Year 8 Mock Exam Comprehensive Study Guide: Algebra, Geometry, and Measurement

Fundamentals of Algebra

  • Algebraic Terminology

    • In the algebraic expression 7a+6x\mathbf{7a + 6x}, the coefficient of xx is the numerical value combined with the variable, which is 66.

    • In the algebraic expression 7a−6\mathbf{7a - 6}, the constant term is the part of the expression that does not contain a variable, which is −6-6.

  • Algebraic Substitution

    • Given specific values for variables, an expression can be evaluated. For example, if x=2x = 2 and y=−3y = -3:

      • Expression A: 4x−2y4x - 2y is evaluated as 4(2)−2(−3)=8+6=144(2) - 2(-3) = 8 + 6 = 14.

      • Expression B: −3x2+4y-3x^2 + 4y is evaluated as −3(2)2+4(−3)=−3(4)−12=−12−12=−24-3(2)^2 + 4(-3) = -3(4) - 12 = -12 - 12 = -24.

  • Identifying the Highest Common Factor (HCF)

    • The HCF is the largest numerical and algebraic factor shared by multiple terms.

      • For the terms 63y63y and 184184, the HCF must be calculated based on prime factors of the numeric values.

      • For the terms 56x256x^2 and 64x64x, the HCF is 8x8x.

Algebraic Expansion, Factorisation, and Simplification

  • Expanding Algebraic Expressions

    • Expansion involves distributing the term outside the parentheses to every term inside.

    • 4(a+2)=4a+84(a + 2) = 4a + 8

    • −2(a−3)=−2a+6-2(a - 3) = -2a + 6

    • Complex expansion and simplification:

      • 3p(2+2p)+4(p+5)=6p+6p2+4p+20=6p2+10p+203p(2 + 2p) + 4(p + 5) = 6p + 6p^2 + 4p + 20 = 6p^2 + 10p + 20

      • 8(x+6)−3(x+2)=8x+48−3x−6=5x+428(x + 6) - 3(x + 2) = 8x + 48 - 3x - 6 = 5x + 42

      • 4+3(x+2)=4+3x+6=3x+104 + 3(x + 2) = 4 + 3x + 6 = 3x + 10

      • 7−3(2y+5)=7−6y−15=−6y−87 - 3(2y + 5) = 7 - 6y - 15 = -6y - 8

  • Factorising Algebraic Expressions

    • Factorisation is the reverse of expansion, identifying common factors to rebuild brackets.

    • 4a+8=4(a+2)4a + 8 = 4(a + 2)

    • 3x−9=3(x−3)3x - 9 = 3(x - 3)

    • 16pq+8q=8q(2p+1)16pq + 8q = 8q(2p + 1)

    • 10x2+25xy=5x(2x+5y)10x^2 + 25xy = 5x(2x + 5y)

    • 14ab2+21ab=7ab(2b+3)14ab^2 + 21ab = 7ab(2b + 3)

  • Simplification Procedures

    • Algebraic Fractions: Divide the numerator and denominator by their HCF.

      • 84x=2x\frac{8}{4x} = \frac{2}{x}

      • 3x6xy=12y\frac{3x}{6xy} = \frac{1}{2y}

    • Collecting Like Terms: Group terms with identical variable components.

      • 3x+7−2x−5=x+23x + 7 - 2x - 5 = x + 2

      • −4p2+7pq+p2+8qp=−3p2+15pq-4p^2 + 7pq + p^2 + 8qp = -3p^2 + 15pq

  • Expression Transcription

    • Converting verbal scenarios into algebraic expressions:

      • Two jumpers at $x\$x and four pants at $y\$y each results in the expression: 2x+4y2x + 4y.

Geometry of Circles and Sectors

  • Circumference and Perimeter Formulas

    • Full Circle: C = 2\times \text{\pi} \times r (where rr is the radius) or C = \text{\pi} \times d (where dd is the diameter).

    • Arc Length of a Sector: This is the length of the curved portion, calculated as L = \frac{\theta}{360} \times 2 \times \text{\pi} \times r.

    • Perimeter of a Sector: This includes the arc length plus the two straight edges (radii), calculated as P = \frac{\theta}{360} \times 2 \times \text{\pi} \times r + 2r.

  • Area Formulas for Circular Shapes

    • Full Circle: A = \text{\pi} \times r^2

    • Sector: A = \frac{\theta}{360} \times \text{\pi} \times r^2

    • Semicircle (Half Circle): A = \frac{1}{2} \times \text{\pi} \times r^2

    • Quadrant (Quarter Circle): A = \frac{1}{4} \times \text{\pi} \times r^2

    • Three-Quarters of a Circle: A = \frac{3}{4} \times \text{\pi} \times r^2

  • Area Calculation Examples

    • A circle with a diameter of 10 cm10\,cm (radius r=5 cmr = 5\,cm): A = \text{\pi} \times 5^2 = 25\text{\pi}\,cm^2.

    • A circle with a radius of 8 cm8\,cm: A = \text{\pi} \times 8^2 = 64\text{\pi}\,cm^2.

  • Specific Circumference and Perimeter Cases

    • Case A: A full circle with a radius of 8 cm8\,cm.

    • Case B: A semicircle with a diameter of 4 cm4\,cm.

    • Case C: A sector with a radius of 7 cm7\,cm and an internal angle of 110∘110^{\circ}.

Measurement Conversions: Area and Volume

  • Area Unit Conversions

    • Relationship: 1 cm2=102 mm2=100 mm21\,cm^2 = 10^2\,mm^2 = 100\,mm^2.

    • Relationship: 1 m2=1002 cm2=10,000 cm21\,m^2 = 100^2\,cm^2 = 10,000\,cm^2.

    • Relationship: 1 km2=10002 m2=1,000,000 m21\,km^2 = 1000^2\,m^2 = 1,000,000\,m^2.

    • Conversions:

      • 10 cm210\,cm^2 to mm2mm^2: 10×100=1000 mm210 \times 100 = 1000\,mm^2.

      • 180 m2180\,m^2 to cm2cm^2: 180×10,000=1,800,000 cm2180 \times 10,000 = 1,800,000\,cm^2.

      • 750 cm2750\,cm^2 to m2m^2: 750×110,000=0.075 m2750 \times \frac{1}{10,000} = 0.075\,m^2.

  • Volume Unit Conversions

    • Relationship: 1 cm3=103 mm3=1000 mm31\,cm^3 = 10^3\,mm^3 = 1000\,mm^3.

    • Relationship: 1 m3=1003 cm3=1,000,000 cm31\,m^3 = 100^3\,cm^3 = 1,000,000\,cm^3.

    • Relationship: 1 km3=10003 m3=1,000,000,000 m31\,km^3 = 1000^3\,m^3 = 1,000,000,000\,m^3.

    • Conversions:

      • 650,000 cm3650,000\,cm^3 to m3m^3: 650,000×11,000,000=0.65 m3650,000 \times \frac{1}{1,000,000} = 0.65\,m^3.

      • 540 km3540\,km^3 to m3m^3: 540×1,000,000,000 m3540 \times 1,000,000,000\,m^3.

Volume and Capacity Calculations

  • Geometric Volume Formulas

    • Cube: V=s3V = s^3 (where ss is side length).

    • Rectangular Prism: V=L×W×HV = L \times W \times H.

    • Cylinder: V = \text{\pi} \times r^2 \times h.

    • Triangular Prism: V=Area of Triangle×Length=(12×b×h)×LV = \text{Area of Triangle} \times \text{Length} = (\frac{1}{2} \times b \times h) \times L.

  • Volume Problems

    • Cube: Side length = 50 cm50\,cm. V=503=125,000 cm3V = 50^3 = 125,000\,cm^3.

    • Rectangular Prism: 10 cm×8 cm×4 cm10\,cm \times 8\,cm \times 4\,cm. V=320 cm3V = 320\,cm^3.

    • Cylinder: Radius = 14 cm14\,cm, Height = 30 cm30\,cm. V = \text{\pi} \times 14^2 \times 30\,cm^3.

    • Triangular Prism: Triangle base = 8 cm8\,cm, Triangle height = 11 cm11\,cm, Prism length = 13 cm13\,cm. V=(12×8×11)×13=44×13=572 cm3V = (\frac{1}{2} \times 8 \times 11) \times 13 = 44 \times 13 = 572\,cm^3.

  • Capacity Conversions

    • 1 cm3=1 mL1\,cm^3 = 1\,mL

    • 1000 mL=1 L1000\,mL = 1\,L

    • To find capacity in mLmL and LL, apply these ratios to the volumes calculated above.

Properties of Geometric Figures and Angles

  • Angle Relationships

    • Complementary Angles: Angles that sum to 90∘90^{\circ}.

    • Supplementary Angles: Angles that sum to 180∘180^{\circ} (angles on a straight line).

    • Angles in a Triangle: The interior angles of any triangle sum to 180∘180^{\circ}.

    • Angles around a Point: The sum of angles around a single vertex is 360∘360^{\circ}.

    • Example Calculations:

      • In a right-angled scenario with one angle given as 35∘35^{\circ}, the unknown is 90−35=55∘90 - 35 = 55^{\circ}.

      • In a triangle with angles 40∘40^{\circ} and 60∘60^{\circ}, the third angle x=180−(40+60)=80∘x = 180 - (40 + 60) = 80^{\circ}.

      • Linear pair: Given 130∘130^{\circ}, the adjacent unknown is 180−130=50∘180 - 130 = 50^{\circ}.

      • Revolution: Given 85∘85^{\circ} and 35∘35^{\circ}, other unknowns must satisfy the 360∘360^{\circ} total.

  • Identifying Polygons and Triangles

    • Shapes are identified by side lengths, such as an isosceles triangle (5 cm,5 cm,4 cm5\,cm, 5\,cm, 4\,cm) or an equilateral triangle (5 cm,5 cm,5 cm5\,cm, 5\,cm, 5\,cm).

    • Quadrialterals are classified by specific properties:

      • All sides equal length: Square, Rhombus.

      • Two pairs of sides with equal length: Rectangle, Parallelogram, Kite.

      • Two pairs of parallel lines: Square, Rectangle, Rhombus, Parallelogram.

      • One pair of parallel lines: Trapezium.

      • Opposite angles are equal: Parallelogram, Rhombus, Square, Rectangle.

      • Only one pair of opposite equal angles: Kite.