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Study Notes & Materials

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Prep Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 Grade 9 Grade 10 Grade 11 Grade 12
Prep / Foundation — Study Notes
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123
Counting 1-20
Count objects one by one. Numbers go in order 1 to 20.
Key Facts
After 7 comes 8. Before 5 is 4.
5 is more than 3 (bigger number)
Count: 1,2,3,4,5 stars = 5 stars
Worked Examples
What comes after 12? → 13
Which is more: 4 or 7? → 7
Count 8 dots: 1,2,3,4,5,6,7,8
Shp
Basic Shapes
Shapes have different numbers of sides and corners.
Key Facts
Circle: 0 sides, 0 corners
Square: 4 equal sides, 4 corners
Triangle: 3 sides, 3 corners
Worked Examples
Pizza = circle
Book cover = rectangle
Yield sign = triangle
+5
Adding to 5
Adding means putting numbers together.
Key Facts
2+1=3
3+2=5
4+1=5
0+5=5
Worked Examples
2 cats + 3 cats = 5
1+4=5
3+0=3
Grade 1 — Study Notes
← Back to Grade Select
+10
Adding to 10
Single-digit numbers that make totals up to 10.
Key Facts
5+5=10 (doubles)
3+7=10 (make-10 pairs)
If 3+4=7 then 4+3=7
Worked Examples
4+6=10
8+2=10
7+__=10 → 3
-
Subtracting
Taking away. The answer is the difference.
Key Facts
10-3=7
8-5=3
Subtraction = opposite of addition
Worked Examples
9-4=5
10-7=3
6 apples eat 2 → 4 left
10s
Tens and Ones
Numbers like 23 = 2 tens + 3 ones.
Key Facts
23 = 20+3
45 = 4 tens + 5 ones
Tens digit is LEFT, ones is RIGHT
Worked Examples
36 → 3 tens 6 ones
50 → 5 tens 0 ones
19 → 1 ten 9 ones
Grade 2 — Study Notes
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+100
Adding to 100
Adding two-digit numbers, including carrying.
Key Facts
Add tens first: 34+25=50+9=59
Carrying: 47+36=83
Round to estimate: 47+36 ≈ 50+40=90
Worked Examples
38+45=83
67+24=91
50+50=100
Grp
Groups
Multiplication is adding equal groups.
Key Facts
3 groups of 4 = 4+4+4=12=3×4
2×5=10
Skip count by 2s, 5s, 10s
Worked Examples
5 bags × 3 apples = 15
2×6=12
4×5=20
cm
Measuring
Standard units for length, mass and capacity.
Key Facts
10mm=1cm, 100cm=1m, 1000m=1km
1000g=1kg
1000mL=1L
Worked Examples
Pencil ≈ 15cm
Door ≈ 2m
Bottle = 600mL
Grade 3 — Study Notes
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×
Multiplication
Repeated addition. 3×4 = three groups of four.
Key Facts
6×7=42, 7×8=56, 8×9=72 (learn these!)
Any number × 0=0, × 1=itself
3×4=4×3 (commutative law)
Worked Examples
5×6=30
4×4=16
6 eggs × 4 boxes = 24
½
Fractions
Part of a whole. Bottom = total parts, top = your parts.
Key Facts
½ = 1 out of 2
¾ > ½
1/3 of 12 = 12÷3 = 4
Simplify: 4/8 = ½
Worked Examples
½ of 10 = 5
¼ of 20 = 5
1/2+1/4 = 3/4
hr
Time
Reading clocks and calculating time.
Key Facts
60 min = 1 hr, 24 hr = 1 day
Half past = 30 min past
Quarter past = 15 min past
Worked Examples
2:30 = half past two
3:15 = quarter past three
4:45 = quarter to five
Grade 4 — Study Notes
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÷
Division
Splitting into equal groups. Inverse of multiplication.
Key Facts
48÷6=8 (because 6×8=48)
Dividend ÷ Divisor = Quotient
Remainder: 17÷5=3 r2
Worked Examples
36÷9=4
100÷4=25
63÷7=9
.
Decimals
Parts of a whole using base-10 place value.
Key Facts
0.1 = 1/10, 0.01 = 1/100
3.47 = 3 ones + 4 tenths + 7 hundredths
Compare: 0.6 > 0.59
Worked Examples
0.5+0.3=0.8
1.25+2.4=3.65
3.7-1.2=2.5
P=
Perimeter
Total distance around the outside of a shape.
Key Facts
Rectangle: P=2(l+w)
Square: P=4s
Triangle: P=a+b+c
Worked Examples
Rect 8×5: P=26cm
Square s=7: P=28cm
Triangle 3,4,5: P=12cm
Grade 5 — Study Notes
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%
Percentages
Per cent means out of 100.
Key Facts
% to decimal: ÷100 (25%=0.25)
10% of X = X÷10
Finding %: (part÷whole)×100
50%=½, 25%=¼, 75%=¾
Worked Examples
20% of 80=16
15% of 60=9
$40 with 25% off = $30
A=
Area
Space inside a 2D shape, in square units.
Key Facts
Rectangle: A=l×w
Square: A=s²
Triangle: A=½×base×height
Worked Examples
Room 5×4m: A=20m²
Square 6cm: A=36cm²
Triangle base 8, h 5: A=20cm²
bar
Graphs
Displaying data visually. Different types for different data.
Key Facts
Mean=sum÷count
Median=middle value (sorted)
Mode=most frequent
Range=max-min
Worked Examples
Mean of 4,6,8,10 = 7
Median of 3,5,7,9,11 = 7
Range of 2,5,8,11 = 9
Grade 6 — Study Notes
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r:
Ratios
Comparing quantities. Written a:b.
Key Facts
Simplify 12:8 = 3:2 (÷HCF 4)
Share $20 in 3:2: 5 parts, $4 each → $12:$8
Equivalent: 1:2=2:4=3:6
Worked Examples
3:1 blue to yellow = green
200mL in 3:2 → 120+80
Scale 1:50000, 2cm=1km
-n
Negative Numbers
Less than zero. Left of 0 on number line.
Key Facts
5+(-3)=2
5-(-3)=8
Same signs multiplied/divided → positive
Different signs → negative
Worked Examples
-3+7=4
-4×-2=8
6-(-5)=11
-12÷3=-4
P
Probability
How likely an event is. 0=impossible, 1=certain.
Key Facts
P(event)=favourable÷total
P(not A)=1-P(A)
P(A and B)=P(A)×P(B) if independent
Worked Examples
Die P(even)=3/6=½
Coin P(heads)=½
P(not rain)=1-0.3=0.7
Grade 7 — Study Notes
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x=
Algebra Basics
Pronumerals represent unknowns. Solve with inverse operations.
Key Facts
x+5=12 → x=7
3x+2x=5x (like terms)
2(x+3)=2x+6 (expand)
y=3x+1, x=4 → y=13 (substitute)
Worked Examples
2x=14 → x=7
4a+3b-2a=2a+3b
3(2x-5)=6x-15
x+7=12 → x=5
/
Linear Patterns
Constant difference d. nth term: T(n)=a+(n-1)d.
Key Facts
a=first term, d=common difference
Positive d=increasing, negative d=decreasing
Graph = straight line
Worked Examples
3,7,11,15: d=4, T(n)=4n-1
T(10)=4(10)-1=39
20,15,10: d=-5, T(n)=25-5n
ang
Angles
Types of angles and key angle rules.
Key Facts
Triangle angles sum to 180°
Angles on straight line sum to 180°
Vertically opposite = equal
Co-interior angles sum to 180°
Worked Examples
50°+70°+?=180° → 60°
120°+?=180° → 60°
35°+?=90° → 55°
Grade 8 — Study Notes
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=
Linear Equations
Solve by doing the same to both sides.
Key Facts
Goal: isolate the variable
Two-step: 2x+3=11 → 2x=8 → x=4
Variables both sides: 3x+2=x+10 → x=4
Worked Examples
5x-7=18 → x=5
3(x+2)=21 → x=5
2x+5=x+9 → x=4
c²
Pythagoras
Right triangles: a²+b²=c². c is hypotenuse.
Key Facts
c=√(a²+b²)
Shorter side: a=√(c²-b²)
Triples: (3,4,5),(5,12,13),(8,15,17)
Worked Examples
Legs 3,4 → c=5
Legs 5,12 → c=13
Hyp 10, leg 6 → other=8
avg
Statistics
Collecting and analysing data.
Key Facts
Mean=sum÷count
Median=middle when sorted
Mode=most frequent
Range=max-min
Worked Examples
3,5,5,7,10: mean=6, median=5, mode=5
IQR=Q3-Q1
Outlier=value far from rest
Grade 9 — Study Notes
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x²
Quadratics
Has x². Graph is a parabola.
Key Facts
Expand (x+3)(x+2)=x²+5x+6 (FOIL)
Factorise x²+5x+6=(x+2)(x+3)
Difference of squares: x²-9=(x+3)(x-3)
Worked Examples
(x+4)(x-1)=x²+3x-4
x²-7x+12=(x-3)(x-4)
x²-16=(x+4)(x-4)
sin
Trigonometry
SOH CAH TOA for right triangles.
Key Facts
sin=Opp/Hyp, cos=Adj/Hyp, tan=Opp/Adj
Find side: multiply by trig ratio
Find angle: use inverse sin⁻¹, cos⁻¹, tan⁻¹
Worked Examples
Hyp=10, 30°: Opp=10×sin30°=5
Opp=7, Hyp=10: θ=sin⁻¹(0.7)=44.4°
Adj=8, 40°: Opp=8×tan40°=6.7
~
Similarity
Same shape, different size. Sides in same ratio.
Key Facts
Scale factor k=new÷original
Area scales by k²
Volume scales by k³
Worked Examples
3,4,5 ~ 6,8,10: k=2
Scale factor 3: area factor 9
Map 1:50000, 2cm=1000m
Grade 10 — Study Notes
← Back to Grade Select
=
Quadratic Equations
Solve ax²+bx+c=0.
Key Facts
Formula: x=(-b±√(b²-4ac))/2a
Discriminant Δ=b²-4ac: >0 two solutions, =0 one, <0 none
Completing square: x²+6x+5=(x+3)²-4
Worked Examples
x²-5x+6=0 → (x-2)(x-3)=0 → x=2 or 3
2x²+x-3=0 → x=1 or x=-1.5
Δ of x²+2x+5: 4-20=-16 < 0, no real roots
cos
Trig Functions
Extend trig to any angle. Graphs are waves.
Key Facts
sin,cos repeat every 360°
tan repeats every 180°
CAST rule: which ratios positive in each quadrant
Sine rule: a/sinA=b/sinB
Worked Examples
sin(150°)=sin(30°)=0.5
cos(120°)=-0.5
45°=π/4 radians
SA
Surface Area
Total area of all faces of a 3D solid.
Key Facts
Box: SA=2(lw+lh+wh)
Cylinder: SA=2πr²+2πrh
Sphere: SA=4πr²
Worked Examples
Box 4×3×2: SA=52cm²
Cylinder r=3,h=10: SA≈245cm²
Sphere r=5: SA≈314cm²
Grade 11 — Study Notes
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f(x)
Functions
One output for each input.
Key Facts
Domain=valid inputs, Range=outputs
f(3) means substitute x=3
f(x)+k shifts up k
f(x-h) shifts right h
Worked Examples
f(x)=2x+1: f(3)=7
g(x)=x²: range y≥0
h(x)=√x: domain x≥0
log
Logarithms
Inverse of exponential. logₐ(x)=y means aʸ=x.
Key Facts
log(AB)=logA+logB
log(A/B)=logA-logB
log(Aⁿ)=n logA
logₐ(a)=1, logₐ(1)=0
Worked Examples
log₂(8)=3 (2³=8)
log(100)=2
log(6)=log2+log3
int
Intro Calculus
Rates of change and accumulation.
Key Facts
Derivative=gradient of tangent
Power rule: d/dx(xⁿ)=nxⁿ⁻¹
Integral reverses differentiation
∫xⁿdx=xⁿ⁺¹/(n+1)+C
Worked Examples
d/dx(x³)=3x²
d/dx(5x²+3x)=10x+3
∫2x dx=x²+C
Grade 12 — Study Notes
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f'
Differentiation
Instantaneous rate of change.
Key Facts
Power rule: d/dx(xⁿ)=nxⁿ⁻¹
Chain rule: d/dx[f(g(x))]=f'(g(x))×g'(x)
Product rule: d/dx[uv]=u'v+uv'
Stationary: set f'(x)=0
Worked Examples
f(x)=3x⁴: f'(x)=12x³
f(x)=sin(2x): f'(x)=2cos(2x)
y=x²-4x: dy/dx=2x-4=0 → x=2
Int
Integration
Reverse of differentiation. Area under curve.
Key Facts
∫xⁿdx=xⁿ⁺¹/(n+1)+C
∫sinx dx=-cosx+C
∫eˣdx=eˣ+C
Area=∫ₐᵇf(x)dx
Worked Examples
∫3x²dx=x³+C
∫₀²(x²+1)dx=14/3
∫cosx dx=sinx+C
σ
Statistics (Advanced)
Probability distributions, hypothesis testing.
Key Facts
Normal: bell curve, mean μ, std dev σ
68-95-99.7 rule
z=(x-μ)/σ
Binomial: P(X=k)=C(n,k)pᵏ(1-p)ⁿ⁻ᵏ
Worked Examples
z=(75-70)/5=1: 1 std dev above mean
P(X=2) from B(5,0.3)=0.309
r=0.95: strong positive correlation
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