Foiled Proof
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange answerers point people to it as a quick visual intuition for the distributive law and for expanding (a+b)(c+d), and a questioner cites it for a geometric view of the product rule.
Recommended by 2 people across 2 different months, 2013 to 2014.
What it is
A 1m20s minutephysics animation by Henry Reich that gives a geometric, area-based picture of the distributive law, using rectangles split into parts. It later extends the same picture to the product rule from calculus.
What people said
Recommends Shaun · 2013-12 · Mathematics Stack Exchange · on “How to calculate $(a+b)(c+d)$”
Remember that, say, $\color{red}{(a+b)}$ is just another number, so you can apply the distributive law to $\color{red}{(a+b)}(c+d)$ to give $\color{red}{(a+b)}c+\color{red}{(a+b)}d$ but then the distributive law (in the other direction) applied twice gives $a\color{blue}{c}+b\color{blue}{c}+a\color{purple}{d}+b\color{purple}{d}$. Watch this MinutePhysics video [this video] for another perspective.
Recommends Anant · 2014-05 · Mathematics Stack Exchange · on “Prove the distributive law $a(b+c)=ab+ac$ for real numbers?”
Henry Reich - creator of minutephysics [LINK] - gave a geometrical intuition for the distributive law in this video [this video] . I've pasted a snapshot from that below. For a formal proof, I defer to Christian Blatter's answer :)
Cites Ovi · 2013-05 · Mathematics Stack Exchange · on “Geometric representation of product rule?”
At time 1:06 of this video by minutephysics [this video] , there is a geometric representation of the product rule: However, I don't understand how the sums of the areas of those thin strips represent $d(u\cdot v)$. The only way I can see it is that $d(u\cdot v)$ is a small change in the area of the square, and those thin strips do represent that; however, I'm not sure if this is correct and if it is, how formal of a proof is this? Thanks!