• To fully engage students in the story behind this lesson, it’s worth spending the time to play the videos of the atmosphere before the game and the miracle moment. After seeing the emotions of the announcers and Fernández’s teammates, students will be fully invested in the story and in answering the lessons’ Key Question. Just make sure to have some tissues on hand as Dee Gordon rounds the bases.
• This lesson positions students to use z-scores and the standard normal curve (µ = 0, σ = 1) for all their normal calculations. However, students may be tempted to perform all their calculations using a normal curve that matches the original scale of the data (µ = 89.4, σ = 1.7). Then, z-scores would no longer be necessary. So, why ask students to calculate z-scores? Because when students develop an intuitive understanding of what constitutes an “unusual” z-score (usually more than 2σ away from the mean), they’ll better internalize what constitutes an “unusual” z-test statistic in later inference units. By having students work with z-scores and normal curves now, we’re positioning them to more easily grasp the logic and formulas of hypothesis tests later in the year.
• Establish the expectation that students sketch a normal curve to make sense of each problem first, before turning to technology. A sketch helps students identify the mean, scale, and relevant region under the curve before performing any calculations. This not only reduces common setup errors, such as incorrect bounds or tail selection, but also helps students interpret whether their answers are reasonable. In addition, drawing and labeling a curve provides a clear way for students to show their thinking, which is expected on the AP exam.

First, download this lesson's Handout Key and read through its Discussion Question section. Then, check out our model discussion norms and the additional background notes below.

• A natural way to extend the discussion is to ask: “How low would this probability have to be to convince you that Colón purposefully threw an easy pitch? What’s your cutoff?” Allowing students to share their own cutoff values – e.g. 1%, 5%, 10% – helps build intuition for establishing a significance level (the cutoff for rejecting a null hypothesis) in later units.
• Another extension to build intuition for later units is to ask students: “Do these results convince you that Colón purposefully threw an easy pitch? Do the results provide irrefutable proof that Colón purposefully threw an easy pitch?” Although the conclusion that Colón purposefully threw an easy pitch is convincing, it’s not irrefutably proven. There is a probability (1.5%) that he would throw a pitch this slow (or slower) by chance alone. Even though that’s a small chance, the probability of throwing a pitch this slow is not 0%. Building this intuition will also help students in later inference units, in which they’ll be expected to draw conclusions about hypothesis tests in non-determinative language.
• It’s helpful to conclude the discussion with the note that Colón’s “easy” pitch doesn’t take away from the power of the Marlins Miracle. In fact, it adds to it. We’re at our best when we collaborate and cooperate across lines of difference – including across opposing teams. The power of the moment comes from people working together for a greater cause: in this case, to honor a fallen friend.

• Purposefully throwing an easy pitch isn’t new in baseball. In fact, it has a name: grooving a pitch. Pitchers are known to groove pitches when the moment calls for it. For example, grooved pitches are often used as “parting gifts” during the last at-bats of great hitters retiring after a long career. There is an unwritten rule though: if you groove a pitch, you never talk about it. Otherwise, you may diminish the hitter’s accomplishment. By never discussing his pitch to Gordon, Colón upheld this unwritten rule.
• This context highlights how statistics can be used to interpret a single unusual event. Encourage students to reflect on how an improbable outcome, such as Dee Gordon’s home run, can lead people to question whether something occurred by chance, while recognizing that rare events can still occur.
• If time permits, consider asking students to propose other scenarios in sports or real life where unusually rare events might lead people to question whether something occurred by chance. This can help reinforce the broader applicability of normal calculations.

• Reinforce that the letter Z is used only for the standard normal distribution. For other normally distributed random variables, letters such as X, Y, or W are typically used. Highlighting this distinction helps students interpret probability statements more precisely and helps set them up for interpreting z-test statistics in later inference units.
• Emphasize that the sign of the z-score is critical. A positive z-score indicates a value above the mean, while a negative z-score indicates a value below the mean. The absolute value gives the distance from the mean. Students sometimes focus only on magnitude and overlook direction, which can lead to misinterpretation.
• When solving z = (x − μ) / σ, some students may find it more intuitive to think in terms of distance from the mean. For example, they can calculate how many standard deviations correspond to a given z-score (e.g., 1.64 × σ) and then add or subtract that distance from μ. This approach can feel more concrete and reduce algebraic errors.