In a fourth down situation in football, most coaches in the NFL decide to punt the ball to the other team. But is giving the ball away with such regularity really a good idea, and should teams punt as often as they do?

In this lesson, students use quadratic functions to develop a model of expected points, which measure how many points a team can expect to score from different field positions. They then apply this model to determine when teams should punt the ball, and more importantly, when they shouldn’t.

Students will

Read graphs to estimate the probability of scoring a touchdown or field goal from different field positions

Add quadratic functions to develop a model of a team’s expected points at different field positions

Evaluate a quadratic function to model the probability that a team will convert on fourth down

Calculate a team’s total expected points if they go for it on fourth and if they punt

Develop strategies for when to punt on fourth down

Before you begin

Students should be able to evaluate quadratic functions and add two quadratic functions together. Some previous experience with piecewise functions will also be helpful. This lesson also makes use of expected value; students should feel comfortable with the idea that, for example, if a team has a 50% probability of successfully kicking a 3-point field goal, then the expected value of attempting the kick is 0.50 × 3 = 1.50 points.

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Building Functions (BF), Interpreting Functions (IF), Linear, Quadratic, and Exponential Models (LE)

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Building Functions (BF), Creating Equations (CED), Interpreting Functions (IF)

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Building Functions (BF), Interpreting Functions (IF)

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Conditional Probability and the Rules of Probability (CP)

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Building Functions (BF), Creating Equations (CED), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI), Similarity, Right Triangles, and Trigonometry (SRT)

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Conditional Probability and the Rules of Probability (CP), Interpreting Categorical and Quantitative Data (ID)

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Building Functions (BF), Linear, Quadratic, and Exponential Models (LE), Seeing Structure in Expressions (SSE)

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Congruence (CO), Modeling with Geometry (MG)

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Building Functions (BF), Creating Equations (CED), Linear, Quadratic, and Exponential Models (LE)

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Building Functions (BF), Creating Equations (CED), Interpreting Functions (IF)

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Interpreting Functions (IF), Linear, Quadratic, and Exponential Models (LE)

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Building Functions (BF), Interpreting Functions (IF)

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Building Functions (BF), Functions (F), Interpreting Functions (IF)

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Conditional Probability and the Rules of Probability (CP), Making Inferences and Justifying Conclusions (IC)

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Creating Equations (CED), Building Functions (BF), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI)

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Mathalicious lessons provide teachers with an opportunity to teach standards-based math through real-world topics that students care about.

How have video game console speeds changed over time? Students write an exponential function based on the Atari 2600 and Moore's Law, and see whether the model was correct for subsequent video game consoles.

Topic:
Building Functions (BF), Interpreting Categorical and Quantitative Data (ID), Interpreting Functions (IF), Linear, Quadratic, and Exponential Models (LE)