Algebra 2 Workbook

Purpose of Studying the Course

Course Structure

The Algebra course focuses on five core content areas:

• Numbers and Computations
• Algebraic Expressions
• Equations and Inequalities
• Functions
• Logic and Sets

These topics develop over three years of study. They naturally connect and build upon each other.

Integrated Learning Approach

Students use logical reasoning throughout the course. Set-theoretic language appears in all sections. This creates a universal mathematical foundation. The Algebra course is truly integrated in nature.

Functional Approach

Teaching Functions in Grades 7-8

We introduce functions through practical problems. Students see real-world applications first.

Advanced Function Study (Grade 9+)

Functions are introduced through mathematical theory. Students develop deeper understanding.

Why This Matters

Functions model real processes in nature and society. This makes math relevant to students.

Progressive Learning Path

• Elementary school: Early function concepts begin
• Grade 6: Functional dependency understanding
• Grade 7: Conscious work with functions
• Grades 7-9: Repeated refinement of concepts
• Grade 9: Abstract understanding as element matching

Properties Development

Students first learn properties for individual functions. In Grade 9, properties are generalized. These are applied to graph construction.

Importance of Algebra

Algebra is foundational to secondary school education. It supports study in both sciences and humanities. Mastery is essential for further education and daily life.

Building Scientific Thinking

Students develop understanding of algebraic abstractions. They learn how math reflects natural and social processes. Mathematical modeling becomes a practical tool.

Skills Development

Algebra naturally builds these abilities:

• Observation and comparison
• Pattern recognition
• Critical thinking
• Justifying conclusions
• Clear statement formulation

Logical Thinking Development

Mastering algebra helps students develop logical thinking. They practice deductive and inductive reasoning, generalization, specification, abstraction, and analogy.

Teaching algebra requires students to work independently. This independent problem-solving aligns with active learning principles.

This approach fosters intellectual honesty and objectivity. It helps students overcome mental stereotypes from everyday experiences. It also shapes personal qualities that promote social mobility and independent decision-making.

Numbers and Computations Line

The "Numbers and Computations" line lays the groundwork for further math studies. It boosts logical thinking, helps students use algorithms, and builds practical skills for everyday life. In basic school, students learn about rational and irrational numbers, leading to an understanding of real numbers. They also explore more complex topics like the Euclidean algorithm and the fundamental theorem of arithmetic. Mastery of the numerical line is completed at the upper level of general education.

Algebraic Expressions and Equations

Building Mathematical Tools

This course develops tools for solving real-world problems. Students apply algebra to mathematics and related subjects.

Expression Types Covered

• Rational expressions (primary focus)
• Irrational expressions (introduced this course)
• Trigonometric expressions and transformations

Why Algebra Matters

Algebra is the language of mathematical models. It describes real-world processes and phenomena.

Thinking Skills Development

Algebra builds essential abilities:

• Algorithmic thinking for computer science
• Deductive reasoning skills
• Mathematical creativity through symbolic forms

BYOM Practical Approach

Students start with real problems, not abstract theory. They discover the need for new mathematical tools. This approach applies to all major equation types:

• Linear Diophantine equations
• Linear equations with two unknowns
• Quadratic equations
• Fractional-rational equations
• Systems and sets of inequalities

Connecting Math to Reality

Students see algebra solving physical problems. Letters transform from "lifeless" symbols to living tools. This motivates high school students effectively.

Mathematical Language Mastery

Students learn to express ideas through:

• Verbal explanations
• Symbolic notation
• Graphical representations

They understand math's role in civilization and culture.

Logic and Sets Focus

This section develops clear communication skills. Students express thoughts accurately and concisely.

Main Goals

The main goal of this course is to help students develop learning skills. It focuses on their intellectual and moral growth, health preservation, and support. Each student will follow a unique path for self-development. They will gain deep and solid knowledge in mathematics. This will prepare them for further education in high school and vocational institutions.

Course Content

Numbers and Computations

The square root of a number. The concept of an irrational number. Decimal approximations for irrational numbers. Properties of arithmetic square roots help in transforming numerical expressions and calculations. Real numbers. The identity of the form.

a²=a(√a)², where a ≥ 0; a²=a√(a²)=|a|. Exponents with integer exponents and their properties. Standard notation of a number.

Algebraic Expressions

Quadratic trinomial; factoring a quadratic trinomial. Algebraic fraction. The basic property of an algebraic fraction. Addition, subtraction, multiplication, and division of algebraic fractions. Rational expressions and their transformation.

Equations and Inequalities

Quadratic equation, incomplete quadratic equations. Quadratic equation roots formula. Vieta's theorem. Solving equations reducible to linear and quadratic. Biquadratic equations. Simple fractional-rational equations. Quadratic equations with parameters. Graphical interpretation of equations with two variables and systems of linear equations with two variables. Examples of solving systems of nonlinear equations with two variables. Systems of two linear equations with absolute values.

Solving word problems algebraically and arithmetically. Problems reducible to solving quadratic equations.

Numerical inequalities and their properties. Inequality with one variable. Equivalence of inequalities. Proof of inequalities. Some notable inequalities. Linear inequalities with one variable. Systems of linear inequalities with one variable. Sets of linear inequalities with one variable. Linear inequalities with two variables and their systems. Graphical representation of the set of their solutions. Quadratic inequalities. Solving rational inequalities. Interval method.

Functions

Concept of a function. Domain and range of a function. Ways of defining functions. Graph of a function. Reading properties of a function from its graph. Examples of graphs of functions reflecting real processes. Functions describing direct and inverse proportional relationships and their graphs. Functions y=x, y=x², y=x³, y=√x, y=|x|. Graphical solution of equations and systems of equations.

Piecewise-defined functions. Functions y=ax², y=ax²+b, y=k(x−d)², y=ax², y=ax²+b, y=k(x-d)² and their graphs. Quadratic function y=ax²+bx+c.

Logic and Sets

Necessity and sufficiency. Properties and characteristics. Criteria. Complex statements.

Set-theoretical concepts. Set, element of a set. Defining sets by listing elements by characteristic property. Standard notation of numerical sets. Empty set and its notation. Subset. Union and intersection of sets. Illustration of relationships between sets using Euler-Venn diagrams.

Planned Subject Outcomes

Numbers and Computations

The student will learn to:

• use initial concepts of the set of real numbers for comparison, rounding, and calculations
• represent real numbers by points on the number line
• apply the concept of arithmetic square root
• find square roots, using a calculator if necessary
• perform transformations of expressions containing square roots using the properties of roots
• use notation for large and small numbers using decimal fractions and powers of 10

The student will have the opportunity to learn:

• apply the identity a²=a(√a)², where a ≥ 0; a²=a(√a)²=|a|, for transformations of expressions with roots
• calculate the approximate value of the square root using Newton's Method
• prove properties of arithmetic square roots
• transform expressions of the form a+bc√(a+b√c)

Algebraic Expressions

The student will learn to:

• Use exponentiation with integer exponents and transform expressions with them
• Manipulate rational expressions based on polynomial and algebraic fraction rules
• Factor quadratic trinomials
• Transform expressions to solve math problems and real-life scenarios

The student will have the opportunity to learn:

• Use long division to divide polynomials
• Transform rational expressions by isolating the integer part of the fraction

Equations and Inequalities

The student will learn to:

• Solve linear and quadratic equations, and rational equations that can be reduced to them. This includes systems of two equations with two variables.
• Conduct basic investigations of equations and systems. This involves using graphs to see if an equation or system has solutions and how many.
• Move from a verbal problem to an algebraic model. This means creating an equation or system of equations and interpreting the result in context.
• Use properties of numerical inequalities for comparison and estimation.
• Solve linear inequalities with one variable and their systems.
• Provide graphical illustrations of the solution set for systems of inequalities.

The student will also:

• Independently create and use algorithms for solving word problems.
• Tackle non-standard problems using an initial plan.
• Transition from verbal problem statements to algebraic models by creating systems of linear equations with three or more variables.
• Analyze the number of solutions of a system.
• Investigate systems of equations with two variables that have letter coefficients.
• Solve systems of two linear equations with two variables that include absolute values.
• Use addition and algebraic methods for systems with three or more variables.
• Solve systems of linear and quadratic inequalities.
• Solve systems of linear inequalities with one variable that have absolute values.
• Plot points on a coordinate plane defined by inequalities with two variables and their systems.
• Apply the method of completing the square to derive the formula for the roots of a quadratic equation.
• Use Vieta's theorem for various tasks.
• Apply techniques for computing the roots of a quadratic equation.
• Investigate linear and quadratic equations with letter coefficients.
• Explore quadratic inequalities with letter coefficients.
• Solve fractional-rational equations using substitution and extracting the integer part.
• Solve integer and fractional-rational inequalities using the interval method.
• Use the mean inequality to find the maximum or minimum value of a polynomial.
• Prove inequalities using various methods.

Functions

The student will learn:

• to understand and use functional concepts and language (terms, symbolic notations)
• to determine the value of a function given the value of the argument
• to determine the properties of a function from its graph
• to plot graphs of elementary functions such as y=x², y=x³, y=x, y=x, y=x², y=x³, y=√x, y=|x|
• to describe the properties of a numerical function from its graph

The student will have the opportunity to learn:

• to transition from one method of defining a function to another
• to compare the properties of different functions
• to plot and interpret graphs of piecewise-defined functions
• to plot and interpret graphs of functions of the form y=ax², y=ax²+h, y=k(x−d)², y=ax², y=ax²+h, y=k(x-d)²
• to find the minimum and maximum values of a quadratic trinomial on a given interval
• to plot and interpret the graph of a quadratic function y=ax²+bx+c

Logic and Sets

The student will have the opportunity to learn:

• to compose, read, and write complex statements (implications) and their inverses
• to distinguish between attributes and properties
• to differentiate between properties, attributes, and criteria
• to determine and justify the truth or falsity of complex statements
• to compose, read, and write complex statements using logical connectives "and," "or"
• to find the intersection or union of numerical intervals when solving systems and sets of inequalities
• to find the intersection and union of sets, the complement and difference of sets
• to define sets by enumerating their elements by characteristic properties
• to use standard notations for numerical sets
• to apply the concepts of equal sets, correspondence between sets, one-to-one correspondence between sets, and equivalent sets
• to illustrate relationships between sets using Euler-Venn diagrams
• to construct the conjunction and disjunction of statements and use mathematical symbolism to represent them
• apply De Morgan's laws
• prove the countability or uncountability of sets
• prove properties of operations on sets, including De Morgan's laws formulas

Quizzes and Tests

In lessons that introduce new topics, we focus on success. Errors are noted and fixed by identifying their causes, like misunderstood rules or definitions. During reflection lessons, students assess themselves and can choose their marks in the journal.

All students receive marks for tests. The test difficulty is set so that about 75% of the class can earn grades of 'A' and 'B'. From 8th grade onward, students prepare for test-based assessments, which include express tests during the course. After each test, students can check their progress with a self-assessment sample and a success scale provided.

This control method helps develop subject-specific skills. It also fosters meta-subject skills, like self-monitoring and self-assessment in learning. Additionally, it encourages personal growth, such as a responsible attitude toward learning and readiness for self-development. Students learn to make conscious choices and plan their educational paths.

Homework

Homework has two parts:

• The compulsory part includes 2–3 tasks suitable for each student. It takes about 30 minutes of independent work.
• The optional part has 1–2 extra tasks. The teacher chooses the compulsory tasks.

Given the students' age, they should help pick homework tasks. For the optional part, tasks marked with an asterisk (*) can be used.

Students can check their compulsory homework at the start of the lesson. They will use a sample provided by the teacher in presentations for self-checking. During notebook checks, the teacher will only evaluate the self-checking accuracy. The additional tasks should be checked individually. Students who solve creative tasks correctly, as assigned, will submit their answers on sheets. These will be displayed in the classroom with their surnames. Only positive marks are given for these tasks.

Express tests at the end of each section can also count as homework. The optional tasks (*) can be included in this.

Language and Logic

Teaching mathematical language as a clear form of communication is a key part of the BYOM program. Mastery of this language shows organized thinking. Understanding precise meanings and logical connections in math enhances proficiency in natural language. This supports the overall development of students' thinking. By 8th grade, students learn about statements, their negations, types of statements, implications, and equivalences. They are also familiar with definitions and theorems, and they know methods to prove statements, including proof by contradiction. They can use quantifiers, implication signs, and equivalence signs.

In 8th grade, students continue to master mathematical language. In the first chapter, they explore property, attribute, and criterion statements. They learn about "necessity" and "sufficiency" in implications. This material is crucial for geometry. The logical concepts introduced connect well with previous topics. Eighth graders expand their understanding of complex statements using logical connectives like "and" and "or." They learn about conjunction, disjunction, and basic logic formulas. Understanding how to construct complex statements helps organize their thinking. Skills gained here are useful in algebra, geometry, and other subjects.

At the start of the school year, the main goal is to review past material. The first chapter introduces new content while revisiting topics from grades 5-6 and algebra 7. This helps students recall lessons and fill knowledge gaps. It also expands their understanding of complex ideas. Repetition should bring new insights; otherwise, it can lead to "intellectual laziness."

Students studying mathematical language revisit key topics from 7th grade. They convert fractions and mixed numbers into repeating decimals and vice versa. They recall how to solve linear equations and inequalities. They practice formulas to simplify expressions. They also solve word problems, explore functions and their domains, and plot graphs for linear, direct proportion, and piecewise functions. They review algorithms for solving linear equations with integers and methods for absolute value equations and inequalities.

During this chapter, special focus should be on plotting linear function graphs. This prepares students for upcoming topics, including "Linear equations with two variables" and "Systems of two linear equations." They will also study "Systems of linear inequalities with two variables involving absolute values." To get ready for these topics, it's important to emphasize absolute values and spend adequate time on solving equations and inequalities with absolute values.

Characterization of Students' Activities

During the study of the content of the first chapter, students:

• review and systematize previously acquired knowledge
• apply the learned methods of action to solve problems in typical and exploratory situations
• justify the correctness of the performed action by referring to the general rule, theorem, property, or definition
• identify true and false statements, determine and justify their truthfulness and falsity
• compose, read, and write complex statements
• construct conjunction and disjunction of statements and use mathematical symbols for their notation