This printable worksheet may be useful:Fibonacci Surprises.
This problem gives students a chance to explore the Fibonacci sequence and make conjectures about the patterns they find. By using algebra to represent consecutive terms in the sequence, students can understand and prove their conjectures while developing fluency in forming expressions and collecting like terms.
Start by introducing the Fibonacci sequence if students have not met it before:
"Here is a sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21...
Can you work out the rule for finding the next term?
Can you work out the next 3 terms?"
Take some time to discuss and make sure everyone understands that each term is the sum of the two previous terms.
Then introduce the three explorations from the problem. You might like to hand out this worksheet.
Students could work in pairs and discuss what they notice. Once they have noticed patterns and begun to make conjectures, you could bring the class together to discuss ways of proving their conjectures.
For the first of the three questions, it is enough to observe that the Fibonacci sequence goes Odd, Odd, Even, Odd, Odd, Even..., to explain why this pattern continues, and to explain why any three consecutive terms will always include two Odds and an Even term.
For the other two prompts, it's useful to choose letters such as $a$ and $b$ to represent two adjacent terms of the sequence, and then use these to write expressions for the next few terms and the combinations suggested in the problems. This is a good opportunity to discuss the power of algebra for proving general statements about numbers!
If $a$ and $b$ are consecutive terms of the Fibonacci Sequence, what comes next?
And what comes after that?
Take time to explore the algebraic representations of each term in the Fibonacci sequence.
Perimeter Expressions offers a good opportunity to introduce or consolidate collecting like terms.
Invite students to create some Fibonacci Surprises of their own. They could write out the algebraic representations of sets of consecutive terms, and explore different ways to combine them in search of interesting results. Finally, they could test their conjectures numerically before presenting their Fibonacci Surprises to their classmates.