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" ... Here the green rectangle on the right has the correct proportions: a/b = φ, the golden ratio. This number is defined by the property that if you place the blue square of side length a adjacent to the green rectangle, the resulting rectangle has the same proportions; that is, (a+b)/a = a/b. This equation allows us to find the value of φ exactly. Take b=1 and then simplify to get the equation a² = a + 1. Via the quadratic formula, we then get two solutions, one positive and one negative. The positive solution is what we call φ: ... "
" ... Most of the arguments against algebra come down to the utilitarian: the old "When am I ever going to use this?" that teachers of every discipline abhor. And it's true that if you ask me to give you an explicit example of how you'll use the quadratic formula in your daily life I will struggle. I will retreat into standard arguments about how mathematical thinking is important for intellectual development, etc. and that algebra actually has practical applications involving ratios and scaling. ... "
" ... Then, while I was walking around the room helping them with the quadratic equation work one to one, I discovered that Pat was just crying, so I talked to Pat for a bit. Mom is sick and it has made for a tough time at home. I told the rest of the class to keep working quietly while I gave Pat a chance to talk it out (it's a Wednesday, so there's no nurse or guidance counselor on duty to handle this kind of issue). At the same time, Sam needed extra extra help on the assigned work, so I kept Sam after and then missed doing my socialist indoctrination with both fifth and sixth period. ... "