This is “Kock Snowflake Fractal”, a lovely woven rug from Donna Loraine Contractor, on display at the 2012 Bridges Math and Art Conference at Towson University.
Read more about the artist here.
Application
The Algebra of Coffee Consumption
Upon discovering that I had been paying much more for my coffee than I realized, I was faced with a dilemma only algebra could solve. Or, at least help analyze.
You see, my local coffee shop offers a simple reward program for regular customers: purchase 10 bags of coffee, and your eleventh bag is free. At the time of my discovery, I had credit for four bags of coffee. Thus, the dilemma: do I continue to buy the over-priced coffee six more times in order to get the free bag? Or do I just start buying cheaper coffee somewhere else?
The coffee at my local shop was costing me around $12 for 12 ounces. I knew I could get comparable coffee at another shop for around $10 a pound. So should I stay or switch?
If I bought 6 more bags from the local shop, I’d spend $72 and get seven 12-ounce bags of coffee (the six I bought, and the free one). Thus, I would get 84 ounces of coffee for $72.
Now $72 dollars at the new place would get me 7.2 pounds of coffee, or around 115 ounces. So I’d be getting an additional 31 ounces, or almost two pounds of coffee, by switching!
Needless to say, I swtiched. But it got me thinking: at what point would it have been better to continue buying the over-priced coffee? I turned to algebra for the answer.
Let x = the number of bags of coffee that have already been purchased. In order to get a free bag of coffee from my local shop, I’d need to buy ten total bags; thus, I’d need to purchase (10 – x) more bags of coffee. After that, I’d receive one free bag, so in the end I’d get (10 – x + 1) = (11 – x) total bags of coffee.
Each 12-ounce bag of coffee costs $12, so I’d spend a total of $12 * (10 – x) to get (11 – x) * 12 ounces of coffee.
This works out to a price of
for the coffee I’d get from the old place.
I know I can get coffee at the other place for $10 per pound, or 
which would mean that the coffee from the old place would be cheaper per ounce than the coffee from the new place.
Now we simplify our inequality:
Thus, I should continue buying coffee at the old place if I’d already bought more than eight bags of coffee there. Otherwise, I’d be better off switching.
I probably didn’t need algebra to tell me to stick with the old place if I’d already bought 9 bags, but algebra did show me just how hopeless my situation was!
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Teaching
Regents Recap — June 2012: Some Improvement
Here is another installment from my review of the June 2012 New York State Math Regents exams.
I tend to be rather critical in my evaluation of these exams, pointing out poorly constructed, poorly phrased, and mathematically erroneous questions. However, there have been some minor improvements of late.
First, it seems as though, in general, the wording of questions has improved slightly. To me, questions on the June 2012 exams were more direct, specific, and clear than in the recent past.
There were also some specific mathematical improvements. For example, although graphs were often unscaled, they seemed generally more precise, avoiding issues like this asymptote error.
There were considerably fewer instance of non-equivalent expressions being considered equivalent. The problem below avoids the domain-issues that plagued recent exams.
Perhaps it’s just luck, but we’ll give the exam writers the benefit of the doubt for now.
And the Algebra 2 / Trig exam definitely demonstrated a more sophisticated understanding of 1-1 and inverse functions, which is good to see in the wake of this absolute embarrassment from last year.
Perhaps someone has been reading my recaps?
Let’s hope we see continued improvement in the clarity and precision of these exams. If these exams are going to be play such an important role in today’s educational environment, it seems of utmost importance that they be accurate and well-constructed.
Art Photography
Math Photo: Sculpture of Spheres
This is a photo of the Sonja Henie Ice Fountain, which stands outside the historic hockey arena in Lake Placid, New York. I suppose most people like it best when it’s covered in ice, but the mathematician in me thinks it looks just fine like this.
Art Photography
Math Art: Kolam Spirals
This is “Kolam – Brown: Four Spirals”, by Shanthi Chandrasekar, on display at the 2012 Bridges Math and Art Conference at Towson University.
Read more about the artist here.