## Tuesday, June 21, 2011

### The Linear Relations of Hamburgers

Maybe you have seen the Burger King Stacker commercial where "Meat Scientists" work on an interesting problem. Needless to say, it piqued my curiosity the second I saw it; it was not long until I was trying to suck every ounce of mathematical value from the video. I am sure that I did not accomplish this goal, but I did manage to find some interesting problems and questions.

First off, the division of cow by pig seems very contrived. Their result (\$) would seem to suggest that (pig)(\$) = cow. Are cows some sort of expensive swine? The representations of mathematicians as old, mustachioed men only reinforces the gender inequality in math today. (to BK's credit, one woman is included... barely) The burger abacus also reeks of mathematical stereotype. With those minor distractions now off my chest, we can get down to the heart of the task:

"How can we achieve maximum meat flavour, for minimum money?"

Sounds like calculus to me. Unfortunately, they give us no flavour-money function, and we don't really have a way (barring a wide-scale taste test) to plot one. There were, however, 2 more interesting pathways that a teacher could take this problem. One is a look at linear relations, and the other is a more open, problem solving take. Both could have value and fit curricular outcomes.

Let's start by examining the different linear relationships in the problem. I want to re-iterate that I have in no way exhausted all the possibilities. For example, I have only begun to scratch the surface of the possibilities of the "bacon" constant. The situation presents numerous variables. Some will be obvious to the students, others will be less obvious. Here is my preliminary list:

Number of Buns
Number of Patties
Cost of Burger
Strips of Bacon
Slices of Cheese
etc.

Combining these together, a teacher could begin to develop the relationships between them. For example, what is the relationship between the cost of the burger and the number of patties it has? This famously straight-forward example will begin the descent into the mathematics of the situation. Have them choose two other variables; Can they create that relationship? Table of values, graphs, and written word can all be employed. Group discussion and teacher modeling is also a great way to get the ball rolling.

It would be interesting to develop and graph the relationship between the number of buns and number of patties necessary to construct single, double, and triple burgers. Graphing these on the same plot brings up all kind of questions. Do we connect the dots? Why? What does the steepness of the line mean? etc.

I would be particularly interested in the Cheese Slice to Patty ratio. Go back and examine the clip again, and you will get my drift. It is not until close examination that these little nuances appear. The Burger Graphs could all be devised and presented in the class room. This will bring more meaning to their interpretation. Interpolation, extrapolation, and estimation all nicely follow this exercise.

Second, there could be many interesting problems that accompany this commercial. I have thought of one, and if you have another, a comment would be greatly appreciated. Consider the following situation:

You are the owner of Burger King, and an order comes in for 19 burger patties. The customer does not care how many burgers are created from the patties, but they want exactly 19 patties used. If you can make single, double, and triple burgers, how many of each type minimizes the number of buns? How many maximizes the number of buns?

This question could be a great starting point into an investigation of pattern. A teacher could choose to start here and then move into the examination of linear relationships. Can students explain their strategy for solving the problem?

Can they generalize their strategy for "n" ordered patties?
Will their final order ever include both a single and a double burger?
Will the minimizing solution always be unique?
Will it ever be unique?
In how many ways can 19 patties be sold?
In how many ways can 'n' patties be sold?

All these questions can lead into the listing of variables, and relationships between them.

Ironically, Burger King has created a fairly rich mathematical environment from which to study linear relations. I am sure they did not expect this when they founded the "Meat Mathematics Institute". As teachers, it is our job to create an inquisitive environment where the mathe-meat-ics can be encountered, teased out, and understood.

NatBanting