Why do this problem?
sets the scene for some important algebraic ideas. It shows that if three unknowns have a particular total, there are potentially different possibilities for these unknowns. As well as giving children practice in addition and subtraction, this problem could be used as a focus for introducing the
idea of working systematically.
It would be good to have the balance up on the interactive whiteboard and, with the class watching, put a weight on the $10$ on the left-hand side and a weight on the $8$ on the right-hand side. Ask them to talk in pairs about what will happen when you put a weight on the $2$ on the right-hand side. Share some ideas, in particular their reasons for their predictions, then test them out on
the interactivity. Repeat this a few times, with different numbers of weights on each side until you feel as if the children are happy with the set-up.
Challenge them to investigate the problem, preferably with pairs working at a computer. Alternatively, you may have "real" balances so that you do not need to use the computer for long periods of time. After inviting children to try out some ideas, it would be good to spend some time talking to the group about how they are finding solutions and what they are recording. You may want to draw
attention to any systems that the children are using which help them find all possibilities. For example, they might keep one weight the same on the right hand side and find all combinations that can go with it; then increase this fixed weight to the next one along and find all combinations etc. You may find that some children are not using the balance itself and are just recording on
Where are you going to hang the weights on the left-hand side?
What will you need to hang on the right-hand side?
Can you find another weight to balance them with the same weights on the left?
How will you know you have found all the possibilities?
How are you recording what you've done?
Ask the pupils what they think the total of the two weights on the left would be to give the biggest number of possibilities for rearrangements on the right-hand side. How do they know and how could they prove this? Learners could be challenged to find all the different ways that two weights can balance two weights. Alternatively, children could investigate what happens if you are allowed to
have three weights, but any two weights may be on one numeral.
It might help if children record each possibility on a different strip of paper. Once they have found a few for a particular pair of weights on the left, you could help them order the strips which would reveal any gaps.