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The B/2 term places the center of the parabola, which you can verify by either inspecting the derivative or fiddling with a graphing calculator. Then, because t
by saulrh 5y ago
The B/2 term places the center of the parabola, which you can verify by either inspecting the derivative or fiddling with a graphing calculator. Then, because the parabola is symmetric, the zeros must be a pair of points mirrored across the center, with the distance from the center determined by the vertical offset (the C term) versus the narrowness (the A term). Which I definitely agree is an easier way to deal with the problem! But I think that, if you're going to do it that way, it might be easier to work from geometric intuition instead of an algebra trick that represents but obscures the same relationship.
- jonsen 5y agoI taught it this way in homeschooling with heavy support of graph-geometric intuition, drawing and looking at various parabolas. Works great.
- civilized 5y agoIt's only obvious that x = -B/2A is the parabola's axis of symmetry if you complete the square in the traditional way, or use calculus to locate the extremum. So it seems to me that we don't need a new method, we need better teaching of the tried-and-true method. If children were exposed to this new thing, even more of them would be traumatized and develop a lifelong loathing for quadratic equations. The ideal method, IMO, is to complete the square AND provide a geometric interpretation of the expression that results.
- jonsen 5y agoIt can be obvious in other ways. For example if you can see how a parabola is the product of two straight lines. And you can teach it that way.
- civilized 5y agoI think, like the OP, this just dances around completing the square without actually eliminating the need for it. How do we know that all parabolas can be expressed as the product of two symmetrical straight lines?
- jonsen 5y agoThis is not about deep math. It's about pedagogics. How to generate some understanding and good feelings for solving quadratic equations.
- civilized 5y agoI just don't get why making some unsupported postulate about how parabolas can be generated from straight lines is better pedagogically than teaching completing the square and providing a geometric interpretation for completing the square.
- jonsen 5y agoThat's ok. I can't explain it here in a few lines. It's rather a pedagogic principle that permeates most of my teaching. It's not just a postulate. It's demonstrated. A style used from the start of algebra.
- SilasX 5y agoI honestly don’t see how this version is any more intuitive than regular completing the square.
- jonsen 5y agoPedagogically there's more to it than just the algebraic manipulations. The algebra here is basically as dull as all other algebra. It's the teacher's job to make it come alive.
- SilasX 5y agoYour previous comment just suggested that there's something pedagogically superior about this approach. If you don't believe that, you shouldn't have made the comment.
- ivansavz 5y ago> complete the square AND provide a geometric interpretation of the expression that results. +1 for this. This is the approach I would recommend too, see Figure 2.2 for the picture behind the completing the square trick: https://minireference.com/static/excerpts/noBSmath_v5_preview.pdf#page=23 https://minireference.com/static/excerpts/noBSmath_v5_previe...
- civilized 5y agoYes, and it has a coordinate geometry interpretation too. Completing the square represents the original quadratic function as a basic x^2 parabola that's been scaled vertically, shifted vertically, then shifted horizontally. If you understand what each of these three operations do to the roots of the parabola, you can derive the quadratic formula. (It's basically just the algebraic way with diagrams, but may help the geometry-loving/algebra-hating types out there.)