Introduction to Pythagorean Theorem

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Objective

SWBAT prove that equality between the sum of short sides of a triangle squared and the longest side squared only occurs with right triangles.

Big Idea

Hands-on manipulatives help students to prove how, why, and when the Pythagorean Theorem shows relationships within triangles.

Bell Ringer

15 minutes

Clarify with your students that you are about to begin a unit on the Pythagorean Theorem and ask them if they have any idea who created the Pythagorean Theorem or what the theorem says.  Brainstorm any relevant ideas on the board by scripting useful information.  Tell your class that it is important to understand who developed the theorem and why it works so that they can prove why, how, and when to use it in math problems.  Read the story What’s Your Angle Pythagoras aloud to your students or even better, allow the students to read the book aloud to the class.  If you use the linked PowerPoint it is more engaging for students.  After reading the story aloud, Turn the PowerPoint to slide number 14 and tell the class this picture is the inspiration for today’s activity. 

 

Preparing Materials

Preparation Work Ahead of Time:

            •   Print, copy, and cut the triangles sketched in handout “Triangles for Opening Pythagorean Theorem Activity.”  If you want, sketch and copy your own variety of triangles from obtuse, to acute, to right triangles.  Make sure all side lengths are whole numbers; do not have side lengths that are 3.25 inches.  All sides should be whole number lengths because the students are going to tile squares along each side length and many tiles are 1 inch by 1 inch or 1 cm by 1 cm in length.

            •    Cut or purchase square tiles that are either one inch square or one centimeter square.  I use craft foam from the Dollar Tree and a die cut to create many tiles in a variety of colors.

            •   Create a poster using a large sheet of poster paper from a roll or sheet of sticky pad and divide the poster in half length wise using a marker.  Label one side of the poster as, “Yes, a2 +  b2 = c2” and label the other side as, “No, a2 + b2 ≠ c2”  Underneath the “No” column, it is optional to divide and label two columns for a specific type of inequality: “ a2 + b2 > c or , a2 + b2 < c2” Including the inequalities allows for good discussion around categorizing triangles by angle measures – equivalent means right triangle, less than means the triangle was obtuse, and greater than means the triangle was acute.  This type of naming a triangle by the length of sides could be a typical ACT/SAT question.

The Activity

40 minutes

Group students into cooperative groups of two to three students and give each group a triangle that is obtuse, acute, or right.  Try to evenly distribute all three types of triangles giving the small triangles to groups who typically work slowly and larger triangles to groups who generally work very quickly.  Have the students record the length of each side and the measure of each angle directly on the triangle itself.  Then ask them to use a marker and name the triangle in the center according to its angle measures.

Remind students of the page where Pythagoras tiles red and blue squares on the two shortest sides of the triangle and then combined the tiles to make one mixed color square on the longest side.  I would even have this part of the story on the board.  Tell them they will be given two colors of square tiles with which they should tile a square along the two shortest sides.  Once they finish, they should call you over to check and take a picture of their “before diagram.”  At this point, also take back any spare tiles that were left.  Remind them now to take all the tiles in both squares and try to create a perfect square on the longest side without having any left-over or running short and not having enough to finish.  When finished, you should be called over again to take an after picture.

As each group finishes direct their attention to a poster on the wall that is split down the middle.  One side says “Yes, the squares along the two shortest sides a and b combined perfectly to make a square along the longest side c.”   The other side says, “No, the squares along the two shortest sides a and b do not combine perfectly to make a square along the longest side c.”  Have students tape their triangle on the correct side for classroom discussion.  Taping the triangles to the post is as far as you will probably get today, but push to at least make it this far.  Some groups will try to take a long time with the tiles, so make sure you are moving them forward as they are working with verbal queues. 

The handout Pythagorean Theorem Unit Opening Activity is another set of step by step instructions for this activity that you can print and follow.