Grade 9 Mathematics Ethiopian Student Textbook Study Notes - Federal Democratic Republic of Ethiopia

Page 1

Mathematics Student’s Textbook Grade 9. Published by the Federal Democratic Republic of Ethiopia Ministry of Education.

Page 2

School Property Notice and Textbook Care Instructions: This textbook is the property of your school. Proper care should be taken to ensure it is not damaged or lost. Ten specific ideas for book maintenance are provided:

1. Cover the book with protective materials (plastic, old newspapers, or magazines).

2. Store strictly in a clean, dry place.

3. Ensure hands are clean before use.

4. Prohibit writing on the cover or inside pages.

5. Use paper or cardboard as bookmarks.

6. Never tear or cut out pictures or pages.

7. Repair torn pages immediately with paste or tape.

8. Pack carefully in your school bag.

9. Handle with care when passing to others.

10. Initial use of a new book: Lay it on its back, open only a few pages at a time, and press lightly along the bound edge to preserve the cover condition.

Page 4

Authors:

• Gurju Awgichew Zergaw (PhD)

• Adem Mohammed Ahmed (PhD)

Editors:

• Mohammed Yiha Dawud (PhD): Content Editor

• Akalu Chaka Mekuria (MA): Curriculum Editor

• Endalfer Melese Moges (MA): Language Editor

Supporting Staff:

• Illustrator: Bahiru Chane Tamiru (MSc)

• Designer: Aknaw H/mariam Habte (MSc)

• Evaluators: Matebie Alemayehu Wasihun (MED), Mustefa Kedir Edao (BED), Dawit Ayalneh Tebkew (MSc), Tesfaye Sileshi Chala (MA)

Affiliations: Federal Democratic Republic of Ethiopia Ministry of Education and Hawassa University.

Page 5

Publication Information:

• Project: General Education Quality Improvement Program for Equity (GEQIP-E).

• Supporters: World Bank, UK’s FCDO (formerly DFID), Finland Ministry for Foreign Affairs, Royal Norwegian Embassy, UNICEF, Global Partnership for Education (GPE), and Danish Ministry of Foreign Affairs via a Multi-Donor Trust Fund.

• Copyright: © 2023 by the Ministry of Education. Rights reserved under Proclamation No. 410/2004 - Copyright and Neighboring Rights Protection.

• Collaborators: Special thanks to Hawassa University, Addis Ababa University, Bahir Dar University, Jimma University, and the JICA MUST project.

• Date: August 2023.

• Printer: Gravity Group Ind LLC, Sharjah, United Arab Emirates.

• Contract No: M OE/GEQIP-E/LICB/G-01/23.

• ISBN: 978-99990-0-024-6.

Page 6

Welcoming Message: Grade 9 marks the first grade of secondary education, describing it as a "golden stage" in an academic career and a transition from primary school Mathematics.

Textbook Structure: The book contains 9 units:

1. Further on Sets

2. The Number System

3. Solving Equations

4. Solving Inequalities

5. Introduction to Trigonometry

6. Regular Polygons

7. Congruency and Similarity

8. Vectors in Two Dimensions

9. Statistics and Probability

Lesson Components (ADEE):

• Activity: Revises prior knowledge and introduces new topics.

• Definition/Theorem/Note: Presents and explains new concepts.

• Example and Solution: Teacher-led or self-study examples to improve understanding.

• Exercise: Practice problems to be solved individually, in pairs, or in groups.

Page 7

Symbols and Methodology: A specific symbol is used to indicate time for reflection on previous learning or steps to be performed mentally to connect past and future lessons.

Page 8-10

Table of Contents Summary:

• Unit 1: Sets and Elements (p.2), Set Description (p.4), Notion of Sets (p.7), Operations (p.12), Application (p.20).

• Unit 2: Natural Numbers/Integers revision (p.28), Rational Numbers (p.45), Irrational Numbers (p.51), Real Numbers (p.61), Application (p.95).

• Unit 3: Linear Equations Revision (p.106), Systems of Linear Equations (p.109), Non-linear Equations (p.123), Applications (p.143).

• Unit 4: Linear Inequalities (p.152), Systems of Inequalities (p.157), Absolute Value Inequalities (p.166), Quadratic Inequalities (p.171), Applications (p.176).

• Unit 5: Right-angled triangles (p.186), Ratios (p.190).

• Unit 6: Interior Angles (p.208), Exterior Angles (p.216), Regular Polygon measures (p.221), Properties (p.224).

• Unit 7: Congruency (p.240), Similar Figures (p.245), Theorems (p.249), Perimeters/Areas ratio (p.259-262), Applications (p.267).

• Unit 8: Vector/Scalar quantities (p.279), Representation (p.282), Operations (p.288), Position Vector (p.299), Applications (p.303).

• Unit 9: Statistical Data (p.313), Probability (p.346).

Page 12-13

Unit 1: Further on SetsUnit Outcomes: Fact explanation, set description, operation definition, Venn diagram usage, and practical applications.

1.1 Sets and Elements:

• Definition: A set is a collection of well-defined objects or elements.

• Well-defined: An object given must be determinable as either in or out of the set.

• Key Terms: Subset, symmetric difference, set description, Venn diagram, empty set, intersection, absolute complement, union, proper subset.

Page 14

Notation:

• If $a$ is an element of set $A$, write $a \in A$ (read as "$a$ belongs to $A$").

• The symbol $\in$ is the Greek epsilon.

• If $b$ is not an element, write $b \notin A$.

• Convention: Sets are denoted by capital letters ($A, B, C$), elements by small letters ($a, b, c$).

Example 2: If $A$ is the set of positive even numbers:

• $4 \in A$

• $5 \notin A$

• $-2 \notin A$

• $0 \notin A$ (Note: 0 is not positive).

Page 15

1.2 Set Description Methods:

1. Verbal Method (Statement form): Described in ordinary English. Example: "The set of whole numbers between 1 and 20."

2. Listing Methods:

   • a) Complete Listing (Roster Method): All elements are listed within braces ${ }$ and separated by commas. Example: {$a, e, i, o, u$}.

Page 16

Listing Methods (Continued):

• b) Partial Listing Method: Used for large sets where a pattern is clear. Example: The set of natural numbers less than 100 is $A = \{1, 2, 3, \dots, 99\}$. The set of whole numbers is $\mathbb{W} = \{0, 1, 2, 3, \dots \}$.

1. Set-Builder Method (Method of defining property): Writing the condition satisfied by members. Format: $\{x | x \text{ satisfies condition } P\}$.

Page 17

Example 4:

• Set $A = \{1, 2, 3, \dots, 10\}$ in set-builder: $A = \{x | x \in \mathbb{N} \text{ and } x \lt 11\}$.

• Set $B = \{0, 2, 4, \dots \}$ in set-builder: $B = \{2x | x \in \mathbb{W}\}$.

Standard Notations:

• $\mathbb{N} = \{1, 2, 3, \dots \}$ (Natural numbers).

• $\mathbb{W} = \{0, 1, 2, 3, \dots \}$ (Whole numbers).

• $\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots \}$ (Integers).

Page 18-20

1.3 The Notion of Sets:

• Empty Set: Contains no elements. Denoted by $\emptyset$ or ${ }$. Also called null or void set.

• Finite Set: Contains a definite (limited) number of elements.

• Infinite Set: Not finite (unlimited elements).

• Equal Sets: Sets with exactly the same elements ($A = B$). Identity matters, not order.

• Equivalent Sets: Finite sets $A$ and $B$ with the same number of elements ($n(A) = n(B)$). Denoted $A \leftrightarrow B$ or $A \sim B$.

• Universal Set ($U$): A set containing all elements under consideration in a particular context, without repetition.

Page 21

Subsets and Proper Subsets:

• Subset ($\subseteq$): $A \subseteq B$ if every element of $A$ is also in $B$.

• Properties:

  1. Every set is a subset of itself.

  2. The empty set is a subset of every set.

  3. If $n(A) = n$, then the number of subsets is $2^n$.

• Proper Subset ($\subset$): $A \subset B$ if $A \subseteq B$ and $A \neq B$.

• Properties:

  1. A set is not a proper subset of itself.

  2. Number of proper subsets is $2^n - 1$.

• Super Set: If $A \subseteq B$, then $B \supseteq A$.

Page 23-28

1.4 Operations on Sets:

• Venn Diagram: Schematic representation using circles within a rectangle ($U$).

• Union ($A \cup B$): Elements in $A$, in $B$, or in both. $A \cup B = \{x | x \in A \text{ or } x \in B\}$.

• Intersection ($A \cap B$): Elements in both $A$ and $B$. $A \cap B = \{x | x \in A \text{ and } x \in B\}$.

• Disjoint Sets: If $A \cap B = \emptyset$.

Laws of Intersection:

1. Law of $\emptyset$ and $U$: $\emptyset \cap A = \emptyset$; $U \cap A = A$.

2. Commutative: $A \cap B = B \cap A$.

3. Associative: $(A \cap B) \cap C = A \cap (B \cap C)$.

Complement of a Set ($A'$):

• Absolute Complement: Elements in $U$ not in $A$. $A' = \{x: x \in U \text{ and } x \notin A\}$.

• De Morgan’s Laws:

  1. $(A \cup B)' = A' \cap B'$

  2. $(A \cap B)' = A' \cup B'$

Relative Complement (Difference):

• $A - B$ or $A \setminus B$: Elements in $A$ but not in $B$. $A - B = \{x | x \in A \text{ and } x \notin B\}$.

• Note: $A - B = A \cap B'$ and $A' = U - A$.

Page 29-30

Symmetric Difference ($A \Delta B$):

• Defined as $(A \setminus B) \cup (B \setminus A)$ or $(A \cup B) \setminus (A \cap B)$.

Cartesian Product ($A \times B$):

• The set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

• $A \times B = \{(a, b): a \in A \text{ and } b \in B\}$.

• If $n(A) = 6$ and $n(B) = 4$, then $n(A \times B) = 24$.

Page 31-32

1.5 Applications (Cardinality):

• For finite sets $A$ and $B$:   $n(A \cup B) = n(A) + n(B) - n(A \cap B)$.

• If $A \cap B = \emptyset$, then $n(A \cup B) = n(A) + n(B)$.

Example Exercise: 60 people meeting. 42 drink tea, 27 drink coffee. If everyone drinks at least one, how many drink both? $n(\text{Tea} \cup \text{Coffee}) = n(\text{Tea}) + n(\text{Coffee}) - n(\text{Both})$ $60 = 42 + 27 - n(\text{Both})$ $n(\text{Both}) = 69 - 60 = 9$.

Page 38-44

Unit 2: The Number System2.1 Revision on Natural Numbers and Integers:

• Euclid’s Division Lemma: Given a non-negative integer $a$ and a positive integer $b$, there exist unique non-negative integers $q$ (quotient) and $r$ (remainder) such that $a = b \times q + r$ where $0 \leq r \lt b$.

• Example: 38 divided by 4: $38 = 4(9) + 2$. Here $q=9, r=2$.

Prime and Composite Numbers:

• Prime: Exactly two distinct factors (1 and itself). 2 is the only even prime.

• Composite: More than two factors. 1 is neither prime nor composite.

• Fundamental Theorem of Arithmetic: Every composite number can be unique factorized as a product of primes.

Divisibility Tests:

• 2: Unit digit is even.

• 3: Sum of digits divisible by 3.

• 4: Last two digits divisible by 4.

• 5: Unit digit is 0 or 5.

• 6: Divisible by 2 and 3.

• 8: Last three digits divisible by 8.

• 9: Sum of digits divisible by 9.

• 10: Unit digit is 0.

Page 51-60

GCF and LCM:

• GCF: Greatest common factor. Product of common prime factors with least powers.

• LCM: Least common multiple. Product of prime factors with highest powers.

• Relation: $GCF(a, b) \times LCM(a, b) = a \times b$.

2.2 Rational Numbers ($\mathbb{Q}$):

• Numbers in form $\frac{a}{b}$ where $a, b \in \mathbb{Z}, b \neq 0$.

• Proper Fractions: $a \lt b$.

• Improper Fractions: $a \geq b$.

• Mixed Fractions: $c \frac{a}{b}$.

• Decimals: Rational numbers are either terminating or repeating ($0.333\dots = 0.\bar{3}$).

2.3 Irrational Numbers:

• Decimals that are neither terminating nor repeating.

• If $a$ is not a perfect square, $\sqrt{a}$ is irrational.

• Example: $\sqrt{2} \approx 1.4142\dots$, $\pi$, $0.12345\dots$.

Page 61-80

2.4 Real Numbers ($\mathbb{R}$):

• Union of rational and irrational numbers. $\mathbb{R} = \mathbb{Q} \cup \mathbb{I}$.

• Intervals:

  • Open: $(a, b) = \{x | a \lt x \lt b\}$

  • Closed: $[a, b] = \{x | a \leq x \leq b\}$

  • Half-open: $[a, b)$ or $(a, b]$.

• Absolute Value: $|x| = x$ if $x \geq 0$; $-x$ if $x \lt 0$. Represents distance from zero.

Exponents and Radicals:

• $a^n = a \times a \dots \times a$ ($n$ times).

• $a^0 = 1$ ($a \neq 0$).

• $a^{-n} = \frac{1}{a^n}$.

• Principal nth Root ($\sqrt[n]{b}$): Positive nth root if $b \gt 0$; negative if $b \lt 0$ and $n$ is odd; 0 if $b = 0$.

• Rational Power: $b^{\frac{1}{n}} = \sqrt[n]{b}$.

Page 84-106

Laws of Exponents ($a, b \in \mathbb{R}; n, m \in \mathbb{N}$):

1. $a^n \times a^m = a^{n+m}$

2. $\frac{a^n}{a^m} = a^{n-m}$

3. $(a^n)^m = a^{nm}$

4. $(ab)^n = a^n b^n$

5. $(\frac{a}{b})^n = \frac{a^n}{b^n}$

Rounding and Accuracy:

• Significant Figures (s.f.): Number of digits from left to right starting at first non-zero digit.

• Bounds: Lower bound (given value - 0.5 × accuracy unit); Upper bound (given value + 0.5 × accuracy unit).

Scientific Notation: $a \times 10^n$ where $1 \leq a \lt 10$ and $n \in \mathbb{Z}$.

Rationalizing the Denominator: Removing radicals from the denominator by multiplying by a rationalizing factor (often the conjugate).

• $\frac{1}{\sqrt{a} - \sqrt{b}} \times \frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} + \sqrt{b}} = \frac{\sqrt{a} + \sqrt{b}}{a-b}$.

Page 105-120

Unit 3: Solving Equations3.1 Linear Equations in One Variable:

• Form: $ax + b = 0, a \neq 0$.

• Steps: Clear fractions with LCM, simplify sides, isolate variable, verify.

3.2 Systems of Linear Equations in Two Variables:

• Form: $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$.

• Methods:

  1. Table: Finding ordered pairs that satisfy both.

  2. Substitution: Solving for one variable and plugging into the other equation.

  3. Elimination (Addition): Adding equations to cancel one variable.

  4. Graphical: Slope-intercept form $y = mx + b$. Intersection point is the solution.

Page 121-137

System Solution Types:

• Unique Solution: Lines intersect at one point (different slopes).

• Infinite Solutions (Dependent): Lines coincide (same slope, same y-intercept).

• No Solution (Inconsistent): Lines are parallel (same slope, different y-intercept).

3.3 Solving Non-linear Equations:

• Absolute Value Equations: $|x + a| = p$ implies $x + a = p$ or $x + a = -p$.

• Quadratic Equations: $ax^2 + bx + c = 0, a \neq 0$.

Page 140-147

Methods for Quadratic Equations:

1. Factorization: Using the zero product property ($ab = 0 \implies a=0 \text{ or } b=0$).

2. Completing the Square: $(x + \frac{b}{2a})^2 + \frac{4ac - b^2}{4a^2} = 0$.

3. Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Discriminant ($D = b^2 - 4ac$):

• If $D \gt 0$: Two distinct real roots.

• If $D = 0$: One real root.

• If $D \lt 0$: No real roots.

Relationships between Roots ($r_1, r_2$) and Coefficients:

• Sum: $r_1 + r_2 = -\frac{b}{a}$

• Product: $r_1 \cdot r_2 = \frac{c}{a}$

Page 148-157

Exponents and Radical Equations:

• Exponential: $a^x = a^y \iff x = y$ for $a \gt 0, a \neq 1$.

• Radical: Equations with variables in the radicand. Must isolate the radical, square both sides (if square root), and check for extraneous solutions.

Page 162-181

Unit 4: Solving Inequalities4.1 Revision:

• Addition/Subtraction: Sign remains same.

• Multiplication/Division by Positive: Sign remains same.

• Multiplication/Division by Negative: Sign reverses.

4.3 Absolute Value Inequalities ($c \gt 0$):

• $|k| \lt c \iff -c \lt k \lt c$

• $|k| \gt c \iff k \gt c \text{ or } k \lt -c$

4.4 Quadratic Inequalities:

• General form: $ax^2 + bx + c \gt 0$ (or $\lt, \leq, \geq$).

• Methods: Product properties ($mn \gt 0 \implies (m,n \gt 0) \text{ or } (m,n \lt 0)$), Sign Charts, or Graphical analysis.

Page 185-201

Unit 5: Introduction to Trigonometry5.1 Right-angled Triangles:

• One angle is $90^\circ$. Longest side is the hypotenuse ($h$).

• Pythagoras Theorem: $b^2 + p^2 = h^2$ ($b$ = base, $p$ = perpendicular).

5.2 Trigonometric Ratios:

• $\sin A = \frac{\text{Opposite}}{\text{Hypotenuse}}$

• $\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

• $\tan A = \frac{\text{Opposite}}{\text{Adjacent}}$

Special Angles values:

• $30^\circ: \sin=1/2, \cos=\sqrt{3}/2, \tan=1/\sqrt{3}$

• $45^\circ: \sin=1/\sqrt{2}, \cos=1/\sqrt{2}, \tan=1$

• $60^\circ: \sin=\sqrt{3}/2, \cos=1/2, \tan=\sqrt{3}$

Page 207-234

Unit 6: Regular Polygons

• Polygon: A simple closed plane figure formed by 3+ line segments joined end-to-end.

• Convex: Each interior angle < 180^\circ.

• Concave: At least one interior angle > 180^\circ.

• Sum of Interior Angles: $(n - 2) \times 180^\circ$.

• Sum of Exterior Angles: Always $360^\circ$.

• Regular Polygon: Equiangular and equilateral.

• Interior Angle of Regular n-gon: $\frac{(n - 2) \times 180^\circ}{n}$.

• Exterior Angle level: $\frac{360^\circ}{n}$.

Properties:

• Every regular n-gon has $n$ lines of symmetry.

• Apothem ($a$): Perpendicular distance from center to midpoint of a side. Radius of incircle.

• Radius ($r$): Distance from center to vertex. Radius of circumcircle.

• Area ($A$): $A = \frac{1}{2} aP$ (where $P$ is perimeter).

• Formulas: $s = 2r \sin(\frac{180^\circ}{n})$, $a = r \cos(\frac{180^\circ}{n})$.

Page 239-273

Unit 7: Congruency and Similarity

• Congruent ($\equiv$): Identical in size and shape. Criteria: SSS, SAS, ASA, RHS.

• Similar ($\sim$): Identical shape, corresponding angles congruent, sides proportional. Criteria: AA, SSS, SAS.

• Ratios:

  • Ratio of Perimeters = Ratio of side lengths ($\frac{s_1}{s_2}$).

  • Ratio of Areas = Square of the ratio of side lengths ($(\frac{s_1}{s_2})^2$).

Page 277-306

Unit 8: Vectors in Two Dimensions

• Scalars: Magnitude only (Mass, Density, Temperature, Distance, Area, Speed).

• Vectors: Magnitude and Direction (Force, Acceleration, Velocity, Displacement, Weight).

• Operation:

  • Addition: Triangle law (head-to-tail) or Parallelogram law (co-initial).

  • Column Vector: $\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}$. Magnitude $|\mathbf{a}| = \sqrt{x^2 + y^2}$.

  • Position Vector: Starts at Origin $(0,0)$. For points $A(x_1, y_1), B(x_2, y_2)$, $\vec{AB} = (x_2 - x_1, y_2 - y_1)$.

Page 311-360

Unit 9: Statistics and Probability

• Statistics: Collection, organization, presentation, analysis, and interpretation of data.

• Data Types: Qualitative vs Quantitative; Discrete (counts) vs Continuous (measurable range).

• Central Tendency:

  • Mean ($\bar{x}$): $\frac{\sum x_i}{n}$.

  • Median: Middle value when arranged in order.

  • Mode: Most frequent value.

• Dispersion:

  • Range: Max - Min.

  • Variance ($\sigma^2$): Mean of squared deviations from the mean.

  • Standard Deviation ($\sigma$): $\sqrt{\text{Variance}}$.

Probability:

• Sample Space ($S$): Set of all possible outcomes.

• Event ($E$): Subset of $S$.

• Theoretical Probability: $P(E) = \frac{n(E)}{n(S)}$.

• Tree Diagrams: Used to map outcomes of repeated experiments (e.g., tossing multiple coins).