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…
Author: Clio
Isosceles Triangles in a Quadrilateral
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…
Carole and the Common Core
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…
Complex Numbers making Real Numbers
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…
What is Multiplication, Anyway?
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…
Transformation Rules
We’re working on rigid transformations in my Geometry classes. The basic transformation rules for translation and reflection over a vertical or horizontal line are straightforward; here, they’re written as functions, rather than the briefer vector notation. Translation of \(h\) horizontally and \(k\) vertically: \[(x, y) \rightarrow (x + h, y + k)\] Reflection over a…
Dan Meyer’s Really Really Really Difficult Puzzle
Dan Meyer offered this puzzle. The essence of it is, given an arbitrary number for volume, can we build an algorithm that will always generate the integer side lengths that give us the least surface area? He put it in more “real world” terms than that, but that’s the gist. If we did it in two…
Choosing a Strategy
(Reposted from my blog for my students) I will often tell my students to select strategies that work best for them to solve a problem, rather than focusing on a single specific strategy. What do I mean by this? I don’t mean that students should use inefficient strategies. One of the things that successful students…
Filling in Blanks: The General Case
Discussion about the puzzle behind the previous post led to this question: Is it possible to construct a similar question, where there are three of five values and a mean given, and where the median is half one of the missing values, with has exactly two solutions? That is, given the values {a, b, c,…
Filling in Blanks
Here’s an interesting mean/median question. Assume your lab assistant took five readings, but only recorded three of them: -3, 5, and 7. The assistant also recorded that the mean was 6, and (for reasons lost on you) that the median was half of one of the missing readings. What were the readings? Actually, the problem…