Build the by-hand fluency the calculator-free session demands: integer and fraction arithmetic, factoring, simplifying radicals, and exact answers, and apply efficient mental-math strategies.

What this topic is asking

The Grade 10 MCAS has two sessions, and one of them is taken without a calculator. That session checks the fluency you should have by hand: integer and fraction arithmetic, factoring, simplifying radicals, and giving exact answers. This section is about building that fluency and using efficient strategies so the calculator-free session is not a weakness.

Why a calculator-free session exists

The calculator-free portion exists to confirm that a student can do core mathematics without a machine: the kind of fluency that underpins all later work. So the questions there are deliberately solvable by hand, with manageable numbers, and the answers are usually exact rather than decimal. Preparing for it means practicing the arithmetic and algebra you might otherwise hand to a calculator.

Fraction and integer arithmetic

The most common no-calculator slips are in fractions and signed numbers.

• Adding or subtracting fractions requires a common denominator first. $\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}$. Never add across numerators and denominators.
• Multiplying fractions multiplies straight across: $\frac{2}{3} \cdot \frac{3}{5} = \frac{6}{15} = \frac{2}{5}$, simplifying where possible.
• Signed numbers: a negative times a negative is positive; subtracting a negative adds. $-3 - (-7) = -3 + 7 = 4$.

Practicing these until they are automatic protects easy points in the calculator-free session.

Factoring and radicals by hand

Two algebra skills carry much of the calculator-free session:

• Factoring: pulling out a GCF, recognizing a difference of squares, and factoring a trinomial by finding two numbers with the right product and sum. These let you solve quadratics by the zero-product property without a calculator.
• Simplifying radicals: removing the largest perfect-square factor, so $\sqrt{72} = 6\sqrt{2}$, and combining like radicals. The session expects $7\sqrt{3}$, not $12.12$.

Efficient strategies

Beyond raw fluency, a few habits save time and reduce error in the calculator-free session:

• Estimate to eliminate. If an answer should be a little more than 6 and an option is 14, rule it out without exact work.
• Simplify before computing. Reduce fractions and pull out factors first; smaller numbers are easier and less error-prone by hand.
• Recognize patterns. Pythagorean triples (3-4-5, 5-12-13), perfect squares, and perfect cubes let you write an answer immediately.
• Keep answers exact. Leave $\frac{1}{3}$, $\sqrt{2}$, or $-2 + \sqrt{10}$ as they are unless a decimal is explicitly asked for.

Mental-math shortcuts worth knowing

A few quick facts speed up the calculator-free session and reduce slips:

• Percent conversions: 50% is $\frac{1}{2}$, 25% is $\frac{1}{4}$, 10% is $\frac{1}{10}$, so 15% is 10% plus half of 10%. To find 15% of 80, take 8 plus 4 = 12.
• Squaring small numbers and powers: know the squares to 15 and the small powers of 2 and 3, so $2^5 = 32$ and $13^2 = 169$ are instant.
• Distributing for products: $6 \times 23 = 6 \times 20 + 6 \times 3 = 120 + 18 = 138$, breaking a product into friendly parts.

These habits let you handle the arithmetic the calculator-free session expects without long written computation.

Evaluating expressions by hand

A common calculator-free task is evaluating an expression at given values, where order of operations governs every step. For $3x^2 - 2x + 1$ at $x = -2$: square first, $(-2)^2 = 4$, then $3(4) = 12$; next $-2(-2) = 4$; then $12 + 4 + 1 = 17$. Work parentheses and exponents before multiplication, and multiplication before addition, and keep careful track of negative signs. Doing this fluently by hand is exactly what the no-calculator session checks.

Try this

Q1. Compute $\frac{3}{4} - \frac{1}{6}$.

• Cue. $\frac{9}{12} - \frac{2}{12} = \frac{7}{12}$.

Q2. Simplify $\sqrt{50}$ by hand.

• Cue. $5\sqrt{2}$.