7th Grade Math: Ratios and Proportionality

• 7th grade math focuses on building a strong foundation in key mathematical concepts.
• Ratios and Proportional Relationships, Integers and Rational Numbers, Expressions and Equations, Geometry and Measurement, and Statistics and Probability are important.

Ratios and Proportional Relationships

• A ratio compares two quantities.
  • Can be written as a fraction, using a colon, or with the word "to".
  • Example: 3 apples to 5 oranges can be written as 3/5, 3:5, or "3 to 5."
• A proportion is an equation stating that two ratios are equal.
  • Can be used to solve problems involving scaling and equivalent ratios.
  • Example: If 2 apples cost $1, then 6 apples cost $3 because 2/1 = 6/3.
• Unit rate is a ratio that compares a quantity to one unit of another quantity.
  • Helps in comparing different rates or prices.
  • Example: If a car travels 120 miles in 2 hours, the unit rate is 60 miles per hour.
• Constant of proportionality represents the constant ratio between two proportional quantities (often denoted as k in the equation y = kx).
  • Shows the multiplicative relationship between two variables.
  • Can be found by dividing y by x for any corresponding pair of values.
• Proportional relationships can be represented in tables, graphs, and equations.
  • Tables show corresponding values that maintain a constant ratio.
  • Graphs are straight lines passing through the origin (0,0).
  • Equations are in the form y = kx, where k is the constant of proportionality.
• Percent increase and percent decrease are used to describe changes in quantities.
  • Percent increase = ((New Value - Original Value) / Original Value) * 100%.
  • Percent decrease = ((Original Value - New Value) / Original Value) * 100%.
• Scale drawings involve creating accurate representations of objects or spaces at a different size.
  • Utilize proportions to maintain accurate relationships between lengths in the drawing and the actual object.
  • Scale is the ratio between the drawing's measurements and the corresponding actual measurements.

Integers and Rational Numbers

• Integers include positive whole numbers, negative whole numbers, and zero.
  • Examples: -3, -2, -1, 0, 1, 2, 3.
• Rational numbers are numbers that can be expressed as a fraction p/q, where p and q are integers and q is not zero.
  • Includes integers, fractions, and terminating or repeating decimals.
  • Examples: -2/3, 0.5, 7, -1.25.
• Adding integers:
  • If the signs are the same, add the absolute values and keep the sign.
  • If the signs are different, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
• Subtracting integers:
  • Add the opposite of the second integer to the first integer.
  • Example: 5 - (-3) = 5 + 3 = 8.
• Multiplying and dividing integers:
  • If the signs are the same, the result is positive.
  • If the signs are different, the result is negative.
• Converting rational numbers between fractions and decimals is a key skill.
  • To convert a fraction to a decimal, divide the numerator by the denominator.
  • To convert a terminating decimal to a fraction, write the decimal as a fraction with a denominator that is a power of 10, then simplify.
• Operations with rational numbers follow the same rules as with fractions and decimals.
  • Need to find common denominators when adding or subtracting fractions.
  • Multiply fractions by multiplying the numerators and the denominators.
  • Divide fractions by multiplying by the reciprocal of the divisor.
• Real-world contexts often involve applying operations with rational numbers.
  • Examples: Calculating changes in temperature, balancing a checkbook, or determining the distance between two points on a map.

Expressions and Equations

• An expression is a combination of numbers, variables, and operations.
  • Does not contain an equals sign.
  • Examples: 3x + 5, 2(y - 1).
• A variable is a symbol (usually a letter) that represents an unknown value.
  • Used to write expressions and equations that represent real-world situations.
• Combining like terms involves adding or subtracting terms that have the same variable raised to the same power.
  • Simplify expressions by combining like terms.
  • Example: 3x + 2x - y + 4y = 5x + 3y.
• The distributive property states that a(b + c) = ab + ac.
  • Used to remove parentheses in expressions.
  • Example: 2(x + 3) = 2x + 6.
• An equation is a statement that two expressions are equal.
  • Contains an equals sign.
  • Example: 2x + 3 = 7.
• Solving equations involves finding the value of the variable that makes the equation true.
  • Use inverse operations to isolate the variable.
  • Example: To solve 2x + 3 = 7, subtract 3 from both sides (2x = 4), then divide by 2 (x = 2).
• Writing and solving equations to represent real-world problems is a key application of algebra.
  • Identify the unknown, define a variable, write an equation based on the problem, and solve.
  • Example: If a movie ticket costs $8 and you have $30, how many tickets can you buy? (8x = 30, x = 3.75, so you can buy 3 tickets).
• Solving inequalities is similar to solving equations, but with an inequality sign instead of an equals sign.
  • The solution to an inequality is a range of values.
  • Example: x + 3 > 5, so x > 2.
• When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality sign.
  • Example: -2x < 6, so x > -3.

Geometry and Measurement

• Area is the amount of space inside a two-dimensional shape, measured in square units.
  • Formulas for areas of basic shapes:
    • Rectangle: Area = length * width.
    • Triangle: Area = 1/2 * base * height.
    • Circle: Area = πr², where r is the radius.
• Circumference is the distance around a circle.
  • Formula: Circumference = 2πr or πd, where d is the diameter.
• Volume is the amount of space inside a three-dimensional object, measured in cubic units.
  • Formulas for volumes of basic shapes:
    • Rectangular prism: Volume = length * width * height.
    • Cylinder: Volume = πr²h, where h is the height.
• Surface area is the total area of all the surfaces of a three-dimensional object.
  • For a rectangular prism: Surface Area = 2(lw + lh + wh).
• Relationships between angles are important in geometry.
  • Complementary angles add up to 90 degrees.
  • Supplementary angles add up to 180 degrees.
  • Vertical angles are equal.
• Scale drawings involve changing the size of a figure while maintaining its shape.
  • Corresponding angles are equal, and corresponding sides are proportional.
• Geometric constructions involve creating accurate geometric figures using tools like a compass and straightedge.
  • Example: Constructing an equilateral triangle or bisecting an angle.

Statistics and Probability

• Statistics involves collecting, organizing, analyzing, and interpreting data.
  • Purpose is to draw conclusions and make predictions.
• Measures of central tendency describe the typical value in a data set.
  • Mean is the average of the numbers.
  • Median is the middle value when the numbers are arranged in order.
  • Mode is the value that appears most often.
• Range is the difference between the largest and smallest values in a data set.
  • Indicates the spread of the data.
• Probability is the chance that an event will occur.
  • Expressed as a fraction, decimal, or percentage.
  • Probability of an event = (Number of favorable outcomes) / (Total number of possible outcomes).
• Simple events are events with only one outcome.
  • Example: Rolling a 3 on a six-sided die.
• Compound events are events with more than one outcome.
  • Example: Rolling an even number on a six-sided die (2, 4, or 6).
• Experimental probability is based on the results of an experiment.
  • Experimental probability = (Number of times the event occurs) / (Total number of trials).
• Theoretical probability is based on mathematical calculations.
  • Theoretical probability = (Number of favorable outcomes) / (Total number of possible outcomes).
• Simulations can be used to estimate probabilities when it is difficult or impossible to calculate them theoretically.
  • Involve performing repeated trials of an event and recording the results.
• Making predictions based on data involves using statistical measures and probabilities to estimate future outcomes.
  • Example: If a basketball player makes 80% of their free throws, you can predict that they will make about 8 free throws out of 10 attempts.