Section	Teacher-facing learning goals
Side Lengths and Areas of Squares	Lesson 1 Calculate the area of a tilted square on a grid by using decomposition, and explain (orally) the solution method. Estimate the side length of a square by comparing it to squares with known areas, and explain (orally) the reasoning. Lesson 2 Comprehend the term “square root of ” \(a\) (in spoken language) and the notation \(\sqrt{a}\) (in written language) to mean the side length of a square whose area is \(a\) square units. Create a table and graph that represents the relationship between side length and area of a square, and use the graph to estimate the side lengths of squares with non-integer side lengths. Determine the exact side length of a square and express it (in writing) using square root notation. Lesson 3 Comprehend the term “irrational number” (in spoken language) to mean a number that is not rational and that \(\sqrt{2}\) is an example of an irrational number. Comprehend the term “rational number” (in written and spoken language) to mean a fraction or its opposite. Determine whether a given rational number is a solution to the equation \(x^2=2\) and explain (orally) the reasoning. Lesson 4 Calculate an approximate value of a square root to the nearest tenth, and represent the square root as a point on the number line. Determine the exact length of a line segment on a coordinate grid and express the length (in writing) using square root notation. Explain (orally) how to verify that a value is a close approximation of a square root. Lesson 5 Comprehend that \(\text-\sqrt{a}\) represents the opposite of \(\sqrt{a}\). Determine a solution to an equation of the form \(x^2=a\) and represent the solution as a point on the number line. Identify the two whole number values that a square root is between and explain (orally) the reasoning.
The Pythagorean Theorem	Lesson 6 Comprehend the term “Pythagorean Theorem” (in written and spoken language) as the equation \(a^2+b^2=c^2\) where \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse of a right triangle. Describe (orally) patterns in the relationships between the side lengths of triangles. Determine the exact side lengths of a triangle in a coordinate grid and express them (in writing) using square root notation. Lesson 7 Calculate an unknown side length of a right triangle using the Pythagorean Theorem, and explain (orally) the reasoning. Explain (orally) an area-based algebraic proof of the Pythagorean Theorem. Lesson 8 Calculate unknown side lengths of a right triangle by using the Pythagorean Theorem, and explain (orally) the solution method. Label the “legs” and “hypotenuse” on a diagram of a right triangle. Lesson 9 Determine whether a triangle with given side lengths is a right triangle using the converse of the Pythagorean Theorem. Generalize (orally) that if the side lengths of a triangle satisfy the equation \(a^2+b^2=c^2\) then the triangle must be a right triangle. Justify (orally) that a triangle with side lengths 3, 4, and 5 must be a right triangle. Lesson 10 Describe (orally) situations that use right triangles, and explain how the Pythagorean Theorem could help solve problems in those situations. Use the Pythagorean Theorem to solve problems within a context, and explain (orally) how to organize the reasoning. Lesson 11 Calculate the distance between two points in the coordinate plane by using the Pythagorean Theorem and explain (orally) the solution method. Generalize (orally) a method for calculating the length of a line segment in the coordinate plane using the Pythagorean Theorem.
Side Lengths and Volumes of Cubes	Lesson 12 Comprehend the term “cube root of \(a\)” (in spoken language) and the notation \(\sqrt[3]{a}\) (in written language) to mean the side length of a cube whose volume is \(a\) cubic units. Coordinate representations of a cube root, including cube root notation, decimal representation, the side length of a cube of given volume, and a point on the number line. Lesson 13 Determine the whole numbers that a cube root lies between, and explain (orally) the reasoning. Generalize a process for approximating the value of a cube root, and justify (orally and in writing) that if \(x^3=a\), then \(x=\sqrt[3]{a}\).
Decimal Representation of Rational and Irrational Numbers	Lesson 14 Comprehend that a rational number is a fraction or its opposite, and that a rational number can be represented with a decimal expansion that “repeats” or “terminates”. Represent rational numbers as equivalent decimals and fractions, and explain (orally) the solution method. Lesson 15 Compare and contrast (orally) decimal expansions for rational and irrational numbers. Coordinate (orally and in writing) repeating decimal expansions and rational numbers that represent the same number.
Let's Put it to Work	Lesson 16 Apply ratios and the Pythagorean Theorem to solve a problem involving the aspect ratio of screens or photos, and explain (orally) the reasoning. Describe (in writing and using other representations) characteristics of rectangles with the same aspect ratio or with different aspect ratios.

8.8 Pythagorean Theorem and Irrational Numbers