10.7 Math Conventions & Figure Assumptions

10.7 Math Conventions & Figure Assumptions

Quick Answer: GRE math uses real numbers, planar figures, and straight lines. Geometric figures are NOT drawn to scale - derive from given information, not the picture. Coordinate planes and data graphs ARE drawn to scale. A prime is a positive integer > 1 divisible only by 1 and itself; sqrt denotes the nonnegative root.

Why Conventions Matter

The GRE Math Conventions PDF (free on ETS.org) fixes the assumptions underlying every Quant question. Knowing them prevents two costly mistakes: reading too much into a geometry figure, and ignoring the legitimate information a graph provides. The conventions are short, testable, and several appear directly as question traps.

The Full Convention List

The ETS Math Conventions document specifies the following. Each is a rule, not a suggestion.

The Figure-Assumption Rule (Critical)

This is the single most-tested convention. Geometric figures are not drawn to scale. That triangle that looks equilateral might be scalene. That angle that looks like 90 degrees might be 89 or 91. That circle that looks like it has radius 5 might have radius 50.

What You CANNOT Assume from a Geometry Figure

• Angle measures (a drawn right angle is not necessarily 90 degrees unless marked).
• Length proportions (a segment that looks twice as long is not necessarily twice as long).
• Parallelism (two lines that look parallel may not be).
• Perpendicularity (two segments that look perpendicular may not be, unless the right-angle square marker is shown).
• Shape regularity (a polygon that looks regular may not be).
• Area or relative area.

What You CAN Assume from a Geometry Figure

• Straight lines are straight (if drawn straight).
• Points are in the relative positions shown (a point labeled inside a circle is inside the circle).
• The order of points on a line is as drawn.
• If a right-angle square marker is drawn at a vertex, the angle is 90 degrees.
• If parallel arrows (>>) are drawn on two lines, they are parallel.
• If equal-length tick marks appear on segments, those segments are equal.

The Trap in Action

A triangle is drawn with two sides appearing equal, but no tick marks. You cannot assume it is isosceles. Compute from given information: if the stem says AB = AC, then it is isosceles; if the stem is silent, treat it as a general triangle.

Another classic: a quadrilateral is drawn to look like a square, but no right-angle markers and no equal-length tick marks. You cannot assume it is a square. If the stem calls it a rectangle, you may assume four right angles but not equal sides. If the stem calls it a rhombus, you may assume four equal sides but not right angles. Only a square - or a rectangle with tick marks - gives both.

When You CAN Read a Figure

Coordinate planes, number lines, and data graphics are drawn to scale. You may estimate from them.

Coordinate Plane

• The x and y axes are perpendicular and to scale (though the units per gridline may differ between axes).
• A plotted point's coordinates can be read off the axes.
• The slope of a line can be estimated visually as positive, negative, zero, or undefined, and approximated by rise/run from the gridlines.
• A parabola's vertex can be located approximately.

Number Line

• Relative positions correspond to magnitude: a point further right is larger.
• Equal spacings correspond to equal differences (if gridlines are shown).

Bar, Line, and Pie Charts

• Bar heights correspond to values on the y-axis.
• Line graph points correspond to values at the labeled x-positions.
• Pie wedge angles correspond to proportions (a 25% slice is a 90-degree wedge).

The Prime Convention

ETS defines a prime as a positive integer greater than 1 divisible only by 1 and itself. Consequences:

• 1 is not prime. This is the most-tested prime convention; the GRE loves the trap of listing 1 as prime.
• 2 is the smallest prime and the only even prime.
• Negative numbers are not prime. Primes are positive by definition.
• 0 is not prime. It is not greater than 1.
• The first ten primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

The Square Root Convention

The radical symbol sqrt denotes the nonnegative root only. So sqrt(25) = 5, not +/-5. If the GRE wants both roots, it writes +/-sqrt(25) or asks for the solutions to x^2 = 25 explicitly.

Consequence for QC questions: if Quantity A is sqrt(16) and Quantity B is -4, Quantity A is greater (4 > -4). A common trap is treating sqrt(16) as +/-4 and answering D (cannot be determined).

Decimal and Comma Conventions

The GRE uses the American convention:

• Decimal point separates integer from fractional part: 3.5 = 3 + 5/10.
• Comma is a thousands separator: 1,000 = one thousand.
• The GRE does NOT use the European convention (comma for decimal, period for thousands).

If you are an international test-taker, train yourself to read 3,5 as 35 (mistaken) and 3.5 as three and a half (correct). The on-screen calculator uses the decimal point for entry; do not type a comma.

Interval Notation

Infinity always uses a parenthesis, never a bracket, because infinity is not a number.

Worked Example: Geometry Figure Trap

A triangle ABC is drawn with AB and AC appearing equal and angle A appearing to be 60 degrees. The stem gives AB = 5 and BC = 5 but says nothing about angle A.

Question (QC): Quantity A: the measure of angle A. Quantity B: 60 degrees.

• You CANNOT assume angle A is 60 degrees just because it looks like it - figures are not drawn to scale.
• But AB = 5 and BC = 5 means triangle ABC is isosceles with AB = BC (given).
• The stem does not give AC, and angle A is not determined by the givens.
• Different values of AC yield different values of angle A (within triangle inequality constraints).
• Answer: D - the relationship cannot be determined from the information given.

The trap is assuming angle A is 60 degrees from the drawing and answering C. The convention - figures are not drawn to scale - is exactly what protects you.

Worked Example: Reading a Coordinate Plane

A line is plotted on a coordinate plane passing through (1, 2) and (4, 8).

Question: What is the slope of the line?

• Coordinate planes ARE drawn to scale, so you can read the points from the axes.
• Slope = (8 - 2) / (4 - 1) = 6 / 3 = 2.
• Visual estimate: rise of 6 over run of 3 confirms slope 2.

Here, reading the figure is legitimate because the convention says coordinate systems are to scale.

Common Convention Traps to Memorize

1. Primes include 2 but not 1.
2. sqrt(x^2) = |x|, not x. The nonnegative root means sqrt((-3)^2) = sqrt(9) = 3, not -3.
3. A drawn right angle is not 90 degrees without the marker.
4. A drawn equilateral-looking triangle is not equilateral without tick marks or a stem statement.
5. Coordinate and graphical data are to scale; geometry figures are not.
6. Decimal point, not comma, separates integer from fraction.
7. Infinity gets a parenthesis, not a bracket.

Strategic Implications

• For geometry questions, write the givens on scratch paper and ignore the picture for measurements. Use the picture only for relative position and for marked information (right-angle squares, tick marks, parallel arrows).
• For coordinate and graph questions, use visual estimation freely as a first pass, then verify with computation.
• For QC questions involving sqrt, remember the nonnegative convention: sqrt(25) is 5, never -5.
• For prime-number questions, immediately rule out 1, 0, and negatives.
• For interval-notation questions, check whether endpoints are included; the bracket-versus-parenthesis distinction is the answer to many QC items.