Session 1 Notes: Algebraic Thinking

A B C

Notes for Session 1, Part B

Note 3

The Eric the Sheep problem challenges us to describe patterns and use them to predict for large numbers of sheep. The problem is a good choice for this session because the pattern is more easily described in words than in symbols. Some people, however, may become frustrated that they can't find a symbolic rule for describing the pattern. At this stage, it may be difficult to accept the notion that we're thinking algebraically, even though we're not using symbolic notation.

Groups: Begin the activity by having eight or nine people come to the front of the room. These people are sheep, and one of them, Eric, is at the end of a line waiting to be shorn. (Be sure to designate both an Eric and a sheep shearer.)

Here is the story: It's a hot summer day, and all of the sheep are standing in line to be shorn. Eric is at the end of the line, and in this case there are __ sheep in front of him (as many as there are standing in line). But Eric is impatient, and every time the shearer takes a sheep from the front to be shorn, Eric then sneaks up the line 2 places. (Groups: Act this out. First move a sheep up to be shorn, then have Eric move up 2 places. It's important that you sequence the shearing first, and then Eric moving up second.)

Then consider: How many sheep will be shorn before Eric? Guess the answer first, then act out the remaining steps.

Next, find some way of predicting how many sheep will be shorn before Eric if there are 50 sheep in front of him.

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Note 4

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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Note 5

Groups: Discuss Problems B10 and B11 as a whole group. Although everyone may not be aware of this, we are in fact choosing a representation to describe a real-world situation. Unfortunately, many problems that we have encountered do not give us a choice about representation.

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