A visual proof is supposed to appeal to our visual intuition - I don't know about you, but negative areas are not something that is visually intuitive to me.
If you're bothered by the idea of negative-area rectangles, there's no need to justify the assumption that b < a, because that's the only way you can assign any meaning to either side of the equation.
One easy place to get some intuition for signed areas is in the context of integrals.
You know position is the integral of velocity right? So say you walk in a straight line from your starting point, then you keep slowing down until you come to a stop and walk backwards past your starting point.
If you were to graph your velocity vs time at some point it would dip below the t axis because your velocity would be negative. Ok cool. If you integrate from the point you came to a stop and started walking backwards you’re calculating the area above the negative velocity curve(between it and the time axis). You’ll find it is a negative area. You know it has to be negative because you walked backwards past your starting point so it gets so negative that it cancels out all the positive area from when you were walking forwards.
This sort of thing is more useful for teaching than being an actual proof. You teach the same concept in different ways and the students can form a more solid understanding of the underlying concept independent of symbols or shapes, or at least they may understand it one way if they don't get the other ways.
You can do all sorts of factorizations the same way and handle negative areas by drawing them in a different color.
FartyMcFarter|1 year ago
A visual proof is supposed to appeal to our visual intuition - I don't know about you, but negative areas are not something that is visually intuitive to me.
thaumasiotes|1 year ago
mithametacs|1 year ago
Maybe if you viewed an animation where `a` starts larger than `b`, and steps till it's smaller.
Then you could see where the negativity happens. And seeing is nice.
unknown|1 year ago
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Out_of_Characte|1 year ago
seanhunter|1 year ago
You know position is the integral of velocity right? So say you walk in a straight line from your starting point, then you keep slowing down until you come to a stop and walk backwards past your starting point.
If you were to graph your velocity vs time at some point it would dip below the t axis because your velocity would be negative. Ok cool. If you integrate from the point you came to a stop and started walking backwards you’re calculating the area above the negative velocity curve(between it and the time axis). You’ll find it is a negative area. You know it has to be negative because you walked backwards past your starting point so it gets so negative that it cancels out all the positive area from when you were walking forwards.
foxglacier|1 year ago
You can do all sorts of factorizations the same way and handle negative areas by drawing them in a different color.