To Do + Math on Vacation

The Mathniños 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.

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)

Rules
• 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.

Remember:
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.

Cards:

Obtuse Triangle	Has an obtuse angle	Has NO congruent sides	Scalene Triangle
Right Triangle	Has two or more acute angles	Has two or more congruent sides	Isosceles Triangle
Acute Triangle	Has a right angle	Has three congruent sides	Equilateral Triangle
Has at least two congruent angles	Has three acute angles	Has three sides	Has a line of symmetry
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 triangle	STEAL! If you can find someone with an acute scalene triangle.	STEAL! If you can find someone with a right isosceles triangle.

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!

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 mathcasts.org: 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.

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.

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.)

Net Result

I really enjoy designing nets (2-D plans that fold up into 3-D objects), and I love designing them in dynamic geometry, where you can design all nets. If I ever got the time to do math research again, I can see going in that direction.

I designed these four sketches for my geometry class, which is working on a project to design their own package with a few constraints. Each sketch is available as a dynamic webpage or the geogebra file. Here's 5 nets that were made with the sketches, in a printable pdf format. Geogebra actually has very nice priniting controls, so if you're interested in designing your own, choose that option. Remember you can install it, or run it from your browser at geogebra.org.

General tetrahedron: webpage or geogebra file

Square pyramid: webpage or geogebra file

Convex oblique pentagonal prism: webpage or geogebra file

I was very disappointed that the above sketch, though intuitive, couldn't make concave prisms. This next sketch is the answer, but would be a muddle to try to figure out how it was made. The net is pretty though, for designing solids, and I'm proud of it as work.

General Oblique Pentagonal Prism: webpage or geogebra file

Note that you can use the pentagonal prism nets to make quadrilateral and triangular prism nets by making some of the base vertices collinear.

Let me know what you think, and send me your dynamic geometry design challenges!

PS> I also wrote up a memoir for my class of how I made the pyramids (which really sounds egotistical; reminds me of a Tom Lehrer line about "even the Pharoahs, had to import, Hebrew braseros" Listen at the link. If you do, check out Lobachevsky, a great math song.) Sorry for the ramble - I'm tired. Here's the memoir.

Math Teachers at Play 20

The new Math Teachers at Play Carnival is up at Denise's Let's Play Math! blog. Besides the interesting puzzle she starts with, and the hilarious math FAIL images, there were a few posts I found especially interesting:

• Ashley Allain's post on computing fractions using Scratch, a new, kid-friendly programming language.
• Rachel Lynnette documents her use of a free symmetry game that looks promising.
• I'll be using this cone problem on a test, I think. Pretty elegant. Wonder if my students read this?

Due to the new monthly structure, there was a heap of submissions this time around. Check it out!

Constructivism and WatchKnow

Two interesting web links to pass on:

1) Interesting essay found by Michael Paul Goldenberg (no relation) at Education Notes Online: a teacher and parent commenting on constructivism and the math wars in NYC. There is some really interesting math happening in the city classrooms, in particular the Math in the City project, and the essay is a good read. Check it out: The Math Wars Revisited: Lisa, Why Doth I Love Thee....


2) New web resource: one of the founders of Wikipedia has started a collection of educational videos that is rapidly growing and becoming comprehensive. It's called WatchKnow. Everything from Carl Sagan addressing the shape of space-time (I was just talking about that with my students...) to a geometric solids story starring Sniffer Bob (which is what we were discussing...). The rating system will become really helpful as the site matures, I think.

Pick On Someone Your Own Size

Definitely the longest game name I've ever used. But the fourth graders who piloted the game have decided, and who am I to argue.

Pick on Someone Your Own Size
A Math Mystery Game

Fourth grade students in Mrs. Bruckbauer’s class described this as a math mystery game, because you’re trying to find out what the other player’s hiding.

Materials: game sheet, calculator (if needed for checking), scrap paper for making your plays.

How to Play: Each player comes up with 3 numbers that add up to 1000. When both players are set, they fill in the fight boxes from largest to smallest. You get a point when you have the greater number.

For example, 800, 100, 100 and 500, 300, 200; player 1 gets 1 point and player 2 gets 2 points. It’s okay to have two of your numbers be the same. If your number is equal to an opponent's, no points.

Player1 Player 2
800 > 500
100 < 300
100 < 200
1 point 2 points

You play for five rounds, and the person with the most points wins.

Player 1 Score ....................................................................... Player 2 Score

PDF of the game.

Teaching with the game: I started with a guessing game. I picked a number between 0 and 1000, and they tried to guess it, using only my more or less answer as clues. I picked 673, and the students took turns guessing. I wrote down on the resulting comparison. using x for my number, like x>89. They guessed: 89, 587, 1000, 998, 823, 650, 657, 760, 700, 699, 689, 666, 680, 676 and 673. (I wrote 673, then x then filled in the =, to great rejoicing, x=673.) Just refreshing use of the comparison relation, and seeing how they were with ordering numbers in the hundreds. They really enjoyed this, and I think that could be turned into a lesson of its own.

Then I played me vs. the students, as I often like to teach games. It solidifies rules and shares some strategies. My triples were (900,99,1), (500, 250, 250), (501, 498, 1), (350, 350, 300), (501, 301, 198). As the students played later I saw them using several variations on these, and also the idea of modifying a previous guess.

The game was really engaging, and students took their guesses seriously. I watched at first, and a couple hadn't really gotten the adds to 1000 idea, and several had to be reminded to put their guesses in order. I went back and forth on that in design, but it really worked well for game play.

Rules they wanted to add were having 1/2 point or 1 point each for ties, and playing an extra set if it ends in a tie. The ties were nice for getting the students to modify their guesses up or down a few ticks.

The math involved encourages students to think about sums to nice numbers, as well as how to partition 1000. I shared with students how they could pick two numbers and do 1000-sum to find the third. We had calculators for them to quickly check their opponents picks, but mostly they did it in their head. Several students, not too familiar with the calculators, really enjoyed playing with them to try different numbers.

Game analysis: it's a little more interesting than you might think, kind of a nuanced Roshambo. (Rock-Paper-Scissors) The three basic strategies are big-small-small, medium-medium-small, and third-third-third. Of course, you can win by a point, and that's where some nuance and psychology comes in. Much like Rock-Paper-Scissors.

Let me know if you give this a try and what you think! Thanks to Denise at Let's Play Math! for catching the missing instruction.

Golden Math Cartoon & Math Teachers at Play 19

I had to post this. I'm trying to help edit a collection of extension activities for Geometry being put together by teams of students under Char Beckmann for the Michigan Council of Teachers of Mathematics. (The Sum More books - more as they become available.) One of the activities involves cutting up a cartoon and then trying to arrange it in order to make the idea of arranging arguments in order in a proof. Long ramble, here's the cartoon: (Click for full size)

Their take on a pi joke. Pretty good for 9 and 10, eh? I inked it, and was told I boffed Fiddlestick's mouth in the last frame. They did a great job putting in clues as to what comes next.

Math Teachers at Play 19 (back on numbering) came out last Friday. My contribution was Area Block. The carnival is moving to a monthly schedule, which I think will be good. What was most fascinating to me:

• Maria Anderson, just up the road in Muskegon, is doing amazing work with technology in a math for elementary teachers class. I've got to meet her sometime!
• Jason Dyer's post on a reading experiment was interesting. (I wonder if it has to do with this cognitive miser issue. (Not a MTAP link.)
• Kendra at the Pumpkin Patch had a cute, quick sum game.

Lots more interesting stuff, including Sue (the host, Math Mama) interviewing a micro-photographer about his math.

An Ad?

While Math Hombre is traditionally ad free, I do owe a link to my daughter Ysabela. In exchange for her review of Babymouse: Dragonslayer, the excellent math/fantasy graphic novel by Jennifer Holm, I promised to plug her market craft.

Yzzy's Sticky Silverware has recycled silverware affixed with two powerful magnets and artistic embellishments to make a beautiful and practical refrigerator decoration. A steal at $1.25 decorated ($1 plain), you might pay up to $5 for a similar product at a craft show. Check it out! Mailing is at cost.

Some of the math involved was: figuring out the unit cost from the total materials cost, and also balances, profit margin, the length of the total wire based on the wire per silverware, (which was a pretty good upper el measurement problem). What is her break even point? That's the next calculation!

Area Block

New game! Blokus meets Nim, this game works on area and strategy. Tested with fourth and fifth graders, but flexible upwards with sophistication in area computation techniques. Get it as a pdf here: AreaBlock.pdf

Materials: Game Board, 2 different color pens, pencils, crayons or markers.

How to play: Players take turns making a single shape on the board that has an area of 10 or less. The game is done when the board is filled, and the player with the most squares covered wins. The table is for recording how many squares you cover each turn.

The first player to go can only color up to 8 squares on their first shape. (Otherwise it would always be best to go first.) After that the limit is always 10.

The squares colored in to make your shape need to share a side, not just touch at a corner. (The shape you make has to be a polygon.) A player can even use slanted lines – as long as they can figure out the area for their shape! When you are making a new shape, it does not have to touch your old – you can put it anywhere there’s room.

After you make or shade in your shape, record the area of your block – you don’t want to miss any points. The next player then colors a polygon of up to 10 squares of area in a different color.

When the board is completely filled, the game is done. Total up your squares and see who won!

Notice you can check if all the squares are counted by adding both scores – you should get 100.

Whoever lost gets to choose the next time if they want to go first or second.

Variations:
• The board can be changed to have holes or blocks or be a different shape. As long as there are 100 squares. After trying these boards, make your own.
• The game can be played with dice. Roll 2 dice, and you can fill in the total rolled.
• Advanced players could try where their score is not the area – but the perimeter. (You might want to try that with no slanted sides.)

Board 0:

Board 1:

Board 2:

Board 3:

Board 4:

It's pretty fun! I haven't been able to spot a degenerate strategy yet. Give it a try, and let me know what you think.

GeoGebra: Triangle Tuning

What can we deduce from the side lengths of a triangle?

This is a preservice teacher activity, easily adaptable to middle or high school use. GeoGebra is a free dynamic geometry program available as an online applet or downloadable program, from geogebra.org.

Objective: TLW explore triangle properties relating type and side length.

Schema Activation: What are the 7 triangle types? Fill them in in the ‘Type’ column below. You'll fill in the other columns as you go through the activity.

.....Type ....................... Examples [Like (3,5,5) ] .......................What do you notice?
1.

2.

3.

4.

5.

6.

7.

Focus: One of the reasons dynamic geometry is so powerful is the support it allows the teacher (or curriculum designer) to give students for finding examples. Lots of examples. As we’ve discussed in class, the natural way we reason is to go from lots of specific examples to the general.

Activity:
1) Open the sketch TriangleBySide.ggb. (Online at faculty.gvsu.edu/goldenj/TriangleBySide.html)

a. Collect at least 2 examples of each type of triangle. Where possible, try not to have the triangles be similar, where all the sides are multiples of another triangle.
b. Which were hardest to find? Was it something to do with the type or how you were looking?

2) Open the sketch PythagoreanData.ggb. (Online at faculty.gvsu.edu/goldenj/PythagoreanData.html)

a. Look at your right triangle examples on this triangle. How do you think ancient mathematicians noticed this cool pattern with the squares

b. Check your other examples on this sketch. For each, record whether the sum of the areas of the smaller squares is <, =, or > the area of the large square.

c. What pattern(s) do you notice?

d. Why do you think your pattern(s) might be true?

3) Go to http://teachers.henrico.k12.va.us/math/GeoGebra_Site/pythagoras/pythaPrblm1.html (link on Blackboard) for the Ladder sketch. Answer the questions there.
a. How high will he get if he places the ladder 3 m off the wall? Drag the ladder point and find a solution. Sketch the solution with all its lengths (s, r, h) on paper.

b. Now, calculate the solution of task (a) on paper. Which lengths are given, which are sought? Do you get the same solution?

c. At what distance of the wall should Pythagoras place his ladder in order to reach the window 4.50 m above the ground? Sketch the solution with all its lengths (s, r, h) on paper.

d. Now, calculate the solution of task (c) on paper. Compare your solution to your sketch?

e. As a teacher, what do you notice about the difference between doing the tasks in sketch and on the paper?

4) Open the sketch AdjustableLadder.ggb (Online at faculty.gvsu.edu/goldenj/AdjustableLadder.html)

a. What young Pythagoras might not have known is that a safe ladder ratio is 4:1 for the height to distance from the wall. What length ladder does he need to reach the window safely? Is there one solution or more?

b. How would you solve this problem on paper?


Reflection:
a. What confirmed, new or deepened understanding did you develop regarding triangles through these activities?

b. How did the dynamic environment affect your understanding?

EDIT:
Be sure to look at the comments. Scott Farrar has some notes on using this with 9th grade geometry, and has added a complimentary activity.

• http://scottfarrar.googlepages.com/triangleineq.html
• The worksheet http://scottfarrar.googlepages.com/TriangleCategories.pdf

Math Resources

Moodle
Moodle Math documentation
http://docs.moodle.org/en/Mathematics
Pretty technical information, but accurate and detailed.

Book about teaching math with Moodle
http://www.packtpub.com/moodle-1-9-math/book

Content
http://illuminations.nctm.org/
NCTM’s free lesson and resource center, with many online applets. Note the Calculation Nation link leads to some excellent 1 and 2 player games, but require a free login account. (Guest access to try it.)

http://ohiorc.org/for/math/
An Ohio Resource Center with links to many topics by grade and content. Some high quality resources.

http://nrich.maths.org
From the British equivalent of the NCTM. Once you get past how they say ‘maths,’ there’s an excellent supply of interesting problems, some clever student solutions, and open challenges. Also features some interactive applets.

Internet Resources for in class use
http://math.hippocampus.org/
An extensive collection of online exposition of topics from algebra and calculus. (As well as Bio, Chemistry, Physics, etc. Some support for Spanish.) Includes a voice delivering reading of the text (with a skip forward feature), and assessment along the way. Not clear what the purchase of a license adds, but everything I’ve seen was free.

http://www.imagination3.com/
GE’s interactive whiteboard, open to collaboration, optional graph paper background. Colors and some shapes are possible. You can save files.

http://www.wolframalpha.com/
Wolfram|Alpha, a free computational engine online. Amazing and worth exploring. Also some encyclopedia function. Supercedes some of the older online utilities.

http://geogebra.org
GeoGebra, a free dynamic geometry program that can be used as a web app or by downloading and installing. Some powerful algebraic applications, too.

http://www.plu.edu/~heathdj/java/
A collection of java applets, from many different mathematical content areas.

http://www.univie.ac.at/future.media/moe/onlinewerkzeuge.html
Powerful collection of online tools. One of the best online function plotters (graphing calculator) is here.


Games
http://www.funbrain.com/brain/MathBrain/MathBrain.html
K-8 math games. Not very challenging, but there are students for wom this will be just right. Connected with Poptropica, which my kids love, and features some decent problem solving.

http://gprime.net/game/
Huge collection of flash games – not all kid appropriate. Some excellent math and reasoning games, though. Try 3D logic, a maze game on 3 cube faces (http://gprime.net/game.php/3dlogic), gridlock, the classic game (http://gprime.net/game.php/gridlock) or grow island, a deductive reasoning game (http://gprime.net/game.php/growisland), and one of a family of grow games (http://www.eyezmaze.com/grow/v3/index.html).

http://www.logicmazes.com/
Very challenging puzzles by a very interesting man, Robert Abbott. Also links to some games of his creation.

Utility
http://incompetech.com/graphpaper/lined/
Free graph paper pdf generator. Very handy.

Teacher
http://www.learner.org/resources/browse.html?discipline=5
Annenberg Teacher Resource Center. Extensive collection of lessons, videos of students and teachers, and professional development materials. All free. Sortable by grade level.

Money Problems

Excellent variation on the "how many ways to make change problem" at the New York Times today. The Freakonomics column is reporting the work of Patrick DeJarnette. I can see giving this problem from 4th grade to linear algebra!

Click on the money tag to see my previous money games. Click on the cartoon to go to the excellent Non Sequitur website.

Math Teachers at Play #16

The new carnival is up.

A clock face activity for learning how to tell time supports students with a nice simple representation. The biographies reviewed here look like must buys. Very nice math teaching memoir from Math Mama. We should all write more of these. Denise, carnival originator, starts a good division discussion. Some interesting online math games are highlighted at the Innovative Educator.

The order of operations conversation reminded me of a student teacher I just saw who has picked up the nickname Aunt Sally. ("It's Not Funny," she said in response to my giggle.) I'm hugely in favor of a four step Order of Operations. It's also a great opportunity to discuss with students the difference between a result or theorem and a convention.

Anchor Charts

While we did this as a teacher prep activity, I think it would be interesting with K-12 students as well. I first heard about anchor charts in Mosaic of Thought (I think), and like how they serve as both assessment and culture building. I've had students make charts about a particular concept, about what to do when you're stuck, and for what it means to do, learn and teach mathematics.

For this most recent lesson I had the students read "Mind mapping As a Tool in Mathematics Education", by Astrid Brinkman from Mathematics Teacher, Feb 2003. (They had previously expressed curiosity about and a lack of experience with concept maps.) One reason I love these is that they are frequently surprising. I expect one like this, that echoes the NCTM process standards we emphasize in the first unit: (Remember you can click images to see them full size.)

But then they go and make these completely original things like:
with different processes emphasized

and

Highlighting the difference and connection between
school mathematics and real mathematics.

Instructions to the students:

Activity: Anchor Charts
The following example comes from Ellin Keene’s (2008) To Understand: "Teachers generate anchor charts to capture and celebrate increasing sophistication in oral language use." (p. 278)
If you substitute ‘understanding learning in mathematics’ for ‘oral language use,’ you have the purpose of this activity. (Follow a link for a free pdf of the first chapter, under samples.)

Create an Anchor Chart for Learning in Mathematics
1. Identify the concepts and ideas that you want to remember as they relate to doing mathematics.
2. Develop an anchor chart that captures and celebrates your increasing sophistication in understanding “Doing Mathematics.” This might be a list, or a mind map, or a representation of your own creation. You decide – just be prepared to share your chart during our next class.
3. Be sure you leave 10 minutes to reflect.

Reflection: How well does your group’s anchor chart capture what you want your future students to think of hen you ask them “what does it mean to do mathematics?”


Home Extension: You might want to check out how math teachers use anchor charts at books.google.com - Integrating Literacy and Math. by Ellen Fogelberg, Carole Skalinder, Patti Satz, Barbara Hiller, and Lisa Bernstein.

Other interesting examples:

(OK, the last one's mine.
Fair's fair if I'm putting up my students.
No, I don't know what's up with that guy's hair.)

Send me yours or your students' and I would be happy to post that!

Geogebra

I know I've gone post crazy. I will stop soon, really.

I'm trying Geogebra this semester with my Geometry K-8 class, first computer session tomorrow. It's a freeware java implementation of dynamic geometry quite similar to Geometer's Sketchpad, with an even stronger algebraic interface. In addition to playing with it to get a sense of what's going on, I've got two sketches to look at:

Square Not Square Where you adjust quadrilaterals by vertices
to check the 'real' type, as all 9 look like squares.

Quad By Angle Where you determine a quadrilateral

by the angles rather than by edges or diagonals.

Links lead to webpages with the java sketches, as geogebra has a great export to webpage feature. They are java, so some browsers don't handle them as well as Firefox. If you prefer the actual geogebra sketches, it's SquareNotSquare.ggb and QuadByAngle.ggb. Most of the sketches I've seen so far have been for HS or university, so if you're interested in K-8 uses, please let me know!

EDIT: Followup - the students were mostly positive, and some extremely positive. Definitely a better reaction than to the old computer labs with sketchpad. There was more interest in the program because it is so accessible; many more of them saw themselves as potential users of this time. It also connected with people's desire for visual representation. Makes me wonder, have I been underusing manipulatives this semester?

Book Review: Babymouse

Read a very cute book recently, Babymouse: Dragonslayer, by Jennifer Holm and her brother Matthew. I had never read Babymouse before, but it was recommended by Jen Robinson (who herself was recommended by Kathy Coffey, my guru in all things about teaching reading). Jen Robinson's blog is a great resource for anyone interested in kid's lit as a parent, teacher or reader.

Babymouse is a math-hater, because she is bad at it. Her teacher, shockingly, starts her on her adventure by sending her to be a mathlete. While there is no math per se in the book, it is all about immersion and expectations by the Mathlete coach, and employment and approximation on Babymouse's part. And all with excellent connections and fantasies of our great fantasies, Narnia, Lord of the Rings and Potter. A homerun all around. My only unrequited wish - more actual math. All that for only $6 at Amazon.

The italicized terms here are from one of the best frameworks for teaching and learning I've ever seen. It's really the lens through which I evaluate and work on my teaching. Brian Cambourne's article "Toward an educationally relevant theory of literacy learning: Twenty years of inquiry" is available online here.


My daughter Ysabela's review is here. She says the title misspelling is intentional...

Math Teachers at Play 15?!

The new carnival is up at mathfuture. One of the most interesting things about it is the host site, which looks at Web 2.0 math applications. For example, their GameGroup, which I may be seeking to join.

Other than that the 2 posts that were most interesting to me are a description of a role-playing site for math-centric careers and 25 uses of Wolfram Alpha (which I love).

Denise has a good starter on Mental Math, but I don't see these as techniques you teach as much as ways of thinking that you demonstrate and can grow out of kids' number sense. I was saying to Xavier (my 4th grader) last night, knowing 6x9 is not as important as being able to find it efficiently, and knowing things like it's connected to 6x10 - 6 or 6x5+6x4 is much more important.

Angles

What do you see as the big ideas with respect to teaching angles?

To me:

• filling around a point, no gaps or overlaps between two boundaries
  
• connection with a circle (filling all the way around a point) - important for units
• size of the angle corresponds to how open

So I love to begin teaching angle with pattern blocks. The activity I start with is adapted from one taught at GVSU by Jan Shroyer, don't know where she got it or whether she wrote it. The activity as a Word .doc is here. If printing from the web, try to size the pattern blocks so they are life size. (Doesn't affect the angles, of course, but makes it much more natural.)

A Very Special Blossom

A blossom is a special pattern in mathematics when copies of the same shape are arranged to fit together all the way around a point. Try to blossom the following shapes. Record how many fit around a point. Sketch either the shapes or the edges that meet at the point.

What do you notice?

Would the blue or red pattern blocks blossom?

Teaching notes: the white rhombus is chosen especially, since the wide angle doesn't blossom. The narrow angle will provoke a little discussion, 11 or 12 to blossom, because if they're tracing one block, the thickness of the drawn lines add up. The wide angle will draw responses of 1, 2, 3, 4, and 8. 4 will usually be two wide and two narrow angles (tessellating the rhombus) and the 8 is from filling the rest with the narrow angle. The red trapezoid and blue rhombus will sometimes have the students seeing blossoms with the narrow angle but not the wide.

After the connection of 360 degrees with filling all the way around, these blossoms can be used to deduce the measures of the pattern block angles. This is nice in conjunction with measuring practice with a protractor or angle ruler.



Filling Time

Use pattern blocks to measure the following clockwise angles. (Start at 12, and then measure in clockwise direction to the other edge.) Use all of the same unit for each angle. Measure each angle twice using different units, if possible.

Teaching Notes: You will see students make a lot of connections with congruent angles doing this. Also, there will probably develop an appreciation for the smaller angles as units. There is a natural tendency to measure the smallest direction, so that will bring up the clockwise/counter-clockwise thing, which is a nice connection with rotations, which will be a great way to teach angle to kinesthetic learners. The middle left angle brings up the idea of partial units, as it is not a whole number of pattern blocks for any of the shapes. The scientific standard is to measure to half of the smallest unit, so a good answer is 1 1/2 white rhombus (small angle) or 1/2 square. How many green triangles is a nice discussion.

The next activity I'm including the way I work it for preservice or inservice teachers. Easy to adapt for 5-12th grade students, though. The Word .doc version is here.

Telling Angles

Objective: TLW expand their understanding of angles, connect with the angles on a clock face, and use reasoning to find angle measures by comparison with known angles.

Schema Activation: What do you know about angles, measuring them and degrees? List your top 3 facts or bits of relevant knowledge.







Activity:
1) Forget the time, what angle is it? Record the angles made by the clock hands below. Add one sentence of justification for how you know.

2) Teacher question: why might I have sequenced the clocks the way I did?




3) Record the angles made by the clock hands below. Add one sentence of justification for how you know. Notice the hour hands are no longer pointing directly at a number.

4) 11:50, 1:10, 3:20, 7:40. For each time, draw in the hands precisely, and then determine the angle between the hands. Describe your process for each time. Start with the one you think would be easiest.

Reflection: What 3 ideas do you most want your students to understand about angles and angle measure?







Extensions: Challenge questions:
a. Is there a time for any angle?
b. Is the clock more likely to have an acute, right, or obtuse angle when you look at it?
c. How many degrees does the angle change in 1 minute? 5 minutes? 10 minutes? Does it depend on what time it is at the start?