Ohio Algebra I: a complete guide to expressions and structure

What this category demands

This guide covers expressions and structure, the Ohio Algebra I block drawn from A-SSE (seeing structure in expressions), A-APR.1 (polynomial operations), N-RN (rational exponents), and N-Q (quantities and units). These standards make up much of the Number and Quantity, Expressions and Equations reporting category, and they are the algebraic fluency every later topic depends on. Many of them appear on Part 1, where no calculator is allowed. Each dot-point page has its own practice: interpreting expressions, rewriting using structure, polynomial operations, factoring polynomials, exponents and radicals, and units and quantities.

Reading an expression: terms, factors, coefficients

A-SSE.1 is about reading an expression. A term is separated by a plus or minus; a factor is multiplied within a term; a coefficient is the numerical factor. In context, a constant term is usually a fixed amount and a coefficient is usually a rate. For a cost model $C = 30 + 0.20m$, the $30$ is the flat fee and $0.20$ is the cost per mile. The same reading connects to a graph: the constant is the $y$-intercept and a rate coefficient is the slope.

Rewriting and factoring

A-SSE.2 and A-SSE.3 ask you to rewrite an expression by using its structure, and to choose the form that reveals a property. The order of factoring is fixed:

1. Greatest common factor (GCF) first.
2. Difference of squares $a^2 - b^2 = (a - b)(a + b)$.
3. Trinomial: for $x^2 + bx + c$, two numbers that multiply to $c$ and add to $b$; for $ax^2 + bx + c$, the $ac$-method with grouping.

Polynomial operations

A-APR.1 covers adding, subtracting, and multiplying polynomials, and the idea that polynomials are closed under these operations. Add and subtract by combining like terms, distributing the minus sign over every term when subtracting. Multiply with the distributive property; for binomials use FOIL, and add exponents when multiplying variables.

Exponents and rational exponents

N-RN.1 and N-RN.2 require the exponent rules, none of which are on the reference sheet: product (add), quotient (subtract), power of a power (multiply), zero ($x^0 = 1$), and negative ($x^{-a} = \frac{1}{x^a}$). A rational exponent is a radical: $x^{1/n} = \sqrt[n]{x}$ and $x^{m/n} = \sqrt[n]{x^m}$. So $8^{2/3} = (\sqrt[3]{8})^2 = 4$.

Units, quantities, and accuracy

N-Q.1 to N-Q.3 ask you to convert units, choose units, and report a sensible accuracy. Use dimensional analysis: multiply by a conversion factor arranged so the unwanted unit cancels. The reference sheet supplies the conversions ($1$ mile $= 1.609$ km, $1$ cup $= 8$ fluid ounces). A rate's units are the output over the input, so a slope in a cost-per-mile model is in dollars per mile. Report a final answer to a precision that fits the data, and round only at the end.

How this category is examined

• Multiple choice and multiple-select. Identify a coefficient, pick equivalent forms, simplify with exponent rules, or convert a unit; choose all true statements.
• Equation and numeric entry. Type a factored form, a simplified product, a rational-exponent value, or a converted quantity in standard form.
• Drag and drop. Match parts of an expression to their meaning, or match forms to the property they reveal.

Check your knowledge

Work these as you would for credit on the Ohio test.

1. In $C = 12n + 50$, interpret the $12$ and the $50$. (2 points)
2. Factor $x^2 + 5x - 14$ completely. (1 point)
3. Factor $4x^2 - 100$ completely. (2 points)
4. Simplify $(3x^2 + 2x - 5) - (x^2 - 6x + 1)$. (2 points)
5. Expand $(2x - 1)(x + 4)$. (1 point)
6. Simplify $\dfrac{x^9}{x^4}$ and evaluate $27^{2/3}$. (2 points)
7. Convert $4$ gallons to quarts using $1$ gallon $= 4$ quarts. (1 point)