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Square Dancing

What secrets are hidden in squares?

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Square Dancing

What secrets are hidden in squares?

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What secrets are hidden in squares? While the Pythagoreans are best known for their triangle theorem, they were in fact religious cult who believed in the god of mathematics. Pythagoras and his followers believed that everything in the universe could be described as a ratio of two numbers...and tenet that was quickly challenged.

In this lesson, students use concrete models to explore square numbers and square roots and confront the philosophical and moral questions posed by the existence of irrational numbers.

REAL WORLD TAKEAWAYS

• Sometimes humans are persecuted for proclaiming their truths because it threatens a status quo.
• For the Pythagoreans, numbers were a religion and they expected them to reveal the truth of the universe.
• The Pythagoreans persecuted Hippasus when he shared his discovery of irrational numbers, because that “truth” was inconveniently in conflict with the Pythagoreans' understanding at the time.

MATH OBJECTIVES

• Develop a geometric understanding of square root; reason about and find square roots of rational numbers
• Identify the "square root of 2" as irrational
• Use rational approximations of irrational numbers and locate them on a number line

Great anytime, including at the beginning of a unit before students have any formal introduction to the topic.
Grade 8
Squares, Roots, Irrationals
Grade 8
Squares, Roots, Irrationals
Content Standards 8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. 8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., &pi;<sup>2</sup>). For example, by truncating the decimal expansion of &radic;2, show that &radic;2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. 8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x<sup>2</sup> = p and x<sup>3</sup> = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that &radic;2 is irrational.
Mathematical Practices MP.2 Reason abstractly and quantitatively. MP.7 Look for and make use of structure. MP.1 Make sense of problems and persevere in solving them. MP.6 Attend to precision.