Position Rational Numbers on Diagrams - 6th Grade Math

1. 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)?
   Answer: $(-4,\,1)$ 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).
2. 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 $0$?​​
   Answer: $-2.5$ On a vertical number line, going down means the numbers get smaller, so they become negative. Going up means positive. This question asks where to place a point that is $2.5$ units below $0$. Let's think step by step. The word "below" tells us to move down, so the value must be negative. We move exactly $2.5$ units, so the distance from $0$ is $2.5$. Put those together: down $2.5$ units lands you at $-2.5$. Picture a diver starting at the surface, which is $0$. As she sinks $2.5$ meters, her elevation drops to $-2.5$. The deeper she dives, the more negative the number gets! Try this at home: Draw a vertical number line. Place $-1$, $-2.5$, and $-4$. Which is lowest? Saying each value out loud helps the idea stick. You've got this!
3. A point $R$ is plotted on a coordinate plane (1 unit per grid square). The point is 3 units right of the $y$-axis and 2 units down from the $x$-axis. What are the coordinates of $R$?
   Answer: $(3,-2)$ 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 coordinates for 3 units right of y-axis (x=3) and 2 units down from x-axis (y=-2) are (3,-2). A common error is reversing signs, like (-3,2) for left and up as in B, or swapping values like (2,-3) in C. 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.
4. On a horizontal number line from $-10$ to $10$ with tick marks every 1 unit, which value is farther from $0$?
   Answer: $-6$ This question tests positioning integers, fractions, and decimals (positive and negative) on number lines (horizontal or vertical) and coordinate planes, understanding that signs indicate direction from zero or the axes. On a number line, negative numbers are to the left of 0 on horizontal lines or below 0 on vertical lines, positive numbers to the right or above, and fractions or decimals are placed between integers, such as 1/2 between 0 and 1 at 0.5, or -1.5 between -2 and -1. On a coordinate plane, an ordered pair (x,y) is plotted by moving x units horizontally from the origin (negative to left, positive to right), then y units vertically (negative down, positive up), with signs determining the quadrant (I: both positive, II: x negative y positive, III: both negative, IV: x positive y negative). For example, position -3, 0, 1/2, 2.5 on horizontal number line: -3 3 left of 0, 0 at center, 1/2 halfway between 0 and 1, 2.5 halfway between 2 and 3; or plot (2,3) 2 right 3 up (QI), (-4,1) 4 left 1 up (QII), (-2,-3) 2 left 3 down (QIII), (3,-2) 3 right 2 down (QIV). The value farther from 0 is -6 (distance 6) compared to 3/2 (distance 1.5), matching choice A on the 1-unit scale. A common error is thinking they are the same (choice C), or selecting 3/2 (choice B), or claiming not enough info (choice D), miscalculating absolute distances. To position on a number line: (1) identify horizontal or vertical, (2) locate zero, (3) determine direction from sign (negative left/down, positive right/up), (4) measure distance (|number| from zero), (5) mark position; for fractions/decimals estimate; mistakes include ignoring absolute value for distance.
5. 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?
   Answer: $-3,\,-1.5,\,0,\,\tfrac{2}{3},\,3.25$ 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.
6. A student is labeling a horizontal number line from $-2$ to $2$ with tick marks every $\tfrac{1}{4}$. Where should $\tfrac{3}{2}$ be placed?
   Answer: At $1.5$ (two ticks right of $1$). 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 position for 3/2 (1.5) is at 1.5, two ticks right of 1 (1.25 and 1.5, since ticks every 0.25). A common error is miscounting ticks, like one tick to 1.25 as in A or three to 1.75 as in C. For number line positioning: (1) identify if horizontal or vertical, (2) locate zero (center/origin), (3) determine direction from sign (negative→left/down, positive→right/up), (4) measure distance (|number| from zero: |-3|=3 units, |2.5|=2.5 units), (5) mark position (3 units left for -3, 2.5 units right for 2.5). Fractions/decimals: estimate position (1/2=0.5 midway between 0 and 1, 3/4=0.75 three-quarters to 1); mistakes include fractions/decimals at wrong positions or scale not respected.
7. Looking at the number line shown, what is the total distance between point J and point L if you travel through point K?
   Answer: $2.7$ units, calculated by adding the individual segment lengths Point J is at -1.2, K is at 0.3, and L is at 1.5. Distance from J to K is |0.3 - (-1.2)| = 1.5. Distance from K to L is |1.5 - 0.3| = 1.2. Total distance through K is 1.5 + 1.2 = 2.7. Choice B gives the same numerical result but represents direct distance. Choice C incorrectly adds coordinate values. Choice D uses incorrect subtraction method.
8. On a horizontal number line from $-4$ to $4$ with tick marks every $0.25$, where is $0.75$ located?
   Answer: Between $0$ and $1$, at the third tick mark after $0$ This question tests positioning integers, fractions, and decimals (positive and negative) on number lines (horizontal or vertical) and coordinate planes, understanding that signs indicate direction from zero or the axes. On a number line, negative numbers are to the left of 0 on horizontal lines or below 0 on vertical lines, positive numbers to the right or above, and fractions or decimals are placed between integers, such as 1/2 between 0 and 1 at 0.5, or -1.5 between -2 and -1. On a coordinate plane, an ordered pair (x,y) is plotted by moving x units horizontally from the origin (negative to left, positive to right), then y units vertically (negative down, positive up), with signs determining the quadrant (I: both positive, II: x negative y positive, III: both negative, IV: x positive y negative). For example, position -3, 0, 1/2, 2.5 on horizontal number line: -3 3 left of 0, 0 at center, 1/2 halfway between 0 and 1, 2.5 halfway between 2 and 3; or plot (2,3) 2 right 3 up (QI), (-4,1) 4 left 1 up (QII), (-2,-3) 2 left 3 down (QIII), (3,-2) 3 right 2 down (QIV). The correct positioning for 0.75 is between 0 and 1 at the third tick mark after 0 (ticks at 0.25, 0.5, 0.75), matching choice A with 0.25 increments. A common error is placing it at the first tick (0.25 like choice B), or between 1 and 2 (choice C), or in negatives (choice D), ignoring decimal value or scale. To position on a number line: (1) identify horizontal or vertical, (2) locate zero, (3) determine direction from sign (negative left/down, positive right/up), (4) measure distance (|number| from zero), (5) mark position; for fractions/decimals estimate like 0.75 three-quarters to 1; mistakes include scale ignored, wrong positions for decimals.
9. A student plots point $P(-2,-3)$ on a coordinate plane where each grid square is 1 unit. Which description matches the correct location of $P$?
   Answer: Move 2 units left and 3 units down from the origin. 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 description for (-2,-3) is move 2 units left and 3 units down from the origin. A common error is mixing directions, like moving right and up for negatives as in A, or reversing the order of movements like in C. 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.
10. On a coordinate plane, point $P$ is plotted at $(-2,\,3)$. Which description correctly tells how to get to $P$ from the origin?​​
    Answer: Move 2 units left, then 3 units up. 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.
11. 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?​​
    Answer: $-\tfrac{3}{2}$ 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).
12. 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 $0$?
    Answer: $-2.5$ 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.
13. On a horizontal number line with tick marks every $0.25$, which point is located exactly halfway between $-2$ and $-1$?
    Answer: $-1.5$ 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 point halfway between -2 and -1 is -1.5, matching choice C. A common error is miscalculating the midpoint, like averaging wrong to get -1.25 (A) or -1.75 (B), or confusing with positive side like -2.5 (D). 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 (midpoint as average); mistakes include scale not respected (every 0.25), wrong direction, or placing at integers instead of between.
14. On a horizontal number line marked in increments of $0.5$ from $-5$ to $5$, which list shows the numbers in the correct order from left to right? Numbers: $-3$, $-1.5$, $0$, $\frac{2}{3}$, $5$
    Answer: $-3,\,-1.5,\,0,\,\frac{2}{3},\,5$ This question tests positioning integers, fractions, and decimals (positive and negative) on number lines (horizontal or vertical) and coordinate planes, understanding that signs indicate direction from zero or the axes. On a number line, negative numbers are to the left of 0 on horizontal lines or below 0 on vertical lines, positive numbers to the right or above, and fractions or decimals are placed between integers, such as 1/2 between 0 and 1 at 0.5, or -1.5 between -2 and -1. On a coordinate plane, an ordered pair (x,y) is plotted by moving x units horizontally from the origin (negative to left, positive to right), then y units vertically (negative down, positive up), with signs determining the quadrant (I: both positive, II: x negative y positive, III: both negative, IV: x positive y negative). For example, to position -3, -1.5, 0, 2/3, 5 on a horizontal number line: -3 is 3 units left of 0, -1.5 is 1.5 units left of 0 (between -2 and -1), 0 at center, 2/3 (about 0.67) between 0 and 1, 5 is 5 units right of 0. The correct positioning is the order from left to right as -3, -1.5, 0, 2/3, 5, matching choice A on the number line with 0.5 increments. A common error is misordering negatives like putting -1.5 before -3 (more negative is further left), or placing 0 before -1.5, or 2/3 before 0, ignoring signs and values. To position on a number line: (1) identify horizontal or vertical, (2) locate zero, (3) determine direction from sign (negative left/down, positive right/up), (4) measure distance (|number| from zero), (5) mark position; for fractions/decimals estimate like 2/3 ≈0.67 between 0.5 and 1; mistakes include wrong direction, reversed order, ignoring scale.
15. A coordinate plane uses a scale of 1 unit per grid square. A student starts at the origin and moves 3 units right and 2 units down. What point did the student plot?
    Answer: $(3,\,-2)$ 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). Moving 3 right (x=3) and 2 down (y=-2) plots (3,-2), matching choice D. A common error is reversing coordinates like (2,-3) in C or confusing directions like (-3,-2) in B. 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 confusing up/down with left/right.
16. A student is graphing points on a coordinate plane with 1 unit per grid square. The student accidentally swaps the coordinates and plots $(2,\,3)$ as $(3,\,2)$. If the correct point is $(2,\,3)$, where should it be located?
    Answer: 2 units right and 3 units up from the origin This question tests positioning integers, fractions, and decimals (positive and negative) on number lines (horizontal or vertical) and coordinate planes, understanding that signs indicate direction from zero or the axes. On a number line, negative numbers are to the left of 0 on horizontal lines or below 0 on vertical lines, positive numbers to the right or above, and fractions or decimals are placed between integers, such as 1/2 between 0 and 1 at 0.5, or -1.5 between -2 and -1. On a coordinate plane, an ordered pair (x,y) is plotted by moving x units horizontally from the origin (negative to left, positive to right), then y units vertically (negative down, positive up), with signs determining the quadrant (I: both positive, II: x negative y positive, III: both negative, IV: x positive y negative). For example, position -3, 0, 1/2, 2.5 on horizontal number line: -3 3 left of 0, 0 at center, 1/2 halfway between 0 and 1, 2.5 halfway between 2 and 3; or plot (2,3) 2 right 3 up (QI), (-4,1) 4 left 1 up (QII), (-2,-3) 2 left 3 down (QIII), (3,-2) 3 right 2 down (QIV). The correct location for (2,3) is 2 units right and 3 units up from origin, matching choice A. A common error is swapping to 3 right 2 up (choice B like the student's mistake), or using negatives incorrectly (choices C, D). To plot 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 direction from signs.
17. 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 $C$?
    Answer: $-\frac{1}{16}$ 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.
18. On a horizontal number line with tick marks every $0.5$ unit, a student plots $-2.5$, $-1$, $0$, $\tfrac{1}{2}$, and $3$. Which statement is correct?
    Answer: $-2.5$ is halfway between $-2$ and $-3$. 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 statement is -2.5 is halfway between -2 and -3, as on 0.5 unit ticks, it's midway on the negative side. A common error is placing positives left of 0 like in A or negatives right like in C. For number line positioning: (1) identify if horizontal or vertical, (2) locate zero (center/origin), (3) determine direction from sign (negative→left/down, positive→right/up), (4) measure distance (|number| from zero: |-3|=3 units, |2.5|=2.5 units), (5) mark position (3 units left for -3, 2.5 units right for 2.5). Fractions/decimals: estimate position (1/2=0.5 midway between 0 and 1, 3/4=0.75 three-quarters to 1); mistakes include direction from sign wrong, fractions/decimals at wrong positions, or scale not respected.
19. A thermometer is shown as a vertical number line where each tick mark is $5^\circ\text{C}$. The center tick is $0^\circ\text{C}$, values increase upward, and decrease downward. Which placement is correct?
    Answer: $-10^\circ\text{C}$ is 2 ticks below $0^\circ\text{C}$ and $20^\circ\text{C}$ is 4 ticks above $0^\circ\text{C}$. 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 placement is -10°C at 2 ticks below 0°C (since each tick is 5°C, -10 is two ticks down) and 20°C at 4 ticks above 0°C (20 divided by 5 is 4 ticks up). A common error is confusing directions on vertical lines, like placing negatives above 0 as in A, or miscalculating ticks like in C and D by halving or doubling the count. For number line positioning: (1) identify if horizontal or vertical, (2) locate zero (center/origin), (3) determine direction from sign (negative→left/down, positive→right/up), (4) measure distance (|number| from zero: |-3|=3 units, |2.5|=2.5 units), (5) mark position (3 units left for -3, 2.5 units right for 2.5). Fractions/decimals: estimate position (1/2=0.5 midway between 0 and 1, 3/4=0.75 three-quarters to 1); mistakes include direction from sign wrong, vertical number line direction reversed, or scale not respected.
20. A horizontal number line is labeled at every integer from $-5$ to $5$, with one tick mark halfway between each pair of integers. Which number should be placed at the tick mark halfway between $1$ and $2$?
    Answer: $\tfrac{3}{2}$ 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 tick mark halfway between 1 and 2 is 1.5 or 3/2, matching choice A. A common error is picking wrong fractions like 1/2 (B) or 5/2=2.5 (C), or unrelated 2/3 (D). 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 (3/2=1.5 midway between 1 and 2); mistakes include fractions at wrong positions, scale not respected, or confusing with other intervals.

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