Part B:
Reasoning About Situations (45 MINUTES)
Eric the Sheep
Here’s a simple number puzzle that leads to a surprisingly challenging investigation. [SEE NOTE 3]
It’s a hot summer day, and Eric the Sheep is at the end of a line of sheep waiting to be shorn. There are 50 sheep
in front of him. Being an impatient sort of sheep, though, every time the shearer takes a sheep from the front of
the line to be shorn, Eric sneaks up 2 places in line.
Problem B1. Without working out the entire problem, predict how many sheep will get shorn before Eric.
[SEE NOTE 4]
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.
NOTE
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.
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.
The Eric the Sheep problem is taken from Maths 300. Permission has been given by the publisher, Curriculum Corporation, PO Box
177, Carlton South,Victoria, 3053, Australia - http://www.curriculum.edu.au - email: sales@curriculum.edu.au - Tel: # 61 (3) 9207 9600
- Fax: #61 (3) 9639 1616
Session 1
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Patterns, Functions, and Algebra
Part B, cont’d.
Problem B2. One way to help solve a complicated problem is to solve a smaller version of the same problem. In
this activity, you will solve a version of Eric’s problem, one with a shorter line. Notice any patterns you find—they
will help you understand and solve the larger problem.
You can use counters, plastic chips, or coins to try out various smaller versions of this problem. Complete the following table:
Number of sheep
in front of Eric
Try It Online!
This problem can be explored
online as an Interactive Activity.
Go to the Patterns, Functions,
and Algebra Web site at
www.learner.org/learningmath
and find Session 1, Part B, Problem B2.
Number of sheep
shorn before Eric
4
5
6
7
8
9
10
11
Problem B3. Use the table from Problem B2 to predict how many sheep will get shorn before Eric if there are 50
in line in front of him. [SEE TIP B3, PAGE 27]
Patterns, Functions, and Algebra
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Session 1
Part B, cont’d.
Write and Reflect
Problem B4. Describe the strategies you used to find the answer to Problem B3 and how you could predict the
answer for any number of sheep in the line. Is your method for predicting “algebraic”? Why or why not?
VIDEO SEGMENT (approximate times: 7:21-11:01):You can find this segment on
the session video approximately 7 minutes and 21 seconds after the Annenberg/CPB logo. Zero the counter on your VCR clock when you see the Annenberg/CPB logo.
In this video segment, Professor Cossey and the participants discuss strategies for solving a problem with sheep shearing and then discuss the involvement of the number three in all the solutions.
Were you surprised by the different ideas presented in solving this problem? Which of these ideas reflect algebraic
thinking?
Problem B5. Work out the solutions for the boxes left blank in the table below:
Number of sheep
in front of Eric
Number of sheep
shorn before Eric
37
296
1,000
7,695
13
21
Help: You will have to work backwards for the last two entries. Some problems may have more than one answer.
Session 1
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Patterns, Functions, and Algebra
Part B, cont’d.
Changing the Rules
Problem B6. Eric gets more and more impatient. Explore how your rule changes if Eric sneaks past 3 sheep at a
time. How about 4 sheep at a time? 10 sheep at a time?
Problem B7. When someone tells you how many sheep there are in front of Eric and how many sheep at a time
he can sneak past, describe how you could predict the answer. [SEE TIP B7, PAGE 27]
Take It Further
Problem B8. What if Eric sneaks past 2 sheep first, and then the shearer takes a sheep from the front of the line?
Does this change your rule? If so, how?
Problem B9. The farmer hires another sheep shearer. There is still one line, but the 1st and 2nd sheep in line get
shorn at the same time, then Eric sneaks ahead. Explore what this does to your rule.
Write and Reflect [SEE NOTE 5]
Problem B10. There are several ways to represent a problem situation: as a written rule, in words or symbols, as a
graph, as an equation, or as a table. What representation (or representations) did you use for Eric’s problem? Why
did you choose those representations?
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.
Patterns, Functions, and Algebra
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Session 1