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Introduction to Algebra in 7th Grade BYOM


2023-05-26 |    0

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The mathematics (algebra) course for the 7th grade of secondary school is part of the continuous mathematics curriculum of the educational system developed by L.G. Peterson. It ensures the continuity of students' mathematical preparation from preschool to their transition to high school or attainment of secondary vocational education. The main objective of this course is to develop students' ability to learn, their intellectual and moral-spiritual development, and to maintain and support children's health. It aims to equip each student with a deep and solid foundation of mathematical knowledge, skills, and abilities tailored to their trajectory of self-development. These are necessary for continuing education in any profile of the high school and institutions of secondary vocational education in terms of technology and didactics, content, and methodology.

The pedagogical instrument for achieving the set goals in the course at all stages of education, taking into account the age-related psychological characteristics of children's development, is the educational system of the activity-based teaching method developed by L.G. Peterson. It implements the methodological version of the systemic-activity approach.

Content Features of the Course Construction

Implementing the activity-based teaching method in the course allows for the organization of comprehensive mathematical activities for students in studying all sections of the course. These activities involve acquiring new knowledge, transforming it, and applying it, encompassing all three stages of mathematical modeling:

1.    The stage of mathematizing reality involves constructing a mathematical model of a certain fragment of reality.

2.    The stage of studying the mathematical model involves constructing a mathematical theory that describes the properties of the model.

3.    The stage of applying the obtained results to the real world.

In constructing mathematical models, students gain experience in using mathematical knowledge to describe objects and processes in the surrounding world, explain the causes of phenomena, and assess their quantitative and spatial relationships.

While studying the mathematical model, they develop mathematical language, and logical, algorithmic, and creative thinking. They learn to investigate and identify properties and relationships, visually represent the obtained data, and construct and execute algorithms.

Furthermore, when applying the obtained results to the real world, students apply mathematical knowledge to solve problems. Here, they refine their ability to execute algorithms for solving equations and inequalities and their systems when solving word problems. Students work with diagrams, tables, and charts, analyze and interpret data, and acquire proficiency in mathematical communication.

A characteristic feature of the program is that, unlike other programs, students are initially presented with a practical problem to solve. In constructing a mathematical model for the problem, they need to expand their existing mathematical tools. This approach introduces all major equations (linear Diophantine equations, linear equations with two variables and their systems, quadratic equations, fractional-rational equations) and systems and sets of inequalities. This allows students to see the connection between the "lifeless" algebraic symbols and the "living" world around them and serves as a way to motivate high school students to study mathematics.

With the same purpose, the course includes many physics problems, the solutions of which can be reduced to the techniques and methods just learned. This helps students understand mathematics as a powerful tool for understanding real processes in the world.

The same approach is used to introduce the concept of a "function." The introduction to any new function in 7th grade begins with considering practical problems for which the function provides a generalized description. A key feature of the program is that the problem-solving section of each topic is aimed not only at practicing specific new knowledge (which was a traditional task of textbooks) but also at organizing independent activities for students to discover new concepts or methods.

The problem-solving section begins with tasks designed to facilitate students' independent discovery of new knowledge within the course program. Initially, students are assigned to activate the previously learned methods of action, which are sufficient for constructing new knowledge and activating the corresponding cognitive operations. Then, an assignment reveals the absence of the knowledge planned to be discovered by the students.

Subsequently, questions or tasks are provided to help students discover new knowledge through learning activities (such as observation, experimentation, analogy, applying and adapting existing methods to new situations, formulating hypotheses, and justifying them). Afterward, the student's results are compared with the text in the theoretical part of the topic, which serves as a reference example.

It is worth noting that the educational texts in the theoretical part are also structured based on the method of reflexive self-organization. Their structure can be presented as follows:

• Presenting a new and interesting problem to the students; the solution is impossible with known methods.

• Clarifying what exactly is currently inaccessible for solving the problem.

• Searching for an idea (approach) to solve the specific new problem based on the student's existing knowledge and applying the found approach to its solution.

• Generalizing this approach as a method that allows solving a whole class of similar problems.

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• Providing a detailed analysis of many examples applying the method, ranging from the simplest to substantial and highly complex problems.

Such a tutorial structure helps students work independently with the theoretical material, which is important for their subsequent learning and further professional self-development. It is worth noting that the problem-solving section of the tutorial, in addition to providing a set of tasks for organizing discovery, includes many exercises for students to independently practice the acquired knowledge (concepts, methods of action).

The course presents tasks of varying levels of difficulty, including Olympiad-level problems. This approach corresponds to the psychological characteristics of adolescents. The "sense of adulthood" not yet reinforced by real responsibility is a special form of self-awareness that emerges during the transitional period and determines the primary relationships of adolescents with the world. This sense manifests in the need for equality, respect, and independence and the demand for serious, trusting relationships with adults. The course provides a place and means to fulfill the "sense of adulthood" for the students.

Such a structure, adapted to the implementation of the activity-based teaching method by L.G. Peterson, considers other characteristics of the adolescent period, such as a tendency towards fantasizing and uncritical planning of their future. The outcome of the action becomes secondary, and their creative idea takes the forefront. If the teacher primarily evaluates the quality of students' "products" in their schoolwork and does not find room for nurturing their ideas, it devalues the learning process for the students. The organization of education, incorporated in the problem-solving section of the textbook, allows students to experiment with their abilities, which is one of the most prominent characteristics of adolescents.

The independent attempts of students to discover mathematical theory are a form of such experimentation. It is worth noting that a differentiated approach was used in selecting the educational content. Starting from the 7th grade, work begins to prepare students for the pre-professional level of education. To achieve this, the course includes optional sections for study in general education classes. The course content expands by covering topics such as mathematical logic, divisibility theory, linear equations and inequalities (solving equations in integers, solving inequalities with absolute value), as well as practical applications of acquired knowledge, including the topic of "Functional Dependency and Information Encoding," among others.

Considering the modern level of development of mathematical theory, the educational content is presented in the form of seven main content-methodological lines, the study of which begins at the preschool stage and continues through all stages of education, up to the final years of secondary school: modeling, logical, numerical, algebraic, geometric, functional, and data analysis lines. The coherence of the course is achieved through constant comparison and interpretation of the results obtained in different content-methodological lines.

While learning mathematics from grades 1 to 6, conditions were created for the students to be well-prepared for studying all topics covered in the 7th-grade algebra course. Using activity-based and new teaching methods has significantly expanded the topics covered. In grades 5-6, logical concepts were studied, and general methods of mathematical activity were mastered, establishing a solid foundation for studying mathematics in the 7th grade and higher levels of school.

Starting from the 7th grade, the study of the algebraic line becomes the main goal of the course, and this line, along with the functional line, occupies a substantial part of it. The data analysis line is important, with its material being introduced "incrementally" throughout the course rather than as a separate block. The other lines now serve a supportive function for these main lines.

Within the framework of studying "algebra," the geometric line, starting from the 7th grade, is not further developed in terms of content and serves as a background for studying the other course lines.

Numerical line

In elementary school, the numerical line was constructed based on counting objects (elements of a set) and measuring quantities. Students acquired an understanding of the concept of natural numbers and zero, principles of writing and comparing non-negative integers, the meaning and properties of arithmetic operations, interconnections between them, mental and written calculation methods, estimation, evaluation, and verification of arithmetic results, dependencies between their components and results, and methods of finding unknown components. On the other hand, they were introduced to various quantities and the general principle of measurement, learning to perform operations with the values of quantities.

The use of the activity-based teaching method not only preserved the full content of the traditional school mathematics curriculum but also enriched it, taking into account the sensitive periods of children's development. For example, in the 3rd grade, they studied Numeration methods and operations with non-negative integers within 12 digits; in the 4th grade – fractions, addition, and subtraction of fractions with the same denominators, and mixed numbers. In grades 5-6, students studied common and decimal fractions (including repeating decimals) and negative numbers. Thus, by the beginning of the 7th grade, students will grasp the concept of rational numbers and perform computations with rational numbers. Students also learned about the history of the concept of a number and the method of expanding number sets.

At a fairly advanced level, students studied topics related to the divisibility of numbers: the concept of divisibility, properties of divisibility, and divisibility tests for 2, 5, 10, 25, 4, 125, and 8, for 3 and 9, as well as their combinations, greatest common divisor (GCD), least common multiple (LCM), prime and composite numbers.

In the 7th grade, students revisit the concepts of prime and composite numbers, and become acquainted with the fundamental theorem of arithmetic (any composite number can be expressed as a product of prime factors, and two factorizations of the same number into prime factors may differ only in the order of the factors), learn about the canonical prime factorization of a number, and enhance their knowledge of finding the GCD using the Euclidean algorithm. They refine their understanding of known divisibility properties and learn new ones (at this stage of learning, these topics can already be considered part of the course's algebraic line).

In the 7th grade, with the introduction of the axiomatic method, students build the theory of divisibility in the set of integers. They can become familiar with congruences and their properties and construct the arithmetic of remainders. The constructed theory of divisibility is valuable not only for its aesthetics and the systematic review of known divisibility properties but also for its application of the learned axiomatic method in constructing mathematical theories. Here, students master methods for solving problems related to divisibility, which can be useful in various mathematical competitions and Olympiads.

In the 7th grade, students refine their understanding of rational numbers and learn to convert repeating decimals into fractions.

Algebraic Line

As part of the study of the algebraic line in the 5th and 6th grades, students learned to use symbolic notation to formulate and prove general statements. This enabled them to prove properties and criteria of divisibility, properties of proportions, and more. Students gained an understanding of numerical and symbolic expressions, their reading, writing, and the purpose of using variables. They found the values of symbolic expressions for given values of variables and performed transformations when solving equations. Thus, a solid foundation was laid for students to study the algebraic content of the 7th-grade curriculum.

Let's consider how the algebraic line develops in the 7th grade, highlighting the following directions:

• Performing equivalent manipulations on expressions.

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• Understanding the concept of exponentiation and applying its properties.

• Solving equations.

• Solving inequalities.

In the 7th grade, students review and systematize the familiar laws of arithmetic operations and their understanding of equivalent expressions and transformations. Based on their knowledge of the laws of arithmetic operations, students independently establish basic rules of equivalent manipulations. Once the rules of equivalent manipulations are formulated, students perform manipulations on symbolic expressions they had previously performed, but now they justify them in a new way.

Due to the strong algebraic preparation provided in the course, most seventh-grade students should not have difficulty with such tasks. Unlike the previous stage of education, performing equivalent manipulations on algebraic expressions at this stage of learning is a necessary skill.

Furthermore, students are introduced to transformations of algebraic expressions involving products and quotients. With the formulation of rules for equivalent manipulations of products, students can learn how to transform algebraic fractions and expressions containing division.

In the 7th grade, students develop the concepts of a monomial and a polynomial, their standard form, and their power. They acquire the ability to perform arithmetic operations on monomials, add and subtract polynomials, multiply a monomial by a polynomial, and multiply a polynomial by a polynomial.

Also, in the 7th grade, students understand the formulas of reduced multiplication, allowing them to streamline the process of algebraic multiplication-related transformations. They become familiar with formulas such as the square of a sum and the square of a difference, the difference of squares, the cube of a sum and the cube of a difference, and the sum and difference of cubes. Students learn to apply these formulas of reduced multiplication to algebraic transformations involving multiplication and to rationalize calculations. More advanced students can be introduced to using Pascal's triangle for raising a binomial to an arbitrary natural power.

Also, in the 7th grade, students learn how to factor polynomials using the following methods: factoring out the common factor, grouping, and using formulas of reduced multiplication. They apply various auxiliary techniques when factoring polynomials, such as rearranging terms, representing a polynomial term as a sum or difference of similar terms, adding or subtracting the same term, and identifying a complete square.

Furthermore, they apply factoring to algebraic manipulations, solving equations (including quadratic equations), and rationalizing calculations. As part of advanced learning, seventh-grade students are encouraged to use factoring to simplify algebraic fractions.

By the 7th grade, students have developed an understanding of the concept of exponentiation with natural exponents. They can find the values of powers with natural exponents in simple cases and perform operations on basic numerical expressions involving exponents.

In the 5th grade, the concept of exponentiation with natural exponents was introduced on the set of natural numbers. However, students also understand exponentiation with rational numbers because, as they become familiar with numbers in the 5th and 6th-grade courses, they are given simple tasks involving raising fractions, decimals, and negative numbers to powers. This work aimed to develop initial experience in students or as advanced learning for more prepared students. Therefore, knowledge of exponentiation and the ability to apply it to the set of rational numbers were not obligatory learning outcomes for all students.

In the 7th grade, the tasks of forming the concept of natural exponentiation of rational numbers and the ability to apply the properties of exponents for transforming expressions and rationalizing calculations become mandatory for all students. In addition, the definition of exponentiation with rational numbers as exponents is introduced, including the concept of zero exponents for rational numbers. Students become acquainted with the properties of exponents and use them for expression manipulations.

By the 7th grade, students possess the following knowledge about equations: the concept of an equation, the variable in an Equation, the solution to an equation, and the method of solving equations using equivalent manipulations. In addition to the equations traditionally offered for solving in the 5th and 6th grades, students become acquainted with solving simple equations involving absolute values.

In the 7th grade, students learn the definition of equivalent equations and equivalent manipulations of equations. They refine the rules for equivalent manipulations of equations. Students become familiar with the concept of a linear equation in one variable, and they derive the algorithm for solving linear equations in one variable. They learn to solve equations with absolute values of the following types: |kx + b| = c (where k ≠ 0), |ax + b| = |cx + d|, and equations containing multiple absolute values. More advanced students can learn how to solve linear Diophantine equations in two variables.

In the 7th grade, students refine their understanding of equations, and the concept of a system of linear equations in two variables is introduced. Initially, students familiarize themselves with the concept of a linear equation in two variables, learn to plot its graph and find its solutions. Afterward, students become acquainted with the concept of a system of equations and the graphical method of solving them. More advanced students can learn how to apply the theorem on integer points of a graph to solve systems of equations. Furthermore, students become familiar with algebraic methods of solving systems of equations.

In the 7th grade, students consider other types of rational equations. Initially, students develop the concept of a quadratic equation: they learn to solve quadratic equations and equations that can be reduced to quadratic form using variable substitution (the concept of a biquadratic equation is introduced).

Once students become familiar with the concept of an algebraic fraction and its properties and learn to perform arithmetic operations on algebraic fractions, they proceed to solve fractional-rational equations. When solving fractional-rational equations, students employ several methods to transform fractional expressions into integers while considering the domain of validity, the condition of equating the algebraic fraction to zero, and the fundamental property of proportion. More advanced students can explore other methods of solving fractional-rational equations, such as variable substitution, extraction of the integer part of an algebraic fraction, and their combination.

By the 7th grade, students can already solve simple inequalities and represent their solutions on a number line. In the 7th grade, their knowledge of inequalities is refined (understanding what inequality is, solving an inequality, and what it means to solve it), and the concepts of strict and non-strict inequalities are introduced. When studying piecewise linear functions, students examined various numerical intervals, their names, notations, and geometric representations on a simplified number line. Therefore, when studying inequalities, they review such numerical intervals as open and closed rays and also become familiar with intervals of the form (–∞, +∞).

In the 7th grade, students become acquainted with the definition of equivalent inequalities, equivalent transformations of inequalities, and the rules of equivalent transformations of inequalities. Afterward, they learn about linear inequality in one variable and derive an algorithm for solving linear inequalities in one variable. Students also have the opportunity to learn how to solve equations involving absolute values. However, developing this skill is not mandatory for studying three hours of algebra per week.

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Modeling Line

Most of the studied algorithms for solving equations and inequalities are applied by students when solving word problems. The particular feature of the course is that the motivation for studying a new type of equation (or inequality) arises from the need to solve practical problems.

After acquiring a general method for solving a particular mathematical model, students return to solving the problem that necessitated the construction of the new mathematical theory (introduction of new concepts and algorithms). Thus, attention is given to all three stages of mathematical modeling (the stage of mathematizing reality, the stage of studying the mathematical model, and the stage of applying the obtained results to the real world). As a result, students realize the practical significance of mathematics and its place in the world around them. Within the modeling line (the line of word problems), students acquire various forms of mathematical activity, understand the practical value of mathematical knowledge, develop universal learning skills, and enhance their thinking, imagination, and communication.

To solve problems in the 7th grade, students use the toolkit they have acquired from grades 1-6 (such as diagrams and tables) and apply the problem-solving algorithm through mathematical modeling, refining it in the process. They learn that the mathematical model can be represented not only by an equation but also by an inequality and multiple relationships describing the interconnections between quantities stated explicitly or implicitly in the problem conditions.

Functional Line

Let's consider how the functional line of the course develops in the 7th grade, highlighting the following directions:

• The concept of a function;

 • Types of functions studied;

• Properties of functions studied;

• Numerical sequences as functions of a natural argument;

By the 7th grade, due to functional introduction, students are familiar with the concept of a variable, can work with a coordinate plane, and have experience in graphing based on formulas and tables. They know that variables can represent dependencies between quantities and can be recorded using formulas, tables, and graphs. They have an understanding of direct and inverse proportionality and their corresponding graphs. Additionally, in the sixth grade, students initially understood generalized functional relationships between quantities as a specific type of dependency. Therefore, in the 7th grade, when introducing one of the central mathematical concepts - the concept of a function, students only need to clarify its practical significance once again (prediction of real events, encoding) and become familiar with its new name. As a result, students develop the concept of a function as a rule that associates elements of two sets of arbitrary nature.

Following the general methodological approach adopted in this course, introducing any new function in the 7th grade begins with considering practical problems for which it provides a generalized description. Thus, the following functions are studied: direct proportionality, linear, and piecewise-linear functions. It should be noted that the study of piecewise-linear functions in the 7th grade was prepared by working with graphs of variable motion (students analyzed and plotted motion graphs starting from the 4th grade). Each function studied is examined, its graph is constructed, and its properties are identified.

Logical line

Significant attention is devoted to developing the logical line in the course. In grades 5-6, the logical line unfolds as a chain of interconnected questions: mathematical language - statements - proof - proof methods - definitions - equivalent statements – negation - logical implication - theorem, and so on. In the 7th grade, students revisit some of these questions. They refine the structure of definitions, become familiar with proof by contradiction, and have the opportunity to study the concept of logical reasoning based on Euler-Venn diagrams. They also explore the reasons and types of logical errors.

Stochastic Line

Starting from the early school years, the stochastic line aims to develop information literacy among students. This includes the ability to independently gather information from observations, reference books, encyclopedias, internet sources, and discussions; work with acquired information by analyzing, systematizing, and presenting it in the form of diagrams, tables, summaries, and graphs; draw conclusions; identify patterns and essential features; classify information; and systematically explore different possibilities.

In the 7th grade, students revisit the question of organizing information, systematizing their accumulated knowledge, and applying it to various practical tasks. It is recommended to continue this work during lessons and extracurricular project activities, such as creating their own informational materials like presentations, collections of problems and examples, informational leaflets, and more. Through this activity, students can develop computer skills necessary for learning in school and modern life.

In the 7th grade, students are also introduced to elements of combinatorics, statistics, and probability theory.

In the early grades and later in grades 5-7, students develop the experience of systematic exploration of possibilities using logical selection, tables, and decision trees. They use these methods to justify reasoning through exploration and solve tricky problems. As usual, the usefulness of constructing new mathematical tools is revealed through reflective analysis of practical problems. During the solution process, the inadequacy of existing exploration tools becomes apparent.

In the 7th grade, students encounter the issue of statistical characteristics of processes and become familiar with the following statistical measures: mean, mode, median, and range of a data set.

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General Recommendations for Teachers

For a secondary school teacher who starts working with the 7th-grade materials, it is important to be familiar with the curriculum of the 5th and 6th grades for the same subject. Therefore, it is necessary to become acquainted with the materials designed for the 5th and 6th grades and the system of benchmarks (methods of action) that students have learned in primary school.

Assessment System

During lessons introducing new knowledge, conducting independent learning activities, and completing creative tasks, only success is evaluated, and errors are identified and corrected based on determining their causes (i.e., rules, algorithms, and definitions that have not been sufficiently mastered). Self-assessment is used during reflection lessons, and marks are given at the student's discretion.

Marks for tests are given to all students, with the difficulty level adjusted so that approximately 75% of the class can achieve grades A and B.

It should be noted that the goal of the course is not for every student to complete all the tasks in the course. The minimum required learning outcomes according to the curriculum are determined by educational standards.

Homework

Homework consists of two parts:

• The mandatory part includes 2-3 manageable tasks for each student and requires approximately 30 minutes of independent work.

 • The optional part consists of 1-2 additional tasks.

Considering the age characteristics of the students, it is recommended to involve them in selecting their homework assignments.

Self-assessment of the mandatory part of the homework, error correction, and marking in the notebooks can be done by the students themselves at the beginning of the lesson using a provided sample or template presented by the teacher through presentations. In this case, the teacher evaluates only the accuracy of the self-assessment. The additional part of the homework is recommended to be assessed individually. Only positive marks are given when evaluating these tasks. 

 

Conclusion

 

 

In conclusion, the mathematics (algebra) course for the 7th grade of secondary school, developed by L.G. Peterson, aims to provide students with a solid foundation of mathematical knowledge and skills tailored to their self-development trajectory. The course focuses on the activity-based teaching method, which allows students to engage in comprehensive mathematical activities and develop their mathematical modeling, problem-solving, and critical thinking skills. The curriculum covers various topics, including numeration, divisibility, algebraic expressions, equations, inequalities, and exponentiation. The course structure promotes independent learning and discovery, with students actively participating in the learning process. By incorporating practical problems, physics applications, and real-world examples, the course motivates students to see the relevance and applicability of mathematics in their lives. Overall, the course provides a comprehensive and well-structured approach to teaching algebra, preparing students for further education and future endeavors in mathematics and related fields.


 

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