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* *Reflection: Developing a Conceptual Understanding
A thought is an idea in transit - Section 3: Application Task 1

Students in task 1 must write their conclusion. Some may simply write sentences like.."all the tiles form another square on the hypotenuse", or similar statements without really demonstrating in their writing that the concept was understood. I "poke" these students and have them rewrite the conclusion until the concept is correctly expressed.

Yet, even those that may correctly write what is expected, may not have made the connection between the geometric idea and the algebraic equation, a^2 + b^2 = c^2. Some may be outspoken enough to express this, but others may not. I try and make sure the connection is made. Task 2 may help strengthen their understanding of the concept.

*Did they make the connection?*

*Developing a Conceptual Understanding: Did they make the connection?*

# A thought is an idea in transit

Lesson 1 of 10

## Objective: SWBAT understand why that famous (and useful) Pythagorean equation a^2 + b^2 = c^2 works.

*55 minutes*

#### Accessing Prior Knowledge

*10 min*

As we begin our unit on the Pythagorean Theorem, it's a good idea to review squares and square roots. The objective of this Warm-up activity is straightforward. I want my students to access their prior knowledge so that we are starting in a good place (and I know where that place is). Instead of using a worksheet, I have students use this applet (Full URL listed below). As they work, students shade unit squares to indicate a target area of a larger square. This is a kind of visual prelude to one of the Pythagorean area proofs; where squares are drawn or placed on each of the sides of a right triangle.

**Applet URL **(accessed Dec 2 2104): http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.NUMB&ID2=AB.MATH.JR.NUMB.SQUA&lesson=html/object_interactives/square_roots/use_it.html

I ask that each student open the_Applet and explore it for a while. There are a few things that I am looking out for as I observe my students working.

- The blinking dot at the origin (0,0), must be dragged with the pointer to match the Target Area.
- Students need to enter the square root, which is the side length, but may not be obvious at first.

My students quickly catch on to this simple applet, so few instructions are needed. I try never to underestimate how quick young students are to pick up on how an app or online game functions, but these days they usually surprise me with how quickly they figure things out.

Once students have fiddled with the app for a few minutes, I ask that they think about what the creator of this app's objective was, and I randomly call on students to explain. This makes students go a bit further than just "playing" with the tool.

#### Resources

*expand content*

#### New Info

*5 min*

For today's New Info section, I show my students the following video:

**Source URL**: http://www.youtube.com/watch?v=1LWE18gHfXU

What I want is to simply give students a bit of historical background on Pythagoras, and not the actual theorem or proof of the theorem. There are many interesting things about Pythagoras' life and his followers of the times and I invite students to search and read about Pythagoras and his school of thought.

*expand content*

#### Application Task 1

*20 min*

After we watch the video, I plan to separate my students into groups of three who will work together during the remainder of the lesson. Each group will be given a set of tiles that they cab use to demonstrate one of the **Area Proofs of the Pythagorean Theorem**. The sets contain 25 tiles to be used to create perfect squares on each of the legs of a **3-4-5 right triangle**. Each group of students also gets a copy of the triangle where they will place tiles and form the squares.

**Teacher's Note:** Before the lesson, I make sure I get my triangle side´s measurements on this sheet correct by previously measuring the tiles. The 3-4-5 Right Triangle student sheet is a right triangle you can use with tiles that are exactly 1 inch by 1 inch. In other words, the legs of this triangle exactly 3 and 4 inches long, making the hypotenuse 5 inches. When working with this set, the 5x5 square tiles may go beyond the paper area. I let students know this is fine in advance. Here is a triangle indication diagram.

Once I distribute the 3-4-5 right triangle and the tiles to each group of students, I show the following short video and ask that all students watch it before beginning to work.

**Source URL**: http://www.youtube.com/watch?v=CAkMUdeB06o (Accessed Dec 2 2014)

My experience is that most of my students, even those that have seen the Pythagorean theorem equation (a^2 + b^2 = c^2), don't know or have not seen the area proof of it. This video gives them a notion of the task they will be carrying out in the next few minutes. I ask the groups to start while I go around guiding any confused student.

Since we're working with trios, it is very unlikely that at least one student in the group will not know what to do or how to start out. If there is a group that has no idea how to begin, I just indicate that they should start by building a perfect square on each of the two legs of the triangle. You may have to point out that the two legs are those sides that intersect to form the right angle.

There is always one group that realizes that all the tiles used on both legs can be used to make a perfect square on the hypotenuse. This soon spreads and other groups will do the same. Click below to see a student performing the task.

Each group should write their conclusions on their respective triangle sheets. Students should show in their writing that they understand the connection between the areas of the squares and the Pythagorean Equation.

*expand content*

#### Application Task 2

*15 min*

I ask students to remain in their groups while I hand each student the Application task 2 handout. I want to reinforce the Pythagorean Concept geometrically first, by asking students to construct a third square given two squares, then algebraically by having the students take measurements and write the corresponding Pythagorean equation.

For this second part, students should use calculators because their measurements will probably not involve a pythagorean triplet. Students should carefully cut out both squares and use the space provided on the sheet to place these squares and draw the third square. I expect students to flip and twist the squares around so as to form the legs of a right triangle with their sides, then draw a connecting line, the hypotenuse. Students must then carefully draw the third sqaure on the hypotenuse using the corners of their rulers to make as perfect a square as they can.

I added the link to a video on Pythagorean theorem which can reinforce student´s understanding of the concept. It goes through the geometric concept without mentioning the actual equation. A bit long but worth giving to students to see, especially struggling students.

https://www.youtube.com/watch?v=9755B-aLI2E&feature=c4-overview&list=UUKZYfQiLaYwyM-zVsTIbuZg

#### Resources

*expand content*

#### Closure

*5 min*

To end the lesson I hand each student an exit pass, but not before asking every group to return to their regular seating arrangement so I can address the class as a whole. I tell the class that they have 3 minutes to complete the exit pass.

There are 4 triangles on dotted graphs. For triangles A, B, and D, the student needs to state the area of the square above the hypotenuse. For triangle C, the student is asked to state the sum of the areas of the squares on both its legs.

For students more advanced students, I walk over and indicate that I want them to also state the side length of side of the square in question. In other words, for triangle A, B, and C, the student should state the length of the hypotenuse, and for triangle C, the student should state the lengths of both legs. (this can be previously written on the exit pass and handed to these more advanced students discretely along with the rest)

#### Resources

*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)