Notes on Embedding – L’s and Plusses

L’s and Plusses is what one student called the shape of graphs.  I still didn’t know what she meant, until she showed me.  Here is an L, she said, and sketched it:


And this is a Plus:


I got that what she wanted to distinguish is not about the red lines, but about the way the axes are shown.  In secondary school math vocabulary, we would say that his L graph shows a single quadrant of the graph, and the Plus graph shows all the quadrants.

What I realized, and why I didn’t immediately catch on to what she meant, is that I tend to think of the graph as the red thing.  Yes, axes are there, and scales are there, but I had only thought of them as “anchoring” the red line, and the red line was the “real” graph for me.

Yet I’ve since encountered more students for whom the difference between the L’s and the Plusses is very significant and real.  The jump from drawing L graphs (mostly in elementary school) to Plus graphs (mostly in secondary school) can be a real hurdle, and one that is aggravated by teachers often not getting what the big deal is.  One student literally didn’t know where to start when asked to draw a graph, being given a worksheet that showed a Plus.  The teacher thought to have made the problem easier and quicker by pre-drawing the axes and the scales – but not for this student.  When asked if she could do the problem by ignoring the worksheet, she performed fine.

Looking back at an earlier discussion about embedding, we can say that the L universe is embedded in the Plus universe.  The L is just one piece, one quadrant, of the Plus.  By making room for the other three quadrants, we didn’t intend to muck with the L quadrant.  We intended the opposite: we intended to honor and confirm the L quadrant, but allow room for parts of the graph that didn’t fit in the L quadrant.

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