Start with the exact ADA section contract, then connect every displayed scope line to an observable study outcome. The map makes calculator use, units, assumptions, and evidence limits visible before timed practice begins.
The ADA publishes a 40-question total and a 30 mathematical / 10 applied-mathematics split. Inside the mathematical group it lists Algebra, Data Analysis, Interpretation, and Sufficiency, Quantitative Comparison, and Probability and Statistics.
It does not publish item quotas for those mathematical subtopics. DAT TRAIN’s 18 objectives are a study organization—not an estimate of test frequency. Subtopic weights published: no.
The calculator can accelerate arithmetic. It cannot choose the relationship, restore a dropped unit, or prove that a comparison is determined.
1
Model
Name the target. Translate the relationship. Attach units and restrictions before entering numbers.
2
Solve
Choose the shortest valid path: algebra, estimation, a bound, a table, a graph, or a controlled calculator step.
3
Check
Test sign, scale, unit, domain, and substitution. Reject precision that the prompt cannot support.
4
Decide
Answer the exact question asked—especially for comparison and sufficiency, where a full numerical solution may be unnecessary.
estimate≈ 24confirm23.8
CALCULATOR-AWARE, NOT CALCULATOR-LED
Set up first. Estimate second. Key in last.
The official interface supplies a digital calculator for this section. Use it after the model is stable, then compare the display with an expected sign and scale. A polished decimal produced by the wrong equation is still wrong.
THE COMPLETE OFFICIAL SCOPE
Two groups. Ten displayed leaves.
Open a problem group, then a topic. Each topic keeps the official wording and adds bounded DAT TRAIN outcomes, prerequisite links, common traps, representations, and free learning references.
I
Mathematical Problems
30 official questions · 9 displayed leaves · 16 DAT TRAIN objectives
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A1
Equations and expressions
2 observable outcomes
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01
Translate structure before simplifying
Convert verbal or nested algebraic relationships into an expression or equation while preserving grouping, sign, and unit meaning.
Must know
Operation order follows the written structure, including grouping symbols and fraction bars.
A variable expression represents a relationship; an equation adds an equality constraint.
Evidence boundary
Do not infer a missing domain restriction, unit, or relationship that the prompt does not supply.
Common trap
Words can be translated in the order they appear. Relational phrases determine grouping and operation direction before arithmetic begins.
02
Solve and verify equations
Solve linear, quadratic, rational, or simultaneous equations at the stated level and verify candidates in the original constraints.
Must know
Equivalent transformations preserve the solution set only when their conditions are respected.
Substitution into the original equation catches extraneous or excluded values.
Evidence boundary
No advanced mathematics or calculus; choose methods supported by the supplied algebraic structure.
Common trap
Every value produced by an algebraic manipulation is a solution. Squaring, clearing denominators, and other conditional moves can create candidates that fail the original equation.
A2
Inequalities
2 observable outcomes
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01
Preserve order through transformations
Solve one-variable inequalities and express the complete solution set using symbols, intervals, or a number line.
Must know
Multiplying or dividing by a negative reverses the inequality direction.
Strict and inclusive boundaries produce different endpoint notation.
Evidence boundary
Report the full solution region rather than selecting a convenient test value as the answer.
Common trap
An inequality sign behaves exactly like an equals sign under every operation. Order reverses under multiplication or division by a negative quantity.
02
Resolve compound and absolute inequalities
Combine intersection or union conditions and test boundary regions for compound, rational, or absolute-value inequalities.
Must know
‘And’ usually describes an intersection; ‘or’ usually describes a union.
Critical points partition the number line into regions whose signs can be tested.
Evidence boundary
A sign chart establishes interval behavior; it does not authorize cancellation across a zero denominator.
Common trap
Two boundary values imply every value between them works. The operator and sign pattern determine whether the middle, outside, or separate regions satisfy the condition.
A3
Exponential notation
1 observable outcome
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01
Control exponential notation
Apply exponent and scientific-notation rules while preserving sign, base, reciprocal, and order-of-magnitude meaning.
Must know
Exponent rules apply to matching bases and to complete products or quotients, not arbitrary sums.
A negative exponent creates a reciprocal; it does not make the value negative.
Evidence boundary
Keep exact form until the requested precision is known; do not manufacture significant digits.
Common trap
A negative exponent means a negative result. It indicates reciprocal power; the base determines the sign.
A4
Absolute value
1 observable outcome
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01
Read absolute value as distance
Interpret absolute value as distance and solve equations or comparisons by separating the required cases.
Must know
Absolute value is a nonnegative distance from zero or another reference point.
An equation |u| = c with c > 0 produces two cases; a negative c produces no real solution.
Evidence boundary
Case splitting must preserve the entire expression inside the absolute-value bars.
Common trap
Absolute value means delete a minus sign. It measures distance and may require two algebraic cases.
A5
Ratios and proportions
2 observable outcomes
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01
Normalize ratios and rates
Compare ratios, unit rates, percentages, and proportional quantities only after aligning the same base and units.
Must know
A ratio compares ordered quantities; swapping numerator and denominator changes its meaning.
Percent change uses the original quantity as the denominator unless the prompt defines another base.
Evidence boundary
A similar-looking fraction is not comparable until both quantities and units refer to the same basis.
Common trap
Percent change is final minus initial divided by final. The ordinary comparison base is the original value.
02
Scale proportional models
Solve direct, inverse, and multistep proportion problems and verify that the direction of change matches the model.
Must know
Direct proportionality keeps a ratio constant; inverse proportionality keeps a product constant.
A reasonableness check should confirm whether increasing one quantity should increase or decrease the other.
Evidence boundary
Do not assume proportionality merely because two quantities change together.
Common trap
Every rate relationship scales directly. Some models are inverse or nonproportional; the stated relationship controls the equation.
A6
Graphical analysis
2 observable outcomes
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01
Extract quantities from graphs
Read axes, scale, units, points, intervals, slope, and intercept from a graph before calculating or comparing.
Must know
Axis labels and tick spacing define the quantitative meaning of every mark.
Slope is a change ratio with units; an intercept is a modeled value at a reference input.
Evidence boundary
Do not extrapolate beyond the displayed or stated domain without an explicit model.
Common trap
The steepest-looking line always has the greatest slope. Slope comparisons require the axis scales and units, not visual angle alone.
02
Move between models
Match equations, tables, and graphs by testing invariant features such as intercepts, slope, curvature, symmetry, and domain.
Must know
A small set of structural features can reject most mismatches before point-by-point calculation.
Equivalent representations must agree on both values and allowed inputs.
Evidence boundary
A few matching sample points do not prove two models identical over every input.
Common trap
One shared point means a graph and equation are the same model. Many different models intersect; compare defining behavior and domain.
B
Data Analysis, Interpretation, and Sufficiency
2 observable outcomes
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01
Read data before summarizing
Identify variables, groups, units, totals, trends, and anomalies in tables or displays before selecting a numerical summary.
Must know
A total, percentage, rate, and percentage-point difference answer different questions.
The display’s population and denominator define what can be concluded.
Evidence boundary
Describe association or displayed change without converting it into an unsupported causal claim.
Common trap
A larger percentage always represents a larger count. Counts depend on both the percentage and the group size.
02
Test information sufficiency
Decide whether each supplied statement, alone or together, determines the requested quantity without solving beyond what the decision requires.
Must know
Sufficiency asks whether a unique answer follows, not whether one compatible answer can be found.
Each statement must be tested independently before testing their combination.
Evidence boundary
Do not import unstated positivity, integrality, geometry, or domain assumptions.
Common trap
A statement is sufficient if it lets me construct one answer. It must exclude every alternative answer allowed by the prompt.
C
Quantitative Comparison
2 observable outcomes
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01
Compare without over-solving
Compare two quantities by simplifying a difference or ratio, bounding values, or selecting valid test cases.
Must know
The sign of A − B directly identifies their order when it can be determined.
Bounds and monotonic behavior can settle a comparison without exact values.
Evidence boundary
Division-based comparisons require a known nonzero denominator and sign-aware reasoning.
Common trap
Exact numerical values are required before two quantities can be compared. Structure, bounds, or a sign test may determine the relationship directly.
02
Prove indeterminacy with counterexamples
Choose admissible values that produce different comparison outcomes when the relationship cannot be determined.
Must know
One valid case can disprove an ‘always’ claim; two valid cases with different outcomes prove indeterminacy.
Every test value must satisfy all stated restrictions.
Evidence boundary
A counterexample outside the prompt’s domain is not evidence.
Common trap
If two convenient examples agree, the relationship is determined. Agreement among examples does not replace a proof; search boundary and sign-changing cases.
D
Probability and Statistics
2 observable outcomes
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01
Build the sample space
Model simple, conditional, complementary, and multi-stage probabilities from a complete sample space or event structure.
Must know
Probability is favorable measure divided by total measure only when the modeled outcomes use the appropriate weighting.
‘At least one’ is often cleanly computed with a complement; conditional probability changes the denominator.
Evidence boundary
Do not assume independence, equal likelihood, or replacement unless the experiment supports it.
Common trap
Separate events can always be multiplied. Multiplication requires conditional probabilities or justified independence.
02
Choose and interpret statistics
Compute and interpret center, spread, weighted averages, percentiles, and simple distribution comparisons from the supplied data.
Must know
Mean, median, and mode respond differently to skew and outliers.
A weighted mean uses the count or weight attached to each value, not an unweighted average of subgroup means.
Evidence boundary
A summary statistic does not reveal the full distribution or establish causation.
Common trap
The mean of group means is always the overall mean. Subgroup means must be weighted by their group sizes unless those sizes are equal.
II
Applied Mathematics (Word) Problems
10 official questions · 1 displayed leaf · 2 DAT TRAIN objectives
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A
Applied Mathematics (Word) Problems
2 observable outcomes
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01
Translate the situation into a model
Name the requested quantity, define variables, align units, and translate rates, percentages, mixtures, motion, work, finance, or measurement constraints into equations.
Must know
A unit ledger exposes missing conversions and distinguishes a rate from a quantity.
The model should state what each variable measures before numbers are substituted.
Evidence boundary
The official label is broad; DAT TRAIN examples may organize common contexts but do not claim unpublished ADA context quotas.
Common trap
The operation can be chosen from the numbers alone. The relationships and units determine the model before arithmetic begins.
02
Solve, estimate, and audit the result
Solve an applied model, use the digital calculator selectively, and reject results that fail magnitude, unit, sign, domain, or context checks.
Must know
An estimate predicts the answer’s sign and scale before exact arithmetic.
A calculated value must be interpreted in the requested unit and feasible context.
Evidence boundary
Calculator output is arithmetic evidence, not proof that the chosen model or interpretation is valid.
Common trap
A calculator display is automatically the answer. Round, label, and test the result against the original situation.
A DEPENDABLE STUDY ORDER
Build proof before pace.
The official hierarchy defines what may be tested, not how it must be taught. This order makes later decisions depend on representations and checks already in place.
01
Algebraic grammar
Preserve grouping, sign, domain, and equivalence before speed. Equations, inequalities, exponents, absolute value, and ratios share one symbolic language.
02
Representations
Move deliberately among equations, tables, graphs, units, and words. The same relationship should survive every change of view.
03
Decision problems
Practice sufficiency and quantitative comparison as proof tasks: determine what must be true without doing work the question does not require.
04
Uncertainty and data
Build sample spaces, choose statistics, preserve denominators, and keep association separate from causation.
05
Applied translation
Name the target, define variables, align units, model the relationship, estimate, calculate, and audit the result in context.
SEVEN PRACTICE ROUTES
Build the proof. Switch the method. Protect time.
This study map is not a question bank, a calibrated form, or a score converter. Choose a focused topic to read a lesson and work through explained examples, or mix algebra, graphs, comparison, probability, statistics, and applied mathematics in one practice circuit.
The question bank includes 1,046 Quantitative questions. Build a complete 40-question form with 30 Mathematical and 10 Applied Mathematics questions, using a 45-minute clock or untimed practice. Explanations appear after submission; a keyboard-and-touch calculator, saved-session recovery, and raw results are available. Automated validation is complete; independent mathematics and assessment review, representative-device accessibility testing, and student pilot analysis remain open, so the items remain draft and uncalibrated.