Understand Opposites on Number Line - 6th Grade Math

Evaluate: $-(-(-2))$ .
Answer: $-2$ This question tests evaluating multiple applications of the opposite operation, requiring careful tracking of sign changes. Starting with 2, we apply negation three times: -(-(-2)) means first find opposite of 2 (which is -2), then opposite of -2 (which is 2), then opposite of 2 (which is -2). For example, -(-(-5)) = -(-5) = -5 (three negations), while -(-5) = 5 (two negations); the pattern shows odd numbers of negations give the opposite, even numbers return to original. Evaluating step by step: -2 (first negation), then -(-2) = 2 (second negation), then -(2) = -2 (third negation), so -(-(-2)) = -2. Common errors include losing track of negations (getting 2), thinking three negatives always make positive (getting 2), or arithmetic mistakes (getting 0 or -4). The key insight: count the negations - odd number gives opposite of original (here 3 negations: 2 → -2), even number gives original back. This demonstrates that negation operations can be simplified by counting: odd number of negatives = negative result, even number = positive result.
Compare the distances from 0: $|7|$ and $|-7|$ . Which statement is true?
Answer: $|7| = |-7|$ because opposites are the same distance from 0. This question tests recognizing opposites on a number line, where opposite signs indicate opposite sides of 0 and they are equidistant, understanding that the opposite of the opposite returns to the original like −(−3)=3, and that zero is its own opposite. Opposites are numbers with opposite signs that are equidistant from 0 on the number line; for example, 5 is 5 units to the right of 0, and -5 is 5 units to the left, so they are opposites on opposite sides. The opposite of the opposite involves flipping the sign twice, which returns to the original number, such as −(−3) flips -3 to 3, and −(−10)=10, always following -(-a)=a. Zero is special because the opposite of 0 is 0, as it's the only number equal to its own opposite and is neither positive nor negative. Distance is key, as opposites have equal distance from zero, like |5|=5 and |-5|=5, both 5 units from 0. The statement |7| = |-7| is true because opposites are the same distance from 0, as in choice B. A common error is claiming |-7| = -7 or that negatives are farther, but absolute value gives the positive distance for both.
Which statement is always true for any integer $a$ (from $-20$ to $20$ ) on a number line?
Answer: $-(-a)=a$ . This question tests recognizing opposites on a number line, where opposite signs indicate opposite sides of 0 and they are equidistant, understanding that the opposite of the opposite returns to the original like −(−3)=3, and that zero is its own opposite. Opposites are numbers with opposite signs that are equidistant from 0 on the number line; for example, 5 is 5 units to the right of 0, and -5 is 5 units to the left, so they are opposites on opposite sides. The opposite of the opposite involves flipping the sign twice, which returns to the original number, such as −(−3) flips -3 to 3, and −(−10)=10, always following -(-a)=a. Zero is special because the opposite of 0 is 0, as it's the only number equal to its own opposite and is neither positive nor negative. Distance is key, as opposites have equal distance from zero, like |5|=5 and |-5|=5, both 5 units from 0. The statement -(-a)=a is always true for any integer a, as in choice B, because double opposites return to the original. A common error is thinking opposites of negatives stay negative or that 0 has no opposite, but these violate the properties of opposites.
On a number line, which pair of numbers are opposites?
Answer: $-5$ and $5$ This question tests recognizing opposites on a number line, where opposite signs indicate opposite sides of 0 and they are equidistant, understanding that the opposite of the opposite returns to the original like -(-3)=3, and that zero is its own opposite. Opposites are numbers with opposite signs that are equidistant from 0 on the number line; for example, 5 is 5 units to the right of 0, and -5 is 5 units to the left, so they are opposites on opposite sides. The opposite of the opposite involves flipping the sign twice, which returns to the original number, such as -(-3) flips -3 to 3, and -(-10)=10, always following -(-a)=a. Zero is special because the opposite of 0 is 0, as it's the only number equal to its own opposite and is neither positive nor negative. Distance is key, as opposites have equal distance from zero, like |5|=5 and |-5|=5, both 5 units from 0. The pair -5 and 5 are opposites because they have opposite signs and the same distance from 0, so choice A is correct. A common mistake is pairing numbers like -4 and 5, which have different distances from 0, or thinking opposites are the same number like -5 and -5.
On a number line, three points are marked: $X$ at $-4$ , $Y$ at the opposite of $X$ , and $Z$ at the opposite of the opposite of $X$ . If you walk from point $Y$ to point $Z$ , in which direction do you move and how far?
Answer: $8$ units to the left, because you move from $4$ to $-4$ When you see a question about "opposites" on a number line, you're working with the concept that opposite numbers are the same distance from zero but on different sides. The opposite of any number $a$ is $-a$ . Let's find each point's location step by step. Point $X$ is at $-4$ . The opposite of $X$ means the opposite of $-4$ , which is $4$ . So point $Y$ is at $4$ . Now for point $Z$ : it's at the opposite of the opposite of $X$ . The opposite of $X$ is $4$ , so the opposite of that is $-4$ . Therefore, point $Z$ is at $-4$ . Walking from point $Y$ at $4$ to point $Z$ at $-4$ means moving from $4$ to $-4$ . This is $8$ units to the left, confirming answer D. Let's examine why the other answers miss the mark. Choice A incorrectly assumes $Y$ and $Z$ are at the same location, but $Y$ is at $4$ while $Z$ is at $-4$ . Choice B places point $Y$ at $0$ instead of $4$ , misunderstanding what "opposite of $-4$ " means. Choice C has the right distance of $8$ units but gets the direction backwards—it describes moving from $-4$ to $4$ instead of from $4$ to $-4$ . Remember: when finding the opposite of a negative number, you get a positive number, and vice versa. Also, "opposite of the opposite" brings you back to where you started—so $X$ and $Z$ end up at the same location.
Evaluate the expression $-(-8)$ . (This means "the opposite of the opposite of 8.")
Answer: $8$ This question tests recognizing opposites on a number line, where opposite signs indicate opposite sides of 0 and they are equidistant, understanding that the opposite of the opposite returns to the original like $-(-3)=3$ , and that zero is its own opposite. Opposites are numbers with opposite signs that are equidistant from 0 on the number line; for example, 5 is 5 units to the right of 0, and -5 is 5 units to the left, so they are opposites on opposite sides. The opposite of the opposite involves flipping the sign twice, which returns to the original number, such as $-(-3)$ flips -3 to 3, and $-(-10)=10$ , always following $-(-a)=a$ . Zero is special because the opposite of 0 is 0, as it's the only number equal to its own opposite and is neither positive nor negative. Distance is key, as opposites have equal distance from zero, like $|5|=5$ and $|-5|=5$ , both 5 units from 0. Evaluating $-(-8)$ means the opposite of -8 is 8, so choice B is correct. A common mistake is thinking $-(-8) = -8$ or -16, but flipping the sign twice brings you back to the positive original.
Which statement is always true for any integer $a$ ?
Answer: $ -(-a) = a $ This question tests recognizing the fundamental property of double negation: the opposite of an opposite always returns to the original number. The expression $-(-a) = a$ is always true because taking the opposite twice brings you back to where you started: if a is any number, $-a$ is its opposite, and the opposite of $-a$ is a again. For example, $-(-5) = 5$ , $-(-(-3)) = -3$ , $-(-0) = 0$ ; this pattern holds for all integers, fractions, and real numbers. Option A correctly states $-(-a) = a$ , which is the double negation property that always holds true. Common errors include thinking $-(-a) = -a$ (not recognizing the sign flip), $-(a) = a$ (single negation doesn't return original), or $|a| = -|a|$ (absolute values are never negative). To verify: pick any value for a, like a = 7: then $-(-7) = 7$ ✓; or a = -4: then $-(-(-4)) = -(-4) = 4$ ✓. This property is fundamental to algebra and shows that the opposite operation is its own inverse: applying it twice cancels out.
Evaluate: $-(-7)$ . (This is the opposite of the opposite of $7$ .)
Answer: $7$ This question tests evaluating the opposite of an opposite, which always returns to the original number due to the property -(-a) = a. The expression -(-7) means "the opposite of the opposite of 7": first we find the opposite of 7 (which is -7), then find the opposite of that result (-7), which gives us 7. For example, -(-3) = 3 (opposite of -3 is 3), -(-10) = 10 (opposite of -10 is 10), and this pattern -(-a) = a works for any number. Evaluating -(-7): the inner negative makes 7 become -7, then the outer negative makes -7 become 7, so -(-7) = 7. Common errors include thinking double negatives stay negative (-(-7) = -7), adding the values (-(-7) = -14), or confusing with zero. The key insight is that applying the opposite operation twice returns you to where you started: if you flip a number's sign twice, you get back the original number. This fundamental property helps simplify expressions and understand that negation is its own inverse operation.
Sarah claims that $0$ is the opposite of itself because it's the only number that equals its own opposite. To verify this, she checks whether $-(0) = 0$ . Which reasoning best explains why Sarah's claim is correct?
Answer: Zero has no sign, so applying the opposite operation leaves it unchanged The opposite of $0$ is $-0$ , which equals $0$ because zero has no sign to change. This makes $0$ unique among all numbers as being its own opposite. Choice A incorrectly describes distance relationships. Choice C describes zero's location but not why it's its own opposite. Choice D creates a circular definition that doesn't explain the concept.
A classroom number line shows 0 in the center. Point $P$ is at $-3$ . Which statement correctly describes $-3$ and its opposite?
Answer: The opposite of $-3$ is $3$ , and they are the same distance from 0 on opposite sides. Every number has an opposite, which is the number that sits on the other side of $0$ but exactly the same distance away. Let's find the opposite of $-3$ . Point $P$ is at $-3$ , which is $3$ units to the left of $0$ . To find its opposite, hop the same $3$ units to the right of $0$ . That lands you on $3$ . So the opposite of $-3$ is $3$ , and they are the same distance from $0$ on opposite sides. Think of $0$ as a mirror. Whatever stands $3$ steps to one side has a reflection standing $3$ steps to the other side. Same distance, opposite direction! Try this at home: Draw a number line and pick any number, like $5$ or $-2$ . Count its steps from $0$ , then count the same steps the other way to find its opposite. You'll spot the pattern fast!
A student marks $3$ on a number line. Where should the student mark $-3$ so that $3$ and $-3$ are opposites?
Answer: Three units to the left of $0$ This question tests understanding how to position opposites on a number line: they must be on opposite sides of 0 and equidistant from it. If 3 is marked (3 units right of 0), then -3 must be 3 units left of 0 to be its opposite, creating symmetry around 0. For example, if 5 is marked 5 units right of 0, then -5 goes 5 units left of 0; if -7 is marked 7 units left, then 7 goes 7 units right; opposites mirror each other across 0. To place -3 as the opposite of 3, it must be three units to the left of 0, making them equidistant but on opposite sides. Common mistakes include placing -3 on the same point as 3 (same position, not opposite), three units right of 0 (same side as 3), or at 0 (confusing 0's role as center point with being everyone's opposite). The key visualization: 3 is three steps right from 0, so -3 is three steps left from 0, creating perfect symmetry. This demonstrates that opposites are reflections across 0 on the number line, maintaining equal distance but opposite direction.
On a number line, point $P$ is located at $-8$ and point $Q$ is located at the opposite of the opposite of $-8$ . What is the distance between points $P$ and $Q$ ?
Answer: $0$ units because both points are at the same location Point $P$ is at $-8$ . The opposite of $-8$ is $8$ , and the opposite of $8$ is $-8$ . So point $Q$ is also at $-8$ . Since both points are at the same location, the distance is $0$ . Choice B incorrectly assumes $Q$ is at the opposite of $-8$ . Choice C incorrectly assumes $Q$ is at $8$ . Choice D uses an incorrect distance formula.
Compare the distances from $0$ : $|5|$ and $|-5|$ . Which statement is true?
Answer: $|5| = |-5|$ This question tests understanding that opposites have equal absolute values, meaning they're the same distance from 0 on the number line. The absolute value |a| represents the distance from 0, regardless of direction: |5| = 5 (distance of 5 from 0) and |-5| = 5 (also distance of 5 from 0), so opposites always have equal absolute values. For example, |7| = |-7| = 7 (both are 7 units from 0), |-3| = |3| = 3 (both are 3 units from 0), demonstrating that opposite numbers have the same distance from 0. The correct answer is |5| = |-5| because both equal 5, representing the same distance from 0 on the number line. Common errors include thinking negative numbers have greater distance (|5| < |-5|), positive numbers have greater distance (|5| > |-5|), or that |-5| equals -5 (confusing absolute value with the number itself). Key concept: absolute value removes the sign and gives distance, so |a| = |-a| for any number a. This property confirms that opposites are equidistant from 0, just on different sides of the number line.
Which number is the opposite of 0 on a number line?
Answer: $0$ This question tests recognizing opposites on a number line, where opposite signs indicate opposite sides of 0 and they are equidistant, understanding that the opposite of the opposite returns to the original like −(−3)=3, and that zero is its own opposite. Opposites are numbers with opposite signs that are equidistant from 0 on the number line; for example, 5 is 5 units to the right of 0, and -5 is 5 units to the left, so they are opposites on opposite sides. The opposite of the opposite involves flipping the sign twice, which returns to the original number, such as −(−3) flips -3 to 3, and −(−10)=10, always following -(-a)=a. Zero is special because the opposite of 0 is 0, as it's the only number equal to its own opposite and is neither positive nor negative. Distance is key, as opposites have equal distance from zero, like |5|=5 and |-5|=5, both 5 units from 0. The opposite of 0 is 0, as in choice C, because it's at the center and flipping its sign doesn't change it. A common error is thinking the opposite of 0 is undefined or a different number like 1, but zero is uniquely its own opposite.
On a number line, the opposite of a number is the same distance from $0$ but on the other side. What is the opposite of $-12$ ?
Answer: $12$ This question tests understanding that the opposite of a number is found by changing its sign while maintaining the same distance from 0. Opposites are numbers equidistant from 0 but on opposite sides of the number line: if -12 is 12 units left of 0, its opposite is 12 units right of 0, which is 12. For example, the opposite of -5 is 5 (changes from negative to positive), the opposite of 8 is -8 (changes from positive to negative), and the opposite of 0 is 0 (special case). The opposite of -12 is 12 because we flip the sign from negative to positive while keeping the distance of 12 units from 0. Common mistakes include keeping the same sign (thinking opposite of -12 is -12), changing the magnitude (like -11), or thinking 0 is involved when it's not. To find opposites: (1) identify the sign (negative for -12), (2) flip to opposite sign (positive), (3) keep same distance from 0 (12 units), resulting in 12. This reflects the symmetry of the number line where every negative number has a positive counterpart the same distance from 0.
Use the number line shown. Points $M$ , $N$ , and $P$ represent three consecutive applications of the opposite operation, starting with $M$ at $2$ . Which statement correctly describes the relationship between these points?
Answer: Points $M$ and $P$ are at the same location, while $N$ is equidistant from both Starting at $M = 2$ : First opposite gives $N = -2$ . Second opposite gives $P = 2$ . So $M$ and $P$ are both at position $2$ , while $N$ is at $-2$ , equidistant from both $M$ and $P$ . Choice B incorrectly calculates $P$ 's position. Choice C incorrectly assumes equal spacing. Choice D incorrectly places $P$ at $0$ .
On a number line, a value of $+6$ is 6 units to the right of 0. Which number is the opposite of $+6$ ?
Answer: $-6$ This question tests recognizing opposites on a number line, where opposite signs indicate opposite sides of 0 and they are equidistant, understanding that the opposite of the opposite returns to the original like -(-3)=3, and that zero is its own opposite. Opposites are numbers with opposite signs that are equidistant from 0 on the number line; for example, 5 is 5 units to the right of 0, and -5 is 5 units to the left, so they are opposites on opposite sides. The opposite of the opposite involves flipping the sign twice, which returns to the original number, such as -(-3) flips -3 to 3, and -(-10)=10, always following -(-a)=a. Zero is special because the opposite of 0 is 0, as it's the only number equal to its own opposite and is neither positive nor negative. Distance is key, as opposites have equal distance from zero, like |5|=5 and |-5|=5, both 5 units from 0. For +6, which is 6 units to the right of 0, its opposite is -6, 6 units to the left of 0, so choice B is correct. A common mistake is thinking the opposite is twice the distance, like -12, but opposites are always the same distance on the opposite side.
A pattern on a number line shows: $5 \to -5 \to 5 \to -5 \to ?$ If this pattern continues by repeatedly applying the opposite operation, what number will appear in the 10th position of this sequence?
Answer: $-5$ , because the 10th position corresponds to an even-numbered step When you encounter a repeating pattern question, you need to identify the rule and determine where in the cycle a specific position falls. This sequence shows numbers alternating between positive and negative by applying the "opposite" operation (multiplying by -1). Let's trace the pattern: Position 1 is $5$ , position 2 is $-5$ , position 3 is $5$ , position 4 is $-5$ , and so on. The pattern repeats every 2 positions. To find the 10th position, you need to determine whether 10 corresponds to an odd position (which gives $5$ ) or an even position (which gives $-5$ ). Since 10 is even, the 10th position will be $-5$ . Answer A is wrong because you don't multiply by position numbers—you're applying the opposite operation repeatedly. The sequence doesn't grow larger; it just alternates. Answer B incorrectly identifies what happens at even positions. While it's true the pattern repeats every two steps, position 10 doesn't return to the starting value of $5$ —it lands on $-5$ . Answer D misunderstands what "opposite operation" means. Taking the opposite of a number (multiplying by -1) doesn't make values approach zero; it just flips the sign. For repeating pattern problems, always identify the cycle length first, then use division to find where your target position falls within that cycle. Here, odd positions give the original number, even positions give its opposite.
A number line is shown from $-10$ to $10$ with 0 in the center. Point $A$ is at $5$ . Where is the opposite of point $A$ located?
Answer: At $-5$ , 5 units to the left of 0 This question tests recognizing opposites on a number line, where opposite signs indicate opposite sides of 0 and they are equidistant, understanding that the opposite of the opposite returns to the original like $ -(-3)=3 $, and that zero is its own opposite. Opposites are numbers with opposite signs that are equidistant from 0 on the number line; for example, 5 is 5 units to the right of 0, and -5 is 5 units to the left, so they are opposites on opposite sides. The opposite of the opposite involves flipping the sign twice, which returns to the original number, such as $ -(-3) $ flips -3 to 3, and $ -(-10)=10 $, always following $ -(-a)=a $. Zero is special because the opposite of 0 is 0, as it's the only number equal to its own opposite and is neither positive nor negative. Distance is key, as opposites have equal distance from zero, like $ |5|=5 $ and $ |-5|=5 $, both 5 units from 0. The opposite of 5 is at -5, 5 units to the left of 0, so choice A is correct. A common mistake is confusing directions, like saying -5 is to the right, but negatives are always left of 0 on a standard number line.
A student writes the expression $-(-(-(-6)))$ and claims it equals $6$ because 'four negatives make a positive.' Which error in reasoning should be corrected?
Answer: The student should work from inside out: $-6 \to 6 \to -6 \to 6$ , confirming the result is $6$ When you encounter nested negative signs like $-(-(-(-6)))$ , you need to evaluate them step by step from the inside out, just like with parentheses in order of operations. Let's work through this expression systematically: Start with the innermost value, $-6$ . Then apply each negative sign one at a time moving outward. The first negative sign gives us $-(-6) = 6$ . The next negative sign gives us $-(6) = -6$ . Finally, the outermost negative sign gives us $-(-6) = 6$ . So the correct sequence is $-6 \to 6 \to -6 \to 6$ , and the final answer is indeed $6$ . Choice C correctly describes this inside-out process and confirms the result is $6$ , making it the right answer. Choice A is wrong because working "outside in" isn't how we evaluate mathematical expressions—we always work from the innermost operations outward. Choice B incorrectly suggests there are only three negative signs when there are actually four, and even if there were three, the student's reasoning about "counting negatives" is flawed. Choice D is incorrect because nested negatives do follow consistent mathematical rules—you just need to apply them systematically rather than trying to count them all at once. The key insight is that while the student got the right answer, their reasoning of "four negatives make a positive" oversimplifies the process. Remember: always evaluate nested operations from the inside out, applying one operation at a time.

Start this quiz