Mar 11

Pi/Pie Day

pi day puz

Jay sent this over to me from The Puzzler. It took me a second to get it, but ended up giving me a big smile. Happy Pi Day everyone!

Feb 11

SADMEP instead of PEMDAS

We write our expressions with the understanding that a reader will order the operations involved using PEMDAS, or as I like to write with my students P/E/MD/AS. This is a convention. On Mars though a very different convention is observed. They use SA/DM/E/P to parse their expression.
Question 1: How would a Martian evaluate the following expression?

We, of course, get 50 using PEMDAS, but what would the Martians get?

Question 2: How could we translate into Martian, so they end up doing the operations in same order as we do? How can we rewrite the expression so they evaluate it to 50? (Note: in my answer I took the Martian use of nested parentheses to be outside in, instead of inside out like ours–they’re Martians after all, and I couldn’t get it to work any other way :) )

Dec 10

Math + Syntax

Some times mistakes are more interesting than right answers. I remember a Prealgebra class I was teaching a few semesters back. We were reviewing for a test and doing a problem something like this…

Evaluate the expression -2x + 13, for x = 3.

Most of the students were getting the correct answer, 7. A few of them were making sign mistakes and getting answers like 19, -19. One student answered -10. I was so surprised and curious by the answer, I put him on the spot and asked how he got -10?

His explanation was simply that he’d replaced x with 3,
-23 + 13
and then added

“Cool!,” I said. He’d understood the syntax of mathematical expressions incorrectly, but had an alternate syntax all of his own.

Now, here’s the challenge for the week.

(1) If we are to understand, mathematical expressions in his way. What would the solution to the equation 2x + 10 = 243 be?

(2) Can you find an equation with no solution under this syntax?

(3) Can you find where the solution is the same under both the usual and the alternate syntax?

Apr 10

Teaching Problem

I use MyMathLab to run my classes and I have all this great info on each assignment called an “Item Analysis”. Below is an item analysis for a sample exam in one of my algebra classes. Here’s my question, which are the most urgent problems to cover when I go to give my class the review? I’m not looking for an intuitive listing of the top problems, but a spreadsheet function that I can use to sort the problems. Note, something that makes this problem harder is that the questions aren’t given with the same frequency. (Hint: the most obvious is calculating the percent of time a question is gotten correct, but there are certain problems with that.)

Jan 10

Pieces of Eight Challenge

Lee has thought up another devious puzzle along the lines of Solve my t-shirt and Denise’s New Years math challenge.

The object of this challenge is to develop ten mathematical expressions that equal 8. You must use the digits 2, 7 and one other. This other digit must be a 0 in the first expression, a 1 in the next expression, a 2 in the next expression and so on, up to 9. You must use the 2, the 7 and the other digit once and only once in each expression. (Note: For the analitically challenged, solution(s) when the “other digit” is 2 will have two 2’s, solution(s) when the “other digit” is 7 will have two 7’s.)

You may use +, -, *, /, exponentiation, decimal points and parentheses.

Here’re the two known solutions when using 2, 7 and 1.

8=2+7-1=1^2+7 (2nd solution by John Stokes)
pieces of eight

Dec 09

New Year Math Fun

I found this great New Years math activity almost exactly a year too late. Check out this post from Lets Play Math. I’m going to check tomorrow to see if the fun rolls into the new year.

Dec 09

Generalizing the concept of prime

primesTeaching a lot of developmental classes, I spend a lot of time with basic concepts like the fundamental theorem of arithmetic. If you are as geeky as me, it charges your card to no end that every natural number can be written uniquely as a product of primes. It is so strange (and wonderful) that a certain subset of numbers, the primes, are the multiplicative backbone of all the natural numbers. I think of the fundamental theorem as a decompostion theorem in the sense that shows us that all the naturals can be broken down in a particular way. Another of these “decompostion” theorems is Lagrange’s four square theorem, that any number can be expressed as the sum of four or less perfect squares. Why this theorem is cool but not as cool as the fundamental theorem is that the decompostion of a number in squares is non-unique. Think 64 = 8^2 and 64=4^2+4^2+4^2+4^2. Also Lagrange’s decompostion is a decomposition in two operations addition and exponentiation. So the fundamental theorem is one of many, but also special.

Lately, I have been wondering what “prime” would be for operations other than multiplication? This is what I was thinking about: a natural number n is prime in operation @ if the only way to decompose n in @ for the naturals is n@e, where e is the identity for @. Applying this to addition there is only one prime number, 1. Every other number is composite. Take 4, it can be written as 3+1, 2+2, or 1+1+1+1. The latter is in fact the additive prime factorization of 4. Addition is kind of boring though. Not enough primes.

Note, I left the identity on the right-side in the definition, so that we could apply the definition to exponentiation. Here we have a lot more primes. In fact the only composites are the perfect squares, perfect cubes, perfect….. The first ten exponential primes are 2, 3, 5, 6, 7, 10, 11, 12, 13, 14. Compare that to the first ten multiplicative primes 1, 2, 3, 5, 7, 11, 13, 17, 19, 23. We can’t really say that there are more exponential primes than multiplicative primes. (Showing that there are an infinite number of exponential primes is, I think, just a matter of adapting the same old proof we had in multiplication.) What we can say is that the exponential primes definitely seem more dense. In fact, I think it stands to reason that (other than 1) every multiplicative prime is by necessity an exponential prime. Now imagine primes for super/hyper-exponentiation. The primes get more and more dense.

Now let’s play. What is the exponential prime factorization of 16?
What about the hyper-exponential prime factorization of 16?

pic by Adrian Leatherland

Dec 09

Circle Sums

circle puzzle002
My mom sent this one over. It’s fun, but don’t be fooled by the Venn-Diagram-ness of it all. Read the directions. The numbers actually have a very different meaning than they do in a standard Venn Diagram.

Dec 09

Solve my t-shirt

wrd prob t-shirt
This week’s problem of the week was found…at a t-shirt shop? Math really is everywhere.

Nov 09

Conic sections anyone?

Nancy stopped by today and gave me a fun problem from her college algebra class. I had forgotten all the conic sections except for the circle and had to look around online a bit. Anyway, I thought you might also like a little trip down memory lane. Here’s the problem:
Give the equation of a hyperbola with a center at (0,1) and one of its foci at (0,5) that has an asymptote at