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[mathjax]I was thinking about the third scenario described in al-Khwarizmi’s al-Jabr: \(x^2 = 3x + 4\). I was curious about the integer solutions of the general pattern, \(x^2 = ax + b\). It’s easy enough to demonstrate that this will hold if \(x = b = a + 1\), since that means \((a + 1)^2…
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[mathjax]First of all, let me get this out of the way: “Hey, you kids! Get off my lawn!” In this post, I comment on the notational shifts from what I was trained in back in the 1980s and what textbooks do now. I was reading Power Puzzles 2 by Philip Carter and Ken Russell when…
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[mathjax]Today’s lesson in my Geometry class was on the use of the geometric mean when finding missing values of right triangles. For every right triangle, two of its altitudes are the legs and the third is perpendicular to the hypotenuse. The length of the altitude is the geometric mean of the lengths of the two…
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[mathjax]In my previous post, I looked at the first two detailed examples provided by al-Khwarizmi in his compendium, the title of which gives us the word “algebra”. Al-Khwarizmi discussed three types of mathematical objects: Numbers (N, constants), roots (R, unknowns), and squares (S, squares of unknowns). Because he was limited to positive solutions, he was…
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[mathjax]The word “algebra” comes to us from the title of a book, al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabalah written by Mohammed ben Musa al-Khwarizmi (there are variations in the transliterations of both the title and the author) around AD 825. He was not the inventor of algebra; indeed, his book was a compendium and extension…
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[mathjax]In my last post, I noted that it’s possible to create an isosceles trapezoid from four isosceles triangles, but I wasn’t sure if there was a way to construct a quadrilateral from isosceles triangles such that the quadrilateral was neither a rectangle nor an isosceles trapezoid. Now I know that it is not. Let’s reconsider…
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[mathjax]In this post, I’ll discuss two issues. First, I’ll look at a problem taken from a major textbook, and explain why the solution is wrong. Then, I’ll discuss why this particular problem bothers me in the greater context of mathematics education. First, the problem. This is a question from Pearson’s Common Core Geometry supplemental materials…
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[mathjax]Here’s another meme criticizing the Common Core: The criticism is that the student has provided a fully correct answer and gotten dinged for not providing an estimate. This is, of course, taken as yet another illustration of why Common Core is ruining America’s future. In other cases of Common Core outrage, the problem has been…
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[mathjax]As a point of curiosity, I found myself wondering when a complex number to an integer power creates a real number. For the sake of completeness, I also looked at when the result is a fully imaginary number. More rigorously, define \(\mathbb{C}^*\) as the set of complex numbers \(a + bi\) where both \(a\) and…
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Yet again, the internet has seized upon an elementary student’s math homework and has decided to argue about Common Core. This time, it’s about a test question. The student was asked to “Use the repeated addition strategy to solve : 5 x 3” (Reddit, via The Telegraph, via Greg Ashman); the student answered “5 + 5…