Wednesday, December 30, 2009

To Do + Math on Vacation

The MathniƱos at the Modern Wing

Calder Mobile

Math Teachers at Play 22
I'm hosting this months blog-carnival, and I'd love your submissions from your own blog or nominations from blogs you read. Submit nominations at the blogcarnival site, or using the widget at the bottom of the right column on this page.

I never had a chance to write about Carnival 21 at Math Mama Writes... but Jonathan (JD2718) had a nice post about a subtle shift that produced a good effect.

An aside: I got to meet Math Mama over break and Sue was as nice and interesting as you'd think from her blog. We were going to meet up with Maria from Teaching College Math, but the weather prevented it (sorta).

Biggest Math Story of the Year
Time Magazine has submitted that the proof of Langlands' Fundamental Lemma was the 7th Biggest Scientific Discovery of 2009. A quite technical exposition of the Lemma (not the proof) is here. (Hat tip Peter Woit of Not Even Wrong). I do not have an understanding of the math, but it concerns a connection between Galois group theory (about solutions to algebraic equations) and general linear algebra representation theory (think matrices on steroids). One of the most famous theorems that uses the Lemma has Femat's Last Theorem as a corollary. Powerful stuff.

It's nice for students to know that math is ongoing, and also to hear terms like lemma, theorem and corollary used genuinely. I also like how this emphasizes the power of connections in mathematics, and how even (or especially) the world's most powerful mathematician's start problem solving by solving simpler problems or finding another way to put the problem. It also should be noted that Ngo's proof was submitted in 2008, but took until this year to verify.

Math (Art) on Vacation
My wife and I recently won tickets to the Cirque de Soleil show Banana Shpeel (which I would recommend.) On the quick trip to Chicago this week sans kids (hat tip: grandparents!) we had more time in the Art Institute than we would have had otherwise.

In the geometry class I've been writing about, we do one constructive project in which students build a polyhedron for the following context:
Hoity Toity, the upscale chain of Haute Couture for the masses, is having a competition to design new and original knick-knack boxes. Being an accomplished mathematician, you have an unfair advantage, which you intend to exploit to the maximum. Boxes must have a volume of at least 1 liter. (1000 cm3).
Students do a terrific job in general. One student this semester made a dog-shaped polyhedron. Cool. Some classic Archimedean solids, as well as prisms and antiprisms and such.

But at the Art Institute I had two ideas for new variations. The Modern Wing in itself is an inspiration as it is filled with beautiful proportions and rectangles with more connections and relationships than you could ever hope to count.

The Chess Set

This wonderfully sculpted set filled me with visions of two people working together to make a set of polyhedra vs non polyhedra. They could do found or constructed objects. It will require connections (what makes the difference between the king and queen, how will players know this is a rook, what's the difference in the bishops between the sides...) and analysis of the solids and their volume.

The Building

I so wanted to see the isometric drawings for this. I wouldn't insist that the building be built, which gives me pause, but I would want two different representations that would help someone build it. Students should design their own advanced building and think about it in scale

I'm curious to see where these ideas go, and what students make of them. I would like to hear more from other teachers about from where their ideas come, and the process they follow to implement them and refine them in the classroom. I think we teachers do a lot of our sharing as a product exchange.

Sunday, December 13, 2009

Triangle Detective

This game is a variation of a variation of a game. There's a terrific game called Polygon Capture which originally came from a Mathematics Teaching in the Middle School article (Oct 98, William M. Carroll). The idea is to turn over a sides property card and an angle property card and capture the polygons that fit the description. For my classes I'm on about version 3 of it, and it's a solid middle school game. (Contact me if you want my copy.) At one point a class was interested in adapting it to triangles, so we did a version of the game that focused on them.

Michigan has moved a lot of the triangle identification objectives down to 4th grade, though, so I thought I should adapt that to be a 4th grade game, and that's what I'm posting today. Mrs. Bruckbauer's students decided it should be Triangle Detective, because you inspect the triangles to see if they fit the card. They are so right. As usual. They also suggested the three points per triangle scoring, and preferred it to the 'most triangles wins' rule.

Triangle Detective
(click here for a PDF version, with cards and triangles to cut out)

• Put the triangles into the middle and shuffle or mix up the cards.
• On a player's turn they flip over a card. They catch a triangle if they can find a triangle that matches the card's description. All players have to agree it fits the description.
• Master Cards are special challenges. Are you a triangle master?
• STEAL cards are the only way to take a triangle someone else has captured.
• Play until the deck is empty. Players get three points for each triangle, and the most points wins.

Two things are congruent in geometry if they are exactly the same size and shape.

An angle is right if its sides are perpendicular, like the corner of a square.
An angle is obtuse if it is bigger than a right angle.
An angle is acute if it is smaller than a right angle.

A triangle is acute if it has ALL acute angles.
A triangle is right if it has ONE right angle.
A triangle is obtuse if it has ONE obtuse angle.

A triangle is equilateral if it has THREE congruent sides.
A triangle is isosceles if it has TWO congruent sides.
A triangle is scalene if it has NO congruent sides.


Has an
obtuse angle
Has NO
congruent sides
Has two or more
acute angles
Has two or more
congruent sides
Has a
right angle
Has three
congruent sides
Has at least two
congruent angles
Has three
acute angles
Has three
Has a line of
Master Card
Take a triangle IF
you can name
its side type AND
its angle type
Master Card
Take a triangle IF
you can explain
the type of each
angle in the
If you can find
someone with an
acute scalene
If you can find
someone with a
right isosceles

Triangles: (Click for full size image) (3 of each type... including some borderline cases.)

Teaching Notes: The game worked really well for review. The students were engaged and asking good questions. Students were motivated to ask about vocabulary they didn't know, got to see other people apply vocabulary, and see triangles in many different orientations. The statements on the cards led to them thinking about alternative ways to say the same thing, and to think about properties in combination.

Acute triangles are an issue, because it seems to them that one acute angle should be enough. Which is actually nice parallel reasoning from how right and obtuse triangles are explained. We talked about how each angle has a type and each triangle has one angle type. So if it's not obtuse nor right...

We used square and triangle pattern blocks to help check the triangles angles, and talked about how equilateral triangles have all the same angle as well, and that's a way to check them.

The students got quite expert at checking side lengths, and quickly weren't satisfied with 'they look the same.'

They were generally quite helpful to each other, to the point where one student asked them to stop helping because she wanted to find her own.

Good luck to if you try it. And, of course, I'd love to hear how it goes!

Monday, December 7, 2009

Other People's Geogebra

Transformations on a Graph
I've been looking for Geogebra applications for function transformations, and wanted to share a couple of the neat sketches I've found.

Michael Higdon, a math teacher at Kincaid, a college prep school in Texas, has a quadratic function in vertex form, y=a(x-h)^2+k, with a, h and k as sliders to study transformations.
Geogebra webpage: Transformation of Functions

Mike May, a Jesuit math teacher at St. Louis University, has a beautiful applet where you can input the function, and control vertical and horizontal shifts and scaling with sliders.
Geogebra webpage: Translation Compression

An overall great collection of interactive webpages appears at The Interactive Mathematics Classroom. It has a nice search feature and a good breakdown by area of mathematics.

My first attempt at a transformations sketch is with a cubic as the starting function. Although you can change the function.

As a webpage, and as the geogebra file.

Slope in Linear Equations
A nice collection of middle school or Algebra I activities for linear equations from Slope Explorations
Mathcasts are screencasts of writing with voice-overs. They have mathcasts for K-12, and a nice collection of interactive math activities.

Here's my first slope sketch. It tries to get at the idea of the slope being constant on a line regardless of what points are selected.

As a webpage or a geogebra file.

Sunday, December 6, 2009

A Bigger Hex

As a webpage, and as a geogebra file.

Made a simple hexagon dilation sketch in geogebra for my geometry class. Let's you vary the objects, measures area and perimeter, and control scale factor. Algebra view let's you see individual side lengths also.

It's part of a set of four similarity problems for stations. Here's the pdf.

Family Mathematics Videos

A former student at GVSU, Jamie Trost, is showing a lot of initiative, and posting videos to youtube providing instruction on elementary topics. They could be used as support for students who are struggling, extension for students succeeding in one method by exposure to another, or as support for parents trying to make sense of their children's math class.

She's just getting started, and is very open to suggestions, for topics and for improving the videos. If you get a chance, please check out Jamie's Family Mathematics, and give her some feedback. Her initial series is on multiplication.

Friday, December 4, 2009

Why Pi?

Quick book review.

Why Pi? is a DK picture book/encylcopedia by Johnny Ball. The author sounds as if he was the Mr. Wizard of Great Britain in the 70s, and is the author of many books, several of which are math and science related. But this is my first of his.

Xavier (9) says: it's really good. I bow to whoever suggested it. It was fun and funny.

Ysabela (10) says: it's kind of funny, but in an interesting way. It's got a lot of information, but is definitely enjoyable reading. And it has Egyptians, awesome.

John (45) says: the book was unexpectedly thorough. It has a lot of excellent math and science history, from several ancient cultures, and deeply explores the concept of measurement as it pertains to many topics. It is a great read, and a solid reference. Despite this exposure coming from the library, I will be looking to purchase it. As the kids have pointed out, it is long on humor, and deep in breadth. (Both difficult quantities to measure.)