Rational Numbers and Exponents
Students will deepen their understanding of rational numbers, as they investigate irrational numbers and their place in the number system. Students will also consider exponents and how solving for a base can yield a rational or irrational number.
Essential Outcomes
The Number System
- NY-8.NS.1: Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion eventually repeats. Know that other numbers that are not rational are called irrational.
- NY-8.NS.2: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions.
Expressions, Equations and Inequalities
- NY-8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. e.g.,3^2 x 3 ^ – 5 = 3 ^ – 3 = ⅓^3 = 1/27
- NY -8.EE.2: Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Know square roots of perfect squares up to 225 and cube roots of perfect cubes up to 125. Know that the square root of a non-perfect square is irrational. e.g., The √2 is irrational.
Other Standards Addressed in this Unit
- 8.EE.3: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.
- 8.EE.4: Perform multiplication and division with numbers expressed in scientific notation, including problems where both standard decimal form and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology.
Essential Questions and Big Ideas
- What differentiates rational and irrational numbers?
- Rational numbers can be expressed as a fraction a/b or as a decimal that ends or repeats.
- Irrational numbers cannot be represented as a fraction and as a decimal, there is no pattern.
- How can I compare irrational numbers?
- Rational and irrational numbers can be compared.
- Irrational numbers can be placed between rational numbers based on place value.
- What are the properties of integer exponents?
- When multiplying powers with the same base, the exponents are added.
- When dividing powers with the same base, the exponents are subtracted.
- When a power is raised to an exponent, the exponents are multiplied.
- When multiplying different bases with the same exponent, the bases can be multiplied.
- A negative exponent equals the power as the denominator of a unit fraction.
- What is a square root?
- A square root represents a number to the exponent ½.
- A square root requires finding a number that multiplied by itself equals that amount.
- What is a cube root?
- A square root represents a number to the exponent ⅓. Square root of a = b where b x b = a
- A square root requires finding a number that multiplied by itself three times equals that amount. The cube root of a = b where b x b x b = a
- How can I use roots to solve equations?
- If you know an amount to the second power equals another amount, you can use the square root to find the amount. a^2 = 36 a = square root of 36 = 6.
- What is scientific notation and why do I use it?
- A number is written in scientific notation when it is represented as the product of a factor and a power of 10.
- Scientific notation is used to make calculations with unusually large or small numbers.
Download the complete Grade 8 Math Unit 1 framework to customize for your own planning.