Math Grade 8: Real Numbers and Ordering

• Math Grade 8 Study Sheet

Real Numbers

• Convert all numbers to the same form when ordering, decimals are easiest
• Fractions can be converted to decimal form by dividing the numerator by the denominator
• Mixed numbers can be converted to improper fractions and then divided
• Percents can be converted to decimals by moving the decimal point two places to the left
• Compare numbers using place value

Ordering Real Numbers

• Order numbers from least to greatest
• Example: 2√2.7, 36/11, -3.13, 328%, -3
• Conversion examples: 2√2.7 ≈ 3.286, 36/11 = 3.27, 328% = 3.28, -10/3 = -3.3
• Order of the given numbers: -3.3, -3.13, 36/11, 328%, 2√2.7

Sets of Real Numbers

• Real Numbers include rational and irrational numbers
• Counting numbers are whole numbers from 1 up, also known as natural numbers: {1, 2, 3, 4,...}
• Whole numbers include counting numbers and zero: {0, 1, 2, 3,...}
• Integers include whole numbers and their opposites: {...-2, -1, 0, 1, 2,...}
• Rational numbers can be written as a fraction: {6, -0.25, 75%, 5/6}
• Irrational numbers are non-repeating & non-terminating decimals: {√28, π, 2.345...}

Scientific Notation

• Format: a × 10ⁿ where 1 ≤ a < 10 and n is an integer, with 10 as the required base

Converting Between Standard Decimal Notation & Scientific Notation

• Converting to scientific notation involves moving the decimal until one digit is in front
  • Moving left results in a positive exponent
  • Moving right results in a negative exponent
  • 53,700 = 5.37 × 10⁴, 0.0000098 = 9.8 × 10⁻⁶
• Converting from scientific to standard notation depends on the exponent
  • Positive exponent = move right
  • Negative exponent = move left
  • 6.3 x 10⁻⁷ = 0.00000063, 4.922 × 10⁵ = 492,200

Slope/Rate of Change

• Slope-intercept form: y = mx + b, where m = slope and b = y-intercept
• Slope formula: RISE/RUN
• Given 2 points (x₁, y₁) and (x₂, y₂): m = (y₂ - y₁) / (x₂ - x₁)
• Example: Points (1,9) and (2,12) results in a slope of m = 3/1 = 3
• Marco spends y dollars on x bags of chips at $0.50 each and a $1.95 drink, the equation: y = 0.5x + 1.95

Proportional vs. Non-Proportional Relationships

• Proportional relationship is linear and passes through the origin
• Equation form: y = mx, e.g., y = 2x
• If cross products are equal, the relationship is proportional
• Non-proportional relationship is linear but doesn't pass through the origin
• Equation form: y = mx + b, e.g., y = 2x + 4
• Cross products are not equal in a non-proportional relationship

Functions

• Function: no x-values repeat; graph passes vertical line test
• Not a function: x-values repeat; graph fails vertical line test
• Function example: {(-2, 1), (-1, 1), (0, 5), (1, 3), (2, -3)}
• Non-function example: {(-2, -1), (-1, 4), (-1, 2), (0, 3), (1, 4)}

Multi-Step Equations

• Solving the multi-step equation -13x - 53 = 2(2x - 14) leads to x = -9

Equations and Inequalities

• Ian has $50 and deposits $25 weekly, Eva starts with $0 and deposits $45 weekly
• To find when they have the same amount: 50 + 25x = 45x
• To find when Eva has as much or less money than Ian: 50 + 25x ≤ 45x

Intersections of Graphed Equations

• Intersection point/solution: (-3, 3)
• Verifying the solution with equations y = 3x + 12 and y = (1/3)x + 4

Pythagorean Theorem

• Theorem: a² + b² = c²
• Used to solve for missing measures of right triangles
• Example 1: Sides 5 and 12, solve for c, c = 13 cm
• Example 2: Side 9 and hypotenuse 15, solve for a, a = 12 in

Surface Area

• Lateral surface area is the area, excluding bases
• Total surface area is the sum of the lateral surface area and the area of the bases
• P = perimeter of the base, h = height of the prism, B = area of the base

Surface Area Formulas

• Prisms: Lateral Surface Area = Ph, Total Surface Area = Ph + 2B
• Cylinders: Lateral Surface Area = 2πrh, Total Surface Area = 2πrh + 2πr²

Volume Formulas

• Volume formula for a cylinder: V = Bh
• Volume formula for a cone: V = 1/3Bh
• Volume formula for a sphere: V = 4/3πr³

Simple & Compound Interest

• Simple interest: I = prt
• Compound interest: A = p(1 + r)^t
  • I = interest, A = final amount, p = principal, r = rate, t = time (years)

Transformations

• Translation: slide
• Reflection: flip over a line
• Rotation: turn a certain degree around a point
• Dilation: enlarge or reduce by a scale factor, k
• Translation: slide (x, y) → (x + 9, у – 5) preserves congruence & orientation
• Across x-axis (x, y) → (x, -y)
• Across y-axis (x, y) → (-x, y)
• Dilation (x, y) → (kx, ky) preserves orientation

Rotation Effects

• Effects on (x, y):
  • 90° = (y, -x),
  • 180° = (-x, -y),
  • 270° = (-y, x)
• Preserves congruence

Scatterplots

• Trend line represents data in a scatterplot
• Positive linear association: as x increases, y increases
• Negative linear association: as x increases, y decreases
• No association: points are scattered randomly
• Non-linear association: points follow a curve