📘 Grade 9 → Unit 1: Further on Sets · Unit 2: The Number System
Sets (operations, Venn diagrams), Natural numbers, Integers, Rational, Irrational, Real numbers, Exponents, Radicals, Scientific notation.
📗 Grade 10 → (builds on number system)
Number properties used in polynomial, exponential and logarithmic functions.
English: Real numbers = rational + irrational. √2, π, e are irrational. Set operations: ∪ (union), ∩ (intersection), ′ (complement). Scientific notation: a × 10ⁿ where 1 ≤ |a| < 10.
Set operations፡
• A ∪ B = union (ሁለቱም ውስጥ ያሉት ሁሉ)
• A ∩ B = intersection (በሁለቱም ያሉት ብቻ)
• A′ = complement
Scientific notation: 3.2 × 10⁴ = 32000።
Rational = can be written as p/q. Irrational = cannot (√2, π). All integers are rational.
Operations on sets + distinguishing rational vs irrational በentrance ላይ ብዙ ጊዜ ይመጣሉ!
2. Equations እና Inequalities
📘 Grade 9 → Unit 3: Solving Equations · Unit 4: Solving Inequalities
Linear equations, systems of linear equations (2 variables), absolute value equations, quadratic equations. Linear, absolute value and quadratic inequalities.
📗 Grade 10 → Unit 2: Polynomial Functions
Polynomial equations, factor theorem, zeros of polynomials.
📙 Grade 11 → Unit 2: Rational Expressions and Rational Functions
Equations involving rational expressions; solving rational equations.
English: Linear: ax + b = 0 → one solution. Quadratic: ax² + bx + c = 0 → discriminant D = b² − 4ac.
D > 0 → two real roots; D = 0 → one root; D < 0 → no real roots.
For inequalities, flip the sign when multiplying or dividing by a negative number.
x = −b ± √(b² − 4ac)2a
በጣም ግልፅ አማርኛ (ከEnglish ጋር):
Linear equation = አንድ መፍትሔ (solution) አለው። Quadratic equation (ax² + bx + c = 0) → discriminant D = b² − 4ac ይመልከቱ፡
• D > 0 → ሁለት የተለያዩ real roots
• D = 0 → አንድ (repeated) root
• D < 0 → real root የለም
Inequality ሲፈቱ በnegative number ካባዙ ወይም ከከፈሉ ምልክቱን ይገልብጡ (flip the inequality sign)።
Discriminant: “Positive → Two, Zero → One, Negative → None”.
Always check extraneous solutions when solving equations with absolute value or radicals.
Quadratic formula, discriminant እና systems of linear equations በentrance ላይ በጣም ብዙ ጊዜ ይመጣሉ!
3. Relations እና Functions
📗 Grade 10 → Unit 1: Relations and Functions · Unit 2: Polynomial · Unit 3: Exponential & Logarithmic
Definition of relation vs function, domain, range, types of functions, graphs. Polynomial, exponential and logarithmic functions.
📙 Grade 11 → Unit 1: Relations and Functions · Unit 2: Rational Functions
Deeper study of functions, inverse functions, rational functions and their graphs.
English: A function assigns exactly one output to each input. Vertical line test.
Domain = possible x-values; Range = possible y-values.
Exponential: y = aˣ. Logarithmic: y = loga x is the inverse of exponential.
loga(xy) = loga x + loga y.
Function = ለእያንዳንዱ input አንድ ብቻ output ይሰጣል። Vertical line test በgraph ላይ ይጠቀሙ። Domain = x ሊወስደው የሚችለው ዋጋ። Range = y የሚወጣው ዋጋ።
Exponential function: y = aˣ (a > 0, a ≠ 1)። Logarithmic function: y = loga x — የexponential inverse ነው።
ዋና ባህሪያት፡ log(xy) = log x + log y፣ log(xⁿ) = n log x።
“Log turns multiplication into addition, division into subtraction, powers into multiplication.”
📘 Grade 9 → Unit 5: Introduction to Trigonometry
Trigonometric ratios in right-angled triangles (sin, cos, tan), solving right triangles, height and distance problems.
📗 Grade 10 → Unit 4: Trigonometric Functions
Trigonometric functions of any angle, unit circle, graphs of sin, cos, tan, identities.
📙 Grade 11 → (applications & further identities)
Trig identities and equations used in more advanced problems.
English: In a right triangle:
sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent
Pythagorean identity: sin²θ + cos²θ = 1. Unit circle: coordinates (cos θ, sin θ).
📘 Grade 9 → Unit 6: Regular Polygons · Unit 7: Congruency and Similarity
Interior/exterior angles of polygons, properties of regular polygons, congruence of triangles, similarity theorems.
📗 Grade 10 → Unit 5: Circles · Unit 6: Solid Figures
Circle theorems (angles, chords, tangents), arc length, sector area. Surface area and volume of prisms, cylinders, pyramids, cones, spheres.
English: Sum of interior angles of an n-gon = (n − 2) × 180°.
Congruent triangles: SSS, SAS, ASA, RHS. Similar triangles: corresponding angles equal, sides proportional.
Circle: angle at centre = 2 × angle at circumference.
📘 Grade 9 → Unit 8: Vectors in two dimensions
Vector representation, addition, subtraction, scalar multiplication, magnitude.
📙 Grade 11 → Unit 3: Matrices · Unit 4: Determinants · Unit 5: Vectors
Matrix operations, determinants, inverse of a matrix, vectors in plane and space, scalar product.
English: Vector has magnitude and direction. a + b is the diagonal of the parallelogram.
Matrix multiplication is not commutative (AB ≠ BA in general).
Determinant of 2×2 matrix [a b ; c d] = ad − bc. Inverse exists only if det ≠ 0.
|a| = √(x² + y²) | det [a b ; c d] = ad − bc
በጣም ግልፅ አማርኛ (ከEnglish ጋር):
Vector = magnitude + direction አለው። ሁለት vectors መደመር = parallelogram ዲያጎናል። Matrix multiplication commutative አይደለም (AB ≠ BA)።
2×2 matrix determinant = ad − bc። Inverse የሚኖረው det ≠ 0 ከሆነ ብቻ ነው።
For 2×2 inverse: swap a and d, change signs of b and c, then divide by determinant.
📗 Grade 10 → Unit 7: Coordinate Geometry
Distance formula, midpoint, equation of a straight line (slope-intercept, point-slope), parallel and perpendicular lines.
📙 Grade 11 → Unit 6: Transformations of the plane
Translations, reflections, rotations, enlargements on the coordinate plane.
English: Distance between (x₁, y₁) and (x₂, y₂) = √[(x₂ − x₁)² + (y₂ − y₁)²].
Slope m = y₂ − y₁x₂ − x₁.
Parallel lines → same slope. Perpendicular → product of slopes = −1.
Equation: y − y₁ = m(x − x₁).
d = √[(x₂ − x₁)² + (y₂ − y₁)²] | M = (x₁ + x₂2, y₁ + y₂2)
Distance, midpoint, slope እና equation of a line በentrance ላይ በጣም ብዙ ጊዜ ይመጣሉ!
8. Statistics እና Probability
📘 Grade 9 → Unit 9: Statistics and Probability
Data presentation (tables, charts), measures of central tendency (mean, median, mode), basic probability.
📙 Grade 11 → Unit 7: Statistics · Unit 8: Probability
More advanced statistics (variance, standard deviation), conditional probability, independent events, tree diagrams.
English: Mean = sum ÷ n. Median = middle value. Mode = most frequent.
Probability of event A: P(A) = number of favourable outcomes ÷ total outcomes.
Independent events: P(A and B) = P(A) × P(B).
P(A) = n(A)n(S) | P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
በጣም ግልፅ አማርኛ (ከEnglish ጋር):
Mean = ድምር ÷ ብዛት። Median = መሃከለኛው ዋጋ። Mode = በብዛት የሚደገመው። Probability P(A) = favourable ÷ total።
Independent events፡ P(A and B) = P(A) × P(B)።
“AND → multiply, OR → add (then subtract intersection if not mutually exclusive).”
Mean/median/mode, basic probability rules እና tree diagrams በentrance ላይ ብዙ ጊዜ ይመጣሉ!
9. Sequences, Series እና Limits
📕 Grade 12 → Unit 1: Sequences and Series · Unit 2: Introduction to Limits and Continuity
Arithmetic and geometric sequences/series, sigma notation, infinite series, limits of sequences and functions, continuity.
English: Arithmetic: aₙ = a + (n − 1)d. Sum Sₙ = n2 [2a + (n − 1)d].
Geometric: aₙ = arⁿ⁻¹. Sum Sₙ = a(rⁿ − 1)/(r − 1) (r ≠ 1).
Infinite geometric |r| < 1 → S∞ = a1 − r.
Limit: as x → a, f(x) approaches L.
aₙ = a + (n − 1)d | aₙ = arⁿ⁻¹ | S∞ = a1 − r (|r| < 1)
በጣም ግልፅ አማርኛ (ከEnglish ጋር):
Arithmetic sequence: እያንዳንዱ ቀጣይ term በተመሳሳይ d (common difference) ይጨምራል።
aₙ = a + (n − 1)d። Sₙ = n2 × [2a + (n − 1)d]።
AP → common difference (add). GP → common ratio (multiply). For infinite GP, only converges if |r| < 1.
AP/GP formulas, sum of series እና basic limits በentrance (especially natural science) ላይ በጣም አስፈላጊ ናቸው!
10. Differential እና Integral Calculus
📕 Grade 12 → Unit 3: Differential Calculus · Unit 4: Applications · Unit 5: Integral Calculus
Derivatives (power rule, product, quotient, chain), applications (max/min, rates of change). Integration as reverse of differentiation, definite integrals, area under curve.
English: Derivative = slope of tangent = rate of change.
Power rule: d/dx (xⁿ) = n · xⁿ⁻¹.
Integral is the reverse process.
Definite integral ∫ from a to b of f(x) dx gives signed area between curve and x-axis.
ddx (xⁿ) = n · xⁿ⁻¹ | ∫ xⁿ dx = xⁿ⁺¹n + 1 + C (n ≠ −1)
በጣም ግልፅ አማርኛ (ከEnglish ጋር):
Derivative = የtangent line slope = rate of change።
Power rule፡ d/dx (xⁿ) = n · xⁿ⁻¹። Integral = derivative ተቃራኒ ሂደት ነው። Definite integral = ከ a እስከ b ያለው area (signed)።
“Derivative brings the power down and subtracts 1. Integral adds 1 to the power and divides by the new power.”
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