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.

## Monday, November 23, 2009

## Saturday, November 21, 2009

### 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?

## Wednesday, November 18, 2009

### 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.

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.

## Friday, November 13, 2009

### 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.

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.

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.

Pick on Someone Your Own Size

A Math Mystery Game

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.

## Friday, November 6, 2009

### 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:

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.

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