Pivot Points Facilitation Guide Pivot Points
Explicit Facilitation Guide • 25-Minute High-Impact Lesson
NYS Geometry
Lesson Plan
Learning Target
"At the end of 25 minutes, students will be able to rotate a polygon around a center that is not the origin."
NYS Next Gen Standard
NY-GEO.G.CO.5: Know and apply rotations, reflections, and translations to polygons.
Academic Vocabulary
Pivot Point Center of Rotation Rigid Motion Congruence Translation
Math Practices (SMP)
MP.7: Look for and make use of structure by deconstructing a complex rotation into simpler steps.
25 Min Regents Mastery
ENL Strategies
Visual Anchors: Show the arc on the slide to map physical motion.TPR: Use hand motions for "Shift (move body left), Spin (rotate arm), Shift Back (move body right)".Sentence Frames: Use the reflective prompt on worksheet page 4.
0-4m
Activation: Do Now
Direct Instruction:
Say: "We know origin rotations. Turn to Worksheet Page 1. You have 2 minutes to fill in the rules. These are our anchors."
Action: Show Slide 2 (Blank) for brainstorm, then Slide 3 (Rules) to verify.
4-12m
Modeling: The Pivot
The Method:
"The math only spins around zero. To rotate around (4, 2), we move the world so (4, 2) is at zero. Subtract. Rotate. Add. "
Action: Show Slide 4 (Problem) to prompt student strategy. Then Slide 5 (Solution) to walk through arithmetic.
12-22m
Practice & Discourse
Say: "Turn to Problem 1 on Page 3. Start with Vertex A. Subtract the pivot. Apply the rule. Add back. Check your A' with your neighbor before doing B and C."
Discourse: Ask: "Did the triangle's area change after we spun it? Why not?"
22-25m
Assessment
Action: Students complete the "Mastery Check" on Page 4 of the packet. Collect to verify Step 3 mastery.
Support Scaffolding
Vector Paths: Have students draw an L-shaped path from pivot to vertex. Example: "3 right, 0 up". Rotate that vector 90° CCW: "0 right, 3 up" to find the new relative location.
Advanced Extension
Challenge students to rotate a figure 180° about (1,1) and then translate it by (-2, -2). Is the result equivalent to a 180° rotation about the origin?
Pivot Points Slides PIVOT POINTS
Non-Origin Rotations
Learning Target:
"At the end of 25 minutes, students will be able to rotate a polygon around a center that is not the origin."
Recall: Origin Rules
2m
90° (x, y) → (__ , __)
180° (x, y) → (__ , __)
Think:
Why do we need rules for (0,0) if we are spinning around other points today?
Verify Your Anchors
Check
90° (x, y) → (-y, x)
180° (x, y) → (-x, -y)
The Key:
Standard rules are only for rotations about the origin. Today, we "cheat" by moving our pivot to (0,0)!
Modeling The Pivot
Task
The Model Problem:
Rotate Point A(1, 2) about Pivot P(4, 2) by 90° CCW.
1. SUBTRACT?
2. ROTATE?
3. ADD?
Step-by-Step Logic
Solve
Step 1: Subtract P(4, 2)
(1 - 4, 2 - 2)
= (-3, 0)
Step 2: Rotate 90° CCW
(-3, 0) → (0, -3)
Step 3: Add Back (4, 2)
(0 + 4, -3 + 2)
A'(4, -1)
Regents Practice
JUNE 2016 #28
"ΔRST has vertices R(-2, 3), S(4, 4), and T(2, -2). Rotate ΔRST by 180° about the point (1, 1)."
Strategy
"Move the Pivot to zero by subtracting (1, 1) from every coordinate first."
Go!
Open your worksheet to Page 3. Solve for all three vertices.
Pivot Points Worksheet Pivot Points: Part 1
Lesson Activation
Name:
Full Learning Target
"At the end of 25 minutes, students will be able to rotate a polygon around a center that is not the origin."
Step 1: Do Now
Recall the origin rotation rules for any point (x, y):
Rotation Degree Coordinate Rule 90° CCW 180° 270° CCW (90° CW)
Pivot Points: Part 2
The Precision Pivot Method
"The math only spins around (0,0). So if we want to spin around (h, k), we shift the world there first."
1
SUBTRACT (h, k)
Move the center point to (0,0)
2
APPLY RULE
Use your origin rules from Page 1
3
ADD BACK (h, k)
Move back to the true location
Pivot Points: Part 3
Regents Practice
1. Triangle ABC has vertices A(1, 2), B(-4, 3), and C(-3, -5). Rotate ΔABC 90° CCW about the point (4, 2).
Calculation Workspace (Steps 1, 2, 3)
Vertex A'
Vertex B'
Vertex C'
x y 10 -10 10 -10
Pivot Points: Part 4
Reflective Thinking
"We used a 3-step shortcut to rotate around the point (4, 2). Explain why the triangle image must still be congruent to the original. Use property-based evidence."
Mastery Check
Rotate segment AB with A(-1, 3) and B(0, 7) by 180° about (1, 1).
A':
B':
Pivot Points Key Answer Key
Pivot Points • Teacher Resource
Part 1: Origin Rules
90° CCW: (-y, x)
180°: (-x, -y)
Regents Practice 1: 90° CCW about (4, 2)
Vertex A(1, 2)
1. Sub (4,2): (-3, 0)
2. Rule: (0, -3)
3. Add (4,2): A'(4, -1)
Vertex B(-4, 3)
1. Sub (4,2): (-8, 1)
2. Rule: (-1, -8)
3. Add (4,2): B'(3, -6)
Vertex C(-3, -5)
1. Sub (4,2): (-7, -7)
2. Rule: (7, -7)
3. Add (4,2): C'(11, -5)
Exit Ticket Mastery Check
Rotate A(-1, 3) about (1, 1) by 180°:
Sub: (-2, 2) → Rule: (2, -2) → Add: A'(3, -1)
Rotate B(0, 7) about (1, 1) by 180°:
Sub: (-1, 6) → Rule: (1, -6) → Add: B'(2, -5)
Justification Key:
"Yes, segment AB is parallel to A'B'. A 180° rotation maps a line onto a parallel line if the center of rotation is not on the original line."
Pivot Points Exit Ticket Exit Ticket
Mastery Check • Rotations Off-Origin
Name:
Target: Pivot around (h, k)
"Segment AB has endpoints A(-1, 3) and B(0, 7). Rotate segment AB by 180° about the point (1, 1). Solve for the image A' and B'."
Vertex A' Calculation
Step 1: Sub • Step 2: Rot
A':
Vertex B' Calculation
Step 1: Sub • Step 2: Rot
B':
Reflective Justification
"Is segment A'B' parallel to the original segment AB? How do you know using the properties of a 180° rotation?"
