Algebra - Notation, vocabulary and manipulation: where appropriate, interpret simple expressions as functions with inputs and outputs. (Foundation tier)

The idea

A function is a rule that takes an input, does something to it, and gives one output. A simple expression can be read that way: 3x − 4 is the rule ‘multiply by 3, then subtract 4’, so an input of 5 gives an output of 11. A function machine draws the same rule as boxes in a row, read from left to right. To find an input from a known output, work backwards through the boxes and undo each operation, so an output of 20 needs 20 + 4 = 24, then 24 ÷ 3 = 8. One input always gives one output, though two inputs can share an output, as 3 and −3 both do for the rule x². Questions give you a machine with a blank box, or a table of inputs and outputs to complete.

What earns the marks

• Applying the operations in the order the machine or the expression sets out.
• A correct output from a given input, even if a later part of the question is wrong.
• Undoing the operations in reverse order to recover an input.
• A table of inputs and outputs completed correctly throughout.
• A correct inverse chain written down, such as ‘add 4, then divide by 3’ for 3x − 4.

Common mistakes and how to put them right

• Working a machine backwards in the same order as the forward operations.

  Put it right: Reverse the order as well as each operation.

• Reading 3x − 4 as subtract 4 first, so getting 3 from an input of 5.

  Put it right: Multiplication by 3 comes first.

• Using the wrong inverse, such as subtracting where the box multiplies.

  Put it right: Each operation is undone by its inverse.

• Forgetting that both 3 and −3 are inputs giving an output of 9 for the rule x².

  Put it right: One input gives one output, but two inputs may share an output.

Exam tips

• Write the operations as a chain before you calculate, so the order is fixed on paper and not in your head.
• Going backwards, reverse both the order and each operation, then feed your answer forwards to check it.
• Fill in every cell of a table you are given; a value that breaks the pattern usually shows a slip.