# Substiting into Formulae

By Andy | January 29, 2005 | 2 Comment

I was reading through September 2004 issue of MT (Mathematics Teacher – published by the ATM), and came across an article on teaching “Subtituting into Formula”

In the article Colin Foster (from our local Henry VII’s) talks about trying to find ways to make the task of subtituting in to formulas seem more interesting and relevant.

Whilst looking at the some sequnces with his Y9 group he stumbled across the idea that any finite sequnce can be expressed by a sufficiently complicated equation…

So he decided to set the students a more complicated task. He devised a formula which when you substitute in the integers 1,..,8 produces the schools phone number!!

The formula is quite complicated:

$$f(x) = \frac{107n^2}{5040}- \frac{241n^6}{360} + \frac{1547n^5}{180}+\frac{4157n^4}{72}+\frac{156179n^3}{720}-\frac{81017n^2}{180}$$$$+\frac{197387n}{420}- 180$$

So subsituting in you get:$$f(1) = 7, f(2) = 6,$$ … , $$f(8) = 1$$

He talk about the way in which he uses the task in the lesson by allocating pairs of students to look at substitutin in each number and then trying to build up a coherent sequnce and seeing if anyone can guess what it is… He also talks about the fact that their are inaccuracies from the claculator and how he asks students to think about why these might be the case…

I found the whole idea quite interesting and so decided to try to find the corresponding formula for Ruth’s school.

The article gave no clues as to how you might find your own formula… So with a but of hunting around and a couple of failed experiments I releasied that it is _just_ a matter of solving a system of 8 simultaneous equations. Now being the lasy person I am I decided to do this by forming a Matrix and getting excel to claculate its invervse and hence solve the system. (excel file attached at the end of the post)

So now we have a Formula, Ruth said she would have a go at using it with her Year 9 group. When she has had a go I will report back as to how well it worked.

*Note:* This will produce the decimal co-efficents of your equation, however you will need to turn these into fractions yourself!

#### Andy

06 March, 2005 Reply

Hi there! I was doing a search to see whether anyone had cited my articles and it was very nice to stumble across your maths "blog"! You've got some interesting stuff there that I'm going to go back and look at after I've typed this. I gather from what you've written that you're not that far away - are you based in Coventry as well?

#### Andy

06 March, 2005 Reply

Yes! I am currently based in Coventry. I am in the process of finishing a PGCE in secondary Mathematics (retraining after spending a couple of years teacher FE Business Studies and ICT).

I have just realised from looking at your website that we met at the Gifted and Talented day at Elm Bank earlier this year. I co-lead the session on proof in the morning.

I am currently on placement at Cardinal Wiseman School in Coventry, but have accepted a job teaching Maths at Warwick School, in Warwick (whch would be obvious were it not for Warwick University!!)

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