04/19/2023 Kathleen M. Sylla Summmer Institute 2007
Objectives
• The students will investigate the mathematical patterns found in quilts.
• The students will investigate special right triangles through measurement and the use of the Pythagorean Theorem.
• The students will investigate special right triangles using radicals.
04/19/2023 Kathleen M. Sylla Summmer Institute 2007
In a Radical Way
Pythagoras of Samos lived from about 569 BC to about 475 BC
04/19/2023 Kathleen M. Sylla Summmer Institute 2007
In a Radical Way
Pythagoras was a Greek philosopher who made important developments in mathematics, astronomy, and the theory of music. The theorem now known as Pythagoras's theorem was known to the Babylonians 1000 years earlier but he may have been the first to prove it.
04/19/2023 Kathleen M. Sylla Summmer Institute 2007
Pythagorean Theorem Building squares on the sides of a right triangle.
Pythagoras' Theorem states that the sum of (the areas of) two small squares equals (the area of) the big one.
In algebraic terms, a2 + b2 = c2 where c is the hypotenuse while a and b are the legs of the triangle.
What is the length of the hypotenuse?
222 cba
222 ca
ca 2
222 caa
222 ca
ac /2
41.12
2/ leghypotenuse45, 45, 90
04/19/2023 Kathleen M. Sylla Summmer Institute 2007
04/19/2023 Kathleen M. Sylla Summmer Institute 2007
Square Root of 2
• The square root of 2 is the length of the hypotenuse (longest side) of a right triangle whose other two sides are each one unit long.
• The square root of 2 has the approximate value 1.41421356
04/19/2023 Kathleen M. Sylla Summmer Institute 2007
30, 60, 90
222 2aba
222 cba
222 4aba 22 3ab 22 3ab
3ab
3/ ab
2
11
60
30 30
60 60
3 / short long
04/19/2023 Kathleen M. Sylla Summmer Institute 2007
Behold Quilt
Behold
What is the ratio of the area of the white square to whole square?
What is the side length of the white square?
What are the length of the legs of the triangles?
04/19/2023 Kathleen M. Sylla Summmer Institute 2007
Sunshine State Standards
• Number Sense, Concepts and Operations Benchmarks– MA.A.1.4.4 Students apply their understanding that
numbers can be represented in a variety of equivalent forms.
– MA.A.3.4.3 Students evaluate the hypotenuse and leg length of a triangle using the 45,45,90 degree Triangle Theorem.
• Geometry and Spatial Sense Benchmark– MA.C.1.4.1 Students understand and prove the
properties of 45, 45, 90 and 30, 60, 90 degree triangles.
04/19/2023 Kathleen M. Sylla Summmer Institute 2007
Algebra- Prentice Hall
• Introduce inSection 10-3• Finding and Estimating Square Roots
• 11-1 Simplifying Radicals• 11-2 The Pythagorean Theorem• 11-4 Operation with Radical
Expressions• 11-5 Solving Radical Equations
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