Mathematics K–6 · Years 5–6
Slide, flip, turn: transformations with tracing paper and tessellations
Measurement and space
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The idea
A translation, a reflection or a rotation moves a shape without changing its size or angles, and shapes tessellate when the angles meeting at each point add to 360 degrees.
What you need
- tracing paper, a pin and a pencil
- card templates: an equilateral triangle, a square, a regular pentagon, a regular hexagon, a regular octagon and a scalene triangle
- a protractor
- grid paper with axes
How to do it
- Trace the scalene triangle. Slide the tracing 4 squares right and 2 up and draw the image. Measure the sides and angles of both and compare.
- Fold the tracing along a line and draw the reflected image. Check that corresponding points are the same distance from the fold.
- Pin the tracing at one vertex and turn it a quarter turn, then a half turn, drawing each image. Measure a side length after each turn.
- Measure one interior angle of each regular template. Try to tile the desk with copies of each template, meeting at a point with no gaps. Record which ones tile.
- Make a tessellation with the scalene triangle using half turns about the midpoint of a side, and continue it across the page.
What you should see
After a slide, a flip or a turn the triangle has the same side lengths and angles as before; only its position and orientation change. Interior angles are 60 degrees (triangle), 90 (square), 108 (pentagon), 120 (hexagon) and 135 (octagon). Triangles, squares and hexagons tile alone because 6 times 60, 4 times 90 and 3 times 120 each make 360; pentagons leave a 36 degree gap (3 times 108 is 324) and octagons leave 90 degrees (2 times 135 is 270), so they do not tile alone. Any triangle tessellates by half turns because its three angles add to 180 degrees and each point of the tiling gathers every angle twice, 2 times 180 = 360.
What changes
This activity lists no variables to change, measure and keep the same.
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- A shape gets smaller when it is rotated because it looks different.
- Any regular shape can tile a floor on its own.
Safety card
Hazards
- pin
Controls
- an adult places the pin; use a cork mat
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package is not needed; ordinary classroom supervision under the school's usual risk management.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K-10 Syllabus (2022), NSW Education Standards AuthorityMA3-2DS-01MAO-WM-01
- Australian Curriculum v9AC9M5SP03AC9M6SP03
Sources
The pages the author read to write this activity.