# Enumerating possibilities of combinations of two variables

With Year 6 children expected to work on the objective ‘enumerate possibilities of combinations of two variables’, we should be clear on the difference between the underlying concept and the algebraic representation of it.

2g + w = 10

For questions such as this, children should first have a secure understanding of the part, part, whole model.  We can show that 2 lots of something add one lot of something else is equal to 10 by using a concrete manipulative such as Numicon.  First, children represent the whole, in this case 10. Then they can speculate on the two equal parts (2g), trying out g=1 before finding the Numicon piece that fills the gap and therefore is equal to w: Having found one solution, they can continue to work systematically to find alternative solutions.  Trying g = 2 is logical: Lining up solutions beneath the whole reinforces the idea that the expressions are equivalent.  Children can continue to work systematically: This also provides a scaffold for questions of greater depth, such as ‘What is the greatest number that g can represent?  Explain…’
Subtraction?  Not a problem, although in this case, children must know that for subtraction, you always do so from the whole.

10 = 3g – w

In this question, the whole is 3g and the parts are 10 and w: What is not clear from this model is the trial and error that went into it.  Children may well try 3 ones and quickly realise that it is already less than 10, so subtracting from it will not give a valid solution.  There is lots of scope here for discussion about the smallest number that g could represent.

The use of Numicon leads nicely into children representing problems as bar models.  Here are the two examples used so far: