Groups: Answer Problems B1-B9 in pairs or small groups.
It may be challenging to find the underlying function, which is a step function. It is important to reflect on what representations (table, graph, equation) were most helpful in thinking about how to predict down the line.
A typical answer to describing the function is, "Take the number of sheep in front of Eric, divide by 3, and round up." This is a perfectly reasonable description, although some people may feel it is not as "legitimate" as a rule with symbolic notation. In fact, there is a way to represent this symbolically using the ceiling function notation:
n. This denotes the smallest integer greater than or equal to n. In the case of this problem, the number of sheep shorn before Eric would be n/3, where n is the number of sheep in front of Eric. Do not focus on this notation, though. We don't want the emphasis here to turn to symbolic notation.
It's important to understand where the "three-ness" appears in the situation: 1 sheep is shorn, and Eric cuts in front of 2 sheep. Look at the three possible situations that Eric can be in when at the end of the line, and how they relate to the remainders when dividing by 3.
When we're completing the table in Problem B5, we will have to work backwards for the last two entries. In fact, there are multiple answers for these because the function is not one-to-one.
The beauty of this problem is that it at first seems so simple, yet the extensions are quite challenging, even for sophisticated learners.
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