This week, as I relaxed on my veranda soaking up the unusual Melbourne sunshine, I spotted something quite extraordinary. Meet this little fella (although 'little' might not be the best word)! This impressively sized cicada, enjoying the midday sun alongside me, stretched a solid 5-6 cm...and that's not counting his wingspan. A surprising visitor, to add a touch of wild intrigue to my sunny afternoon.
But why a surprise? Well, because I wasn't expecting to see one of these guys for another 4-5 years. This is because cicadas appearance correlates with quite a unique mathematical pattern. One that is directly and purposefully linked to prime numbers.
Down here on the South-Eastern coast of Australia, cicadas usually make a big entrance, with the last one being in the summer of 2020/2021. In fact, they spend most of their life underground in their larval (nymph) stage, periodically emerging in large numbers every prime-numbered year. For most Australian species, this cycle lasts around 7 years, but for some North American species, the cycle most commonly ranges between 13 and 17 years. Why prime numbered years though? It seems odd (pun intended) that primes should be of any interest, or usefulness at all to such a simple creature's breeding cycle, right? Wrong!
The use of prime numbers here is actually a fascinating, and ingenious evolutionary strategy. You see, by emerging at intervals of 7, 13, or 17 years (prime numbers not being divisible by smaller numbers), cicadas cleverly evade synchronisation with the life cycles of their predators. These predators often have shorter, non-prime life cycles. If cicadas emerged at regular intervals, predators could easily adapt and feast during these emergence years. But with prime-numbered cycles, cicadas outsmart these predators, ensuring their survival and continued breeding success. It's a beautiful dance of numbers, survival, and evolution.
This phenomenon is just one example of how prime numbers play a fascinating role in our world. Yet, these numbers often don't get the attention they deserve in mathematics classes. To make this intriguing connection more meaningful to students, I thought up a playful classroom game called 'Are You Smarter Than A Cicada?'. Here's how it works:
A deck of ‘Game Cards’ with numbers ranging from 1 to 17 (to represent years).
A 'Cicada Emergence Game Board' with numbers 1-17 marked in a circular 'clock' formation.
Tokens or small figurines to represent cicadas and predators. (*Cicada predators include birds, small mammals, insects (e.g. wasps), spiders and amphibians).
Students play in pairs or groups of 3.
Each player begins with their own game board
Each player chooses a token to be their cicada and places it at number 1 on the cicada emergence chart.
Each player selects a predator and assigns a cycle (e.g. birds may be 2 years, insects 3 years, and small mammals 4 years).
Players then place the selected predator on their respective starting points.
Cicada Emergence Race Game Board
Players draw a card to determine the number of years that pass in a turn.
Move the cicada token forward by the drawn number on the chart using modular arithmetic (e.g., if you draw a 10 and the cicada is on 7, it moves to (7+10) year 17 on the board, which means it has emerged).
Predators move automatically every 2, 3, or 4 spaces each turn, simulating their own cycles.
The goal is to complete as many cycles of 17 years without landing on the same number as a predator.
If a cicada lands on the same number as a predator, it gets 'eaten' and has to start over.
The cicada that completes the most cycles before getting 'eaten' wins the game.
Suggestions for learning:
Begin with one predator and see how adding more predators affects game play.
Before beginning the game ask students to make predictions about how long it will take for the cicada to get eaten.
Use game boards with different emerging cycles (7 or 13).
Use game boards with all three emerging cycles and determine which cycle poses the greatest likelihood of the cicada getting eaten.
Students will learn how the prime number 17 ensures cicadas rarely meet predators.
They'll practice adding numbers and applying modular arithmetic in a fun way.
You've probably noticed that this game is designed around modular arithmetic, a number concept not explicitly introduced to school-aged students, but one that they use regularly when reading time or using calendars. Often compared to 'clock maths', modular arithmetic operates on a principle where numbers loop around a circular number line, resetting after reaching a specified limit. For instance, in a 12-hour clock, adding 4 hours to 11 o’clock brings us back to 3 o’clock.
This 'wrap-around' effect is modular arithmetic at work. It's a fundamental concept used in various real-world applications, such as determining the day-of-the-week, computer science algorithms, cryptography, music theory and as we've seen here, some natural phenomena like cicada emergences.
But the cicada's life cycle is much more than a peculiar quirk of nature. It's a vivid testament to how mathematics is intricately woven into the fabric of the natural world. These little creatures, from our own backyards, are just one of the countless examples showcasing nature's mathematical genius. From the spirals of a seashell to the constellations in the night sky, the universe speaks in numbers and patterns. Witnessing a cicada's emergence is like getting a glimpse into this grand design, a design that mirrors the very principles we explore in our maths classrooms. It's a reminder that sometimes, the greatest marvels of mathematics are not found in textbooks, but in the living, breathing world around us.