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).

Numerical, Algebraic, Graphical and Diagrammatic representations in the new KS3 maths

Recently we have been reviewing the KS3 programme of study in relation to the new KS3 National Curriculum and also how our KS3 programme relates to the changes in the KS2 curriculum to see whether we need to make changes due to overlap. In my perusal of the KS3 curriculum document this phrase jumped out at me:

• move freely between different numerical, algebraic, graphical and diagrammatic representations

I must admit my immediate thought was 'ambitious' at least in terms of the "move freely" part; in my experience it is quite difficult to get pupils seeing different representations as representing the same thing. But then I thought how much fun it would be looking at all of the different representations at once and started thinking about how many I could come up with; here is my (not at all definitive) list:

• Sequences: 1, 3, 5, 7, 9, 11, ....
• Coordinates:  (1, 1), (2, 3), (3, 5), (4, 7), (5, 9), ...
• Table of values  1     2      3       4      5

                                     1     3      5       7      9

• Algebraic Sequence notation: Tn = 2n - 1
• Line equation y = 2x - 1

• One-dimensional diagrammatical representation:

• Two-dimensional diagrammatical representation:

I definitely think it will be interesting to work with pupils and find ways of working with algebra in different representations - choosing a suitable representation for a problem and transferring from one representation to another as the need allows.

Expanding and Factorising - Using areas to support understanding

Recently I have been musing over the visualisation of algebra, and have been working with my pupils over viewing algebra as different dimensions. In particular the different visualisations that one can use to interpret algebraic multiplication. For example understanding that 2n can be be visualised like this:

or like this:

A lot of the work has therefore focused on when different visualisations (including 3 dimensions, and understanding if not drawing when we go beyond 3 dimensions) are useful, and when they lead to a more efficient approach for generalising approaches to multiplication. A really poignant example of this has been in expanding and factorising.

One of the first examples we explored was 5(2x + 3), and how this could be viewed like this:

or like this:

But that the first representation breaks down when expanding something like 2p(7p + 5), because we cannot create "2p" repeats. However the area representation still holds for this:

We were even able to explore how negatives could be handled with this representation, for example 5(2x - 4) being viewed like this:

With the logic here being that the blue shaded area is the area represented by 5(2x - 4), and that it is the whole area of 5 x 2x (=10x) subtract the 5 x 4 area (=20) and so we have 10x - 20 shaded.

An interesting example to explore was -6(2 - p), and see how to end up with -12 + 6p, for this we looked at this representation:

by thinking about what the negative of 6(2 - p) would look like (we did consider making the 6 point downwards, and showed that this can work as well as it would lead to -12 - (-6p), but the above was considered a more efficient representation).

What was really interesting was that having thoroughly explored this representation, how quickly the pupils took to factorising. To introduce factorisation all I did was put this image up on the screen and we talked about what would go next to each arrow:

Pupils all pretty much immediately saw the required lengths for the first two. The last one did lead to an interesting discussion about partial factorisation, and so I told the pupils that one of our aims is to maximise the shared height - this quickly led to this height being 3x. Even switching the positives for negatives didn't lead to difficulties as we were used to examining it as the difference of a large area and a smaller area:

I am beginning to think there is a real power in this representation to provide a consistent approach to visualising algebra, provided part of the work we do with pupils is showing them the different ways multiplication can be interpreted (as repeating a shape in the same dimension or extending into the next dimension). 

This week I will be tackling the expansion of two binomials and the factorisation of a quadratic into two binomials using a similar approach; I am really looking forward to seeing how that goes.

Why can't you simplify a + b?

Here we go, Year 7 bottom set teaching simplifying algebra, what does it mean to add together letters, or subtract letters. Why can't we write a + b as ab? Why is -2d + d equal to -d and not -3d? All of these are more I am trying to tackle. So we spent last lesson learning to translate "maths language" i.e.:

and completing activities around decoding expressions. This lesson then I started with this:

and got pupils to re-draw the picture so that all of the a arrows were together, and then the b arrows; invariably getting this picture:

We then moved on to this picture:

The interesting part of this of course being that some of the d's now move in the opposite direction, which cancel out d's above. We re-drew this to show that we don't actually need 4 d's.

We finished then with this:

which became this:

which was used to illustrate that we although it looks like we only need to go backwards, we cannot because we don't know how far backwards to go, so we have to go forward 2e before going backwards 5f.

I really like this representation of using lengths to represent variables, as it then generalises nicely into area when multiplying for example:

and then into other relationships between area and algebra.

The kids seem to be getting there with understanding, I think it will take them two or three more lessons before they are secure with approaching algebra like this, but I am convinced it will be worthwhile to start this process now in Year 7.

Prime factorisation, indices and standard form - some great questions

As we all know by now (in England anyway) the new GCSE in maths is going to require pupils to make links between areas of maths, and challenge pupils to apply understanding in ways they might not have previously. In writing the new homework booklets for my department I have been challenging myself to ask questions in this vein, and have found a rich source in linking prime factorisations, indices and standard form. Now admittedly there are already related topics, however I think that I have developed some questions that challenge pupils understanding of these topics in ways that perhaps haven't been used as frequently before now. Here I am sharing a run-down of my top seemingly straightforward questions (in no particular order):

1) 108 = 2² × 3³. 108² = 11664. Find the prime factorisation of 11664.

2) 9216 = 2¹⁰ × 3². Find the value of √9216.

3) Calculate (3 x 10⁴)³, giving your answer in standard form.

4) Find the prime factorisation of 6 x 10⁴, giving your answer in index form.

5) Find the prime factorisation of 3.2 x 10⁷, giving your answer in index form.

6) Find √(1.6 x 10⁵), giving your answer in standard form.

7) Calculate √(1/25), giving your answer in standard form.

8) Calculate (1.25 x 10⁸)^(2/3), giving your answer in standard form.

These and more will be in my term 2 homework booklet for the pupils aiming at grades 7+ on the new GCSE, and I think are precisely the sort of skills that the new GCSE is aimed at ensuring pupils develop.

The missing link in Adding and Subtracting fractions

Ok, I hadn't realised it had been so long since I had blogged - I have been insanely busy writing homework booklets and test papers over half term so that has left little time for much else. One thing it did leave time for however was for me to make a nice mental leap in my teaching of adding and subtracting fractions.

As I explored in one of my more recent blog posts, the use of bar models for adding and subtracting fractions I felt really did help a number of my pupils see why fractions added and subtracted in the way they did. There was a minority though that still struggled, and at first I couldn't really see what else I could do - I couldn't think of a better way to explain the concept. Then (and I am not really sure how) I realised something crucial: these pupils didn't really understand the concept of equal area in fractions. The pupils couldn't see that a common denominator was necessary because they were thinking about how many parts each fraction had and didn't realise that the total area of the diagrams had to be the same for both fractions. Of course my earlier diagrams had been the same size, but I hadn't stressed it, believing it to be obvious. So the next time I introduced the topic of adding and subtracting fractions, I started with this picture:

Something so simple ended up being a watershed moment for me in teaching adding and subtracting fractions, because it forced pupils to see that quarters and thirds had to be turned into twelths (or a multiple of 12). We followed this up with this slide:

and this slide:

For those pupils that had continued to struggle, this was a revelation. They didn't need to break the picture up into 12, it already was! They had drawn it that way themselves, because the rectangles needed to be the same size. We could talk about why 8 square long rectangles, or 9 square long rectangles, wouldn't be useful, and everyone could see it. We came out with these pictures:

When I finally set this related challenge:

Everyone did it - 100% completion. Some within a minute. Most pegged 18 square long rectangles pretty much immediately, and only a few had an initial struggles about then representing 4/9 and 1/6 - but once we got past those they were there. In the end every book had a picture like this in:

and every pupil understood.

The full prezi can be found here, and it is definitely an approach I will be using again.

My new favourite vector resource - via Back to Back activities!

Walking around my department towards the middle of last week (which I try and do whenever I get the chance, which unfortunately is not as often as I would like) I spied a fantastic image that one of teachers was using as a part of a "no pens day"; having pupils sit back to back whilst one describes and the other draws this picture:

perhaps it was because I was due to teach it the following week, but my mind raced immediately to this picture, which I promptly designed at the end of last week

of course the topic being...Vectors!

I love vector mathematics - it is such a useful way of visualising so many key concepts in maths and science; I use them to conceptualise negatives, translations (I actually draw on the vector arrows), all sorts of things. What particularly struck me about this image is the way it ties vectors nicely with similar triangles and scale factors, For the top end pupils the discussion as to why B to E is 2a and why K to G is 2b and building up the whole picture from there, is a great discussion to come out of this picture, along with then all of the other vectors is as good a top end vector resource I have seen - eventually it is possible to generate this picture:

In terms of trying to define vectors in terms of other vectors, what a great activity! It won't stop there either - tomorrow I will be using the image to explore ideas like:

(a) Are the points KLJ on a straight line? What would the vector be? What about FDE?
(b) Do the points KHE divide the diagonal MC into 5 equal sections?
(c) If the line from N to D is extended so that it intersects the line segment between A and B, into what ratio does it divide the line segment AB?

Of course that is not the only way to use this image - over the half term I will likely create something that uses it for trigonometry, scale diagrams, Pythagoras' Theorem etc as well - all out of a simple image that an NQT was using for drawing.

My resources around this image can be here and here.