Lesson 7 Which offer shall I take

# Episode 1

Reasoning Resources: None
Whole class preparation: story set context
Generating numeric expressions for related situations. Begin with questions about CD and DVD players or similar fashionable technologies. Lead to subscriptions to CD/DVD clubs and the offers that are possible. Talk about having two clubs to choose from that have the same subscription cost for two years, say.
Paired work: ways of finding and expression a relationship
Generalising work through a table of values, the rule in words, or the rule in symbols.

Pupils move from ‘all words’ rules onto a mixture of words and symbols or symbols only.

A common source of errors in symbolisation is the confusion between using a letter as a code (shorthand for an object name) or as a number variable.
They now have to make some mathematical sense of the offer that ‘With Club 1 you get 8 CDs free when you join, then 2 at the end of every month.’ They should work out how many CDs they get in five or eight months, after any number of months, and improve their first rule by making it more concise. Pupils who have found values for different months should be encouraged to organise them in lists or tables.
Once you are confident that there is a range of representations (in sentences, tables, symbols and mixtures of these), encourage pairs to compare their ideas with other pairs and prepare to explain them to the whole class.
Whole class sharing: comparing representations of a rule
This can be compounded by using the first letters of the names of the objects we are referring to and mentioning the idea of using ‘shorthand’ to make things simpler in maths.

Letters in algebra are ‘holders for any number; ‘represent any number, or ‘stand in for any number’.

Algebraic convention of omitting the multiplication sign creates some confusion when we translate rules in abbreviation to algebra.
Typical formulations would be in terms of expressions and equations using a mixture of abbreviations and numbers.
Allow time for the class to realise the advantages of the algebraic convention, but that it can also be the source of errors and misconceptions.

One way to highlight this is that where letters are present in
both sides of the equals sign, they mean different things. In the example highlighted here, CD stands for ‘any number of CDs" is on one side and C just for the words Compact Disk’ on the other.

Allow each group time to share and justify their ideas, with others commenting upon them or suggesting improvements. Once a good range of ideas is presented, pupils could discuss the advantages and disadvantages of each. This is important as this lesson is about the pupils progressing naturally towards linear equations.

Episode 1 should end with the class focusing upon a formulation such as 2 * m+ 8 =CD, or simpler C = 8 + 2 X M with each letter described as ‘number of CDs/months. The algebraic convention for omitting the multiplication sign (i.e. 2M meaning 2 x M) should be presented as a convenience.Pupils could debate their preferences when you extract a few correct but different formulations. They can also read, or accept, that the algebraic expression can be a meaningful sentence, e.g.‘This says the total number of CDs in any month equals twice the number of months plus 8!