NCEA Pāngarau Co-requisite (NZQA Standard 32412) cheat sheet

Order of operations: brackets, exponents, then × and ÷ left to right, then + and − left to right. 2 + 3 × 4² = 2 + 3 × 16 = 50.

Integer sign rules: (+)(+)=+, (−)(−)=+, (+)(−)=−. Subtracting a negative adds: 7 − (−3) = 10.

Fraction operations: a/b + c/d = (ad + bc)/bd; a/b ÷ c/d = a/b × d/c. Simplify by cancelling common factors before multiplying.

Percentages: part = percent/100 × whole. Increase by 15% means multiply by 1.15; decrease by 15% means multiply by 0.85.

Ratio sharing: total parts = sum of ratio numbers; one share = total ÷ total parts; each share = ratio number × one share.

Linear pattern: y = mx + c, where m is the constant difference between consecutive terms and c is the value when x = 0.

Rearrange formulas by inverse operations: to make x the subject of y = 3x − 5, add 5 then divide by 3, giving x = (y + 5)/3.

Index laws: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1 for a ≠ 0.

Standard form: a × 10ⁿ where 1 ≤ a < 10. 0.0042 = 4.2 × 10⁻³; 35 000 = 3.5 × 10⁴.

Perimeter is the distance around a shape; area is the space inside, measured in square units.

Area formulas: rectangle A = l × w; triangle A = ½ × b × h; parallelogram A = b × h; trapezium A = ½(a + b)h.

Circle: circumference C = πd = 2πr; area A = πr². Use π ≈ 3.14 or the calculator value.

Volume: prism V = area of cross-section × length; cylinder V = πr²h; cone V = ⅓πr²h; sphere V = 4/3 πr³.

Pythagoras: in a right triangle a² + b² = c², where c is the hypotenuse (the side opposite the right angle).

Angle facts: angles on a straight line sum to 180°; angles at a point sum to 360°; angles in a triangle sum to 180°; angles in a quadrilateral sum to 360°.

Parallel lines: corresponding and alternate angles are equal; co-interior angles sum to 180°.

Unit conversions: 1 km = 1000 m; 1 m = 100 cm; 1 cm = 10 mm; 1 kg = 1000 g; 1 L = 1000 mL; 1 m² = 10 000 cm²; 1 m³ = 1 000 000 cm³.

Scale: map distance ÷ scale factor = real distance. A scale of 1:200 means 1 cm on the plan represents 200 cm (2 m) in reality.

Mean = sum of values ÷ number of values. Median = middle value of ordered data. Mode = most frequent value. Range = highest − lowest.

For grouped or frequency data, mean = Σ(f × x) ÷ Σf, where x is the class midpoint and f is the frequency.

The median is the (n+1)/2-th value when n is odd; when n is even it is the average of the n/2-th and (n/2+1)-th values.

Probability scale runs from 0 (impossible) to 1 (certain). P(event) = favourable outcomes ÷ total equally likely outcomes.

P(not A) = 1 − P(A). For independent events, P(A and B) = P(A) × P(B).

Two-way tables: find the required cell or row/column total, then divide by the grand total for a probability.

A random sample is chosen so every member of the population has an equal chance of selection; a biased sample over-represents some group.

Correlation describes the direction and strength of a linear relationship; correlation does not prove causation.

Graph choice: bar charts compare categories, line graphs show change over time, histograms show distribution of numerical data, scatter plots show relationships between two variables.

Problem-solving cycle: understand the problem, choose a strategy, carry out the plan, check the answer against the context.

Estimate first to check reasonableness: round to convenient numbers (e.g. 48 × 21 ≈ 50 × 20 = 1000).

Show working line by line; each step must be justified by a rule, formula, or given information.

Use a diagram, table, or list to organise information when a problem has several conditions or cases.

Work backwards from the answer using inverse operations to verify a result.

Check units and magnitude: an answer of 0.5 m for a room length signals an error; 5 m is plausible.

Generalisation: test a rule on several cases, state it in words, then express it algebraically (e.g. nth term = 3n + 1).

When comparing, calculate a common measure (percentage, rate, or per-unit value) before concluding which is larger.

Justify with evidence: quote the calculated values, not just a claim, and state any assumptions made.

Identify the given information, the quantity asked for, and the units required before calculating.

Best-buy comparisons: unit price = total price ÷ quantity; the lowest unit price is the best value, provided quality and quantity suit the need.

Percentage change = (new − old) ÷ old × 100. A positive result is an increase; a negative result is a decrease.

GST in Aotearoa New Zealand is 15%: GST amount = price × 0.15; GST-inclusive price = price × 1.15; GST-exclusive price = price ÷ 1.15.

Speed, distance, time: d = s × t, s = d ÷ t, t = d ÷ s. Keep units consistent (km and hours give km/h).

Wage and pay: total pay = hours × hourly rate; overtime = hours × rate × multiplier; gross pay before deductions, net pay after deductions.

Compound interest: A = P(1 + r)ⁿ, where P is principal, r the rate per period as a decimal, and n the number of periods.

Budgeting: total income − total expenses = surplus (positive) or deficit (negative); express each category as a percentage of income.

Rounding in context: money is usually rounded to 2 decimal places; check whether the context requires rounding up (e.g. number of boxes needed).