The Pythagorean Theorem and the Distance Formula

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SWBAT find the distance between two points of an oblique line segment on the coordinate plane using both the Pythagorean Theorem and the Distance Formula.

Big Idea

The main point of this lesson is for students to recognize the similarities between the Pythagorean Theorem and the Distance Formula.

Warm Up

15 minutes

This Warm up is intended to take about 15 minutes for the students to complete, and for me to review with the class.  I allow students to work on the Warm Up to see what they already know from their prior knowledge of working with the Pythagorean Theorem.  I then start reviewing finding the lengths of the sides of the triangle using Pythagorean Theorem before I ask students if any of them remember the Distance Formula to find the distance between two points on a coordinate plane.  Some students try to state the formula, depending on the class, I help them remember it.  I write out the Distance Formula and model how the Pythagorean Theorem is derived from the Distance Formula.  I review the Warm Up in the video below.


Independent Practice

25 minutes

The Independent Practice (Apply Pythagorean Theorem or Distance Formula) is intended to take about 25 minutes for the students to complete, and for us to check in class.  Some of the questions ask for approximations, while others ask for the exact answer.

I warn students to read the directions carefully.  Some directions want the answers to be in exact form, and some in approximate form.  If the instructions do not state which form, it is the student's choice.  I work a few examples shown below before students work on the Independent Practice.

I have students hand in this assignment as a formative assessment.  It is a variety of problems that allows me to check for a variety of skills:

  • Can the student draw the triangle correctly from the given information
  • Can the student apply the distance formula correctly
  • Can the student apply the Pythagorean Theorem correctly
  • Does the student know the difference between the approximate and the exact answer
  • Does the student know how to find the approximate or the exact answer

Some mistakes that students made are listed below in the two samples.

In Sample 1 this common mistake is made by several students.  The student in this example, gives an approximate answer instead of an exact answer.  Also leaving the radical around the answer when the square root had already been taken.

In Sample 2 the student substitutes 17 for a leg instead of the hypotenuse.  The student also does not finish evaluating for c.  The answer shows c squared equals 353, and the student needs to take the square root to find c. 

This shows that students need more practice with Radicals.  That is my intention in this unit is to provide students with a deeper understanding of squaring and taking the square root to apply in the Quadratic Unit.

Exit Slip

10 minutes

I intend for the Exit Slip to take about 10 minutes, and I have the students hand it in for a quick formative assessment for me to see their progress.  The Exit Slip is similar to the Warm Up to reinforce the goal of the lesson, again to be able to apply the Pythagorean Theorem or the Distance Formula.  This Exit Slip is easily used as a self-check also because the answers to the length of the oblique line segment should be the same, no matter what method is used.

This gives students the opportunity to attempt both formulas and look for and correct their mistakes if the answers do not match.  It is much more effective for students to figure out their own mistakes.  Even with a partner, it is better than me just telling them the answer.

There were some students that had to rework one of the problems because the formulas did not match.  Students do not receive credit for the Exit Slip unless the work is shown.