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# Algebra 1 Workbook

2023-07-05 |    0

The mathematics (Algebra 1) course for the 7th grade of secondary school is part of the continuous mathematics curriculum developed by L.G. Peterson. This program ensures students' math skills grow from preschool through high school or vocational training. Its main goal is to improve learning, boost intellect, and nurture moral and spiritual growth. It aims to keep kids healthy and build a strong math foundation, critical for their future in high schools or vocational schools.

## L.G. Peterson's Teaching Method

L.G. Peterson's teaching method meets course goals across all educational levels. It also considers children's mental growth. It uses the methodological version of the systemic-activity approach.

### Main Course Features

The course's activity-based method presents full mathematical activities. Students cover all course sections, learning, applying, and using new knowledge. This method also includes all three stages of mathematical modeling:

• The stage of mathematizing reality involves constructing a mathematical model of a specific fragment of reality.

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

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

Students create math models to explain real-world objects and processes. They also study phenomena and their connections. This boosts their skills in math, logic, algorithms, and creativity. Moreover, they learn to check properties and relationships. They can then visually represent data and build and apply 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. Students in this program learn to analyze data and communicate math ideas well. A unique aspect is starting with a real problem to solve. To model the problem, they have to expand their math skills. This introduces them to key equations and inequalities. It helps them see how algebra connects to the real world. This approach also motivates high school students to study math.

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." In 7th grade, any new function is introduced by considering practical problems for which the function provides a generalized description.

The program's key feature is its problem-solving sections. Unlike textbooks, these don't just review new knowledge but also encourage students to explore new ideas and techniques.

First, students use what they've learned to solve problems, which helps them learn and think critically. Then, an assignment shows what they still need to discover.

Next, questions and tasks guide students. They can learn through observation, experiments, and more. They apply and adapt past methods to new challenges. They also make and support hypotheses. Finally, their results are compared with the topic's theoretical material.

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:

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

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

3. 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.

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

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

This tutorial helps students study on their own. It's key for their learning and professional growth. The problem-solving section offers tasks for exploration. The course also offers exercises to apply new knowledge. It features tasks of different difficulties, including Olympiad-level problems. This method aligns with adolescents' psychological traits. It highlights a unique self-awareness called the "sense of adulthood." This self-awareness arises during their transition to adulthood. 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.

This structure, designed for the activity-based method by L.G. Peterson, considers teens' traits. These include their love for daydreaming and hasty future planning. It shifts focus from results to creativity. A teacher who just grades work, ignoring student ideas, lowers the value of learning. The problem-solving section in the textbook promotes hands-on learning for 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 covers topics like math logic, divisibility, and solving linear equations and inequalities. It also includes practical applications such as "Functional Dependency and Information Encoding."

Modern math theory divides education into seven lines. These start in preschool and continue through high school. The lines are modeling, logical, numerical, algebraic, geometric, functional, and data analysis. The course stays coherent by comparing and interpreting results across these 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, we studied logical concepts and the general methods of math. We mastered these methods, which set a solid foundation for studying math in 7th grade and higher grades.

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 study of "algebra," the geometric line is not developed past 7th grade. It is a background for studying other courses.

### Numerical line

In elementary school, the numerical line was constructed based on counting objects (elements of a set) and measuring quantities.

Students learned natural numbers, zero, and how to write and compare non-negative integers. They also explored arithmetic operations, their properties, and mental and written calculation methods. Further, they learned to estimate, evaluate, and verify results. Finally, they discovered how to solve for unknowns. They also tackled quantities and measurement, performing operations with them.

The teaching method was activity-based, keeping the traditional math curriculum, which made it richer for children's development. For example, in 3rd grade, they focused on methods and operations for integers up to 12 digits. In 4th grade, they moved on to fractions, adding and subtracting 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 number concept and the method of expanding number sets.

In advanced classes, students learn about numbers' divisibility. They study the concept, properties, and tests for numbers like 2, 5, 10, and more. They also delve into GCD, LCM, prime, and composite numbers.

In 7th grade, students review prime and composite numbers. They learn the fundamental theorem of arithmetic, which states any number can be expressed as a product of primes. They also study canonical prime factorization and improve their skills with the Euclidean algorithm for finding the GCD. Additionally, they explore known and new divisibility properties. These lessons align with the course's algebraic focus.

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 theory of divisibility is valuable for its beauty and methodical study of known properties. It also teaches the axiomatic method for building theories. Students learn to solve divisibility problems, handy for math contests 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.

• Understanding the concept of exponentiation and applying its properties.

• Solving equations.

• Solving inequalities.

In 7th grade, students review arithmetic laws, equivalent expressions, and transformations. With this knowledge, they create rules for similar operations. Then, they apply these rules to symbolic expressions. Now, they justify these steps with a new approach. The course provides strong algebra training. So, most seventh-graders should not have trouble with such tasks. Unlike the earlier stage of education, doing the same manipulations on algebra at this stage is a necessary skill.

Furthermore, students are introduced to transformations of algebraic expressions involving products and quotients. The rules let students do equal operations on products. They can learn to change algebraic fractions and expressions with division.

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In the 7th grade, students develop the concepts of a monomial and a polynomial, their standard form, and their power.

In 7th grade, students learn to work with monomials and polynomials. They add, subtract, and multiply them. They also simplify algebraic multiplication by understanding key formulas. These include square and cube formulas, as well as the product and sum of cubes. Additionally, students tackle challenges with binomials and Pascal's triangle.

Furthermore, they dive into polynomial factoring. Techniques range from grouping to recognizing common factors. Students also learn to simplify by rearranging terms and identifying complete squares. Factoring becomes a key tool in solving equations and simplifying 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 grasp exponents with fractions. They learn them as they get to know numbers in the 5th and 6th-grade courses. They do this through 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 7th grade, students must master two key skills. First, they learn the natural exponentiation of rational numbers. Second, they grasp using exponents to simplify expressions and solve problems. Also, they discover how to raise rational numbers to powers. In 7th grade, students learn about zero exponents and exponent properties. They use these to manipulate expressions. Additionally, they grasp the concept of equations. They understand variables and how to find solutions. Students also learn to solve equations using equivalent steps. Moreover, they tackle equations with absolute values for the first time.

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.

First, students learn about quadratic equations and how to solve them. They also explore biquadratic equations.

Then, they move on to algebraic fractions, learning their properties and operations. Next, they solve fractional-rational equations, aiming to turn fractions into whole numbers. They consider the domain, setting fractions to zero, and the property of proportion. Advanced students explore methods like substitution and simplification.

By 7th grade, students can solve simple inequalities and show the solutions on a number line. They then dive deeper into inequalities, including strict and non-strict forms. They also study piecewise linear functions and their representations on a number line.

Therefore, when studying inequalities, they review numerical intervals such as open and closed rays and also become familiar with intervals of the form (x,y).

In 7th grade, students learn about equivalent inequalities and their transformations, as well as the rules for these transformations. 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.

### 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.

In 7th grade, students use tools learned from grades 1-6, like diagrams and tables, to solve problems. They apply a problem-solving method through mathematical modeling and improve it as they go. They also discovered a mathematical model can be more than just an equation. It can also be an inequality or a set of relationships showing connections between quantities.

### 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 7th grade, due to an intro, students know what a variable is. They can use a coordinate plane and have graphed from 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. In sixth grade, students learned about relationships between quantities. They saw these as specific dependencies. Now, in seventh grade, they encounter a key math concept: functions. They'll revisit the concept's real-world uses, like predicting events and encoding information. Then, they'll learn its name... As a result, students develop the concept of a function as a rule that associates elements of two sets of arbitrary nature.

In this course, we use a methodical approach. Firstly, we introduce functions in 7th grade by looking at the practical problems they solve. These include direct proportionality, linear, and piecewise-linear functions. Notably, students start working with motion graphs in 4th grade. They then move on to study piecewise linear functions. The approach is simple. We analyze each function, draw its graph, and identify its properties.

### Logical Line

The course focuses on building logical thinking. In grades 5-6, students learn a sequence of ideas. They start with math terms and move to proofs and definitions. They also explore negation, implication, and theorems. In 7th grade, they review these topics. They refine their definitions and learn proof by contradiction. They also study logical reasoning using Euler-Venn diagrams. Additionally, they look at common logic mistakes and their types.

### Stochastic Line

The stochastic line aims to develop information literacy among students starting from the early school years. In 7th grade, students learn to gather and organize information. They then use it for practical tasks. Teachers suggest they practice this in class and in projects. For example, they can create presentations, problem collections, and informational leaflets. 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 in grades 5-7, students learn to explore options systematically. They use logical selection, tables, and decision trees 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 7th grade, students encounter process statistics and become familiar with the measures mean, mode, median, and range.

### 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. So, you must get to know the materials for 5th and 6th grades and the benchmarks students learned in primary school.

### Assessment System

In lessons, we focus on new topics, self-study, and creative tasks. We only assess success. Mistakes are corrected by understanding their causes, such as unmastered rules or definitions. Then, we use self-assessment during reflection sessions. Finally, students decide their grades.

All students are given marks for tests, and the difficulty level is 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. Educational standards determine the minimum required learning outcomes according to the curriculum.

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### Homework

Homework consists of two parts:

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

2. 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.

Students do self-assessment, error correction, and marking. They use a sample provided by the teacher. This happens at the start of the lesson. 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

L.G. Peterson's 7th-grade algebra book aims to build a strong math foundation for students. It tailors the teaching method to their development. The book uses activities to enhance skills in modeling, problem-solving, and critical thinking. It covers topics like numbers, divisibility, algebra, equations, inequalities, and exponents. The course structure encourages self-learning. It also includes practical problems, physics applications, and real-world examples. This approach shows the importance of math in daily life. In short, the book offers a solid, organized teaching method. It prepares students for higher math education and careers.

Would you like your child to be challenged in math?

Join our exclusive online Russian Math BYOM classes!

 Private Lessons | Group Lessons