Variables, Expressions, and Integers Study Notes

Variable: A letter used to represent one or more numbers.
Numerical Expression: Consists of numbers and operations, such as $4 \times 10$ . Other forms include $4 \cdot 10$ or $4(10)$ .
Variable Expression: Consists of numbers, variables, and operations (e.g., $4 \times d$ or $4d$ ).
Evaluate: To evaluate a variable expression, substitute a number for each variable and evaluate the resulting numerical expression.
- Example: A blue whale eats about $4$ tons of food daily. To find total food for $d = 120$ days: $4 \times 120 = 480$ tons.
Verbal Model: Describes a problem using words as labels and math symbols to relate the words.
Operation Indicators:
- Addition: plus, the sum of, increased by, total, more than, added to.
- Subtraction: minus, the difference of, decreased by, fewer than, less than, subtracted from.
- Multiplication: times, the product of, multiplied by, of.
- Division: divided by, divided into, the quotient of.
Order Importance: In subtraction and division, order matters. "The difference of a number and $7$ " is $n - 7$ . "The quotient of a number and $5$ " is $\frac{n}{5}$ .
Study Strategy: When writing multiplication, avoid using $\times$ to prevent confusion with the variable $x$ .
Watch Out: Use a fraction bar for division (e.g., $\frac{n}{12}$ instead of $n \div 12$ ).

Power: The result of repeated multiplication of the same factor (e.g., $125 = 5 \times 5 \times 5$ ).
Base: The number used as a factor.
Exponent: Shows the number of times the base is used as a factor.
- Example: In $5^3$ , $5$ is the base and $3$ is the exponent.
Reading Powers:
- $12^1$ : " $12$ to the first power" ( $12$ ).
- $(0.5)^2$ : " $0.5$ to the second power" or " $0.5$ squared" ( $0.25$ ).
- $4^3$ : " $4$ to the third power" or " $4$ cubed" ( $64$ ).
- $8^4$ : " $8$ to the fourth power" ( $4096$ ).
Geometric Formulas:
- Area of a Square ( $A$ ): $A = s^2$ .
- Volume of a Cube ( $V$ ): $V = s^3$ .
Case Study: A cube-shaped block of ice with side length $20\,\text{in.}$ has a volume of $V = 20^3 = 8000\,\text{in.}^3$ .

The Rules:
1. Evaluate expressions inside grouping symbols (parentheses $( )$ , brackets $[ ]$ , fraction bars).
2. Evaluate powers.
3. Multiply and divide from left to right.
4. Add and subtract from left to right.
Mnemonic: PEMDAS (Please Excuse My Dear Aunt Sally).
Nested Grouping: When grouping symbols appear inside others, work from the innermost symbols outward.
Problem Solving Plan:
1. Read and Understand: Identify what you know and what you need to find.
2. Make a Plan: Write a verbal model.
3. Solve the Problem: Evaluate the expression.
4. Look Back: Check if the answer is reasonable (e.g., using estimation).
Case Study: Flower Flag totaling $1,387,800$ plants: $(50 \times 2000) + (7 \times 64,100) + (6 \times 106,700) + 198,900$ .

Integers: The numbers $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$ .
Negative Integers: Integers less than $0$ .
Positive Integers: Integers greater than $0$ .
Zero: Neither negative nor positive.
Graphing: Use a number line to compare. Numbers to the right are greater.
Absolute Value: The distance from $0$ on a number line, written $|a|$ . Distance is never negative.
Opposites: Two numbers with the same absolute value but different signs (e.g., $-10$ and $10$ ). The term $-a$ is read as "the opposite of $a$ ."

Number Line Rules:
- To add a positive integer, move to the right.
- To add a negative integer, move to the left.
Absolute Value Rules:
1. Same Sign: Add the absolute values and use the common sign (e.g., $-6 + (-4) = -10$ ).
2. Different Signs: Subtract the lesser absolute value from the greater absolute value and use the sign of the number with the greater absolute value (e.g., $5 + (-8) = -3$ ).
Additive Inverse Property: The sum of a number and its opposite (additive inverse) is $0$ : $a + (-a) = 0$ .

Rule: To subtract an integer, add its opposite: $a - b = a + (-b)$ .
Change in Quantity: To find the change in a variable (like elevation or temperature), subtract the original value from the value after the change: $\text{Change} = \text{Value After} - \text{Original Value}$ .
Example: Volcano Kick-’em-Jenny elevation changed from $-235\,\text{m}$ in $1962$ to $-182\,\text{m}$ in $2002$ . Change: $-182 - (-235) = 53\,\text{m}$ .

Multiplication Rules:
- Same sign product is positive: $2(4) = 8$ , $-2(-4) = 8$ .
- Different signs product is negative: $2(-4) = -8$ , $-2(4) = -8$ .
- Any integer times $0$ is $0$ .
- Product sign rule for multiple factors: The product of an even number of negative factors is positive; an odd number is negative.
Division Rules:
- Same sign quotient is positive: $-8 \div (-4) = 2$ .
- Different signs quotient is negative: $8 \div (-4) = -2$ .
- Quotient of $0$ and any nonzero integer is $0$ .
- Constraint: Division by $0$ is undefined.
Mean (Average): The sum of the values in a data set divided by the number of values.

Coordinate Plane: Formed by the intersection of the horizontal $x$ -axis and vertical $y$ -axis.
Origin ( $O$ ): The intersection point $(0, 0)$ .
Quadrants: The axes divide the plane into four regions (Quadrants I, II, III, IV).
Ordered Pair: Represents a point $(x, y)$ .
- $x$ -coordinate: Horizontal position.
- $y$ -coordinate: Vertical position.
Scatter Plot: A graph showing paired data as points. It is used to identify relationships between data sets.
Case Studies:
- Fish: Scatter plot of rainbow trout shows gathered data: as length increases, mass tends to increase.
- Fuel Economy: Relationship between car engine size (liters) and highway mileage ( $\text{mpg}$ ).