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 substituting 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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Monitoring Questions:

• How many tiles would be needed if the length of the pool was 18 feet? 21 feet?

• Can you write a rule using _______'s method for counting the tiles?

• What's the same about these different counting methods? What's different?

• What is the input of this function? What is the output?

• How does the side length of the pool affect the cost?

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Formalize Later:

Note how the sequence of this lesson and the next two are purposefully placed between exponential functions and logarithmic functions. Instead of covering all function concepts in the first unit, compositions and inverses are saved until right before students dive into one of the most crucial inverse relationships: exponential and logarithmic functions. You may have noticed that each unit of the CED introduces one or two major function concepts while also introducing one or two function types. Unit 1 of the CED (Units 1-3 in Calc Medic) introduces the major function concepts of rate of change and transformations. Unit 2 of the CED (Units 4-5 in Calc Medic) introduces the major function concept of inverses, which are based on compositions. We will later see how Unit 3 of the CED (units 6-8 in Calc Medic) introduces the major function concept of periodicity. Once a major function concept is introduced, it is applied to all subsequent function types in the course.

 

In today’s lesson about compositions, 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. Questions assessing composite functions should mix representations such as question 4 of the Check Your Understanding (graphical and analytical). 

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By the end of this lesson, students should be able to construct a composite function by composing two or more other functions and decompose a composite function into two or more functions. Note that there are several ways a function can be decomposed. For example, in question 3 of the Check Your Understanding, f could be the square root function or f(x) could be the square root of the quantity x-5 or f(x) could be the square root of the quantity 1/2x (in which case g(x) would be the quantity 4x-10). The possibilities are endless.

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In question 4 of the CYU, students consider how the order of a composition impacts the output values and thus also the graph. This becomes especially salient when composing the function g(x)=x+k or g(x)=kx with a function f. Depending on the order in which the functions are applied, the composition could represent a horizontal or a vertical transformation! This is a nice tie back to Lesson 3.2 when students studied transformations.