Distance in Geometry

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Objective

Students will be able to derive the distance formula by making connections to the Pythagorean Theorem.

Big Idea

Understanding why the distance formula works with respect to the "change in x or y," enables students to solve novel problems requiring the measurement of distance.

Homework Review

20 minutes

Since I encouraged my students to show high-quality work and write high-quality explanations when completing Mr. Herrmann's Square Problem for homework, I'll give students the opportunity to show off at the start of today's lesson. We'll take time for students to share their solutions and to compare their work. I'll encourage students to read each other's explanations carefully. To encourage participation, we'll use a round robin format to structure sharing opportunities. 

  1. In the first round, starting with the Facilitator and then going in a clockwise fashion, each student shares out his/her answer without explaining or justifying. Group members should refrain from agreeing or disagreeing with each answer.  
  2. In the second round, again starting with the Facilitator, all students justify their answers by explaining his/her problem solving process, which should include how they used formal definitions and reasoned their way through the problem.  In this round, students may agree or disagree, as well as add on to other’s explanations by saying things like, “I agree/disagree with you because…” or “that’s similar to what I did, but I also…”  Ultimately, the goal is for all students in the group to come to consensus regarding the solution as well as clear ways to explain their thinking.

Since some groups may finish discussing before others, I am ready to pose a question like, “what was challenging or surprising in the problem?” to encourage groups to reflect more deeply and to reinforce the notion that problems worth solving often require persistence and a number of strategies.  

Notes: The Pythagorean Theorem in Coordinate Geometry (Distance Formula)

15 minutes

In the past, I have seen my students struggle to make sense of the Distance Formula for a variety of reasons.  First, the formula tends to look complex to students with all of its symbols. I have found that without making a specific connection to the Pythagorean Theorem, a lesson on the distance formula can sends a message to students that they should memorize and use the formula without making sense of why and how it works.

In today's notes, I give my students the opportunity to think about "the unknown distance as the hypotenuse of a right triangle". As I proceed with my presentation I consistently use the phrase “change in x or y” to refer to the length of a leg of a right triangle that can be sketched from this hypotentuse. I find that this approach helps my students to internalize the meaning of the symbols in the distance formula. 

As usual, I make sure to use a concrete example before having students generalize the process. I want to give my students the opportunity to derive the distance formula on their own, explaining each part of the formula, which is one way that students look for and make use of structure to produce a mathematical argument (MP7, MP3).

Practice Using the Distance Formula

15 minutes

After our presentation and exploration of the Distance Formula, I give my students the opportunity to to practice using it. I want them to self-assess the extent to which they understand the process underlying this formula. 

The Distance Formula Practice sheet that asks students to find the distance between points on a coordinate plane as well as in space. It asks students to apply the Distance Formula for different purposes, which include determining the perimeter of a triangle or classifying a triangle as scalene, isosceles, ore equilateral given its coordinates.