Diagrammatic views of sequences

Expanding on my 'recent' (haven't blogged in ages admittedly) post about different views in algebra, I have been looking at the idea of showing different expressions using diagrammatic views of sequences, and thought I would outline a few thoughts here:

1) Linear or quadratic?

A really nice thing to do with pupils is to look at patterns that arise from (or generate depending on your point of view) linear sequences, compared to quadratic. In particular what is the difference between the way a linear pattern grows, compared to a quadratic pattern? Have a look at these patterns and see if you can decide whether they show linear or quadratic sequences without writing down the numbers:

Most people that know about sequences will be able to identify that the 1st, 3rd and 4th sequences are linear, because the same number of squares are added each time (the colours make this quite easy to identify), whereas in the second and third there are more squares of each colour - what is interesting though is to explore these views of the different sequences:

Linear

Each of the sequences 1, 3, and 4 can be rearranged to give these lines, , showing that they only grow in a 'linear' fashion, which doesn't work with the quadratic sequences as the number of squares is different each time (although you can technically rearrange them to make lines, they don't grow in a linear way).

Another interesting way to look at the linear sequences is using a graph:

Quadratic

Or if you prefer:

What is nice here is that these sequences illustrate that quadratic sequences are the two dimensional extension to linear sequences. The graphs can also be used to illustrate the difference to a quadratic and the quadratic shape:

Showing the curved nature of the quadratic graph as opposed to the straight line nature of a linear graph.

2) Different forms of an expression

Another possible use of these pictures is to illustrate the different ways of writing identical expressions, for example if we take sequence 1 from above without the colours:

It shouldn't be too hard to show pupils that the calculations for the number of squares in each successive pattern is 4 x 2, then 4 x 3, then 4 x 4 then 4 x 5, so in general 4(n+1). Consider the same picture with some slightly different colouring:

and we should be able to demonstrate that this is also 4n + 4 (the yellow squares given by 4n, and then 4 green squares on the end of each pattern). This is also true in quadratic sequences, taking sequence 2 from above:

Similar to above, the calculations this time are 1 x 2, 2 x 3, 3 x 4, 4 x 5, or in general n(n+1), if we then compare to the image below:

We can show quite clearly that this is also n² + n.

I am sure there are other uses I haven't yet thought of (I think it may be applicable to geometric and Fibonacci sequences as well, and possibly series at A-Level). When I get chance to explore more I will try and remember to write about it!

P.S. - of course if you have multi-link cubes or similar then pupils can actually build these sequences, graphs etc. as well as just seeing or drawing the pictures.

Ratios, Fractions and Linear Functions

Back at #mathsconf6 (or was it #mathsconf5?) Luke Graham (@BetterMaths) led a sessions about teaching the new GCSE. One of the most popular topics to come out of the sessions in terms of required support was R8, which is about the foundation content "relate ratios to fractions and to linear functions." I would like to show how this can be achieved using one of my favourite tools, the bar model.

For those who haven't seen a bar model before - this is one way of representing it (and my preferred way, although I have seen others). Now from this picture we can ask a number of questions:

1) What fraction is shaded blue?
2) What fraction is shaded green?
3) What is the ratio of blue to green?
4) How many times bigger is the green area than the blue area?
5) What fraction of the green area would the blue area represent?

These questions basically highlight the relationships between the three different representations as well as the different ways fractions can be thought of from a ratio (i.e. considering the fraction of the whole, or the fraction one part represents of another). The answers to the questions are the mathematical ways of relating the different representations i.e.:

1) ⅕
2) ⅘
3) 1:4
4) 4
5) ¼

i.e. we can say that the ratio 1:4 represents ⅕ and ⅘ of the whole, or the function G = 4B [i.e. the green area is 4 x the blue function] or B = ¼G [i.e. the blue area is a quarter the size of the green area].

This can also be done with more complicated ratios, particularly non-unit ratios, such as:

Answering the 5 questions this time leads to:

1) ⅖
2) ⅗
3) 2:3
4) 1½
5) ⅔

Which can be seen as the ratio 2:3 being equivalent to the fractions ⅖ and ⅗ of the whole, the function G = 1½B or the function B = ⅔G.

I have found that getting pupils to go through this process of writing down these equivalent representations definitely helps, and reinforcing them whenever we work with ratio and proportion to remind the pupils of the different ways of viewing the relationship. An interesting one recently was as a nice way of illustrating percentage changes, and in particularly that you cannot reverse a percentage change using the same percentage: i.e. in the example above you can see that a reduction of 40% (i.e. removing the two blue bars) would be reversed by an increase of 66.666...% (i.e. ⅔ of the three bars is needed to get back to where we were). Obviously there are also some nice links with reciprocity of fractions and the like which can also be useful. My big advice though would be to set aside some time to explore these relationships explicitly, give pupils different images, ratios, fractions and functions and get pupils to re-write using the equivalent representations (and in my opinion all linking through the bar model).