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Sequences - finding the nth term

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richard 12 March 2007 21:28
Posts344
Joined28/06/06

I've been thinking about this quite a bit recently. I don't want to say to the students "so, to find the nth term you look at the difference and write down that number followed by n, then you adjust accordingly."

It seems very easy to tell them how to do it without them developing any real understanding.

Any thoughts and/or activities about this would be very welcome.

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HeatherNorth 13 March 2007 23:58
Posts243
Joined07/09/06
I've looked at it in two ways in a few ways in the past, depending upon the way pupils are able to do mathematics.  For a GCSE piece of coursework we asked students to investigate difference tables - from this starting point one student discovered differentiation!  For much younger pupils I ask them to use a starting number and a generating rule to produce a sequence - and then we look at the generalisation for each sequence (some of which we can find easily, some of which we can find with a bit of work, and some of which we choose to leave until we have a bit more experience.  I tend to always look at the generating rule and the generalisation together for each sequence and even only last week a pupil called out in surprise 'Oh if the difference is 2 it will always be times by 2' !!! For me allowing pupils to discover things from the opportunities we give them means they will understand what they are doing.  I am also interested in other people's ideas.  Thank you, Heather
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Helene_Huille 26 March 2007 13:35
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Joined27/11/06

I also like to let them discover how to generalise, and usually, I ask them as homework, so they have some time to think about it.... (or research it...) (usually a sequence such as 3n+2)

Could you give the 1 000 000 000th term of that sequence.... and how would you find it? What if I gave you a bigger number???? another sequence?? then we spend nearly a whole lesson discussung generalisation, what we could use it for etc....

You need to choose a number so big that they don't attempt it by hand.... (on a fun point I once asked for the 312th term in the sequence, and the boy gave me the answer with a letter of his parents stating that he had spent hours doing it, and they didn't really see the point OOOPSS!)

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Helen 27 March 2007 13:18
Posts203
Joined19/02/07

Do you know any good websites that show pratical uses or visual problem solving questions

Helen

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Carlos 28 March 2007 12:24
Posts1
Joined28/03/07

With the sequence 7, 11, 15, 19 ... I tend to say something like "it goes up in fours, so it's 4n" then "if there was a number in the sequence before the 7, what would it be?" This "zeroth term" does the same job as the intercept on a linear graph. I thus link the work to graphs and co-ordinates. The word "linear" itself suggests a straight line (graph) and so I draw their attention to the lingo.

Carlos

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richard 28 March 2007 13:59
Posts344
Joined28/06/06

Hi all,

Thanks for the ideas - I think you're right, Carlos, about linking it to the graph.

What other images do people use for sequences? I'm making a cardsort which includes

the first few terms of the sequence
a description of it in words (eg - the odd numbers)
A graph
A set of coordinates
The equation of the straight line
an expression for the nth term
a set of coordinates
A 'the 243rd term is...' card
A 'the gradient is...' card
A matchstick sequence

I want to try to help my students to develop as many images of a linear sequence as I can - and I hope the cards will help me to see which they're comfortable with and which they're not.

I'm sure I'm missing some - what other images do people use?

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Helen 28 March 2007 16:30
Posts203
Joined19/02/07

Hi Carlos - thank you for that

What a good way to teach it (I am a primary teacher so generally only teach finding the nth term practically)

So is this right 7,11,15,19

It goes up in fours so it's 4n, 7 - 4 = 3   So the nth term is 4n + 3

 

Let me try another one   3, 8, 13, 18

It goes up in fives so it's 5n, 3 - 5 = -2   So the nth term is 5n - 2

 

In my younger days I did A Level maths and Further Maths - I can't remember anyone giving me a system to find out the nth term - you just had to work it out.  

I like the idea of linking this straight to graphs.

 

Helen

Richard - linking another tread (Bridging Units) - this would make a great bridging unit for transistion from Year 6 to Year 7]

 

Helen

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barbarawmasters 29 March 2007 17:08
Posts9
Joined06/03/07
I do a slight variation of the above.

I write out the times table appropriate for the sequence eg for the sequence

                                   7    11   15   19   where the difference is 4

            4 x table is     4    8      12  16

They usually notice that the sequence is the 4 times table with 3 added each time to form the sequence hence the nth term is 4n + 3.

it also helps them to grasp the notion of 1st term etc as well as making it easier to cope with sequences involving negatives, decimals and fractions.

Barbara
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Helen 29 March 2007 18:25
Posts203
Joined19/02/07

Thanks for that Barbara

Do you have any good problems for finding the nth term

Helen

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HeatherNorth 30 March 2007 13:57
Posts243
Joined07/09/06

Hi Barbara, i haven't come across your idea before I really like that - as it give them a step towards noticing the nth term by thinking about it.

Re ideas for sequences .... I use a folded sheet of A3 paper - folded to make 8 columns and 4 rows.  In the first row pupils decide on their starting number and generating rule. (For younger pupils this can be a starting situation and generating rule e..g. I put one dinosaur in the first box - and the generating rule is to add one more dinosaur to the next box.)

Then in the second row they do the same.  and so on until they have generated four sequences.

Then for each sequence, we find the nth term.  It may be that some sequences are not soluble for their ability at that time .... so we make a chart of what we have solved, what we are still solving and the sequences we think are going to be too difficult for us to solve at the moment.

With best wishes, Heather

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