Explanation: Given terminal image \(K'(1, 5)\) after shift \(\langle -3, 8 \rangle\), reverse the vector to trace back to \(K\): \(x = 1 - (-3) = 4\); \(y = 5 - 8 = -3\). The pre-image is \(K(4, -3)\).
Teacher Insight: Distractor A (\(-2, 13\)) is the most frequent error: applying the forward rule directly onto the image.
6 Triangle Translation Rule from Grid
Correct: A
Explanation: Track vertex \(A(-4, 1)\) to \(A'(2, -2)\). Horizontal: \(2 - (-4) = +6\) (6 right). Vertical: \(-2 - 1 = -3\) (3 down). Confirm with \(B(-1, 1) \to B'(5, -2)\). Rule is \((x, y) \to (x + 6, y - 3)\).
Common Misconception: Distractor B reverses the direction of movement (moving from image to pre-image).
7 Forward Point Translation on Grid
Correct: A
Explanation: Starting at \(P(3, 4)\) with rule \((x - 7, y - 6)\): \(x' = 3 - 7 = -4\); \(y' = 4 - 6 = -2\). The image is \(P'(-4, -2)\).
Common Misconception: Distractor B adds instead of subtracting (\(3+7=10, 4+6=10\)).
8 Inverse Pre-Image Vertex Recovery
Correct: B
Explanation: Given image \(H'(-2, 3)\) after rule \((x + 5, y - 4)\). Invert operations to find pre-image \(H\): \(x = -2 - 5 = -7\); \(y = 3 + 4 = 7\). Thus, \(H(-7, 7)\).
Teacher Insight: Distractor A applies forward operations to the image. Students must reverse the arrows when working backward.
Coordinate Shifts Unit • Teacher Solution Guide Continue to Page 3 (Items 9–12) →
Part III: Advanced Polygon Translations & Coordinate Mapping
Page 3 of 4
Quick Key (9–12):
9: C 10: A 11: B 12: D
9 Segment Translation Rule from Grid
Correct: C
Explanation: For segment \(\overline{CD}\) to \(\overline{C'D'}\), compare endpoint \(C(-5, -2)\) to \(C'(1, 3)\): \(\Delta x = 1 - (-5) = +6\) (right 6); \(\Delta y = 3 - (-2) = +5\) (up 5). The rule is \((x, y) \to (x + 6, y + 5)\).
Common Misconception: Distractor A inverts the signs by calculating pre-image minus image.
10 Rectangle Vertex Translation
Correct: A
Explanation: From the grid, vertex \(L\) is at \((5, -1)\). Applying translation rule \((x - 4, y + 5)\): \(x' = 5 - 4 = 1\); \(y' = -1 + 5 = 4\). The translated vertex is \(L'(1, 4)\).
Common Misconception: Distractor B inverts the signs (\(5+4=9, -1-5=-6\)).
11 Triangle Vertex Translation on Grid
Correct: B
Explanation: Pre-image vertex \(E(-3, -2)\) translates under \((x + 4, y - 3)\): \(x' = -3 + 4 = 1\); \(y' = -2 - 3 = -5\). The resulting image is \(E'(1, -5)\).
Common Misconception: Distractor C adds 3 to \(y\) instead of subtracting.
12 Parallelogram Translation Rule
Correct: D
Explanation: Vertex \(S(2, -1)\) maps to \(S'(-3, 3)\). Horizontal shift: \(-3 - 2 = -5\) (5 left). Vertical shift: \(3 - (-1) = +4\) (4 up). The rule is \((x, y) \to (x - 5, y + 4)\).
Teacher Insight: Distractor C correctly identifies 5 left but errs on the vertical direction.
Coordinate Shifts Unit • Teacher Solution Guide Continue to Page 4 (Items 13–15 & Matrix) →
Part IV: Multi-Step Visual Solutions & 15-Point Diagnostic Rubric
Page 4 of 4
Quick Key (13–15):
13: A 14: B 15: C
13 Triangle Vertex Translation
Correct: A
Explanation: Pre-image vertex \(T\) is at \((4, 1)\). Under rule \((x - 6, y + 2)\): \(x' = 4 - 6 = -2\); \(y' = 1 + 2 = 3\). Thus, \(T'(-2, 3)\). Distractor B (\(10, -1\)) adds 6 and subtracts 2 (inverting both operations).
14 Inverse Point Recovery on Grid
Correct: B
Explanation: Given terminal image \(W'(1, -2)\) after rule \((x - 5, y - 4)\). Reverse the operations to find pre-image \(W\): \(x = 1 + 5 = 6\); \(y = -2 + 4 = 2\). Pre-image is \(W(6, 2)\). Distractor A applies forward operations to the image (\(1-5=-4, -2-4=-6\)).
15 Multi-Quadrant Triangle Translation Rule
Correct: C
Explanation: Track vertex \(X(-5, 4)\) to \(X'(2, -1)\). Horizontal: \(2 - (-5) = +7\) (7 right). Vertical: \(-1 - 4 = -5\) (5 down). Verify with \(Y(-2, 4) \to Y'(5, -1)\): \(-2 + 7 = 5, 4 - 5 = -1\). The rule is \((x, y) \to (x + 7, y - 5)\). Distractor A reverses signs.
14 – 15 Correct Exceeds Standard
12 – 13 Correct Meets Standard
9 – 11 Correct Approaching
≤ 8 Correct Needs Intervention
Targeted Remediation Groups:
• Rule Identification (#3, #6, #9, #12, #15): Focus on calculating terminal coordinate minus initial coordinate \((x_2 - x_1, y_2 - y_1)\).
• Inverse Pre-Image (#5, #8, #14): Emphasize reversing directional vector arrows and applying opposite algebraic operations.
• Vertex Mapping (#4, #7, #10, #11, #13): Reinforce integer addition/subtraction skills on the coordinate plane.
Teacher Answer Key • Coordinate Shifts Assessment • End of Key •
