Crease Proofs Lesson Plan Teacher Resource
Crease Proofs
Bridging Origami and Rigorous Geometric Proofs
Grade Level Grades 8–12
Duration
60–90 Minutes
Materials
Origami sheets, rulers, pencils, worksheet
Focus
Similarity, Coordinate Proofs
Learning Objectives
Identify and map reflections, perpendicular bisectors, and congruent angles generated by folding paper.Prove mathematically that folding a corner to a midpoint creates similar right triangles (Haga’s First Theorem).Formulate formal two-column or paragraph proofs translating physical operations into geometric axioms.
Standards Alignment
CCSS.MATH.CONTENT.HSG.CO.A.5: Given a geometric figure and a transformation, describe the rotations and reflections that carry it onto itself.
CCSS.MATH.CONTENT.HSG.CO.C.9: Prove theorems about lines and angles.
CCSS.MATH.CONTENT.HSG.SRT.B.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Suggested Lesson Pacing
Phase Pacing Key Activities & Instructional Focus 1. Spark 10 Min Introduce Origami as an axiomatic math system. Quick fold to bisect angles. 2. Fold 15 Min Execute Haga's folding step: corner of square to the top edge's midpoint. 3. Map 15 Min Measure sides, identify overlapping triangles, notice the 3-4-5 ratio. 4. Prove 25 Min Draft formal geometric proofs establishing AA similarity and solving side lengths. 5. Debrief 10 Min Discuss the power of origami over compass. Wrap up with exit tickets.
Curriculum Development: Hands-on Mathematics Page 1 of 2
Teacher Instruction & Facilitation Guide
Step-by-Step Lesson Execution Plan
Haga's Theorem
1
The Spark: Folding vs. Drawing (10 mins)
Ask students: "Can you divide a segment into exactly 3 equal parts without a ruler?" Explain that while ancient Greek geometry (compass and straightedge) cannot easily trisect angles or find exactly 1/3 of a segment without complex constructions, origami can do it in two folds. Introduce the Huzita-Hatori Axioms as the baseline mathematical rules of paper.
2
The Fold: Corner to Midpoint (15 mins)
Distribute square origami sheets. Guide students to first fold the paper in half vertically to find the midpoint of the top edge , crease lightly, and unfold. Then, fold the bottom-left corner precisely to that top midpoint. Crease firmly. Trace the crease lines with a pencil and label vertices following the Activity Sheet .
3
The Discovery: Triangles & Similarity (15 mins)
Folds create three distinct, non-overlapping right triangles on the paper. Ask students to identify them. Let them measure the sides of the triangles in centimeters.Critical Observation: The triangles are similar, and they all have side lengths in a perfect 3:4:5 ratio ! Haga's Theorem proves this algebraically.
4
The Proof: Analytical Rigor (25 mins)
Students proceed to write two formal proofs on their activity sheets:
Proof A (Similarity): Use Angle-Angle (AA) Similarity. Since the corner of the folded sheet remains a 90° angle, the adjacent angles on the line of the fold must sum to 90° with the angle of the adjacent triangle, proving similarity.Proof B (Side Lengths): Set the side of the square to \( 1 \). If the top midpoint is at \( 0.5 \), use the Pythagorean Theorem (\(a^2 + b^2 = c^2\)) to solve for the exact dimensions of the remaining triangles to confirm the 3-4-5 ratio.
Classroom Facilitation Tips
Common Misconceptions
Students struggle to see that the folded segment is congruent to the original unfolded segment. Remind them that folding is an isometry (preserves length and angle measure).
Pacing & Differentiation
For advanced students, ask them to find where to fold the corner to divide the top edge into custom ratios (e.g., 1:2 or 1:4). For struggling students, use pre-drawn grids on origami paper to aid measurement.
Curriculum Development: Hands-on Mathematics Page 2 of 2
Crease Proofs Slide Deck Geometry in Action Unit: Spatial Reasoning & Proofs
Crease Proofs
Translating physical paper-folding transformations into rigorous mathematical arguments.
Origami Geometry
Slide 1 of 8
The Spark A Classic Math Mystery
The Trisection Problem
Ancient Greek mathematicians proved you cannot trisect an arbitrary angle or split a line into perfect thirds using only a standard compass and unmarked straightedge.
But with a single square of paper and two folds, origami achieves this effortlessly. How?
The Origami Axioms
Folding paper defines points, lines, reflections, and intersections as a rigorous, axiomatic system.
Can paper beat ancient Greek tools? Slide 2 of 8
Step-by-Step Haga's First Theorem
The Midpoint Fold
1
Fold your square in half vertically, pinch the top edge to find the exact midpoint , and unfold.
2
Bring the bottom-left corner up and place it directly on that top midpoint.
3
Make a sharp, flat crease across the bottom left of your paper. Unfold and inspect the lines.
Midpoint (M) Corner A
Notice the three distinct right triangles that form. Slide 3 of 8
Transformation Math Reflections & Symmetry
Rigid Motions
Every fold of a paper represents a reflection over the crease line.
Because reflections are rigid motions , they are isometries: they preserve distances and angle measures.
Crease Line Theorem:
The crease line is the exact perpendicular bisector of the segment connecting the original corner point to its new landed position on the top edge.
Angle Preservation
The 90° corner remains exactly 90° after being folded. It just rests in a new position!
Distance Preservation
The lengths of any lines drawn on the folded flap are congruent to their corresponding unfolded parts.
Folding is not drawing; it is a mechanical isometry. Slide 4 of 8
Discovery The Secret Triangles
Three Right Triangles
The single corner fold partitions the rest of the square sheet into three distinct right triangles on the outer edges:
Triangle I: Left side of the top edgeTriangle II: Right side of the top edge
Crease Proofs Activity Sheet Crease Proofs: Folding Midpoints
Discovering Haga's First Theorem through Physical Isometries
Geometry • Secondary Education
Student Name
Date
Class Period
Part 1: Fold & Measure
Follow the instructions below with your square sheet of paper. Trace all resulting crease lines with a pencil, then label the vertices and triangles according to the diagram.
Step 1: Fold the paper in half vertically to find the midpoint of the top edge. Pinch the top edge, then unfold.
Step 2: Fold the bottom-left corner precisely to the top midpoint. Crease firmly and lay flat.
Step 3: Identify the three right triangles created on the exterior border of the folded paper (Triangles I, II, and III).
M (Midpoint) Tri I Tri II Tri III
Part 2: Empirical Measurements
Using a metric ruler, measure the sides of each triangle in centimeters to the nearest millimeter. Record your data and calculate the ratio.
Triangle Short Leg (a) Long Leg (b) Hypotenuse (c) Ratio (a / b) Triangle I (Left) Triangle II (Right) Triangle III (Bottom)
Part 3: Formulating Hypotheses
1. What relationship do you notice between the side ratios of Triangles I, II, and III? How do they compare to standard Pythagorean Triples?
2. If folding paper is a rigid motion (isometry), what properties of lines, segments, and angles are preserved when the paper corner is folded over?
Origami Mathematics Unit Page 1 of 2
Crease Proofs: Formalizing Origami
Rigorous Proof Scaffold & Algebraic Proofs
Part 4: Proofs
Task A: Prove Triangle Similarity
Write a formal two-column or paragraph proof to establish that Triangle I (Left) is similar to Triangle II (Right) using Angle-Angle (AA) similarity.
Task B: Solve the Side Lengths Algebraically
Let the side length of the square paper be \( 1 \). If the top midpoint is at coordinate \( (1/2, 1) \) on a Cartesian plane, set the vertical leg of Triangle I as \( x \). Write and solve the Pythagorean equation to find the exact side lengths of Triangle I and show that they form a \( 3:4:5 \) ratio.
Crease Proofs Rubric Assessment Guide
Crease Proofs Assessment Rubric
Evaluation Matrix for Origami Folding & Mathematical Proofs
Total Weight 16 Points
4 - Exemplary
Exceeds standards, fully complete
3 - Proficient
Meets standard, minor minor flaws
2 - Developing
Approaching standard, minor gaps
1 - Beginning
Incomplete, needs extra support
Criterion 4 - Exemplary 3 - Proficient 2 - Developing 1 - Beginning Folding Accuracy & Execution Creases are highly precise; alignment matches the top edge midpoint exactly. Creases are precise; alignment is within 1–2mm of top midpoint. Folds are slightly misaligned (>3mm offset) or show minor extra creases. Folds are incorrect, incomplete, or paper is damaged. Data & Ratio Mapping All side lengths correctly measured. Ratios solved and 3-4-5 connection identified. Measurements are mostly accurate. Ratios solved with minor arithmetic error. Some measurements missing. Ratios do not clearly identify the 3-4-5 triple. Data table is incomplete or contains widespread measurement errors. Similarity Proof (AA) Rigorous logical sequence proving AA similarity with precise math notation. Proves similarity but misses 1-2 minor justifications or steps. Recognizes similarity but proof is disorganized or lacks geometric postulates. Proof is absent or contains major mathematical errors. Algebraic Precision Pythagorean equation is correctly set up, expanded, and solved (\(x = 3/8\)). Correct algebraic setup but includes a minor computational error. Incomplete setup. Has difficulty setting up algebraic variables. Algebraic proof is missing, incomplete, or contains major errors.
Total Score
/ 16
Grade Equivalent
Teacher Feedback & Learning Next Steps
Origami Mathematics Unit Page 1 of 1
