The prompt, in giving rise to different lines of inquiry, is suitable for students in years 7 up to 12 (grades 6 to 11). If using the prompt with the younger end of the age range, the teacher might replace the generalisation with a particular case - for example, The sum of the first eight terms of the Fibonacci sequence equals the tenth term subtract one. The students might then create their own generalisation after exploring other cases.

The slides support the following lines of inquiry:

• Generating the terms in the Fibonacci sequence;

• Summing different terms to find properties of the sequence;

• Expressing the terms algebraically and proving properties of the sums are always true;

• Proving the generalisation in the prompt and others by mathematical induction; and

• Solving problems related to missing terms in Fibonacci-type sequences, including by forming and solving simultaneous equations.

Question, notice, and wonder

During the question, notice, and wonder phase of the inquiry, the teacher must draw out the students' existing knowledge of the Fibonacci sequence and how it is generated. The teacher's aim is for all students to have a clear understanding before exploring the prompt.

In explaining the sequence, the teacher might refer to Leonardo of Pisa's original problem about the growth of a population of rabbits and the assumptions within his model. Alternatively, students might research the origins of the sequence outside the classroom.

Exploration

The initial phase of exploration sees students choose different values for n to verify the prompt is true for particular cases. 

An interesting line of inquiry that arose in a year 7 class involves testing Fibonacci-type sequences that start with different numbers. For example, the sum of the first five terms of the sequence 1,  4,  5,  9, 14, 23, 37 is 33, which is the seventh term subtract the second one. (In the prompt, you subtract one because that is the second term of the Fibonacci sequence.) 

The result can be proved for the first five terms of any Fibonacci-type sequence using algebra (where a and b are the first two terms): a, b, a + b, a + 2b, 2a + 3b, 3a + 5b, 5a + 8b. The sum of the first five terms is 5a + 7b, which is the seventh term subtract b (the second term).

March 2025