Position Rational Numbers on Diagrams
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6th Grade Math › Position Rational Numbers on Diagrams
In the coordinate plane, point Q is located at $$\left(-1\frac{1}{3}, 2\frac{2}{3}\right)$$ and point R is located at $$\left(\frac{5}{6}, -\frac{1}{6}\right)$$. If you plot point S such that it has the same x-coordinate as point Q and the same y-coordinate as point R, in which quadrant will point S be located?
Quadrant IV, where x is positive and y is negative
Quadrant II, where x is negative and y is positive
Quadrant III, where both coordinates are negative numbers
Quadrant I, where both coordinates are positive numbers
Explanation
Point S has the same x-coordinate as Q, which is $$-1\frac{1}{3}$$ (negative), and the same y-coordinate as R, which is $$-\frac{1}{6}$$ (negative). Since both coordinates are negative, point S is in Quadrant III. Choice A would require both coordinates positive. Choice B would require negative x and positive y. Choice D would require positive x and negative y.
A vertical number line has markings every $$\frac{1}{4}$$ unit. If the number $$-1\frac{1}{2}$$ is located 3 markings below the number $$-\frac{3}{4}$$, and you move up 7 markings from $$-1\frac{1}{2}$$, at which number will you be positioned?
$$1\frac{1}{4}$$, which is five markings above zero on the number line
$$\frac{1}{4}$$, which is one marking above zero on the number line
$$-\frac{3}{4}$$, which is three markings below zero on the number line
$$\frac{3}{4}$$, which is three markings above zero on the number line
Explanation
Moving up 7 markings from $$-1\frac{1}{2}$$ means adding $$7 \times \frac{1}{4} = \frac{7}{4} = 1\frac{3}{4}$$. So: $$-1\frac{1}{2} + 1\frac{3}{4} = -\frac{3}{2} + \frac{7}{4} = -\frac{6}{4} + \frac{7}{4} = \frac{1}{4}$$. Choice B would be if you moved up 3 markings. Choice C results from incorrectly calculating $$1\frac{1}{2} - \frac{3}{4}$$. Choice D results from adding instead of accounting for the negative starting position.
On a number line, point $$A$$ is located at $$-\frac{3}{4}$$ and point $$B$$ is located at $$\frac{5}{8}$$. If point $$C$$ is positioned exactly halfway between points $$A$$ and $$B$$, what is the coordinate of point $$C$$?
$$\frac{1}{16}$$
$$-\frac{1}{16}$$
$$\frac{7}{32}$$
$$-\frac{7}{32}$$
Explanation
To find the midpoint, use the formula: $$\frac{a + b}{2}$$. First convert to common denominators: $$-\frac{3}{4} = -\frac{6}{8}$$. Then: $$\frac{-\frac{6}{8} + \frac{5}{8}}{2} = \frac{-\frac{1}{8}}{2} = -\frac{1}{16}$$. Choice B incorrectly uses positive sign. Choice C results from using $$\frac{-3 + 5}{4 + 8} = \frac{2}{12}$$ incorrectly. Choice D combines the error in C with wrong sign.
On a coordinate plane, you start at the origin and move $$2\frac{1}{4}$$ units right, then $$1\frac{3}{4}$$ units down, then $$\frac{1}{2}$$ unit left, and finally $$\frac{3}{4}$$ unit up. What are your final coordinates?
$$\left(2\frac{3}{4}, -\frac{1}{2}\right)$$, after combining movements in the given sequence
$$\left(1\frac{1}{2}, -1\frac{1}{4}\right)$$, after calculating net displacement in each direction
$$\left(1\frac{3}{4}, -1\right)$$, after completing all horizontal then vertical movements
$$\left(\frac{3}{4}, -1\frac{1}{2}\right)$$, after applying movements relative to previous positions
Explanation
Starting at (0,0): Move right $$2\frac{1}{4}$$: x = $$2\frac{1}{4}$$. Move down $$1\frac{3}{4}$$: y = $$-1\frac{3}{4}$$. Move left $$\frac{1}{2}$$: x = $$2\frac{1}{4} - \frac{1}{2} = 1\frac{3}{4}$$. Move up $$\frac{3}{4}$$: y = $$-1\frac{3}{4} + \frac{3}{4} = -1$$. Final coordinates: $$(1\frac{3}{4}, -1)$$. Choice B doesn't account for leftward movement. Choice C miscalculates horizontal net movement. Choice D makes errors in both coordinate calculations.
A horizontal number line is marked in increments of $\tfrac{1}{2}$ from $-4$ to $4$. Which list shows the numbers in the correct left-to-right order on the number line?
Numbers: $-3,,-1.5,,0,,\tfrac{2}{3},,3.25$
$-3,,-1.5,,0,,\tfrac{2}{3},,3.25$
$-3,,-1.5,,0,,3.25,,\tfrac{2}{3}$
$-1.5,,-3,,0,,\tfrac{2}{3},,3.25$
$3.25,,\tfrac{2}{3},,0,,-1.5,,-3$
Explanation
This question tests positioning integers, fractions, and decimals (positive/negative) on number lines (horizontal/vertical) and coordinate planes, understanding signs indicate direction from zero/axes. On a number line, negative numbers are left of 0 (horizontal) or below 0 (vertical), positive numbers right/above, fractions/decimals between integers (1/2 between 0 and 1 at 0.5, -1.5 between -1 and -2). For example, position -3, 0, 1/2, 2.5 on horizontal number line: -3 is 3 units left of 0, 0 at center, 1/2 halfway between 0 and 1, 2.5 halfway between 2 and 3. The correct left-to-right order is -3, -1.5, 0, 2/3 (about 0.666), 3.25, which matches choice B. A common error is reversing the order like in choice D, placing positives left of negatives, or misordering decimals and fractions like swapping 2/3 and 3.25 in choice A. To position on a number line: (1) identify if horizontal or vertical, (2) locate zero, (3) determine direction from sign (negative→left/down, positive→right/up), (4) measure distance from zero, (5) mark position. For fractions/decimals, estimate position (2/3≈0.666 between 0 and 1); mistakes include wrong direction, ignoring scale (increments of 1/2), or misordering values.
A scuba diver’s elevation relative to sea level is shown on a vertical number line (up is positive, down is negative). Which value should be placed 2.5 units below $0$?
$-25$
$-0.25$
$2.5$
$-2.5$
Explanation
This question tests positioning integers, fractions, and decimals (positive/negative) on number lines (horizontal/vertical) and coordinate planes, understanding signs indicate direction from zero/axes. On a number line, negative numbers are left of 0 (horizontal) or below 0 (vertical), positive numbers right/above, fractions/decimals between integers (1/2 between 0 and 1 at 0.5, -1.5 between -1 and -2). For example, on a vertical number line, position -3, 0, 1/2, 2.5: -3 is 3 units below 0, 0 at center, 1/2 halfway above 0 and 1, 2.5 halfway above 2 and 3. The value 2.5 units below 0 is -2.5, matching choice B. A common error is placing positives below like 2.5 (A) or wrong decimals like -0.25 (C) or -25 (D), or reversing vertical direction. To position on a number line: (1) identify if horizontal or vertical, (2) locate zero, (3) determine direction from sign (negative→left/down, positive→right/up), (4) measure distance from zero, (5) mark position. Mistakes include vertical direction wrong (negatives up), scale ignored, or confusing below with above.
A game score change is shown on a horizontal number line where each tick mark is 1 point. Which value is farther from $0$ on the number line?
$1.4$
$-\tfrac{3}{2}$
They are the same distance from $0$.
Not enough information to tell.
Explanation
This question tests positioning integers, fractions, and decimals (positive/negative) on number lines (horizontal/vertical) and coordinate planes, understanding signs indicate direction from zero/axes. On a number line, negative numbers are left of 0 (horizontal) or below 0 (vertical), positive numbers right/above, fractions/decimals between integers (1/2 between 0 and 1 at 0.5, -1.5 between -1 and -2). For example, position -3, 0, 1/2, 2.5 on horizontal number line: -3 is 3 units left of 0, 0 at center, 1/2 halfway between 0 and 1, 2.5 halfway between 2 and 3. Comparing distances, |-3/2|=1.5 > |1.4|=1.4, so -3/2 is farther from 0, matching choice A. A common error is thinking they are the same (C) or picking the positive (B), or claiming not enough info (D) without calculating absolute values. To position on a number line: (1) identify if horizontal or vertical, (2) locate zero, (3) determine direction from sign (negative→left/down, positive→right/up), (4) measure distance from zero, (5) mark position. For fractions/decimals, estimate position; mistakes include ignoring absolute value for distance, wrong positions, or scale not respected (each tick 1 point).
A coordinate plane uses a scale of 1 unit per grid square on both axes. Which ordered pair is in Quadrant II (left of the $y$-axis and above the $x$-axis)?
$(4,,1)$
$(-4,,-1)$
$(-4,,1)$
$(1,,-4)$
Explanation
This question tests positioning integers, fractions, and decimals (positive/negative) on number lines (horizontal/vertical) and coordinate planes, understanding signs indicate direction from zero/axes. Coordinate plane: ordered pair (x,y) plotted by moving x units horizontally from origin (negative→left, positive→right), then y units vertically (negative→down, positive→up), signs determine quadrant (I: both +, II: x- y+, III: both -, IV: x+ y-). For example, plot (2,3), (-4,1), (-2,-3), (3,-2): (2,3) is 2 right, 3 up (Quadrant I), (-4,1) is 4 left, 1 up (Quadrant II), (-2,-3) is 2 left, 3 down (Quadrant III), (3,-2) is 3 right, 2 down (Quadrant IV). The ordered pair in Quadrant II (x negative, y positive) is (-4,1), matching choice B. A common error is confusing quadrants, like picking (4,1) in I (A), (-4,-1) in III (C), or (1,-4) in IV (D). To position on coordinate plane: (1) start at origin (0,0), (2) move x (right if +, left if -), (3) move y (up if +, down if -), (4) mark point. Mistakes include coordinates reversed, wrong signs for quadrants, or ignoring scale (1 unit per grid).
On a coordinate plane, point $P$ is plotted at $(-2,,3)$. Which description correctly tells how to get to $P$ from the origin?
Move 2 units left, then 3 units up.
Move 2 units right, then 3 units up.
Move 3 units left, then 2 units up.
Move 2 units left, then 3 units down.
Explanation
This question tests positioning integers, fractions, and decimals (positive/negative) on number lines (horizontal/vertical) and coordinate planes, understanding signs indicate direction from zero/axes. Coordinate plane: ordered pair (x,y) plotted by moving x units horizontally from origin (negative→left, positive→right), then y units vertically (negative→down, positive→up), signs determine quadrant (I: both +, II: x- y+, III: both -, IV: x+ y-). For example, plot (2,3), (-4,1), (-2,-3), (3,-2): (2,3) is 2 right, 3 up (Quadrant I), (-4,1) is 4 left, 1 up (Quadrant II), (-2,-3) is 2 left, 3 down (Quadrant III), (3,-2) is 3 right, 2 down (Quadrant IV). For (-2,3), move 2 left then 3 up, matching choice C. A common error is swapping distances like 3 left and 2 up (B) or wrong directions like down instead of up (D). To position on coordinate plane: (1) start at origin (0,0), (2) move x (right if +, left if -), (3) move y (up if +, down if -), (4) mark point. Mistakes include direction from sign wrong, coordinates reversed, or mixing horizontal/vertical moves.
On a coordinate plane, each grid square represents 1 unit on both axes. Which point is located 4 units left of the $y$-axis and 1 unit above the $x$-axis?
$(-4,1)$
$(1,-4)$
$(-1,4)$
$(4,1)$
Explanation
This question tests positioning integers, fractions, and decimals (positive/negative) on number lines (horizontal/vertical) and coordinate planes, understanding signs indicate direction from zero/axes. Coordinate plane: ordered pair (x,y) plotted by moving x units horizontally from origin (negative→left, positive→right), then y units vertically (negative→down, positive→up), signs determine quadrant (I: both +, II: x- y+, III: both -, IV: x+ y-). For example, plot (2,3), (-4,1), (-2,-3), (3,-2): (2,3) is 2 right, 3 up (Quadrant I), (-4,1) is 4 left, 1 up (Quadrant II), (-2,-3) is 2 left, 3 down (Quadrant III), (3,-2) is 3 right, 2 down (Quadrant IV). The correct point is (-4,1), as 4 units left of y-axis means x=-4 and 1 unit above x-axis means y=1. A common error is reversing signs, like choosing (4,1) for right instead of left as in A, or swapping coordinates like (1,-4) in B. For coordinate plane: (1) start at origin (0,0), (2) move x (first coordinate: right if +, left if -), (3) move y (second coordinate: up if +, down if -), (4) mark point. Mistakes: direction from sign wrong, coordinates reversed, or vertical direction wrong.