A Novel Take on Graphing

Here’s the pitch:

Draw a horizontal axis and label it with the numbers $1,10,100,...$ (What numbers should go to the left of 1?)
Draw a vertical axis, labelled similarly.
Draw a downward sloping line at a 45 degree angle through the “origin” (which seems to now be the point $(1,1)$ . The equation $y=-x$ no longer seems appropriate. Can you write an equation that fits this graph?
Try going in the other direction. If you write down the equation that would normally give a circle and try to graph it on these axes, what do you get?
Try lines, parabolas, circles, exponential functions, logarithmic functions, rational functions, etc.
Try other ways of labelling your coordinate axes.

Note on Notation: My students and I reserve $x$ and $y$ for the standard axes labelled in the standard way. We generally use $a$ to represent the strange values along the horizontal axis and $b$ to represent the strange values along the vertical axis.