💡 6th Grade Math: Integers and Absolute Value Practice Questions
1
Solved Example
Easy Level
Question 1: 💡 Identify all the integers from the following list of numbers: \( -7, \frac{1}{2}, 0, 3.5, 12, -100, \sqrt{25} \)
Solution & Explanation
Integers are whole numbers and their opposites (negative whole numbers). They do not include fractions or decimals.
\( -7 \) is a negative whole number, so it is an integer. ✅
\( \frac{1}{2} \) is a fraction, so it is NOT an integer. ❌
\( 0 \) is a whole number, so it is an integer. ✅
\( 3.5 \) is a decimal, so it is NOT an integer. ❌
\( 12 \) is a positive whole number, so it is an integer. ✅
\( -100 \) is a negative whole number, so it is an integer. ✅
\( \sqrt{25} \) simplifies to \( 5 \), which is a positive whole number, so it is an integer. ✅
Answer: The integers in the list are \( -7, 0, 12, -100, 5 \) (from \( \sqrt{25} \)).
2
Solved Example
Easy Level
Question 2: 👉 Write an integer to represent each situation: a) A submarine is 500 feet below sea level. b) You deposited 75 into your bank account. c) The temperature is 10 degrees above zero.
Solution & Explanation
When representing real-world situations with integers, we use positive numbers for increases or above zero, and negative numbers for decreases or below zero.
a) "500 feet below sea level" indicates a decrease or a position lower than zero. Answer: \( -500 \)
b) "Deposited 75" means adding money, which is an increase. Answer: \( 75 \)
c) "10 degrees above zero" indicates a temperature higher than zero. Answer: \( 10 \)
3
Solved Example
Medium Level
Question 3: 📌 Draw a number line and plot the following integers: \( -4, 0, 3, -1 \). Then, order them from least to greatest.
Solution & Explanation
To plot integers on a number line, locate each number's position relative to zero. Numbers to the left are smaller, and numbers to the right are larger.
First, let's visualize the number line. Zero is in the middle. Positive numbers are to the right, and negative numbers are to the left.
Imagine a line with marks for \( -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 \).
Plot \( -4 \): Four units to the left of zero.
Plot \( 0 \): At the origin.
Plot \( 3 \): Three units to the right of zero.
Plot \( -1 \): One unit to the left of zero.
Looking at their positions from left to right on the number line gives us the order from least to greatest.
Answer: The integers plotted are \( -4, -1, 0, 3 \). Ordered from least to greatest: \( -4, -1, 0, 3 \).
4
Solved Example
Medium Level
Question 4: What is the opposite of each integer? a) \( 8 \) b) \( -15 \) c) \( 0 \)
Solution & Explanation
The opposite of an integer is the number that is the same distance from zero on a number line but in the opposite direction. It's also called the additive inverse.
a) For \( 8 \): \( 8 \) is \( 8 \) units to the right of zero. Its opposite will be \( 8 \) units to the left of zero. Answer: The opposite of \( 8 \) is \( -8 \).
b) For \( -15 \): \( -15 \) is \( 15 \) units to the left of zero. Its opposite will be \( 15 \) units to the right of zero. Answer: The opposite of \( -15 \) is \( 15 \).
c) For \( 0 \): \( 0 \) is at zero. It is not in any direction. Answer: The opposite of \( 0 \) is \( 0 \).
5
Solved Example
Medium Level
Question 5: Calculate the absolute value of each number: a) \( |-9| \) b) \( |20| \) c) \( |-0| \)
Solution & Explanation
The absolute value of a number is its distance from zero on a number line. Distance is always a positive value or zero.
a) \( |-9| \): The number \( -9 \) is \( 9 \) units away from zero on the number line. Answer: \( |-9| = 9 \)
b) \( |20| \): The number \( 20 \) is \( 20 \) units away from zero on the number line. Answer: \( |20| = 20 \)
c) \( |-0| \): The number \( 0 \) is \( 0 \) units away from zero on the number line. Answer: \( |-0| = 0 \) (Note: \( |-0| \) is the same as \( |0| \))
6
Solved Example
Medium Level
Question 6: Compare the following pairs of numbers using \( <, >, \text{ or } = \). a) \( -6 \text{ ___ } 2 \) b) \( |-5| \text{ ___ } |5| \) c) \( -10 \text{ ___ } -3 \)
Solution & Explanation
When comparing integers, remember that numbers to the left on a number line are smaller, and numbers to the right are larger. For absolute value, calculate the distance from zero first.
a) Compare \( -6 \) and \( 2 \): On a number line, \( -6 \) is to the left of \( 2 \). Therefore, \( -6 \) is less than \( 2 \). Answer: \( -6 < 2 \)
b) Compare \( |-5| \) and \( |5| \): First, find the absolute values: \( |-5| = 5 \) and \( |5| = 5 \). Now compare \( 5 \) and \( 5 \). They are equal. Answer: \( |-5| = |5| \)
c) Compare \( -10 \) and \( -3 \): On a number line, \( -10 \) is further to the left than \( -3 \). Therefore, \( -10 \) is less than \( -3 \). Answer: \( -10 < -3 \)
7
Solved Example
Real World Example
Question 7: A diver is exploring a coral reef at an elevation of \( -25 \) feet relative to sea level. A bird is flying directly above the diver at an elevation of \( 40 \) feet. a) Represent the diver's and the bird's elevations as integers. b) What is the absolute value of the diver's elevation? What does it mean in this situation?
Solution & Explanation
This problem uses integers to describe positions relative to sea level and applies the concept of absolute value to distance.
a) Representing elevations as integers:
For the diver: "an elevation of \( -25 \) feet relative to sea level" is already an integer. Diver's elevation: \( -25 \) feet.
For the bird: "an elevation of \( 40 \) feet" means \( 40 \) feet above sea level. Bird's elevation: \( 40 \) feet.
b) Absolute value of the diver's elevation:
The diver's elevation is \( -25 \) feet.
The absolute value is \( |-25| \).
\( |-25| = 25 \).
In this situation, the absolute value of \( 25 \) means the distance the diver is from sea level, which is \( 25 \) feet. It tells us how far down the diver is, regardless of direction.
8
Solved Example
Medium Level
Question 8: An integer has an absolute value of \( 7 \). What are all the possible values for this integer? Explain your reasoning.
Solution & Explanation
The absolute value of a number is its distance from zero on the number line.
If a number's absolute value is \( 7 \), it means the number is \( 7 \) units away from zero.
On a number line, there are two numbers that are \( 7 \) units away from zero:
One is \( 7 \) units to the right of zero, which is \( 7 \).
The other is \( 7 \) units to the left of zero, which is \( -7 \).
Both \( |7| = 7 \) and \( |-7| = 7 \).
Answer: The possible values for the integer are \( 7 \) and \( -7 \). Reasoning: Both \( 7 \) and \( -7 \) are exactly \( 7 \) units away from zero on the number line.
6th Grade Math: Integers and Absolute Value Practice Questions
Example 1:
Question 1: 💡 Identify all the integers from the following list of numbers: \( -7, \frac{1}{2}, 0, 3.5, 12, -100, \sqrt{25} \)
Solution:
Integers are whole numbers and their opposites (negative whole numbers). They do not include fractions or decimals.
\( -7 \) is a negative whole number, so it is an integer. ✅
\( \frac{1}{2} \) is a fraction, so it is NOT an integer. ❌
\( 0 \) is a whole number, so it is an integer. ✅
\( 3.5 \) is a decimal, so it is NOT an integer. ❌
\( 12 \) is a positive whole number, so it is an integer. ✅
\( -100 \) is a negative whole number, so it is an integer. ✅
\( \sqrt{25} \) simplifies to \( 5 \), which is a positive whole number, so it is an integer. ✅
Answer: The integers in the list are \( -7, 0, 12, -100, 5 \) (from \( \sqrt{25} \)).
Example 2:
Question 2: 👉 Write an integer to represent each situation: a) A submarine is 500 feet below sea level. b) You deposited 75 into your bank account. c) The temperature is 10 degrees above zero.
Solution:
When representing real-world situations with integers, we use positive numbers for increases or above zero, and negative numbers for decreases or below zero.
a) "500 feet below sea level" indicates a decrease or a position lower than zero. Answer: \( -500 \)
b) "Deposited 75" means adding money, which is an increase. Answer: \( 75 \)
c) "10 degrees above zero" indicates a temperature higher than zero. Answer: \( 10 \)
Example 3:
Question 3: 📌 Draw a number line and plot the following integers: \( -4, 0, 3, -1 \). Then, order them from least to greatest.
Solution:
To plot integers on a number line, locate each number's position relative to zero. Numbers to the left are smaller, and numbers to the right are larger.
First, let's visualize the number line. Zero is in the middle. Positive numbers are to the right, and negative numbers are to the left.
Imagine a line with marks for \( -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 \).
Plot \( -4 \): Four units to the left of zero.
Plot \( 0 \): At the origin.
Plot \( 3 \): Three units to the right of zero.
Plot \( -1 \): One unit to the left of zero.
Looking at their positions from left to right on the number line gives us the order from least to greatest.
Answer: The integers plotted are \( -4, -1, 0, 3 \). Ordered from least to greatest: \( -4, -1, 0, 3 \).
Example 4:
Question 4: What is the opposite of each integer? a) \( 8 \) b) \( -15 \) c) \( 0 \)
Solution:
The opposite of an integer is the number that is the same distance from zero on a number line but in the opposite direction. It's also called the additive inverse.
a) For \( 8 \): \( 8 \) is \( 8 \) units to the right of zero. Its opposite will be \( 8 \) units to the left of zero. Answer: The opposite of \( 8 \) is \( -8 \).
b) For \( -15 \): \( -15 \) is \( 15 \) units to the left of zero. Its opposite will be \( 15 \) units to the right of zero. Answer: The opposite of \( -15 \) is \( 15 \).
c) For \( 0 \): \( 0 \) is at zero. It is not in any direction. Answer: The opposite of \( 0 \) is \( 0 \).
Example 5:
Question 5: Calculate the absolute value of each number: a) \( |-9| \) b) \( |20| \) c) \( |-0| \)
Solution:
The absolute value of a number is its distance from zero on a number line. Distance is always a positive value or zero.
a) \( |-9| \): The number \( -9 \) is \( 9 \) units away from zero on the number line. Answer: \( |-9| = 9 \)
b) \( |20| \): The number \( 20 \) is \( 20 \) units away from zero on the number line. Answer: \( |20| = 20 \)
c) \( |-0| \): The number \( 0 \) is \( 0 \) units away from zero on the number line. Answer: \( |-0| = 0 \) (Note: \( |-0| \) is the same as \( |0| \))
Example 6:
Question 6: Compare the following pairs of numbers using \( <, >, \text{ or } = \). a) \( -6 \text{ ___ } 2 \) b) \( |-5| \text{ ___ } |5| \) c) \( -10 \text{ ___ } -3 \)
Solution:
When comparing integers, remember that numbers to the left on a number line are smaller, and numbers to the right are larger. For absolute value, calculate the distance from zero first.
a) Compare \( -6 \) and \( 2 \): On a number line, \( -6 \) is to the left of \( 2 \). Therefore, \( -6 \) is less than \( 2 \). Answer: \( -6 < 2 \)
b) Compare \( |-5| \) and \( |5| \): First, find the absolute values: \( |-5| = 5 \) and \( |5| = 5 \). Now compare \( 5 \) and \( 5 \). They are equal. Answer: \( |-5| = |5| \)
c) Compare \( -10 \) and \( -3 \): On a number line, \( -10 \) is further to the left than \( -3 \). Therefore, \( -10 \) is less than \( -3 \). Answer: \( -10 < -3 \)
Example 7:
Question 7: A diver is exploring a coral reef at an elevation of \( -25 \) feet relative to sea level. A bird is flying directly above the diver at an elevation of \( 40 \) feet. a) Represent the diver's and the bird's elevations as integers. b) What is the absolute value of the diver's elevation? What does it mean in this situation?
Solution:
This problem uses integers to describe positions relative to sea level and applies the concept of absolute value to distance.
a) Representing elevations as integers:
For the diver: "an elevation of \( -25 \) feet relative to sea level" is already an integer. Diver's elevation: \( -25 \) feet.
For the bird: "an elevation of \( 40 \) feet" means \( 40 \) feet above sea level. Bird's elevation: \( 40 \) feet.
b) Absolute value of the diver's elevation:
The diver's elevation is \( -25 \) feet.
The absolute value is \( |-25| \).
\( |-25| = 25 \).
In this situation, the absolute value of \( 25 \) means the distance the diver is from sea level, which is \( 25 \) feet. It tells us how far down the diver is, regardless of direction.
Example 8:
Question 8: An integer has an absolute value of \( 7 \). What are all the possible values for this integer? Explain your reasoning.
Solution:
The absolute value of a number is its distance from zero on the number line.
If a number's absolute value is \( 7 \), it means the number is \( 7 \) units away from zero.
On a number line, there are two numbers that are \( 7 \) units away from zero:
One is \( 7 \) units to the right of zero, which is \( 7 \).
The other is \( 7 \) units to the left of zero, which is \( -7 \).
Both \( |7| = 7 \) and \( |-7| = 7 \).
Answer: The possible values for the integer are \( 7 \) and \( -7 \). Reasoning: Both \( 7 \) and \( -7 \) are exactly \( 7 \) units away from zero on the number line.