Common Core State Standards - TheNumber System and Expressions & Equations
The Number System
8.NS Know that there are numbers that are not rational, and approximate them by rational numbers.
1. Understand informally that every number has a decimal expansion; the rational numbers are those with decimal expansions that terminate in 0s or eventually repeat. Know that other numbers are called irrational.
2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions
For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
Expressions & Equations
8.EE Work with radicals and integer exponents.
1. Know and apply the properties of integer exponents to generate equivalent numerical expressions.
2. Use square root and cube root symbols to represent solutions to equations of different forms. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
8.NS Know that there are numbers that are not rational, and approximate them by rational numbers.
1. Understand informally that every number has a decimal expansion; the rational numbers are those with decimal expansions that terminate in 0s or eventually repeat. Know that other numbers are called irrational.
2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions
For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
Expressions & Equations
8.EE Work with radicals and integer exponents.
1. Know and apply the properties of integer exponents to generate equivalent numerical expressions.
2. Use square root and cube root symbols to represent solutions to equations of different forms. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
Enduring UnderstandingThe students will understand that in certain situations, an estimate is as useful as an exact answer. Students will understand the scale of increase or decrease of a quantity in relation to exponents.
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KnowledgeStudents will know:
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SkillsStudents will be able to...
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Common Misconceptions
- Students may see irrational numbers as truncated rational numbers, seeing pi as the rational number 3.14.
- Students may confuse the radical symbol with the division symbol.
- Students may think that the number line only contains the numbers that are labeled.
- Students may have difficulty understanding the relative magnitude of numbers.
- Students may confuse the operations for the properties of integer exponents, particularly if these properties were taught as rules to memorize rather than being developed conceptually.
- Students may inappropriately apply exponent laws to coefficients, including confusing negative exponents and negative coefficients.
- Students may struggle with simplifying expressions with negative exponents.
- When solving equations with exponents, students may not realize that there may be multiple solutions.
- Students may miscount decimal places when converting to or from scientific notation.