SAT Math Shortcuts • Topic 4 of 4

Pattern Recognition

Recognising structure lets you skip work. Spot special forms: differences of squares (a² − b² = (a+b)(a−b)), perfect-square trinomials, and common Pythagorean triples (3-4-5, 5-12-13) you can read off instantly. In sequences, find the constant difference (arithmetic) or ratio (geometric) and project forward. In repeating cycles — units digits of powers, remainders, calendar days — find the cycle length and use the remainder. Many “hard” questions are quick once you see the pattern they’re built on. Train yourself to pause and ask “what structure is this?” before grinding through computation; the SAT rewards the student who sees the shortcut.

✅ Solved examples

1. Factor 49 − x² using a pattern.
Difference of squares: (7 − x)(7 + x).
2. A right triangle has legs 6 and 8. Hypotenuse (by pattern)?
A 3-4-5 triple ×2 → 10.
3. Sequence 4, 7, 10, 13, … What is the 6th term?
Add 3 each time: 13, 16, 19 → 6th term is 19.
4. Units digit of 2⁵ given the cycle 2,4,8,6?
Cycle length 4; 5 mod 4 = 1 → 1st in cycle = 2.

✏️ Practice — try these, take hints as needed

1. Factor x² − 25.
Difference of squares.
(x−5)(x+5).
—
(x − 5)(x + 5).
2. Right triangle legs 9 and 12; hypotenuse (pattern)?
3-4-5 ×3.
—
—
15.
3. Sequence 5, 9, 13, … common difference?
Subtract terms.
—
—
4.
4. Next term after 2, 6, 18, … (pattern)?
×3 each time.
—
—
54.
5. Units digits of powers of 2 repeat every how many?
2,4,8,6,…
—
—
4.
← Answer Choice Testing Try a full-length mock →