Unit 8: Pythagorean Theorem and Irrational Numbers

Students develop the concepts of square roots, cube roots, irrational numbers, and the Pythagorean Theorem, applying them to two- and three-dimensional problems and to distances in the coordinate plane.

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Unit

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Concept

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Practice

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Set 1

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Advanced

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What this unit covers

A Side Lengths and Areas of Squares · 5 lessons

Goals

• Comprehend the term “irrational number” to mean a number that is not rational and that √2 is an example of an irrational number.
• Comprehend the term “square root of n” and the notation √n to mean the side length of a square whose area is n square units.
• Use the square root symbol to represent solutions to equations of the form x^2=n and represent the square root as a point on the number line.

Lessons

1. The Areas of Squares
2. Side Lengths and Areas
3. Square Roots
4. Rational and Irrational Numbers
5. Square Roots on the Number Line

B The Pythagorean Theorem · 6 lessons

Goals

• Calculate the distance between two points in the coordinate plane by using the Pythagorean Theorem.
• Explain an area-based algebraic proof of the Pythagorean Theorem.
• Use the Pythagorean Theorem to calculate unknown side lengths of right triangles and to solve problems within a context.

Lessons

1. Finding Side Lengths of Triangles
2. A Proof of the Pythagorean Theorem
3. The Converse
4. Applications of the Pythagorean Theorem
5. More Applications of the Pythagorean Theorem
6. Finding Distances in the Coordinate Plane

C Representations of Rational and Irrational Numbers · 3 lessons

Goals

• Coordinate representations of a cube root, including cube root notation, decimal representation, the edge length of a cube of given volume, and a point on the number line.
• Represent rational numbers as equivalent decimals and fractions.

Lessons

1. Edge Lengths, Volumes, and Cube Roots
2. Decimal Representations of Rational Numbers
3. Infinite Decimal Expansions

D Let's Put It to Work · 1 lesson

Lessons

1. When Is the Same Size Not the Same Size? (Optional)