Grade 8 Math Review Flashcards

Acknowledgements

• Ato Zelalem Mulatu (AAEB Head): Acknowledged for outstanding leadership in the textbook and teacher's guide preparation.
• Ato Admasu Dechasa, Ato Dagnew Gebru, Ato Samson Melese, W/ro Abebech Negash, Ato Desta Mersha, Ato Sisay Endale: Commended for efforts in addressing challenges and providing feedback.
• School Principals: Acknowledged for allowing textbook writers time off from teaching and for moral support.

Table of Contents - Summary

• Unit 1: Rational Numbers
  • Concepts, representation on a number line, relationship among number sets, absolute value.
  • Comparison, ordering, operations, properties, and real-life applications.
• Unit 2: Squares, Square Roots, Cubes, and Cube Roots
  • Squares and square roots of rational numbers; use of tables and calculators.
  • Cubes and cube roots; applications.
• Unit 3: Linear Equations and Inequalities
  • Review of the Cartesian coordinate system.
  • Graphs of linear equations, solving linear inequalities, and applications.
• Unit 4: Similarity of Figures
  • Similar plane figures and triangles.
  • Tests for similarity (AA, SSS, SAS).
  • Perimeter and area of similar triangles.
• Unit 5: Theorems on Triangles
  • Angle sum property of triangles (180°).
  • Exterior angle theorem.
  • Theorems on right-angled triangles (Euclid's and Pythagoras' theorems, and their converses).
• Unit 6: Circles
  • Lines and circles; central and inscribed angles.
  • Angles formed by intersecting chords; applications of circles.
• Unit 7: Solid Figures and Measurements
  • Prisms, cylinders, pyramids, and cones.
  • Surface area and volume of solid figures; applications.
• Unit 8: Introduction to Probability
  • Concept of probability; simple events.
  • Applications in business, climate, road transport, accidents, and drug effects.

Chapter 1: Rational Numbers

• Learning Outcomes: Define, represent, and order rational numbers; perform operations and solve related practical problems.
• Main Contents: Concept, comparison, ordering, operations, properties, and applications.
• Introduction: Importance of understanding number classifications and their properties.

1.1. The concept of Rational numbers

• Competencies: Describe rational numbers practically and express them as fractions.
• Group Work 1.1:
  • Examples of Natural, Whole, and Integer numbers.
  • Inclusion relationships among number sets.
  • Definition of a rational number.
  • Practical problem involving fractions (Solomon's pet toys).

1.1.1. Representation of Rational Numbers on a Number line

• Competency: Represent rational numbers as a set of fractions on a number line.
• Revision on Fractions:
  • Definition of a fraction as part of a whole.
  • Numerator and denominator.
  • Examples using DVDs and rectangular fields, Numerator 3 and Denominator 5.
• Types of Fractions:
  • Proper fraction: Numerator < Denominator.
  • Improper fraction: Numerator ≥ Denominator. Expressed as a whole number + proper fraction (mixed fraction).
• Integers on a Number Line:
  • Representation of integers.
  • Example: Identifying a non-integer point x between 2 and 3.
• Definition 1.1: Rational number is a number that can be written in the form of \frac{a}{b}, where a and b are integers and b \neq 0.
• Example 1.3: \frac{1}{5}, \frac{3}{7}, \frac{3}{5}, -\frac{8}{2}, -\frac{14}{4}, and \frac{5}{9} are rational numbers.
• Note: The set of rational numbers is denoted by \mathbb{Q}.
• Locating Rational Numbers on a Number Line:
  • Positive on the right, negative on the left of zero.
  • Positive proper fractions exist between zero and one.
  • Improper fractions are converted to mixed fractions before representation.
• Example 1.4:
  • a) \frac{2}{5} : Divide between 0 and 1 into 5 equal parts, the second part represents \frac{2}{5}.
  • b) \frac{3}{2} = 1\frac{1}{2} : Lies between 1 and 2 at the 1/2 point.
  • c) -\frac{3}{4} : Lies between -1 and 0, divide into 4 equal parts, the third part will be -\frac{3}{4}.
  • d) -\frac{5}{2} = -2\frac{1}{2} : Lies between -3 and -2, divide into 2 equal parts, the first part is -\frac{5}{2}.
• Note: Two rational numbers are said to be opposite, if they have the same distance from 0 but in different sides of 0. For instance \frac{3}{2} and -\frac{3}{2} are opposites.
• Exercise 1.1 This part is a self-assessment for the student.

1.1.2. Relationship among \mathbb{N}, \mathbb{W}, \mathbb{Z} and \mathbb{Q}

• Competency: Describe the relationship among the sets \mathbb{N}, \mathbb{W}, \mathbb{Z} and \mathbb{Q}.
• Recall:
  • A collection of items is called a set.
  • The items in a set are called elements and is denoted by \in.
  • A Venn diagram uses intersecting circles to show relationships among sets of numbers.
• The Venn diagram shows how the set of natural numbers, whole numbers, integers, and rational numbers are related to each other.
• When a set is contained within a larger set in a Venn diagram, the numbers in the smaller set are members of the larger set.
• Example 1.5:
  • a. -13: integer, rational number
  • b. \frac{1}{7}: rational number
  • c. -\frac{5}{76}: rational number
  • d. 10: natural number, whole number, integer, rational number.

1.1.3. Absolute value of Rational numbers

• Competency: Determine the absolute value of a rational number.
• Activity 1.1: The absolute value of a rational number describes the distance from zero that a number is on a number line without considering direction.
• Definition 1.2: The absolute value of a rational number ‘𝑥𝑥’, denoted by │𝑥𝑥│, is defined as:
  • \begin{equation} │𝑥𝑥│ =\left{ \begin{array}{l} 𝑥𝑥, \quad if \quad x \geq 0 \ −𝑥𝑥, \quad if \quad x < 0 \end{array} \right. \end{equation}
• Example 1.7:
  • a. │6│ = 6
  • b. │0│ = 0
  • c. │−15│ = −(−15) = 15
• Example 1.8: Simplify absolute value expressions.
• Definition 1.3: An equation of the form │𝑥𝑥│=𝑎𝑎 for any rational number 𝑎𝑎 is called an absolute value equation.
• Geometrically the equation │𝑥𝑥│ = 8 means that the point with coordinate 𝑥𝑥 is 8 units from 0 on the number line.
• Note : The solution of the equation │𝑥𝑥│= 𝑎𝑎 for any rational number 𝑎𝑎, has
  • i. Two solutions 𝑥𝑥 = 𝑎𝑎 and 𝑥𝑥 = −𝑎𝑎 if 𝑎𝑎 > 0.
  • ii. One solution, 𝑥𝑥 = 0 if 𝑎𝑎 = 0 and
  • iii. No solution, if 𝑎𝑎 < 0.
• Example 1.9: Solving absolute value equations.

1.2. Comparing and Ordering Rational numbers

• Competency: At the end of this sub-topic students should:
  • Compare and order Rational numbers.

1.2.1. Comparing Rational numbers

• In day to day activity, there are problems where rational numbers have to be compared.
• Activity 1.2: Insert
• A rational number \frac{a}{b} can be expressed as a decimal number by dividing the numerator a by the denominator b.
• Comparing Fractions
  • Comparing fractions with the same denominator
  • Fractions that represent the same point on a number line are called Equivalent fractions.
  • Comparing fractions with different denominators, Method 1 and Method 2.
• Comparing Rational numbers using number line.

1.2.2. Ordering Rational numbers

• Ordering rational numbers means writing the given numbers in either ascending or descending order.
• Example 1.14. Arrange the following rational numbers in:
  • i) Increasing (ascending) order
  • ii) Decreasing (descending) order
• Exercise 1.5: This part is a self-assessment for the student

1.3. Operation and properties of Rational Numbers

• Competencies: At the end of this sub-unit students should:
  • Add rational numbers.
  • Subtract rational numbers.
  • Multiply rational numbers
  • Divide rational numbers.

1.3.1. Addition of rational numbers

• Activity 1.3. Adding rational numbers with same denominators, Adding rational numbers with different denominators
• Rule 1: To find the sum of two rational numbers where both are negatives
• Rule 2: To find the sum of two rational numbers, where the signs of the addends are different, are as follows
• Properties of Addition of Rational Numbers:
  * Commutative: 𝑎𝑎 + 𝑏𝑏 = 𝑏𝑏 + 𝑎𝑎,
  * Associative: 𝑎𝑎 + (𝑏𝑏 + 𝑐𝑐) = (𝑎𝑎 + 𝑏𝑏) + 𝑐𝑐
  * Properties of 0: 𝑎𝑎 + 0 = 𝑎𝑎 = 0 + 𝑎𝑎
  * Properties of opposites : 𝑎𝑎 + (-𝑎𝑎) = 0
• Exercise 1.6: This part is a self-assessment for the student

1.3.2. Subtraction of rational numbers

• The process of subtraction of rational numbers is the same as that of addition. Subtraction of any rational numbers can be explained as the inverse of addition.
• Example 1.22: Compute the following difference.

1.3.3. Multiplication of rational numbers

• To multiply two or more rational numbers, we simply multiply the numerator with the numerator and the denominator with the denominator
• Method 1: Area Model
• Note: The product of two rational numbers with different signs can be determine in three steps
• The product of two negative rational numbers is a positive rational number.
• Properties of multiplication of rational numbers
  • Commutative: 𝑎𝑎 × 𝑏𝑏 = 𝑏𝑏 × 𝑎𝑎
  • Distributive: 𝑎𝑎 × (𝑏𝑏 + 𝑐𝑐) = 𝑎𝑎 × 𝑏𝑏 + 𝑎𝑎 × 𝑐𝑐
  • Properties of 0: 𝑎𝑎 ×0 = 0 = 0 × 𝑎𝑎
  • Properties of 1: 𝑎𝑎 ×1 = 𝑎𝑎 = 1× 𝑎𝑎
• Note:
  * The product of an even number of negative factors is positive.
  * The product of an odd number of negative factors is negative.
  * The product of a rational number with at least one factor 0 is zero.
• Exercise 1.8: This part is a self-assessment for the student

1.3.4. Division of Rational Numbers

• Rules for Division of Rational numbers
  • When dividing rational numbers:
    1. Determine the sign of the quotient:
       a) If the sign of the dividend and the divisor are the same, then sign of the quotient is (+).
       b) If the sign of the dividend and the divisor are different, the sign of the quotient is (−).
• Note: *aaa ÷ bbb \,is\ read as aaa is divided by bbb. * In aaa ÷ bbb = ccc, ccc is called the quotient, aaa is called the dividend and bbb is called the divisor. * aaaa, bbbb and cccc are integers, bbb ≠ 0 and aaa ÷ bbb = ccc, if and only if aaa = ccc ×
  • For any two rational numbers \,aaaa\bbb and \cccc\ddd
  • For any rational number \,aaaa\bbb$$ where

1.4. Real life applications of rational numbers

• Rational numbers are used in sharing and distributing something among a group of friends.
• Rational numbers are used to express many day to day real life activities.

1.4.1. Application in sharing something among friends

• Example 1.33: There are four friends and they want to divide a cake equally among themselves. Then, the amount of cake each friend will get is one fourth of the total cake.
• Example 1.34: Three brothers buy sugarcane. Their mother says that she will take over a fifth of the sugarcane. The brothers share the remaining sugarcane equally. What fraction of the original sugarcane does each brother get?

1.4.2. Application in calculating interest and loans

• Simple interest interest is a payment for the use of money or interest is the profit return on investment. Interest can be paid on money that is borrowed or loaned.
• Interest can be calculated by I = PRT.
• Exercise:1:9

SUMMARY FOR UNIT 1

• A rational number is a number that can be written as where a and b are integers and 𝑏𝑏 ≠ 0.The set of rational numbers is denoted by ℚ.
• Rules of signs of Addition
  The sum of two negative rational numbers is negative.
  The product of two negative rational numbers is positive.
  Subtraction of any rational numbers can be explained as the inverse of addition.

REVIEW EXERCISE FOR UNIT 1

• This Review exercise helps students to test their understanding about the unit.