Enlargements and scale factors

problem
Hex

Age

11 to 14

Challenge level

Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.
problem
Conical bottle

Age

14 to 16

Challenge level

A right circular cone is filled with liquid to a depth of half its vertical height. The cone is inverted. How high up the vertical height of the cone will the liquid rise?
problem
Arrow arithmetic 1

Age

14 to 16

Challenge level

The first part of an investigation into how to represent numbers using geometric transformations that ultimately leads us to discover numbers not on the number line.
problem
Golden triangle

Age

16 to 18

Challenge level

Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.
problem
Matter of scale

Age

14 to 16

Challenge level

Can you prove Pythagoras' Theorem using enlargements and scale factors?
problem
Sierpinski triangle

Age

16 to 18

Challenge level

What is the total area of the triangles remaining in the nth stage of constructing a Sierpinski Triangle? Work out the dimension of this fractal.
problem
Squareflake

Age

16 to 18

Challenge level

A finite area inside and infinite skin! You can paint the interior of this fractal with a small tin of paint but you could never get enough paint to paint the edge.
problem
Flower show

Age

14 to 16

Challenge level

How long will it take six gardeners to dig six circular flower beds?
problem
Von Koch curve

Age

16 to 18

Challenge level

Make a poster using equilateral triangles with sides 27, 9, 3 and 1 units assembled as stage 3 of the Von Koch fractal. Investigate areas & lengths when you repeat a process infinitely often.
problem
The rescaled map

Age

14 to 16

Challenge level

We use statistics to give ourselves an informed view on a subject of interest. This problem explores how to scale countries on a map to represent characteristics other than land area.