Students will learn how to convert between decimal and binary. Then they'll learn that a number is even in binary if it ends in 0, and odd if it ends in 1. Also, to multiply a number in binary by 2, you just have to append a 0. Likewise, to multiply by any power of 2, such as \(2^m,\) you just have to append \(m\) zeros. Exploring these ideas should help students realize that divisibility tests are dependent on the base chosen, and that multiplication by the base always amounts to appending 0s. Once this is understood, give your students the Josephus problem, as seen here. Just as seen in the video, start by making a table with columns \(n\) and \(W(n),\) for small \(n.\) Hopefully then, students will make a conjecture for \(W(1),\) \(W(2),\) \(W(4),\) etc. That is, the value of \(W(n)\) when \(n\) is a power of 2. Then challenge students to figure out why \(W(n)\) jumps up by two each time before resetting. If students are unable to conjecture the formula \(W(n) = 2\ell + 1,\) as seen in the video, make the conjecture and ask students why it's true. Once students have solved the problem, ask them for a procedure to quickly calculate \(W(n)\) from \(n.\) The solution is to take the leading 1 and place it at the tail of the number, in the units place. For example, if \(n = 41_{10} = 101001_2,\) then \(W(n) = 10011_2 = 19_{10}.\)

Conclude by leading this investigation:

Speed Guess Fractions (estimate, division)

by MathPickle