Eleventh Graders Collaborating and Recognizing Isomorphisms in their Problem Solving

PurposesStudent collaboration; Student engagement; Student model building; Reasoning
DescriptionInitial Problem: A local pizza shop has asked us to help design a form to keep track of certain pizza choices. They offer a plain pizza that is cheese and tomato sauce. A customer can then select from the following toppings: pepper, sausage, mushrooms, and pepperoni. How many different choices for pizza does a customer have? List all the choices. Find a way to convince each other that you have accounted for all possible choices. Suppose a fifth topping, anchovies, were available. How many different choices for pizza does a customer now have? Why?

A group of eleventh grade students, Robert, Stephanie, Shelly, and Amy Lynn, were given the pizza problem and asked to first to find how many different possible pizza choices they would have if they were given four toppings. The students initially recalled two previous problems they had encountered, (1) the pants and shirts problem which dealt with making outfits from a number of shirts and number of pants, represented by a tree diagram and (2) the tower problem that dealt with building towers “n” high selecting from two different colors. Shelly tried, unsuccessfully, to figure out how many combinations there were by using a formula she recalled from a math class to find the combinations. When Shelly got an answer, Stephanie suggested that they make tree diagrams in order to find the number of combinations and compare their answers. The students began by creating tree diagrams and figuring out what would be the best way to organize them. Stephanie and Shelly shared the different combinations they got from their tree diagram and once they both agreed that there were sixteen different combinations, the group moved onto the next problem that dealt with five toppings.

The students began to approach the problem the same way as they approached the first problem with four toppings, by making a tree diagram. Once the students began working they recognized some of the number patterns from the tree diagram. The students noticed that the numbers matched patterns that are found in Pascal’s Triangle. When questioned by Researcher Maher about why they appeared in Pascal’s Triangle, the students hesitated to produce an explanation. Then Stephanie tried to explain what each number in the triangle represented and why, but was not yet able to explain how the numbers moved from one row to the next in reference to pizzas.

A teacher joined the group, listened to their explanations and offered some support to encourage the students to continue to work together to understand where the numbers of different types of pizzas are represented on rows of Pascal’s Triangle. Stephanie was then able to successfully understand the relationship of numbers of toppings and rows of Pascal’s Triangle and how increasing the number of topping choices is represented on different rows of Pascal’s Triangle, thereby gaining an intuitive understanding of the Addition Rule.
Another member of the group, Robert, had a different approach. He worked on the problem using the knowledge he had from the tower problem:
“You have been invited to participate in a TV Quiz Show and have the opportunity to win a vacation to Disneyworld. The game is played by choosing one of the four possibilities for winning and then picking a tower out of a covered box. If the tower matches your choice, you win. You are told that the box contains all possible towers three tall that can be built when you select from cubes of two colors, red and yellow. You are given the following possibilities for a winning tower: a. All cubes are exactly the same color; b. There is only one red cube; c .Exactly two cubes are red; d. At least two cubes are yellow. Question 1.Which choice would you make and why would this choice be any better than any of the others? Question 2. Assuming you won, you can play again for the Grand Prize which means you can take a friend to Disneyworld. But now your box has all possible towers that are four tall (built by selecting from the two colors, yellow and red). You are to select from the same four possibilities for a winning tower. Which choice would you make this time and why would this choice be better than any of the others?”
Since this problem gave him the equation 2^n, he was able to quickly figure out how many different combinations there were by using n as the number of toppings. Robert uses a drawing to explain what each tower would look like and explained where he got the equation from. Other students chimed in and helped Robert explain the tower problem in reference to pizzas by showing one color representing choosing a topping and another color represent not choosing a topping.

When asked to explain their final solution to Researcher Maher, the students were able successfully explain the link between the problem and Pascal’s Triangle by showing how each number of the Triangle represented a type of pizza with a certain number of toppings and what happens when you add another topping to the pizza. The students were also able to explain successfully how the tower problem also connected to the pizza problem. Stephanie drew a diagram to show Researcher Maher how the solution to the tower problem can be used to explain a certain type of pizza with the two colors representing having a topping or the absence of a topping. By explaining the link between the solution to the pizza problem, Pascal’s Triangle and the tower problem, the students found a final solution and were able to explain their reasoning. The recognition of the isomorphism among the problems allowed the students to think back about the problems they had dealt with previously to help them solve a new problem.

The students all demonstrated their ability to work together and have discussions about solving the problem they were given. In the course of making sense of their observations and of what their peers were saying and doing, they moved back and forth between their representations, which had become less concrete and more abstract and symbolic (Tarlow, 2010). The students were able to take their knowledge of problem they have preciously solved as a class and apply it to the new problem they were given. Students demonstrated their ability to collaborate and have abstract discussions in order to come to a conclusion and a valid argument about the solution of their problem.

Tarlow, L. D. (2010). Pizzas, Towers and Binomials. In C. A. Maher, A. B. Powell, & E. B. Uptegrove (Eds.), Combinatorics and Reasoning: Representing, Justifying, and Building Isomorphisms (pp. 121-131). Springer: New York, NY.
Created on2015-04-19T14:44:41-0400
Published on2015-07-31T08:57:37-0400
Persistent URLhttp://dx.doi.org/doi:10.7282/T3HH6MXR