## Thursday, January 14, 2010

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

1. Great Carnival! Thanks for putting it together so nicely.

2. Here is a short list of consecutive semi-primes past 21,22:
21 22
33 34
34 35
38 39
57 58
85 86
86 87
93 94
94 95
118 119
122 123
133 134
141 142
145 146
146 147
154 155
158 159
177 178
182 183
201 202
202 203
205 206
213 214
214 215
217 218
218 219
253 254
273 274
286 287
298 299
301 302
302 303
326 327
334 335
338 339
362 363
381 382
385 386
393 394
394 395
445 446
446 447
453 454
454 455
481 482
501 502
514 515
526 527
537 538
538 539
542 543
553 554
565 566
622 623
633 634
634 635
694 695
697 698
698 699
706 707
717 718
745 746
766 767
778 779

3. Thanks you guys!

4. Didn't get a chance to post this earlier, but in relation to a chain, or in fact the different lengths which have been called a chain within semi-recent history, I have the following in my notes...
"Edmund Gunter was a professor of astronomy at Gresham College in London and a friend of Briggs. He published tables of the logs of Sines and Tangents and is credited with the creation of the terms Cosine and Cotangent. Gunter also invented a surveying instrument called Gunter's Chain which was 22 yards long and divided into 100 links. An acre of land is equal to ten square chains and a mile was 80 chains long. The chain is now obsolete as a unit of measure but was once very common in laying out townships and mapping the US along the train routes in the 19th century. In America there was a federal law passed in 1785 that all official government surveys must be done with a Gunter Chain. It was also called the Surveyor's Chain. On a visit to Stratford on Avon while at Hall's croft, the home of Shakespeare's daughter Susanna and her husband, Dr John Hall, I came across an early map of the town and the only legend shown was in Gunter's Chains. Watching an English Cricket match the other day (Dec of 2006) I realized that the length of the bowling area (between the two wickets) is one chain also.

There were other types of chains used for surveying. The image at the right is from Milton B. Goff's First Book in Arithmetic from 1876. The lengths shown clearly indicate that Gunter's chain is used. Another popular type was Ramsden's chain which is also called the engineer's chain and has a length of 100 feet." S

5. 22 can be seen as the sum of 44 consecutive integers, from -21 to 22!