transformations

Grade 8 Module 7: Introduction to Irrational Numbers Using Geometry

STANDARDS:

CCSS.MATH.CONTENT.8.G.A.1

Verify experimentally the properties of rotations, reflections, and translations:

CCSS.MATH.CONTENT.8.G.A.1.A

Lines are taken to lines, and line segments to line segments of the same length.

CCSS.MATH.CONTENT.8.G.A.1.B

Angles are taken to angles of the same measure.

CCSS.MATH.CONTENT.8.G.A.1.C

Parallel lines are taken to parallel lines.

CCSS.MATH.CONTENT.8.G.A.2

Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

CCSS.MATH.CONTENT.8.G.A.3

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

CCSS.MATH.CONTENT.8.G.A.4

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Module 7 begins with work related to the Pythagorean Theorem and right triangles. Before the lessons of this module are presented to students, it is important that the lessons in Modules 2 and 3 related to the Pythagorean Theorem are taught (M2: Lessons 15 and 16, M3: Lessons 13 and 14). In Modules 2 and 3, students used the Pythagorean Theorem to determine the unknown length of a right triangle. In cases where the side length was an integer, students computed the length. When the side length was not an integer, students left the answer in the form of x²=c, where c was not a perfect square number. Those solutions are revisited and are the motivation for learning about square roots and irrational numbers in general.

Topic A: Square and Cube Roots (8.NS.A.1, 8.NS.A.2, 8.EE.A.2)

Lesson 1: The Pythagorean Theorem

Lesson 2: Square Roots

Lesson 3: Existence and Uniqueness of Square Roots and Cube Roots

Lesson 4: Simplifying Square Roots (Optional)

Lesson 5: Solving Equations with Radicals

Topic B: Decimal Expansions of Numbers (8.NS.A.1, 8.NS.A.2, 8.EE.A.2)

Lesson 6: Finite and Infinite Decimals

Lesson 7: Infinite Decimals

Lesson 8: The Long Division Algorithm

Lesson 9: Decimal Expansions of Fractions, Part 1

Lesson 10: Converting Repeating Decimals to Fractions

Lesson 11: The Decimal Expansion of Some Irrational Numbers

Lesson 12: Decimal Expansions of Fractions, Part 2

Lesson 13: Comparing Irrational Numbers

Lesson 14: Decimal Expansion of π

Topic C: The Pythagorean Theorem (8.G.B.6, 8.G.B.7, 8.G.B.8)

Lesson 15: Pythagorean Theorem, Revisited

Lesson 16: Converse of the Pythagorean Theorem