Shape Shifters: Transformations and Tessellation

You need: Paper · Ruler · Coloured pencils · Scissors · Protractor

1. Draw a large triangle with a ruler. Make it any shape you like.
2. Colour each corner a different colour and cut the triangle out.
3. Tear off the three corners.
4. Place the three torn points together so they touch at one spot, edges side by side.
5. Check: do they make a straight line (180°)? Compare with triangles made by others.
6. Measure each angle of a new triangle with a protractor and add them. Explain any small differences.

How you'll know: You can show with torn corners, and explain in words, why every triangle's angles make a straight line.

Safety: Carry and use scissors carefully.

Go further: Use the same trick on a quadrilateral. What do its four corners make? Why?

You need: Tablet or camera · Clipboard and recording sheet · Tape measure · Protractor

1. Walk the school grounds in pairs within the agreed area.
2. Photograph or sketch at least five repeating patterns: pavers, bricks, fences, grates, leaves, bark.
3. For each, record the shape, how many meet at one corner, and a measurement of one tile.
4. Look for one natural pattern too (a leaf, a flax weave, a spider web, a shell).
5. Back in class, sort your patterns: regular (one shape), mixed (more than one shape), or not a tessellation.

How you'll know: Your sheet shows at least five patterns with shapes named and the corner count recorded.

Safety: Stay in the agreed area, away from car parks and roads. Wear a hat and sunscreen. Do not touch insects or spider webs.

Go further: Measure the angles at one corner of a paver pattern. Do they add to 360°?

You need: Grid paper · Ruler · Coloured pencils · Table template

1. Draw a quadrilateral, a pentagon, a hexagon and an octagon.
2. In each, pick one corner and draw straight lines to every other corner you can reach without crossing a side.
3. Count the triangles. Each has 180°.
4. Fill in the table: sides (n), triangles, angle sum.
5. Find the pattern and write the rule: angle sum = 180(n − 2)°.
6. Test the rule on a 12-sided shape. Then find each angle of a regular 12-gon.

How you'll know: Your table and rule predict the angle sum of a shape you have not drawn yet.

Go further: Walk around a chalk polygon on the court, turning at each corner. Why do the exterior angles of every polygon add to 360°?

You need: Card templates of regular triangles, squares, pentagons, hexagons and octagons (same side length) · Scissors · Glue · Large sheet of paper

1. Cut out at least six of each shape.
2. Try to fit each shape around one point with no gaps or overlaps.
3. Record how many fit and the size of each angle.
4. Check: does the total at the corner equal 360°?
5. Try mixing shapes (for example octagons and squares, or hexagons and triangles).
6. Glue down your best mixed tessellation and label the angles at one corner.

How you'll know: You can explain with numbers why triangles, squares and hexagons tessellate but pentagons and octagons do not on their own.

Safety: Carry and use scissors carefully.

Go further: Find a pentagon that does tessellate. (Hint: it does not have to be regular.) Check your corners add to 360°.

You need: Four-quadrant grid paper · Tracing paper · Small mirror · Ruler · Coloured pens

1. Draw a triangle with corners at A(1, 1), B(4, 1) and C(1, 3).
2. Translate it 3 left and 4 down. Record the new corners.
3. Reflect the original in the y-axis. Check with a mirror. What happens to each x-coordinate?
4. Rotate the original 90° clockwise about (0, 0) using tracing paper. Record the new corners.
5. Enlarge the original by scale factor 2 from (0, 0). Measure the new sides and compare the areas.

How you'll know: Your coordinate table shows a clear rule for each transformation, and your enlarged triangle has sides twice as long and an area four times as big.

Go further: Find a combination of two reflections that gives the same result as a 180° rotation about (0, 0).

You need: Printed images from Te Ara, museum collections and the gallery on this page · Tracing paper · Recording sheet

1. Choose one pattern: tukutuku, tāniko, siapo or ngatu, Islamic tilework, an Escher print, or honeycomb.
2. Read what the source says about who made it, its name and its meaning.
3. Lay tracing paper over the pattern. Find and mark a translation, a reflection line and a centre of rotation.
4. Record the symmetry: lines of symmetry and order of rotational symmetry.
5. Write two sentences about the geometry and one about what the pattern means to the people who made it.

How you'll know: You can point to each transformation in the pattern and explain it, and you can say who made it and what it means.

Go further: Tāniko weavers could not weave curves, so their designs use triangles, diamonds and steps. How does the way something is made change the geometry you can use?

You need: Card squares (about 6 cm) · Scissors · Sticky tape · A3 paper · Coloured pencils or felt pens

1. On the top side of the square, draw a curvy or zig-zag line and cut it out.
2. Slide (translate) the piece straight down to the bottom side and tape it on. Do not flip it.
3. Repeat with the left side, sliding the piece to the right side.
4. Trace the tile, slide it along, trace again. Fill the A3 sheet with no gaps.
5. Decorate so that every tile looks like your own original motif.

How you'll know: Your traced tiles fit with no gaps or overlaps across the whole sheet, and you can name the transformation you used.

Safety: Carry and use scissors carefully.

Go further: Make a second tile where the nibble is rotated 90° about a corner instead of translated. How does the pattern change?

You need: Card · Ruler · Protractor · Scissors · Glue or tape

1. Choose a prism or pyramid shape for your stand (a triangular prism makes a good sloping stand).
2. Sketch the net: all the faces laid flat and joined along edges.
3. Measure and draw the net accurately on card, with tabs for gluing.
4. Cut, score the fold lines lightly and fold.
5. Test: does it close with no gaps? Does it hold your design board?

How you'll know: Your net folds into a closed 3D shape that stands up and holds your board.

Safety: Carry and use scissors carefully. Teachers use craft knives for scoring if needed.

Go further: There are 11 different nets for a cube. How many can your group find? How do you know two are really different?