Students will be introduced to discrete unary recursive functions. They'll start with function like the following:

$$\begin{align} & f(0) = 0 \\ & f(x + 1) = f(x) + 2 \end{align}$$This is a funny way of writing \(f(x) = 2x.\) Students should also learn these functions can be expressed as piecewise functions. For example, here's another way of expressing \(f:\) $$f(x) = \begin{cases} 0 & \text{if }x = 0 \\ f(x - 1) + 2 & \text{if }x \gt 0 \end{cases}$$

Then students will learn how to express linear functions. Finally, students will learn how to convert between the explicit formula for an arithmetic series:

$$a_n = a_1 + (n - 1)d$$and the recursive formula:

$$a_n = \begin{cases} a_1 & \text{if }n = 1 \\ a_{n - 1} + d & \text{if }n \gt 1 \end{cases}$$Khan Academy jumps right into expressing arithmetic series as piecewise functions. I think this is a mistake. The power of pattern matching is obvious from the existence of functional languages like Standard ML. The best way to approach pattern matching and recursive functions, in my opinion, is to start simple, with functions like \(f(x) = 2x,\) then graduate to more complicated functions, such as the recursive formula for arithmetic series.

Next, give your students these challenges:

Conclude by leading this investigation:

Venn Diagram Puzzles

by MathPickle