Experience First

While the content of these early lessons is review from previous years, use the familiar ideas to introduce the Experience First, Formalize Later (EFFL) format. Explain that students will be working in their small groups on the front page of the activity and that a class debrief will follow. Assign one student in each group to be the reader. Tell students that you will be passing out whiteboard markers for groups to write up one of their answers on the board.

 

In this lesson, students use a map of the U.S. to reason about the Cartesian plane, calculating the distance between airports and unpacking the idea of a midpoint. The sequence of questions has students use the idea of horizontal distance and vertical distance to reason about the true distance between two points from a right triangle perspective. After students have found the halfway point on the graph, encourage students to notice patterns in the actual ordered pairs. Ask: “What’s the relationship between these x-values?” “How could you tell which one is the midpoint and which one is an endpoint?” We anticipate that students in Precalculus will arrive at the idea of an average, though it might not be intuitive for all students. For students who immediately draw on the midpoint formula they used in previous years, probe deeper by asking why this formula works and what it means.

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Formalize Later

In the debrief, have groups explain how they got their answers while having other students summarize their peers’ ideas. Have students pull out a different colored writing utensil (we prefer red pens) to add the margin notes to their paper.

 

It is important that students understand why the distance formula works and how it is really the Pythagorean Theorem in disguise. Be as explicit as possible about how the legs of a right triangle correspond to the horizontal and vertical components of the flight path. You may wish to color code the legs of the triangle with their components in the distance formula.

 

For many students, the midpoint formula actually obscures the conceptual understanding of a half-way point. Be sure students understand that counting spaces (on a grid or in your head) is just as valid as plugging numbers into the formula. This method may even be preferred when finding the second endpoint, given the midpoint and first endpoint.

 

After the debrief, record the “Important Ideas” together as a class, then launch students into the “Check Your Understanding” problems, either individually or in pairs.