Fantasy High School Part 4 PreCalc
I’ve been looking around at sample curriculum sequences due to odd seeming topics I’ve seen at home via the Pre-Calculus class my older son is taking, In t...
digression
Everything is Connected (yet again)
The other week I was checking out the geometric puzzles at https://sciencevsmagic.net/geo/ as part of a MathsJam evening. A small part of the process requir...
Walkthrough of Geometry Problem from Michael Pershan
This post is motivated by a conversation I had with Michael where I asked if he would be willing to document his problem solving process and if so I would d...
Dot in a Box: Addendum
This is a continuation to my last post: dot-in-a-box
Dot in a Box
[@fleonsotelo]
Deep Dive on a Problem
Once again the Math Circle didn’t meet this time due to MLK Day rather than snow. In fact, next week, I have an offsite at work which is going to interfere ...
Another example of a problem that’s much harder one way
We had a snow day this Monday and so there was no Math Club. Instead, I’ve written a continuation in my series of posts on the curious way geometry problems ...
Trig Angle Addition Two Ways
I’m really enjoying contrasting these two approach right now to deriving the since and cosine angle addition formulas. Just like in the normal pedagogy for t...
Cool walk through for geometric transforms
[Since its AMC 8 today - here’s a geometry walkthrough instead for the week]
Heron’s Formula vs Trigonometry
This random pedagogical thought occurred to me today: Both Heron’s Formula and the Law of Cosines provide ways to find the area of a triangle with just its ...
Fantasy High School Part 3 (Statways)
I’m returning here to a perennial topic of mine: High School Curriculum Reform.
Summer ‘19: What did I do last summer?
Six years ago I really viewed many Mathematical topics as cut and dry. How hard can it be to learn everything there is to know about say Algebra I? Is the...
Angle Pairs
Visual proof that $ \angle{BCA} + \angle{DCE} = \pi / 4 $ or alternatively $ \arctan \left( \dfrac{u-v}{u+v} \right) + \arctan \left(\dfrac{v}{u}\right) = \p...
Regular Heptagon puzzle walk through
There are too many wildly different an interesting ways to attack this problem to not document.
Pentagon + Quadrature of the Parabola
Last week, I saw this really fun parabola problem from @diegorattaggi and I became interested for two reasons:
Langley’s Adventitious Angles
I’ve been working through “Geometry Revisited” and have come to a section of old chestnuts one of which was Langley’s Adventitous Angles.
Noticing and Wondering is for Everyone
One very common prompt seen online is “What do you notice and wonder?” I like the frame of mind it suggests and often use it or variants with the kids in Ma...
Euler Line Walkthrough
http://www.gogeometry.com/problem/p742-circumradius-orthocenter-centroid-midpoint-distance-square.htm
Roots of Unity: Pursuing Extra Roots
Background: This piece all started with my last post thinking about equalities of the form $ \cos (nx) = \cos(mx) $
Trigonometry Walk Through Three Ways
I saw the following trigonometry problem the other day and decided it would make another good walk through since it hits several themes I’ve been exploring.
Spring Break - Geometry Walk Through
Note: with Spring I really have no kids. Even my own are with my parents so here’s an old walk through I had laying around. On reread after a significant ga...
The curious case of the equilateral triangle and 3 concentric rings or the hidden angle bisector
Each of the vertices of an equilateral triangle lie on one of the three concentric circles with radii 1, 2 and 3. Find the length of the side of the equilat...
Let’s use the Angle Bisector Theorem extensions!
What is the ratio of the ellipse’s width to its height?#math #maths #mathchat #mathschat #nerdsniping #MTBoS #iteachmath pic.twitter.com/CGntV3U1f3 — Matt En...
Triangle in a Square: Areas
So this happened today:
Trig Walk Throughs
I’m in the process of writing a bigger piece on the role of trig formulas but I was looking for an example of their use during a geometry proof and came to t...
Reexamining the Law of Sines
I’ve been working my way through Geometry Revisited over the last few days and so far I’m really happy with the purchase.
Factorization Redux
Factoring must be in the air …
Proof of Pentagon Construction
I was walking through the following construction of a regular pentagon from the AoPS Geometry textbook recently.
Dot Product Discovery
I was reading twitter yesterday and saw this tweet:
Everything old is new again
This will be a short post but I’m excited enough to write this all down. Yesterday I had one of those moments where you go through a range of feelings. It al...
Roots of Unity and Cyclotomic Polynomials
Most of the treatments of this topic are fairly grounded in Abstract Algebra and for this post I wanted to record my hopefully simpler conceptual framework.
Fantasy High School Part 2
[See: fantasy-high-school for part 1]
Fantasy High School
@carloliwitter was tweeting recently about re-imagining High School mathematics and asking for ideas. The topic has caught my fancy even though its a bit off...
Sometimes representation does matter
This is a small observation based on a post from @samjshah on the topic of the trig double angle formulas:
More connections
I’ve been thinking alot about polynomial deltas recently. See: polynomial-differences. It turns out, that there are a variety of problems where its fun to u...
Exponent Tower Redux
I saw a different version of the tower of 7’s problem in a book I’m reading on number theory. This is the mostly rambling thought process I’ve been going thr...
In praise of the Rational Roots Theorem
First some personal historical background. In my school district, you could do Algebra in middle school but unlike a standard class it only covered linear eq...
CoCa Photo Diary - Art Math Intersection
I had a chance during lunch to look at Dan Finkel’s brainchild at the Center on Contemporary Art.
Fun with Pentagons
I’m in the mood for a geometry walk-through. I’ll start out by saying this one has tons of solutions. I’ve thought of 3 or 4 and seen several additional ones...
11/14 2017 AMC 8 and a digression
Math Club was super easy for me today. I paced outside the classroom while everyone took AMC8.
Elegant Solutions vs. Hacking
This is a study in contrasts around a fun problem by @eylem:
Angle bisector
I’ve officially reached the point of the Summer where I’m missing interacting with kids besides my own. In the meantime, this is another geometry walk-throu...
Open Ended Problems
I’ve been thinking more about open ended problems after reading a couple of different posts recently. Full disclosure: I actually engage in problem solving e...
Double angles
I’ve been thinking about a generalization of the 15-75-90 construction over the last few days and have realized there are a lot more interesting consequences...
And yet more 15-75-90 fun
Continuing on the theme of 15-75-90 triangles (See: Last time and First Time) several interesting riffs on 15-75-90’s in a box have come up recently.
15 - 75 Triangle Redux
I came up with this problem after looking at the original one from @five_triangles (Find the area of the trapezoid ABCD) That’s a lot of fun but along the way…
Not so Innocuous Quartic
$x^2 - 16\sqrt{x} = 12$
Mid-Winter break Geometry
By tradition, I’m going off on some problem solving walk-throughs:
Making Explorations Successful
In a fit of perhaps excessive caution, the district cancelled all after school activities today despite the snow being almost completely melted. So I’m tabl...
Technology
I saw a funny ignite talk “Algebra Inferno” the other day comparing disliked teaching practices to the various circles of hell a la Dante.
Winter break cyclic 3-4-5
This is another exercise in documenting geometry problem solving. I chose this problem because again it has a 3-4-5 triangle within it and the overall setup ...
1:2 triangles and their link to Pythagorean triples
I’ve mentioned before how instinctively it feels like the 1:2 triangle ought to have a more natural angle measure. In fact its in a 90 - 26.57 - 63.43 degree...
More Geometry in a Box (Trisection)
Continuing an occasional topic, I saw another great simple box construction.
Another Fundamental Square Construction
Its amazing how much for want of a better word beauty is lurking in very simple constructions. I’ve talked about some square variants before: sometimes-one-d...
2016 Year in Review
With another year under my belt, its time to look back and think about what I’ve learned over the process. (Here’s my review from last year: the-year-in-rev...
Two problems I’m looking at
I was reading a fun post over @ http://eatplaymath.blogspot.com/2016/06/my-first-problem-set-for-my-problem.html where Lisa is brainstorming problem sets. Sh...
Memorial Day Miscellania
Pedagogy Riff
2 Fun 3-4-5 Pythagorean Triples
I’m going to warehouse these problems from @five_triangles here. I really like how they both show constructions for a 3-4-5 Pythagorean Triple. My plan is t...
Geometry - Similar Triangles again
For spring break here’s a geometry walk through I wrote up a while back but never got around to publishing.
Looking for a more elegant approach
This is a continuation of my geometry problem solving posts. I spent the last few days thinking about the above problem from @five_triangles. This ends up…
President’s Day Break Geometry
There’s no school on Tuesday for President’s day and therefore no Math Club. If I had realized this more fully I would have perhaps picked an extra problem o...
Sometimes its Harder the other way Part 2
I spent some time thinking about what initially looked like a very simple triangle congruence problem last night which I’ve outlined below. Given a perpendic...
Blog Anniversary.
Why Blog? Its been about a year, 75 posts and 3900 hits since I first started blogging. So it seems appropriate to step back and ask some bigger questions. ...
That article in Atlantic about explaining your work
The Atlantic recently published an interesting article about requiring students to explain their work http://www.theatlantic.com/education/archive/2015/11/ma...
8/17/15 Pythagorize Seattle Photos
Today MoMath celebrated the Pythagorean Triple comprised of the date: **8^2 + 15^2 = 17^2 **with a math happening at South Lake Union. In case its not obvio...
Random Geometry Recursion
You can recursively keep halving the length of the diagonals in this fashion. Creating a series of smaller 30-60-90 triangles in the process. (I promise I ...
On Hints
I’ve been reading the following thought provoking post by Michael Pershan @
Digression: Is this problem broken?
Update: As Dan pointed out I made an incorrect assumption in my sequence generation. The better technique is to generate the 2 lowest integers find the third...