A cool sequence problem…

Our oldest son (nearly 10) posed the following challenge:

What comes next in this list?

1, 1, 1, 2, 2, 3, 2, 4, 3, …

Answer and rationale (his and mine) after the jump…

Root extraction, part II: cube roots

As you might guess, this post builds on “Root extraction, part I“, which gave a way to visualize the traditional square root algorithm geometrically, an approach that has the advantage that each step appears natural and easily motivated.

Our goal herein is to do much the same for cube roots. The point is to find a geometric construction, ideally one well-suited to physical manipulatives, in which the steps in building the successive digits of the cube root of a number are transparent.  As with the post on square roots, I make no claims to originality in what follows.

Example:  Find . 

Root extraction, part I: square roots

I recently discussed the traditional algorithms for computing square and cube roots in my History of Math class.   Our reading, on mathematics in ancient China, gave both algorithms as a set of rules for manipulating number rods.  For me, it was fascinating to see past the text: the rules as given would transfer directly to an abacus/soroban calculation, and were essentially the same as the rules that prior generations of American schoolchildren would have been drilled on in school.

My students (mostly high school math teachers) found the book’s explanation of the method obscure;  the key is to view the process geometrically, rather than as a mechanical set of rules for manipulating digits.

I make no claim of originality in what follows; I offer it here in part because I can’t find any lucent discussions along these lines on the web. 

Math Helps College Students

According to the August 4, 2007 issue of Science News (Vol. 172, No. 5, p.78), the more math and science classes students take in high school, the better they perform overall in college (emphasis mine):

Apparently, high school math is the key to good grades in college science classes.

A survey of more than 8,000 students from 74 colleges found that each additional year of high school math correlated with a 1-to-2-point advantage, on a 100-point scale, in college chemistry, physics, and biology grades. For example, 2 additional years of high school math typically corresponded to a 3-point improvement in college biology—the difference between, say, a B+ and an A-.

Kids who took more high school classes in chemistry, physics, or biology gained a similar edge when they took a class within the same discipline at the college level. However, no significant benefit crossed a line between science disciplines. Only math seemed to boost grades in other subjects. The study appears in the July 27 Science.

Coauthor Philip Sadler of Harvard University says that students who take advanced high school math classes are better able to handle the more basic math required in college science classes.

The results are “not surprising,” says James Milgram, a Stanford University mathematician and a member of a presidential panel advising the U.S. Department of Education. He points out that decadal surveys by the department have shown that as more students have taken advanced high school math classes, their chances of graduating from college have improved. “There is overwhelming evidence that the single most important factor that correlates with success in college is what is done in high school math,” says Milgram.

(Hat tip to Brian Witz.)

Grocery Store Polygons: Cookie Cake Tops

The post two days ago (Junk Food Geometry) focused on edible polygons, but perhaps my favorite examples of grocery store polygons are inedible: the cookie cake tops at Wegmans. The aspect that stood out initially to me is that they are non-standard polygons. The medium sized one, shown to the left, is a heptagon! This cookie top and a pillbox we once found are the only two real-life examples I’ve seen of regular heptagons.  Edited to add :  of course, within days I found heptagons in a Harry Potter game and in coins.

Junk Food Geometry

I was about to post about the interesting shapes in the cookie cake tops at our grocery store (which I’ll do shortly), and I found myself getting distracted thinking about all of the interesting shapes in the snack aisle itself.

“Double Down” Dumbs It Down

Double Down is a quiz show for New York State high school students that airs on PBS (WCNY to be specific). I happened to catch a rerun last night, and one of the categories was “Math”. Here are a couple of the questions:

1. A polygon with 5 sides is called what?
2. A polygon with 8 sides is called what?

These are high school students, remember. My 20-month old daughter knows what an octagon is. Can we give these kids some credit?!

Counting on your Fingers

Three Finger Tricks for Multiplying also made me think of another finger trick: using both hands to count to 99. It’s not as clever as multiplication, but it’s one I use regularly for keeping track of numbers and for simple addition or subtraction.

Three finger tricks for multiplying

Many math sites teach the following method of using your fingers to remember the multiples of nine: to find the product of 9 times n, hold your hands out in front of you and fold down your n^th finger from the left to separate the tens and the ones. For example, to find 9×4, you would hold down your 4^th finger from the left as in the above photograph. The bent finger separates the tens and ones digits, so the configuration of 3 fingers (folded finger) 6 fingers gives the answer of 36.

While this method has enjoyed great popularity among students and teachers, there are two other lesser-known finger tricks for multiplying numbers.