Model Simple Contexts with Quadratic Functions
• Construct a simple quadratic model. • Use the model to solve problems and make predictions.
Now that students have constructed quadratic functions and compared them with linear and exponential functions, they start to explore contexts that can be modeled by quadratic functions. Different contexts naturally lead to different forms, and students use the skills developed in Section 3 to convert between forms. They also consider what shape graph is suggested by the context and use that information to narrow down the possibilities for expressing the function. This section could revisit the context from Section 1 in a deeper way, or bring in completely new contexts as listed here.
WHAT: This advanced activity builds on the work with visual patterns from early on in the unit in a more advanced situation. Students are presented a picture of a three-dimensional tower and asked to find how many cubes to build it, how many cubes in a larger version of the tower, and to model the relationship between height and number of cubes with a quadratic function F-BF.1.
WHY: Skeleton tower provides another, more challenging visual pattern for students to investigate, and gives students an opportunity to persevere in solving a problem MP.1 and to construct a general expression out of a repeated calculation MP.8.
WHAT: In 2004, the social network Facebook launched to little fanfare. Today over 10% of the world’s population have a Facebook account, and the company is worth billions of dollars. But where, exactly, does the value of a social network come from? In this lesson, students construct a quadratic function to model how a network’s value (measured in terms of the number of connections between users) grows with the number of people using the network. They find a quadratic function that gives the number of connections in a complete graph with n nodes, using both equations and graphs F-IF.C.7a, F-BF.1 MP.4. The sequence is $1, 3, 6, 10, \cdots$, the sequence of triangular numbers they saw earlier in the unit.
WHY: The value of Facebook based on the number of connections is an authentic and engaging application of a simple quadratic sequence. Students discuss who gets the most value out of a social network: the users themselves, or the advertisers vying for likes, comments, and dollars.
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