Lesson 26 Chunking in algebra
|The formal algebraic language of bracketing (and the concept of distribution of multiplication ever addition) scale is very high in terms of difficulty (near mature formal, in Piagetian terms). Research evidence shows only a small proportion (around 15%) of 15-year-olds are secure in handling bracketing in this way.
The lesson focuses on the processes that underlie the use of brackets. This is carried out first by grouping of two or more terms together.
|In most cases the following tasks can be worked through one at a time, with a minute or two for pairs or groups consulting each other and coming up with ideas for a class discussion. Freely use loops, boxes, colour or underlining of terms that are being treated together in chunks, conveying to pupils that there is nothing ‘sacred’ about the order in which terms are written. Allow pupils to come up to the board to show their suggestions. Use prompts such as:
With low achieving pupils this episode can be made into a whole lesson, largely in instruction mode, to ensure that the skills involved in handling simple algebraic terms are practised, and to introduce the ideas of chunking with familiar number expressions.
|A round on scanning the whole expression and grouping of terms|
|‘Chunking’ is a special form of grouping, with the reason for putting things together being the convenience rather than some common feature of the terms.
Only addition and multiplication are used, to avoid issues of order of operations, which form the agenda for many instruction and practice lessons. The words ‘scanning; ‘rearranging; ‘breaking-up’ and ‘chunking’ are used, but the teacher and the pupils themselves may use other ways to explain their meanings.
|In ‘average’ classes you could start by writing on the board an addition sum of five numbers such as 19 + 98 + 1 + 3 + 2. Then discuss the idea of scanning, or looking at the whole thing first, before rushing to do the sum. This overall look is vital, although rarely emphasised in lessons. Ask: What do you notice? Pupils should see that putting the 1 next to 19, and the 2 next to 98 is a better arrangement. The idea of grouping into easy numbers can then be called ‘chunking!
Pose the following problem: My uncle won a maths competition in the Andes... He can do sums like this very, very easily: 199 + 87 + 46+ 24+1+10+3+24
Ask the children to use the ‘scanning and chunking’ method to attempt this question. Then try a decimal addition such as:9.5 + 8.2+15+18+02+ 65 + 05
Discuss the different methods pupils use to approach the task, down to the simplest level. Explore the possibility of different pairings, or ‘threeings! and some of the different ways of explaining the terms.
|A round on breaking up a number to ease multiplication by parts|
|Intuitive example of multiplying a bracket by a number outside it, written up in mathematical form.
Intuitive example of multiplying two brackets, each with an addition sum, written up in mathematical form.
|Ask the children how they work out in their head a multiplication such as 3 times £5.15. The pupils should realise that they do actually break up the money into parts, multiply each part separately, then add the parts again.
Write this process on the board using brackets: 3X £5.15 =3 (£5 + 10p + 5p) =£154+30p+15p = £15.45
Explain that this is easier to do in their heads because it is familiar, and that when writing it down it is better to use brackets so that everybody else can follow the working. Show where to put in the brackets: (3 X £5) + (3 X 10p) + (3 X 5p)
Ask them now to work out 6 X £5.22 in the same way and to explain how they are doing it. In discussion, allow pupils to restate the same ideas in their own phrases. Then ask them to do 6 and a half x £5.22. Extract the idea that they will need two pairs of brackets for this question: (6 + 1/2) x £5.22 =(6 + 1/2) X (£5 + 20p + 2p)
Show that to work it out, everything in the first bracket multiplies everything in the second bracket, then everything is added. In passing, note that the £ sign and the penny sign are codes and not variables — and that we can actually use words for them.