##
* *Reflection: Diverse Entry Points
Pythagorean Theorem Converse - Section 1: Accessing Prior Knowledge

Open questions like today's Launch change the nature of the game in ways that can level the playing field in the classroom. There are so many possible answers that all students have the opportunity to participate. And, they sometimes tie-up the thinking of students who are very proficient with procedures and work more quickly than their peers.

As discussed in the narrative below this link, today's Launch question gives way to various correct answers. It´s quite important to change the notion in student´s minds that in math there is only one answer for everything and that either you get it or don´t get it. The type of questions asked can help change this idea and make it a ¨safer¨ atmosphere for participation.

With this particular introduction question, I plan to accept all answers from the students, including those with no direct application to right triangles. An important goal is for students to communicate a way in which they understand the square root of 18 as a number.

As my students share, I will ask questions and make comments about my student´s contributions. I will encourage my students to do the same. For today, it´s not about students coming up with their own ideas and exposing them so that others can see and learn from how they think about numbers.

*Open questions to differentiate*

*Diverse Entry Points: Open questions to differentiate*

# Pythagorean Theorem Converse

Lesson 7 of 10

## 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.

*60 minutes*

#### Accessing Prior Knowledge

*15 min*

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: 3
^{2}+ 3^{2}= c^{2 }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.

*expand content*

#### New Info

*15 min*

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.

#### Resources

*expand content*

#### Exploration Activity

*25 min*

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:

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.

*expand content*

#### Generalization

*5 min*

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).

*expand content*

##### Similar Lessons

###### Introduction to Pythagorean Theorem

*Favorites(72)*

*Resources(21)*

Environment: Suburban

###### Day Four & Five

*Favorites(11)*

*Resources(8)*

Environment: Urban

###### The Pythagorean Theorem

*Favorites(3)*

*Resources(14)*

Environment: Suburban

- UNIT 1: Number Sense
- UNIT 2: Solving Linear Equations
- UNIT 3: Relationships between Quantities/Reasoning with Equations
- UNIT 4: Powers and Exponents
- UNIT 5: Congruence and Similarity
- UNIT 6: Systems of Linear Equations
- UNIT 7: Functions
- UNIT 8: Advanced Equations and Functions
- UNIT 9: The Pythagorean Theorem
- UNIT 10: Volumes of Cylinders, Cones, and Spheres
- UNIT 11: Bivariate Data

- LESSON 1: A thought is an idea in transit
- LESSON 2: Reasoning with Pythagoras
- LESSON 3: Fluency with Pythagorean Triples
- LESSON 4: Hypotenuse Hype
- LESSON 5: Missing a Leg (Day 1)
- LESSON 6: Missing a Leg (Day 2)
- LESSON 7: Pythagorean Theorem Converse
- LESSON 8: Draw a Right triangle! You can´t go wrong.
- LESSON 9: Applying Pythagoras' Theorem with 7 "Choice" Problems
- LESSON 10: Round Robin Review (Unit 9/L1-7)