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Hi all! Author here. Didn't expect to see this on HN so early this morning :). Lots of good feedback, I'll try to answer some of it collectively here. HN read

by kalid 14y ago

Hi all! Author here. Didn't expect to see this on HN so early this morning :). Lots of good feedback, I'll try to answer some of it collectively here. HN readers who can spit out the quadratic formula in their sleep aren't the real audience for this article :). It's someone googling "Why are we factoring all the time?". A high-level question needs a high-level answer before diving in. I'm not a fan of "Because that's where the graph touches the x-axis" because it again leads to... Why does touching the x-axis matter? My intuition is that we have a system and we have something we want it to become. The trick is to track the difference as its own system, write it as interlocking parts, and break any of the parts. "How do we break a chain?" => we break any of the links. Intuitively, that's what we're enabling when we factor an equation into a series of multiplications. I'm probably going to continue to reword the explanation, but that is fundamentally why we bother rewriting as a series of multiplications. On rigor and simplicity: guilty as charged. I believe you need to become intuitively comfortable with an idea, even if slightly incorrect, before getting into the nuance. We tell kids a cat is a furry animal that has claws and a tail. Later, we refine to say they're all descendants of a common ancestor. Later we say there's this thing called DNA which holds genetic information, and all cats have DNA in common. See this article for more: http://betterexplained.com/articles/developing-your-intuition-for-math/ http://betterexplained.com/articles/developing-your-intuitio... I go into the 4 rigorous and common definitions of e, something which confused me (and many, many engineering students) for years. But approaching with intuition it all snaps together. Again, that's my teaching style though! It's been very successful for me and other students. Rigor is always available on Wikipedia and Mathworld if you need it.

klochner 14y ago
I applaud your effort to describe why we factor equations, I just don't think you've done that here. A high-level question needs a high-level answer before diving in. I'm not a fan of "Because that's where the graph touches the x-axis" because it again leads to... Why does touching the x-axis matter? Because that's where the the system "becomes what you want it to become", i.e, where: x^2 + x - 6 = 0 x^2 + x = 6 You're trying to describe factoring to someone without the basic fundamentals to even need to do factoring. By analogy, that's like saying "I want to explain why we need piston rods, but I don't want to mention anything about engines." Prerequisites are meaningful. [Edit] I just verified that graphing is typically taught after factoring, so I at least sympathize with the challenge you're facing.
- kalid 14y ago
  Thanks for the feedback! Yep, graphing usually comes much later, so I'm trying to find an explanation that works without it. For the piston rods, I'd say something like "They help capture the power of an explosion. And an engine is a really, really cool device to make this power useful. Want to see how it works?" :)
batista 14y ago
I'm a fan, I've even bought the "Math, better explained" ebook in the past, but this article, while nice wasn't on par with the material there. Not to be scrapped though, just needs some rewording and some more meat.
- kalid 14y ago
  Thanks! Yep, this was one of the faster ones I've written. I'm doing a few experiments to see how to increase my article output :).
csense 14y ago
The way I'd explain this is the reverse of the way you're going about it. You're starting with a complicated equation and simplifying it. I'd start with the simplified equations and make them gradually more complicated, showing carefully at each stage how to get back to the simpler case. 0. We can solve linear equations. 1. If you have a factored quadratic polynomial equal to zero, you can find its solutions by solving F_i = 0 for each factor F_i (which we can do, by 0, since F_i is linear). A solution to either of those linear equations must be a solution to the original equation by zero product property. 2. If you have a quadratic expression equal to zero, you can find the solutions by factoring, then applying (1). 3. If you have an equation L = R with quadratic expressions on both sides, you can find its solution by subtracting R from both sides to get L-R = 0, then applying (2). Both rigorous and simple, and also introduces one of the standard design patterns of higher mathematics: Find a solution to a simpler problem, then see what kind of more-complicated problems you can solve by reduction to the simpler case.

by kalid 14y ago

klochner 14y ago

I applaud your effort to describe why we factor equations, I just don't think you've done that here. A high-level question needs a high-level answer before diving in. I'm not a fan of "Because that's where the graph touches the x-axis" because it again leads to... Why does touching the x-axis matter? Because that's where the the system "becomes what you want it to become", i.e, where: x^2 + x - 6 = 0 x^2 + x = 6 You're trying to describe factoring to someone without the basic fundamentals to even need to do factoring. By analogy, that's like saying "I want to explain why we need piston rods, but I don't want to mention anything about engines." Prerequisites are meaningful. [Edit] I just verified that graphing is typically taught after factoring, so I at least sympathize with the challenge you're facing.

kalid 14y ago
Thanks for the feedback! Yep, graphing usually comes much later, so I'm trying to find an explanation that works without it. For the piston rods, I'd say something like "They help capture the power of an explosion. And an engine is a really, really cool device to make this power useful. Want to see how it works?" :)

batista 14y ago

I'm a fan, I've even bought the "Math, better explained" ebook in the past, but this article, while nice wasn't on par with the material there. Not to be scrapped though, just needs some rewording and some more meat.

kalid 14y ago
Thanks! Yep, this was one of the faster ones I've written. I'm doing a few experiments to see how to increase my article output :).

csense 14y ago

The way I'd explain this is the reverse of the way you're going about it. You're starting with a complicated equation and simplifying it. I'd start with the simplified equations and make them gradually more complicated, showing carefully at each stage how to get back to the simpler case. 0. We can solve linear equations. 1. If you have a factored quadratic polynomial equal to zero, you can find its solutions by solving F_i = 0 for each factor F_i (which we can do, by 0, since F_i is linear). A solution to either of those linear equations must be a solution to the original equation by zero product property. 2. If you have a quadratic expression equal to zero, you can find the solutions by factoring, then applying (1). 3. If you have an equation L = R with quadratic expressions on both sides, you can find its solution by subtracting R from both sides to get L-R = 0, then applying (2). Both rigorous and simple, and also introduces one of the standard design patterns of higher mathematics: Find a solution to a simpler problem, then see what kind of more-complicated problems you can solve by reduction to the simpler case.