Some of my students do the ICAS Mathematics papers with the University of New South Wales. They just like the challenge and it’s a choice for them if they want. Those that are good at Mathematics generally like to give it a go.
The following question in the Yr 7 and Yr 8 paper had the better students in my class stumped. As their teacher I was totally lost as well. Teegan went home and asked her sister and she came back with an explanation that we couldn’t follow so being the end of term and year we left it.
However when the holidays came around it was something I wanted to find out about so I took the problem to my friend and a past Math’s teacher to help me. Here was the problem.
35. 4! = 4x3x2x1
5! = 5x4x3x2x1
Jess wrote the expression 20! – 19! on the board.
Which of the following has the same value as this expression?
(A) 1!
(B) 20
(C) 19x19!
(D) 20x19!
When first looking at it I thought maybe (A). I was applying the only background knowledge I had which was taking 20! – 19! = and thinking of it as 20a – 19a =. However having peeked at the answers I knew this wasn’t correct. I was worried by the top piece of information but I couldn’t find any pattern and in the end chose to ignore it.
However now I have new information! The ! is a special symbol, it stands for factorial. I hadn't known that. So 4! stands for 4x3x2x1. 5! as above. 6! would be 6x5x4x3x2x1. Factorial is related to the word factor. So 6,5,4,3,2,1 are all the factors of 6!
I now had some helpful information but it took me awhile longer. So….
20! – 19! = 20 x (19 x 18……x1) – 19x(18x17….x1)
= 20 x 19! – 19!
= 19 x 19! which is (C)
Factorial numbers get bigger quickly. 20! is 20 x 19! so if we take away one 19! we have 19 x19!
20! is 20 times larger than 19!
The part in blue took me a little while to get. Later in the day I went over it in my head and I said to my friend awhile later, “So I’m going over the factorial thing and this is what I think…
As he listened he said “I don’t think you have got this part, and he named the place where 20! is 20 times greater than 19!, so we chatted over that and finally I saw it. Or at least I did, until I sat down to write this and then I had to go to his paper to look at the notes he had jotted down and then it came back to me.
Now to those that did Maths for high school and college this probably is a no brainer. But for me who was told in Year 10 that girls didn’t need Maths (1960’s) this was very new learning. My love is reading and writing, however I need to teach Maths to Year 8 and I actually enjoy it too.
Why did I want to solve this problem? Well firstly I was driven by the fact that I wanted to be able to teach my students about it. That was my why. It has no other relation to my life, and I have to say it probably has little relevance to a large part of the population. So why would my students need it? Well obviously they are not going to get the message that girls don’t need Maths so it is a small step in the bigger picture. Now that I knew the word factorials I was empowered to search further about them. A video on You Tube showed me a little more.
I hadn’t seen them mentioned in Algebra at the Year 8 level but what’s the harm in going further? I was partly driven also by my need not to have my students say half way through the year “I haven’t learned anything new in Maths this year!” Now while it wasn’t Teegan who said this, or indeed anyone in my class in 2011, I know I will get great joy in sharing my knowledge with her and the others who are ready for it. Students who on the whole are showing through testing they are at Level 5 in the NZ curriculum but are still at primary school.
And learning for the sake of learning is actually very satisfying. Now I need to share it in a way my students will get it, so that one day very soon they can go beyond me. One thing I can predict is they are going to grasp this far quicker than I did.
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