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Modeling Math Ideas with Ratio and Proportional ReasoningText Box 9.1 A Math Happening 9a: The Leaky Bathtub Problem Doug’s bathtub, when full will drain in 12 minutes. The hot water tap takes 6 minutes to fill the tub and the cold water tap takes 4 minutes to fill the tub. If Doug opens both taps but forget to put in the drain plug, in how many minutes will the tub be filled? LESSON STUDY VIGNETTE: THE LEAKY BATHTUBThis rate problem was presented by teachers to their students as part of a lesson study. In their strategies to solve the task, students demonstrated different ways of understanding ratios and a variety of approaches that can be connected to the learning progressions of the CCSSM Standards and to the eight Common Core Standards of Mathematical Practice (MP). This particular task prompted students to think about the relationship of two quantities (the fullness of the bathtub and elapsed time). The student approaches shown below began with recognizing and defining the rates involved in this task using rate language 6.RP.1, 6.RP.2, and unit fractions to describe the rates (e.g., 1/6 tub of hot water per minute or 2 gallons of hot water per minute). Recognizing these rates involved looking for structure (MP7), while describing and interpreting the descriptions of the rates involved precise use of language (MP6). The students used the term “per” to state the rates involved in this task reflecting an understanding of the ratios used in this problem. Student groups built on this understanding to structure tables and create visual representations to examine the relationships between the quantities in the task (see Figure 9.1). One of the groups solved the problem by assigning 12 as the total number of gallons that the tub would hold, and used all whole numbers. They created a model using unifix cubes to show that 5 units in (2 units of hot water plus 3 units of cold water), minus 1 unit out is a net of 4 units coming into the tub per minute. Participants tried to make connections between this and 5/12 in, minus 1/12 out are 4/12 in. Another group was able to recognize the proportional relationship between the fullness of the Figure 9.1 Teachers anticipated possible student strategies for the Leaky Bathtub lesson. Source: Authors. bathtub and the elapsed time and represents this relationship with an equation in the form of y = cx where c is the constant of proportionality 7.RP.A.2, 7.RP.A.2.C. For this task, the equation was y = 1/3x, where c = 1/3. This group used the strategy of representing the relationship as a unit ratio 6.RPP.B, 6.RP.A.3.B and provided multiple representations connecting the structure of the problem with a table, a graph, and an equation. One group, which jumped immediately to using a formula (rate x time), struggled to make connections and find a solution for this task. Throughout the process of solving this problem, students demonstrated an ability to use reasoning with rates to solve realworld and mathematical problems. 6.RP.A.3 below are some samples of the student work from the lesson study. 
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