This unit introduces students to irrational numbers with a focus on connecting geometric and algebraic representations of square roots, cube roots, and the Pythagorean Theorem. 

In the first section, students extend work from grade 6, composing and decomposing shapes to find the areas of tilted squares. They see “square root of ” and  to mean the side length of a square with area square units, and understand that finding the solution to equations of the form  means determining which values of make the equation true. Students learn and use definitions for “rational number” and “irrational number,” learn (without proof) that  is irrational, and plot square roots on the number line.

Three right triangles are indicated. A square is drawn using each side of the triangles. The triangle on the left has the square labels “a squared equals 16” and “b squared equals 9” attached to each of the legs. The square labeled “c squared equals 25” is attached to the hypotenuse. The triangle in the middle has the square labels “a squared equals 16” and “b squared equals 1” attached to each of the legs. The square labeled “c squared equals 17” is attached to the hypotenuse. The triangle on the right has the square labels “a squared equals 9” and “b squared equals 9” attached to each of the legs. The square labeled “c squared equals 18” is attached to the hypotenuse.

In the second section, students continue using tilted squares as they investigate relationships between side lengths of right and non-right triangles. Students are encouraged to notice patterns among the triangles before being shown geometric and algebraic proofs of the Pythagorean Theorem. They use the Pythagorean Theorem and its converse to solve problems in two and three dimensions, for example, to determine lengths of diagonals of rectangles and right rectangular prisms, and to estimate distances between points in the coordinate plane. 

In the third section, students see that “cube root of ” and  mean the side length of a cube with volume cubic units. They also represent a cube root as a decimal approximation and as a point on the number line.

In the fourth section, students consider the decimal expansions of rational and irrational numbers. They learn how to rewrite fractions as a repeating decimal, how to rewrite a repeating decimal as a fraction, and reinforce their understanding that irrational numbers have a place on the number line even if they cannot be written as a fraction of integers.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as explaining, justifying, and comparing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Explain

• Strategies for finding area (Lesson 1).
• Strategies for approximating and finding square roots (Lesson 5).
• Strategies for finding triangle side lengths (Lesson 6).
• Predictions about situations involving right triangles and strategies to verify (Lesson 9).
• Strategies for finding distances between points on a coordinate plane (Lesson 11).
• Strategies for approximating the value of cube roots (Lesson 12).

Justify

• Which squares have side lengths in a given range (Lesson 2).
• Ordering of irrational numbers (Lesson 5).
• Ordering of hypotenuse lengths (Lesson 8).

Compare

• Rational and irrational numbers (Lesson 4).
• Lengths of diagonals in rectangular prisms (Lesson 9).
• Strategies for approximating irrational numbers (Lesson 14).

In addition, students are expected to use language to generalize about area of squares, square roots, and approximations of side lengths and about the distance between any two coordinate pairs; critique reasoning about square root approximations; and critique a strategy to represent repeating decimal expansions as fractions. Students also have opportunities to describe observations about the relationships between triangle side lengths and between hypotenuses and side lengths for given triangles; interpret diagrams involving squares and right triangles; interpret equations and approximations for the value of square and cube roots; and represent relationships between side lengths and areas.

The table shows lessons where new terminology is first introduced in this course, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.