Lesson Plan
Happy Numbers are a fun investigation that lead to some interesting results. It is a great lesson when covering powers and indices and involves simple mental arithmetic.
The process is simple, take a number (two digit is best) such as 13, and square each digit (1 squared = 1, 3 squared = 9) add them together to form a new number (10) then square the digits again and add them again (1 squared + 0 squared) = 1.
If you keep going, every number will either land on 1, meaning it is happy, or end up in a loop of eight numbers that will never reach one (unhappy). It is so simple to check of a number is happy, and involves quite a bit of squaring practice. After a while students begin to remember the square numbers through repetition.
The lesson is more than just squaring numbers – students will spot patterns, make assumptions and guesses, and find quick strategies to get to the answers. All great skills to improve as mathematicians.
It leads into some more difficult projects, including powerful numbers and narcissistic numbers. I also love the numberphile video with Matt Parker, who eloquently creates a mapping system to show all of the happy numbers and how they link together.
My Experience
I’ve only done Happy numbers once, with a small Y4 high ability group. I taught them briefly about square numbers and then set out the happy number example.
We then worked together on another number and it took a while but they recognised we were in a loop. To accentuate the fact we were in a loop I pointed out a fact about one of the numbers in the recurring loop (ooh 37 was my first house number), so that when we ended back up on it again, they recall it more easily.
I then wonder aloud who would be the first person to find another happy number, and away we go, each student chose a two digit number and began working on it. Some of them would go for a while and assume their number was not happy, but I pushed them to keep going and often they would land on 37 andat that point tell me it was unhappy.
What is interesting is that after fifteen minutes, my students could tell me if a number was unhappy or happy much sooner, but struggled to explain why they knew. I had to really eek it out from them that once they stumbled on a pattern that they already have, they knew the outcome. Like I said, I think they knew that, but were not confident to explain it.
Mixed into this lesson is powerful numbers – which I have only done with older students. I find powerful numbers a gateway into prime decomposition. most students quite like the topic but once they’ve done a couple they get bored. This investigation gives them a focus, trying to find powerful numbers by checking the decomposition of prime factors, and it also has the hook that it is an unsolved maths problem – no one can prove that there are never three consecutive powerful numbers.
Powerful numbers sound more interesting than they are really. Basically if each prime factor can be squared and still be a factor then the number is powerful, so any power of 2 or 3 etc is powerful, but the easiest way to list them is to use decomposition. 22 x 32 x 52 for example is powerful, but 22 x 52 x 7 is not, as 7 appears only once in the factorisation.
Maths Beyond the Curriculum 