NCEA Level 1 Pāngarau cheat sheet

Order of operations: work through brackets, then exponents, then multiplication and division left to right, then addition and subtraction left to right. Example: 3 + 4 × 2 = 11, not 14.

Rounding: to round to a given place, look at the next digit; 5 or more rounds up, 4 or less stays. 4.376 to 2 d.p. is 4.38.

Significant figures: count from the first non-zero digit. 0.00456 has 3 s.f.; 2500 rounded to 2 s.f. is 2500 but written 2.5 × 10³ to show precision.

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

Percentages: percentage change = (new − old)/old × 100. Increase by 15% means multiply by 1.15; decrease by 15% means multiply by 0.85.

Ratios: to share an amount in the ratio a:b, divide by (a + b) to find one part, then multiply by a and b. Total parts method always works.

Standard form: a × 10ⁿ where 1 ≤ a < 10. 0.00032 = 3.2 × 10⁻⁴; 47 000 = 4.7 × 10⁴. Multiply by adding indices, divide by subtracting indices.

Negative indices and powers: x⁻ⁿ = 1/xⁿ; x⁰ = 1 for any non-zero x; x^(1/2) = √x; x^(1/3) = ∛x.

Estimation check: round each number to 1 s.f. to sanity-check a calculator answer. If 48.7 × 19.3 gives 940, the estimate 50 × 20 = 1000 confirms the order of magnitude.

Expanding: (x + a)(x + b) = x² + (a + b)x + ab. Factorising reverses this; look for a common factor first, then difference of two squares a² − b² = (a − b)(a + b).

Linear equations: isolate the variable by doing the same operation to both sides. For 3x − 7 = 14, add 7 then divide by 3 to get x = 7.

Straight line: y = mx + c, where m is the gradient and c is the y-intercept. Gradient m = (y₂ − y₁)/(x₂ − x₁).

Pythagoras: in a right-angled triangle, a² + b² = c², where c is the hypotenuse. Use it to find a missing side, and check the converse to test for a right angle.

Area and perimeter: rectangle A = lw; triangle A = ½bh; circle A = πr², C = 2πr; parallelogram A = bh; trapezium A = ½(a + b)h.

Volume: cuboid V = lwh; prism V = area of cross-section × length; cylinder V = πr²h. Capacity 1 cm³ = 1 mL; 1000 cm³ = 1 L.

Trigonometry (right triangles): sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj. Use inverse functions to find an angle.

Angle rules: angles on a straight line sum to 180°; angles at a point sum to 360°; interior angles of a triangle sum to 180°; co-interior angles on parallel lines sum to 180°.

Statistics: mean = sum ÷ count; median = middle value when ordered; mode = most frequent. The range = maximum − minimum and measures spread, not centre.

Probability: P(event) = favourable outcomes ÷ total outcomes; 0 ≤ P ≤ 1; P(not A) = 1 − P(A). For independent events multiply probabilities.

Choosing a strategy: use proportional reasoning (unit rate or scale factor) when two quantities vary together; use additive reasoning only when the same amount is added or subtracted.

Unit rates: divide to find the cost or quantity per one unit, then multiply by the required number. 6 items cost $8.40, so 1 item is $1.40 and 10 items are $14.00.

Scale drawings: scale 1:200 means 1 cm on the drawing represents 200 cm (2 m) in real life. Convert units before multiplying.

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

Reverse percentage: if a price after a 20% discount is $64, the original is 64 ÷ 0.8 = $80, not 64 × 1.2.

Time and rate problems: distance = speed × time; rearrange to time = distance ÷ speed. Keep units consistent (km and hours, or m and seconds).

Checking reasonableness: estimate first, compare with the exact answer, and confirm the answer is in the correct units and sensible in context.

Presenting working: show the formula or strategy, substitute values, state the answer with units, and round only at the final step to avoid rounding error.