Georgia Milestones Algebra: a complete guide to numerical reasoning and modeling

What this part of the course demands

This guide covers Numerical Reasoning and Modeling, the opening strand of Algebra: Concepts & Connections. It draws on three domains: Numerical Reasoning (A.NR) for numbers and radicals, Patterning and Algebraic Reasoning (A.PAR) for expressions and factoring, and Mathematical Modeling (A.MM) for units and quantities. These standards are part of the Algebra domain that, together with Functions, makes up about 60 percent of the EOC, so the fluency you build here is foundational. Each dot-point page carries its own worked Milestones-style questions: rational and irrational numbers, radicals and rational exponents, units, quantities, and accuracy, interpreting and structuring expressions, and polynomial operations and factoring.

Rational and irrational numbers

A rational number is a ratio of integers; its decimal terminates or repeats. An irrational number's decimal never terminates and never repeats. The fast classifier on the EOC is the perfect-power test: $\sqrt{n}$ is rational exactly when $n$ is a perfect square, so $\sqrt{16} = 4$ is rational but $\sqrt{20} = 2\sqrt{5}$ is irrational. The closure rules matter for justification items: rational $+$ rational is rational, rational $+$ irrational is irrational, and a nonzero rational times an irrational is irrational, while two irrationals can combine to either ($\sqrt{2} \cdot \sqrt{2} = 2$).

Radicals and rational exponents

Put a radical in simplest form by factoring out the largest perfect power: $\sqrt{72} = 6\sqrt{2}$, $\sqrt[3]{54} = 3\sqrt[3]{2}$. Convert between notations with $\sqrt[n]{a^m} = a^{m/n}$, the root is the denominator, the power is the numerator. Then use the exponent rules. To evaluate $a^{m/n}$, take the root first to keep the numbers small: $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$.

Units, quantities, and accuracy

Treat units as algebra. Write conversion factors as fractions so unwanted units cancel: $\frac{60\,\text{mi}}{1\,\text{hr}} \cdot \frac{5280\,\text{ft}}{1\,\text{mi}} \cdot \frac{1\,\text{hr}}{3600\,\text{s}} = 88$ ft/s. A rate's units tell you whether to multiply or divide. Report answers to an accuracy that fits the context: whole units for countable things, the cent for money, and a "how many needed" answer rounded up.

Interpreting and structuring expressions

Read an expression by its parts: terms (added or subtracted), factors (multiplied), coefficients (the number on a variable), and the constant term. In a model like $C = 30 + 0.10t$, the constant 30 is the fixed fee and the coefficient 0.10 is the rate. Use structure to rewrite, most often factoring out the GCF ($2w^2 + 6w = 2w(w + 3)$), because equivalent forms answer different questions.

Polynomial operations and factoring

Add and subtract by combining like terms, distributing the minus sign across the whole second polynomial. Multiply with FOIL: $(x + a)(x + b) = x^2 + (a + b)x + ab$. Factor in order: GCF, then difference of squares $a^2 - b^2 = (a - b)(a + b)$, then trinomial (two numbers that multiply to the constant and add to the middle coefficient), then grouping if the leading coefficient is not 1.

How this strand is examined

• Multiple choice and numeric entry. Classify a number, simplify a radical or rational-exponent power, convert units, or multiply and factor a polynomial.
• Drag and drop. Sort numbers into rational and irrational bins, match radicals to exponent forms, or assemble factors.
• Constructed response. Justify a rationality claim, set up and solve a modeling problem with units, or factor completely with reasoning shown.

Check your knowledge

Work these as you would for credit on the EOC.

1. Classify $\sqrt{32}$, $\frac{5}{6}$, and $0.\overline{7}$ as rational or irrational. (1 point)
2. Simplify $\sqrt{75}$. (1 point)
3. Evaluate $27^{2/3}$. (1 point)
4. Convert 90 km/hr to m/s (1 km $= 1000$ m). (2 points)
5. In $P = 50 + 8h$, what does the 8 represent? (1 point)
6. Expand $(x - 4)(x + 7)$. (1 point)
7. Factor $x^2 - 5x - 14$ completely. (1 point)
8. Factor $4x^2 - 25$ completely. (1 point)