are the whole numbers, their opposites, and zero:

…, -3, -2, -1, 0, 1, 2, 3, …

• Negative integers are less than zero, such as -1, -5, and -20.
• Positive integers are greater than zero, such as 1, 5, and 20.
• Zero is neither positive nor negative.

Integers are useful for describing temperatures, elevations, bank balances, floors below ground, and changes in position.

2. Comparing Integers on a Number Line

On a horizontal number line, numbers increase as we move to the right and decrease as we move to the left.

Therefore, the number farther to the right is always greater.

Use the symbols:

• < means “less than”
• > means “greater than”
• = means “equal to”

3. Absolute Value

The absolute value of a number is its distance from zero on a number line. Absolute value is written using vertical bars.

Absolute value is never negative because distance is never negative.

4. Opposites

Two numbers are opposites if they have the same absolute value but different signs.

The opposite of a positive number is negative, and the opposite of a negative number is positive. The opposite of zero is zero.

If a is a number, then -a means “the opposite of a.”

Worked Example: Comparing Temperatures

On one day, the temperature was -12°C. On another day, it was -7°C. Which temperature was greater?

-12 < -7

Therefore, -7°C was the greater temperature. It was also warmer because it is farther to the right on the number line.

Practice Questions

Use number lines when helpful. Explain your thinking in the application problems.

1. Skill Practice

1. Which number is not an integer: -31, 74, 22.5, -7, or 192?
2. Explain why the absolute value of a number is never negative.
3. Write the integers in order from least to greatest: -9, 12, 6, -3, 0, -5.
4. State the absolute value of each number:
   1. 1
   2. -9
   3. 15
   4. -12
5. State the opposite of each number:
   1. 14
   2. -33
   3. 0
   4. 81
6. Evaluate each expression when x=-3:
   1. |x| + 8
   2. |x| + |-1|
   3. 20 – |x|
   4. |50| – |x|
7. Insert <, >, or =:
   1. -8   ___   3
   2. -9   ___   -12
   3. 0   ___   -4
   4. -15   ___   -7
   5. -6   ___   -6
8. Write each set of integers in order from least to greatest:
   1. -12, 4, -6, 0, -1
   2. 15, -8, -4, 7, -5
   3. 35, 60, -10, -5, 40
   4. -22, -30, -25, -16
9. Graph the integers -5, -2, 0, 3, and 6 on a number line. Which integer is farthest from zero?
10. Which number has the greater absolute value: -18 or 12? Explain your answer.

2. Applications

20. Supercooled insects: The table shows the temperatures at which different insects freeze.

20. Which insect has the lowest freezing point?
21. Order the freezing points from lowest to highest.

21. Planetary temperatures: A moon has a surface temperature of -392°F according to one measurement and -389°F according to another measurement. Which temperature is greater? Which measurement is warmer, and by how many degrees?
22. Weekly temperatures: The daily high temperatures were: Sunday -19°C, Monday -17°C, Tuesday -14°C, Wednesday -9°C, Thursday -13°C, Friday -18°C, and Saturday -21°C.
    1. Did the temperature increase or decrease from Sunday to Monday?
    2. Did the temperature increase or decrease from Friday to Saturday?
    3. Which day had the highest temperature? Which day had the lowest?
23. Bank balance: A bank account has a balance of -€18. A deposit of €25 is made.
    1. Represent the starting balance as an integer.
    2. What is the new balance?
    3. Is the new balance positive or negative?
24. Elevator: An elevator moves from floor -3 to floor 5.
    1. How many floors does the elevator travel?
    2. What is the distance between floor 5 and floor -4?
25. Diving: A diver is 18 metres below sea level, represented by -18. The diver rises 7 metres.
    1. What integer represents the diver’s new position?
    2. Is the new position above or below sea level?
26. Challenge: Find two different integers that have an absolute value of 13. Explain why they are opposites.

Answers