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Quick Write-
What do you notice happening?
I noticed that ________.

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The Pythagorean Theorem: Interactive Puzzle

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An interactive demonstration of the Pythagorean theorem.

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Interactive Maths - The Interactive Way to Teach Mathematics

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In this section there are activites relating to Pythagoras Theorem, which is one of the most fundamental results in geometry. It has been known for thousands of years, by most ancient civilisations, but it is named after a Greek mathematician and philosopher.

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In this section there are activites relating to Pythagoras Theorem, which is one of the most fundamental results in geometry. It has been known for thousands of years, by most ancient civilisations, but it is named after a Greek mathematician and philosopher.

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Using a discovery lab to introduce students to the Pythagorean Theorem was an amazing way to kick off my Pythagorean Theorem unit. I'm excited to share with you exactly how we got hands-on with the Pythagorean Theorem proof and how it helped my students really understand this geometry concept. I'll also share with you how I used foldable notes to introduce this unit to my students.

Why discovery labs?

Having students begin a unit with a discovery lab activity gets them involved in creating the mental model of what's happening mathematically right from the start. They turn from passive recipients of information to more involved learners. As I've shared before, this is an easy to implement way to get students to use math talk, search for and identify patterns, draw conclusions, and make mathematical arguments, all before the formal instruction. And the Pythagorean Theorem discovery lab directly addressed our first learning goal for the unit (see how I used "I Can" statements to plan the learning and help students track their learning here.)

Calling these activities "labs" reminds students that they are expected to experiment and try things in a low stress environment. It encourages students to ask questions, make predictions, and test their ideas. And with my students I've seen it provide a great shared experience we can refer back to throughout the whole unit of study. Students have more context for the formal instruction when we get to it, and they ultimately understand it in a deeper way than I've seen before. The Pythagorean Theorem in so many ways is especially perfect for this kind of lesson because it's based in understanding a proof. Given the right direction, students can come to the same conclusions as Pythagoras. That's pretty cool!

For several years I've seen all over Pinterest different ways people model the mathematical argument of the Pythagorean Theorem. The use of square numbers represented with boxes for the numbers (as seen below) is a physical way of showing what the equation a ² + b ² = c ² means. In this discovery lab, I wanted to use that same proof, but also have students look at examples and non-examples of the Pythagorean Theorem with right triangles so they had a richer understanding of why it works with just specific numbers.

Step 1- Building background

To get students ready to dive in and play around with the Pythagorean theorem, I started with vocabulary terms. We quickly defined the following terms:

My students knew absolutely nothing about the Pythagorean Theorem. They were truly starting at ground zero. But I wanted them to get warmed up looking at shapes and making any observations they could.

Step 2- Trials and observations

Students had to be precise in how they lined up pieces on the work mat. Then, it became clear which right triangles worked, and which ones didn't. (Last year, my attempt at this discovery lab was a major flop. Mostly due to the fact that one of the squares was missing a row. Oops. Precision really matters here. But with that fix made, and emphasizing to students to carefully have the corners meeting, this year went sooooo much better.)

Again, students sketched the models they built and then reported whether or not they made a right triangle. Then, they answered the question, "Is there a relationship between the squares of the side lengths?"

Finding the pattern of the Pythagorean Theorem

The conversations were good, and students realized that only certain number combinations make a right triangle. However, they weren't seeing any patterns between the numbers, their squares, and what makes a right triangle. So I decided to drop some specific hints.

The first student who realized it quietly came to me and whispered the answer. "I think that the two numbers squared is the same as the third number." When I told her she had it, her eyes lit up. It was especially awesome to see because she has traditionally struggled in math and doesn't always have a lot of confidence. After she figured it out, other partnerships found the pattern too. As each of them told me what they were thinking, I told them they were right on. Then, I challenged them to test the other triangle we made from set 2 to see if it followed the same pattern.

Step 3- Drawing conclusions about the Pythagorean Theorem proof

To wrap up the discovery lab, I gave students three reflection prompts to respond to:

1. Write a rule about the lengths of the sides of a triangle.

2. Write a formula for this rule.

3. Explain what the Pythagorean Theorem is. How could the Pythagorean Theorem help you find the third side of a right triangle?

The rest of this unit went far faster than I expected. While discovery labs are typically a great way to start a unit before getting into the formal instruction, what I noticed with this unit is that we flew through the note taking. Students also showed proficiency very quickly in our practice with this concept. I think a big part of that was the solid understanding they created themselves through this activity.

Taking notes in interactive notebooks

Then, we walked through a worked problem with Pythagorean Theorem together on the back. We annotated the problem to make sure students knew exactly what was happening.

Reflecting on discovery labs and foldable notes

Using discovery labs and math interactive notebooks has revolutionized my classroom over the past two years. Together they make a powerful way to combine student centered learning and meaningful class notes that students will actually look back at and use. They get students more involved in the learning process, with relatively small adjustments for me as a teacher.

The foldable notes provided my students a simple, clear way to organize the new information about using the Pythagorean Theorem. This was a great resource for students. They referred back to them whenever they had a question. I don't remember them looking back in a text book in the same way in past years 🙂 There's something about students creating their own reference that's really powerful.

Want everything I used in my classroom for this unit? The entire Pythagorean Theorem discovery lab (student pages, work mat, and prepared squares) can be found here.

This interactive notebook foldable is also ready to just print and go here

Looking for more practice ideas, activities and resources for teaching about the Pythagorean Theorem? Check out this post here.

Thanks so much for reading. Until next time!

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Teaching the Pythagorean Theorem Proof through Discovery

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Using a discovery lab to introduce students to the Pythagorean Theorem was an amazing way to kick off my Pythagorean Theorem unit. I'm excited to share with you exactly how we got hands-on with the Pythagorean Theorem proof and how it helped my students really understand this geometry concept.

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A Right-Angled Triangle (named as right triangle) is a triangle which has one of its angles equal to 90 degrees.

There are properties associated with a right triangle.

A Hypotenuse is the line segment opposite to the right-angle. The Hypotenuse is also the longest side of a Right Triangle.

An Opposite is the line segment opposite to the angle ϴ.

An Adjacent is the line segment next to the angle ϴ.

Pythagorean Theorem

In a Right Triangle, the sum of the squares of the lengths of adjacent and opposite side is equal to the square of the length of hypotenuse.

Where,

Problem 1: Find the unknown length in the right triangle shown.

By Pythagorean theorem,

This means that,

Converse of Pythagorean Theorem

If the sum of the squares of the lengths of adjacent and opposite sides is equal to the square of the length of hypotenuse, then the triangle is a right triangle.

Where,

Problem 2: Identify if the triangle shown is a right triangle or not.

To show if the triangle is a right triangle, we need to check whether its lengths satisfy the Pythagorean theorem:

If this triangle is right then this also must be true,

So, the triangle is a right triangle.

If you are still struggling to get your students engaged with your Pythagorean Theorem Lesson we have another lesson available on PreAlgebraCoach.com. The lesson teaches the students to derive the Pythagorean Theorem on their own rather than just giving it to them and then some sample problems. The lesson below is an extremely engaging way to teach the Pythagorean Theorem and its Converse. I would recommend teaching it this way then go through the above material.
Here is the Link to Access the Lesson: https://prealgebracoach.com/the-pythagorean-theorem/
This lesson is based around creating (Deriving) the Pythagorean Theorem together in class using GOLF!

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A Right-Angled Triangle (named as right triangle) is a triangle which has one of its angles equal to 90 degrees. A Hypotenuse is the line segment opposite to the right-angle. The Hypotenuse is also the longest side of a Right Triangle. An Opposite is the line segment opposite to the angle ϴ.

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Calculating Distance Using the Pythagorean Theorem

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Create their own star and calculate distance

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Create their own star and calculate distance