Scale of the Universe

Amazing flash animation where you choose the scale with a slider and it zooms from quantum foam to the entire universe. Spectacular.

Reminds me of the classic logscale comics at XKCD. Making their own scale picture is an excellent assignment.

Problem Solving Video

Thought this little video had some excellent problem solving advice. It's an ad for a scholarship competition for a design program. Reminds me of a series of images called Advice to Sink in Slowly. Great graphic design and some pretty solid advice.

Hey! Now the Sink in Slowly people have their own site and a store for the Advice.

Multiple+Representation=Multiplication

Working with the 4th graders last week, the objective was just to develop multiplication facts, as they are struggling with the multi-digit multiplication.

I'm a big believer in automacity vs strict memorization, as I believe it leads to fluency and solid pre-algebraic thinking, as well as deepening operation understanding. The lesson was pretty simple, but a good place to start.

Objective: TLW see connections between adjacent multiplication facts, and use those connections to help computation.

Materials: unifix cubes, graph paper with a 5x5 structure (link goes to a 2 page pdf graph paper, so it can be printed both sides easily), blank multiplication chart.

Lesson:

Cubes 25-30 min

Start out with a small set of cubes, such as two stacks of three cubes. What multiplication problem is this? (You might choose if you're going to make an issue of order or not. To me, this is 2 of 3, making it 2x3.) This is 2x3 and 2x3 is 6. We're going to pass the cubes around our group.

When you have the cubes, each person can either add a cube to each stack, or add a stack of the same height. Then you say the new multiplication and what the answer is. I add a cube to each stack and say it is 2x4, which is 8.

As the stacks went around, I saw students slowly gaining an idea of figuring out the next problem by adding on to what they knew before. It took a little bit to get the idea of what multiplication computation it was, but they got the idea of what moves were allowable immediately. Soon, several of the students were adding to get the next fact.

We restarted with 3 stacks of 1. The students were much more fluid. There was a bit of an issue with the cubes being distracting with them. If I had thought about working with students who hadn't used the cubes much, I would have given them time to play first, setting up multiplication problems of their choice.

Graph Paper 15 min

The next phase of the lesson was to move to graph paper. We drew a 2x3 rectangle, and the students were comfortable with thinking about that as 2x3. We did one together, where the group decided which side to add squares to. 2x4, 2x5, 3x5, ...

Then each student got their own graph paper and started building a chain of rectangles, with the new dimensions filled in and the result. I saw several students using the adding strategy. One student didn't get the idea of what the connection was, and just drew rectangles and filled in the area. But maybe that was what she needed to attend to.

Multiplication Chart 5-10min

As it was time for students to go, we summarized by looking at a multiplication chart. Filled in a fact they agreed on, 6x5. I led them through how to use that to go on, by adding to get to 6x6 or 7x5. They went back to class with their own chart and a page of graph paper. As I saw them working on the charts in their free time, some were using patterns as they had seen them before, some were using them for the first time, and one student asked for how that worked. A couple of examples got her started.

The next week: The week after this I tried to get the students to help me develop a game. A couple of them found it less than engaging, but Mrs. B mentioned that all the students were antsy. Day before a long weekend? Cabin fever? The game is designed to become obsolete, but I don't think that's the issue. I picked it because I've wanted to work this out, and the other thing they've been working on in class is area and perimeter of compund rectangular shapes.

Break Up (In development)

Two players or teams.
5-structure graph paper, pen, optional dice.

Game play: Determine the size of a starting rectangle. This can be done through choice, each team choosing a side length, or rolling three dice, or rolling four dice, or rolling 2 dice plus 10. If dice rolling, each team should roll one side.

On your team's turn, you either divide a rectangle, or calculate an area, or do both. Your team gets a point whenever an area is filled in. After all the areas are filled in, the team's whose turn it is next gets to try to find the total area. To emphasize using known facts, you can only fill in a rectangle if you know the area as a fact.

Examples: you determine a 12x15 rectangle. The first time divides the 12 into 10 and 2, and fills in 10x15=150. The second team divides 15 into 10 and 5, and fills in 2x10=20. The first team fills in 2x5=10. The second team finds the total, 150+20+10 and gets 210. So 12x15=180.

You determine a 9x12 rectangle. The first team sections off a 6x9, and doesn't know that as a fact. (There was one student who loved dividing in half.) The second team split off 5x9 and filled in 45. The first team filled in 1x9. The second team filled in 6x6 as 36. The first team (fudging a bit) figured 3x6 with 12+6. The second team mis-added 45+9+36+18 (hard sum!) and the other team got 108.

Notes: I thought the game would be better as a cooperative game, but the kids wanted to try it with points. They thought about scoring the area (as I have) but that gives the first team too big an advantage. They looked forward to scoring points, but didn't seem to care much about winning. They thought maybe you should keep track of points across multiple games. It did strongly encourage mental computation.
There's not much strategy to this game. It's about tic-tac-toe level that way. I could see this turning into kids designing their own board of compounded rectangles, that might be interesting. But it definitely encourages relational thinking for multiplication facts, which is worthwhile. If anyone has ideas for improving the gameplay, I'd love to hear them.

EDIT:
Sue VanHattum, from Math Mama Writes, was reminded of a game called Raging Rectangles from a North Carolina instructional resource packet. See the comments for details.

Math Teachers at Play #22

Welcome to Math Teachers at Play #22!

Titanium Edition!

Question 1: Do you know how many yards in a chain? Hint: 1 acre =1 chain x 1 furlong. OK, that's not a hint so much as a taunt. Hint2: a cricket pitch is a chain long. Blimey!

Elementary
Rachel Lynette presents Send your Kids on a Multiplication Scavenger Hunt posted at Minds in Bloom. She writes on a range of topics and you're sure to find other things of interest to you, too. For example, the Creative Classroom.

TIC presents Free Math Numberline Activities posted at Technology In Class. This blog features free resources of interest to teachers in all content areas, not just math.

Sue VanHattum presents Challenge: Write a Kids' Poem about Math posted at Math Mama Writes.... She has a fun comment thread started already as people start to investigate and respond with poems.

I once gave a class the challenge to write a number that when you say it out loud properly has haiku form. Got some fun answers. E.g. 22,220,220.

Crewton Ramone presents The Importance of Addends. posted at Crewton Ramone's Blog of Math. Crewton mixes an opinion piece with some advice about the importance of and how to compute addends.

Deb at math.about.com had a quick post linking to 10 100's Chart activities. I also quite like this representation.


Question 2: This edition is pentagonal. But where's the 22?

Secondary
Sue VanHattum presents Pythagorean Triples posted at Math Mama Writes.... This is the start to a fun investigation into the triples by doing some nice problem posing.

Denise presents 2010 Mathematics Game posted at Let's play math!. This is an interesting challenge that involves some non-standard arithmetic. Commenters have made some good inroads on it, but without giving anything away to keep you from trying the problem. As Denise said, "Did you know that playing games is one of the "Top 10 Ways To Improve Your Brain Fitness"? So slip into your workout clothes and pump up those mental muscles with the 2010 Mathematics Game!"

yofx has moved to its new site, and issued a nice Pieces of Eight challenge. Great order of operations problem. (G.o.o.o.p.?)

The Exponential Curve has been posting a series on graphing lines, including a nice one on multiple representations.

John Cook presents Roots of integers — The Endeavour posted at The Endeavour. John has a nice intuitive proof of an understandable and interesting theorem, and there's another in the comments.

Archimedes proof that 22/7 is greater than Pi involved circumscribing a circle with a 96-gon and then taking a ratio of its perimeter to the circle's diameter and showing that is less than 22/7.

The enlarged picture on the right shows the circumference of the blue circle to be less than the perimeter of the red 96-gon.

Question 3: 21 and 22 are the 2nd pair of consecutive semiprimes. (A semiprime is the product of two primes.) Are there more?
UPDATE: Matt has a lot of information about this in the comments. Check out his post visualizing complex domain for functions - a dance!

Higher math
Pat Ballew presents Lotteries and Math posted at Pat'sBlog. Pat notes, "Lotteries seem to be more than just a tax on the mathematically illiterate.. they are a great source of math problems." Certainly these combinatorics can be confounding, but the context is so engaging to students that they will fight for it.

Math~Blog posted an interview with the author of the very interesting book Number Freak, Derrick Niederman.

Question 4: This is the first MTAP of 2010, a pleasing year because of its 2x*100+x structure. When was the last year with that pattern? When is the next?

Math Teaching
Maria H. Andersen presents How to Grade a Student Blog posted at Teaching College Math. She says, "I've been having students blog as one of their learning projects in Math for Elementary Teachers. This feels a lot more like play than work!" Maria's really doing some innovative things, and this is worth checking out.

Kate at f(t) has a good post on a simple assessment technique using red, yellow and green cards.

The Fun Math Blog had a neat caption contest with a still from A Serious Man, the new Coen brothers movie. It gave me the idea to make an assessment where you get at students' attitudes by having them caption some math images. 63 pretty fun captions were submitted.

Maria Miller presents Choosing a homeschool math curriculum posted at Homeschool Math Blog. Maria points to a resource for guiding a homeschooler through this decision.

For elementary, I can't resist plugging Contexts for Learning Mathematics here. Brilliant, and integrates well with literacy learning.

Question 5: 22 is expressible as a sum of 4 consecutive integers. (Which?) But it is not the sum of any other string of consecutive integers. Is that true for any sum of four consecutive integers?

End Note
This edition almost didn't come to pass because of the rare but deadly Blogcarnival Catch 22. Math teachers improve their teaching by sharing with colleagues. But you have to know there's room for improvement to be open to sharing. But if you seek improvement that's already proof of quality teaching. In more logical language:

Improve your teaching => Know you need improvement AND Are willing to share with colleagues
Not a good teacher => Not be willing to share with colleagues
By definition of an implication, this is equivalent to (Are a good teacher OR Not be willing to share)
By DeMorgan, that is equivalent with Not (Not a good teacher AND Be willing to share)
By the contrapositive of original statement, implies Not improving teaching.

So we were almost shut down by the Internet Blog Overlords. Luckily, we were able to make a case that Good teacher => Desiring improvement, thus escaping the Catch 22.

Question 6: The 12 pentominos are justifiably famous and interesting mathematically. If you used hexagons (like, say the yellow hexagon pattern blocks) how many non-congruent configurations can you make? (With the same edge-sharing matching rules as for pentominoes.)

Next Time
See Math teachers at Play #23 at MATHRECREATION. Dan posts some pretty serious mathematics, like recent explorations of Catalan numbers, and some other mathematical diversions, too, like his recent post More Folds.

Remember that you can submit posts from other people's blogs if you see something interesting or worthwhile. Submit your ideas at BlogCarnival.com.

Also be sure to check our sister carnival out: most recently the 61st edition of the Carnival of Mathematics.

Miscellanea
A few submissions seemed sortapseudorelevant, so I don't want to leave them out...

• OnlineCollege presents 100 Best Education Blogs of 2009 posted at Associate's Degree. You'll see some familiar blogs in the ranks, I'm sure.
• Herbert Aitken presents 50 Free Online Educational Games That Are More Fun Than You’d Think posted at How To E-D-U. A few of my kids favorites are in here.
• Scott Palat presents Online Math Games and Lessons |. This quick post by Meaghan Montrose links to 2 reasonable looking resources.
• Peggy Stoppelmoor presents 25 Tips for Students & Teachers Using Google Wave posted at Top Online University Reviews.

We'll consider that last one ... the wave goodbye!

Recommended Reading

From the always entertaining Speed Bump cartoon.

I'm trying to develop a recommended reading list for my students. Not comprehensive, like for a library, but the place to start, first books to buy, etc. I thought with the carnival coming up, it might be a good time to beg suggestions, either for categories or books. Links lead to the best previews I could find, mostly Google Books.

Please leave your suggestions in the comments and I'll incorporate them and make note in the post. Books I've read I'll add to the main list, and the others I'll list below. Sue VanHattum (Math Mama Writes...) immediately added a bunch, and has a post on this topic, too.

The List

updated April 2010

Picture Books
Math Curse, Lane and Scieska: just the best math book ever written.
Anno's Mysterious Multiplying Jar, or anything by Mitsumasa Anno. Just charming books, and lovely besides.
Spaghetti and Meatballs For All, Marilyn Burns: my favorite of the eplicitly mathematical genre. Tang and Murphy have their place but Burns is the queen of the genre. Princess - Elinor Pinczes.

Of course there are many possible for this category. See a School Library Journal article about this. My colleague Char Beckmann has (co-)written several neat articles in Mathematics Teaching in the Middle School on using these. And Teaching Children Mathematics frequently has articles on the topic.

Math Fiction
The Phantom Tollbooth, by Norman Juster
The Man Who Counted, by Malba Tahan
Flatland, by Edwin Abbott (full text!)
The Number Devil, by Hans Magnus Enzensberger
Dec 2010: Zachary Shiner at Irrational Cube has a nice list going for these.

Elementary Math Teaching
Making Sense: Teaching and Learning Mathematics with Understanding, Fennema, Carpenter, Hiebert, Fuson, et al.
Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction, Fosnot and Dolk

Secondary Math Teaching
The Teaching Gap, Stigler and Hiebert

Math Teaching, General
What’s Math Got to Do With It? by Jo Boaler, motivates teaching significant mathematics and advises on teaching methodology, relating it to research.
A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginitive Life Form, Paul Lockhart.  Original essay available from Keith Devlin at the MAA.


Math Ed Research
Knowing and teaching elementary mathematics, Liping Ma
Experiencing school mathematics: traditional and reform approaches, Jo Boaler
Adding it up: helping children learn mathematics, Kilpatrick, Swafford, Findell
Mathematical Problem Solving by Alan Schoenfeld. Quite the academic tome, it's best to get from libraries. However Schoenfeld has a lot of the relevant materials available at his website.

Literacy Learning
Mosaic of Thought, 2nd ed., Keene and Zimmerman
To Understand, Keene
Teaching with Intention, Miller
The Whole Story, Brian Cambourne (out of print but still available used). If not reading the book, please at least read my favorite article on teaching. (Generously made freely available by The Reading Teacher.)

Thanks for any ideas you have. And I hope you maybe found one new book in the bunch, at least!

Reader Additions

Picture/Young Reader
Quack and Count by Keith Baker:board book
Carry On, Mr. Bowditch, by Jean Lee Latham: fictionalized account of the 19th century life of Nathaniel Bowditch, actual author of the American Practical Navigator
How Hungry Are You?, by Donna Jo Napoli and Richard Tchen: sharing division context
Number Stories of Long Ago by David Eugene Smith (full text!)

Teachers
Hannah, Divided by Adele Griffin: story of a twice exceptional student
The Art and Craft of Problem-Solving, by Paul Zeitz 
Math Power: How to Help Your Child Love Math, Even If You Don't, by Patricia Kenschaft
Math: An American Phobia, by Marilyn Burns
The Art of Problem Posing, by Stephen Brown
What's Happening in Math Class? (2 volumes) edited by Deb Schifter
Math for Smarty Pants and The I Hate Math Book by Marilyn Burns
Reconstructing Mathematics Education by Schifter and Fosnot (immediately on my wishlist!)

About Math 
Uncle Petros and Goldbach’s Conjecture by Apostolos Doxiadis:lots of (slightly twisted) history of math.
The Cat in Numberland by Ivar Ekeland: story of the cat who lives in the Hotel Infinity
Mathematics: A Human Endeavor by Harold Jacobs: delightful textbook
The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel: biography
Chances Are: Adventures in Probability by Michael and Ellen Kaplan: History, philosophy, science, and statistics all come together
Surreal Numbers by Donald Knuth: higher math based on a Conway problem
Euclid in the Rainforest by Joseph Mazur: Logic, infinity and probability
Powers of Ten, by Philip and Phylis Morrison: famous series of photographs/images

Mindstorms, by Seymour Papert
Out of the Labyrinth: Setting Mathematics Free by Robert and Ellen Kaplan
Anything by Martin Gardner

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!