## Archive for April, 2009

### Gum and Math: Is there a connection? REALLY?

April 28, 2009

Last week I was all excited about an article showing a connection between schools with playgrounds and higher math scores.  As it happens, at the very same time there was another news report about higher test scores.  This one linked them to gum.

The first article I read, from the LA Times, explained:

The study was conducted by researchers at Baylor College of Medicine and was sponsored by the Wrigley Science Institute. The study included 108 students, ages 13 to 16, who were assigned either to chew sugar-free gum during math class, while doing math homework and during math tests, or to refrain from gum-chewing. After 14 weeks, the students took a math test and their grades were assessed.

The result was increased test scores for the gummy kids on the Texas Assessment of Knowledge and Skills achievement test.  The increase was small — only 3% — but the National Post says it was statistically significant.  Several articles drew the connection that chewing gum can lower stress, and that may have helped with the testing.

The National Post article turned out to be one of the more detailed ones that I ran across (explaining, for example, that the gum-chewers chewed gum 86% of the time in school and 36% of the time while doing homework) and it shared a fact that I didn’t see in many other articles:  there were TWO standardized tests given, not just one.  The other standardized test was the Woodcock Johnson III Tests of Achievement, and on this test there was no statistically significant difference in the scores of those who chewed gum and those who didn’t.

Furthermore, when I looked further, I found  this nutrition site that revealed that the course grades for both groups actually decreased over time, although the gum chewers had less of a decrease.    So all in all, it’s kind of interesting but perhaps not worth the hype of headlines like “Chewing Gum Improves Math Scores… No Lie” and “Teenage Brain Power Boosted by Chewing Gum:  Wrigley Study”.

Isn’t that a gross photo at the top?  That’s Bubblegum Alley in my hometown [photo taken by Samir and posted on wiki under Creative Commons License].  When I was about 10 a friend and I wrote “Sit on It” in gum on the wall of Bubblegum Alley.  We felt very sneaky.

### Carnival of Mathematics #51: The Return

April 25, 2009

The Carnival is Back…it was just taking a little vacation.  This issue, the Resurrection Carnival,  is being hosted by squareCircleZ and comes complete with a reference to Area 51.

Incidentally, I was browsing his sitemap and, learned that three years ago he wrote an article that answered a question I’ve been wondering about for a while.  I heard a while back that there are more people alive today than dead.  This didn’t really make sense, but I wasn’t certain what the numbers would even be.  It turns out to be false, and in reality about 5 to 6% of the people who have ever lived are alive today.   There’s more information here.

### Tetrahedral Pyramids (part 2 of 3)

April 24, 2009

The first post in this series, Pascal’s Pyramid (part 1 of 3), explored number patterns that arise in a 3 dimensional version of Pascal’s Triangle: a pyramid with square cross sections.  Another way to do a 3D version of Pascal’s Triangle would use triangular cross sections.   If you think of building a tetrahedron by stacking oranges in a pile, each orange is in contact with (up to) three oranges that lie in the level above it.

As before, we set the top number in the pyramid to be 1, and assume that at the lower levels, each number will be the sum of the (up to) three adjacent numbers on the previous level.

Also as before, we know to expect the binomial coefficients to appear on each of the three triangular faces of our new pyramid.

In the interior, the numbers generated satisfy the recursion

f(n+1, a+1, b+1) = f(n, a, b) + f(n, a, b+1)+ f(n, a+1, b+1)

where f(0,0,0) = 1, and f(n,a,b) is understood to be zero if any of the parameters go out of range (that is, if a or b is either negative or greater than n, or if b>a).

The resulting pyramid:

And for example, the numbers on the seventh level from the top are:

Examining this triangle, we recognize an odd variation on Pascal’s triangle:  each row is a multiple of a row of the usual Pascal triangle, where the multiplier is the left-most entry in the row.  Since those terms are also terms on an outer face of the pyramid, we know they too are binomial coefficients, and we are led to conjecture that  $f(n, a, b) = { n \choose a } {a \choose b}$.

As before, we can prove this result by induction on n.  The base case, when n=0, works, since ${0 \choose 0} {0 \choose 0} = 1$.

If we assume (for some specific value of n)  that each $f(n, a, b) = {n \choose a} { a \choose b}$, then it follows that

$f(n+1, a+1, b+1) = {n \choose a}{a \choose b} + {n \choose a}{a \choose {b+1}}+ {n \choose {a+1}}{{a+1} \choose {b+1}}$

$= {n \choose a} \bigg( {a \choose b} + {a \choose {b+1}} \bigg) + {n \choose {a+1}}{{a+1}\choose {b+1}}$

$= {n \choose a}{{a+1} \choose {b+1}} + {n \choose {a+1}}{{a+1} \choose {b+1}}$

$= {{n+1} \choose {a+1}}{{a+1}\choose {b+1}}$

And thus we have our identity by induction on n.

This suggests a direction for exploration:  what interesting things do we know about Pascal’s Triangle, that might generalize to three dimensions?

Coming up:  Sierpinski’s Pascal Pyramid (3/3)

### The Playground/Math Association

April 23, 2009

Russ  Lopez and his two buddies are the Defenders of the Playground.  I picture them with capes and swords, but actually they’re profs (Lopez from Boston University and the others from Tufts) who just studied the association between elementary school playgrounds and test scores.  According to BU Today today:

When Lopez studied the 2003 results of the fourth-grade English language MCAS (Massachusetts Comprehensive Assessment System), standardized tests that almost all public school students must take, he saw no discernible differences between children at the 70 schools with new playgrounds and children at schools with old playgrounds.

But when he looked at math scores, he saw a very different picture. In schools where fourth graders had new playgrounds, 25 percent more kids passed the math MCAS. And that remained true after he and his team controlled for factors such as demographics and the number of students receiving free or reduced-price lunches.

Of course, as the article goes on to explain, that doesn’t mean that building more playgrounds will automatically raise test scores — there could be other factors in play (so to speak).  But, especially in comparison to the English tests results, it’s certainly an interesting finding and I look forward to reading the follow-up.

The playground photo was taken by drk_faerie.

### Traveling the Lower 48

April 22, 2009

I just started reading The English Major by Jim Harrison.  It’s about Cliff, a retired high school teacher and farmer, whose wife has just left him.  He  decides to travel to the lower 48 states in the US using a puzzle as a  guide.

He starts in Michigan, then goes to Wisconsin, and in Minnesota he hooks up with a former student of his, Marybelle,who decides to ride with him through North Dakota to Montana. At this point Cliff relates:

After breakfast in Wahpeton and before she fell asleep Marybelle had said it would be nice to do some north and south zigzagging on the way to Bozeman.  I didn’t say anything but this distressed me as I had intended to enter and exit each state exactly once.

When I read this, I was immediately distracted by the question of whether or not that was even possible.

As a follow up question, if Cliff had started anywhere he wanted instead of his hometown, how close could he end to where he started?

For reference here’s a map of the United States, with Hawai‘i and Alaska conveniently resized and located in Mexico. Click for a bigger version.

### Pascal’s Pyramid (part 1 of 3)

April 20, 2009

In an earlier post, Ξ had mentioned a question from Who Wants To Be A Millionaire that made reference to “a famous “pyramid” of numbers that starts with the number one on top”.  The intended answer was “Pascal’s Triangle” (my emphasis).

That got me wondering what a three dimensional analogue of the Pascal Triangle might look like.

Today we’ll explore a Pascal-like construction of a square pyramid.   The top-most point on the pyramid will be assigned the value 1.  As we move down the pyramid,   let’s assign each point a value  by taking the sum of the 4 terms directly above and adjacent to each point.

If you think about the 4 outer faces, each point along a face only has two points that lie above it.  Thus if we restrict our attention just to one outer face, our construction is identical to the two dimensional Pascal triangle, and for  each face we should get the familiar binomial coefficients.

One can describe  how a given entry depends on those above it:  If we let f(n, a, b) be the entry in the nth layer down from the top, in row a and column b, then the entries that lie above will be in layer n-1, their row will be a or a-1, and their column will be b or b-1.  Thus

f (n, a, b) = f(n-1, a-1,b-1) + f(n-1, a, b-1) + f(n-1, a-1, b) + f(n-1, a, b)

with f(0,0,0)=1, and f understood to be zero if any of the parameters get out of range (e.g. if a or b > n, or is negative).

Numerically, we get the following pattern (for 0 ≤ n ≤ 6):

For example, the entries in the fifth layer are:

Each entry appears to be the product of the first entry in its row, with the first entry in its column.  Since we know the row and column values are binomial coefficients, we conjecture that $f(n,a,b) = {n \choose a} {n \choose b}$.

To prove this, we start by verifying that when n=0, the product of the binomial coefficients is equal to 1.  Proceeding inductively, we assume that the result holds for n-1, and we compute f(n,a,b):

${ {n-1} \choose {a-1}} {{n-1} \choose {b-1}} + {{n-1} \choose {a}} { {n-1} \choose {b-1}} + {{n-1} \choose {a-1}}{{n-1} \choose {b}} + {{n-1} \choose {a}} {{n-1} \choose {b}}$

$= \bigg( {{n-1} \choose {a-1} } + {{n-1} \choose {a}} \bigg) \bigg( {{n-1} \choose {b-1}} + {{n-1} \choose {b}} \bigg)$

$= { {n}\choose {a}} {{n} \choose {b}}$

QED by induction; each entry in Pascal’s Square Pyramid is a product of two binomial coefficients, and is equal to the sum of the four terms immediately above it.

Coming upTetrahedral pyramids (2/3);  Sierpinski’s Pascal Pyramid (3/3)

### Math Teachers at Play and some other stuff

April 17, 2009

Math Teachers at Play #5 is up at Let’s Play Math! It’s organized by topic, with pictures and neat quotations thrown in for good measure.    And as usual, it is full of interesting posts!

Speaking of Carnivals, apparently the Carnival of Mathematics was just hibernating and it will reappear next week.  At least, that’s the rumor, where “rumor” means I read it on jd2718.

And speaking of reading things, after reading Keith Raskin’s comment I headed over to Natural Blogarithms to look for Pythagorean Triple stuff, and I was immediately distracted by the following puzzle:

The four numbers A, B, A+B and A-B are all prime.  The sum of these four numbers is

A) Even
B) Divisible by 3
C) Divisible by 5
D) Divisible by 7
E) Prime

(from the 2002 AMC 10/12B #15).

I quickly convinced myself of one answer, then decided there was a misprint and talked to Batman about it at work, and then we he realized that exactly one of the possible answers was correct.  At that point the problem seemed a lot more interesting.

If you need something a little more lighthearted this Friday afternoon, here’s the 1-20 roll call from Sesame Street.

### Faulty Unit Conversion

April 16, 2009

Language Log has a post up (via HeadsUp the blog) about a Fox News story that had some metric issues:

The tests involved head-on crashes between the fortwo and a 2009 Mercedes C Class, the Fit and a 2009 Honda Accord and the Yaris and the 2009 Toyota Camry. The tests were conducted at 40 miles per hour (17 kilometers per liter), representing a severe crash.

On a personal note, I have a 1999 Camry that does 23.4 km/l highway, 12.8 km/l city.

### Counting Chickens

April 13, 2009

Watch out — those cute little chicks will know if you’ve eaten one of their chocolate eggs.

At least, that’s the word according to the folk in Italy at the Universities of Padova and Trento.  They did experiments and found that newly hatched chicks can tell More versus Less:

In tests the chicks were shown a set of objects, in this case identical small balls, in groups of either two or three.

In one of the experiments the chicks choose consistently to walk towards a group of three balls rather than a group of two.

You can sort of see a video here, although the video doesn’t seem to show actual experiments:

The one aspect that bothers me a little is this:

When the ball were hidden behind a screen, but one of the balls could be seen being passed from the larger group to the smaller one, the chicks were still able to identify which group now contained three objects.

It wasn’t clear from my source (The Telegraph) if they ever moved one from a group of, say, 4 to a group of 1 so that the group that received the ball was still the smaller group.  In other words, maybe the chicks just were following the action.  (But it seems possible according to the BBC news article that the researchers took that into account — I can only access the journal Proceedings of the Royal Society. B where this was published for articles that are at least a year old, so I can’t verify it.)

There’s no word yet on whether the Easter Bunny has these same addition skills.

Chick photo by Fir0002 [cropped and flipped], published under GNU-FDL.

### GraphJam Fun

April 11, 2009

I read FailBlog regularly (there’s also a G-rated version), and this morning I happened to notice a site called GraphJam, which was instantly interesting because it had the word Graph in it.  “Is this new?” I wondered.  Well, no — it’s apparently been around for over a year.  I’m just not very observant.

But in case there’s someone else who hasn’t seen it as well, here’s a few examples for this fine Saturday afternoon.

There are Venn Diagrams

and pie charts

Flow charts

and bar graphs

see more Funny Graphs

Enjoy!  [Note:  most of these are fine, though like FailBlog they aren’t all G-rated.]

### Summer Jobs Puzzle

April 9, 2009

I’m in the mood for a puzzle.  There are about a million on the web, but I liked this one from  Puzzlers Paradise because it talked about summer and because there was a cool grid that you can fill online here.  [You click once in a square to X it out, twice to put in the Yeah-I-got-it! circle, and three times thrice to clear it.].  The same site has a whole bunch of these, and a lot more puzzles on their main page.

This summer, five friends each found a different job to earn money. As it happened, they all started their new jobs in the same week; each on a different day and each earning a different amount. Determine the type of job each got, the day of the week each started working, each friend’s mode of transportation to their new job, and how much per hour each friend was earning.

1. Chris didn’t have a newspaper route but he did use the bus. The boy who mowed lawns took the train to work.

2. The boy who used a bicycle started working on Friday. Joe didn’t get a job as a painter.

3. The five friends are represented, in no particular order, by the following: the boy who made \$6.00 per hour, the boy who started work on Thursday, the boy who used a car to get to work, Chris, and the boy who mowed lawns.

4. Peter started work the day before the boy who worked as a cashier but two days after the boy who made \$7.00 per hour. The boy who worked as a cashier made \$0.50 per hour more than Greg made.

5. The boy who started work on Tuesday made \$6.25 per hour. Rob didn’t start work on Monday. Peter made more per hour than the painter but less than the boy who used a car.

6. Rob walked to work. The person who started on Wednesday didn’t make \$6.50 per hour. The golf caddy used a car to get to work.

Good Luck!

The photo of the beach has nothing to do with summer jobs, but it was taken during the summer [by TwoPi, in Oregon].

### Why is April 15 Tax Day?

April 8, 2009

We’re one week away from April 15, and if you’re wishing you had a little more time to finish your taxes then be glad that you don’t live 50 years ago.

In 1913 Congress passed the Sixteenth Amendment to the Constitution, which said that Congress can collect income tax.  They’d already done this before, but were making it all official (although apparently when it was introduced it was expected to fail).  At that point, they declared March 1 to be Tax Day.  March 1, as you may recall, was New Year’s Day for the Ancient Romans, but that’s probably just a coincidence since quite a few days were New Year’s Day at one time and place or another.

After 1918 the Tax Day was changed to March 15, which probably resulted in a host of “Beware the Ides of March” jokes.

Finally, in 1954, the date was changed to April 15 [although that didn’t take effect until the following year].  According to this site (where I got most of the info so far), this was because the IRS was getting swamped with so many returns at the last minute, and they hoped that having more time would spread that out a bit.

In searching around, I also discovered the original 1040 form from 1913, which doesn’t look as simple as I would have hoped.    You can see all of them through the years here, although just looking at all those tax forms doesn’t exactly give one a feeling of peace and relaxation.

### Tax Math: Mesopotamia Stye

April 6, 2009

We’ve talked about Aztec Taxes and Inca Taxes; what about taxes in Mespotamia?

There are many many tablets that can attest to the fact that they had them.  According to Dr. Tonia Sharlach at the University of Pennsylvania, lots and lots of things were taxed year round [Here’s a whole article about the salt tax!], with payment of objects instead of money.  This apparently led to a lot of smuggling, and Dr. Sharlach’s article quotes a letter from about 3900 years ago written by an office to an employee:

Irra’s son sent smuggled goods to Pushuken but his smuggled goods were intercepted. The Palace then threw Pushuken in jail! The guards are strong…please don’t smuggle anything else!

Apparently one of the worst taxes was the labor tax (sometimes called the “burden”).   While it sometimes feels like a certain amount of my job amounts to a labor tax, it’s admittedly quite a bit nicer than being somewhere else to work for a few months.

Upcoming…why is tax day April 15 anyway?

The map was posted under GNU-FDL by Nareklm.

### Math Teachers at Play #4

April 4, 2009

It’s been two weeks since the last Mathy carnival (oh Carnival of Mathematics, where aaaaaaarrrrrrrrrreeeeeeee you?), and on this snowy (seriously) April day nothing could be better than a nice long list of great blog posts.  And that’s exactly what there is at  Math Teachers at Play #4!    It’s being hosted over at Homeschool Bytes, a blog that shares a whole bunch of resources for homeschooling.  (Although they’re not limited to homeschooling, of course.  Our older kiddo is learning multiplication, and Timez Attack looks like a great way to  help him get the number facts at his fingertips.)

But back to the Carnival.  The carnival divided into different steps/levels, from counting money to the SATs [both the modern version and some questions that could have occurred if there were SATs in 1557], so there’s something for everyone.  Time to play!

### Phonetic Phun on a Phriday

April 3, 2009

Last August, Ξ and I were inspired by a post about the International Radiotelephony Spelling Alphabet at our favorite non-math blog, puntabulous, to create a satirical version, one that would be completely confusing and virtually useless.

We posted a preliminary version of this to the comments of that original post.  Since then, the list has evolved, with help from our fellow commentator friends at puntabulous, as well as help from Batman, NP, and several other of Ξ’s coworkers.  What follows is the fruit of that community effort.
Note:  As with the Official version, this is meant to be read aloud.

• A is for aye
• B is for bdellium
• C is for czar
• D is for djinni
• E is for eye
• F is for fyce  (??)
• G is for gnu
• H is for hour
• I is for iajo
• J is for jicama
• K is for knight
• L is for llama
• M is for mnemonic
• N is for night
• O is for one
• P is for philter
• Q is for Quran
• R is for roister    (say it fast)
• S is for Sea
• T is for tsar
• U is for uighur
• V is for vrouw
• W is for why
• X is for Xi
• Y is for you
• Z is for zwieback