Stacking oranges - mathematics activities

Description

This picture shows oranges stacked in the shape of a pyramid with a rectangular base. There are many numerical and spatial patterns in the display. You need to imagine each face of the stack, and of each of its layers, in order to conceptualise the geometry and the inherent numerical patterns of the stack.

Activities

1. Find a general rule for how many oranges there are in each layer in the photograph. How many oranges form the stack in the photograph?
2. Consider stacks that have a different number of oranges in the bottom layer. For example, the bottom layer (or 'base') might be 4 x 6; 5 x 7; 6 x 9 etc. Choose a specific bottom layer (eg 5 x 7) and predict how many levels the stack will have or how high the stack will be. How many oranges in total? Explore at least four different bottom layers.
3. What if the base of the stack is an equilateral triangle? How do the numbers and rules from the first two activities change?
4. Explore and analyse, perhaps using physical materials, different types of stacks with bases that are not rectangular or triangular but some other polygonal shape. You may notice further connections between the patterns. Being systematic is a good strategy in such explorations.
5. Try to imagine, then draw, the stack in the photograph from above, that is in an aerial or bird's-eye view. Using different colours for different layers would be helpful. What are some of the patterns in other non-rectangular stacks, when viewed from above? Using a digital camera to photograph actual stacks you make would be helpful, as might a computer with good graphics software.
6. Find out what you can about the famous astronomer and mathematician Johannes Kepler who, in 1611, proposed the 'Kepler conjecture'. What is this conjecture (or prediction)? Has it ever been proved (or resolved)? You need to rely on only reputable sources here!
7. Stacking is an example of 'packing'. In industrial warehouses, packing efficiently is important for the manufacturer who wishes to minimise wasted space in the building and of the size of the 'footprint' of the various stacks, which, in turn, depends on the shape of the objects in the stack. Find out what you can about pallet stacking and large-scale storage stacks.