• Real Numbers

• MP5. Use appropriate tools strategically.

• MP7. Look for and make use of structure.

• MP8. Look for and express regularity in repeated reasoning.

1. Number and Quantity

2. Algebra and Functions

• Investigate rational and irrational numbers and their relative approximate positions on a number line. (Entrepreneurial Skills: Inquiry/Analysis)

• Use technology to look for repetition in decimal expansions and use number lines to order and compare irrational numbers relative to rational numbers. (MP5)

• Apply understanding of rational and irrational numbers to describe and work within the structure of the real number system effectively and efficiently. (MP7)

• Recognize repetition in decimal expansions of rational numbers and recognize when a decimal expansion cannot be represented by a rational number. (MP8)

• Reason about unusually large or small quantities. (Entrepreneurial Skills: Inquiry/Analysis)

• Use calculators or computers to compute with and approximate radicals and roots, and understand how such tools represent scientific notation. (MP5)

• Explore the structure of numerical expressions with integer exponents to generate equivalent expressions. (MP7)

• Look for how positive integer exponents are equivalent to repeated multiplication by the base and how negative integer exponents are equivalent to repeated division by the base. (MP8)

• How many irrational numbers exist?

• Why is there no real number closest to zero?

• Can you accurately plot an irrational number on the number line? How do you know?

• How is performing operations on numbers in scientific notation similar to or different from performing operations on numbers in standard notation?

• Why does a positive number raised to a negative exponent not equal a negative number?

• This expectation (8.NS.A) supports the major work of the grade.

• This expectation (8.EE.A) represents major work of the grade.

• In Grade 5, students use whole-number exponents to denote powers of ten. In Grades 6 and 7, they work with algebraic and numerical expressions containing whole-number exponents.

• In Grade 7, students apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

• In Grade 8, this expectation supports working with radicals and integer exponents. This concludes the introduction of all numbers that comprise the real number system.

• In Grade 8, this expectation connects with knowing that there are numbers that are not rational and approximating them by rational numbers, understanding and applying the Pythagorean Theorem, and solving real-world and mathematical problems involving volume of cylinders, cones, and spheres. 

• In high school, students use properties of rational and irrational numbers, work with rational exponents, and extend their understanding of number systems to include complex numbers.