Saturday, September 8, 2018

Geometric Construction of the Regular Pentagon

I never particularly enjoyed geometric constructions when I was younger. It may be because I had a tendency to press down too hard, and mess up the width my compass was set to.  Like so many things, technology has a fix for that.

You can do geometric constructions on a number of sites, and in geogebra. My favorite site, which I discovered about 5 years ago, is I having been telling students about it for years, and decided this summer to play with it again, to see how much I still like it. (A lot, it turns out.)

There are 40 challenges, in sets of 4. This time around I got every shape but the pentagon. I asked for hints (from math friends on facebook) which I then avoided reading because I wanted to do it on my own. I got nowhere that way, and finally followed instructions for how to construct it. Every other shape I have constructed makes sense on its own, but the construction of a regular pentagon must be proved with algebra. Ok, I don't feel so bad about 'cheating'.

I kept constructing it over and over, as I tried to understand it better. The first time (which you see above) I used these instructions (which I found by following links from wikipedia) and constructed it at sciencevsmagic, achieving my final shape challenge. Then I did it again using geogebra. I needed to figure out how it worked, and was at a loss. Cut-the-knot, a site I've gone to many times with my math questions, had a different construction - which I followed on both sciencevsmagic and geogebra - and an explanation (by Scott Brodie) that I kept working through as I wrote this.

[Sadly, Alexander Bogomolny, the creator of the cut-the-knot site, has died. I hope the math blogging community can find a way to maintain his amazing site. We miss you, Alexander.]

I will explain this as I understand it. My explanation will mostly just restate what I learned from these sites. But it may help some people, since some of the reasoning steps in Scott's explanation were hard for me to follow.

The constructions are worth doing first, so you can get a feel for what's happening as you do it. But when you're done, the question is still open - is this really a regular pentagon? The following explanation proceeds in two major steps. First it looks at relationships in the regular pentagon and the pentagram (which is a regular pentagon with a five-sided star inside it). Then it looks at the relationships built by the construction method, and we finally see that the side length given in the two ways is the same. Bingo!

The Construction
If you're ready to follow this, I trust that you already know how to construct the perpendicular bisector of a line segment. I will leave out the construction details for each of those.
  • Starting with a circle with center O, construct a horizontal diameter, AB.
  • Now construct its perpendicular bisector, CD.
  • Construct the perpendicular bisector of AO, intersecting it at E. 
  • Construct the circle with center at E, through C, labeling its intersection with AB as F.
  • Construct the circle with center at C, through F, labeling its intersection with the original circle (above B) as G.
  • Go around the original circle, constructing this same size circle from each new point. (Center at G, through C, gives H at the other intersection with the original circle. Etc.)
The question remains: Does this really construct a regular pentagon, or is it maybe just pretty close?

Relationships in a regular pentagram
  • The angles in a regular pentagon are each 108°, so each external angle, like ∠HMN, is 72°. That makes each angle at a point, like ∠NHM, 36°. Since ∠HMG is 108°, ∠MHG and ∠MGH must each be 36°. 
  •  Since angles ∠GLH and ∠LGH are each 72°, ΔHGL has two equal sides, HL and HG (which also equal CM, HI, etc).
  • ΔCHI is similar to ΔIHM. So corresponding sides are in proportion. We get CH/HI=HI/HM, giving us HI2 = CH*HM = CH(CH-CM) = CH(CH-HI).
  • Let CH = x*HI. Then the equation above gives us HI2 =x*HI(x*HI-HI), which gives us a (perhaps) familiar equation: 1 = x2-x, whose solution (in this case, where x is clearly greater than 1) is x = (1+√5)/2 aka φ. (At this point, I get excited. φ, aka the golden ratio, shows up in such different contexts!)
  • So CH = φ*HI, and (similar triangles) HI = φ*HM and HM = φ*MN.
  • We now have relationships between all the sides of the pentagon, star, and line segments on the star, but these are not yet connected to the radius of the circle.
  • Now consider ΔODH. Two sides are radii, and the angle at O is 36°. Another triangle similar to all the others we've found. So OD = φ*DH.
  • Also note that ∠DHC (inscribed angle on the diameter) is a right angle.
  • Let's switch to simple variables names now. Let r=OD, s=CG, t=DH, and d=CI=CH.
  • Then d= φ*s, r= φ*t, and t2+d2 = (2r)2, giving (r/φ)2 +(φ*s)2 = 4r2.
  • This gives us s2 = 1/φ2*(4r2 - (r/φ)2) = (4/φ2 - 1/φ4)*r2.

Analyzing our construction
  • We will find the length of s = CG = CF.
  • OE=1/2*r. And EC = √(OE2+OC2) = √(5/4)*r = √5/2*r = EF. OF = (√5/2- 1/2)*r.
  • So s2 = CG2 = CF2 = OC2 + OF2 = (1+(√5/2- 1/2)2)*r2.
  • We have two very messy expressions for s2, one from the regular pentagon/pentagram and one from our construction. Are they equal? It may help to write out positive and negative powers of φ to help simplify the first expression. Yes! They are both equal to (5-√5)/2*r2, making the side length of the regular pentagon inscribed in a circle of radius 1 equal the square root of (5-√5)/2.
  • And that proves that our construction created a regular pentagon. (Whew!) 

I've completed 32 of the 40 challenges at sciencevsmagic. I still haven't figured out how to make some of the shapes within the original circle, and still haven't found the least moves for some of the shapes. I'm so glad there's still something to work toward the next time I revisit this.

[Blogger doesn't deal well with superscripts and square roots. Once again, I am noticing that I ought to learn some LaTex. Sigh.]

Sunday, July 1, 2018

Math Teachers at Play #118

In two more months, we'll hit MTAP #120, which should be the ten-year anniversary mark. Is it?

The number 118 ...
... factors to 59*2.
... is 1110110 in base two, 11101 in base three, 1312 in base four, 433 in base five, and 226 in base 7.
Not particularly exciting...

Aha! Can you make 118 using four 4's? (I did.) I wonder if you can make it using five 5s, or ...?

Summertime Learning
Summer Math Resources from Math Mammoth's Maria Miller. 

Denise Gaskins offers us one of her FAQs, on forgetting what they learned. My favorite of all her suggestions? Play games!

On Routines and Lessons
I like Geoff's perspective: "What’s the shortest amount of time you could possibly do the talking? Go with that. And maybe subtract a few more minutes." After 30 years of teaching, I am still learning, and I still get excited when I see something that might make a difference for my students. This looks helpful, and perfect for my summer meditations on teaching.

Young Math
Can you imagine doing a college math lesson with 2nd graders, and having it work out well? Manan Shah did it! His lesson was on set theory.

Ahh... A whole blog on magical math books. And she wrote about Christopher Danielson's new book, How Many? If you like this, she also has a list of every book she has posted about. Thank you, Kelly Darke!

Remember playing the card game War? If you're a parent, those memories may not be so fond. Kids love it and parents can get sooo bored. Kent wrote a great post about his interactions with his son around Addition, Subtraction, and Multiplication War. (If you want a good summary of even more mathy variations on war, check out Denise Gaskins' classic post.)

The first ever Global Math Week went well, with its exploding dots exploding around the world. Thank you, James Tanton, for getting people all over the globe into math.

How do you talk about numbers with young children? There are so many ways! Here's another: Counting with Dice, from Dave Martin.

Which is bigger, eπ or πe? Sure, you can check it on a calculator. Or you can use areas to see why it's true. Lovely! (from Glenn Waddell)

The Fundamental Theorem of calculus says that you can figure out areas by using anti-derivatives. I do a project to help students understand it. Sam Shah does even more. This is Part III of his lessons for this topic. I recommend clicking on the links to Parts I and II first.

A Few More Goodies...
Dan McKinnon shares his notes from his Origami Workshop. For those who enjoy doing origami, can you find something new here?

Logic and Math go together so well. Check out this blog full of Venn Diagrams to fill in.

In his post From Surds to Ab-surds, Pat Bellew looks at an interesting relationship, which looks like bad simplifying but is still correct. He suggests it as a challenge in an algebra course to produce more correct 'bad simplifying' equations. Hmm, I want to think about how to do that.

Until Next Month...
Our sister blog carnival, The Carnival of Mathematics, often includes posts that are above my head. This month, our big sister is quite approachable. Enjoy!

If you have suggestions for next month's MTAP, share via the form at the Carnival home page.  Sharing in the carnival, or hosting, is a great way to increase connections in the #mtbos/#iteachmath community.

Sunday, January 7, 2018

Logic Puzzle - What Does Your Friend See?

I love logic puzzles, and was drawn by the title saying this was a hard one. Usually Nautilus is well-written, but their version of this puzzle isn't as good (in my opinion) as the original, blogged about by Presh Talwalkar.
The Nautilus version of the puzzle says to imagine your brightest friend. I imagined a friend I know likes logic puzzles (Sharon), but since I wanted a different initial than mine for notation purposes, I imagined another smart friend who might like logic puzzles (Linda). And I began scribbling away with S's (for Sue) and L's.
The answer I got is different than the answer the author got because we made different assumptions. Mine were based on his wording, his were based on Presh's wording.

Puzzle #1 (with Sue's interpretation):
You’ve been caught snooping around a spooky graveyard with your best friend. The caretaker, a bored old man fond of riddles (and not so fond of trespassers), imprisons each of you in a different room inside the storage shed, and, taking your phones, says, “Only your mind can set you free.”

To you, he gestures toward a barred window. Through it, you can see 12 statues. Out of your friend’s window, which overlooks the opposite side of the graveyard, she can see eight. Neither of you know the other’s count.The caretaker tells you each, individually, that together you can see either 18 or 20 statues. Unfortunately, there’s no way to tell your friend how many you can spot.

The only way for you both to escape is for one of you to give the total number of visible statues. Get it wrong, and neither of you ever leave. The caretaker asks you each once a day [Sue assumed neither person knew who was asked first], and you can choose to answer or to pass. If you both pass on a given day, the question—are there 18 or 20?—is posed to each of you again the next day, and the next, and so on, until you get it right or wrong.
Puzzle #2 is at Presh's blog. (No way to copy-paste that one.) It's now Alice and Bob, and they know that Alice gets asked first, so they'd both be released before Bob is asked if she has it right.
I think I have found a solution to (my version of) Puzzle #1. I would love to hear other folks reasoning before posting anything.

Sunday, December 31, 2017

Arithmetic, by Paul Lockhart

This book is part history and part philosophy of arithmetic. He also includes a few exercises in italics (one every few pages).

I love his easygoing tone (arithmetic is not mathematics, it's an art or craft, something like knitting, I enjoy it, and I hope you will), his low-key sense of humor, and his perspective.

I loved his book Measurement, and look forward to someday working through all the challenging problems in it. This one is much easier, and yet it's not boring for me.

It would be a lovely book to read to kids and think through together.

(It came out this year in hardcover. $22.95. I love the cover, though I know I shouldn't judge a book by that...)

[Edited to add:
I wrote this review before finishing the book, because I was so excited. I've read a few more pages now. Sadly, Lockhart is sometimes sexist. On page 45 he mentions the (Japanese) emperor's concubines, and says "Now, this is why people do arithmetic!" (To please the concubines.) No. It's not. And I thought better of you, Paul Lockhart.

I still think I'm going to love 99% of this book...]

Saturday, November 4, 2017

Re-reading Archimedes Codex

I am re-reading The Archimedes Codex. It is fabulous. A detective story of history, science, and math. (I wrote my first review of it two years ago.)

Both authors have a sweet nerdy guy sense of humor, gently self-deprecating, piercing when it should be.
"What are readers today [of science] afraid of? They are afraid of equations. With good reason: they were force-fed such equations for several, terrible years of their childhood and adolescence." 

I actually think he (Netz) is wrong here. I love math, and I love the beauty of some equations, but equations are still intimidating when you don't yet see what they're saying.

Here's another math-related quote, pointing in a different direction:
"Archimedes wrote out this problem in verse! A poet-mathematician! - the thought seems to us absurd, but it was natural for Archimedes, whose entire science was built on a sense of play and beauty, on hidden meanings."
Not surprisingly, I disagree with him here too. Some of us like mixing poetry and math. And the problem written up this way was a silly one. Archimedes was kind of like Martin Gardner with a few of his problems, and it makes sense he'd be playful in his presentation of it. Kind of like this one by Mike Shenk (sorry for the xmas reference this early).

Monday, August 14, 2017

First Day, Again

First day of classes. I was not as excited as usual. But I had my prep done, and once I got in the classroom, I loved it like always.

25 people in beginning algebra. I tell them how math is not about memorizing but about making sense. I get them talking in groups about the first time math didn’t make sense and got past them.

I ask them to estimate the percentage of the population that’s uncomfortable with math. First one says 100%. I almost laugh, but manage not to. Percentages all over, 32% up to another at 95%. I tell them I don’t know either but I guess 70 to 80%. So that means most elementary teachers are nervous when they teach fractions, and then they pass it on. I think I see a few nods.

I ask how to make meaning of 3 -5. Someone says you go past 0. I say “You’re talking about a number line, right?” And I draw it. I say I like that, but how can we give some real world meaning to this problem. Someone else says “Debt”, but I hear “Death” at first. I shake my head at my bad hearing. (I hope it doesn’t interfere with my teaching.) And I flesh it out. “Yeah, you’ve got $3 in your pocket. And you want to buy a $5 something. What does the 3 -5 tell you?” A student says, “How much you need to borrow.” I add in the temperature model, which I tell them might have worked better for my students back in Michigan who had experienced 3 degrees, and going down 5 degrees from there.

Then I give them 31 – 52. They discuss in their groups. It’s easier than I meant it to be.

We discuss the syllabus in between other things. I give them a sheet full of magic squares that use negative numbers. Some use fractions. I’ll need to check in tomorrow to see how they did with those.

I think the class went well. If they really feel good about it, they’ll end up thinking I’m their best teacher ever. That only happens about a third of the time with this course. I can hope…

Then I had to run to my next class. Statistics. I had them average the ages of 5 other students, and type their average onto my computer. I averaged their averages. Which is not the same as just averaging everyone. But it often comes close. Then I put a number line up, and we each put a dot at our age. That’s a dot plot. I showed how it’s skewed right. And talked about how median is a better way to show the center than mean (which is the average they had done).

I had an hour to eat lunch and chill.

Then I taught calculus. I love that class! Every time. Draw y = x squared. Draw a tangent line at x=3. Estimate the slope of the tangent line. What makes this different from algebra is that we need the idea of infinity here. Ahh… Happy me.

Friday, June 30, 2017

Math Teachers at Play #109 (a blog carnival)

Who is 109?     109 is a twin prime, twinned with 107.    (from

+ If 109 is written in Roman notation (CIX), then it becomes reflectable along the line it is written on.
+ The pipe organ at the Cathedral of Notre Dame in Paris has 109 stops.
+ When chilled below minus 109°F, CO2 becomes a solid, called dry ice.
+ 109 equals the square root of 11881 or 118 - 8 - 1.
+ The only three-digit prime formed by concatenation of consecutive numbers. [Silva]
+ 109 = 1*2+3*4+5*6+7*8+9. [Silva
+ The Sun is just over 109 times the diameter of the Earth. [Friedman]  


A Puzzle: Can you make 109 from four 4's? (I don't promise that it's possible...)

At age 109, Augusta Bunge became the youngest living great great great great grandmother. Is that mom to the fifth power?

Math Teachers at Play is a monthly blog carnival, hosted at a different blog each month. I was hoping to give you 109 math links this month, but life intervened (parenting...) long before I got through my storehouse of cool stuff. There are plenty of goodies here, but not as many as I'd hoped.

There has been an explosion of super cool mathy books since I last hosted MTAP. Here are some I know about. I am embarrassed to admit that I haven't read most of them, and so I can't guarantee how cool they are. Let me know in the comments.


Puzzles & Games 

Early Math


Probability & Statistics

Writing in Math Class 

Math and ...

On Teaching 

Random Stuff

I have more but it's bedtime and June is ending. Would you like to see your favorite blog post in next month’s playful math blog carnival? Submissions are always open!

(Note: Edited on 7/1 to add a few forgotten links, and fix a few broken links.)

Tuesday, June 6, 2017

What I learned at CAP's Community of Practice

CAP is California Acceleration Project. Check out their publications. The first time I attended one of their conferences, I struggled with the word acceleration. It does not mean getting through the material faster. It means getting to the good stuff faster - shortening the pathway of required prerequisite courses students must take before taking a college level course. Their work is mainly with math and English, the two subjects that generally hold students back.

In math, the college level course for someone not interested in STEM is statistics. Students take a placement test, and the large majority (86% at my college) are placed in remedial courses, anywhere from 1 to 4 levels below the statistics course. Imagine a student starting 3 levels below, at pre-algebra, which is where over half of our students are put by the placement test. If we had phenomenal success rates, with 90% passing each course, and phenomenal persistence rates, with 90% going on to the next course, we'd still only get 43% of these students finishing statistics (.9^8 = .43). What happens to the other 57%? Usually they give up on college, for at least a while.

Because housing is pretty segregated in the U.S., and that makes k12 education pretty segregated, with people of color getting less resources dedicated to their schools, this becomes a civil rights issue. CAP is dedicated to:  changing the way we place students (many who do badly on the placement test can still pass a college statistics course), developing models for co-requisite courses that students can take with statistics to improve their success rates, and developing radically shortened and improved remedial pathways (creating a pre-statistics course that prepares students with just enough algebra and lots of data analysis).

I have been attending their workshops whenever I can for the past few years. This past weekend I went with two other math faculty and 5 English faculty. Even though I've seen much of the information before, I still got a lot out of it. (Maybe I'm a slow learner!)

Here's something I put together yesterday at the request of our dean for equity, which summarizes some of the important points I learned...

Planning a High-Impact Course

More important than any one course are these 3 principles:
  • Create separate pathways for STEM and non-STEM.
  • Place students as high in the sequence as possible.
  • Shorten the sequence as much as possible.

CAP’s 5 design principles
  1. Backward design
  2. Low-stakes, collaborative practice
  3. Relevant, thinking-oriented curriculum
  4. Just-in-time remediation
  5. Intentional support for affective needs

High Performing Math Classrooms (Internationally)
James Stigler on high performing math countries.
All have these things in common:
  • Productive struggle
  • Explicit connections
  • Deliberate practice, increasing variation and complexity over time

Lesson Planning (CAP)
Given a topic you want students to learn through groupwork,
  • Identify the prerequisite skills needed,
  • Decide whether these will be addressed through productive struggle (ie not addressed overtly), targeted group activity, or just-in-time mini-lecture, and how you’ll do that,
  • Plan main activity,
  • Plan closure (vital for making explicit connections)
  • Note: Over-scaffolding brings down the thinking level required.
  • (Sue has a form from CAP. If this link works, it’s to all the CAP materials: )

Thinking Levels
The thinking levels mentioned above come from a study by Quasar. Here’s the relevant info:
“This research yielded two major findings: (1) mathematical tasks with high-level cognitive demands were the most difficult to implement well, frequently being transformed into less-demanding tasks during instruction; and (2) student learning gains were greatest in classrooms in which instructional tasks consistently encouraged high-level student thinking and reasoning and least in classrooms in which instructional tasks were consistently procedural in nature.” (Stein p. 4)

QUASAR Task Analysis Guide (adjusted slightly to address statistical thinking as well as mathematical thinking)
Lower-Level Cognitive Demand
Memorization Tasks  
Involve either reproducing previously learned fact, rules, formula, or definitions;  
Cannot be solved using procedures because a procedure does not exist or because the time frame in which the task is being completed is too short to use a procedure;  
Not ambiguous; clear and direct instructions to reproduce previous material;  
No connection to the concepts or meaning that underlie the fact, rules, formula, or definitions.

Procedures Without Connections Tasks  
Algorithmic; direct instructions to use a procedure or the use of the procedure is evident based on prior instruction, experience, or placement of the task.  
Require limited cognitive demand for successful completion.
There is little ambiguity about what needs to be done and how to do it.  
No connections to concepts or meaning that underlie the procedure being used.  
Focused on correct answers rather than developing mathematical or statistical understanding.  
Require no explanations, but may require students to “show work”.

Higher-Level Cognitive Demand
Procedures With Connections Tasks  
Focus students’ attention on the use of procedures or concepts for the purpose of developing deeper levels of understanding of mathematical or statistical concepts and ideas.  
Suggest pathways to follow (explicitly or implicitly) that are broad general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts.  
Usually are represented in multiple ways (e.g. graphs, tables, numerical summaries, verbal descriptions).
Making connections among multiple representations helps to develop meaning.  
Require some degree of cognitive effort.
Although general procedures may be followed, they cannot be followed mindlessly.
Students need to engage with the conceptual ideas that underlie the procedures in order to successfully complete the task and develop understanding.

Doing–Mathematics or Doing–Statistics Tasks  
Require complex and non-algorithmic thinking (i.e. there is not a predictable, well-rehearsed approach or pathway explicitly suggested by the task, task instructions, or a worked-out example).  
Require students to explore and understand the nature of mathematical or statistical concepts, processes, or relationships.  
Demand self-monitoring or self-regulation of one’s own cognitive processes.  
Require students to access and make appropriate use of relevant knowledge and experiences  
Require students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions.  
Require considerable cognitive effort and may involve some level of anxiety for the student due to the unpredictable nature of the solution process required.

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