Numbers

Unit Review Sheet

Facts and Definitions

Lesson 1: Positive and Negative Rational Numbers

• When you combine an equal but opposite quantity to a given quantity, you make zero. A positive number and its negative counterpart make what we call a zero pair.
• When you multiply or divide two positive numbers, the answer is a positive number.
• When you multiply or divide a positive and negative number, the answer is negative.
• When you multiply or divide two negative numbers, the answer is positive.

Lesson 2: Fractions and Decimals

• A terminating decimal stops after a certain number of digits. (ex: 0.5)
• A repeating decimal has a pattern that goes on forever. (ex: 0.3333333)
• Use a bar over the repeating number(s) to indicate the repeating pattern.
• To convert a fraction into a decimal, you divide the numerator by the denominator.
• To convert a terminating decimal into a fraction, you write the decimal as a fraction and simplify.
• To convert a repeating decimal into a fraction, you follow these four steps:
  1. Set the decimal as x,
  2. Multiply to move the repeating digits,
  3. Subtract one x from both sides, and
  4. Solve for x.

Lesson 3: Properties of Exponents

• Positive exponents are repeated multiplication of the same number. For example, 3³ = 3 x 3 x 3 = 27.
• For negative exponents, you flip the base to create a reciprocal. The formula is a⁻ⁿ = 1/aⁿ.
• Any base (except zero) with a zero exponent is 1. The formula is a⁰ = 1.
• The Product of Powers Rule states that when multiplying powers with the same base, you add the exponents: a^m x aⁿ = a^m+n.
• The Quotient of Powers Rule states that when dividing powers with the same base, subtract the exponents: a^m/aⁿ = a^m-n
• The Power of a Power Rule states that you start with the base and multiply the exponents: (a^m)ⁿ = a^mxn
• The Power of a Product Rule states that when a product is raised to a power, the power is applied to each factor inside the parentheses: (a ⋅ b)^m = a^m ⋅ b^m
• The Power of a Quotient Rule states that when raising a fraction to a power, apply the exponent to both the numerator and the denominator: (a/b)ⁿ = aⁿ/bⁿ

Lesson 4: Square and Cube Roots

• An inverse operation is an operation that "undoes" another operation.
• A root operation is the inverse of an exponent operation.
• The 2nd root is called the square root (√). The root symbol with no index number is always the square root.
• The 3rd root is called the cubed root (∛).
• A perfect square is a number that can be made by multiplying a whole number by itself.

Lesson 5: Irrational Numbers

• A rational number is any number that can be expressed as a fraction where the numerator and denominator are integers, and the denominator is not zero.
• An irrational number is any number that cannot be written as a fraction.
• When you square a square root, you get the original number under the root. For example, (√25)² = 25.
• To compare and order a set of numbers when one or more is a square root: Square each number in the set; Use the squared values to determine the correct order of the numbers; and write your answer in the format: a < b < c.
• To approximate a square root: Identify the two closest perfect squares; use these landmarks to estimate the square root; and write your answer as an approximate value (e.g., √30 ≈ 5.5).
• To approximate a square root to two decimal places: Identify the closest perfect squares; test decimal values to refine the approximation; and narrow it down to two decimal places.

Lesson 6: Scientific Notation

• Scientific notation is a way of writing numbers as a number between 1 and 10 multiplied by a power of 10.
• An order of magnitude is a power of 10, meaning one order of magnitude represents a number that is 10 times larger or smaller. Two orders of magnitude indicate a change by a factor of 10 × 10, or 100 times, and so on.
• To convert any number to scientific notation, move the decimal point until there is one digit to the left of the decimal. Count how many times you moved the decimal point. That is the exponent you use for your 10. (21,000 —> 2.1 x 10⁴)
• To convert scientific notation to standard form, move the decimal point in the base number by the exponent's value: right for positive exponents and left for negative exponents.
• To add or subtract numbers in scientific notation, ensure the exponents are the same, adjust if needed, then add or subtract the base numbers while keeping the common exponent.
• To multiply numbers in scientific notation, multiply the base numbers and add the exponents. Combine the results in scientific notation.
• To divide numbers in scientific notation, divide the base numbers and subtract the exponents. Combine the results in scientific notation.

Lesson 7: Arctic Marine Research

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Lesson 8: Unit 1 Test

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Final Project: Mars Station Test Mission

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