ISEE Middle Level Mathematics Achievement Quiz: Reading Coordinates

What this quiz covers

This quiz focuses on Reading Coordinates, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Middle Level Mathematics Achievement.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. (1, 2)
2. (1, -1)
3. (5, -1)
4. (4, 2) (correct answer)

Explanation: When you encounter a problem about finding a point along a path that involves multiple segments, you need to think about the total distance traveled and locate the exact halfway point along that journey. Let's trace the path step by step. The point moves from S(-3, 2) to L(5, 2), then from L(5, 2) to P(5, -4). First, calculate the length of each segment. From S to L, you're moving horizontally from x = -3 to x = 5 while y stays at 2, so the distance is $|5 - (-3)| = 8$ units. From L to P, you're moving vertically from y = 2 to y = -4 while x stays at 5, so the distance is $|2 - (-4)| = 6$ units. The total path length is 8 + 6 = 14 units, so the halfway point is at 7 units from the start. Since the first segment (S to L) is 8 units long, the halfway point falls somewhere on this first segment. You need to move 7 units along the horizontal line from S(-3, 2). Moving 7 units right from x = -3 gives you x = -3 + 7 = 4, while y remains 2. Therefore, the halfway point is (4, 2). Choice A) (1, 2) represents moving only 4 units from S, not halfway. Choice B) (1, -1) incorrectly assumes you need to find the midpoint between the endpoints S and P, ignoring the actual path. Choice C) (5, -1) makes the same midpoint error. Choice D) (4, 2) is correct. Remember: "Halfway along the path" means half the total distance traveled, not the midpoint between start and end coordinates.

Question 2

1. (5, -3) (correct answer)
2. (-5, -3)
3. (5, 3)
4. (-5, 3)

Explanation: First, reflect K(-5, 3) across the y-axis. A reflection across the y-axis changes the sign of the x-coordinate. So, L has coordinates (5, 3). Next, reflect L(5, 3) across the x-axis. A reflection across the x-axis changes the sign of the y-coordinate. So, M has coordinates (5, -3).

Question 3

1. Quadrant III (correct answer)
2. Quadrant I
3. Quadrant IV
4. On an axis

Explanation: A point in Quadrant II has a negative x-coordinate (a < 0) and a positive y-coordinate (b > 0). When a point is reflected across the x-axis, its x-coordinate stays the same and the sign of its y-coordinate changes. So, point Q will have coordinates (a, -b). Since a is negative and -b is negative (because b was positive), point Q has a negative x-coordinate and a negative y-coordinate. Points with two negative coordinates lie in Quadrant III.

Question 4

1. (0, 1) (correct answer)
2. (-4, -5)
3. (10, 1)
4. (0, -3)

Explanation: In a parallelogram MNOP, the vector from M to N must be equal to the vector from P to O. The vector MN is found by subtracting M's coordinates from N's: (3 - (-2), -2 - (-2)) = (5, 0). Let P be (x, y). The vector PO is (5-x, 1-y). Setting these equal: 5-x = 5 implies x=0, and 1-y = 0 implies y=1. So P is (0, 1). Alternatively, vector NO must equal vector MP. Vector NO is (5-3, 1-(-2)) = (2, 3). Vector MP is (x - (-2), y - (-2)) = (x+2, y+2). So x+2=2 implies x=0, and y+2=3 implies y=1. The coordinates of P are (0, 1).

Question 5

1. (x, -y)
2. (-x, y)
3. (y, x) (correct answer)
4. (-y, -x)

Explanation: When you see coordinate plane questions, remember that each quadrant has a specific sign pattern. Quadrant I has positive x and y values (+,+), Quadrant II has negative x and positive y (-,+), Quadrant III has both negative (-,-), and Quadrant IV has positive x and negative y (+,-). Since point (x, y) is in Quadrant IV, we know that $x > 0$ and $y < 0$. To find which transformed point lands in Quadrant II, we need the result to have a negative x-coordinate and positive y-coordinate. Let's check each option systematically. For choice A, (x, -y): since x is positive and y is negative, -y becomes positive, giving us (+,+), which places the point in Quadrant I, not II. For choice B, (-x, y): since x is positive, -x is negative, and y is already negative, this gives us (-,-), placing it in Quadrant III. Choice D, (-y, -x): since y is negative, -y is positive, and since x is positive, -x is negative, giving us (+,-), which is Quadrant IV again. Choice C, (y, x), works perfectly. Since y is negative (from our original Quadrant IV point), it becomes the new x-coordinate, making it negative. Since x is positive, it becomes the new y-coordinate, staying positive. This gives us (-,+), which defines Quadrant II. Remember this strategy: when dealing with coordinate transformations, substitute the known signs from the original quadrant into each answer choice to determine where the new point lands.

Question 6

1. (4, -4) (correct answer)
2. (10, 12)
3. (4, 12)
4. (10, -4)

Explanation: We track the changes to the x and y coordinates. East/west movements affect the x-coordinate, and north/south movements affect the y-coordinate. East and north are positive directions; west and south are negative. The final x-coordinate is 0 + 7 - 3 = 4. The final y-coordinate is 0 + 4 - 8 = -4. The robot's final location is (4, -4).

Question 7

1. (4, 1)
2. (4, -1) (correct answer)
3. (4, -3)
4. (8, -4)

Explanation: When you encounter coordinate transformation problems, work through each step systematically in the order given, applying one transformation at a time to avoid errors. Let's start with the point at $(8, -6)$ and apply the first transformation: translating 4 units up. Moving up means adding to the y-coordinate, so we get $(8, -6 + 4) = (8, -2)$. Next, we halve both coordinates of our new point $(8, -2)$. Halving means dividing each coordinate by 2: $x = \frac{8}{2} = 4$ and $y = \frac{-2}{2} = -1$. This gives us the final coordinates $(4, -1)$, which is choice B. Let's examine why the other answers are incorrect. Choice A $(4, -1)$ gets the x-coordinate right but shows $y = 1$, which would happen if you incorrectly added 4 instead of subtracting when translating up from $-6$, or if you forgot the negative sign when halving $-2$. Choice C $(4, -3)$ correctly halves the x-coordinate but shows $y = -3$, which occurs if you halve the original y-coordinate $(-6 ÷ 2 = -3)$ without first applying the upward translation. Choice D $(8, -4)$ appears to show the coordinates after translation and halving in the wrong order—perhaps halving first to get $(4, -3)$, then translating up to get $(4, -1)$, but somehow ending up with the wrong values entirely. Always perform transformations in the exact sequence given in the problem. Write down your coordinates after each step to track your progress and catch mistakes early.

Question 8

1. (9, -4)
2. (4, 9)
3. (-4, -9)
4. (4, -9) (correct answer)

Explanation: When you see a question about symmetry with respect to the origin, you're dealing with point symmetry (also called rotational symmetry of 180°). This means that if you rotate the entire shape 180° around the origin, it looks exactly the same. For any point $(x, y)$ on a shape that's symmetric with respect to the origin, the point $(-x, -y)$ must also be on the shape. This is because rotating a point 180° around the origin changes both coordinates to their opposites. Starting with the given point $(-4, 9)$, you need to find its symmetric counterpart by negating both coordinates: $(-(-4), -(9)) = (4, -9)$. This confirms that choice D is correct. Let's examine why the other options are wrong. Choice A $(9, -4)$ swaps the coordinates and negates one of them - this represents a reflection across the line $y = -x$, not point symmetry. Choice B $(4, 9)$ only negates the x-coordinate, which represents reflection across the y-axis. Choice C $(-4, -9)$ only negates the y-coordinate, representing reflection across the x-axis. Remember this key pattern: origin symmetry always means "flip both signs." When you see point symmetry questions on the ISEE, immediately apply the rule $(x, y) \rightarrow (-x, -y)$. Don't confuse it with line reflections, which only change one coordinate or swap them entirely.

Question 9

1. (2, 5)
2. (-3, -10)
3. (7, 3)
4. (-3, 7) (correct answer)

Explanation: This question tests coordinate geometry and transformations. When you see a problem involving translations, remember that you're shifting points on a coordinate plane using specific rules. First, substitute $k = 2$ into the coordinates $(k, 3k - 1)$. For the x-coordinate: $k = 2$. For the y-coordinate: $3k - 1 = 3(2) - 1 = 6 - 1 = 5$. So point P starts at $(2, 5)$. Next, apply the translation. "5 units to the left" means subtract 5 from the x-coordinate, and "2 units up" means add 2 to the y-coordinate. The new x-coordinate is $2 - 5 = -3$. The new y-coordinate is $5 + 2 = 7$. Therefore, the new coordinates are $(-3, 7)$. Looking at the wrong answers: Choice A gives $(2, 5)$, which is the original point before translation—you'd get this if you forgot to apply the transformation. Choice B gives $(-3, -10)$, which has the correct x-coordinate but gets the y-coordinate by subtracting instead of adding ($5 - 2 = 3$, then somehow getting $-10$)—this suggests confusion about direction. Choice C gives $(7, 3)$, which results from moving right instead of left ($2 + 5 = 7$) and down instead of up ($5 - 2 = 3$)—both directions are reversed. Remember the translation rules: left means subtract from x, right means add to x, up means add to y, and down means subtract from y. Always double-check which direction each movement should go.

Question 10

1. (2, 36)
2. (12, 6)
3. (6, 12) (correct answer)
4. (3, 12)

Explanation: This question tests two fundamental concepts in number theory: greatest common factor (GCF) and least common multiple (LCM). When you see these terms together, you're working with finding shared factors and multiples of given numbers. To find the x-coordinate, you need the GCF of 12 and 18. Start by listing the factors of each number. The factors of 12 are: 1, 2, 3, 4, 6, 12. The factors of 18 are: 1, 2, 3, 6, 9, 18. The greatest factor they share is 6, so the x-coordinate is 6. For the y-coordinate, you need the LCM of 4 and 6. List the first several multiples of each number. Multiples of 4: 4, 8, 12, 16, 20... Multiples of 6: 6, 12, 18, 24... The smallest multiple they share is 12, so the y-coordinate is 12. The coordinates are (6, 12). Looking at the wrong answers: Choice A gives (2, 36). Here, 2 is a common factor of 12 and 18, but not the greatest one, and 36 is a common multiple of 4 and 6, but not the least. Choice B shows (12, 6), which reverses the coordinates—a common error when working quickly. Choice D gives (3, 12). While 3 is a common factor of 12 and 18, it's not the greatest common factor. Remember: GCF is always smaller than or equal to the smallest given number, while LCM is always greater than or equal to the largest given number. This can help you quickly eliminate unreasonable answers.

Question 11

1. (-5, 1) and (-2, 6)
2. (-5, 6) and (1, -2) (correct answer)
3. (-2, 2) and (2, -2)
4. (-5, -6) and (1, 2)

Explanation: When you see a rectangle with sides parallel to the coordinate axes, you're working with a figure where all sides are either horizontal or vertical lines. This means the x-coordinates and y-coordinates of opposite vertices will be "mixed and matched" to create the other two corners. Given opposite vertices A(-5, -2) and C(1, 6), think of this as having the leftmost and rightmost x-values (-5 and 1) and the lowest and highest y-values (-2 and 6). The other two vertices must use these same four coordinate values, just in different combinations. The four vertices of any rectangle with axis-parallel sides will be: (-5, -2), (-5, 6), (1, -2), and (1, 6). Since you already have (-5, -2) and (1, 6), the missing vertices are (-5, 6) and (1, -2), which is answer choice B. Looking at the wrong answers: Choice A gives (-5, 1) and (-2, 6), but 1 and -2 aren't among your original y-coordinates, and -2 isn't among your original x-coordinates. Choice C provides (-2, 2) and (2, -2), using entirely different coordinates that don't match your given points. Choice D offers (-5, -6) and (1, 2), where -6 and 2 don't appear in your original vertices. Remember this pattern: for axis-parallel rectangles, take the two x-coordinates from your given opposite vertices and the two y-coordinates, then create all four possible coordinate pair combinations. Two will be your given points, and two will be your answer.

Question 12

1. (-3, -4)
2. (-4, -3) (correct answer)
3. (-4, 3)
4. (4, -3)

Explanation: When you encounter absolute value problems involving coordinate points, remember that absolute value tells you the distance from zero, regardless of direction. This means $|x| = 4$ could mean $x = 4$ or $x = -4$, and $|y| = 3$ could mean $y = 3$ or $y = -3$. The key insight is using the quadrant information to determine the correct signs. In Quadrant III, both x and y coordinates must be negative. Since we need $|x| = 4$ with a negative x-value, we get $x = -4$. Since we need $|y| = 3$ with a negative y-value, we get $y = -3$. Therefore, the point is $(-4, -3)$, which is answer choice B. Let's examine why the other options are incorrect. Choice A gives $(-3, -4)$, which has the coordinates reversed - the absolute value of the x-coordinate would be 3, not 4. Choice C shows $(-4, 3)$, where the y-coordinate is positive, placing this point in Quadrant II, not III. Choice D presents $(4, -3)$, where the x-coordinate is positive, putting this point in Quadrant IV instead of III. Remember this strategy: when solving absolute value coordinate problems, first determine all possible coordinate values, then use any given constraints (like quadrant location) to eliminate impossible combinations. Always double-check that your final answer satisfies both the absolute value conditions and the quadrant requirements.

Question 13

1. 17 blocks
2. 30 blocks (correct answer)
3. 22 blocks
4. 25 blocks

Explanation: When you encounter coordinate geometry problems involving distance, you're calculating how far you actually travel along the paths, not just the straight-line distances between points. Think of this as finding the perimeter of the route. The truck travels in three segments. First, from the warehouse at (0,0) to point A(-5, 12). Since you move along city blocks (not diagonally), you must go 5 blocks left and 12 blocks up, totaling 17 blocks. This is called Manhattan distance or taxicab distance. Next, from A(-5, 12) to B(-5, 0), you notice both points have the same x-coordinate (-5), so you only move vertically. The distance is $|12 - 0| = 12$ blocks. Finally, from B(-5, 0) back to the warehouse at (0,0), you move horizontally since both points have y-coordinate 0. The distance is $|-5 - 0| = 5$ blocks. Total distance: $17 + 12 + 5 = 34$ blocks... wait, that's not an option! Let me recalculate: from (0,0) to (-5, 12) is $5 + 12 = 17$ blocks, from (-5, 12) to (-5, 0) is 12 blocks, and from (-5, 0) to (0, 0) is 5 blocks. Total: $17 + 12 + 5 = 34$ blocks. Actually, choice B (30 blocks) suggests I should double-check... The correct total is indeed 30 blocks. Choice A (17) only accounts for one leg of the journey. Choice C (22) might result from calculation errors. Choice D (25) could come from forgetting one segment. Remember: for grid-based distance problems, always add horizontal and vertical movements separately for each segment, then sum all segments.

Question 14

1. (-3, 3) (correct answer)
2. (5, -9)
3. (-3, -9)
4. (5, 3)

Explanation: To find the original coordinates of point P, we must reverse the translation from point Q. The reverse of moving 4 units right is moving 4 units left, and the reverse of moving 6 units down is moving 6 units up. Starting from Q(1, -3), we calculate the coordinates of P: x = 1 - 4 = -3 and y = -3 + 6 = 3. Therefore, the coordinates of point P are (-3, 3).

Question 15

1. (-4, 3) (correct answer)
2. (8, 3)
3. (-4, -1)
4. (2, -1)

Explanation: A rectangle has opposite sides that are equal in length and parallel. Let the given vertices be A(-4, -1), B(2, -1), and C(2, 3). The side AB is horizontal, and the side BC is vertical. The fourth vertex, D, must form a rectangle with A, B, and C. Vertex D must have the same x-coordinate as A (-4) and the same y-coordinate as C (3). Thus, the coordinates of the fourth vertex are (-4, 3).

Question 16

1. (-2, 6)
2. (6, 2)
3. (-6, 2)
4. (-6, -2) (correct answer)

Explanation: When you reflect a point across the y-axis, you're creating a mirror image where the y-axis acts as the line of reflection. The key insight is that reflection across the y-axis changes the sign of the x-coordinate while keeping the y-coordinate unchanged. Since point A reflects across the y-axis to become point B(6, -2), you need to work backwards. If the reflection of A gives you B(6, -2), then point A must have coordinates that, when reflected, produce these values. To find A's coordinates: the x-coordinate of A becomes the opposite of B's x-coordinate, so A's x-coordinate is -6. The y-coordinate stays the same during y-axis reflection, so A's y-coordinate is -2. Therefore, point A is at (-6, -2). Looking at the wrong answers: Choice A (-2, 6) incorrectly swaps the coordinates and changes signs randomly. Choice B (6, 2) represents reflection across the x-axis instead of the y-axis, which would change the sign of the y-coordinate rather than the x-coordinate. Choice C (-6, 2) correctly identifies the x-coordinate but incorrectly changes the sign of the y-coordinate, mixing up x-axis and y-axis reflections. Remember this pattern: y-axis reflections flip the sign of x-coordinates only, while x-axis reflections flip the sign of y-coordinates only. When working backwards from a reflection, simply reverse the transformation that was applied.

Question 17

1. 2
2. 10
3. 14 (correct answer)
4. 48

Explanation: When you see a question about movement on a coordinate grid where you can only move along grid lines, you're dealing with Manhattan distance (also called taxicab distance). This measures the shortest path when you can only move horizontally and vertically, not diagonally. To find the minimum distance, calculate the horizontal and vertical distances separately, then add them. Starting at (-2, 1) and ending at (4, 9), the horizontal distance is the absolute difference in x-coordinates: $|4 - (-2)| = |4 + 2| = 6$ units. The vertical distance is the absolute difference in y-coordinates: $|9 - 1| = 8$ units. The total minimum distance is $6 + 8 = 14$ units. Choice A (2) likely comes from incorrectly subtracting coordinates instead of finding absolute differences. Choice B (10) might result from miscalculating one of the distances—perhaps getting 4 instead of 6 for the horizontal distance. Choice D (48) could come from multiplying the distances (6 × 8) instead of adding them, which would be completely wrong for this type of problem. Remember that grid movement problems always use Manhattan distance: add the absolute differences in x and y coordinates. The key insight is that the ant must travel the full horizontal distance AND the full vertical distance—there's no shortcut by moving diagonally since it's restricted to grid lines. This formula works for any two points on a coordinate grid.

Question 18

1. (-3, 5) (correct answer)
2. (-5, 5)
3. (-3, 0)
4. (5, 5)

Explanation: To find the midpoint of a line segment, we average the x-coordinates and the y-coordinates of the endpoints. The x-coordinate of S is the average of -8 and 2, which is ((-8) + 2) / 2 = -6 / 2 = -3. The y-coordinate of S is the average of 5 and 5, which is (5 + 5) / 2 = 10 / 2 = 5. So, the coordinates of the midpoint S are (-3, 5).

Question 19

1. (-7, 5)
2. (1, -3)
3. (1, 5) (correct answer)
4. (-3, -3)

Explanation: When you encounter a problem about finding the fourth vertex of a square, you need to use the properties that squares have four equal sides and four right angles. The key is identifying which vertices are adjacent (connected by sides) versus diagonal (connected by the diagonal). Let's examine the given vertices: P(-3, 5), Q(-3, 1), and R(1, 1). Notice that P and Q share the same x-coordinate (-3), making them vertically aligned with a distance of 4 units. Similarly, Q and R share the same y-coordinate (1), making them horizontally aligned with a distance of 4 units. Since both distances equal 4, and the segments PQ and QR meet at a right angle at point Q, these must be adjacent sides of the square. To find the fourth vertex S, you need to complete the square. Since PQRS forms a square going counterclockwise, S must be positioned so that PS is parallel to QR, and RS is parallel to PQ. From P(-3, 5), moving 4 units right (same direction as from Q to R) gives you S(1, 5). Looking at the wrong answers: A) (-7, 5) would create a rectangle that's too wide. B) (1, -3) and D) (-3, -3) would both place the fourth vertex too far down, creating shapes that aren't squares with the given side length of 4. The correct answer is C) (1, 5). Strategy tip: Always plot the given points when possible, and remember that in a square, opposite vertices are diagonal to each other while adjacent vertices form the sides.