Level 2 · Performance Magic

Leapfrog Addition

What if you could beat a calculator in a race to sum ten random numbers? Leapfrog Addition is a stunning performance piece that looks like superhuman computation. In reality, it is a beautiful application of Fibonacci-style sequences combined with our old friend, the Elevens Trick.

The Performance Steps

1. Ask a volunteer to call out any two starting numbers.
2. Have them generate a list of 10 numbers by adding the previous two together (exactly like the Fibonacci sequence).
3. While they are writing, secretly identify the 7th number in the list.
4. Multiply that 7th number by 11 in your head. That is your total sum.
5. Dramatically announce the total before they even finish adding the column!

The Algebraic Engine

This trick works flawlessly because of how variables stack in an additive recursive sequence. Let's replace our two starting numbers with the variables a and b. If we algebraically map out the first 10 terms of this pattern, it looks like this:

Look closely at the algebraic sequence above: 5a + 8b is exactly the 7th term! Therefore, the sum of all 10 numbers will always be exactly 11 times the 7th number.

Cognitive Benefits

Leapfrog Addition is a masterclass in connecting disparate mathematical concepts. It introduces students to recursive functions and sequence generation, while simultaneously forcing them to retrieve and apply a prior mental math skill (the Elevens Trick) under pressure. This builds exceptional cognitive flexibility.

To pull this off smoothly, you need to spot the 7th term and execute the Elevens Trick without hesitating. Use the interactive solver below to run through randomized practice sets.