Lesson: Shortest Distance

Objective: Develop and apply the distance formula

*This lesson appears late in unit two. This unit focuses on understanding numbers and operations. To that end, we work with many situations that allow us to explore how numbers behave and introduce algebraic ideas at the same time. By the time we see this lesson, students have learned signed numbers and their operations, number properties, order of operations, exponent rules, Pythagorean Theorem, and slope.*

To begin the lesson, we measure the classroom and make a scale drawing of it using a scale of 1 unit on a grid = 1 foot in the room. We then decide to place objects from the classroom on the drawing by establishing an origin and axis system based on that origin. This process of setting an origin and determining locations based on it emulates a simplistic version of what surveyors do.

This leads to a good discussion on where in the room the origin should be and what would make it easiest to work with. Eventually students settle on the back left corner of the classroom (from their perspective) since that makes the classroom the 1st quadrant. This choice creates all positive ordered pairs.

Since the measuring tapes are only 25 feet, measuring becomes an interesting challenge. Discussion quickly ensues on logistics and the role of accuracy. We try to measure with the tape on the floor, but are careful to keep the tape taut if measuring off the ground. Students then find ordered pairs for objects in the room they choose. We pick up the lesson at this point being taught by Heather Foes:

*NOTE: This is not a professional quality video.*

The lesson continues, moving between whole group discussion and small groups with students practicing tasks and digging deeper into the ideas of the lesson. I've described below what occurred but was not shown on the video.

After finding a few specific distances, we generalize the process using Pythagorean Theorem with generic ordered pairs. Students need to really understand the order of operations to use the formula accurately.

Students practice using the formula. As a class, we discuss how a graph can be used to avoid using the distance formula but there are limitations to that method as well. Lastly, we connect this idea to slope. Both slope and Pythagorean Theorem can be used on or off a grid. We want students to see how the formulas look with a grid and without as well as the pros and cons of each method. The goal is not just apply the distance formula to two random points. Instead, the goal is to see how the formula can come about and why it is necessary.

A discussion always arises about what can be measured and what cannot. We also discuss the pros and cons of physical measurement vs. math on paper. Students usually surprise us by liking the idea of working on paper. Measuring doesn't require much math, but it is challenging to do it accurately. Also, it requires them to move and they don't always enjoy that element.

This lesson illustrates the depth of the lessons in the MLCS course. The algebra or an algorithm are not the only goals of the lesson. It's the process and discussion of getting there that elicits the most learning and improvement of students' conceptual understanding. Students quickly see that real life problems are almost always much more complicated than traditional textbook problems. But those same real life problems are also very rich and interesting. They are worth the effort necessary to solve them.

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