Pure math: recall the rule, apply it once, get out.
Mathematics Knowledge measures familiarity with mathematical notation and the facts and procedures of algebra and geometry. Where Arithmetic Reasoning asks you to model a situation, MK asks whether you know the rule: how to factor, how to solve an inequality, what the angles of a triangle add to, how to find the slope of a line through two points.
The items are short - often a single expression with no words attached - and there is no calculator and no figure to reason from beyond simple diagrams. Speed comes from recognition. A test-taker who has memorised the standard formulas and the laws of exponents finishes MK with time left; one who rederives each rule under pressure does not.
MK is the second AFQT math subtest, and it is where prepared candidates gain the most points per hour of study, because the content is a closed, finite list of rules that can be drilled.
Section 5 of 10 in our timed mock template. This is one of the four subtests in the AFQT, so it affects enlistment eligibility directly.
Deeper AFQT page
There is a separate page on how Mathematics Knowledge drives the AFQT, with a week-by-week training routine: Mathematics Knowledge and the AFQT.
The shapes below are what our bank is written against, and what the section is built around. None of them are reproduced from the official test.
Specific, not generic: what the items in this topic test, and the one or two moves that solve most of them.
Treat every expression as an instruction to be followed in a fixed order. Substitution with a negative value is the highest-yield drill: f(x) = x^2 - 3x + 1 at x = -2 gives 4 + 6 + 1 = 11, because squaring a negative gives a positive and -3 times -2 gives +6. Both of those signs have their own distractor.
15 original questions in this topic set.
Isolate the variable with inverse operations, and remember the one rule that has no analogue in equations: multiplying or dividing an inequality by a negative flips the direction of the inequality. If the answer set is an interval, check whether the endpoints are included.
15 original questions in this topic set.
The facts that appear most often are the 180-degree triangle angle sum, angles on a straight line, vertical angles being equal, the Pythagorean triples (3-4-5, 5-12-13, 8-15-17, 7-24-25), and circumference and area of a circle. Sketch the figure and label what you know, even when a figure is given.
15 original questions in this topic set.
Write the formula before substituting, and check that the answer is in square units for area and cubic units for volume. The rectangle-versus-triangle trap is real: a right triangle with legs 6 and 8 has area 24, not 48, because it is half the 6 by 8 rectangle.
15 original questions in this topic set.
Know the four laws cold: same base multiplying adds exponents, dividing subtracts, a power of a power multiplies, and anything to the zero power is 1. For radicals, simplify by pulling out perfect squares. And remember that the square root symbol means the principal (non-negative) root.
15 original questions in this topic set.
Look in this order: common factor first, then difference of two squares, then a trinomial of the form x^2 + bx + c where you need two numbers that multiply to c and add to b. Factoring is also the fastest way to evaluate a complicated-looking expression - x^2 - 9 at x = 101 is (101 - 3)(101 + 3) in your head.
15 original questions in this topic set.
Conventions matter more than cleverness here: 1 is not prime, 2 is the only even prime, zero is even and is divisible by every integer, and absolute value is always non-negative. For "which must be true" items, test the edge cases - 0, 1, -1, and a fraction between 0 and 1.
12 original questions in this topic set.
Slope is rise over run: (y2 - y1) divided by (x2 - x1). Keep the subtraction order the same in numerator and denominator or the sign flips. Horizontal lines have slope 0, vertical lines are undefined, and perpendicular lines have slopes that are negative reciprocals of each other.
18 original questions in this topic set.
Sampled from the Mathematics Knowledge bank below, spread across its topics so you can see the range. The correct option is marked, the method is written out, and the tempting wrong options are explained - the same format you get in the app the moment you answer.
If x = 4, what is the value of 3x + 7?
Substitute x = 4 into the expression: 3(4) + 7 = 12 + 7 = 19. Multiply before adding.
Solve for x: x + 9 = 15
Subtract 9 from both sides: x = 15 - 9 = 6. Check: 6 + 9 = 15.
Two angles are complementary. One angle measures 35 degrees. What is the measure of the other angle?
Complementary angles add up to 90 degrees. The other angle is 90 - 35 = 55 degrees.
A rectangle has length 9 cm and width 4 cm. What is its area?
The area of a rectangle is length times width: 9 times 4 = 36 square cm.
On the clock
MK is about 75 seconds per item on our template. These questions are short; the honest split is roughly 15 seconds to identify the rule and 45 to apply it. If you cannot identify the rule in 20 seconds, the item is not going to yield - mark it and come back.
Free Mathematics Knowledge practice on this site, graded instantly with explanations. Timed sections, wrong-answer review and the full library are in the app.