Section 2.7: Algebra, Functions, and Patterns

Expressions, Variables, and Equations

Competency 0003 on Field 222 includes Algebra (3.1), and elementary teachers must model the algebra-readiness that carries into middle grades. A variable is a symbol (usually a letter) that stands for an unknown or changing quantity. A numerical expression contains only numbers and operations (such as 4 + 3 x 2). An algebraic expression also contains variables (such as 3x + 5); it names a value but has no equals sign. An equation states that two expressions are equal (3x + 5 = 20), and an inequality compares them with <, >, less-than-or-equal, or greater-than-or-equal.

Order of Operations (PEMDAS)

To evaluate an expression unambiguously, follow PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right). Multiplication and division rank equally and are done in reading order; the same is true for addition and subtraction.

Worked example. Evaluate 4 + 3 x (2 + 5) - 6 / 2.

1. Parentheses: 2 + 5 = 7, giving 4 + 3 x 7 - 6 / 2.
2. Multiplication and division, left to right: 3 x 7 = 21 and 6 / 2 = 3, giving 4 + 21 - 3.
3. Addition and subtraction, left to right: 4 + 21 = 25, then 25 - 3 = 22.

A student who worked left to right ignoring PEMDAS would wrongly get a different value, which is the most common error to correct.

Solving Linear Equations and Inequalities

To solve a one-variable linear equation, apply inverse operations to both sides to isolate the variable, keeping the equation balanced.

Worked example. Solve 3x + 5 = 20.

1. Subtract 5 from both sides (inverse of + 5): 3x + 5 - 5 = 20 - 5, so 3x = 15.
2. Divide both sides by 3 (inverse of x 3): 3x / 3 = 15 / 3, so x = 5.
3. Check: 3(5) + 5 = 15 + 5 = 20, which matches, so x = 5 is correct.

Inequalities are solved the same way, with one rule: multiplying or dividing both sides by a negative number reverses the inequality symbol. For -2x < 8, divide by -2 and flip the sign to get x > -4.

Functions: Inputs, Outputs, Domain, and Range

A function is a rule that assigns exactly one output to each input. If any input produced two different outputs, it would not be a function. The set of allowed inputs is the domain; the set of resulting outputs is the range. Function notation writes f(x) to mean the output when the input is x; for f(x) = 2x + 1, f(3) = 2(3) + 1 = 7.

Functions can be represented four ways, and Field 222 expects fluency translating among them: a table of input-output pairs, a graph on the coordinate plane, an equation such as y = 2x + 1, and a verbal description ("double the input and add one"). The vertical line test identifies a function graphically: if any vertical line crosses the graph more than once, the graph is not a function.

Patterns and the Rule

Students identify and extend numerical patterns (sequences of numbers) and geometric patterns (shapes or figures) and describe the rule. For the pattern 2, 5, 8, 11, ..., each term increases by 3, so the rule is "start at 2 and add 3"; the nth term is 3n - 1. A geometric pattern of dot arrangements can grow by a fixed number of dots per step (linear) or by a squaring rule (nonlinear).

Linear vs. Nonlinear Growth

A pattern is linear when it changes by a constant amount each step (a constant first difference) and graphs as a straight line. The sequence 2, 5, 8, 11 is linear because the difference is always +3. A pattern is nonlinear when the amount of change itself changes. The sequence 1, 2, 4, 8, 16 doubles each step (exponential growth), and 1, 4, 9, 16, 25 is the squares (quadratic growth); both curve rather than forming a line.

Systems of Two Linear Equations (Conceptual)

A system of two linear equations is a pair of equations considered together, such as y = 2x and y = x + 3. The solution is the ordered pair (x, y) that makes both true at once, which is the point where the two lines intersect on a graph. Here 2x = x + 3 gives x = 3 and y = 6, so the lines cross at (3, 6). Two lines may intersect once (one solution), be parallel (no solution), or be the same line (infinitely many solutions). Elementary teachers introduce this conceptually as "where do the two rules agree?" rather than through formal algebraic manipulation.