A hands-on, inquiry-based geometry lesson focusing on translations, reflections, rotations, and dilations. Students use physical Patty Paper (tracing paper) to understand rigid and non-rigid transformations, leading to a mastery of compound transformations on the coordinate grid.
A compound transformation performs more than one operation in sequence. The output coordinates of the first step become the input coordinates of the second step!
Applying a reflection then rotating is usually not the same as rotating then reflecting. Always evaluate steps in the exact order requested!
Think & Write
Explain why dilating a shape by \(k = 1.5\) and then translating it creates an image that is similar but not congruent to the pre-image.
Teacher Check Stamp Area
SHAPE SHIFTERS GUIDED NOTES PAGE 2 OF 2
Output: A''', B''', C''', D'''
📌 Navigator Checklist & Formatting:
For each coordinate burn, draw the final shape and label the vertices clearly (e.g., \(A''\) or \(C'''\)) on your separate large Cartesian Graphing Sheet. Attach your used physical Patty Paper showing your fold creases and tracing marks directly to the back of this packet!
1. Reflect on the Patty Paper process: Which of the three burns (Translation, Reflection, Rotation) did you find easiest to model physically? Which was hardest? Explain why.
2. algebraic synthesis: Combine the algebraic rules from Burn 1, 2, and 3 into a single compound coordinate mapping formula that sends original point \(A(x,y)\) directly to final point \(A'''(x''', y''')\). Show your algebraic step-by-step substitution work.
3. non-commutative logic: Suppose you decided to rotate the probe \(180^\circ\) BEFORE doing the reflection and translation. Would your spaceship land in the same final position? Use coordinates or geometry vocabulary to prove your answer.
Project Grading Rubric
Criteria
Exemplary (4 pts)
Proficient (3 pts)
Developing (2 pts)
Beginning (1 pt)
Graphing & Coordinates
All points and shapes plotted, connected, and labeled perfectly across all steps.
Plotting is complete; minor coordinate error in labeling or drawing.
Plotted coordinates show major errors, or steps are missing labels.
Plotted shapes do not match coordinate descriptions; incomplete.
Algebraic Rules
All mapping functions written correctly using correct prime notations.
Rules are mostly correct with slight arithmetic or notation slips.
Multiple incorrect transformations or notation mistakes.
Rules are missing, incomplete, or fundamentally incorrect.
Tracing Execution
Patty Paper used extensively, displaying precise crease alignment and traces.
Patty Paper attached but trace lines or folds lack precision.
Tracing paper attached but shows little connection to plotted shapes.
No tracing paper attached; no evidence of hands-on strategy.
Mathematical Reasoning
Inquiry answers are thorough, logical, and provide deep proof-based geometry insight.
Answers are complete and correct but lack complete vocabulary depth.
Answers are partially complete; contains logical flaws or misconceptions.
Incomplete reflection; answers show basic failure of understanding.
*Teacher Sign-Off is required before moving to the next station. Make sure Patty Paper is placed properly and aligned before drawing the coordinate points!
SIGN-OFF STAMP
SHAPE SHIFTERS MOVEMENT CARDS PAGE 2 OF 2
E'(_____, _____) → E''(_____, _____)
F'(_____, _____) → F''(_____, _____)
Section 4: Conceptual Reasoning
Provide mathematically sound explanations using geometry vocabulary.
9. Non-Commutative Proof
Suppose you translate a shape by \(\langle 2, 2 \rangle\), then rotate it \(90^\circ\) CCW. Will you get the same image if you rotate \(90^\circ\) CCW first, then translate by \(\langle 2, 2 \rangle\)? Explain why or why not using coordinates of point \((1, 0)\).
10. Rigid vs. Non-Rigid Isometry
A compound transformation is made of a translation, then a reflection, then a dilation of factor \(k = 1.2\). Is the final image congruent to the pre-image, similar, or neither? Support your answer using definitions.
8. The Shrinking Turn of \(\Delta DEF\): Step 1: D'(-2,2), E'(-4,2), F'(-2,4) | Final: D''(-1,1), E''(-2,1), F''(-1,2)
Section 4: Conceptual Reasoning Answers
9. Non-Commutative Proof:
Let point be \((1, 0)\).
- Translate first: \((1, 0) \rightarrow (1+2, 0+2) = (3, 2)\). Rotate \(90^\circ\) CCW: \((3, 2) \rightarrow (-2, 3)\).
- Rotate first: \((1, 0) \rightarrow (0, 1)\). Translate next: \((0, 1) \rightarrow (2, 3)\).
Since \((-2, 3) \neq (2, 3)\), the order of operations directly impacts the final position.
10. Rigid vs. Non-Rigid Isometry:
The image is similar. A dilation stretches the distance between points by \(k=1.2\), meaning side lengths change. Therefore, congruence is not preserved (isometry is broken). However, dilations preserve relative angles and proportionality, making the final image geometrically similar to the pre-image.
