Tuesday, February 12, 2019

When Math Tells a Story

On the Living Math Forum group, I claimed that Algebra 1 tells a story more than Algebra 2 does. N asked me to explain what I had in mind. Here's my reply (with a few revisions):

Lately I've been saying this sort of thing in my pre-caclulus class, not about the course, but about individual equations. We say math is a language. If it is, then we should be able to tell stories in it. Each equation makes a statement, and sometimes those statements tell stories.

The equation for a circle is (x-h)2 + (y-k)2 = r2 . Many students see each equation like this as separate from any other equations/formulas they know. I try to get them to look at this deeply. From the structure of it (square plus square equals square), I see that it's really the Pythagorean Theorem. Why would something for right triangles show up in the equation of a circle?! (That blew me away a few years back. I've been teaching math for 30 years, but that question seemed deep.)

It's because our coordinate system has the two axes perpendicular to each other. So the distance from the x-coordinate of a point on the circle to the x-coordinate of the center is measured horizontally and the similar y distance is measure vertically. You can build a right triangle from the center to (almost) any point on the circle. The constant radius is the hypotenuse of that right triangle.

So this equation tells a little story.

How does a whole course tell a story?

Algebra is about solving equations and about graphing. We want to see how real life situations (anything with data that has two components, like time and height) can be represented with equations and with graphs. In Algebra 1 students learn to solve simple equations and to graph. And hopefully they learn how the two skills are connected. There's a bit more. Systems of equations allow us to model slightly more complex situations, using more variables. And in my (community college) Beginning Algebra course (which is pretty much equivalent to a high school Algebra 1), our grand finale (after factoring) is graphing and recognizing equations of parabolas.

Near the beginning of the course, I introduce my favorite problem. I bought a tree and planted it. It was one foot tall at first. It grows two feet a year. Let's make a data table for height versus time, and a graph, and an equation. What does the input variable (let's use t instead of x) mean? What does the output variable (let's use h instead of y) mean? What does the slope mean? What does the y-intercept (or h-intercept) mean? The graph and the equation both tell the story of the tree. Linear growth is modeled with lines, which have equations of the form y = mx + b (or, in our tree story, h = 2t + 1). You can come at that from so many angles.

The course can tell the tree's story, or any story that can be told through data, graphs, and/or equations. It tells the story of using math to help us think quantitatively about problems we care about. (And of course there are plenty of things we care about that cannot be quantified.)

Sunday, February 10, 2019

Links to Good Math Posts

I am closing tabs, so that maybe my browser will move faster. Here are all the links I couldn't stand losing:

Thursday, January 10, 2019

Geometry and Visual Thinking Puzzle Collection - Help Needed

Math lovers, I need your help. (Math-likers welcome too. Math-haters who are into torturing themselves are also quite welcome, if you know any.) If you love puzzles, that might be enough to suck you in. I'm hoping...

I am pulling together a collection of problems meant to entice students who aren't all that into math. (So, relatively easy puzzles.) I think we focus too much on algebra and not enough on the visual skills that come along with geometry. So each problem in this collection includes visualizing. Most of them involve geometry.

I'm calling them PPODs - Puzzle Problem Of the Day. I plan to post them in my classrooms and in the math department of the community college I work at.

I have found quite a few good problems in Geometry Snacks, by Ed Southall and Vincent Pantaloni. (I bought the e-version of the book yesterday.) I just now discovered Ed's blog (solvemymaths.com), and found more goodies. My other favorite source is the beautiful by-hand drawings done by Catriona Shearer, @cshearer41. I couldn't find who to credit for about a quarter of the problems I've collected.

I'm aiming for 60 problems, one for (almost) each day (M-Th) of the 16-week semester. I have about 50 already. What I need is help determining the difficulty level of each one. I need volunteers to help me figure that out. I'm using a scale of 1 to 5 myself, and anything I'd call a 4 or 5, I'm throwing out. (I've gotten stuck on a few problems. Those are not in the collection, of course.)

If you're interested, I'll send you my collection. And then you post your ratings here.
  Just let me know in the comments if you'd like to check these out. (Or email me at mathanthologyeditor@gmail.com.) Also, I'm interested in your ratings and thoughts even if you just end up doing a few of the puzzles.

Wednesday, November 28, 2018

Logarithms: How Math Helps Science

I'm teaching my algebra students about logarithms today. It is likely the hardest algebra topic there is. When I started at Contra Costa College, after having taught full-time at Muskegon Community College for 6 years, I think I still was not 100% solid on logs. (I could do the problems, but proving those properties was a tangled mess.)

I had developed my murder mystery project because my officemate at MCC was a chemistry teacher, and she had complained that teachers weren't teaching logs well enough in Intermediate Algebra.I hoped that project would help. I think it does, but I don't have anything solid to base that on.

For many years now, I've told students a bit of history in my introduction to logarithms. John Napier invented them in 1614 as a way of making hard calculations easier. (Can you imagine finding the square root of 192.7 without a calculator? You'd have to guess, check, and modify repeatedly...)

Around that time Kepler came up with his three laws of planetary motion. Over the years, I have repeatedly wondered out loud whether Kepler was using logarithms while he figured these things out.
Today when this came up, I was able to turn to our tutor and ask him to look it up. He found a fascinating article on it.

Turns out Kepler came up with the first two "laws" before playing with Napier's new idea. But his third came after.

I. Planets move in ellipses with the Sun at one focus.
II. The radius vector (line from sun to planet) describes equal areas in equal times.
III. "The proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances."

This is an improper use of the word proportion. He was making a proportion from the logs of the ratios, and saw that log(T1/T2) = 1.5*log(r1/r2). In modern terms, we say something like: The square of the ratio of the times (that it takes two planets to go around the sun) equals the cube of the ratio of their mean distances from the sun. The fact that he was thinking in terms of logs shows how helpful the new (in 1614) concept was for helping scientists see patterns. (Read that article for more on Kepler and Napier.)

I love that math can help us see new things.

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 sciencevsmagic.net. 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.
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