What is a function?

What's the best way to teach functions?

This can be a pretty obscure and abstract concept. I've tried many different things over the years to develop some understanding for students while also establishing some relevance. I'm still looking for something perfect to introduce this concept.

I like Dan Meyer's look at functions. He commented ....
Next Week’s Skill
Determining if a relationship is a function or not.
This is another skill that can become quickly instrumental (run a vertical line over the graph, etc.) and obscure why it is aspirin for a particular kind of headache.
Let us know your ideas for motivating the definition of a function in the comments.

He gathered up feedback from others and put together a nice blog post on this "headache". Check it out at

If Functions Are Aspirin, Then How Do You Create The Headache?

I like his approach to considering that functions provide you with certainty. I think I will try a variation of his "game" to introduce that idea to students.

Then I'll do an more abstract peak at what that means (remember we established what a relation is. Saying something like "what does this mean algebraically?". Perhaps I will use something like you see in this Khan Academy video "What is a  Function".  But then I always like to acknowledge how tricky it is to understand what is meant by "every input has exactly one output" - what does that feel like? How would it not make sense to have one input have multiple outputs? So I like to explore this a bit more concretely with students to create some relevance with the idea of certainty. I like doing the "Ice Cream Shoppe" example (I may use this as a foldable....).It's just a clever little situation that applies the idea of functions to a real life situation.

Ye Old Ice Cream Shoppe

A new ice cream shoppe opens in town and you get a part time job there. I have a visual with one scoop, two scoops & three scoops to go with this. We look at three scenarios.

scenario #1 the regular prices for the ice cream cones are $2 for one scoop, $3 for two scoop and $4 for three scoop. We map this on our diagram. Every input (cone size) has exactly one output (price). So this is a function and in fact it is a one-to-one function. This is clearly well defined, predictable, and "makes sense".

scenario #2 it is the grand opening of the shoppe and for the day all sizes of ice cream cones are $2!  Yay! so we map this on the diagram. Every input (cone size) has exactly one output (price). Yes, they all have the same price, but that's okay. It's still a function. This is also well defined, predictable, and makes sense. It is a function. 

scenario #3  your boss is out of the shoppe for the day. Your best friend comes in and he orders a 3 scoop and you charge him $2 (!) Then your least favorite person comes in and orders a 3-scoop and you charge him $5! Whoa!  Wait a minute, the input of a 3-scoop has two different outputs in this scenario ($2 and $5). Even though this might make sense to you it's really not a good thing to be doing. It is not well defined or predictable (who knows what you will charge the next person who orders a 3-scoop).

the visuals above haven't been "worked through" with the mapping arrows and labeling input/output (domain/range) and commenting on whether they are functions and why or why not. This is all something students would do. 

Interactive Student Notebook Pages for Functions

I do a couple of pages. First there is a frayer diagram on the left and sorted function cards opposite that on the right hand side.

Then we do two pages for domain & range. The left hand side compares continuous vs discrete with information about domain & range.  Here are examples for students to work out on the right side reflect under the DIXROY.  We already learned how to write interval notation, so I have them do that with domain & range.

Then I do a two page spread of "Function Machines & Function Notation" with examples on the right hand side. The "under examples" we just filled in with no foldable.

Links for each of those are contained via the descriptions above. 

Sarah Hagan has a lot of other good ideas on her blog math=love. You'll see I got a lot of my ideas from her!  Thank you again Sarah!

Addendum on domain & range. I found students had trouble with identifying domain and range in graphs especially when it got tricky. So my plan is to add one more foldable to this series here borrowing once again from math=love (this time her algebra 2 page on functions). She's got a great set of foldables and explore cards to visually understand domain and range. Read the whole entry on her blog if interested, but here are some images to give an idea of what she did.

But remembering how long cutting can take with my students and Sarah's comments about that, I will use my trusty cutting board to get those cards cut up ahead of time (or have them cut them at home the night before....)