M&M Pythagorean Theorem

1. M&M Pythagorean Theorem By: Nicole Havrilla

2. Level: Middle school, grades 6-8 • Goal: Students will explore and analyze the Pythagorean Theorem. • *This will allow students to better visual the formula • Materials: • -3 Pre-made M&M structures; a mat that consists of a triangle with squares on each of the three sides: scalene right triangle, isosceles right triangle, and one non-right triangle (counter example for students). • Bag of M&M’s for each work station • Worksheets for all students • Calculators (if needed)

3. **Experimental based (Empirical) - one of the best ways for students to learn (based on research) • **Students are able to visually see and manipulate the truth behind the Pythagorean theorem.

4. Objectives: • Given small groups and various stations ,with differing pre-made layouts and a bag of M&M’s, students will find the areas of each square, with 90% accuracy, based on teacher answer key. • Given class findings on what each group has discovered, students will create inductive arguments about the Pythagorean relationship, with 90% accuracy, based on teacher observation. • Given the hands-on-activity and the Pythagorean theorem formula, students will apply the formula to find the missing side of a right triangle, with 90% accuracy, based on teacher answer key.

5. Purpose: • Students explore the Pythagorean theorem using M&M’s to measure areas. • Deductive thinking and algebraic notation help establish connections as students move from station to station, where the layout of each mat is different but the procedure is similar. • Enjoyable for students and allows them to visually represent a written equation.

6. A little background and insight… Over 2000 years ago there was an amazing discovery about triangles: When the triangle has a right angle (90 degrees) the squares are made of each of the three sides, and the biggest square has the exact same area as the other two squares added together!! This is called Pythagorean Theorem

7. IN THEORY… This is what the students will show through this activity.

8. Definition: Leg- two shorter sides of a triangle Hypotenuse - longest side of the triangle In a right triangle the square of the hypotenuse is equal to the sum of the squares of the two legs.

9. Getting started: • Divide students into small groups • Assign each group a station; students will rotate stations counter clockwise • Students write down observations and predictions on the provided worksheet

10. Procedure: • Insert the cardboard right triangular fence between the 3 squares. Notice that a square is formed on each side of the triangle. • Fill the two smaller squares on the legs of the right triangle with one layer of M&M’s. Completely fill both squares without overlapping any M&M’s. Record data in the given chart. • Remove the cardboard right triangular fence and push all the M&M’sinto the square on the hypotenuse. • Re-insert the cardboard right triangular fence and flatten the layer of M&M’s. Verify whether the M&M’s completely cover the square on the hypogenous. Record data in the given chart.

11. Station #1 Scalene right triangle

12. Station #2 Station #3 Isosceles right triangle Counter example: non-right triangle

13. Data Collection: ** numbers in each square represent the number of M&M’s counted during each trial**

14. Students will use their chart to determine what type of triangle they are looking at

15. *Students begin to notice the Pythagorean theorem station Why is this useful? If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right triangles!)

16. Now we can use algebra to find any missing side. Has to be a right triangle…

17. Conclusions: • Written proofs do not allow students to visually see what they are working with. • This allows them to experiment directly with concrete objects to understand the Pythagorean theorem. • Conceptual learning rather than memorizing formulas. • Less prone to forget which two areas they add together as well.

18. Extension Activity #1: -See if students can work backwards in the equation (Have students fill the square formed on the hypotenuse and then use those M&M’s to fill the smaller two squares. Does it still work? )

19. Extension Activity #2: -Use the same procedure however explore the relationship between areas when other similar shapes are constructed on the sides of a right triangle, such as semicircles, triangle, and quarter circles. Using Semi-circles An equilateral triangle is along each side A quarter circle is along each side

20. References: http://www.mathsisfun.com/pythagoras.html http://www.wou.edu/~burtonl/courses/math494.594/PythagoreanTheoremJellyBeansJohnson.pdf http://www.mathsisfun.com/right_angle_triangle.html http://livelovelaughteach.wordpress.com/2012/04/02/jelly-bean-pythagorean-theorem-proof/