## LUMINAMATH COMPETITION CURRICULAR MATHEMATICS SUBJECT

Chapter 1: Using numbers in elementary calculations
Subchapter Content
1.1. Writing, reading and forming numbers up to 1000
• Reading and writing numbers from 0 to 1000;

• identifying of orders and classes;
• highlighting the units/tens/hundreds digit of a number;
• generating numbers smaller than 1000, whose digits meet given conditions (eg: specifying the digits of units/tens/hundreds);
• counting 1 by 1, 2 by 2, 3 by 3, 100 by 100, etc., in ascending and descending order, specifying the range limits (from ... to)
1.2 Comparing numbers in the 0-1000 range Comparing some groups of objects by placing the elements one under the other, circling the common parts, matching;
• writing the results obtained by comparison, using the signs <, >, =;

• comparing two natural numbers smaller than 1000, when they have the same number of hundreds/tens/units, with the help of the positioning counter;

• placing given numbers in ascending/descending order;

• identifying the "neighbors" of a number from 0 to 1000;
• identification of even and odd numbers from a given string;
• selecting numbers according to a given criterion (eg: "Transcribe the numbers greater than 395 and less than 405");

1.3. Ordering numbers in the 0-1000 range, using positioning on the number axis, estimations, approximations
• ascending/descending ordering of three-digit natural numbers by comparing them two by two;

• identification of numbers smaller than 1000 under specified conditions;

• estimating the order of magnitude of some groups of objects/symbolic representations/numbers;
• rounding to tens and/or hundreds of a given number;
• estimating the result of a calculation without performing the calculation;
• writing a string of even/odd numbers, given the range limits;
• identifying, writing and reading the order relationship between given numbers;
1.4. Carrying out additions and subtractions, mentally and in writing, in the 0-1000 range, using counting and/or grouping whenever necessary
• performing additions/subtractions with numbers less than 1000, without and with passing over the order and checking by the reverse operation;

• highlighting the properties of addition (commutativity, associativity, neutral element), without specifying the terminology;

• identifying the elements of a second set, given the elements of the first set and the correspondence rule;
• rounding to tens and/or hundreds of a given number;
• identifying the "rule" for a correspondence of the following type: 62→68; 63→69; 64→70
• solving additions and subtractions, mentally and in writing, with and without crossing the order, respecting the algorithm and the correct placement of units, tens, and hundreds
1.5. Perform multiplications and divisions in the 0-1000 range by repeated additions/subtractions
• highlighting several ways of grouping the elements of a set to determine its cardinality;

• finding out a sum of equal terms by solving some practical problems;

• performing multiplications in the 0-100 range, through repeated additions or using multiplication properties;
• highlighting some properties of multiplication (commutativity, associativity, neutral element), without specifying the terminology;
• making divisions with remainder 0, in the 0-100 range by repeated subtractions or resorting to multiplication;
• solving exercises with the order of operations;
• solving problems in which operations of the same order/of different orders are required
1.6. Using mathematical names and symbols (sum, total, terms of a sum, difference, remainder, minus, decrement, product, factors of a product, quotient, divided, divisor, <, >, =, +, -, ·, :) in solving and/or composing problems
• formulating and solving problems starting from a given theme/from given numbers/from verbs/expressions that suggest operations;

• solving exercises of the type: "Find out the sum/ difference/ product/ amount/ half/ quarter/ double etc.";

• the use of half and quarter fractions, using concrete support or drawings (pizza, cake, apple, bread, candy box, notebook, etc.);
• finding out an unknown term, using the balance method, the addition/subtraction test or through tests;
• making different ways of grouping the terms/factors, using graphic signs with the meaning of parentheses

Chapter 2: Highlighting the geometric characteristics of some objects located in the surrounding space
Subchapter Content
2.1. Locating objects by establishing coordinates relative to a given reference system, using learned phrases
• the positioning of objects in space, in relation to other specified objects;

• recognizing of the vertical, horizontal or oblique position of some objects from the immediate reality or within some drawings;

• making simple drawings, respecting a given axis of symmetry;
• making and completing some tables following instructions in which the words "row" and "column" are used
• establishing the coordinates of an object in a plane in relation to a given reference system (eg: it is located on the wall with the door, to the left of the cupboard)
• identifying the inside and outside of a figure

2.2. Highlighting some simple characteristics specific to flat geometric shapes and geometric bodies identified in different contexts
• identifying and naming flat shapes: square, triangle, rectangle, circle
• recognizing of some geometric bodies in the immediate environment (cube, cuboid, sphere, cylinder, cone);
• identifying the number of flat geometric shapes from a given drawing/ from a "fragmented" geometric figure;
• grouping some geometric shapes/bodies according to given criteria;
• identification of the axis/axes of symmetry of the geometric figures;
• marking a half/quarter of the surface of a geometric figure with the corresponding fraction: ½, respectively ¼
• identifying of equivalent fractions: 1/2=2/4;

Chapter 3: Identifying of phenomena/relations/regularities/structures in the immediate environment
Subchapter Content
3.1. Solving problems by investigating, observing and generalizing patterns or regularities in the immediate environment
• completing strings of numbers less than 1000 or ordered objects, respecting specified rules;
• filling in blank spaces from a string of objects/symbols/numbers;
• identifying the algorithm for solving some exercises;

Chapter 4: Generating simple explanations using elements of logic
Subchapter Content
4.1. Describing a work plan using some scientific terms, graphical representations and the logical operators "and", "or", "not"
• staging some problems/problematic situations that use the logical operators "and", "or", "not";
Chapter 5: Solving problems starting with sorting and representing data
Subchapter Content
5.1. Solving problems of the type a±b=x; a±b±c=x in the 0 - 1000 range; a·b=x; a:b=x, in the 0 - 100 range
• identifying the meaning of the data of a problem

• identifying words that suggest arithmetic operations (gave, received, broke, distributed equally, for each, etc.)

• solving problems using concrete objects, drawings, or symbolic representations
• making and completing some tables following instructions in which the words "row" and "column" are used
• associating the solution of a problem with a graphic representation/drawing; marking the half/quarter with the corresponding fraction: ½, respectively ¼;
• solving real problem situations by using the operations of addition and subtraction in the 0 - 1000 range, respectively of multiplication and division in the 0 - 100 range
• organizing the data of a problem in tables or simple graphs for the purpose of solving
• solving problems in several ways

Chapter 6: Using Conventional Standards for Measurement and Estimation
Subchapter Content
6.1. Use of non-conventional measures to determine and compare masses, lengths and capacities
• the appropriate choice of non-conventional units for mass measurement;

• comparing the masses of some objects, the mass of one of which is contained by a whole number of times in the mass of the other;

• the ordering of given objects, based on the successive comparison (two by two) of their length / capacity / mass;
• identifying of objects based on characteristics regarding their length/capacity/mass ("longer", "shorter", "full", "empty", "lighter", "heavier", etc.);

6.2. Using units of measure to determine, compare, and order the durations of various events
• finding the correspondence between an event and the season in which it takes place;
• making a correspondence between the time indicated by analogue clock and the electronic one;
• calculating the number of hours/days/weeks in a given interval;
6.3. Completing value-equivalent exchanges through standard and non-standard conventional representations and through the use of money in simple revenue-expenditure-type game-problems, with numbers from the 0 - 1000 range
• recognizing the 1 leu, 5 lei, 10 lei, 50 lei, 100 lei, 200 lei, 500 lei banknotes
• exchanging a group of coins/banknotes with a banknote/another group of banknotes or coins having the same value;
• addition and subtraction within the limits of 0 - 1000, using the banknotes and coins learned;
• comparing sums of money composed of different coins and banknotes;
• x
6.4. Identifying and usage of common units of measurement for length, capacity, mass (meter, centimeter, liter, milliliter, kilogram, gram) and appropriate tools
• measuring the capacity of some objects and expressing it in milliliters;
• measuring the mass of objects and expressing it in kilograms/grams;
• identification of suitable measuring instruments for making measurements (ruler, tailor's tape, carpenter's tape measure, measuring cup, scale, balance)
Chapter 1: Identifying some relationships/regularities in the immediate environment
Subchapter Content
1.1. Observing some patterns / regularities from everyday life, to create own reasonings
• identifying the construction rule of a string of symbols or numbers
1.2. Applying a rule to continue repetitive patterns
• generating/completing strings of symbols or numbers using a given rule

• the use of a calculation formula (for example: to calculate the perimeter, to determine an unknown number from a numerical relationship)

Chapter 2: Using numbers in calculations
Subchapter Content
2.1. Recognizing of natural numbers in the 0-10000 range and subunit or equiunit fractions with denominators less than or equal to 10
• reading a number and writing numbers from 0 to 10000 with numbers / letters

• identifying, in a number, the number of units / tens / hundreds / thousands

• composing and decomposing numbers into / from thousands, hundreds, tens, and ones.
• counting up and down from 1 to 1, from 2 to 2, from 3 to 3, specifying the range limits (from ...to..., less than ... but greater than ...)
• generating numbers less than 10000, whose digits meet given conditions (for example, the units digit is 1, the tens digit is 2 more than the units digit, etc.)
• approximation (rounding) of natural numbers to different orders
• forming, writing, and reading numbers using Roman numerals (I, V, X, L, C, D, M)
• identifying, in familiar situations, fractional writing
• reading and writing subunit and equiunit fractions
• determining a fraction when the numerator and/or the denominator meet certain conditions
• the intuitive representation of a given subunit fraction starting from familiar situations
• writing subunit fractions starting from lots of objects, from a drawing/graphical representation or from a text

2.2. Comparison of natural numbers in the 0 – 10000 range, respectively of subunit or equiunit fractions that have the same denominator, less than or equal to 10
• comparing two numbers less than 10000 using positional counting or representations
• comparing numbers less than or equal to 10000 using the comparison algorithm
• ascending/descending ordering of numbers less than or equal to 10000
• the use of signs <, >, = in comparing numbers or fractions with the help of concrete examples and graphic representations
• comparing fractions with the same denominator using familiar objects or graphic representations
• the ordering of subunit fractions
2.3. Addition and subtraction of natural numbers in the 0 - 10000 range or with fractions with the same denominator
• performing additions/subtractions with passing and without passing over the order, with numbers in the 0 – 10000 range, using calculation algorithms, numerical decomposing, and the properties of operations
• decomposing of numbers in the 0 – 10000 range, using addition and subtraction, without passing and with passing over the order
• using the properties of addition in calculations (commutativity, associativity, neutral element)
• estimating the result of a calculation from the 0 – 10000 range, without performing it
• the use of quick calculation techniques (properties of operations, number decomposing, etc.)
2.4. Performing multiplications of numbers in the 0 - 10000 range and divisions using the multiplication table and the division table respectively
• performing multiplications between two or three-digit numbers and one-digit numbers
• performing multiplications between two-digit numbers
• writing a number as a product of two or three factors
• estimating the order of magnitude of the result of a calculation without performing it (for example, 197x2 will be smaller than 200 x30=600)
• performing the proof of a multiplication/division operation
• solving exercises, with the known operations, respecting the order of performing the operations and the meaning of the round brackets
Chapter 3: Exploring the geometric characteristics of some objects located in the immediate environment
Subchapter Content
3.1. Locating objects in space and in representations, in familiar situations
• describing the position of objects in space, in relation to other objects

• making and completing some tables following instructions in which the words "row" and "column" are used

• establishing the coordinates of an object in a graphic representation in the form of a network
3.2. Exploring simple features of geometric shapes and solids in familiar contexts
• identifying and naming flat geometric figures
• recognizing and description of objects that have the shape of known geometric bodies, from the immediate environment (cube, parallelepiped, cylinder, sphere, cone)
• identifying the number of flat geometric figures from a given drawing/from a "fragmented" geometric figure
• grouping some geometric figures or bodies according to given criteria (number of sides, number of angles, shape/number of faces, number of vertices, number of edges)
Chapter 4: Using Conventional Standards for Measurement and Estimation
Subchapter Content
4.1. The use of standardized tools and measurement units, in concrete situations
• measuring dimensions, capacities/volumes, masses, using appropriate tools

• identifying and comparing the values of coins and banknotes

• choosing the appropriate units to measure durations of time
4.2. Operating standardized units of measurement, without conversions
• recording the activities carried out in the school in a set time interval (for example, in a week)
• ordering some data according to the sequence of their development in time (for example, activities in a day/week)
• performing calculations using units of measurement for length, mass, capacity (volume), monetary units
• solving practical problems involving standard units of measurement

Chapter 5: Solving problems in familiar situations
Subchapter Content
5.1. Use of specific terminology and mathematical symbols in solving and/or composing simple reasoning problems
• solving exercises of the type: "Find out the product/ amount/ half/ quarter/ double etc."
• identifying some fractions, using concrete support or drawings (pizza, cake, apple, bread, candy box, chocolate bars, etc.)
• finding an unknown term, using the balance method or by performing the addition/subtraction test
• the use of symbols (<, ≤,>, ≥,=) to compare numbers or the results of arithmetic operations
• identifying the role of round brackets on the final result of an exercise
• the use of symbols for unknown numbers or figures, in various calculations or for solving problems
• transforming addition problems into subtraction problems, multiplication problems into division problems and vice versa
• formulating problems starting from concrete situations, representations and/or mathematical relationships, images, drawings, schemes, exercises, graphs, tables
5.2. Recording in tables some observed daily data
• selecting and grouping some symbols/numbers/figures/geometric bodies according to several given criteria and recording the data in a table
• extracting and sorting numbers from a table, based on given criteria
5.3. Solving problems with studied arithmetic operations, focusing on 0 - 10000
• identifying and analyzing the data from the hypothesis of a problem
• identifying words/phrases in problem statements that suggest the studied arithmetic operations (gave, received, distributed equally, twice as much, etc.)
• associating the solution of a problem with a graphic representation/drawing or with a given numerical expression
• solving problems through several methods
• identifying concrete situations that can be translated into mathematical language
• checking the results obtained after solving a problem

Chapter 1: Identifying some relationships/regularities in the immediate environment
Subchapter Content
1.1. Generation of repetitive patterns / regularities
• identifying correspondences between two sets of numbers, in practical situations
• description of a rule starting from a given string
• making repetitive models with given objects
• building simple regularities with symbols, numbers, figures, geometric bodies, respecting one or more different rules
• the use of a calculation formula (for example: to calculate the perimeter, to determine an unknown number from a numerical relationship)

Chapter 2: Using numbers in calculations
Subchapter Content
2.1. Recognizing natural numbers in the 0 - 1000000 range and fractions with denominators less than or equal to 10, respectively equal to 100
• writing numbers in the 0 – 1000000 range with numbers / letters

• reading and writing numbers from 0 to 1000000

• identifying the digits of units/tens/hundreds/thousands/tens of thousands/hundreds of thousands of a number
• composing and decomposing numbers from/to hundreds of thousands, tens of thousands, thousands, hundreds, tens, and ones
• counting using a given step, in ascending and descending order, specifying the limits of the range (from ... to ..., less than ... but greater than ...)
• generating numbers smaller than 1000000 that meet given conditions
• forming, writing, and reading numbers using Roman numerals
• the transcription with Roman numerals of some numbers written with Arabic numerals
• the use of Roman numerals in common situations (for example, writing ordinal numerals with Roman numerals)
• identifying numerators and denominators of fractions
• reading and writing subunit, superunit and equiunit fractions, in familiar situations or in representations
• determining a fraction when the numerator and/or the denominator meet certain conditions

2.2. Comparing natural numbers in the 0 – 1000000 range, respectively of fractions that have the same numerator or the same denominator, less than or equal to 10 or denominator equal to 100
• comparing numbers less than or equal to 1000000 using the comparison algorithm
• writing the results obtained by comparison, using the signs <, >, =
• comparing some fractions with the whole, in familiar situations
• comparing two fractions with the same denominator or the same numerator, starting from objects or graphic representations
2.3. Ordering the natural numbers in the 0 – 1000000 range and, respectively, the fractions that have the same numerator or the same denominator, less than or equal to 10 or denominator equal to 100
• specifying the successor and/or predecessor of a number
• ascending/descending ordering of numbers less than or equal to 1000000
• rounding/approximation to tens/hundreds/thousands/tens of thousands/hundreds of thousands of numerical values (prices, distances, etc.)
• determining numbers that comply with given conditions (less than ..., greater than or equal to ... etc.)
• ordering some fractions using examples from everyday life or graphic representations
2.4. Addition and subtraction of natural numbers in the 0 - 1000000 range or with fractional numbers
• composing and decomposing natural numbers in the 0 – 1000000 range, using addition and subtraction, with and without crossing the order
• role-playing games that require the composition/decomposition of numbers from the 0 – 1000000 range
• performing additions/subtractions, with and without passing over order, in the 0 – 1000000 range, using calculation algorithms, numerical decompositions and properties of operations
• carrying out the test of the addition and subtraction operation
• using the properties of addition in calculations (commutativity, associativity, neutral element)
• estimating the result of a calculation from the 0 – 1000000 range, without performing it
• using the calculator to solve additions and subtractions or to check some results
• using some quick calculation techniques (properties of operations, groupings and decompositions of numbers, etc.)
• intuiting the equivalence of a fraction with a sum or a difference of fractions with the same denominator, with the help of graphic representations or familiar examples
2.5. Multiplication of numbers in the 0 - 1000000 range when the factors have at most three digits and division of one-digit or two-digit numbers
• performing multiplications and divisions by 10, 100, 1000
• performing multiplications where the factors have at most three digits
• performing multiplications of a number less than 1000000 by a single-digit number
• the use during calculations of some properties of multiplication
• performing number multiplications in the 0 - 1000000 range, in writing
• writing a number as a product of two or more factors
• dividing by one-digit or two-digit numbers in the 0 – 1000000 range
• estimating the order of magnitude of the result of a calculation, without performing it (for example, 19x27 will be smaller than 20x30=600)
• checking a multiplication/division operation
• solving exercises with the known operations, respecting the order of performing the operations and the meaning of the brackets (round and square brackets only)
• solving problems with operations of the same order/of different orders; graphical representation method, comparison method, going backwards method

Chapter 3: Exploring the geometric characteristics of some objects located in the immediate environment
Subchapter Content
3.1. Locating objects in space and symbols in various representations
• describing the position of objects in space, in relation to other objects (parallel, perpendicular)
• making and completing some tables following instructions in which the words "row" and "column" are used
• the use of a simple representation for orientation in space, in familiar conditions
3.2. Exploring the characteristics, relationships and properties of geometric shapes and solids identified in different contexts
• identifying and naming flat figures
• recognizing in familiar situations/in representations of objects with a geometric shape (cube, parallelepiped, pyramid, cylinder, sphere, cone)
• identifying the component elements of a plane figure: angle, side, vertex
• identifying the number of flat geometric shapes from a given drawing/ from a "fragmented" geometric figure
• identifying perpendicular or parallel line segments
• establishing the axes of symmetry of some geometric figures
Chapter 4. Use of conventional standards for measurements and estimates
Subchapter Content
4.1. The use of standardized tools and measurement units, in concrete situations, including for the validation of transformations
• transformation of the results of some measurements, using known operations
• comparing sums of money composed of different coins and banknotes; games using money
• analyzing and interpreting the results obtained from solving some practical problems, with reference to the studied measurement units
4.2. Working with standardized units of measure, using transformations
• performing transformations with standard measurement units within the scope of the studied operations
• performing calculations using measurement units for length, mass, capacity (volume), monetary units
• solving problems involving standard measurement units (including transformations)
Chapter 5: Solving problems in familiar situations
Subchapter Content
5.1. Usage of specific terminology and mathematical symbols in solving and/or composing problems with various reasoning
• solving exercises of the type: "Find out the half/quarter/double, three-quarters, tenth, hundredth, etc."
• identifying the role of round and square brackets on the result of an exercise
5.2. Organizing of data in tables and their graphical representation
• selecting and grouping some symbols / numbers / geometric figures / geometric bodies according to several given criteria
5.3. Solving problems with the arithmetic operations studied, in the 0 – 1000000 range
• identifying and analyzing data from the hypothesis of a problem
• identifying words/phrases in problem statements that suggest the studied arithmetic operations (gave, received, distributed equally, twice as much, etc.)
• associating the solution of a problem with a graphic representation/drawing or with a given numerical expression
• solving problems through several methods
• identifying concrete situations that can be translated into mathematical language
• checking the results obtained after solving a problem
Chapter 1: Identifying some data, quantities and mathematical relationships, in the context in which they appear
Specific skills Content
1.1. Identifying natural numbers in various contexts
• Identifying a natural number based on conditions imposed on its digits
• Identifying an appropriate arithmetic method for solving a given problem
1.2. Identifying ordinary or decimal fractions in various contexts
• Writing a percentage as an ordinary fraction (for example, 20% is written as 20/100)
1.3. Identifying elementary geometric notions and units of measurement in different contexts
• Observing some geometric figures on physical models/drawings
• Describing and identifying some elements of figures and geometric bodies
• Identifying congruent segments or congruent angles in configurations with axes of symmetry

Chapter 2: The processing of quantitative, qualitative, structural mathematical data contained in various informational sources
Specific skills Content
2.1. Performing calculations with natural numbers using arithmetic operations and their properties
• Performing arithmetic operations with natural numbers
• Performing calculations using the common factor
• Performing power operations using specific calculation rules
2.2. Performing calculations with fractions using properties of arithmetic operations
• Inserting and removing whole numbers from an ordinary fraction
• Multiplying and dividing a decimal fraction with a finite number of non-zero decimals by 10, 100, 1000
• Writing a decimal fraction with a finite number of non-zero decimal places as a product of a decimal number and a power of 10; writing a decimal fraction with a finite number of nonzero decimal places as a quotient of one decimal number and a power of 10
• Calculation of a fraction equivalent to another given fraction, by amplifying or simplifying
• Simplifying an ordinary fraction in order to obtain an irreducible fraction (through successive simplifying, if applicable)
• Performing operations with rational numbers expressed as a decimal and/or ordinary fraction

Chapter 3: Use of specific concepts and algorithms in various mathematical contexts
Specific skills Content
3.1. Using the rules of calculation to perform operations on natural numbers and for divisibility
• Using the division algorithm, with the remainder equal to or different from zero, if the dividend and the divisor have one or more digits

• Approximating results obtained by using the division algorithm

• Calculating numerical expressions containing parentheses (circles, squares and braces), respecting the order in which the operations are performed.
• Determining a natural number based on conditions imposed on its digits (for example, determine numbers of the form a2b5 , knowing that the product of its digits is 120)
3.2. Using algorithms to perform operations with ordinary fractions or decimals
• Application of algorithms for dividing a decimal fraction by a natural number or a decimal fraction with a finite number of non-zero decimal places

• Transformation of ordinary fractions into decimal fractions and vice versa.

3.3. Determining perimeters, areas (square, rectangle) and volumes (cube, rectangular parallelepiped) and their expression in appropriate measurement units
• Conversions of standard units of measurement using decimal fractions

• Calculating the perimeter of a geometric figure, intuitively highlighting the perimeter
• Operations with angle measures (limited to hex degrees and minutes only)
• Determining the volume of a cube, of a rectangular parallelepiped, using the network of cubes with the length of the edge equal to 1 and the deduction of the calculation formula
• Applying the formula for calculating the volume of a cube and a rectangular parallelepiped

Chapter 4: Expressing information, conclusions and solutions for a given situation in the language specific to mathematics
Specific skills Content
4.1. Expressing some properties related to comparisons, approximations, estimates, and natural number operations in mathematical language
• Representing a natural number on the number line, using comparison, and ordering of numbers naturally
• Justifying the estimates of the results of calculations with natural numbers
• Justifying writing a given natural number as a power with the indicated base or exponent
• Expressing two-digit natural numbers as a product of prime numbers
4.2. Using language specific of fractions/percentages in given situations
• Fitting a decimal fraction between two consecutive natural numbers
• Using the specific language for determining a fraction of a natural number n, multiple of the denominator of the fraction
• Use of appropriate language to express monetary transformations (including exchanges currency)
4.3. Translating practical problems related to perimeters, areas, volumes, using the convenient conversion of units of measure, into specific language
• Comparing distances/lengths, perimeters, areas and volumes expressed by different measurement units

Chapter 5: Analyzing the mathematical characteristics of a given situation
Specific skills Content
5.1. Analyzing given situations involving natural numbers to estimate or verify the validity of calculations
• Analyzing whether a number is the square of a natural number (using the last digit, fitting between the squares of two consecutive natural numbers)
• Determining natural numbers that respect certain conditions (for example, determine the prime numbers a and b, knowing that 3a+2b=16 )
• Comparing two natural numbers written as powers using rounding to the same base or to the same exponent
• Expressing two-digit natural numbers as a product of prime numbers
• Applying the divisibility criteria of natural numbers for everyday situations
• Estimating the order of magnitude of numbers of the form 2n, starting from practical problems (e.g. consecutively folded sheets of paper, the story of the chessboard)
• Making estimates using percentages (for example, knowing the number of middle school students in a city and the fact that about 2% of them study a musical instrument, estimate the number of middle school students studying a musical instrument)
• Establishing the truth value of a mathematical statement with natural numbers, using arithmetic methods
5.2. Analyzing given situations involving fractions to estimate or verify the validity of some calculations
• Representing decimal fractions with a finite number of nonzero decimals on the number line using their approximation
• Analyzing schemes, models or algorithms for solving practical problems involving the use of operations with ordinary or decimal fractions and the order in which the operations are performed
• Estimating the measures of some characteristic sizes of some objects in the environment (capacity, mass, price)
• Estimating of the mean of a data set; comparing the estimate with the value determined by calculations
Chapter 1: Identifying some data, quantities and mathematical relationships, in the context in which they appear
Specific skills Content
1.1. Identifying specific notions of sets and the divisibility relation in ℕ
• Recognizing finite or infinite sets (the set of natural numbers, the set of numbers even/odd natural numbers, the set of digits of a number, the set of divisors/multiples of a natural number) .
• Recognizing prime numbers
• Identifying a composite number, from a set of numbers
• Identifying a divisor of a given number
• Writing a two-digit natural number as a product of powers of prime numbers, by direct observation
• Writing the set of divisors of a natural number using prime factorization
• Recognizing pairs of coprime numbers
1.2. Identifying ratios, proportions and directly or inversely proportional quantities
• Identifying, reading, writing reports, percentages.
1.3. Identifying the characteristics of whole numbers in various contexts
• Representing the opposite of a whole number on the number axis; the modulus as the distance from the origin to the represented number.
1.4. Recognizing equivalent fractions, irreducible fractions, and ways of writing a rational number
• Representing rational numbers on the number axis, also using the notions of opposite and modulus
1.5. Recognizing flat geometric figures (straight lines, angles, circles, arcs of a circle) in given configurations
• Identifying relationships between given geometric elements (belonging, inclusion, equality, competition, parallelism, perpendicularity, symmetry).
1.6. Recognizing of elements of planar geometry associated with the notion of a triangle
• Recognizing of isosceles/equilateral/acute/rectangular/obtuse triangles in given geometric configurations.
• Recognizing of congruent triangles in a given geometric configuration.

Chapter 2: The processing of quantitative, qualitative, structural mathematical data contained in various informational sources
Specific skills Content
2.1. Highlighting, in examples, the relations of membership, inclusion, equality and criteria of divisibility by 2, 5, 10n, 3 and 9 in ℕ
• Recognizing and exemplifying elements that belong/do not belong to a given set through diagrams or by listing the elements.
• Recognizing and exemplifying given sets through diagrams or by listing elements; sets that are or are not in the inclusion relation
• Identifying natural numbers that are divisible by 2, 5, 10n, 3 or 9, using the divisibility criteria
• Writing a natural number as a product of powers of prime numbers using prime factorization
• Selecting from a given enumeration of prime/composite natural numbers
2.2. Quantitative processing of data using ratios and proportions to organize data
• Determining a percentage of a given number; determining a number, when a percentage of it is known (for example: reduction/increase in the price of a product, concentration of a solution).
• Calculating an unknown value from a proportion
• Calculating numbers using a series of equal ratios
• Calculate the value of a ratio using a string of equal ratios
• Organization and representation of data in the form of graphs, tables, or statistical diagrams in order to record, process and present them
2.3. Using integer operations to solve equations and inequations
• Comparing whole numbers, starting from their representations on the number line
• Ordering the elements of a finite set of integers
• Using specific rules for performing operations with integers: addition, subtraction, multiplication, division and natural exponent raising
• Validation (by proof) of the solution of an equation or an inequality in the set of integers
2.4. Applying rules of computation for rational numbers to solve equations of the type: x+a=b, x∙a=b, x:a=b (a≠0), ax + b = c, where a, b and c are rational numbers
• Using specific rules for performing operations with rational numbers: addition, subtraction, multiplication, division (calculations involving a maximum of two operations)
• Solving equations using the studied rules of computation
2.4. Applying rules of computation for rational numbers to solve equations of the type: x+a=b, x∙a=b, x:a=b (a≠0), ax + b = c, where a, b and c are rational numbers
• Using specific rules for performing operations with rational numbers: addition, subtraction, multiplication, division (calculations involving a maximum of two operations)
• Solving equations using the studied rules of computation
2.5. Recognizing the collinearity of some points, the fact that two angles are opposite at the apex, adjacent, complementary or supplementary and of the parallelism or perpendicularity of two lines
• Quantitative processing of some information regarding distances, segment lengths or angle/arc measures in order to establish the collinearity of some points, including in the context of the circle (for example: diametrically opposite points, the center of the circle)
• Checking that two angles are supplementary, complementary or congruent
• Applying, in a given configuration, the property of vertical angles and angles around a point to determine angle measures
2.6. Calculation of segment lengths, angle measures in the context of triangle geometry
• Determining the type of triangle by performing numerical calculations with segment lengths and angle measures
• Carrying out numerical calculations for formulating answers regarding the important lines in the triangle

Chapter 3: Use of specific concepts and algorithms in various mathematical contexts
Specific skills Content
3.1. Using appropriate ways of representing sets and determining GCD and LCM
• Performing operations with sets (union, intersection, difference) focusing on practical examples.
• Determining of GCD/LCM by decomposing natural numbers into products of powers of prime numbers
• Checking, by examples, the property (a,b)∙[a,b]=a∙b , where a and b are natural numbers (for example, calculating GCD for coprime numbers)
• Using examples to derive some properties of the divisibility relation in the set of natural numbers
3.2. Applying specific methods of solving problems involving ratios, proportions and directly/inversely proportional quantities
• Determining an unknown term in a proportion.
• Solving problems involving ratios, percentages or proportions.
• Establishing proportionality (direct or inverse) between two quantities and solving problems involving directly or inversely proportional quantities, in practical-applicative or interdisciplinary contexts
• Probability calculation in simple practical applications
3.3. Applying the rules of calculation and using parentheses in performing operations with whole numbers
• Applying the properties of integer operations to optimize numerical calculations.
• Use of exponentiation calculation rules (numerical calculations).
• Effective use of methods for determining an unknown from an equation or inequality (reverse walking method, balance method, transformations of equality/inequality relations)
3.4. Use properties of operations to compare and perform calculations with rational numbers
• Comparing rational numbers, including placing numbers on the number line.
• Ordering the elements of a finite set of rational numbers
• Using properties of rational number operations to optimize numerical calculations
• Use of exponentiation calculation rules (numerical calculations)
• Determining an unknown from an equation (reverse walking method, balance method, transformations of equality relations
3.5. The use of properties related to distances, lines, angles, circles for the realization of geometric constructions
• Construction of the symmetry of a figure with respect to a given line
• Determining of segment lengths using information contained in geometric representations.
• Determining measures of angles/circle arcs using information contained in geometric representations
3.6. Using congruence criteria and properties of particular triangles to determine the characteristics of a geometric configuration
• Establishing the congruence of some triangles by identifying the appropriate congruence criterion
• Using the congruence relationship of triangles to establish the congruence of some segments or angles
• Using the properties of isosceles/equilateral/right triangles to determine segment lengths, distances, angle measures, properties of points on medians, bisectors

Chapter 4: Expressing information, conclusions and solutions for a given situation in the language specific to mathematics
Specific skills Content
4.1. Expressing concrete situations that can be described using sets and divisibility in ℕ
• Expressing some characteristics of the elements of finite sets (for example, the set of even numbers) using mathematical language.
• Formulating simple statements using the words "and", "or", "not" in the context of operations with sets
• Use of terminology specific to divisibility
• Writing the solution of some problems related to the divisibility relation in ℕ
4.2. Expressing in mathematical language the relationships and quantities that occur in problems with ratios, proportions and directly or inversely proportional quantities
• Expressing the relationship of direct or inverse proportionality between quantities in the form of a proportion or an equality of products.
• Expressing in mathematical language the data of a problem that can be solved with the simple rule of three
• Determining the minimum, maximum and average values from a data set
• Organizing information based on criteria, using sorting, classification and graphical representation (with emphasis on the interpretation of the same data set in different contexts and on the use of mathematical software)
4.3. Writing the steps for solving the studied equations and inequalities in the set of integers
• Formulation of logical answers in relation to numerical calculation requirements (intradisciplinary correlations; for example: membership of the result of a calculation to a set, estimation of the result, use of 0 as a factor in products of numbers)
• Writing an equation/inequality equivalent to a given equation/inequality
• Writing the approach to solving some equations or inequalities in the set of whole numbers (including checking the solutions)
• Transposing a problem into an equation that is solved in the set of integers
• Expressing of some characteristics of the module, derived from its definition (|x|=a , |x|<a , |x|≤a , where a and x are integers)
4.4. Writing the stages of solving some problems, using operations in the set of rational numbers
• Formulation of logical answers in relation to numerical calculation requirements (intradisciplinary correlations; for example: belonging of the result of a calculation to a set, estimation of the result)
• Transposing a problem into an equation that is solved in the set of rational numbers
4.5. Expressing, through geometric representations or in specific mathematical language, notions related to straight lines, angles and circles
• The description in mathematical language of given geometric configurations that contain lines, angles, circles
• Transposing given information (mathematically or in a practical context) into geometric configurations containing lines, angles, circles
• Justification of the parallelism of two lines using pairs of angles formed by two lines with a secant
4.6. Expressing in symbolic and figurative geometric language the characteristics of triangles and important lines in the triangle
• Transcription in symbolic language of the characteristics of triangles contained in given geometric figuress
• Transcription, from given geometric figures, in symbolic language of the characteristics of the important lines in the triangle
• Writing the known data (hypotheses) and the unknown (conclusions), in relation to a given situation related to the triangle
• Highlighting some relationships and properties: exterior angle of a triangle, inequalities between sides and relationships between sides and angles of a triangle, etc.

Chapter 5: Analyzing the mathematical characteristics of a given situation
Specific skills Content
5.1. Analysis of given situations in the context of sets and divisibility in ℕ
• The one-to-one association of the elements of two finite sets that have the same cardinality
• Estimating the cardinality of a set in practical-applicative contexts (for example: the number of school students, the number of grades obtained by a student in a semester, the number of cities in a county)
• Analyzing and comparing different methods of solving a divisibility problem
• Application of divisibility properties in ℕ to solve exercises with fractions
5.2. Analyzing practical situations using ratios, proportions and data collections
• Proportionality justification for the application of the simple rule of three.
• Interpretation of data recorded in tables, graphs or diagrams; appraisal
5.3. Interpreting data from problems that are solved using whole numbers
• Analyzing possible consequences arising from changing a set of assumptions in integer problems
• Fitting the solution of an equation to a set of integers without performing calculations
5.4. Determining effective methods in performing calculations with rational numbers
• Analyzing and choosing the optimal method of performing the numerical calculation by using the properties of the studied operations
• Interpreting the answers obtained by solving equations and identifying the set of solutions
5.5. Analyze numerical data sets or geometric representations to optimize calculations with segment lengths, distances, angle measures and arcs
• Establishing the minimum/maximum number of lines determined by a given number of points (without generalization)
• Analyzing a geometric configuration to check properties related to bisectors (for example: bisectors of vertical angles, bisectors of supplementary adjacent angles)
• Analyzing a geometric configuration to check properties related to lengths (for example: ordering points on a line using given segment lengths, chord length at most equal to diameter length)
• Analyzing a geometric configuration to check some properties related to symmetry with respect to a point, symmetry with respect to a line

Chapter 6: Mathematical modeling of a given situation, by integrating acquisitions from different fields
Specific skills Content
6.1. Translation, in mathematical language, of given situations using sets, set operations and divisibility in ℕ
• Deducing immediate consequences that follow from analyzing a data set associated with sets (eg, in general A\B is different from B\A)
• Interpreting practical or interdisciplinary situations (e.g. cardinal/ordinal number) using the language specific to sets and set operations
• Interpretation of some basic notions from geometry (point, segment, half-line, line; relative positions: point-line, line-line) using the language specific to sets
• Identification in practical situations of intersections, meetings or differences of sets (for example: divisibility criteria, two-digit numbers)
• Solving practical problems using the properties of divisibility in ℕ
6.2. Mathematical modeling of a given situation involving ratios, proportions and directly or inversely proportional quantities
• Mathematical modeling of directly or inversely proportional dependencies.
• Interpreting of a set of data described graphically or numerically (for example: if speed is constant, then distance and time are directly proportional; if distance is constant, then speed and time are inversely proportional)
• Interpreting a ratio as a percentage or as a probability
6.3. Transposing, in algebraic language, a given situation, solving the obtained equation or inequality and interpreting the result
• Transposing a given situation into mathematical language, using equations or inequalities
6.4. Mathematical interpretation of practical problems using rational number operations
• Dividing a quantity into parts directly or inversely proportional to several given numbers
• Mathematical interpretation of a proportionality related to segments (for example, the interpretation of the rules of the Fibonacci sequence in geometric constructions with segments, squares and rectangles)
• Transposing, in mathematical language, a given situation, using equations in the context of rational numbers
6.5. Interpreting information contained in geometric representations to determine segment lengths, distances and angle/arc measures
• Description of a problem-situation, with its transposition from current language into symbolic and figurative language
• Estimating the length of a segment, a distance, the measure of an angle or an arc using various data, rules, relationships
• Validation of the result of a calculation/correctness of a geometric representation, using different approaches: estimates, measurements, comparisons
CHAPTER 1. NATURAL NUMBERS
Specific skills Content
1.1. Operations with natural numbers
• Operations with natural numbers; calculation rules with powers.
1.2. Divisibility of natural numbers
• Multiple, divisor, the divisibility by 10, 2, 5, 3, 9 criteria
• Prime numbers and composite numbers.
• Properties of the divisibility relation in N:
• a|a, ∀a∈N; a|b and b|a → a=b, ∀a,b∈N;
• a|b and b|c → a|c, ∀a,b,c∈N;
• a|b → a|k∙b, ∀a,b,k∈N;
• a|b and a|c → a|(b±c), ∀a,b,c∈N.
1.3. Common divisors, common multiples
• Common divisors of two or more natural numbers; greatest common divisor, prime numbers between them.
• Common multiples of two or more natural numbers; least common multiple, the relationship between GCD and LCM
• Simple problems that are solved using divisibility.

CHAPTER 2. THE SET OF RATIONAL NUMBERS
Specific skills Content
2.1. Rational numbers
• Equivalent fractions; irreducible fraction; the notion of rational number; forms of writing a rational number; The absolute value.
• Comparing and ordering rational numbers. The integer part of a rational number.
2.2. Operations with rational numbers
• Operations with rational numbers; properties.
• Order of operations and use of parentheses.
• Weighted arithmetic mean.
2.3. Equations in Q
• Equation of the form ax+b=0, a∈Q, a≠0, b∈Q..
• Problems that are solved with the help of equations.

CHAPTER 3. RATIOS AND PROPORTIONS
Specific skills Content
3.1. Ordinary fractions
• Definition; percentages; problems in which percentages appear.
• Sizes directly proportional; cross-multiplication
• Inversely proportional quantities.
3.2. Proportion
• Proportion; fundamental property of proportions; finding an unknown term in a proportion.
3.3. Statistical elements
• Data organization elements; data representation through graphs; probabilities.

CHAPTER 4. INTEGERS
Specific skills Content
4.1. Integers
• The set of integers Z; the opposite of an integer; representation on the number axis; absolute value (module); comparing and ordering integers.
4.2. Operations with integers
• Addition of whole numbers; properties..
• Subtraction of integers.
• Multiplication of whole numbers; properties; the set of multiples of an integer.
• Dividing integers when the dividend is a multiple of the divisor; the set of divisors of an integer.
• Power of an integer with a natural number exponent; calculation rules with powers.
4.3. Equations
• Equations in Z; inequalities in Z.
• Problems that are solved with the help of equations.

CHAPTER 5. THE SET OF REAL NUMBERS
Specific skills Content
5.1. The square root
• The square root of a perfect square natural number.
5.2. Real number
• Examples of irrational numbers; the set of real numbers R; modulus of a real number: definition, properties; comparing and ordering real numbers; representation of real numbers on the number line by approximations; N included in Z included in Q included in R. Equation of the form x2=a, with a∈Q+.
• Intervals of real numbers.
• Removing the factors from under the radical; introducing the factors under the radical, where a≥0, b≥0, where a≥0, b>0. , unde a≥0, b≥0;
• 5.4. Operations with real numbers
• Addition, subtraction, multiplication, division, exponentiation.
• The geometric mean of two positive real numbers.

CHAPTER 6. ALGEBRAIC CALCULUS
Specific skills Content
6.1. Calculations with real numbers represented by letters
• Addition, subtraction, multiplication, division, exponentiation, reduction of similar terms.
• Algebraic identities, where a, b ∈R.
• (a±b)2=a2±2ab+b2, (a-b)(a+b)=a2-b2; where a, b ∈R.
6.2. Decomposition into factors
• Decomposition into factors using calculation rules in R, the common factor method, algebraic identities

6.3. Equations and inequations
• Properties of the equality relation in the set of real numbers.
• set of solutions of an equation, cu ; equivalent equations, problems that are solved with the help of equations.
• elements of data organization
• Properties of the inequality relation on the set of real numbers.
• Inequations of the form (≥, <, ≤), .
• Problems that are solved with the help of inequations

CHAPTER 7. GEOMETRIC FIGURES
Specific skills Content
7.1. Congruence of triangles
• Triangle: definition, elements; classification of triangles; the perimeter of the triangle.
• Construction of triangles: cases SAS, ASA, SSS. Congruence of certain triangles: criteria for congruence of triangles: SAS, ASA, SSS.
• The congruent triangles method of proof.
7.2. Perpendicularity
• Perpendicular lines (definition, notation, square construction); oblique lines; the distance from a point to a line. The altitude in a triangle (definition, drawing). Concurrency of altitudes in a triangle (without proof).
• Congruence criteria of right triangles: HL, HA, LL, LA.
• Area of the triangle (intuitively on grids of squares).
• Perpendicular bisector of a segment; the property of the points on the perpendicular bisector of a segment; construction of the median of a segment with straightedge and compass; concurrence of the perpendicular bisectors of the sides of a triangle; symmetry to a straight line.
• Property of the points on the bisector of an angle; construction of the bisector of an angle with straightedge and compass; concurrence of the bisectors of the angles of a triangle.
7.3. Area of the triangle
• Area of the triangle (formula) )
7.4. Parallelism
• Parallel lines (definition, notation); construction of parallel lines (by translation); axiom of parallels.
• Parallelism criteria (angles formed by two parallel lines with a secant).
7.5. Properties of triangles
• The sum of the measures of the angles of a triangle; exterior angle of a triangle, exterior angle theorem..
• Median in the triangle; concurrency of the medians of a triangle (without proof).
7.6. Properties of the isosceles and equilateral triangle
• Properties of the isosceles triangle (angles, important lines, symmetry).
• Properties of the equilateral triangle (angles, important lines, symmetry).
7.7. Properties of the right triangle
• Properties of the right triangle (leg opposite to the 30° angle, median corresponding to the hypotenuse-direct and reciprocal theorems).

CHAPTER 8. THE CIRCLE
Specific skills Content
8.1. Elements in a circle
• Circle: definition, elements: center of a circle, radius, chord, diameter, arc; interior, exterior.
• Theorems about arcs and chords in a circle.
8.2. Angles in a circle
• Central angle; measurement of arcs; congruent arcs.
• Angle inscribed in a circle; triangle inscribed in a circle.
8.3. Regular polygons
• Relative positions of a line to a circle; the tangent from a point outside the circle; triangle circumscribed by a circle.
• Regular polygons: definition, drawing.
• Calculation of elements in regular polygons.
8.4. The length of the circle and the area of the disc
• The circumference of the circle and the area of the disc.

CHAPTER 1. THE SET OF RATIONAL NUMBERS
Specific skills Content
1.1. Rational number
• Definition; the representation of rational numbers on the number axis, the opposite of a rational number; absolute value (modulus); N included in Z included in Q.
• Comparing and ordering rational numbers.
1.2. Operations with rational numbers
• Operations with rational numbers. Order of operations and use of parentheses.
1.3. Equations in Q
• The equation of the form ax+b=0, a∈Q*, b∈Q.
• Problems that are solved with the help of equations.

CHAPTER 2. THE SET OF REAL NUMBERS
Specific skills Content
5.1. The square root
• The square root of a perfect square natural number.
2.2. Real number
• Examples of irrational numbers; the set of real numbers R; modulus of a real number: definition, properties; comparing and ordering real numbers; representation of real numbers on the number line by approximations; N included in Z included in Q included in R. Equation of the form x2=a, with a∈Q+.
• Intervals of real numbers.
• Removing the factors from under the radical; introducing the factors under the radical, where a≥0, b≥0, where a≥0, b>0. , unde a≥0, b≥0;
• 2.4. Operations with real numbers
• Addition, subtraction, multiplication, division, exponentiation.
• The geometric mean of two positive real numbers.

CHAPTER 3. ALGEBRAIC CALCULUS
Specific skills Content
3.1. Calculations with real numbers represented by letters
• Addition, subtraction, multiplication, division, exponentiation, reduction of similar terms.
• Algebraic identities, where a, b ∈R.
• (a±b)2=a2±2ab+b2, (a-b)(a+b)=a2-b2 ∈R.
3.2. Decomposition into factors
• Decomposition into factors using calculation rules in R, the common factor method, algebraic identities

3.3. Equations and inequations
• Properties of the equality relation in the set of real numbers.
• set of solutions of an equation, cu ; equivalent equations, problems that are solved with the help of equations.
• elements of data organization
• Properties of the inequality relation on the set of real numbers.
• Inequations of the form (≥, <, ≤), .
• Problems that are solved with the help of inequations

CHAPTER 4. THE TRIANGLE
Specific skills Content
4.1. Properties of triangles
• The sum of the measures of the angles of a triangle; exterior angle of a triangle, exterior angle theorem.
• Important lines in the triangle, their intersection; midline, properties.
• Area of the triangle.
4.2. The isosceles triangle
• Properties of the isosceles triangle (angles, important lines, symmetry).
• Equilateral triangle.
4.3. Right triangle
• Properties of the right triangle. (the leg opposite to the 30° angle, the median corresponding to the hypotenuse, direct and reciprocal theorems)
4.4. Thales' theorem
• Proportional segments.
• Theorem of equidistant parallels; dividing a segment into proportional parts with given numbers.
• Thales' theorem. Reciprocal.
• Bisector theorem. Reciprocal.
4.5. Similar triangles
• Similar triangles.
• The fundamental theorem of similarity. Similar cases of triangles.
4.6. Properties of the isosceles and equilateral triangle
• Orthogonal projections on a line.
• The altitude theorem.
• The leg theorem.
• The Pythagorean Theorem; The Pythagorean Theorem reciprocal.
• Elements of trigonometry in a right triangle: sin, cos, tg, ctg.
• Solving the right triangle.

Specific skills Content
• The sum of the sizes of the angles of a convex quadrilateral.
5.2. Parallelogram
• Paarllelogram; properties.
• Particular parallelograms: rectangle, square, rhombus; properties.
• Areas
5.3. Trapezium
• Definition, classification.
• Isosceles trapezium; properties.
• The midline of a trapezium.
• Area of a trapezium

CHAPTER 6. THE CIRCLE
Specific skills Content
6.1. Elements in a circle
• Circle: definition, elements: center of a circle, radius, chord, diameter, arc; interior, exterior.
• Theorems about arcs and chords in a circle.
6.2. Angles in a circle
• Central angle; measurement of arcs; congruent arcs.
• Angle inscribed in a circle; triangle inscribed in a circle.
6.3. Regular polygons
• Relative positions of a line to a circle; the tangent from a point outside the circle; triangle circumscribed by a circle.
• Regular polygons: definition, drawing.
• Calculation of elements in regular polygons.
6.4. The length of the circle and the area of the disc
• The circumference of the circle and the area of the disc.

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