In my recent post, "Teaching Algebra without algebra", I tried to build algebraic understanding in my students by comparing their intuitive solutions to algebraic ones. I began to wonder how the colleagues in my school would try to solve a similar problem. Would they simply reason their way through it, or would they actually use a system of equations?
So I proposed this problem to them which is completely stolen from Christopher Danielson. Anyone in the mathtwitterblogosphere is probably already familiar with it. Chris is well known within that community. And he made this video of his son which I'll never forget.
Here's the scenario:
My co-teacher, Mrs. Runkle, and I both love to eat Oreo cookies. She especially likes the ones called Double Stuf because they have twice the filling. She loves the filling....but she doesn't like the cookie wafers. They get in her teeth and she looks funny when she smiles. I'm the opposite. I can't get enough of the wafers, but the filling is disgusting. It looks like some kind of heavily processed glop that could double as an adhesive...maybe that's why it's called "Double Stuf". We came up with a solution. We agreed to buy a package of Double Stuf Oreos and share them. She would eat the filling off of each cookie, and then hand the remaining wafers to me to eat.
Note: I actually re-enacted this in a class with her. She scraped off the cream with her teeth, then I took the two cookies and ate them. She didn't know I was going to do that. The kids thought it was gross.
Then I tell the kids that Mrs. Runkle and I had an argument. We can't decide who is consuming more calories. I say the cream is made of stuff that is bad for you because it's mostly sugar and fat and who knows what, and therefore, has more calories. She thinks the two wafers are bigger than the filling and therefore, they have more calories. We look at the nutrition facts to see who is right, but it only says that two double stuff cookies have 140 calories. This didn't seem to help. Luckily, we had a package of regular Oreos, but that didn't help much either. All it told us was that three regular Oreos have 160 calories.
So...who's getting more calories? What can we do to figure this out?
I sent this out as an email to all of my co-workers for them to figure out. I told them that if they came up with a unique solution, they would get one Double Stuf Oreo. (Yeah, just one. I'm cheap.)
Here are some of their responses:
Solution 1 (5 people came up with this method):
160 calories divided by 3 cookies = 53.3 calories per regular cookie
140 calories divided by 2 cookies = 70 calories per Double stuff cookie……
70 – 53.3 calories = 16.7 more calories in a double stuff
16.7 x 2 (double stuff) = 33.4 calories in the double stuff stuffing alone
70 calories total – 33.4 calories in the double stuff stuffing (total) = 36.6 calories in the cookies.
The person who only eats the cookies consumes more calories. 3.2 calories
Solution 2 (1 person came up with this...believe it or not, this was not a math teacher):
3 Regular Oreos (R ) = 160Cal
2 Double Stuff Oreos (D) = 140Cal
R = 160/3
R = 53 1/3Cal
D = 140/2
D = 70Cal
If it is accepted that the components of a regular Oreo are cookies (C ) and filling (F)
If it is accepted that the components of a Double Stuff Oreo are cookies (C ) and Double
R = C + F
D = C + 2F
Therefore the difference between a regular Oreo and a Double Stuff Oreo is a single serving
of filling (F):
D-R = (C+F)-(C+2F)
D-R = C + F – C – 2F = C + F – C – 2F
If D = 70 and R = 53 1/3:
D-R = F
F = 70 – 53 1/3
F = 16 2/3 calories
Since a Double Stuff Oreo contains 2F, the friend who eats only the cookies consumes:
2F = 2X(16 2/3)
2F = 33 2/3Calories per cookie
To find the calorie content of a Double Stuff Oreo Cookie, we need only to go back to the
equation (D = C+2F) and solve for C:
70 = C + 33 2/3
C = 70 – 33 2/3
C = 36 2/3 Calories
The cookie-eating friend consumes 36 2/3 calories per cookie.
The filling-eating friend consumes 33 1/3 calories per cookie.
The friend who eats only the cookie part of the Double Stuff Oreo will consume 3 1/3 MORE
calories per cookie than the friend who eats only the filling part of the same cookie.
Solution 3 (again, not a math teacher - also, this person solved it using a system of equations and two different methods, elimination on the left and substitution on the right), 2 non-math and 1 math teacher used this method:
So about half of the responses used algebra...half didn't. While I think it's great that so many were eager to figure this out, I wonder why some would prefer to use an algebraic method. Did they do that because that's the most natural way for them to solve this problem? Was there something about the problem that tipped them off to think about using a system of equations? Or did they assume that the solution had to be an algebraic one because I teach Algebra?
And finally, can we really say if one solution is better than another? (And what do we mean by "better"?) I can see the benefits of doing it either way. For solution 1, you really have to be thinking about what's going on at each step. In solution 3, once you set it up, your hours and hours of practicing solving systems of equations take over and you don't need to worry about context anymore. You're on auto-pilot. Is that a good thing? And which way was more efficient? Solution one appears to take five calculations. Solution three has about 12. As an Algebra teacher, this concerns me.
1. A big thank you to Marshall Thompson for inspiring me to look into this! He also expressed interest in how people solve these problems differently.
He told me that someone once used this diagram as a solution. I love how simple this makes everything. It is basically using the same method as solution 1, but it practically screams a system of equations.
2. "Double Stuf"? Where the heck is the second "f"?
3. Here is an article about Stuf from The Onion.
4. The amazing Fawn Nguyen has made her own contribution. She actually saw this method first. She is nothing short of genius...or bizarre.