Pythagorean Theorem Converse

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Objective

SWBAT apply the Pythagorean Theorem converse to determine if a triangle is a Right triangle.

Big Idea

Students use Geogebra to explore and find if given side lengths form right triangles or not.

Accessing Prior Knowledge

15 minutes

Beginning a lesson with an open question is a strategy I use to attract my students attention and get them ready to take a journey in a new direction. Today I start the lesson by writing on the board...

The answer is √18, what could the question or problem be?

At first, my students often remain silent. After all, they are used to answering questions, not guessing what they were. But, patience is warranted. If you are not practiced at writing math questions, it takes a minute to orient your thinking in the right direction. 

Open ended questions are also a good differentiation tactic. They really do create a diverse set of entry points and they are likely to result in a diverse set of "questions". I listen carefully as students develop their questions. Their conversations and the problems that they write often give me insight into their depth of understanding (see my Open Questions to Differentiate reflection). 

Here are some possible student responses:

  • Which number is the greatest?     √17,  4,  √18,  2√3 
  • Find the length of a side of a square whose area is 18:
  • Given:  32 + 32 = c2     Solve for c.

Here are two additional sample responses to the Launch question that I might share if no one comes up with a similar problem. I will introduce these problems as a chance to revisit the Pythagorean Theorem.

 

New Info

15 minutes

I explain today's New Info presentation in my New Info Narrative clip video. 

In addition to the video, I would like to add that when students state the converse of the first of the two statements I write on the board, "All people living in NYC, live in New York State", they will usually give, or I will ask for, a counterexample to show that the converse is false. Although I don´t dwell on the terminology at this point, using and knowing what a counterexample is important and helps with the remainder of the lesson.

Exploration Activity

25 minutes

To begin this section of the lesson, I hand one Geogebra Triangles Activity Sheet to each pair of students.This activity works well in pairs.

Teacher's Note: If your are not familiar with Geogebra, it is a free download and quite friendly to use. Here´s a quick Geogebra tutorial video for students:

geogebra triangle construct

In the activity, students explore triangle lengths to see if the square of the largest value, which in a right triangle would be the hypotenuse, equals the sum of the squares of the other two values. Then they construct corresponding triangles on Geogebra. If a triangle is formed, they measure the largest of the 3 angles check to see if the triangle is a Right Triangle. 

Student partners may take roles, one working hands-on with Geogebra and the other writing results in the activity worksheet. This is fine as long as they work together. Grouping pairs with relatively comparable ability is a good way to make this occur.

As students work I will walk around helping out students that need help with the program. However, I typically find that most students pick up on the activity after watching the video. 

Generalization

5 minutes

To bring this lesson to closure, I ask each pair of students to try to generalize the results they obtained in the activity:

Does it seem like the converse of the pythagorean theorem is true? Is further investigation needed?

Each pair should discuss their work and write their response on the backside of their worksheet. After a couple of minutes, I will ask a student to share his/her generalization with the class. Then, I  will ask if there is another student that can add to the response, or, has a different generalization to share. 

Generalizing is a great way to close a lesson and make students reflect on or summarize new learning. It's also a good formative assessment moment where students may reveal unclear grasp of the objectives. This is particularly important for English language learners (see my Lesson Closure... reflection for more about this issue).