DescriptionThis RUAnalytic concentrates on a 7th grade boy, Ariel, in an urban middle school. Ariel, a volunteer in an after-school Informal Math Program, is introduced to the idea of linear functions. During this after school enrichment session, Ariel is presented with the Museum Problem. The Museum Problem states: A museum gift shop is having a craft sale. The entrance fee is $2. Once inside, there is a special discount tale where each craft piece costs $3. What would be the total amount that you would spend if you bought 10 craft pieces? How could you represent the total amount that you would spend if you were to buy any number of craft pieces at the discount price? The RUAnalytic focuses on Ariel’s ability to understand the general solution for the Museum Problem. He works with Researcher Francisco to generate a rule for X amount of items bought, and writes the rule for an inverse function given the amount spent at the museum.

Event 1 shows Researcher Francisco working to engage Ariel in the problem. Ariel and the researcher are discussing the context of going to the museum. They talk about buying gifts at the gift shop and paying an entrance fee before introducing the museum problem.

During Event 2, Ariel answers the first part of the Museum Problem requesting what total amount would be spent if one purchased 10 craft pieces. Ariel answers thirty dollars, and then corrects himself and says thirty two.

In Events 3 and 4, Ariel is asked to solve the second part of the museum problem, “How could you represent the total amount that you would spend if you were to buy any number of craft pieces at the discount price?” Ariel initially struggles with the concept of the variable X in representing a solution. Researcher Francisco folds back to asking Ariel to calculate the total cost for 10 items.

Event 5 depicts Ariel’s understanding of how to calculate the amount spent at the museum for a given predetermined number of item bought at the gift shop. At the end of Event 5, Researcher Francisco engages Ariel, “And let me call the number of … the amount of money that I am going to spend as A. So how I am going to compute that? The amount of money?”

During Event 6, Ariel answers Researcher Francisco by generating a table using the letters A, B, C, D, and E representing the number of items bought. Researcher Francisco helps Ariel build his table to eventually represent the total cost of going to the museum, including the entrance fee. After generating the table, Ariel recognizes the pattern of increasing by three.

In Events 7 and 8, Researcher Francisco asks Ariel to turn his pattern into a general rule. Ariel verbalizes the rule by saying, “You multiply your items by three.... and you got to add your entrance fee.” He then writes down the rule in words followed by symbols: X × 3 + 2 = A. During this event, he also came up with the inverse rule: A – 2 ÷ 3 = X

References
Baldev, P. V. (2009). Urban, Seventh-Grade Students Building Early Algebra Ideas in an Informal After-School Program. Unpublished doctoral dissertation, Rutgers The State University of New Jersey..

Videos
Robert B. Davis Institute for Learning (2005). Early algebra, investigating linear functions, series 6 of 7, Museum problem, Clip 1 of 6: Introduction to the Museum problem [video]. Retrieved from http://dx.doi.org/doi:10.7282/T35D8PTV
Robert B. Davis Institute for Learning (2005). Early algebra, investigating linear functions, series 6 of 7, Museum problem, Clip 2 of 6: Beginning to build a general rule [video]. Retrieved from http://dx.doi.org/doi:10.7282/T31N7Z3J
Robert B. Davis Institute for Learning (2005). Early algebra, investigating linear functions, series 6 of 7, Museum problem, Clip 3 of 6: Finding a rule to give the amount of any number and an inverse rule to find the number [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3S46PX8

The Research on Informal Math Learning program, funded by the National Science Foundation (grant REC-0309062), was directed by Professor Carolyn Maher and implemented by a team of researchers and graduate students from Rutgers University.