Experience First

In this lesson students build their own composite function by expressing regularity in repeated reasoning. First, students consider how the length of a pool determines the number of tiles that are needed to make a border for the pool. Algebraically there are many ways to come up with this expression, so encourage students to use color to demonstrate how they “see” the tiles being added. This is a great opportunity to talk about the equivalence of expressions!

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After determining the number of tiles, students go on to figure out the cost of those tiles with the included delivery fee. As you monitor groups, ask students questions like “what determines the cost of the project?” or “how/why would increasing the length of the pool affect the cost?” Students should articulate that the number of tiles determines the cost, but the length of the pool determines the number of tiles. Be listening for phrases like “increasing the length of the pool increases the number of tiles, which then increases the cost of the project”. This kind of sequential reasoning is critical for developing the students’ understanding of composite functions. 

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When completing the table, it will be helpful if students show their work for calculating the number of tiles and cost of the project. When students see 2(18)+16=52 and in the next column 52(5.75)+9.99, it becomes evident how the output of the first function becomes the input of the second function. Finally we want students to see how this can be stated in one equation, namely by inserting the expression 2x+16 into the cost equation to represent the number of tiles (as determined by the length of the pool).

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Throughout the experience students are asked to attend to the kinds of values that go into a function, and those that come out. Restricted domains for the length of the pool creates a restricted range for the number of tiles; which ultimately determines the price range to complete the project.

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Formalize Later

As always, a lot of formal notation is omitted in the experience and then layered on during the formalization. Support students to see how C(n(x)) demonstrates the sequence of equations and the inputs and outputs of each “stage”. We use letters that represent the context instead of the traditional f(g(x)). 

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Understanding of composite functions is critical for success in AP Calculus. Students must be able to work flexibly with composite functions represented numerically, graphically, or analytically. Consider assigning homework problems that mix representations such as question 3 of the Check Your Understanding (graphical and analytical). 

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Finding the domain of a composite function tends to be the hardest part of the lesson for students. You may wish to spend some time going over the second question of the Check Your Understanding. Make sure that students know that it’s not enough to consider only the domain of the inside function, but that they must think about what outputs are produced by those inputs and if those can be inserted into the output function.