Doubtless, the three-act mathematical task offers a fresh and intriguing way by which to give life application problems a spotlight in the classroom. Students today are accustomed to using technology to interact with the world around them, making the presentation of math problems via digital media a natural progression of their daily lives. Furthermore, three-act tasks pave the way for integrating technology in the classroom. By withholding information in Act I, students must first analyze the problem in order to figure out the appropriate solving strategy. They must then make connections to solve following Act II, and Act III provides an opportunity to reflect on their answer. Likewise, Dick and Hollebrands (2011) point out that effective technology necessitates the same process: analyzation, strategizing, connection formation, and reflection. Perhaps the biggest advantage of a three-act task is its adaptability. Like any narrative, there are no restrictions on the content of Act I, II, or III, and there can always be more than one sequel. A teacher can create a three-act task to fit any lesson and curriculum, if he or she is willing to take the time to do so.
The original problem ultimately requires students to find the price of an individual apple or pear given the price of the fruit per pound and the respective weights of each type of fruit. Thus, students use tools to make these conversions, which aligns with the 7th grade mathematics curriculum outlined by the Alabama State Department of Education (ASDOE), specifically satisfying the following standard:
9. Solve multistep real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form, convert between forms as appropriate, and assess the reasonableness of answers using mental computation and estimation strategies. [7-EE3] (ASDOE, 2015, p. 57).
The sequel to the original problem requires students to recognize the linear relationship between how many apples or pears are purchased and the total cost. In order to create the graph of these linear equations, students must identify the x-axis as the number of fruit purchased, the y-axis as the total cost, and the unit price of the fruit as the slope for the equations. The sequel subsequently finds its place in the Algebra I curriculum outlined by the ASDOE and accomplishes the following standards:
- Write a function that describes a relationship between two quantities.* [F-BF1]
a. Determine an explicit expression, a recursive process, or steps for calculation from a context. [F-BF1a]
- Distinguish between situations that can be modeled with linear functions and with exponential functions. [F-LE1]
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. [F-LE1b]
46. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. [S-ID7] (ASDOE, 2015, pp. 86-87).
Because the first problem draws on knowledge that students learn in 7th grade while the sequel fits into the Algebra I curriculum, this problem would best be implemented in an Algebra I classroom. This way, students could draw on their prior knowledge of solving multistep real life problems to solve the original problem and then use the unit-prices they find to create a linear model for the sequel. From a technological perspective, this is ideal because the original problem can serve as a spreadsheet scaffolding activity in which students’ knowledge of the mathematical material allows them to see how these conversions can be done more efficiently using a spreadsheet. As Niess (2005) encourages, “[M]athematics teachers must redesign their curriculum and instruction to help students learn about the technology they will use to learn mathematics. A spreadsheet is one of these tools.” Given the calculations in the original problem are not complicated, this problem helps students build spreadsheet skills that extend beyond the classroom.
Besides providing the framework for scaffolding technology, this problem serves another important purpose. The nature of this problem gives students the chance to see how given assumptions affect the accuracy of a mathematical model. For example, in Act II students saw that the weight of an individual apple was 6 oz. while the weight of a pair was 4 0z., and these measurements were used to represent the weight of each apple and pair. However, when the apples and pairs were purchased at the checkout, the total cost was more than students would have expected because the fruit was heavier than the assumed weight. As Dick and Hollebrands (2011) note, real-life concerns (such as discrepancies in a scale) “are not to be ignored, but embraced in conversation.” Unlike theoretical mathematical models, the real world is imperfect, and preparing students for this reality will ultimately help them when they face challenges in their future jobs. Beyond realizing potential reasons why the model does not reflect reality, students can take the next step by finding ways to improve the model, such as taking the mean weight from a larger sample of fruit.
In a world becoming increasingly saturated with technology, it is exciting to find methods such as the three-act task that allow teachers to utilize technology throughout the entire teaching process- from presentation of the problem to solving strategies and answer reflection. Besides showing students how to productively use technology, three-act tasks give students the window into reality that so many of them seek. By creating problems that apply to everyone‘s life, teachers engage students who once thought math was just for the future doctors and scientists.
Consider the problem and sequel presented in these pages the first scene in the opening act of a dynamic saga of real life math problems. If you are a teacher, then you are a playwright for all that is yet to come!