7th grade math

1. Variables and Expressions

• Variable: A symbol, usually a letter, representing an unknown value.

• Algebraic Expression: A combination of variables, numbers, and at least one operation.

  • Example: $3x + 5$

• Evaluating Expressions: Substituting a given value for the variable and simplifying.

  • Example: Evaluate $2y - 1$ if $y = 4$

    • $2(4) - 1 = 8 - 1 = 7$

2. Equations and Inequalities

2.1 One-Step Equations

• Use inverse operations to isolate the variable.

  • Addition Property of Equality: If $a = b$, then $a + c = b + c$

    • Example: $x - 7 = 10 \Rightarrow x - 7 + 7 = 10 + 7 \Rightarrow x = 17$

  • Subtraction Property of Equality: If $a = b$, then $a - c = b - c$

    • Example: $y + 3 = 8 \Rightarrow y + 3 - 3 = 8 - 3 \Rightarrow y = 5$

  • Multiplication Property of Equality: If $a = b$, then $ac = bc$

    • Example: $\frac{z}{4} = 5 \Rightarrow 4 \cdot \frac{z}{4} = 5 \cdot 4 \Rightarrow z = 20$

  • Division Property of Equality: If $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$

    • Example: $5p = 30 \Rightarrow \frac{5p}{5} = \frac{30}{5} \Rightarrow p = 6$

2.2 Two-Step Equations

• Undo addition/subtraction first, then multiplication/division.

  • Example: $2x + 3 = 11$

    • $2x + 3 - 3 = 11 - 3$

    • $2x = 8$

    • $\frac{2x}{2} = \frac{8}{2}$

    • $x = 4$

2.3 Inequalities

• An algebraic statement that compares two expressions using symbols like <, >, \le, \ge, \neq

• Solving Inequalities: Same as solving equations, but if you multiply or divide by a negative number, you must reverse the inequality sign.

  • Example 1: x + 5 > 12

    • x + 5 - 5 > 12 - 5

    • x > 7

  • Example 2: $-3x \le 9$

    • $\frac{-3x}{-3} \ge \frac{9}{-3} \quad \text{(reverse the sign)}$

    • $x \ge -3$

• Graphing Inequalities: Represent solutions on a number line.

  • Open circle for < or > (does not include the point).

  • Closed circle for $\le$ or $\ge$ (includes the point).

3. Rational Numbers

• Rational Number: Any number that can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$.

  • Includes integers, fractions, and terminating or repeating decimals.

• Operations with Integers (Positive and Negative Numbers):

  • Addition: Same signs, add and keep the sign. Different signs, subtract and keep the sign of the larger absolute value.

    • $3 + 5 = 8$

    • $-3 + (-5) = -8$

    • $-3 + 5 = 2$

    • $3 + (-5) = -2$

  • Subtraction: Add the opposite ($a - b = a + (-b)$).

    • $3 - 5 = 3 + (-5) = -2$

    • $-3 - 5 = -3 + (-5) = -8$

  • Multiplication/Division: Even number of negative signs = positive result. Odd number of negative signs = negative result.

    • $(-2)(3) = -6$

    • $(-4)(-5) = 20$

    • $\frac{-10}{2} = -5$

4. Ratios, Rates, and Proportions

• Ratio: A comparison of two quantities by division (e.g., $a:b$, $\frac{a}{b}$).

• Rate: A ratio comparing two quantities with different units (e.g., miles per hour).

  • Unit Rate: A rate with a denominator of 1 unit (e.g., $60 \text{ mph}$).

• Proportion: An equation stating that two ratios or rates are equivalent.

  • Solving Proportions (Cross-Multiplication): If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$

    • Example: $\frac{x}{6} = \frac{5}{10} \Rightarrow 10x = 6 \cdot 5 \Rightarrow 10x = 30 \Rightarrow x = 3$

5. Percents

• Percent: A ratio comparing a number to 100 ($x\%$ means $\frac{x}{100}$).

• Conversions: Decimal to percent (multiply by 100, add $\%$), Percent to decimal (divide by 100).

• Solving Percent Problems: Can use proportions ($\frac{\text{part}}{\text{whole}} = \frac{\%}{100}$) or equations.

  • Example: What is $20\%$ of $80$?

    • Equation: $x = 0.20 \cdot 80 \Rightarrow x = 16$

    • Proportion: $\frac{x}{80} = \frac{20}{100} \Rightarrow 100x = 1600 \Rightarrow x = 16$

6. Properties of Operations

• Commutative Property: Order doesn't matter for addition and multiplication.

  • $a + b = b + a$

  • $a \cdot b = b \cdot a$

• Associative Property: Grouping doesn't matter for addition and multiplication.

  • $(a + b) + c = a + (b + c)$

  • $(a \cdot b) \cdot c = a \cdot (b \cdot c)$

• Distributive Property: Multiply a sum by a number.

  • $a(b + c) = ab + ac$

  • Example: $3(x + 2) = 3x + 6$

7. Order of Operations (PEMDAS/BODMAS)

• Parentheses (or Brackets)

• Exponents (or Orders)

• Multiplication and Division (from left to right)

• Addition and Subtraction (from left to right)

  • Example: $10 - 4 \div 2 + 3 \cdot 5$

    • $10 - 2 + 15$

    • $8 + 15$

    • $23$