Work program in mathematics (special course 9th grade)

The school curriculum in mathematics for grade 9 includes the study of methods for solving equations and inequalities. Various analytical and graphical solution methods are used. Equations and inequalities with both one and two variables are considered.

The quadratic function and its properties are considered in detail. It is during the study of this section that various methods of solving second-degree equations are considered. In addition, basic information about arithmetic and geometric progressions is given and elements of combinatorics and probability theory are studied.

• In the 9th grade, the study of the basics of algebra ends, which is accompanied by testing of students' knowledge. After completing the 9th grade in mathematics, the state final certification is taken - GIA.

Below you can find a list of materials for mathematics for grade 9. Each topic is written by our tutor and contains a detailed answer that does not contain anything superfluous - only an explanation of the necessary material.

Topics of the school curriculum 9th grade Mathematics:

Function: domain of definition and domain of values ​​of functions + EXAMPLESProperties of a function: let's look at an exampleQuadratic trinomial and its roots: how to find them, 2 ways to solve Factoring a square trinomial: theorem and formulasQuadratic function: its graph and properties Graphs of a function: what determines the type of graph of a functionBuilding a graph quadratic functions: algorithm and examples Solving inequalities of the second degree with one variable: we give examples Solving inequalities by the method of intervals: we analyze using a specific example The whole equation and its roots: four powers of equations Equations reduced to quadratic: biquadratic and rational Graphical method of solving systems of equations: algorithm and example of solution Sequences: types number sequences and examplesDefinition of a geometric progression: formula for the nth term of the progressionFormula for the sum of the first n terms of a geometric progression + examplesSum of an infinite geometric progression for |q|Even and odd functions: graphs and propertiesFunction y=x^n: linear function, quadratic, cubic and y =1/xDefinition of the nth root: root extractionProperties of the nth arithmetic root: 5 properties with proofDefinition of the power with a fractional exponent: proof and featuresTransformations of expressions containing powers with a fractional exponentDefinition of sine, cosine, tangent and cotangent and examplesProperties of sine, cosine , tangent and cotangent Radian measure of angle: what does it mean, table of correspondences with degrees Relationship between trigonometric functions of the same angle Application of basic trigonometric formulas to transform expressions Reduction formulas: rules and graphs + examples Addition formulas of basic trigonometric functions Double angle formulas: identities and examples Trigonometric functions: properties and their graphs

Mathematics work program for 9th grade

Mathematics work program

for 9th grade

EXPLANATORY NOTE

The work program is based on the following documents:

— Federal component of the state educational standard of basic general education in mathematics;

— Programs for secondary schools, gymnasiums, lyceums. Mathematics. "Bustard". Moscow. 2002. pp. 96-99.

— Programs of general education institutions. Algebra. 7-9 grades / Comp. T.A. Burmistrova. M.: “Enlightenment”, 2012;

Programs of general education institutions. Geometry. 7-9 grades / Comp. T.A. Burmistrova. M.: “Enlightenment”, 2012;

— Federal basic curriculum for secondary (complete) general education

— Textbook “Algebra. Textbook for 9th grade of general education institutions”, ed. Telyakovsky S.A., Makarychev Yu.N. and others, M. “Prosveshcheniye”, 2014.

- Textbook “Geometry Textbook for grades 7 - 9 of general education institutions” authors L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev et al., M. “Prosveshchenie”, 2014.

Teaching mathematics in grade 9 is aimed at achieving the following goals:

Mastering the system of mathematical knowledge and skills necessary for application in practical activities, studying related disciplines, and continuing education.

Intellectual development, continuation of the formation of personality qualities characteristic of mathematical activity: clarity and accuracy of thinking, criticality of thinking, intuition as a collapsed consciousness, logical thinking, elements of algorithmic culture, spatial concepts, the ability to overcome difficulties.

Formation of ideas about the ideas and methods of mathematics as a universal language of science and technology, a means of modeling phenomena and processes.

Cultivating a culture of personality, an attitude towards mathematics as part of universal human culture.

Place of the subject in the federal basic curriculum:

According to the federal basic curriculum for educational institutions of the Russian Federation, 3 hours a week are allocated for studying algebra in grade 9, a total of 105 hours, and for studying geometry - 2 hours a week, a total of 70 hours.

This planning determines a sufficient amount of instructional time to increase the mathematical knowledge of students at the secondary level of school and improve the mastery of other academic subjects.

MODULE “A L G E B A”

PROGRAM CONTENT

1. Properties of functions. Quadratic function

Function. Properties of functions. Square trinomial. Factoring a quadratic trinomial. Solving problems by isolating a squared binomial from a squared trinomial. Function y = ax2 + bx + c, its properties, graph. Power function.

Target -

expand information about the properties of functions
,
familiarize students with the properties and graph of a quadratic function.

Know

basic properties of functions, be able to find intervals of constant sign, increasing, decreasing functions

Be able to

find the domain of definition and range of values ​​of a function, read the graph of a function.

Be able to solve quadratic equations, determine the signs of roots

Be able to factor a quadratic trinomial

Be able to build a graph of the function y=ax2, perform simple transformations of function graphs

Be able to build a graph of a quadratic function, perform simple transformations of function graphs

Be able to build a graph of a quadratic function” and find from the graph the zeros of the function, the intervals where the function takes on positive and negative values.

Be able to graph the function y=ax2 and apply its properties. Be able to graph the function y=ax2 + bx + c and apply its properties

Be able to find the intersection currents of the graph of a Quadratic function with the coordinate axes. Be able to factor a quadratic trinomial.

Be able to graph the function y=xn, know the properties of a power function with a natural exponent, be able to solve the equations xn=a for: a) even and b) odd values ​​of n

Know the definition of the nth root.

2. Equations and inequalities with one variable

Whole equations. Fractional rational equations. Inequalities of the second degree with one variable. Interval method.

Target -

systematize and generalize information about solving integer and fractional rational equations with one variable, develop the ability to solve inequalities of the form ax2 + bx + c> 0 or ax2 + bx + c < 0, where a≠0.

Know

methods for solving equations:

a) factorization;

b) introduction of a new variable;

c) graphical method.

Be able to

solve entire equations by introducing a new variable

Be able to solve systems of 2 equations with 2 variables graphically

Be able to solve equations with 2 variables using substitution and addition

Be able to solve problems “for work”, “for movement” and others by drawing up systems of equations.

Solving third and fourth degree equations with one unknown using factorization and introducing an auxiliary variable.

3. Equations and inequalities with two variables.

Equation with two variables and its graph. Systems of equations of the second degree. Solving problems using systems of second degree equations. Inequalities with two variables and their systems.

Target

— develop the ability to solve simple systems containing second-degree equations. Inequalities with two variables.

Be able to

solve systems containing one equation of the first and another of the second degree. Solve problems using the method of composing systems.

Be able to solve quadratic inequalities algebraically. Be able to solve quadratic inequalities using the graph of a quadratic function

Be able to solve quadratic inequalities using the interval method. Be able to find the set of values ​​of a quadratic function.

Be able to solve the inequality ax2 +in+c.≥0 based on the properties of a quadratic function

4. Progressions

Arithmetic and geometric progressions. Formulas for the nth term and the sum of the first n terms of a progression. Infinitely decreasing geometric progression.

Target -

give the concept of arithmetic and geometric progressions as numerical sequences of a special type.

Achieve

understanding the terms “sequence member”, “sequence member number”, “formula of the nth term of an arithmetic progression”

Know

formula of the nth term of an arithmetic progression, properties of members of an arithmetic progression, methods for specifying an arithmetic progression

Be able to

apply the formula for the sum of the first n terms of an arithmetic progression when solving problems

Know which sequence is geometric, be able to identify whether the sequence is geometric, if so, then find q

Be able to calculate any member of a geometric progression using a formula, know the properties of members of a geometric progression

Be able to apply the formula to solve standard problems

Be able to find the difference of an arithmetic progression

Be able to find the sum of the first n terms of an arithmetic progression. Be able to find any term of a geometric progression.

Be able to find the sum of the first n terms of a geometric progression. Be able to solve problems.

5. Elements of combinatorics and probability theory

Combinatorial multiplication rule. Permutations, placements, combinations. Relative frequency and probability of a random event.

The goal is to familiarize students with the concepts of permutation, placement, combination and the corresponding formulas for calculating their number; introduce the concept of relative frequency and probability of a random event.

Know

formulas for the number of permutations, placements, combinations and be able to use them.

Be able to

use the combinatorics formula when calculating probabilities

7. Final repetition.

Consolidation of knowledge, skills and abilities acquired in lessons on these topics (algebra course for grades 7-9).

Educational and thematic plan

MODULE “GEO METRY”

PROGRAM CONTENT

Vectors

Vector concept. Length (modulus) and direction of the vector. Equality of vectors. Addition and subtraction of vectors. Multiplying a vector by a number. [Collinear vectors. Projection onto the axis. Decomposition of a vector along coordinate axes.] Vector coordinates. Equations of a circle and a line.

The main goal is to formulate the concept of a vector as a directed segment, to show students the application of a vector to solving problems.

Know/understand

concepts

: vector, vector length, equality of vectors, collinear vectors,

codirectional vectors, sum of vectors, difference of vectors, product

vector to number, midline of trapezoid, lemma, coordinate

vectors, radius vector, vector length.

Be able to

: Apply a vector to solve simple problems; calculate the length and coordinates of a vector; perform operations on vectors in geometric and coordinate form.

Relationships between sides and angles of a triangle. Dot product of vectors

Sine, cosine and tangent of an angle. Theorems of sines and cosines. Solving triangles. Relationships between sides and angles of a triangle. Angle between vectors. Dot product of vectors.

The main goal is to introduce students to the basic algorithms for solving arbitrary triangles.

Know/understand

Concepts

: unit semicircle, sine of angle, cosine of angle, tangent of angle,

reduction formulas; angle between vectors; scalar product of vectors.

Theorems

: triangle area theorem, sine theorem, theorem

cosines; theorem on the scalar product of vectors.

Be able to

: use algorithms for solving arbitrary triangles;

calculate the area of ​​a triangle using the learned formulas; calculate the angle between vectors

Circumference and area of ​​a circle

Regular polygons. Circumference and area of ​​a circle.

The main goal is to expand and systematize knowledge about circles and polygons.

Know/understand

Concepts

: regular polygon, inscribed and circumcircle, arc

circles, circular sector.

Formulas

: formula for calculating the angle α of a regular n-gon, the area of ​​a regular polygon, its side and the inscribed radius

circumference, arc length, circumference, area

circle, area of ​​a circular sector.

Be able to: apply formulas when solving problems: calculating areas and sides

regular polygons; radius of inscribed and circumscribed

circles; the length of the arc of a circle and the area of ​​the circle; fulfill

construction of a square, regular triangle, hexagon and 2n-gon.

Movement

The concept of movement. Parallel translation and rotation.

The main goal is to introduce the concept of motion on a plane: symmetries, parallel translation, rotation.

Know/understand

Concepts

: movement, plane display, symmetry, parallel translation, rotation.

Be able to

: construct images of points, segments, triangles

with symmetries, parallel translation, rotation.

6. Repetition. Problem solving.

Educational and thematic plan

Requirements for the level of preparation of 9th grade students:

As a result of studying an algebra course, students should be able to:

know/understand

• the essence of the concept of an algorithm; give examples of algorithms;

• how mathematical formulas and equations are used; examples of their application to solve mathematical and practical problems;

• how the needs of practice led mathematical science to the need to expand the concept of number;

• probabilistic nature of many patterns of the surrounding world;

be able to

solve word problems, including problems involving ratio and proportionality of quantities, fractions and percentages;

perform arithmetic operations, combining oral and written techniques, and the use of computing devices;

find the values ​​of the root of a natural degree, a degree with a rational exponent, using computing devices if necessary;

carry out, according to known formulas and rules, the transformation of literal expressions, including degrees, radicals,

calculate the values ​​of numeric and alphabetic expressions, performing the necessary substitutions and transformations;

determine the value of a function by the value of the argument in different ways of specifying the function;

build graphs of the studied functions;

describe the behavior and properties of functions using a graph and, in the simplest cases, using a formula; find the largest and smallest values ​​from a graph of a function;

solve equations, the simplest systems of equations, using the properties of functions and their graphs;

use acquired knowledge and skills in practical activities and everyday life to:

practical calculations using formulas, including formulas containing powers, radicals, using reference materials and simple computing devices if necessary;

descriptions using functions of various dependencies, representing them graphically, interpreting graphs;

As a result of studying a geometry course, students should be able to:

use geometric language to describe objects in the surrounding world;

recognize geometric shapes, distinguish their relative positions;

depict geometric figures; carry out drawings according to the conditions of the tasks; transform shapes;

calculate the values ​​of geometric quantities (lengths, angles, areas), including: determine the value of trigonometric functions from given angle values; find the values ​​of trigonometric functions by the value of one of them; find the sides, angles and areas of triangles, circular arcs, areas of basic geometric figures and figures composed of them;

solve geometric problems based on the studied properties of figures and relationships between them, using additional constructions, algebraic and trigonometric apparatus, and symmetry considerations;

carry out demonstrative reasoning when solving problems, using known theorems, discovering opportunities for their use;

solve the simplest planimetric problems in space.

Literature

Tutorials:

Algebra: Textbook for 9th grade of general education institutions / Yu.N. Makarychev, N.G. Mindyuk, K.I. Neshkov, S.B. Suvorova; Ed. S.A. Telyakovsky. – M.: Education, 2014.

Geometry, 7 – 9: Textbook for educational institutions / L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev and others – 12th ed. – M.: Education, 2014.

Geometry workbook: To the textbook by L.S. Atanasyan et al. “Geometry 7 – 9”: 9th grade / T.M. Tishchenko. – M.: AST Publishing House LLC: Astrel Publishing House LLC, 2014.

Methodological literature:

G.M. Kuznetsova, N.G. Mindyuk. Programs for secondary schools, gymnasiums, lyceums. Mathematics, 5 – 11 grades. – 4th ed., stereotype. M.: Bustard, 2002.

Educational publication “Programs of general education institutions. Algebra grades 7-9.” Compiled by: Burmistrova T. A., - M.: Education, 2012.

Educational publication “Programs of general education institutions. Geometry grades 7-9. Compiled by: Burmistrova T. A., - M.: Education, 2012.

Atanasyan, L. S., Studying geometry in grades 7-9: methodological recommendations for teachers [Text] / L. S. Atanasyan. - M.: Education, 2005.

Lesson planning