BTT Mathematics / Primary Mathematics Learning Hub / Paper Folding and Symmetry
A paper fold is a geometric reflection. When one point is folded onto another, the crease lies exactly halfway between them and is perpendicular to the segment joining them. When one edge is folded onto a parallel opposite edge, the crease is the midline. When one ray is folded onto another, the crease follows an angle bisector.
This guide turns physical folding into coordinate and symmetry reasoning. A point and its folded image are the same perpendicular distance from the crease. Unfolding reverses the reflection and can create symmetric copies of holes, marks or cut patterns.
This is Primary Mathematics enrichment. It extends Geometry, Angles, Symmetry and Coordinates, Tessellations, Tiling and Spatial Construction Puzzles and Loci, Distance Constraints and Grid Geometry.
Formal origami mathematics is beyond the scope here. The focus is reflection, midlines, bisectors and exact symmetry. Use the MOE Primary curriculum page and the learner’s school programme for required content.
Fold as reflection · Crease as perpendicular bisector · Edge-to-edge folds · Angle folds · Successive folds · Unfolding holes and marks · 24 questions · Worked answers
1. Reflection preserves distance and shape
A reflection flips a point or shape across a line. The image has the same size and shape as the original, but its orientation may reverse.
Worked example A: Reflect across the y-axis
Point (2,3) reflects to (−2,3). The y-coordinate stays fixed while the x-coordinate changes sign.
Worked example B: Reflect across the x-axis
Point (4,−5) reflects to (4,5). The x-coordinate stays fixed while the y-coordinate changes sign.
Worked example C: Reflect across x=4
Point (1,5) is three units left of x=4, so its image is three units right: (7,5).
Worked example D: Reflect across y=2
Point (3,−1) is three units below y=2, so its image is three units above: (3,5).
2. Folding one point onto another creates a perpendicular bisector
If point A is folded exactly onto point B, every point of the crease is equidistant from A and B. Therefore the crease is the perpendicular bisector of segment AB.
Worked example E: Horizontal pair
A=(0,0), B=(6,0). Their midpoint is (3,0), and AB is horizontal. The crease is the vertical line x=3.
Worked example F: Vertical pair
A=(2,1), B=(2,7). Midpoint=(2,4), and AB is vertical. The crease is the horizontal line y=4.
Worked example G: Diagonal pair
A=(0,0), B=(4,4). Midpoint=(2,2). Segment AB has slope 1, so a perpendicular crease has slope −1 and passes through (2,2): y=−x+4.
The crease does not pass through the original point unless special geometry forces it
A point being moved lies away from the crease except when it already lies on the crease and therefore would not move under reflection.
3. Folding one parallel edge onto another creates a midline
For a rectangle whose left edge is x=0 and right edge is x=10, folding left onto right creates crease x=5. Every point of one edge travels across the same perpendicular distance.
Worked example H: Vertical fold
A 10-unit-wide rectangle occupies 0≤x≤10. The vertical edge-to-edge crease is x=5.
Worked example I: Horizontal fold
An 8-unit-high rectangle occupies 0≤y≤8. Folding bottom edge to top edge gives y=4.
Worked example J: Repeated halving
A strip 16 cm long is folded exactly in half twice in the same direction. Its length becomes 8 cm after the first fold and 4 cm after the second.
Layer count under full half-folds
If the whole sheet is folded exactly in half and the layers remain aligned, each full fold doubles the number of layers: one fold gives 2 layers, two gives 4, three gives 8.
Worked example K: Three full half-folds
Starting from one layer, after three such folds there are 8 layers.
4. Folding one ray onto another uses an angle bisector
When two rays meet at a vertex and one is folded exactly onto the other, the crease through the vertex lies along an angle bisector because corresponding rays make equal angles with the fold line.
Worked example L: 60° angle
Folding one side of a 60° angle onto the other places the crease at 30° from each side.
Worked example M: 90° angle
Folding one perpendicular ray onto the other gives a crease at 45° to each.
Worked example N: x-axis to y-axis in the first quadrant
The first-quadrant crease is the line y=x. Reflection across y=x swaps coordinates: (a,b) maps to (b,a).
Worked example O: Coordinate swap
Point (2,5) reflected across y=x becomes (5,2).
5. Successive folds compose reflections
After one fold, use the image as the starting point for the next fold. The order matters.
Worked example P: y-axis then x-axis
Start at (3,2). Reflect across y-axis to (−3,2), then across x-axis to (−3,−2). Two perpendicular reflections through the origin have the same final effect as a 180° rotation.
Worked example Q: x=1 then x=4
Start at x=0. Reflect across x=1 to x=2. Then reflect across x=4 to x=6. Two reflections in parallel vertical lines produce a translation.
Worked example R: Reflection across y=x twice
Any point returns to itself after the same reflection is applied twice. For (2,5): (2,5)→(5,2)→(2,5).
Crease lines can themselves be reflected
When tracking a multi-fold diagram, a previously drawn mark on the paper may move under later folds. Distinguish a physical crease already made in the paper from a coordinate guide drawn on the page.
6. Unfolding mirrors holes and marks across each fold line
If a single hole is punched through all aligned layers after one fold, unfolding produces a reflected pair unless the punch lies exactly on the fold line.
Worked example S: One vertical fold
A sheet is folded along x=5. A hole is punched at (3,2) in the folded coordinate picture, away from the crease. Unfolding across x=5 creates the reflected point (7,2). The two holes are (3,2) and (7,2).
Worked example T: Two perpendicular folds
Fold across x=0 and y=0 so a punch is made at (2,3) away from both creases. Full unfolding gives four symmetric points: (2,3),(−2,3),(2,−3),(−2,−3).
Punch on one crease
If after two perpendicular folds the punch lies on one crease but not the other, reflection across the crease containing the punch creates no new position, while the other reflection does. The full pattern has two distinct points.
Punch at the intersection of both creases
A punch exactly at the intersection remains fixed under both reflections, so it corresponds to one geometric position even though several layers may be pierced.
7. Practice: 24 original questions
Questions 1–8: Reflection and crease lines
1. Reflect (2,3) across the y-axis.
2. Reflect (4,−5) across the x-axis.
3. Reflect (1,5) across x=4.
4. Reflect (3,−1) across y=2.
5. Fold A=(0,0) onto B=(6,0). Find the crease line.
6. Fold A=(2,1) onto B=(2,7). Find the crease line.
7. Fold A=(0,0) onto B=(4,4). Give the crease equation.
8. Explain why the crease for folding one point onto another is a perpendicular bisector.
Questions 9–16: Edges, angles and repeated folds
9. A rectangle spans 0≤x≤10. Fold the left edge onto the right edge. Find the crease.
10. A rectangle spans 0≤y≤8. Fold the bottom edge onto the top edge. Find the crease.
11. A 16 cm strip is folded exactly in half twice. What is the final folded length?
12. How many layers result after three complete half-folds?
13. A 60° angle is folded so one side lies on the other. What angle does the crease make with each side?
14. A 90° angle is folded side-to-side. What angle does the crease make with each side?
15. Reflect (2,5) across y=x.
16. Reflect (−3,4) across y=x.
Questions 17–24: Successive folds and unfolding patterns
17. Reflect (3,2) across the y-axis, then the x-axis. Find the final point.
18. Starting at x=0, reflect across x=1 and then x=4. Find the final x-coordinate.
19. Reflect (2,5) across y=x twice. Find the final point.
20. A sheet is folded along x=5. A hole at (3,2) unfolds to what mirror point?
21. Under folds across x=0 and y=0, a punch at (2,3) unfolds to how many distinct symmetric positions?
22. List those four positions from Question 21.
23. After two perpendicular folds, a punch lies on one crease but not the other. How many distinct geometric positions appear after full unfolding?
24. A punch lies exactly at the intersection of two perpendicular crease lines. How many distinct geometric positions appear after full unfolding?
8. Worked answers
Answers 1–8
1. (−2,3).
2. (4,5).
3. (7,5). It is three units on the opposite side of x=4.
4. (3,5).
5. x=3.
6. y=4.
7. y=−x+4. The crease passes through midpoint (2,2) and is perpendicular to slope-one segment AB.
8. Every point on the crease remains fixed during the fold and must be equally distant from the original point and its image. The equidistant locus of two points is their perpendicular bisector.
Answers 9–16
9. x=5.
10. y=4.
11. 4 cm. 16→8→4.
12. 8 layers. 1→2→4→8.
13. 30°.
14. 45°.
15. (5,2).
16. (4,−3).
Answers 17–24
17. (−3,−2).
18. x=6. 0 reflects to 2 about x=1, then 2 reflects to 6 about x=4.
19. (2,5). Applying the same reflection twice returns the original point.
20. (7,2).
21. 4 distinct positions.
22. (2,3),(−2,3),(2,−3),(−2,−3).
23. 2 positions. Reflection across the crease containing the punch does not move it; the perpendicular fold creates the mirror partner.
24. 1 position. The intersection point is fixed by both reflections.
9. Teaching and transfer
If a learner predicts a fold by eye, mark one point and measure its perpendicular distance to the proposed crease. Its image must land the same distance on the other side.
When midpoint alone is used
A crease folding A onto B must pass through the midpoint and be perpendicular to AB. The midpoint condition alone allows infinitely many lines.
When hole-punch patterns are guessed
Unfold one crease at a time. Reflect every current hole position across that crease, keeping points on the crease fixed. This turns a visual puzzle into repeated exact transformations.
When successive folds are applied in the wrong order
Record each intermediate image. Reflection composition is order-sensitive unless special symmetry makes the results coincide.
When layer count and position count are confused
A punch through several aligned layers can still unfold to fewer distinct positions if it lies on a crease. Count geometric positions separately from physical layers pierced.
Connect to loci
The crease that maps one point to another is exactly the equidistant locus of those two points, linking paper folding directly to Loci, Distance Constraints and Grid Geometry.
Continue through this enrichment collection
For lattice-area counting, use Pick’s Theorem, Lattice Points and Coordinate Area. For remainder-based gcd reduction, use Euclidean Algorithm, GCD and Remainder Reduction. For fraction neighbours and mediants, use Farey Sequences, Mediants and Fraction Neighbours.
Return to the BTT Primary Mathematics Learning Hub.
Original enrichment guide with 24 original practice questions and separate worked answers. Folds are modelled as ideal geometric reflections; real paper thickness and deformation are ignored.