Math Question-Type Playbook

14 question types across 4 domains, each cracked by: identify the form, choose the tool (algebra vs Desmos), solve, then answer the exact ask. ~75% are 4-option multiple choice, ~25% are student-produced "grid-in" (type the answer — negatives allowed, no mixed numbers). The Desmos graphing calculator is available on every question. Examples below are MERIDIAN-original; all calculations verified.

Official mapping: these types sit under College Board's four Math domains (official Table 3 = Assessment Framework v3.01 Table 16). Canonical names + weights live in SIGNAL · Official Taxonomy Map. Source: satsuite.collegeboard.org/media/pdf/digital-sat-test-spec-overview.pdf

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#ALGEBRA (~35%)

#1 · LINEAR EQUATIONS & WORD PROBLEMS

What it is: build and solve a linear equation from a context; or interpret what a slope/intercept means. Method: define the variable, translate words → equation, solve, re-read the ask. Example. A taxi charges a $3 base fare plus $2 per mile. A ride cost $19. How many miles? Exemplar: 3 + 2m = 19 → 2m = 16 → m = 8 miles. Traps: answering the wrong quantity; forgetting the base fare; unit slips. Tips: write the equation explicitly; underline the ask. Hardest (interpret a coefficient). Profit is modelled by P = 45u − 1200, where u = units sold. What does 45 represent? (A) fixed costs (B) profit per unit sold (C) total profit (D) break-even units Answer: (B) — the slope is the change in P per unit. Why hard: (A) is −1200; (D) is where P = 0 (u ≈ 26.7). Crack it: slope = rate of change per 1 unit of x.

#2 · SYSTEMS OF LINEAR EQUATIONS

What it is: solve two linear equations, or interpret the number of solutions. Method: Desmos intersection, or elimination/substitution. Example. 2x + y = 11 and x − y = 1. Find x. Exemplar: add the equations → 3x = 12 → x = 4 (then y = 3). Traps: solving for the wrong variable; sign errors in elimination. Tips: in Desmos, type both equations and click the intersection. Hardest (parameter → number of solutions). For what value of k does 3x − 6y = 9 and x − 2y = k have infinitely many solutions? Answer: divide the first by 3 → x − 2y = 3; for the two to be the same line, k = 3. Why hard: "no solution" (parallel, k ≠ 3) is the confusable case. Crack it: same slope + same intercept = infinite; same slope + different intercept = none.

#3 · LINEAR INEQUALITIES

What it is: solve/interpret inequalities; solution sets; systems as shaded regions. Method: solve like an equation — but flip the sign when you multiply/divide by a negative. Example. Solve −3x + 5 > 14. Exemplar: −3x > 9 → x < −3 (flip on ÷ by −3). Traps: forgetting to flip the sign. Tips: flag every multiply/divide by a negative. Hardest ("which point" in a system). Which point satisfies both y > 2x − 1 and y ≤ −x + 4? (A) (0,5) (B) (3,0) (C) (1,2) (D) (4,4) Answer: (C) — 2 > 2(1)−1 = 1 ✓ and 2 ≤ −1+4 = 3 ✓. Why hard: a point must satisfy all inequalities ((A) fails the second, (B) and (D) fail the first). Crack it: in Desmos, shade both; the answer lies in the overlap.

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#ADVANCED MATH (~35%)

#4 · QUADRATICS (roots, vertex, # of solutions)

What it is: solve quadratics; find zeros/vertex; determine how many real solutions. Method: factor / quadratic formula / Desmos x-intercepts; vertex at x = −b/2a; discriminant b²−4ac for the count. Example. Solve x² − 5x + 6 = 0. Exemplar: (x − 2)(x − 3) = 0 → x = 2 and x = 3. Traps: reporting only one root; sign errors. Tips: for messy quadratics, read the x-intercepts off Desmos. Hardest (discriminant for tangency). For what value of c does y = x² + 6x + c have exactly one x-intercept? Answer: one solution → discriminant 0 → 36 − 4c = 0 → c = 9. Why hard: requires recognising "one x-intercept ⇔ b²−4ac = 0." Crack it: the parabola is tangent to the x-axis (or drag a Desmos slider on c until it just touches).

#5 · EXPONENTIAL FUNCTIONS (growth & decay)

What it is: model or interpret quantities that change by a percentage each period. Method: y = a(1 ± r)^t (or a·b^t); a = initial value, factor = 1 ± r; growth if b > 1, decay if 0 < b < 1. Example. A population of 500 grows 4% per year. Expression after t years? Exemplar: 500(1.04)^t. Traps: using r (0.04) instead of the factor (1.04); confusing growth/decay. Tips: "changes by a percent of itself" ⇒ exponential; factor = 1 ± r. Hardest (period ≠ 1 unit). A 80 mg sample halves every 6 hours. Which models the amount after t hours? (A) 80(0.5)^t (B) 80(0.5)^(t/6) (C) 80(0.5)^(6t) (D) 80 − t/6 Answer: (B) — the exponent is the number of half-life periods, t/6. Why hard: (A) halves every hour; (C) every 1/6 hour; (D) is linear. Crack it: exponent = (elapsed time) ÷ (length of one period).

#6 · EQUIVALENT EXPRESSIONS & FUNCTION NOTATION

What it is: rewrite/simplify expressions, "in terms of," or evaluate composed functions. Method: factor/expand/exponent rules; when choices contain variables, plug in a number to test equivalence; for f(g(x)), work inside-out. Example. Simplify (x² − 9)/(x − 3). Exemplar: = (x+3)(x−3)/(x−3) = x + 3 (for x ≠ 3). Traps: illegal cancelling; ignoring the excluded value. Tips: plug in (say x = 5) into the stem and each choice — keep the one that matches. (Desmos can't do "equivalent" questions for you.) Hardest (composition order). If f(x) = 2x + 1 and g(x) = x², find f(g(3)). Answer: g(3) = 9, then f(9) = 2(9)+1 = 19. Why hard: the trap is computing g(f(3)) = (7)² = 49, or f(3) first. Crack it: evaluate the inner function first.

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#PROBLEM-SOLVING & DATA ANALYSIS (~15%)

#7 · PERCENTAGES

What it is: percent of, percent change, successive percents. Method: multiply sequentially; each percent applies to the current base. Example. An $80 jacket is marked up 25%, then that price is discounted 20%. Final price? Exemplar: 80 × 1.25 = 100; 100 × 0.80 = $80. Traps: adding/subtracting the percents (25 − 20 = "5%"); applying to the wrong base. Tips: "% of" = multiply; successive changes multiply, never add. Hardest (reverse percent). After a 15% raise, a salary is $46,000. What was it before? Answer: x × 1.15 = 46,000 → x = $40,000. Why hard: the trap is 46,000 × 0.85 = 39,100. Crack it: divide by 1.15 — don't subtract 15%.

#8 · RATIOS, RATES, PROPORTIONS & UNITS

What it is: scale quantities; convert units; proportional reasoning. Method: set a proportion; track units and convert before computing. Example. A car uses 6 L per 100 km. How many litres for 250 km? Exemplar: 6 × (250/100) = 15 L. Traps: inverting the ratio; mismatched units. Tips: write the proportion with units; the units should cancel to the answer's unit. Hardest (unit-conversion chain). A printer prints 20 pages per minute. How many pages in 1.5 hours? Answer: 1.5 h = 90 min; 20 × 90 = 1,800 pages. Why hard: forgetting to convert hours → minutes (20 × 1.5 = 30 is the trap). Crack it: convert all units first, then multiply.

#9 · PROBABILITY & TWO-WAY TABLES

What it is: simple and conditional probability from counts/tables. Method: probability = favourable / total; for "given X," the denominator is the X subgroup. Example. Of 50 people: 30 like tea (18 of them also like coffee); 20 don't like tea (5 like coffee). P(likes coffee)? Exemplar: coffee-likers = 18 + 5 = 23 → 23/50. Traps: using the wrong denominator. Tips: read whether the question conditions on a subgroup. Hardest (conditional). Given a person likes tea, what is the probability they also like coffee? Answer: denominator = tea group (30), favourable = 18 → 18/30 = 3/5. Why hard: the trap uses 50 (the whole group). Crack it: "given tea" ⇒ divide by the tea total only.

#10 · STATISTICS & INFERENCE (centre, spread, margin of error)

What it is: mean/median/range/standard deviation; sample inference; margin of error. Method: outliers pull the mean, not the median; margin of error gives an interval = estimate ± MOE. Example. Data: 4, 4, 5, 9, 18. Which is larger, the mean or the median? Exemplar: mean = 40/5 = 8; median = 5 → mean (8) > median (5) (the 18 skews the mean up). Traps: assuming mean = median; ignoring the outlier. Tips: a high outlier ⇒ mean > median (right-skew). Hardest (margin-of-error interpretation). A poll estimates 52% support with a margin of error of ±3%. Which is best supported? (A) Exactly 52% support. (B) Plausibly between 49% and 55% support. (C) Support is definitely above 50%. (D) The value is 52% ± 3 people. Answer: (B) — the interval is 49–55%. Why hard: (C) tempts, but 49% is inside the interval, so support is not "definitely" above 50%. Crack it: build estimate ± MOE and don't over-claim.

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#GEOMETRY & TRIGONOMETRY (~15%)

#11 · LINES, ANGLES & TRIANGLES (incl. similar triangles)

What it is: angle-chasing, triangle relationships, similarity. Method: triangle angles sum to 180°; parallel-line and supplementary rules; similar triangles → equal ratios. Example. Two angles of a triangle measure 40° and 75°. The third? Exemplar: 180 − (40 + 75) = 65°. Traps: forgetting the 180° sum; mixing interior/exterior angles. Tips: mark every known angle on the figure. Hardest (similar triangles). A 6-ft person casts a 4-ft shadow; a nearby tree casts a 30-ft shadow. How tall is the tree? Answer: similar triangles → 6/4 = h/30 → h = 45 ft. Why hard: setting the proportion up correctly (matching person→person, shadow→shadow). Crack it: similar figures share equal side ratios.

#12 · AREA & VOLUME

What it is: area/volume of standard and composite figures (formulas are on the reference sheet). Method: pick the right formula; watch radius vs diameter and units. Example. A cylinder has radius 3 and height 10. Volume? (V = πr²h) Exemplar: π·3²·10 = 90π. Traps: using diameter as radius; forgetting π; wrong formula. Tips: the formula's on the sheet — your job is correct inputs. Hardest (scaling). If a sphere's radius doubles, its volume increases by what factor? Answer: V ∝ r³ → 2³ = 8×. Why hard: the traps are "2×" or "4×." Crack it: volume scales with the cube of the linear scale factor (area scales with the square).

#13 · CIRCLES (equation, arc, sector)

What it is: the circle equation; radius/centre; arc length; sector area; radians. Method: (x−h)² + (y−k)² = r² gives centre (h,k), radius r; arc = (θ/360)·2πr; sector = (θ/360)·πr². Example. Circle: (x − 2)² + (y + 1)² = 16. Radius? Exemplar: r = √16 = 4 (centre (2, −1)). Traps: forgetting the square root; sign of the centre coordinates. Tips: the right side is r², not r; in Desmos, type the equation to see it. Hardest (arc length). A circle has radius 9. An arc subtends a central angle of 40°. Arc length? Answer: (40/360)·2π·9 = (1/9)·18π = 2π. Why hard: arc-vs-sector confusion and the degree fraction. Crack it: arc length = (θ/360) × circumference.

#14 · RIGHT-TRIANGLE TRIGONOMETRY

What it is: sine/cosine/tangent ratios; special triangles; the cofunction relationship. Method: SOH-CAH-TOA (sin = opp/hyp, cos = adj/hyp, tan = opp/adj); sin(x) = cos(90 − x). Example. In a right triangle, the side opposite a 30° angle is 5. Find the hypotenuse. Exemplar: sin 30° = opp/hyp = 0.5 → hyp = 5/0.5 = 10. Traps: mixing up the ratios; calculator in radian mode. Tips: label opp/adj/hyp relative to the angle before choosing the ratio. Hardest (cofunction identity). If sin(x°) = cos(50°) and 0 < x < 90, find x. Answer: sin(x) = cos(90 − x) → 90 − x = 50 → x = 40. Why hard: needs the cofunction identity, not a calculator. Crack it: sine of an angle = cosine of its complement.

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#REFERENCE CARD — MATH TYPES

ALGEBRA: translate→solve→answer the ask · systems by Desmos intersection · flip inequality on ÷(−)
 #solutions: same slope+intercept = infinite; same slope only = none.
ADVANCED: quadratics (factor/formula/Desmos; one solution ⇔ discriminant 0) · exponential factor = 1±r,
 exponent = time/period · equivalent expressions → plug in · f(g(x)) inside-out.
PSDA: successive % multiply (reverse % = divide) · proportions track units · conditional = subgroup denominator
 · outlier pulls the mean · margin of error = estimate ± MOE.
GEO/TRIG: triangle = 180° · similar = equal ratios · volume scales as cube · circle (x−h)²+(y−k)²=r²
 · arc = (θ/360)·2πr · SOHCAHTOA · sin(x)=cos(90−x).
Desmos solves; it does NOT do "equivalent expression"/"in terms of" — plug in instead.

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MERIDIAN · CODEX · MATH QUESTION-TYPE PLAYBOOK | v1.0