##
* *Reflection: ELL Students
Pythagorean Theorem Converse - Section 4: Generalization

Generalizing or summarizing at the closing of a lesson is particularly important for English Language Learners (EEL). As students summarize their thinking, it gives me, as their teacher, the opportunity to highlight important aspects of the content or to rephrase wording.

Students that are struggling with learning English and acquiring a larger vocabulary tend to focus a lot of their energy on these issues. This makes it more difficult for them to evaluate and discriminate between important and not so important aspects of the information given or discovered in class activities.

For some students, everything is new and everything is important. Therefore, the sharing and discussing of the main and most important ideas of the lesson during closure is more crucial to these students. Better yet, these students can be given questions like:

- What did you learn today?
- What do you think the main idea was?
- What do you have questions about?

This re-assessing of their understanding can be greatly appreciated by ELL students.

*Lesson Closure for ELL students*

*ELL Students: Lesson Closure for ELL students*

# 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.

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#### 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*

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- 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)