BYOM 3rd Grade Workbook

Methodological Features

Our Russian Math Books create a complete learning path.

The program spans from preschool through grade 3.

Key Educational Goals

We focus on three main areas:

• Personal growth and character development
• Cross-subject learning skills
• Mathematical understanding

1. Focus on Personal and Cross-Subject Educational Outcomes

Math is a tool, not just a subject.

We use it to develop:

• Critical thinking
• Emotional intelligence
• Creative problem-solving
• Motivation to learn

We use L. Peterson's activity method to achieve this.

The activity-based method technology assumes the following structure for introducing new knowledge in lessons:

• Motivation (self-determination) for activity.
• Activation of prior knowledge and identification of difficulties through a trial learning action.
• Identification of the location and cause of the difficulties.
• Construction of a project to overcome the difficulties.
• Implementation of the constructed project.
• Initial consolidation through verbalization in external speech.
• Independent work with self-assessment based on a standard.
• Integration into the knowledge system and review.
• Reflection on the activity (lesson summary).

The effective use of activity-based teaching relies on key didactic principles. These include activity, continuity, a holistic view of the world, minimax, psychological comfort, variability, and creativity. Each principle plays a vital role in educational work and management that supports children's health. L. Peterson's educational system creates a unified approach to learning and health preservation.

2. Levels of Implementation of the Educational System

L. Peterson's educational system can be applied at three levels: basic, technological, and systemic-technological.

The basic level of the activity-based method (ABM) includes the following 7 steps:

1. Motivation for learning activity.
2. Activation of prior knowledge.
3. Problematic explanation of new knowledge.
4. Initial consolidation through verbalization.
5. Independent work with self-assessment.
6. Integration of new knowledge into the knowledge system and review.
7. Lesson summary.

When using ABM at a basic level, the principle of activity shifts to activating activity in traditional teaching. It's crucial to focus on the principles of minimax and psychological comfort. Using these principles helps each student learn at their own pace, tackling challenges at a "maximum" difficulty they can handle. Ignoring them can overload students.

3. Connection with Practice and Real-World Problems

Students cannot fully learn math without understanding the origins and significance of its concepts. They also need to see math's role in the sciences. Thus, a key task for the school curriculum is to show students the three stages of forming mathematical knowledge.

These stages are:

1. Mathematization: Creating a mathematical model of a part of reality.
2. Studying the Model: Developing a mathematical theory that explains the model's properties.
3. Applying Results: Using the findings in real-world situations.

Natural numbers aren't just abstract ideas; students first learn about groups of objects.

When they study addition and subtraction of natural numbers, students begin by adding and removing items from these groups. Understanding how to add and subtract two-digit numbers starts with visualizing them as dots and shapes, reflecting how these operations developed over time.

4. Continuity Between Preschool, Primary, and Secondary School

Continuity between preschool, primary school, and secondary school involves technology, content, and methods. This ensures a smooth educational journey at all levels.

The choice of content and order of learning key math concepts follows a systematic approach. It builds a multi-level foundation for math concepts, promoting continuity and ongoing development of both content and teaching methods.

Preschool programs help develop critical thinking and cognitive skills. They also foster positive communication and collaborative problem-solving through reflexive self-organization. This preparation lays a strong foundation for adapting quickly and successfully to school learning.

5. Formation of Thinking Style for Technology Use

The computerization of the surrounding world leads to a reevaluation of the importance of many skills and abilities. Of particular importance, for example, is the ability to create and implement an action plan, strictly adhere to given rules and algorithms, assess the plausibility of the obtained answer, consider alternative solutions, organize the search for information necessary to solve a given task, and more.

Thus, in the BYOM curriculum, all tasks in the subject area "Mathematics and Informatics" are successfully solved.

6. Multi-Level Nature of the Textbook

For All Students, No Selection Required

The textbook works with all preparation levels.

Based on minimax and psychological comfort principles.

Teacher preparation matters most, not student selection.

High-Level Teaching Approach

Teaching targets the "maximum" difficulty level.

Lessons operate in the zone of proximal development.

Individual characteristics are always considered.

Every child develops self-belief.

What This Looks Like in Practice

• Challenging tasks from day one
• Fast-paced intellectual work
• All students face productive challenges

How We Evaluate

Progress measured against the child's own past performance.

Errors are learning opportunities for correction.

Causes are identified and addressed together.

Ongoing and final assessments are easier than classwork, which makes low grades rare. Final grades come from "achievements" (only A and B) and test scores. Bad grades show up only when a child neglects tasks or doesn't complete what they can handle. It's better if children assign themselves a negative grade based on class norms.

High-quality material presentation is not mandatory but an opportunity for success. Teachers should notice and support any success, no matter how small. This includes a child's activity, involvement in solving problems, or their own ideas. If a student answers incorrectly, the teacher should avoid negative reactions. Instead, another student can help correct them by asking, "What do you guys think?" The teacher's role is to support the student: "Well done! You helped us understand!" "Do you agree now? Did you understand? Well done!"

The minimax principle helps with multi-level learning, so there's no special selection of children. Engagement and reflection are vital, especially for those with developmental difficulties. High difficulty should come with trust, respect, and friendliness in the classroom. Students need to believe in their abilities: "You can do it!" from the teacher, "I can do it!" from the student, and "They can do it!" from peers. Without this support, learning loses personal meaning, and the school can't help children reach their potential.

The textbook tasks set the level for advanced children. Not every child will complete all tasks. Only 3-4 key assignments on new topics and review tasks are mandatory for everyone. More tasks can be offered to advanced students, but overloading, including homework, should be avoided.

Knowledge reinforcement happens alongside exploring new mathematical ideas (like geometry and algebra). Practice exercises are engaging and often playful (like code decoding and riddles). Each child can practice at their own pace while advanced students receive intellectual challenges. This keeps math lessons fun for everyone, whether strong or less prepared.

Every child should feel the joy of discovery in each lesson. This builds self-confidence and curiosity. Interest and success in learning are key to the child's overall development and the quality of teaching.

7. Creative Tasks in the Work System

Creative Work Builds Potential

Every child can unfold their abilities through creative tasks.

Offer creative work 2-3 times per week.

Types of Creative Tasks

• Create examples for learned techniques
• Design problems from given expressions
• Build problems around specific themes (sports, animals, stories)
• Draw geometric patterns with specified properties
• Encode/decode using computational examples

Assessment Approach

Creative tasks are additional homework.

Never given poor grades - only positive evaluation.

Student-Authored Problem Books

Collect successful works at year-end.

Students become creators, not just performers.

This builds personality and deepens knowledge retention.

8. Volume and Difficulty Level of Homework

It is recommended to provide students with two-level homework assignments, consisting of a mandatory and an optional (additional) part.

The mandatory part should be manageable for the child to complete independently and should not exceed 15-20 minutes of their

independent work time. It is also recommended to give assignments based on the children's own choice, for example: "Choose and

complete one task from # 4-7 that you like."

The optional part, which is done at the student's discretion, may include additional tasks marked with an asterisk, etc.

9. Types and Forms of Work in the Lesson

It is important to diversify work types in lessons. Include collective, group, and individual activities. Use oral work and tasks in squared notebooks. Practice computational skills should be systematic and intensive but last no more than 3-4 minutes. Choose answer numbers for exercises to help students analyze, classify, and find patterns. This approach reinforces counting skills and prepares children for activity-based learning.

When forming concepts, engage all types of memory: visual, auditory, motor, tactile, and others. Work in the exercise book should last no more than 10-12 minutes. During this time, students complete tasks prepared in advance. Independent task completion is typically limited to 1-3 minutes. Afterward, students check their work by comparing it to a model answer, giving themselves a "+" or "-". This encourages self-assessment skills.

Since students check their own tasks, the teacher's focus shifts to developing self-control skills and ensuring neat note-taking.

10. Knowledge Assessment System

The curriculum features a multi-level assessment system. It includes self-assessment for new material, peer assessment during practice, and instructional assessment through independent assignments. Periodic tests occur throughout the year, with a final assessment that has two parts: a transfer test ("minimum") and a final test that includes assessment and self-assessment of mastery.

Independent assignments are tough, so only successful completions are graded. If a child finishes their work without mistakes, which happens for about 3-5 students, they earn an excellent grade. After each assignment, students who make errors focus on correcting them.

If a child corrects their mistakes well, the teacher may give a B or even an A. Low grades are not given for independent work. Instead, "no grade" means the child should be more proactive in fixing their mistakes. The teacher helps each child understand and resolve their errors.

Tests should be easier than independent assignments, and all students must be evaluated. Tasks should be chosen so that about three-quarters of the class can earn an A or B.

In the BYOM program for 3rd grade, students build universal learning actions (personal, regulatory, cognitive, communicative) through key topics. These include numeration, comparison, addition, and subtraction of multi-digit numbers, along with written multiplication and division methods. Students practice mental calculations within 1000 until they become automatic.

Set-theoretical concepts are introduced, unit conversion problems are included, and composite problems using all four operations are provided. Special attention is given to helping students analyze word problems, create geometric representations, and identify relationships and dependencies (topic: "Formulas"). This approach also fosters logical thinking and cognitive skills.

Each lesson should include intensive exercises for computational skills. It's best to choose numbers for these exercises that help children analyze, classify, and identify patterns in the results.

Personal Results

The Student Will Develop:

• an understanding of educational and corrective activities, their similarities and differences;
• an understanding of the general nature of mathematical knowledge, its history of development, and ways of mathematical cognition;
• independence and personal responsibility for their performance in executive activities, their own experience of creative activities;
• the ability to perform self-assessment based on a model, detailed example, and standard;
• experience in reflective self-assessment of their own educational actions;
• the ability to correct errors based on a refined error correction algorithm;
• the ability to apply rules for maintaining and supporting their health in educational activities;
• a desire to make maximum personal contributions to collaborative activities;
• the ability to apply the rules of "author," "comprehender," and "critic" during communicative interactions in pairs and groups;
• motivation to develop speech as a means of successful communication in educational activities;
• activity, friendliness, honesty, and patience in educational activities;
• determination in educational activities based on agreed-upon standards;
• interest in mathematics classes and educational activities as a whole;
• an understanding of friendship, self-confidence, self-criticism, and acceptance of them as values that help the student achieve good results;
• respectful and positive attitude towards oneself and others, aiming for maximum personal contribution to the overall result and striving for common success;
• experience in using constructive behavior strategies in difficult situations and resolving disputes based on reflective methods;
• experience in independent successful mathematical activities according to the 3rd-grade curriculum.

The Student Will Have the Opportunity to Develop:

• the ability to adequately assess their performance and treat negative results as signals prompting them to rectify the situation;
• the ability to establish friendly relationships with classmates and self-assess this ability based on the application of a standard;
• experience in using techniques to mitigate negative emotions when working in pairs or groups;
• experience in distinguishing true and false values;
• positive experience in constructive and creative activities.

Metasubject Results

The Student Will Learn:

• to identify and record the progress of two main stages and steps of educational activities (12 steps);
• to identify individual difficulties in educational activities in various typical situations;
• to determine the place and cause of individual difficulties in educational activities based on the application of a standard;
• to create a plan for their educational activities when encountering new knowledge based on the application of an algorithm;
• to record the result of their educational activities during the lesson on the discovery of new knowledge in the form of a coordinated standard;
• to use the standard to justify the correctness of completing educational tasks;
• to apply the rule of consolidating new knowledge;
• to use given criteria to assess their own work;
• to identify and record the progress of two main stages and steps of corrective activities (12 steps);
• to use the error correction algorithm in their educational activities;
• to apply the refined algorithm for completing homework;
• to use the mathematical terminology learned in the 3rd grade to describe the results of their educational activities.

The Student Will Have the Opportunity to Learn:

• to perform project activities under the guidance of an adult;
• to conduct self-assessment of the ability to apply rules that foster self-confidence based on the application of a standard;
• to conduct self-assessment of the ability to identify and record the progress of two main stages and steps of educational activities (12 steps);
• to conduct self-assessment of the ability to determine the place and cause of difficulty when constructing a new method of action;
• to conduct self-assessment of the ability to plan their own educational activities;
• to conduct self-assessment of the ability to record the result of their educational activities in the form of a standard;
• to conduct self-assessment of the ability to use the standard to justify the correctness of completing educational tasks;
• to conduct self-assessment of the ability to apply the rule of consolidating new knowledge;
• to conduct self-assessment of the ability to apply given criteria to assess their own work;
• to conduct self-assessment of the ability to identify and record the progress of two main stages and steps of corrective activities (12 steps);
• to conduct self-assessment of the ability to determine the place and cause of their own mistakes;
• to conduct self-assessment of the ability to use the error correction algorithm in their educational activities;
• to conduct a self-assessment of the ability to apply the refined algorithm for completing homework.

Cognitive

The Student Will Learn:

• to understand and apply mathematical terminology to solve educational tasks according to the 3rd-grade curriculum;
• to apply algorithms of generalization and classification of a set of objects based on a given property;
• to apply simple techniques for developing their memory;
• to use observation as a method of cognition in their educational activities in simple cases;
• to identify types of models (object models, graphical models, symbolic models, algorithm flowcharts, etc.) and use them as a method of cognition in their educational activities in simple cases;
• to differentiate between the concepts of "knowledge" and "skill";
• to understand and apply basic interdisciplinary concepts in accordance with the 3rd-grade curriculum (set, element of a set, subset, union, and intersection of sets, Euler-Venn diagram, enumeration of options, decision tree, etc.);
• to create and solve their own problems, examples, and equations according to the 3rd-grade curriculum;
• to understand and apply the signs and symbols used in the 3rd-grade textbook to organize their educational activities.

The Student Will Have the Opportunity to Learn:

• to assess their ability to apply algorithms for generalizing and classifying objects by a given property;
• to evaluate their understanding of the stages of the observation method in their studies;
• to assess their ability to identify model types and the stages of modeling in their learning;
• to evaluate their skills in using simple memory techniques;
• to apply learned methods and tools to solve educational tasks;
• to find and correct arithmetic and logical errors in calculations and word problems;
• to use knowledge from the 3rd-grade curriculum in different situations;
• to tackle creative and exploratory problems based on the 3rd-grade curriculum.

Communicative

The Student Will Learn:

• to distribute roles in communicative interactions, formulate the functions of "author," "comprehender," and "critic," and apply the rules of working in these positions;
• to offer their own solutions in collaborative work, and evaluate different options based on the common goal;
• to apply basic techniques of public speaking in order to express their thoughts clearly to others during dialogues;
• to apply the rules of dialogue when working in pairs or groups;
• to apply simple techniques for managing negative emotions in collaborative activities;
• to exercise mutual control, provide assistance, and support classmates when necessary.

The Student Will Have the Opportunity to Learn:

• to conduct self-assessment of the ability to perform the role of "critic" in communication, based on the application of a standard;
• to conduct self-assessment of the ability to express their thoughts clearly to others based on the studied techniques of public speaking;
• to conduct self-assessment of the ability to apply the rules of dialogue when working in pairs or groups, based on the application of a standard;
• to conduct self-assessment of the ability to manage and control negative emotions in collaborative work, based on the application of a standard;
• to conduct self-assessment of the ability to exercise mutual control;
• to demonstrate friendliness when working in pairs or groups.

Subject Outcomes

Numbers and Arithmetic Operations

The Student Will Learn:

• to count in thousands, name place values and classes: units class, thousands class, millions class, etc.;
• to name, compare, add, and subtract multi-digit numbers (up to 1,000,000,000,000), represent a natural number as the sum of its place value parts;
• to multiply and divide numbers by 10, 100, 1000, etc., multiply and divide (without remainder) round numbers within the range of 100;
• to multiply multi-digit numbers (all cases) and write multiplication in columns;
• to divide a multi-digit number by a single-digit number and write division in the form of long division;
• to verify the accuracy of operations with multi-digit numbers using algorithms, inverse operations, and calculators;
• to mentally add, subtract, multiply, and divide multi-digit numbers in cases within the range of 100;
• to perform specific cases of all arithmetic operations with 0 and 1 on the set of multi-digit numbers;
• to apply the properties of arithmetic operations to the set of multi-digit numbers;
• to evaluate the values of numerical expressions involving the studied natural numbers, containing 4-5 operations (with and without parentheses) based on the knowledge of the order of operations;
• to simplify calculations with multi-digit numbers based on the properties of arithmetic operations.

The student will have the opportunity to learn:

• to independently develop and use algorithms for mental and written operations with multi-digit numbers;
• to express multi-digit numbers in various larger units of measurement;
• to see the analogy between the decimal system of representing natural numbers and the decimal system of measurement.

Problem-Solving with Word Problems

The Student Will Learn:

• perform all arithmetic operations with the studied quantities when solving problems.

The Student Will Have the Opportunity to Learn:

• independently construct and use algorithms for solving the studied cases of word problems;
• classify simple problems of the studied types by model type;
• apply a general method of analysis and solution for composite problems (analytical, synthetic, analytical-synthetic);
• analyze, model, and solve word problems involving 5-6 steps using all arithmetic operations within the range of 1,000,000;
• solve non-standard problems related to the topics being studied.

Geometric Shapes and Quantities

The Student Will Learn:

• perform on grid paper the transfer of shapes by a given number of cells in a given direction;
• determine the symmetry of points and shapes with respect to a line based on essential symmetry features;
• construct symmetric shapes on grid paper with respect to a line;
• identify and name shapes that have an axis of symmetry;
• recognize and name rectangular parallelepiped, cube, their vertices, edges, and faces;
• find the volume of a rectangular parallelepiped and a cube using formulas;
• find the areas of shapes composed of squares and rectangles;
• read and write the studied geometric quantities, convert them from one unit of length to another, compare their values, add, subtract, multiply, and divide them by natural numbers.

The Student Will Have the Opportunity to Learn:

• construct nets and physical models of a cube and a rectangular parallelepiped;
• find the surface areas of a rectangular parallelepiped and a cube;
• independently derive properties of geometric shapes being studied;
• use measurements to independently discover properties of geometric shapes.

Quantities and relationships between them

The Student Will Learn:

• Recognize, compare, and order "time" quantities. Use time units—1 year, 1 month, 1 week, 1 day, 1 hour, 1 minute, 1 second—to solve problems and perform arithmetic operations.
• Tell time, name months and days of the week, and use a calendar.
• Use new mass units—1 gram, 1 kilogram, 1 centner, 1 ton. Convert, compare, and perform arithmetic operations with these units.
• Observe relationships between quantities with tables and motion models on a coordinate line. Verbally express these relationships and use formulas like:
• Construct a generalized multiplication formula ( a = b ⋅ c ) for uniform processes.
• Build models of object motion on a number line. Observe relationships describing motion and create related formulas.
• Compose and compare simple expressions with variables. Find their values for given variable inputs in simple cases.
• Apply relationships between the components and results of arithmetic operations to compare expressions.

The Student Will Have the Opportunity to Learn:

• Create and present a project on the history of measuring time. Include calendars, and details about the Julian and Gregorian calendars.
• Observe relationships between variables using tables and number lines. Express these relationships with simple formulas.
• Independently build a scale with a set division value and a coordinate axis. Create formulas for distance between points and the relationship between a moving point's coordinate and time.
• Determine motion parameters: starting point, direction, and speed. Use formulas like x = a + bt and x = a - bt to show how the coordinate x depends on time t.

The Student Will Learn:

• to express the properties of arithmetic operations on the set of multi-digit numbers using algebraic notation;
• to solve simple equations of the form a + x = b, a - x = b, x ⋅⋅ a = b, a ⋅⋅ x = b, a + x = b, x + a = b, providing commentary on the steps of the solution;
• to solve compound equations that can be reduced to a chain of simple equations (2 steps), and provide commentary on the steps of the solution;
• to apply the formula for division with remainder a = b ⋅⋅ c + r, r

The student will have the opportunity to learn:

• to read and write expressions containing 2-3 arithmetic operations, starting from the name of the last operation;
• to independently identify and write in algebraic notation the formula for division with the remainder a = b ⋅⋅ c + r, r
• based on the general properties of arithmetic operations in simple cases:
• to determine the set of roots of non-standard equations;
• to simplify algebraic expressions.

Mathematical Language and Elements of Logic

The Student Will Learn:

• to use symbolic notation for multi-digit numbers, indicate their digits and classes, and represent spatial shapes;
• to recognize, read, and apply new symbols of mathematical language: set notation and its elements, symbols ∈, ∉, ⊂, ⊄, ∅, ∩, ∪.
• to define sets by their properties and by listing their elements;
• to determine the membership of elements in a set, equality and inequality of sets, and to determine if one set is a subset of another set;
• to find the empty set, union, and intersection of sets;
• to represent relationships between sets and their elements, as well as set operations, using Euler-Venn diagrams;
• to differentiate between statements and propositions that are not statements;
• to determine true or false statements in simple cases;
• to construct simple statements using logical connectors and words such as "true/false, that...", "not", "if...then...", "each", "all", "exists", "always", and "sometimes".

The Student Will Have the Opportunity to Learn:

• to justify their judgments using the rules and properties learned in 3rd grade and make logical deductions;
• to justify statements of a general nature and statements of existence in simple cases based on common sense;
• Explore the commutative and associative properties of set union and intersection. Use mathematical symbols to represent these properties. Also, draw an analogy between these properties and the commutative and associative properties of addition and multiplication.
• Solve logical problems using Euler-Venn diagrams.
• Tackle logical problems with help from an adult and also work independently. Master problem-solving techniques in line with the 3rd-grade curriculum.

Working with Information and Data Analysis

The Student Will Learn:

• to use tables for analyzing, representing, and organizing data, and to interpret data from tables;
• to classify elements of a set based on a given property;
• Find information on a topic from various sources, like textbooks, reference books, encyclopedias, and controlled Internet space.
• Carry out project work on topics such as "From the History of Natural Numbers" and "From the History of the Calendar." Plan information searches in reference books, encyclopedias, and controlled Internet space. Present the results of the projects.
• Complete creative projects themed "Beauty and Symmetry in Life."
• Work with materials and tools in primary education, focusing on the subject "Mathematics, Grade 3."

The Student Will Have the Opportunity to Learn:

• Conduct extracurricular project work independently, with adult guidance. Gather information from literature, reference books, encyclopedias, and controlled Internet sources. Present findings using available technology.
• Create problems based on the 3rd-grade curriculum. Become a co-author of the "Problem Book for Grade 3," featuring the best problems invented by students. Compile a portfolio as a 3rd-grade student.

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