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Sheep in Wolf's Clothing

Age 16 to 18
Challenge Level

Why do this problem

This problem gives a taster of abstract representations framed around familiar mathematical concepts. It is useful to prepare the way for students to start thinking about abstract mathematics such as group theory, as concepts such as Identity, Inverse, Equivalence and Closure will emerge during the task. The latter parts of the task are good fun to have ongoing over the course of a week or term.

Possible approach

This problem would be a difficult challenge for keen students to consider individually, perhaps as an extended homework or holiday task.
In lesson time it is suited to a group or class discussion where the problem is gradually solved together -- focus initially only on exhibit A and the related questions, saving exhibits B and C for high-fliers or open problems. All students will know the mathematics for Exhibits A and C, but perhaps skip B (complex numbers) if students are not in their final year.
Start with Exhibit A and write up the rules. Say something like "I've got a set containing pairs of integers, such as (1, 22) and (-34, 8). I have a rule which allows me to combine pairs to give another pair in my set. 
For example $(1, 2) + (2, 3) = (7, 6)$ and $(7, 2) + (3, 5) = (41, 10)$
Oh, and we also have rules such as $(14, 21) \equiv (2, 3)$"
This is a good place to discuss abstract rules and to introduce the concept of equivalence if the class has not met this before.
Allow the class to discuss what might be going on before writing down the general rule
$(a, b) + (c, d) = (ad+bc, db)$
Suggest that small groups attempt to work out what is happening by testing out various numerical examples. When a group feels that they have worked out that the structure is 'addition of fractions' they will then need to convince the rest of the class of their reasoning. 
You can then allow the class to move onto consideration of the other Exhibits if a longer task is desired.

Key questions

Have you tried exploring with small numbers, both positive and negative to get a feel for the structure?
Using our rules, can two pairs be combined to give $(0, N)$?
What happens if you combine two of the same pair together?
Using our rules, which pairs can be combined to give $(N, 0)?$

Possible extension

Exhibits B and C are likely to offer sufficient extension. If more exploration is desired, students can attempt to alter the combination rules and see if any structures emerge.
You could also ask students to devise exhibits of other structures, such as vectors and matrices.

Possible support

You can greatly simplify this tasks by asking 'This is similar to addition of fractions! Can anyone see why?' or 'This is addition of complex numbers' or 'These are the rules of arithmetic'.