Setup Its time for another geometry walkthrough motivated this week by this interesting problem from James Tanton. "Two isosceles right triangles are hinged at corners as shown. Line segment connecting midpoints of their hypotenuses is used as the hypotenuse of yet another isosceles right triangle. Prove A, B, C lie on a straight line. Can anything be said about where B sits on segment AC?" Initial Impressions My first impression was that this reminded me a bit of the 3 hinged squares problem where among the key key observations was that the area of the triangles between the hinges wa ..read more

Its been a long while since my last social media post: https://blog.mathoffthegrid.com/2017/08/how-i-use-twitter.html  and everything is all of a sudden in huge flux.  With all the turmoil on Twitter I've been exploring Mastodon on the math focused server a friend runs: mathstodon.xyz.  A new platform means starting over again You have to rebuild your network of follows and followers.  This is huge and discovering people has made my previous two attempts at using Mastodon unsatisfying.   But this time is different due to the chaos at Twitter. Large en ..read more

Source: @ElyemGercek I saw this post and immediately thought: "What an odd connection to Pascal's Triangle. What's going on?" Due to time constraints I didn't get a chance to look into this for several weeks and while its not quite as geometric as I hoped the connection is fairly natural. First just as in the triangle you need an initial row of values from which the relation will then generate all the rest of the rows.  In this case its the triangle with a single median.  Stewart's Theorem  is helpful here for finding the initial equation.  What's novel is while ..read more

What started this essay was two posts I recently saw. The first started with the idea that no student should take more than five minutes on a problem. If stuck it was unlikely to be productive and they should just move on.  In case, there was any confusion this expanded to the idea that investigations or inquiry based learning was ineffective. Secondly, was a note from an acquaintance about how he was about to start a class for pre-service teachers about the pedagogy of problems in the classroom which piqued my interest about which problems he would be using.  Together these readin ..read more

Graphic scores by John De Cesare (1890–1972). Welcome to the 203rd Carnival. For all the other carnivals future and past, visit The Aperiodical where you can also submit future posts.  This is my fifth go around hosting a carnival and the first time I've done so during March. So hold onto your hats and prepare to be inundated with Pi Day material.  For those jaded souls among you, there will be plenty of interesting math and  no digit reciting contests. (If you're in the other camp I found this  Pi Day Digit Song quite amusing this year) Before we really get go ..read more

I just saw a nice visual proof of the ratios for the 15-75-90 on the internet via @ilarrosac.   This one works via symmetry and the Pythagorean theorem.  ..read more

I really enjoyed working through this problem from Stanley Rabinowitz I saw recently on Twitter. This was one of those problems where I circled around it and after finding the first solution realized I could improve it a lot and simplify the algebra fairly drastically. One first impression,  I saw two things. First there was power of the point at D which could be used to give expressions for DQ and AD.  On the other hand AP looked like  it would be hardest segment to get a purchase on. My second thought was around the midpoints and filling in the similar triangle DEF lookin ..read more

What follows is a walkthrough of the problem above which I had a lot of fun playing around with. Most of the solutions I saw online used trigonometry and so just to be different I'm going to approach this solely via scaled similar triangles relying on the cyclic quadrilateral.  Note: because the answer involves the golden ratio I was hoping to emphasize it more in the proof but so far I don't see a geometric avenue to bring that forward more prominently.  If you happen to find one please comment and tell me about it. Basic Triangle Because of the cycle quadrilateral ACBD and ..read more

Everything I discovered after doing the linear algebra section of PreCalc last year I really liked both 3Blue1Brown's Essence of Linear Algebra series: Youtube link   and Gilbert Strang's OCW series: OCW Link From 3Blue1Brown - I  took away the emphasis on matrices as linear transformations, the focus on basic vectors and the illustration of the geometric interpretation of the determinant. From Strang: I picked up the idea of focusing on column vectors and deriving the determinant via properties first. I thought it was useful to come to stress what was general ..read more

I read a really interesting post by David Bressound  on joining the advisory board for AP PreCalculus: link.  The gist of which was despite misgivings he thought setting up the course would do more good than harm by providing an opportunity to standardize the curriculum.  And in general I think this part of his analysis is probably true.  Given an AP Pre-Calculus test, a lot of high schools that already teach AP Calculus are likely to switch over and replace their own course with one aligned to the AP standard.  I'm much less sure if it will prove attractive to s ..read more