Eagle eyes. Thanks for the note.

I maintain the missing concept is a function that takes a number (or two for a plane) from a number system and produces a point in a space. It is often said a function maps from one to the other. Elaborated and expanded in the last 150 years, the function concept has come to play a central role in mathematics, but nothing has come along to introduce the idea into school education. The result is not pretty: with nothing to get at the heart of the matter, education is left to explain what has happened, not show how to make it happen.

Let me explain a “function” that all us can understand, one that we probably have now. It maps a number to a point in a straight-line space, or the other way around; it is really both a function and its inverse.

A tape measure, or a measuring tape, displays numbers, each associated with its own tick mark. When the tape measure is aligned with the line, and with its tab anchored at the origin mark that I put on the line, a tick mark associated with a number points to a point to the line. It maps one to the other; It is “functioning” in plain sight for all to see, and understand. Extending a tape, one way or the other from the origin provides a function from the signed rational number to all the points on the line that can be located by measurement.

A coordinate system on a line is a function, so is a coordinate system on a surface/plane. A student can put a put one on a line or a plane, can see the association of point and number as physical. In fact a student can put two separate coordinate systems on a line and perform all all of signed rational number arithmetic as with measurement on a line.

There is more to say, but I think one can see how an early introduction of function concept and use can clear a path to more effective teaching and learning. Anyone interested in computer animation of functioneering?

My routine is typically:
1. How many items in the next couple steps? How do you see the pattern growing?
2. How many items would be in stage 46? What would it look like?
3. How many items would be in any stage? How could you describe what any stage would look like?
4. Do you think this is a proportional relationship? Why?— At this point students usually determine if there’s a constant of proportionality or they create a step 0 to see if it would have 0 items. Some notice that the graph doesn’t go up at a constant rate so “the graph must be curved.” By the end of the year almost all students recognize when there’s a squared or cubed in the problem it’s going to be a curved graph.

5. Check by graphing— In the past I had students create their own graph in a shared google doc, but this year I’ve taken to projecting desmos on the screen. I create a table with the points we know and then graph all of the equations students came up with to see if they are the same equation and if they pass through all of the points.

All yours, Dwight.

Nice! At what point in the visual patterns tasks do you have them discuss if it’s proportional or not?

For example, looking at climate change over a short period of time gives one picture, but enlarging the frame to geological scale shows great fluctuations in temperature.

Oo. Let’s load that up in an activity where students frame climate data in different ways to draw different conclusions. What could we do with this?

Thanks, Jen. The visual setup tickled the calculus region of my region but I couldn’t articulate why until your comment. Let me know if you’d like any changes to the activity or visuals.