Lesson 26 Chunking in algebra

# Episode 2

Reasoning Resources: Worksheet 1
Whole class preparation
Pupils approach algebraic expressions not as fixed- shape puzzles to be handled by strict and standard rules, but as malleable representations of some meaningful quantities that can be reformulated in ways convenient to the handler.

Pupils are asked to use compound variables directly in their work (price per unit length, and length itself) and to create expressions in which each of these compound variables is symbolised by a single letter.
This part of the lesson deals with chunking linear algebraic expressions and reversing that process. It introduces the now familiar notion that many things are priced in two parts, a fixed part plus a rate part, i.e.a part that depends on size. As examples you can mention electricity and gas prices (standing charge and price per unit), and also curtain rails (end bits plus price per length).

Use an example for a fixed sum plus a cost per unit for a continuous variable, such as Christmas trees in tubs sold by a shop which charges a tub cost plus a price per foot. Go through a bracketing problem, e.g. suppose the tub is £1.50 and the price per foot is £3. So the price of a tree length L feet in £ will be the compound variable of 3L + 1.50. When a number of trees of the same size are bought, the total cost can be calculated either by chunking or by spreading out in bits. So 20 trees each of length L feet would cost 20 X (3L + 1.50) or (20 X 3L) + (20 X 1.50). The sum total is the same, but the second way allows the shop to separate the money into money for tubs and money for trees.
Pair work and whole class discussion
Thus the time given to working on Worksheet 1 helps pupils become secure in their use of bracketing in a context which can be directly related to their own common sense steps of computation.

There may be disagreement on what the chunks’ in the various expressions are. A working definition may emerge from the discussion: if you can give something a name, then it’s a chunk; e.g. 0.5L can be labelled ‘the weight of the chain’ and is therefore a chunk.
Give out Worksheet 1, which is based on a similar problem. You could have a discussion after most of the class have completed question 3, and then also after question 6.

After question 3 the same options as for the Christmas trees should appear. Conduct a discussion or a vote on preferences, emphasising their equivalence! A different and flexible meaning of a ‘chunk’ would emerge: a multiplication is automatically a chunk, but addition is optional.

The discussion of question 6 should bring out the nested nature of chunking, expressed by the nested nature of pairs of brackets. The expression for the cost of 10 chains and pendants may be 10 X [7 X(0.5L + 12)] where the (0.5L) as a chunk, is then chunked with 12 inside the brackets. Explain that in expressions like 0.5L the pairs of brackets are optional, and often unused. They can check if the pairs of brackets on the calculator are of different shapes or not.

Allow the class to discuss the term ‘nesting; and how appropriate it is for the purpose, linking it to daily language use of ‘inside each other’ and ‘within each other’ or ‘pair within pair‘