Function Diagrams

On BlueSky, Bryan Meyer asks:

Henri, in your experience, what are the pros and cons of using function diagrams with kids (in addition to the more standard Cartesian representation)? 

My BlueSky reply: “Cartesian graphs are a life tool. Function diagrams are a learning tool, so less crucial, though I have found a few activities at various levels irreplaceable.”

Function diagrams represent functions by connecting points on a (usually vertical) x-axis with points on a parallel y-axis. If you’re not familiar with them, see this page on my website and follow the links. (Warning: it’s a bit of a rabbit hole.) Here is a diagram for y = 2x – 3:

I’ll use this post to answer Bryan’s question.

First, the cons:

• As with everything else not already in the curriculum, it does take time, so you have to sort out what it will replace.
• It’s a hard sell to colleagues. Perhaps because it’s not standard fare, I found it difficult to convince some in my department of the benefits of this representation.

And now the pros:

• Working in a different representation helps us reach a broader range of students. Function diagrams are not more difficult than the standard representations, and for some ideas they are more straightforward, so they provide an additional way into the concepts to students who are struggling. At the same time, they are interesting to kids who already understand the threesome of table, formula, and graph. The usual objection to teaching something important in yet one more way is that “it will confuse the students”. I have not found that to be the case — quite the opposite.
• Understanding obtained this way complements and reinforces that which is obtained the traditional way, especially if connections are made between representations. 

Here are four “best of” function diagram activities / lessons which yield a great educational payoff with a minimal time cost — one per grade level. All four of these activities also work well in professional development workshops. In fact, even if teachers do not end up using them with students, that representation expands and deepens their own understanding.

1. Nine Function Diagrams is an excellent conversation starter, as it triggers discussions of many issues that come up in the transition from arithmetic to algebra. The worksheet only involves “one-step” functions, so the focus is really on operations and algebraic notation — two foundational ingredients in that transition. I used it near the start of Math 1, a course my department offered to ninth graders who had not taken algebra in middle school. 
2. Sixteen Function Diagrams takes it a step further: it involves identifying “two-step” linear functions, in other words the m and b in the usual y = mx + b. What’s great about it is that it forces a deeper understanding of those parameters as students figure out and share strategies to speed up the process of recognizing the functions. I used it to review linear functions at the start of Math 2. Because the representation is unfamiliar, it is a way to review these concepts in a non-rote way that is interesting to all students.
3. Name That Function! asks the students to identify standard Algebra 2 / Precalculus functions by watching animated function diagrams. It is quite entertaining, and again makes a great conversation starter.
4. Finally, function diagrams provide a rich environment to discuss rate of change in a way that complements slope. And they allow for a representation of the composition of functions that is much, much clearer than is possible on a Cartesian graph. Putting those two things together reveals why the derivative of the composite function is the product of the two derivatives — the chain rule. They also make the idea of inverse function much more visual. One way to start that conversation is in this worksheet, perhaps also using the images in #2 above for rate of change.

There is much more about these and other activities, a bibliography, and many function diagram applets on my website, starting here: Function Diagrams. And as always, I’m available to answer any questions.

One more thing: on BlueSky, Bryan suggests that this representation can be used to see translations, dilations, and reflections in a 1-D version. That is of course mathematically legitimate. Pedagogically, my experience is that it is not the best way to introduce geometric transformations, because the lack of shapes to transform makes it much less engaging and interesting to kids. However, it is an effective way to test understanding of those transformations in a later activity or assessment, narrowing the focus to 1D after introducing it in 2D. See also this kinesthetic activity, which starts in 1D and introduces rotation and the second dimension.

— Henri

Previous blog posts on function diagrams:  Learning from Teaching  |  Asilomar Report, Part 2

Function diagrams home page.