## Exploring and Creating Mathematics Part 3: Investigating Powers using Pascal’s Triangle

Before reading this post, you might want to see the other two posts in the series:

Post 1 – Creating and Exploring Mathematics: How many different directions can we take a topic in 15 minutes?

Post 2 – Discovering and Creating Mathematics Part 2: Does investigating problems make it easier to pose interesting problems?

Whenever we get a chance as mathematics educators, I think it’s great if we can explore and discover our own mathematics. Of course it’s almost always been created before but I don’t see how can we ask students to investigate and explore if we’re not prepared to do so?

So today I spent a few hours investigating powers using a binomial theorem/Pascal’s triangle approach. Would love to see how other maths teachers would investigate this given the chance. Here’s some interesting stuff I discovered today:

1. The sum of row n of Pascal’s triangle is equal to 2^n.

2. Odd powers of 4 always have final digit ending in 4; Even powers of 4 always have final digit ending in 6.

3. The first four powers of 11 follow exactly the same digits in Pascal’s triangle. E.g.

11, 121, 1331, 14641, …

After that, the pattern doesn’t work. This is a great example to show why proof and reasoning are important in mathematics.

4. The sum of the digits of powers of 11 are always even.

As a side note, one thing that occurred to me is that mathematical writing is generally quite boring. Don’t get me wrong – the mathematics itself is interesting – its just the writing in between. Hence, at certain points I’ve highlighted the “aha” moments by directly writing it. I know that this isn’t convention in mathematics articles but I feel like it would be a great thing to promote with students. Especially during journaling.