Number and Arithmetic
Not publishedof exam
Algebra and Patterns
Not publishedof exam
Measures and Units
Not publishedof exam
Space and Geometry
Not publishedof exam
Chance and Data
Not publishedof exam
Quick Facts
- Assessment
- ICAS Mathematics
- Papers
- Intro and A–J
- Australia
- Years 2–12
- Questions
- Intro 30; A–J 40
- Intro time
- 35 minutes
- A–C time
- 45 minutes
- D–J time
- 60 minutes
- Skill areas
- Five official areas
- Paper map
- Country and year specific
- Universal weights
- Not published
Factor vs Multiple
Factor
- Divides exactly
- Usually smaller
Multiple
- Product with integer
- Usually larger
Divisor vs product
Number Method Picker
- Find percentage amount→Multiply decimal rate
- Recover original amount→Divide by multiplier
- Share in ratio→Find one share
- Compare proportional rates→Find unit rate
- Repeated fixed increase→Simple interest
- Growth earns growth→Compound formula
Fraction + Percent
- Fraction to decimal
- Numerator ÷ denominator
- Decimal to percent
- Multiply by 100%
- Percent to decimal
- Divide by 100
- Percent of amount
- Decimal rate × amount
- Percentage
- Part ÷ whole × 100%
- Percentage change
- Change ÷ original × 100%
- Add fractions
- Use common denominator
- Divide fractions
- Multiply reciprocal
Simple vs Compound Interest
Simple
- Original principal only
- Linear growth
Compound
- Updated balance
- Exponential growth
Fixed base vs growing base
Indices + Finance
- aᵐ × aⁿ
- aᵐ⁺ⁿ
- aᵐ ÷ aⁿ
- aᵐ⁻ⁿ
- (aᵐ)ⁿ
- a raised to mn
- a⁰
- 1 when a ≠ 0
- a⁻ⁿ
- 1 ÷ aⁿ
- Scientific notation
- 1 ≤ |a| < 10
- Simple interest
- I = Prt
- Compound amount
- A = P(1+r)ⁿ
Operation Order
Brackets, Indices, Divide, Multiply, Add, Subtract
B: bracketsI: indicesDM: left-to-rightAS: left-to-right
Expression vs Equation
Expression
- No equals sign
- Simplify or evaluate
Equation
- Has equality
- Solve unknown
Simplify vs solve
Algebra Core
- Distributive law
- a(b+c) = ab+ac
- Common factor
- ab+ac = a(b+c)
- Difference squares
- a²−b² = (a−b)(a+b)
- Solve linear
- Balance both sides
- Substitution
- Replace variable, then calculate
- Quadratic roots
- x = (−b±√D)/(2a)
- Discriminant
- D = b²−4ac
- Linear inequality
- Reverse sign after negative
Lines + Coordinates
- Line equation
- y = mx+c
- Gradient
- Vertical change ÷ horizontal change
- Midpoint
- ((x₁+x₂)/2,(y₁+y₂)/2)
- Distance
- √((x₂−x₁)²+(y₂−y₁)²)
- Parallel lines
- Equal gradients
- Perpendicular lines
- Gradient product −1
- Arithmetic sequence
- aₙ = a₁+(n−1)d
- Origin
- Coordinate (0,0)
Metric Ladder
Kilo, hecto, deka, unit, deci, centi, milli
Each step ×10Reverse step ÷10Square factors twiceCube factors thrice
Perimeter vs Area
Perimeter
- Boundary length
- Linear units
Area
- Region covered
- Square units
Around vs inside
Metric Conversions
- 1 kilometre
- 1000 metres
- 1 metre
- 100 centimetres
- 1 centimetre
- 10 millimetres
- 1 kilogram
- 1000 grams
- 1 litre
- 1000 millilitres
- 1 hour
- 60 minutes
- 1 square metre
- 10,000 square centimetres
- 1 cubic metre
- 1000 litres
Area + Perimeter
- Rectangle area
- A = length × width
- Rectangle perimeter
- P = 2(length+width)
- Triangle area
- A = ½bh
- Parallelogram area
- A = bh
- Trapezium area
- A = ½(a+b)h
- Circle area
- A = πr²
- Circumference
- C = 2πr
- Composite area
- Split, calculate, combine
Volume + Surface Area
- Prism volume
- Base area × length
- Rectangular prism
- V = lwh
- Cylinder volume
- V = πr²h
- Cone volume
- V = ⅓πr²h
- Sphere volume
- V = ⁴⁄₃πr³
- Sphere surface
- SA = 4πr²
- Pyramid volume
- V = ⅓Bh
- Surface area
- Sum exposed face areas
Right-Triangle Ratios
SOH | CAH | TOA
Sine: opposite/hypotenuseCosine: adjacent/hypotenuseTangent: opposite/adjacent
Congruent vs Similar
Congruent
- Same shape
- Same size
Similar
- Same shape
- Proportional sides
Exact match vs scale
Geometry Method Picker
- Right triangle missing side→Pythagoras
- Right triangle side-angle→SOHCAHTOA
- Known opposite pair→Sine rule
- Three sides known→Cosine rule
- Similar figure lengths→Scale factor k
- Composite shape measure→Split then combine
Angle Rules
- Straight line
- Angles total 180°
- Around point
- Angles total 360°
- Triangle
- Interior sum 180°
- Quadrilateral
- Interior sum 360°
- n-gon interiors
- (n−2) × 180°
- Exterior angles
- One-per-vertex sum 360°
- Vertically opposite
- Equal angles
- Parallel co-interior
- Sum 180°
Trigonometry + Scale
- Pythagoras
- c² = a²+b²
- Sine ratio
- sin θ = opposite/hypotenuse
- Cosine ratio
- cos θ = adjacent/hypotenuse
- Tangent ratio
- tan θ = opposite/adjacent
- Sine rule
- a/sin A = b/sin B
- Cosine rule
- c² = a²+b²−2ab cos C
- Length scale
- Multiply by k
- Area scale
- Multiply by k²
- Volume scale
- Multiply by k³
Probability Connectors
AND often multiply; OR often add
Check independenceSubtract overlapUpdate without replacement
Independent vs Mutually Exclusive
Independent
- One leaves other unchanged
- Can occur together
Mutually exclusive
- Cannot occur together
- Intersection equals zero
No effect vs no overlap
Chance Method Picker
- Either event occurs→Addition rule
- Both independent occur→Multiplication rule
- No item replacement→Update next fraction
- Condition already known→Conditional probability
- Predict event count→Expected frequency
- Estimate from trials→Relative frequency
Probability Rules
- Single event
- Favourable ÷ total outcomes
- Complement
- P(not A) = 1−P(A)
- Either event
- Add, subtract overlap
- Mutually exclusive
- Add probabilities
- Independent both
- Multiply probabilities
- Conditional
- P(A∩B) ÷ P(B)
- Expected frequency
- Trials × probability
- Relative frequency
- Successes ÷ trials
- Without replacement
- Update counts and denominator
Mean vs Median
Mean
- Uses every value
- Outlier sensitive
Median
- Uses ordered middle
- Outlier resistant
Balance point vs middle
Data Summaries
- Mean
- Sum ÷ count
- Median
- Middle after ordering
- Mode
- Most frequent value
- Range
- Maximum − minimum
- Interquartile range
- Q3 − Q1
- Cumulative frequency
- Running frequency total
- Frequency density
- Frequency ÷ class width
- Scatterplot
- Correlation, not causation
Common Traps
Paper family scope
Intro through Paper J ≠ Not one universal level
Weight assumptions
Five official skill areas ≠ No universal percentages
Radius vs diameter
Radius reaches centre ≠ Diameter equals 2r
Square vs cubic units
Area uses units² ≠ Volume uses units³
Percent vs percentage points
Relative change uses percent ≠ Rate difference uses points
Negative square notation
(−3)² equals 9 ≠ −3² equals −9
Replacement changes probability
With replacement resets counts ≠ Without replacement updates counts
Correlation vs causation
Correlation shows association ≠ Causation needs stronger evidence
Last Minute
- 1.Choose your exact paper first
- 2.Country mappings are not universal
- 3.Use only paper-appropriate formulas
- 4.Estimate before exact calculation
- 5.Check denominators before adding fractions
- 6.Write units on every measure
- 7.Area uses square units
- 8.Volume uses cubic units
- 9.Label coordinates x then y
- 10.Median needs ordered data
- 11.Probability stays between zero and one
- 12.Check replacement before multiplying probabilities
- 13.Recheck copied graph scales
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