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Packing jelly beans - mathematics activities

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Credits:

With permission of Richard Phillips. Photograph by Richard Phillips.

TLF ID:

R10269

Source:

Australian Association of Mathematics Teachers, http://www.aamt.edu.au/

Digital resource description:

The photograph shows a box containing jelly beans in an array. Looking closely, we see each compartment in this box contains six jelly beans of the same colour. They are arranged in two layers of three beans. There is a different flavour in each compartment. Teachers are encouraged to scan all the ideas suggested here as relevant to the various year level groupings, as there is clearly an increasing mathematical complexity in the activities.

Educational value:

This photograph is a vehicle for practising acute visual perception in discerning the number of jelly beans in each compartment and the number of flavours, as indicated by colour, that are in the box.
There are many flavours of jelly beans. At least one company claimed in 2008 to have 49 official flavours, with more being developed. Finding out how many flavours and how many jelly beans there are in the box uses multiplication (or repeated addition) strategies, where these are familiar to students.
For younger or less mathematically sophisticated students, this activity could help establish the concept of multiplication as repeated addition, by counting the array in different ways: cell by cell (6 + 6 + 6 + …); by columns (18 + 18 + 18 + …) or even by rows (72 + 72 + 72 + …).
The total number of jelly beans in a package depends on the number of cells, and the number of jelly beans in each cell. The compartments in the package form an array; within each compartment the jelly beans are also arranged in an array. Investigation of the situation in the photograph could be used to introduce the notion of multiplication in arrays, and to provide a context for multiplication practice.
This photograph is also a platform for a discussion that leads to an investigation of composite and prime numbers, relating to the shape of the box and its compartments. For example, a rectangular box to hold different numbers of flavours of jelly beans could be made as a box with a single line of compartments.
Other numbers of jelly bean flavours could be arranged in two ways. Still others can be made in more than two ways. The possible arrangements for different numbers of jelly bean flavours are dependent on whether the number of flavours is prime or non-prime. A prime number of flavours will only be able to be arranged in one way, if each compartment is to be filled. For non-prime numbers of flavours the number of possible arrangements will be the number of distinct sets of factors of that number. This could lead to designing different boxes for non-prime number of flavours.
What other shapes of box are possible? Simple multiplication may not be useful when dealing with situations that do not involve rectangular arrays (eg equilateral triangle shaped boxes with similar triangular compartments involves 'triangular' numbers).
Boxes made as different regular polygonal shapes make use of other figurate numbers. These boxes demand other patterning techniques to be used in order to find the total numbers in compartments or in the box as a whole.

Keywords:

Arrays; Counting techniques; Natural numbers; Prime numbers; Regular polygons; Tables (Data)

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