Standard: 8.NS.1 - Rational/Irrational Numbers, Repeating Decimals to Fractions
Description: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Vocabulary: Rational Numbers: Any number that can represented as a fraction or ratio. Examples: 1/2, -0.5 Natural Numbers: Counting numbers {1, 2, 3, 4,...} Whole Numbers: Counting numbers and 0 {0, 1, 2, 3, 4,...} Integers: Positive and Negative whole numbers {-2, -1, 0, 1, 2,...} Terminating Decimal: A decimal that ends. Example: .124 Repeating Decimal: A decimal that will repeat infinitely. Example: .111... *All vocabulary words in red are a type of Rational number. Irrational Numbers: Any number that cannot be represented as a fraction or ratio (includes non repeating and non terminating decimals) Example: pi |
Standard: 8.NS.2 - Approximating Irrational Numbers
Description: 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 (e.g., π2). 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 Vocabulary: Approximation: a value or quantity that is nearly but not exactly correct. Perfect Square: A square root that has an answer that is a whole number. Example: the square root of 25 is 5. Non-perfect Square: A square root that has an answer that is an irrational number. |
Standard: 8.EE.1 - Exponents
Description: Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27 Vocabulary: Exponent: a quantity representing the power to which a given number or expression is raised, usually expressed as a raised symbol beside the number or expression. Example: 2^3 = 2 × 2 × 2 Base: the number that is being multiplied repeatedly. 2 in the above example. Power: the number of times a certain number is to be multiplied by itself. 3 in the above example. Exponent Rules: Product Rule - to multiply two exponents with the same base, you keep the base and add the powers. Quotient Rule - to divide two exponents withe same base, you keep the base the subtract the powers. Negative Rule - negative exponents in the numerator get moved to the denominator and become positive exponents. Negative exponents in the denominator get moved to the numerator and become positive exponents. ***Only move the negative exponents. Zero Rule - anything raised to the zero power is 1. Power - to raise a power to a power you need to multiply the exponents. |
Standard: 8.EE.2 - Square and Cube Roots
Description: 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. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational Vocabulary: Square Root: The square root of a number is a value that, when multiplied by itself, gives the number. Cube Root: The cube root of a number is a special value that, when used in a multiplication three times, gives that number. Radical: An expression that has a square root, cube root, etc. The symbol is √ |
Standard: 8.EE.3 - Powers of 10
Description: 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. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger. Vocabulary: Standard Notation: The normal way of writing numbers Scientific Notation: A way to abbreviate numbers |
Standard: 8.EE.4 - Scientific Notation (adding, subtracting, multiplying, dividing)
Description: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. |
Standard: 8.EE.7a/b - Multi-step Equations
Description: 8.EE.7a: Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). 8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms Vocabulary: Equation: An equation says that two things are equal. Solution: A value we can put in place of a variable (such as x) that makes the equation true. Infinite Solution: An equation where any value will make the equation true. Example: 4x+2=2(x+1); 2=2 all values of x will make the statement true. No Solution: An equation that does not have a solution. Example: 3x + 1 = 3x - 2; 1 cannot equal -2 One Solution: An equation that has only one answer. Example: 2x+1=5; x = 2 Inverse Operation: Opposite operation that undoes the previous operation, such as multiplication and division, addition and subtraction, and squaring and square roots. |
Properties of Equality:
Addition Property: For all real numbers x, y, and z, if x=y, then x + z = y + z. Subtraction Property: For all real numbers x, y, and z, if x=y, then x - z = y - z Multiplication Property: For all real numbers x, y, and z, if x=y, then xz = yz Division Property: For all real numbers x, y, and z, if x=y, and z does not equal 0, then x/z = y/z. Substitution Property: For all real numbers x and y, if x=y, then y can be substituted for x in any expression. Distributive Property: For all real numbers x, y, and z, if x(y+z) = xy+xz |