Understand Positive and Negative Numbers
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6th Grade Math › Understand Positive and Negative Numbers
A robot starts at position $0$ on a number line. Moving forward is positive and moving backward is negative. Which signed number represents moving 9 steps backward from the start?
$-9$
$9$
$+9$
$0$
Explanation
This question tests understanding that positive and negative numbers describe opposite directions or values, such as forward or backward movement on a number line, representing quantities in contexts like robotics, and explaining zero's meaning as the starting point. Positive numbers represent forward steps, like +9 meaning 9 steps ahead, while negative represent backward, like -9 meaning 9 steps back; in temperature, +15°C is above and -8°C below freezing; in money, +$50 is credit and -$30 is debt, with zero as reference like 0 for start, 0°C for freezing, or $0 for balance. For example, +9 is 9 steps forward from start, -9 is 9 steps backward, and 0 is remaining at the starting position. In this case, -9 correctly represents moving 9 steps backward, as the negative sign shows the opposite direction. A common error is omitting the sign or thinking backward is positive, or seeing zero as nothing when it's the specific start, or reversing comparisons like -9 > -5 when -5 > -9. To represent movements, identify forward (positive) or backward (negative), assign the sign like -9 for backward, without units here; zero means the starting point. Understanding signed numbers extends the line to negatives, but mistakes include wrong sign or comparison reversal.
A game uses a score change system: gaining points is positive and losing points is negative. A player’s score change is $+9$. Which situation matches $+9$?
The player gained 9 points.
The player’s score is 0 points.
The player lost 9 points.
The player has 9 points total and cannot change.
Explanation
This question tests understanding that positive and negative numbers describe opposite values, such as gaining or losing points in a game, representing quantities in scoring contexts, and explaining zero's meaning as no change in score. Positive numbers represent gains, like +9 meaning 9 points added, negative losses, like -9 meaning 9 points subtracted; in temperature, +15°C above and -8°C below; in elevation, +450 m above and -230 m below, with zero as no change, freezing, or sea level. For example, +9 is gain of 9, -9 is loss of 9, 0 is no score change. The correct situation for +9 is the player gained 9 points, as positive indicates addition. A common error is thinking positive means loss or total fixed at 9, or confusing with zero score. To represent score changes, identify gain (positive) or loss (negative), assign sign like +9 for gain; zero means balanced no change. Comparing negatives, -5 > -10 as smaller loss; signed numbers extend for both, avoiding sign reversal mistakes.
A hiker’s elevation is measured relative to sea level, where sea level is $0$ meters. The hiker is at $+18$ meters. What does $+18$ mean?
The hiker is 18 meters from the center of Earth
Sea level is 18 meters high
18 meters below sea level
18 meters above sea level
Explanation
This question tests understanding that positive and negative numbers describe opposite directions or values, such as above or below sea level in hiking, representing quantities in contexts like geography, and explaining zero's meaning as sea level. Positive numbers represent above sea level, like +18 m higher, negative below, like -18 m lower; in temperature, +15°C above and -8°C below freezing; in money, +$50 credit and -$30 debt, with zero as 0 m sea level, 0°C freezing, or $0 balance. For example, +450 m is 450 meters above sea level, -230 m is 230 below, and 0 m is at sea level. Here, +18 meters correctly means 18 meters above sea level, as positive indicates above the reference. A common error is thinking positive means below or confusing with distance from Earth's center, or seeing zero as arbitrary when it's sea level specifically. To represent elevations, identify above (positive) or below (negative), assign sign like +18 for above, include units; zero means sea level neutral. Understanding signed numbers models vertical opposites, but mistakes include sign confusion or wrong zero interpretation.
In football, gaining yards is represented with a positive number and losing yards is represented with a negative number. A play is recorded as $-6$ yards. What happened on the play?
The team gained 6 yards
The ball stayed exactly at the line of scrimmage
The team lost 6 yards
The team gained 6 yards and then lost 6 yards
Explanation
This question tests understanding that positive and negative numbers describe opposite directions or values, such as gaining or losing yards in football, representing quantities in contexts like sports, and explaining zero's meaning as no change. Positive numbers represent gains, like +6 yards forward, while negative represent losses, like -6 yards backward; in elevation, +450 m is above and -230 m below sea level; in money, +$50 is credit and -$30 is debt, with zero as reference like 0 yards for no movement, 0 m for sea level, or $0 for balance. For example, +6 yards is a gain of 6, -6 yards is a loss of 6, and 0 yards means the ball stayed put. Here, -6 yards correctly means the team lost 6 yards, as negative indicates backward movement. A common error is thinking negative means gain, or zero as something else like total plays, or confusing with other contexts like temperature. To represent yardage, identify gain (positive) or loss (negative), assign the sign like -6 for loss, and include units; zero means no net change from the line. Understanding signed numbers helps model opposites, but mistakes include sign reversal or ignoring context.
Which temperature is warmer: $-5^\circ\text{C}$ or $-10^\circ\text{C}$?
$-10^\circ\text{C}$ because 10 is greater than 5.
They are the same temperature because both are negative.
$-5^\circ\text{C}$ because it is closer to $0^\circ\text{C}$.
$-10^\circ\text{C}$ because it is closer to 0.
Explanation
This question tests understanding that positive and negative numbers describe opposite values, such as warmer or colder temperatures relative to zero, representing quantities in weather contexts, and explaining zero's meaning as the freezing point. Positive numbers are above zero, like +15°C warmer, negative below, like -8°C colder; in elevation, +450 m above and -230 m below sea level; in money, +$50 gain and -$30 loss, with zero as 0°C freezing, 0 m sea level, $0 balance. For example, +15°C is 15 above freezing, -8°C is 8 below, 0°C is freezing. The warmer temperature is -5°C, as it's closer to zero than -10°C. A common error is thinking the larger magnitude negative is warmer, like -10 > -5, or assuming both negatives are the same, or that closer to zero is colder. To compare temperatures, note that for negatives, less negative (closer to zero) is greater, like -5 > -10; zero is the specific reference. Signed numbers allow full range, avoiding mistakes like reversed comparisons or ignoring signs.
A student tracks elevator floors using signed numbers: floors above the lobby are positive, floors below the lobby are negative, and the lobby is $0$. Which pair of numbers represents opposite locations relative to the lobby?
$0$ and $+5$
$+6$ and $-6$
$+7$ and $+3$
$-4$ and $-9$
Explanation
This question tests understanding that positive and negative numbers describe opposite directions or values, such as above or below a reference point, gain or loss, or credit or debit, while representing quantities in various contexts and explaining zero's meaning as the reference. Positive numbers represent values above or greater than zero, like +15°C for 15 degrees above freezing, while negative numbers represent values below or less than zero, like -8°C for 8 degrees below freezing; in elevation, +450 m is above sea level and -230 m is below; in money, +$50 is a credit and -$30 is a debt, with zero as the reference point—0°C is the freezing point of water, 0 m is sea level, and $0 is a balanced account. For example, in elevator floors, +6 represents 6 floors above the lobby (positive means above), -6 represents 6 floors below the lobby (negative means below), and 0 is the lobby itself, the reference floor. The correct pair for opposite locations is +6 and -6, which are equally distant but in opposite directions from the lobby at 0. A common error is thinking pairs like +7 and +3 or -4 and -9 are opposites when they are both on the same side, or confusing 0 and +5 as opposites when 0 is the reference. To represent opposites, identify equal distances in opposite directions from zero, like +n and -n. Understanding signed numbers shows opposites as additive inverses; mistakes include choosing pairs on the same side or misunderstanding zero.
A science lab tracks electric charge using signed numbers: positive numbers represent positive charge, negative numbers represent negative charge, and $0$ represents neutral. A balloon has a charge of $-5$ (in some units). Which statement best describes the balloon?
It has a negative charge of 5 units
It has a positive charge of 5 units
It is neutral (no net charge)
It has no charge because negative means none
Explanation
This question tests understanding that positive and negative numbers describe opposite directions or values, such as positive or negative electric charge, representing quantities in science contexts, and explaining zero's meaning as neutral. Positive numbers represent positive charge, like +5 units, negative represent negative charge, like -5 units; in elevation, +450 m above and -230 m below; in money, +$50 credit and -$30 debt, with zero as neutral charge, 0 m sea level, or $0 balance. For example, +5 is positive charge, -5 is negative charge, and 0 is no net charge. In this case, -5 correctly means a negative charge of 5 units, as negative indicates the opposite type. A common error is thinking negative means no charge or positive, or zero as something else, or confusing with other signed contexts. To represent charge, identify positive or negative type, assign sign like -5 for negative, include units; zero means neutral balance. Understanding signed numbers applies to science opposites, but mistakes include sign reversal or misinterpreting negative as none.
On a winter morning, the temperature is $-8^\circ\text{C}$. Which statement correctly explains what $-8^\circ\text{C}$ means?
8 degrees below $0^\circ\text{C}$ (freezing point)
The temperature is 0, so there is no temperature
8 degrees above $0^\circ\text{C}$
8 degrees warmer than $+8^\circ\text{C}$
Explanation
This question tests understanding that positive and negative numbers describe opposite directions or values, such as above or below a reference point, gain or loss, or credit or debit, while representing quantities in various contexts and explaining zero's meaning as the reference. Positive numbers represent values above or greater than zero, like +15°C for 15 degrees above freezing, while negative numbers represent values below or less than zero, like -8°C for 8 degrees below freezing; in elevation, +450 m is above sea level and -230 m is below; in money, +$50 is a credit and -$30 is a debt, with zero as the reference point—0°C is the freezing point of water, 0 m is sea level, and $0 is a balanced account. For example, in temperature, +15°C represents 15 degrees above the freezing point, -8°C represents 8 degrees below freezing, and 0°C is exactly at the freezing point where water turns to ice. The correct statement is that -8°C means 8 degrees below 0°C, the freezing point, where 0°C is the reference for above and below temperatures. A common error is thinking -8°C means above zero or warmer than +8°C, or confusing zero as no temperature at all, when actually negative means below the reference and zero is a specific point. To represent temperatures, identify the direction—above freezing is positive, below is negative—and assign the sign accordingly, including units like °C. Zero in temperature means the freezing point, a specific reference, and when comparing negatives, -5°C is warmer than -10°C because it's closer to zero; understanding signed numbers extends the number line to include negatives, allowing representation of both directions from the reference, but mistakes occur if signs are assigned wrong or comparisons are reversed.
A football team starts a play at the line of scrimmage, which is labeled $0$ yards. A gain is positive and a loss is negative. The play result is $-6$ yards. What happened on the play?
The team gained 6 yards
The team gained 0 yards because negative means no yards
The team returned to the locker room (0 yards means stop playing)
The team lost 6 yards
Explanation
This question tests understanding that positive and negative numbers describe opposite directions or values, such as above or below a reference point, gain or loss, or credit or debit, while representing quantities in various contexts and explaining zero's meaning as the reference. Positive numbers represent values above or greater than zero, like +15°C for 15 degrees above freezing, while negative numbers represent values below or less than zero, like -8°C for 8 degrees below freezing; in elevation, +450 m is above sea level and -230 m is below; in money, +$50 is a credit and -$30 is a debt, with zero as the reference point—0°C is the freezing point of water, 0 m is sea level, and $0 is a balanced account. For example, in a game like football, +50 yards represents a gain (positive means forward), -30 yards represents a loss (negative means backward), and 0 yards is the line of scrimmage, no gain or loss. The correct meaning of -6 yards is that the team lost 6 yards, where 0 yards is the line of scrimmage as the reference. A common error is thinking -6 means a gain or that zero means stopping play, or confusing negative as no yards when it represents loss. To represent yards, identify the direction—gain is positive, loss is negative—and assign the sign. Zero means the starting line, and understanding negatives represents backward movement; mistakes include sign reversal or misinterpreting zero.
A diver is $25$ meters below sea level. If sea level is represented by $0$ meters, which signed number represents the diver’s elevation?
$0$
$25$
$+25$
$-25$
Explanation
This question tests understanding that positive and negative numbers describe opposite directions or values, such as elevations above or below sea level, representing quantities in contexts, and explaining zero's meaning as sea level itself. Positive numbers represent elevations above sea level, like +450 m meaning 450 meters above, while negative numbers represent below, like -230 m meaning 230 meters below; in temperature, +15°C is above freezing and -8°C is below; in money, +$50 is a credit and -$30 is a debt, with zero as the reference like 0 m for sea level, 0°C for freezing, or $0 for balance. For example, an elevation of +450 m represents 450 meters above sea level, -230 m represents 230 meters below sea level, and 0 m is exactly at sea level, the ocean's surface. The correct signed number for 25 meters below sea level is -25, as the negative sign indicates below the reference point of zero. A common error is using a positive sign for below, like +25, or omitting the sign altogether, or confusing zero with no elevation instead of the specific sea level reference. To represent elevations, identify if it's above (positive) or below (negative) sea level, assign the sign like -25 for 25 below, and include units m; zero means the specific sea level, neither above nor below. Understanding signed numbers allows representing both directions from the reference, and mistakes include wrong sign assignment or thinking negatives are greater when farther from zero, like assuming -10 is higher than -5 when actually -5 > -10.