This is a neat trick and figuring out how it works and the patterns involved is really kind of fascinating. The fact that this works with any sequence that follows the rule for the Fibonacci sequence: Add the two previous terms to get the next term, makes this a neat problem to solve for middle school students.
How it Works:
The Lesson:
Introduce the lesson by performing the “trick” as shown in the video above. Ask all students to create their own “nacci” sequence and give them time to find the sum of their 10 numbers and tell them their sum as you walk past their tables. When students ask, “How did you do that?” tell them that is the problem of the day. So, the main question to answer is: How can the sum of the first 10 digits of a Fibonacci sequence be found quickly?
All of the Fibonacci-like sequences students create are likely as different as the students themselves, so prompt students with what is the same and what is different about the Fibonacci sequence and one of the student-generated sequences. Post these two sequences for all students to see. Alternatively, you can use the image below.
Possible student responses (if using the image above):
| Same | Different |
| Both Fibonacci | Different sums |
| Both follow the same rule | Start with different numbers |
| Both have 10 terms | All different numbers |
| Both have a multiple of 11 (not all sequences like this have a multiple of 11 in the first 10 terms | The one on the right has 3-digit numbers |
| Both have numbers in the 40s |
Facilitate a discussion about these ideas. Highlight the ideas that they both have 10 terms and they start with any two numbers as a place for students to start. Groups of students should work together to discover the secret to how this trick works and why it works.
Possible questions to keep students thinking:
- What does it mean when we say that the sequence starts with any two numbers?
- What do you notice about the generalized pattern?