## ArchiMath Publishing LLC

Authors: Zvezdelina Stankova, Zdravka Paskaleva, Maya Alashka, Raina Alashka

Pages: 212

ISBN: 978-954-779-259-3

Available: Paperback & PDF

In the third year of the Bulgarian Math curriculum, the 8A textbooks continue to build upon the foundational marriage between algebra and geometry.

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Overview

After a brief review of polynomials, triangle congruence, and constructions, the textbook opens with a systematic treatment of linear inequalities, moving on to the triangle inequality and other classic inequalities in triangles. Students continue their axiomatic study of geometry with the classification of quadrilaterals and careful study of their properties.

In algebra, students learn to work efficiently with square roots and solve quadratic equations, giving a proof of the quadratic formula along the way. This semester concludes with vector geometry and isometries in the plane, topics that most students in the US will not encounter in middle school or high school. However, the Bulgarian curriculum leads students to greater mathematical maturity, and they are entirely prepared for these advanced topics after working through the earlier books.

Chapter 1

1. Integer Expressions. Polynomials (Review)
2. Linear Equations. Algebra on Poly’s (Review)
3. Logic through Geometry and More (Review)
4. Foundations of Geometry (Review)
5. Congruent Triangles (Review)
6. Euclidean Constructions (Review)
7. How Mathematicians Write Solutions
8. Entrance Level: General Tests 1-2
9. How to Study for and How to Take Tests

Chapter 2. Inequalities

Part I: Algebraic Inequalities

1. Numerical Inequalities. Introduction
2. Numerical Inequalities. Properties
3. Equivalent Inequalities
4. Inequalities with One Unknown
5. Linear Inequalities with One Unknown
6. Representing Solutions to Linear Inequalities

by Intervals and Graphs

1. Inequalities that Reduce to Linear Ones
2. Inequalities. Exercises
3. Applications of Linear Inequalities
4. Summary of “Equations vs. Inequalities”
5. Summary of “Algebraic Inequalities”
6. Test on “Algebraic Inequalities”

Chapter 2. Inequalities

Part II: Geometric Inequalities (Optional)

1. Inequalities between Sides and Angles in Triangles
2. Inequalities between Sides and Angles

in Triangles. Exercises

1. The Triangle Inequality
2. The Triangle Inequality. Exercises
3. Constructing a Triangle Given Three Sides
4. Summary of “Geometric Inequalities”
5. Test on “Geometric Inequalities”

Chapter 3. Parallelograms. Trapezoids

1. Parallelogram. Properties of Its Sides
2. Properties of the Diagonals in

a Parallelogram

1. Properties of the Angles in a Parallelogram
2. Parallelogram. Exercises
3. Constructing a Parallelogram
4. Rectangle
5. Rhombus
6. Square
7. Types of Parallelograms. Exercises
8. Trapezoid, Part I. Isosceles Trapezoid
9. Trapezoid, Part II. Right Trapezoid
10. Summary of “Parallelogram. Trapezoid”
11. Test on “Parallelogram. Trapezoid”

Chapter 4. Square Roots

1. Irrational Numbers
2. Square Roots
3. Properties of Square Roots
4. Operations with Square Roots, Part I
5. Operations with Square Roots, Part II
6. Working w/ Expressions w/ Square Roots
7. Rationalizing Expressions w/ Square Roots
8. Summary of “Square Roots”
9. Tests 1-2 on “Square Roots”
10. General Problems on “Square Roots”

2. The Roots of a Quadratic Equation:

2. Equations that Reduce to Quadratic
4. Tests 1-2 on “Quadratic Equations”
5. General Problems on “Quadratic Equations”

Chapter 6. Vector Geometry

Part I: Vectors and Operations

1. Vector. Definitions
2. Sum of Vectors
3. Sum of Vectors. Exercises
4. Product of a Vector with a Number
5. Difference of Vectors
6. Applications of Vectors
7. Summary of “Vectors and Operations”

General Problems on Vector Geometry, I

1. Tests 1-2 on “Vectors and Operations”

Chapter 6. Vector Geometry.

Part II: Midsegments and Centroids (Optional)

1. Midsegment of a Triangle
2. Midsegment of a Triangle. Exercises
3. Centroid of a Triangle
4. Centroid of a Triangle. Exercises
5. Midsegment of a Trapezoid
6. Midsegment of a Trapezoid. Exercises
7. Applications of Midsegments and Centroids
8. Summary of “Midsegments and Centroids”

General Problems on Vector Geometry, II

1. Tests 1-2 on “Midsegments and Centroids”

Chapter 7. Transformations in the Plane: Isometries

1. Translation (along a Vector)
3. Central Symmetry (Point Reflection)
4. Reflection (across a Line)
5. Isometries. Exercises
6. Summary of “Isometries”
7. Test on “Isometries

Appendix

System of Geometric Concepts/Statements

• Points, lines, and angles
• Parallel lines
• Triangle
• Congruent Triangles
• Isosceles Triangle
• Perpendicular Bisector
• Right Triangle
• Angle Bisector
• Circle
• Geometric Inequalities
• Vector Geometry
• Transformation in the Plane
• Construction Problems
• Polyhedral Solids
• Rounds Solids

Summary of Applied Math Concepts/Statements

Summary of Concepts/Statements in Algebra

• Ratio. Proportion
• Integer Expressions. Polynomials
• Algebra on Polynomials
• Equations
• Numerical Inequalities
• Square Roots and Quadratic Formula

Summary of Logic Concepts

Summary of Concepts in Number Theor

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