BYOM Lesson Workbooks
# BYOM 4th Grade Workbook

The proposed mathematics course for primary school is part of a unified continuous mathematics course for preschoolers, primary school, and grades 4-5 of secondary school. It is designed to achieve new educational objectives, including personal, cross-curricular, and subject-specific learning outcomes, as well as readiness for self-development based on fostering students' cognitive motivation, universal learning actions, and overall learning skills. Mathematical knowledge in the course is not considered an end in itself but rather a means of developing specific personal and cross-curricular educational outcomes, mathematical activities, and the development of children's thinking, senses, emotions, creative abilities, and motivational factors. The stated objective is achieved through the use of the didactic system of the activity-based method by L. Peterson. The technology of the activity-based method involves the following structure for introducing new knowledge in lessons:

The educational system can be implemented at different levels: the basic level, the technological level, and the system-technological level. The basic level of the activity-based method (ABM) includes the following 7 steps:

The technological level of ABM implementation refers to the level of teacher's work where the transitional structure (8 steps) and the system of didactic principles of Peterson’s educational system are implemented. Concepts such as benchmark, self-assessment benchmark, and detailed model are included in the practice, and motivation for cognitive activities is organized at the levels of "I want" and "I can". The technological level of ABM implementation allows for the following:

Connection with practice, real-world problems Full-fledged mathematics education is impossible without children understanding the origin and significance of mathematical concepts and the role of mathematics in the system of sciences. Therefore, one of the main tasks of the school curriculum is to reveal to students all three stages of mathematical knowledge formation. These stages are:

The above-mentioned demonstrates
how the first stage of mathematical modeling, the construction of mathematical
models of the surrounding world, is reflected in the mathematics curriculum for
first grade. The second stage, intra-model investigation, is associated with
the study of addition and subtraction operations with single-digit numbers,
constructing an addition table, and studying operations with two-digit numbers.
Finally, the third stage is reflected in solving word problems where the
learned operations with numbers are applied in practical contexts.
Continuity between preschool
education, primary school, and secondary school is achieved in the curriculum
through the integration of technology, content, and methodologies, ensuring a
seamless educational process across all stages of learning. The selection of
content and the sequence of studying key mathematical concepts were based on a
systemic approach.
The development of the thinking
style necessary for the successful
use of ICT (Information and Communication Technology) leads to a reassessment
of the importance of many skills and abilities. For example, the ability to
create and implement an action plan, strict adherence to given rules and
algorithms, assess the plausibility of the obtained answer, explore
various solution options, organize information search required to solve a
given problem, and more, gain particular significance. Therefore, Peterson’s
mathematics course effectively addresses all the objectives of the subject area
"Mathematics and IT.«
The course has a differentiated
approach, allowing children of different levels of preparation in schools and
classes of all types to work with it based on the principles of minimax and
psychological comfort. The selection of children to work with the course is not
assumed based on their level of preparation, but rather on the teacher's level
of preparation.
The instruction is conducted at a
high level of difficulty (the "maximum" level), meaning it is within
the "zone of proximal development" of the most advanced children,
while taking into account their individual characteristics and abilities, and
fostering each child's belief in themselves and their abilities.
Practically, this means that BYOM
presents tasks and the pace of learning that are of a sufficiently high level.
From the very beginning, all children are placed in situations that require
intellectual effort and productive actions. However, in instructional tasks and
independent work, only the child's success and progress relative to themselves
are evaluated. Mistakes are seen as working situation that requires correction,
identification of the cause, and rectification. Ongoing and final assessments
are conducted at a lower level than classroom work, which practically leads
to the disappearance of low grades. Final grades are based on the number of
"achievements" (which are evaluated with high grades only) and marks
for tests. Low scores may appear very rarely, only when a child has shown
negligence or failed to complete an assigned task that is clearly within their
capabilities. It is preferable if the child themselves gives them a negative
grade in accordance with the norms established in the class.
However, the high level of
material presentation is not considered as a mandatory requirement, but rather
as a suggestion and an opportunity for each child to achieve success,
motivating them to take action. Therefore, the teacher should notice and support
any, even the smallest, success of the child - their activity, involvement in
the process of finding a solution, their accurate judgment, or simply an
attempt to propose their own hypothesis. A student's incorrect answer should
not evoke a negative reaction from the teacher, annoyance, or lecture. It is
better if one of the students corrects the answer, saying, "What do you
guys think?" In this situation, it is the teacher's role to provide moral
support to the one who made the mistake: "Well done! You helped us
understand!" or "Do you agree? Do you understand now? Well
done!" and so on.
The principle of minimax is a
self-regulating mechanism for multi-level learning. As mentioned above, it does
not require any special selection of children to work with it. Furthermore,
engagement in educational activities, internal activity, and the development of
a habit of reflecting on each step is particularly important for children with
developmental issues. However, working at a high level of difficulty must be
accompanied by creating an atmosphere of trust, respect, and friendliness in
the classroom, allowing each student to believe in their abilities and truly
"unfold themselves." The teacher must believe in the student, saying,
"You can do it!" The students must believe in themselves, thinking,
"I can do it!" and all the other students in the class should believe
in their classmates, saying, "They can do it!" Otherwise, education
will lose its personal significance for the child, and the school will not be
able to fulfill its main mission - to help the child achieve their individual
maximum.
The volume of tasks in the course
sets the level of the individual educational trajectory for the most prepared
children. Therefore, it is not expected that every child will complete all the
tasks. Only 3-4 key tasks on each new topic and review tasks are mandatory for
everyone, which reinforces the essential learning outcomes. The range of tasks
can be expanded for more advanced children. However, overloading children,
including with homework, should be avoided.
The consolidation and
reinforcement of knowledge in the main content and methodological lines of the
course (numerical line, text-based problem-solving) are conducted in parallel
with the exploration of new mathematical ideas in additional lines (geometry,
algebra, data analysis, etc.). Therefore, the practice exercises do not tire
the children, especially since they are usually presented in a playful form
(code cracking, riddles, etc.). Each child with a lower level of preparation
has the opportunity to gradually master the necessary skills from the essential
learning outcomes, while more advanced children are constantly provided with
intellectual stimulation, making mathematics lessons appealing to all children,
both strong and less prepared.
It is fundamentally important for
each child to experience the joy of discovery and develop confidence in their
abilities and curiosity in every lesson. Interest and success in learning are
the key parameters that determine the comprehensive moral, intellectual, and
physiological development of a child, and therefore, the quality of working
with children.
Some of the topics included in
the 4th-grade curriculum have traditionally been taught in the 5th grade. The
decision to introduce them at an earlier stage is driven by several reasons.
One of the most significant reasons is the need to take into account sensitive
periods, which are the most favorable periods from a psychological perspective
for acquiring certain knowledge. For example, in the 3rd grade, learning about
multi-digit numbers within the range of 12 digits is easier than in the 5th
grade, because children enjoy working with "long" numbers.
By the 4th grade, students
accumulate fatigue from complex calculations, and they are not only logically
but also emotionally ready for the next step - the introduction of fractions.
The new numbers, their properties, and the algorithms that students construct
on a tangible basis evoke surprise, joy, and sometimes even enthusiasm from
their independent victories (which teachers in higher grades do not often
observe when teaching fractions in the 5th or 6th grades). Through this
emotional reinforcement, necessary training in operations with whole numbers is
conducted simultaneously with the study of fractions, and the corresponding
techniques for mental and written calculations are brought to the level of
automated skill.
The transfer of this material
from the 5th grade to the early years of the school became possible thanks to
the new methods proposed in the curriculum, such as graphical modeling of
text-based problems, associative techniques for solving equations, and others,
which significantly reduced the time required for studying many topics in the
curriculum.
In the 4th grade, there are
several successful methodological innovations. Among them, we can mention the
technique for solving fraction (percentage) problems, problems involving
simultaneous motion, solving inequalities with natural numbers, exploring the
properties of geometric shapes through constructions and measurements, reading
and constructing motion graphs, and more. For example, the timely introduction
of the percentage symbol (%) in the 4th grade as one of the notations for a
hundredth part of a quantity helps eliminate the difficulties that students in
higher grades face when studying percentages. It also improves the quality of
solving simple fraction and percentage problems (for cases involving operations
with numbers) in the future.
Thanks to the described transfer
of material from higher grades to the early years of school, time is freed up
in the 5th and 6th grades to address topics that are significant for
11-12-year-old children from both psychological and preparatory perspectives
for their further study of the mathematics curriculum in grades 7-9. These
topics include logic, modeling, the development of geometric concepts,
variational and functional thinking, and more. Moreover, the quality of
learning the middle school mathematics curriculum is significantly enhanced
because the methodologies presented in the primary school course eliminate the
need to translate the conclusions into the language used in higher grades.

Creative assignments in the
textbook's work system
A highly effective means of
allowing each child in the class to unfold and realize their potential is
through creative work. Creative assignments, where children come up with ideas,
create, and invent, should be regularly introduced, preferably 2-3 times a
week. In these assignments, children can come up with examples for a learned
computational technique, create a problem based on a given expression (e.g., 60
+ 15 × 3 or a ÷ 5 − b ÷ 8), solve a problem of a specific type (e.g., related
to motion, cost, work) or based on a given theme (e.g., about sports, animals,
historical events), draw patterns or geometric shapes with specified properties
(e.g., a right triangle, a pattern of circles), decode or encode the name of a
city, book, or movie using computational examples, and so on.
Creative assignments are
typically given as additional homework alongside the mandatory portion and are
never graded poorly. The most successful creative works can be compiled into a
"Problem Book" at the end of the year, with the students themselves
being the authors of their respective works. Such assignments, where children
act not as mere performers but as creators, have a profoundly positive impact
on the development of children's personalities and contribute to a deeper and
more solid assimilation of knowledge.
Volume and difficulty level of
homework
It is recommended to provide
students with two-level homework assignments, consisting of a mandatory part
and an optional (additional) part.
The mandatory part should be
manageable for independent completion by the child 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
completed at the student's discretion, can include additional tasks,
asterisk-marked assignments, and so on.
Types and forms of classroom work
It is necessary to diversify the
types and forms of work in the classroom. The lesson should include collective,
group, and individual forms of work, as well as oral work and work in grid
notebooks. The practice of computational skills should be systematic and
sufficiently intensive during lessons, but should not exceed 3-4 minutes. It is
advisable to give computational exercises a developmental character by
selecting answer numbers in such a way that the resulting series can be
analyzed, classified, and patterns can be identified by the children. This will
help not only to reinforce counting skills but also to prepare children’s
thinking for activity-based methods.
When forming concepts using the
methods adopted in the course, all types of memory are engaged in students –
not only visual and auditory memory but also motor, imagery, tactile, and
others.
Working in the workbook should
not exceed, as a rule, 10-12 minutes. It mainly involves the independent
completion of tasks by the students, which have been prepared beforehand during
frontal work with similar but different tasks.
The time for independent task
completion is usually limited (typically, from one to 3-4 minutes). Then, the
task is checked using a Smartboard. Children compare their solution with the
model answer or sample for self-checking and mark themselves with a "+"
or "-". As a result, the child's ability for self-monitoring is
purposefully developed.
Since the tasks are checked by
the children themselves when completed independently, the teacher focuses
primarily on the developed self-monitoring skills and the accuracy of
record-keeping during their assessment.

The system of knowledge assessment The course includes a multi-level system of knowledge assessment: self-monitoring when introducing new material, peer monitoring during its practice, instructional control through independent assignments, ongoing assessment through periodic tests during the academic year, and final assessment comprising two stages: a transfer test ("minimum") and a final test (to assess the level of program mastery). The independent assignments are set at a high level of difficulty, and therefore only success is evaluated. Specifically, if the entire independent task is completed without errors (typically achieved by 3-5 children in the class), a grade of A is awarded. After each independent task, children who made mistakes work on correcting them. If the error correction is successful and the teacher observes that the child has understood the material, a grade of B or even A can be given for that task. Grades of F or D are not given for independent assignments. "No grade" (indicating that nothing was earned) serves as a more significant signal for the child to be active and correct their own actions than receiving poor grades. The teacher's task is to motivate each child to understand their mistakes and correct them.

The level of tests should be
lower than that of independent assignments (but higher than administrative
control), and all children are evaluated. It is recommended to design test
tasks in a way that approximately three-quarters of the class can handle them
and achieve grades of A or B.
Intensive exercises to practice
computational skills should be included in almost every lesson. It is advisable
to give these exercises a developmental character by selecting answer numbers
in a way that allows children to analyze, classify, and identify patterns in
the resulting series. This approach will help not only consolidate counting
skills but also prepare children for active problem-solving methods.
CONTENT OF THE 4th GRADE MATH
COURSE
Numbers and arithmetic operations
on them
Estimating and approximating
sums, differences, products, and quotients.
Division by two-digit and
three-digit numbers. Division of whole numbers (with remainder). Division of
multi-digit numbers in general. Checking the accuracy of calculations
(algorithm, inverse operation, estimation of results, assessing reliability,
using a calculator for computation).
Measurements and fractions.
Insufficiency of natural numbers for practical measurements. Practical
measurements as a source of expanding the concept of numbers.
Fractions. Visual representation
of fractions using geometric shapes and number lines. Comparing fractions with
the same denominators and fractions with the same numerators. Division and
fractions.
Finding a part of a number, a
number based on its part, and the part that one number represents of another.
Finding a percentage of a number and a number based on its percentage.
Adding and subtracting fractions
with the same denominators.
Proper and improper fractions.
Mixed numbers. Extracting the whole part from an improper fraction.
Representing a mixed number as an improper fraction. Adding and subtracting
mixed numbers (with the same denominators of the fractional part).
Constructing and using algorithms
for the studied cases of operations with fractions and mixed numbers.
Working with word problems
Independent analysis of the
problem, building models, planning, and implementing the solution. Exploring
different solution methods. Relating the obtained result to the problem
statement, assessing its plausibility. Checking the problem.
Composite problems involving 2-5
steps with natural numbers, involving all arithmetic operations, and multiple
comparisons. Problems involving addition, subtraction, and comparison of
fractions and mixed numbers.

Problems involving unit
conversion (fourth proportion).
Problems involving finding a
fraction of a whole and finding the whole based on its fraction.
Three types of fraction problems:
finding a part of a number, finding a number based on its part, and finding a
fraction that one number represents of another. Problems involving finding a
percentage of a number and finding a number based on its percentage.
Problems involving the
simultaneous uniform motion of two objects (approaching each other, moving in
opposite directions, chasing, lagging): determining the distance between them
at a given moment, the time until they meet, and rates of convergence (divergence).
Problems involving calculating
the area of a right triangle and the areas of shapes.
Geometric shapes and quantities
Right triangle, its angles, sides
(legs and hypotenuse), area, and relationship with a rectangle. Reflex angle.
Adjacent and vertical angles. Central angle and inscribed angle in a circle.
Angle measurement. Protractor. Constructing angles using a protractor.
Units of area: square millimeter,
square centimeter, square decimeter, square meter, are hectare, and their
relationships. Estimation of area. Approximate calculation of areas using a
grid. Investigation of properties of geometric shapes through measurements.
Transformation, comparison, addition, and subtraction of homogeneous geometric
quantities. Multiplication and division of geometric quantities by a natural
number.
Quantities and dependencies
between them
Dependencies between components
and results of arithmetic operations. Formula for the area of a right triangle:
A = (a · b) / 2. Scales. Number line. Coordinate line. Distance between points
on the coordinate line. The uniform motion of points on the coordinate line as
a model of uniform motion of real objects. Approaching speed and receding speed
of two objects during simultaneous uniform motion. Formulas for approaching
speed and receding speed. Formulas for distance d between two objects moving
uniformly at time t for motion towards each other (d0 = s - (v1 + v2) · t), in
opposite directions (d0 = s + (v1 + v2) · t), chasing each other (d0 = s - (v1
- v2) · t), and with one object lagging behind (d0 = s + (v1 - v2) · t). The formula for simultaneous motion s = v(approach) · t(meet).
Coordinate angle. Motion graph.
Observing dependencies between quantities and documenting them using formulas,
tables, and graphs (motion). Constructing motion graphs using formulas and
tables. Transformation, comparison, addition, and subtraction of homogeneous
quantities, as well as their multiplication and division by a natural number.
Algebraic concepts
Inequality. The solution is set to an
inequality. Strict and non-strict inequality. Symbols
≥,≤ ≥,≤.
Double inequality. Solving simple inequalities on the set of non-negative
integers using a number line. Using symbolic notation to generalize and
systematize knowledge.

Mathematical language and
elements of logic
Introduction to symbolic notation
for fractions, decimals, percentages, inequality notation, coordinate notation
on a line and in a plane, the language of diagrams and graphs. Definition of the truth
value of statements. Constructing statements using logical connectives and
words such as "true/false that...", "not", "if...
then...", "each", "all", "there exists",
"always", "sometimes", "and/or".
Working with information and data
analysis
Circular, bar, and line diagrams,
motion graphs: reading, interpreting data, construction.
The content of the 4th-grade
mathematics course is aimed at achieving the following personal, meta-subject,
and subject-specific results:
Personal results
The student will develop:
• Motivational foundation for
academic activities:

• Respectful and positive
attitude towards oneself and others, understanding oneself as an individual and
a member of the class collective, a citizen of one's country, awareness and
demonstration of responsibility for the common well-being and success.
• Knowledge of the basic moral
norms necessary for success in learning and orientation towards their
application in academic activities.
• Formation of ethical feelings
(shame, guilt, conscience) and empathy (understanding, tolerance towards the
individual characteristics of others, compassion) as regulators of moral
behavior through learning activities.
• Development of aesthetic
feelings through the perception of harmony in mathematical knowledge, internal
unity of mathematical objects, and the universality of the mathematical
language.
• Acquisition of initial skills
for adaptation in a dynamically changing world based on the method of reflexive
self-organization.
• Experience in independent
successful mathematical activities according to the 4th-grade curriculum.
The student will have the
opportunity to develop:
• Internal position as a student
and a positive attitude towards school and learning, expressed in a
predominance of academic and cognitive motives.
• Stable academic and cognitive
motivation and interest in new general problem-solving methods.
• Positive attitude towards the
results of one’s own and classmates’ academic activities created by the
students themselves.
• Adequate understanding of the
reasons for success/failure in academic activities.
• Manifestation of civic identity
in actions and activities.
• Ability to solve moral problems
based on moral norms, considering the positions of partners and ethical
requirements.
• Ethical feelings and empathy,
expressed in understanding the feelings of others, empathy, and assistance.
• Ability to perceive the
aesthetic value of mathematics, its beauty, and harmony.
• Adequate self-assessment of
one's own actions based on the criteria of a "good student" role,
creating an individual diagram of one's qualities as a student and being
focused on self-development.
Metasubject Results

Regulatory
The student will learn:
• to accept and maintain a
learning task;
• to apply self-motivation
techniques to learning activities;
• to plan, including internally,
their learning activities in accordance with the clarified structure (15
steps);
• to consider the guidelines
provided by the teacher for new learning material in collaboration with the
teacher;
• to apply learned methods and
algorithms for performing key steps in learning activities:
— trial learning action,
— identifying individual
difficulties,
— identifying the location and
cause of difficulties,
— creating a plan to overcome
difficulties (setting goals, selecting methods for implementation, developing
an action plan, choosing resources, determining timeframes),
— implementing the devised plan
and documenting new knowledge as a standard,
— assimilating new knowledge,
— self-monitoring learning
outcomes,
— self-evaluating learning
activities based on success criteria;
• to differentiate between
knowledge, skills, projects, goals, plans, methods, means, and outcomes of
learning activities;
• to perform learning actions in
tangible, media-based, oral, and mental forms;
• to apply learned methods and
algorithms for performing key steps in corrective activities:
— independent work,

— self-assessment (based on
examples, detailed samples, and standards),
— identifying errors,
— determining the cause of
errors,
— correcting errors based on a general error
correction algorithm,
— self-monitoring the outcome of
corrective activities,
— self-evaluating corrective
activities based on success criteria;
• to use mathematical terminology
learned in the 4th grade to describe the results of their learning activities;
• to perceive and take into
account suggestions and evaluations from teachers, peers, parents, and other
individuals;
• to make necessary adjustments
to their actions after completion based on evaluation and considering the
nature of the mistakes made, using suggestions and evaluations to create new,
improved results;
• to apply an algorithm for
reflecting on their learning activities.
The student will have the
opportunity to learn:
• to transform practical tasks
into cognitive ones;
• to independently consider the
guidelines provided by the teacher for new learning material;
• to document the steps of the
clarified structure of learning activities (15 steps) and independently
implement it in its entirety;
• to conduct self-evaluation of
the ability to apply learned techniques of positive self-motivation to learning
activities,
— self-evaluation of the ability
to apply learned methods and algorithms for performing key steps in learning
activities,
— self-evaluation of the ability
to demonstrate responsibility in learning activities,
— self-evaluation of the ability
to apply an algorithm for reflecting on their learning activities;
• to document the steps of the
clarified structure of corrective activities (15 steps) and independently
implement it in its entirety;

• to set new learning tasks in
collaboration with the teacher;
• to determine the types of
projects based on the learning objectives and independently engage in
project-based activities.
Cognitive
The student will learn:
• to understand and apply
mathematical terminology to solve learning tasks according to the 4th-grade
curriculum, using symbolic notation, including models and diagrams, for
problem-solving;
• to perform logical operations
based on learned algorithms — analyzing objects by identifying essential
features, synthesis, comparison, and classification based on given criteria,
generalization, and analogy, and concept formation;
• to establish cause-and-effect
relationships within the scope of studied phenomena;
• to apply learned algorithms of
cognitive methods — observation, modeling, investigation — in learning
activities;
• to engage in project-based
activities using various project structures depending on the learning
objectives;
• to apply rules for working with
text, extracting essential information from different types of messages
(primarily texts);
• to apply basic methods of
integrating new knowledge into their existing knowledge system;
• to search for the necessary
information to complete learning tasks using textbooks, encyclopedias, and reference materials (including electronic and digital resources), in open
information space, including controlled Internet environments;
• to record selective information
about the surrounding world and oneself, including using ICT tools, and
systematize it;
• to be open to diverse
problem-solving approaches;
• to construct oral and written
statements and arguments about an object, its structure, properties, and
relationships;
• to possess a set of general
problem-solving techniques;
• to understand and apply basic
interdisciplinary concepts according to the 4th-grade curriculum (evaluation,
estimation, diagrams: circular, bar, linear, graphs, etc.);
• to create and solve their own
problems, examples, and equations according to the 4th-grade curriculum;

• to understand and use the signs
and symbols used in the 4th-grade textbook and workbook for organizing learning
activities.
The student will have the
opportunity to learn:
• to conduct self-evaluation of
the ability to apply the algorithm of analogical reasoning;
• to conduct self-evaluation of
the ability to apply observation and investigation methods to solve learning
tasks;
• to conduct self-evaluation of
the ability to create and modify models and diagrams for problem-solving;
• to conduct self-evaluation of
the ability to use reading comprehension strategies;
• to establish and apply basic
rules for information retrieval.
• Presenting projects based on
the learning objective;
• Conducting extensive
information searches using library resources and the Internet;
• Presenting information and
documenting it in various ways for communication purposes;
• Understanding that new
knowledge helps solve new problems and is an element of the knowledge system;
• Constructing coherent and
deliberate oral and written messages;
• Choosing the most effective
problem-solving methods based on specific conditions;
• Building logical reasoning that
includes establishing cause-and-effect relationships;
• Voluntarily and consciously
applying learned general problem-solving techniques;
• Applying knowledge from the
4th-grade curriculum in modified conditions;
• Solving creative and
exploratory problems in accordance with the 4th-grade curriculum.
Communicative

The student will learn:
• Distinguishing essential
differences between discussions and arguments, applying rules of conducting
discussions, and formulating their own position;
• Acknowledging the possibility
of different perspectives, respecting others' opinions, and demonstrating
tolerance towards the characteristics of the interlocutor;
• Striving for consensus among
different positions in collaborative activities, negotiating and reaching a
common decision based on communicative interaction (including situations
involving conflicting interests);
• Allocating roles in
communicative interactions, formulating the functions of "author,"
"listener," "critic," "organizer," and
"arbiter," and applying rules while performing these roles
(constructing coherent expressions understood by the partner, asking questions
for comprehension, using agreed-upon standards to justify one's own point of
view, etc.);
• Appropriately utilizing
linguistic means to solve communicative tasks, constructing monologues, and
possessing dialogic speech form;
• Understanding the significance
of teamwork in achieving positive results in collaborative activities and
applying rules of teamwork;
• Understanding the importance of
cooperation in teamwork and applying rules of cooperation;
• Understanding and applying
recommendations for student adaptation in a new collective.
The student will have the
opportunity to learn:
• To conduct self-evaluation of
the ability to apply rules of conducting discussions based on a standard;
• To conduct self-evaluation of
the ability to perform the roles of "arbiter" and
"organizer" in communicative interactions;
• To conduct self-evaluation of
the ability to justify one's own position;
• To conduct self-evaluation of
the ability to consider others' positions in communicative interactions;
• To conduct self-evaluation of
the ability to participate in teamwork and contribute to the team's successful
outcomes;
• To conduct self-evaluation of
the ability to demonstrate respect and tolerance towards others in
collaboration;
• To engage in mutual control and
provide necessary assistance in collaboration.

SUBJECT OUTCOMES
Numbers and arithmetic operations
on them
The student will learn:
• to estimate and approximate
sums, differences, products, and quotients;
• to perform division of
multi-digit numbers by two-digit and three-digit numbers;
• to verify the accuracy of
calculations using the algorithm, inverse operations, estimation, approximation
of the result, and calculation using a calculator;
• to perform mental calculations
with multi-digit numbers, involving operations within the range of 100;
• to calculate the values of
numerical expressions with known natural numbers within the range of
1,000,000,000, involving 4-6 operations (with and without parentheses), based
on the rules of the order of operations;
• to identify fractions, visually
represent them using geometric shapes and a number line, compare fractions,
find a fraction of a number and a number based on a fraction;
• to read and write fractions,
represent them visually using geometric shapes and a number line, compare
fractions with the same denominators and fractions with the same numerators;
• to find a part of a number, a
number based on its part, and the part that one number represents of another;
• to add and subtract fractions
with the same denominators;
• to read and write mixed
numbers, represent them visually using geometric shapes and a number line,
separate the whole part from an improper fraction, represent a mixed number as
an improper fraction, add and subtract mixed numbers (with the same denominators
of the fractional part);
• to apply the properties of
arithmetic operations to a set of fractions.
The student will have the
opportunity to learn:
• to independently develop and
use algorithms for oral and written operations with multi-digit numbers,
fractions, and mixed numbers;
• to perform division of rounded
numbers (with remainder);

• to find a percentage of a
number and a number based on its percentage using general problem-solving
methods;
• to create and present their own
project on the history of fractions and operations with them;
• to solve problems involving
order of operations with numerical expressions;
• to create and solve their own
problems based on the studied cases of numerical operations.
Working with word problems
The student will learn:
• to independently analyze
problems, build models, plan and implement solutions, explain the solution
process, search for different solution methods, compare the obtained result
with the problem statement, assess its plausibility, and solve problems with
questions;
• to solve composite problems
involving 2-5 operations with natural numbers using the concepts of arithmetic
operations, comparison by difference and multiples, and uniform processes (of the
form a = bc);
• to solve problems involving
reducing to unity (fourth proportional);
• to solve simple and composite
problems involving 2-5 operations of addition, subtraction, and comparison by
difference with fractions and mixed numbers;
• to solve problems involving
finding a fraction of a number and a number based on its fraction;
• to solve three types of
problems involving fractions: finding a part of a number, a number based on its
part, and a fraction representing one number as a fraction of another;
• to solve problems involving the
simultaneous uniform motion of two objects approaching each other, moving in
opposite directions, following each other, with one object lagging behind:
determining the rate of approach and the rate of separation, the distance
between moving objects at a given moment, and the time until they meet;
• to solve problems of all
studied types with algebraic data and create word problems based on given
algebraic expressions;
• to independently create their
own problems based on mathematical models such as numerical and algebraic
expressions, diagrams, and tables;
• to perform all arithmetic
operations with the studied quantities when solving problems.
The student will have the
opportunity to:

• independently develop and use
algorithms for solving text problems;
• analyze, model, and solve text
problems involving 6-8 operations with all the studied arithmetic operations;
• solve problems involving
finding a percentage of a number and finding a number based on its percentage,
as a particular case of part problems;
• solve problems involving the
calculation of the area of a right-angled triangle and the areas of figures
composed of rectangles, squares, and right-angled triangles;
• solve non-standard problems
related to the topics studied and use motion graphs to solve text problems.
Geometric shapes and quantities
The student will learn:
• to recognize a right-angled
triangle, its angles, and sides (legs and hypotenuse), and find its area based
on its relationship with a rectangle;
• to find the areas of shapes
composed of squares, rectangles, and right-angled triangles;
• to directly compare angles
using the overlay method;
• to measure angles using various
measuring tools;
• to measure angles using a
protractor and express them in degrees;
• to find the sum and difference
of angles;
• to construct an angle of a
given magnitude using a protractor;
• to recognize a straight angle,
adjacent and vertical angles, a central angle, and an angle inscribed in a
circle, and investigate their basic properties through measurements.
The student will have the
opportunity to:
• independently establish methods
for comparing angles, measuring them, and constructing them using a protractor;

• when exploring the properties
of geometric shapes through practical measurements and physical models,
formulate their own hypotheses (properties of adjacent and vertical angles, the
sum of angles in a triangle, quadrilateral, pentagon; properties of central and
inscribed angles, etc.);
• draw conclusions that the
properties observed in specific shapes cannot be generalized to all geometric
shapes of the same type, as it is impossible to measure each one of them.
Quantities and their
relationships
The student will learn:
• to use the relationships
between the studied units of length, area, volume, mass, and time in
calculations;
• to convert, compare, add, and
subtract homogeneous quantities, multiply and divide quantities by natural
numbers;
• to use new units of area in the
sequence of studied units - 1 mm², 1 cm², 1 dm², 1 m², 1 a, 1 ha, 1 km²; to
convert them, compare them, and perform arithmetic operations on them;
• to estimate the area and
approximate the calculation of areas using a pallet;
• to establish the relationship
between the sides and the area of a right-angled triangle and express it using
the formula A = (a · b) ÷ 2;
• to find the scale division, use
the scale to determine the value of a quantity;
• to recognize the number line,
identify its essential features, determine the position of a number on the
number line, and add and subtract numbers using the number line;
• to identify the essential
features of the coordinate line, determine the coordinates of points on the
coordinate line with non-negative integer coordinates, construct and use the
distance formula between its points to solve problems;
• to construct models of
simultaneous uniform motion of objects on the coordinate line;
• to observe, using the
coordinate line and tables, the dependencies between quantities describing the
simultaneous uniform motion of objects, construct formulas for closing and
separating velocities for all cases of simultaneous uniform motion, and use the
constructed formulas to solve problems;
• to recognize the coordinate
angle, name its essential features, determine the coordinates of points on the
coordinate angle, and construct points based on their coordinates;
• to read and, in simple cases,
construct circular, linear, and bar graphs;
• to read and construct motion
graphs, determine from them: the time of departure and arrival of an object;
the direction of its motion; the location and time of meeting with other
objects; the time, location, duration, and number of stops;
• to create stories based on
motion graphs, reflecting events that could be represented by the given motion
graphs;

• to use the dependencies between
the components and results of arithmetic operations to estimate the sum,
difference, product, and quotient.
The student will have the
opportunity to learn:
• to independently construct a
scale with a given division, a coordinate line, and formulas for the distance
between points on the coordinate line, formulas for the dependency of the
coordinate of a moving point on the time of motion, etc.;
• to observe, using tables and
the number line, the dependencies between variables, and express them in simple
cases using formulas;
• to determine, from formulas of
the form x = a + bt,
x = a - bt,
expressing the dependency of the coordinate x of a moving point on the time of
motion t;
• to construct and use formulas
for the distance d between two uniformly moving objects at a given time t for
motion towards each other (d = s0 - (v1 + v2) · t), in opposite directions (d =
s0 + (v1 + v2) · t), chasing (d = s0 - (v1 - v2) · t), and falling behind (d =
s0 + (v1 - v2) · t); • to encode figures of a coordinate angle using the
coordinates of points, independently create figures using broken lines,
transmit encoded images "at a distance," and decode the codes;
• to determine the velocities of
objects based on the motion graph;
• to independently create motion
graphs and come up with stories based on them.
Algebraic concepts
The student will learn:
• to read and write expressions
involving 2-3 arithmetic operations, starting with the name of the last
operation;
• to write in symbolic form the
commutative, associative, and distributive properties of addition and
multiplication, rules for subtracting a number from a sum and a sum from a
number, dividing a sum by a number, special cases of operations with 0 and 1,
and use all these properties to simplify calculations;
• to extend the properties of
arithmetic operations to the set of fractions;
• to solve simple equations
involving all arithmetic operations of the form a + x = b, a - x = b, x - a =
b, a · x = b, a ÷ x = b, x ÷ a = b mentally, at the level of automated skill,
and be able to justify their choice of operation based on a graphical model,
explain the steps of the solution, naming the components of the actions;
• to solve compound equations
that can be reduced to a sequence of simple steps (3-4 steps) and explain the
steps of the solution in terms of components of the actions;
• to read and write strict,
non-strict, and double inequalities using the symbols >, <, ≤, ≥;
• to solve simple inequalities on
the set of non-negative integers using the number line and mental reasoning,
record the sets of their solutions using set notation.

The student will have the
opportunity to learn:
• based on the general properties
of arithmetic operations in simple cases:

Mathematical language and
elements of logic
Students will learn:
• to recognize, read, and apply
new symbols of mathematical language, including fractional notation, fractions,
percentages (symbol %), strict, non-strict, and double inequalities using the
symbols >, <,
≤ ≤,
≥ ≥,
the approximate equality symbol ≈, coordinate notation on a line and a plane,
circular, bar, and line diagrams, and motion graphs;
• to determine the truth and
falsehood of statements in simple cases, construct simple statements using
logical connectors and words like "true/false," "not,"
"if... then...," "each," "all," "there exists,"
"always," "sometimes," "and/or";
• to justify their judgments
using the rules and properties learned in the 4th grade, and make logical
conclusions;
• to conduct simple logical
reasoning under adult guidance, using logical operations and connectors.
Students will have the
opportunity to learn:
• to justify statements of a
general nature and statements of existence in simple cases, based on common
sense;
• to solve logical problems using
graphical models, tables, graphs, Euler-Venn diagrams;
• to construct (under adult
guidance and independently) and master problem-solving techniques of a logical
nature according to the 4th-grade curriculum.
Working with information and data
analysis
Students will learn:
• to use tables, circular,
linear, and bar diagrams, and motion graphs to analyze, represent, and
systematize data; compare values using these tools, interpret data from tables,
diagrams, and graphs;

• to work with text: identify
parts of the educational text - introduction, main idea, important notes,
examples illustrating the main idea and important notes, check comprehension of
the text;
• to carry out project work on
topics such as "From the History of Fractions," "Sociological
Survey" (on a given or self-selected topic), develop a plan for
information search, select information sources (reference books, encyclopedias,
controlled internet space, etc.), choose methods of information presentation;
• to complete creative tasks on
topics such as "Transmission of Information Using Coordinates,"
"Motion Graphs";
• to work in the material and
informational environment of primary general education (including with
educational models) in accordance with the content of the subject
"Mathematics, Grade 4."
Students will have the
opportunity to learn:
• to summarize educational texts;
• to carry out extracurricular
project work (under adult guidance and independently), gather information from
reference books, encyclopedias, and controlled internet sources, and present
information using available technical means;
• using the information found in
various sources, create their own problems based on the 4th-grade curriculum
and become co-authors of the "Problem Book for Grade 4," which
includes the best problems created by students; compile a portfolio of a 4th-grade
student.

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The proposed mathematics course for primary school is part of a unified continuous mathematics course for preschoolers, primary school, and grades 4-5 of secondary school. It is designed to achieve new educational objectives, including personal, cross-curricular, and subject-specific learning outcomes, as well as readiness for self-development based on fostering students' cognitive motivation, universal learning actions, and overall learning skills. Mathematical knowledge in the course is not considered an end in itself but rather a means of developing specific personal and cross-curricular educational outcomes, mathematical activities, and the development of children's thinking, senses, emotions, creative abilities, and motivational factors. The stated objective is achieved through the use of the didactic system of the activity-based method by L. Peterson. The technology of the activity-based method involves the following structure for introducing new knowledge in lessons:

1. Motivation (self-determination) for the activity.

2. Activation of prior knowledge and identification of difficulties in a trial
learning action.

3. Identification of the place and cause of the difficulty.

4. Construction of a project to overcome the difficulty.

5. Implementation of the constructed project.

6. Initial consolidation with verbalization in external speech.

7. Independent work with self-assessment against a benchmark.

8. Integration into the knowledge system and review.

9. Reflection on the activity (lesson summary).

Lessons of other types follow a
similar structure: reflection (i.e., review and consolidation of knowledge,
self-assessment, and correction of errors), as well as lessons focused on
building knowledge systems and developmental assessment. Such lesson construction
not only helps students develop a solid foundation of mathematical knowledge
but also engages them in performing a comprehensive set of universal learning
actions during each lesson.
The complex of pedagogical
conditions that ensure the implementation of the activity-based method
technology includes the following didactic principles: activity, continuity,
holistic representation of the world, minimax, psychological comfort, variability,
and creativity. These principles maintain their significance in the system of
educational and management support for children's well-being. Thus,
L. Peterson's educational system
enables the establishment of a unified educational, developmental, and
health-preserving process based on an activity-based approach.
The educational system can be implemented at different levels: the basic level, the technological level, and the system-technological level. The basic level of the activity-based method (ABM) includes the following 7 steps:

1. Motivation for learning activities.

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 working at the basic level
of ABM within the framework of didactic principles, the principle of activity
is transformed into the principle of activity activation within the traditional
education system. Special attention should be paid to the principles of minimax
and psychological comfort, as their proper application allows each student to
progress at their own pace, tackling challenges at their "maximum"
but manageable level of difficulty. Ignoring these principles, on the other
hand, can lead to student overload.
The described lesson structure
systematizes the innovative experience of Russian schools, making it an
accessible step for every teacher. It yields rapid results, including positive
dynamics in students' knowledge acquisition, and development of their thinking,
speech, and cognitive interest. The basic level of ABM can be easily mastered
by any teacher during their initial acquaintance with the educational system by
L. Peterson, and it serves as a
starting point for self-development when fully adopting the activity-based
method.
The technological level of ABM implementation refers to the level of teacher's work where the transitional structure (8 steps) and the system of didactic principles of Peterson’s educational system are implemented. Concepts such as benchmark, self-assessment benchmark, and detailed model are included in the practice, and motivation for cognitive activities is organized at the levels of "I want" and "I can". The technological level of ABM implementation allows for the following:

1. Ensure all the outcomes of the basic level of ABM implementation.

2. Create conditions for the development of general learning skills, including the
ability to learn.

The system-technological level of
ABM implementation refers to the level of teacher's work where the
comprehensive structure of educational activities (9 steps) and the system of
didactic principles of Peterson's educational system are implemented. Concepts
such as educational activities and their structure are included in the
practice.
The system-technological level of
ABM implementation allows for the following:
1. Ensure all the outcomes of the basic level of ABM implementation.

2. Develop general learning skills specified in the educational standards.

Connection with practice, real-world problems Full-fledged mathematics education is impossible without children understanding the origin and significance of mathematical concepts and the role of mathematics in the system of sciences. Therefore, one of the main tasks of the school curriculum is to reveal to students all three stages of mathematical knowledge formation. These stages are:

1. The stage of mathematization, which involves constructing a mathematical model
of a fragment of the real world.

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

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

For example, natural numbers are
not initial abstractions, so their study is preceded by familiarizing students
with finite collections of objects. Similarly, the study of addition and
subtraction of natural numbers begins with the consideration of concrete
operations of combining finite collections and removing a part of a collection.
The study of formal operations of addition and subtraction of two-digit numbers
is based on operations with symbolized representations of these numbers using
dots and shapes (following the historical development of these operations).
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The system of knowledge assessment The course includes a multi-level system of knowledge assessment: self-monitoring when introducing new material, peer monitoring during its practice, instructional control through independent assignments, ongoing assessment through periodic tests during the academic year, and final assessment comprising two stages: a transfer test ("minimum") and a final test (to assess the level of program mastery). The independent assignments are set at a high level of difficulty, and therefore only success is evaluated. Specifically, if the entire independent task is completed without errors (typically achieved by 3-5 children in the class), a grade of A is awarded. After each independent task, children who made mistakes work on correcting them. If the error correction is successful and the teacher observes that the child has understood the material, a grade of B or even A can be given for that task. Grades of F or D are not given for independent assignments. "No grade" (indicating that nothing was earned) serves as a more significant signal for the child to be active and correct their own actions than receiving poor grades. The teacher's task is to motivate each child to understand their mistakes and correct them.

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1. Understanding the purpose of learning and adopting the model of a "good
student."

2. Positive attitude towards school.

3. Belief in one's abilities.

• Holistic perception of the
surrounding world, understanding the history of mathematical knowledge
development, and the role of mathematics in the system of knowledge.
• Self-control ability based on
standards, focusing on understanding the reasons for success/failure and
correcting mistakes.
• Reflective self-assessment
based on criteria of academic success, readiness to understand, and consider
suggestions and evaluations from teachers, peers, parents, and others.
• Independence and personal
responsibility for one's results in both executive and creative activities.
• Acceptance of values:
knowledge, creativity, development, friendship, cooperation, health,
responsible attitude towards one's health, ability to apply rules of health
preservation, and support in academic activities.
• Academic and cognitive interest
in studying mathematics and methods of mathematical activities.
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• to
determine the set of roots of non-standard equations;

• to
simplify algebraic expressions;

• to
use symbolic notation to generalize and systematize knowledge.

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