Sheep Talk
Why do this problem?
Doing
this investigation learners notice patterns and make and prove
conjectures. It offers one of the best ways for learners to
discover Fibonacci sequences for themselves. Whether or not they
have already met these sequences it is important to ask whether the
pattern will continue and, more importantly, why it will
continue?
Learners will notice that the same sequences of numbers occur when
they count the number of A's in the words, when they count the
number of B's in the words and also when they count the total
number of letters.The teacher could simply get the learners to
count the letters, fill in a table, make conjectures about how many
A's and B's in the next word and then check if their conjectures
were correct. Then she can choose how long to pursue the reasons
and the proof that the sequences are Fibonacci sequences.
For the proof of why this process gives the Fibonacci sequence you
have to consider how the letters change from one word to the next
and generate copies of themselves. This can lead to a lot of
mathematical talk and it can give excellent practice in
mathematical reasoning and communication.
Possible approach
As with all problem solving, the first step is to understand the
question. If the teacher asks the learners to read the question for
themselves and then to tell
her the rule it will give the learners practice in thinking
for themselves. Asking the learners to write the next word on their
individual 'show-me boards' ensures that every learner is actively
involved. A list of about six words should be written down and then
a table of results set up and the numbers of letters filled in the
table.
Learners can each fill in their own tables and be asked to work in
pairs to see what patterns they can spot. When they see a pattern
(conjecture) they should be asked to use the Sheep Rule to write
down some more words in the sequence and see if the pattern
continues (test the conjecture). If they find it does then learners
should be asked to try to explain why the pattern comes out that
way.
Learners can be asked to write down the rule for the sequence in
words and then perhaps if they can use symbols to give the rule. At
this age the teacher may simply accept the different notations
offered by the class members or choose to discuss with the class
why it is convenient to have an agreed (standard) notation. This
could lead comfortably to using algebraic notation as only counting
is involved and not solving equations, which is perhaps more
demanding.
(The Fibonacci sequence can be defined as the sequence where the
$n$th term is denoted by $F_n$, where $F_0$ and $F_1$ are $0$ or
$1$ and $F_n=F_{n-1}+F_{n-2}$.)
Key questions
Do you notice a pattern in the sequence of numbers?
How many A's do you think there will be in the next word (that you
have not written down yet)?
Would you like to write down the next word in the list and see if
you were right?
Look at an A in a word. How many A's come from it in the next word
and in the word after that?
How many A's in a word come from an AB in the word before it?
How do you find the number of A's in the tenth word? Why?
Would you have to write the whole list of 20 words to find out how
many A's there would be in the twentieth word?
Ask similar questions for B's
Possible extension
Try the problem
1 Step 2 Step
Possible support
It is helpful to write the words underneath each other, with the
letters spaced out so that the learners can see how each letter
comes from one of the letters in the word above it.