9th grade
Students develop deeper skills in algebra and geometry, including being introduced to functions, equations and inequalities, numerical sequences, and concepts of probability.
lesson 1
Quadratic function- 1 . Definition of a quadratic function
- 2 . Finding the value of a quadratic function
- 3 . Finding the argument of a quadratic function
- 4 . Determining whether a point belongs to a quadratic function
- 5 . Creating a table
- 6 . Branches of a quadratic function
- 7 . Increasing and decreasing intervals of a quadratic function
- 8 . Intersection point of the quadratic function with the Oy axis
- 9 . Finding the intersection point of a quadratic function with the Oy axis
- 10 . Zeros of a quadratic function
- 11 . Finding the zeros of a quadratic function
- 12 . Completing the square for a quadratic function
- 13 . Finding the maximum or minimum value of a quadratic function
- 14 . Vertex of a quadratic function
- 15 . Finding the coordinates of the vertex of a quadratic function
- 16 . Axis of symmetry of a parabola
- 17 . Finding the equation of the axis of symmetry of a parabola
- 18 . Graphing a quadratic function
- 19 . Matching given quadratic function graphs
- 20 . Identifying the quadrants of a parabola
- 21 . Finding the domain and range of a quadratic function
- 22 . Constructing a function through points
lesson 2
Irrational equations- 1 . Identifying irrational equations
- 2 . Determining whether a root is an irrational equation
- 3 . Simple irrational equations 1
- 4 . Simple irrational equations 2
- 5 . Irrational equations in the form of products
- 6 . Irrational equations with multiple roots
- 7 . Irrational equations involving differences between roots
- 8 . Irrational equations that can be reduced to quadratic equations
- 9 . Complex irrational equations 1
- 10 . Complex irrational equations 2
lesson 3
Systems of equations- 1 . Determining the solution of a system of equations
- 2 . Solving systems of equations using substitution
- 3 . Solving systems of equations using the addition method
- 4 . Solving systems of equations using the reverse Vieta theorem
- 5 . Solving systems of equations using substitution of variables
- 6 . Solving problems using systems of equations
lesson 4
Inequalities- 1 . Identifying quadratic inequalities
- 2 . Reducing an inequality to a quadratic inequality
- 3 . Finding the solution of a quadratic inequality
- 4 . Solving quadratic inequalities
- 5 . Conditions of quadratic inequalities
- 6 . Solving quadratic inequalities using the graph of a quadratic function
- 7 . Interval method for solving inequalities
lesson 5
Systems of inequalitieslesson 6
Properties of functions- 1 . Finding the value of a function
- 2 . Finding the argument of a function
- 3 . Determining if a point belongs to the graph of a function
- 4 . Domain of a function
- 5 . Finding the domain of a function
- 6 . Range of a function
- 7 . Increasing and decreasing functions
- 8 . Finding the intervals where a function increases or decreases
- 9 . Comparing numbers using function graphs
- 10 . Determining the evenness of a function
- 11 . Determining the oddness of a function
- 12 . Determining the evenness or oddness of a function
- 13 . Determining the evenness or oddness of a function using its graph
- 14 . Drawing the graph of an even function
- 15 . Drawing the graph of an odd function
- 16 . Finding the equation of the symmetry axis of a function
- 17 . Finding the center of symmetry of a function graph
lesson 7
Elements of trigonometry- 1 . Radian measure of an angle
- 2 . Comparing numbers
- 3 . Rotating a point around the origin by a positive angle
- 4 . Rotating a point around the origin by a negative angle
- 5 . Sine, cosine, tangent, and cotangent of an angle
- 6 . Calculating the value of an expression
- 7 . Signs of sine, cosine, tangent, and cotangent
- 8 . Comparing the values of expressions
- 9 . Relations between sine, cosine, tangent, and cotangent
- 10 . Trigonometric identities
- 11 . Sine, cosine, tangent, and cotangent of a negative angle
- 12 . Addition formulas
- 13 . Reduction formulas
- 14 . Double-angle formulas
- 15 . Sum and difference formulas for sine
- 16 . Sum and difference formulas for cosine
- 17 . Simplifying expressions
- 18 . Simple trigonometric equations
lesson 8
Sequences. Progressions- 1 . Methods for defining a sequence
- 2 . Terms of a sequence
- 3 . Finding terms of a sequence
- 4 . Determining if a number is a term of a sequence
- 5 . Identifying arithmetic progressions
- 6 . Common difference of an arithmetic progression
- 7 . Finding the unknown term of an arithmetic progression
- 8 . Formula for the n-th term of an arithmetic progression
- 9 . Determining the order of terms in an arithmetic progression
- 10 . Properties of an arithmetic progression
- 11 . Sum of the first n terms of an arithmetic progression
- 12 . Identifying geometric progressions
- 13 . Common ratio of a geometric progression
- 14 . Finding the unknown term of a geometric progression
- 15 . Formula for the n-th term of a geometric progression
- 16 . Determining the order of terms in a geometric progression
- 17 . Properties of a geometric progression
- 18 . Sum of the first n terms of a geometric progression
- 19 . Identifying infinite decreasing geometric progressions
- 20 . Sum of an infinite decreasing geometric progression
lesson 9
Probability theory and elements of mathematical statisticslesson 10
Geometric transformations in the plane- 1 . Translation
- 2 . Translation: Determining coordinates
- 3 . Symmetry with respect to the Ox-axis
- 4 . Symmetry with respect to the Ox-axis: Determining coordinates
- 5 . Symmetry with respect to the Oy-axis
- 6 . Symmetry with respect to the Oy-axis: Determining coordinates
- 7 . Axis of symmetry
- 8 . Central symmetry
- 9 . Drawing symmetry axes
- 10 . Number of symmetry axes
- 11 . Rotation
- 12 . Rotation: Determining coordinates
- 13 . Similarity transformation
- 14 . Filling in sentences
lesson 11
Similar polygons- 1 . Similar polygons
- 2 . Properties of similar polygons
- 3 . Finding the similarity coefficient
- 4 . Similar triangles
- 5 . Similar triangles: Identifying elements
- 6 . Properties of similar triangles
- 7 . AA criterion for similarity of triangles
- 8 . AA criterion for similarity of triangles: Problems
- 9 . SAS criterion for similarity of triangles
- 10 . SAS criterion for similarity of triangles: Problems
- 11 . SSS criterion for similarity of triangles
- 12 . SSS criterion for similarity of triangles: Problems
- 13 . Criterion 1 for similarity of right-angled triangles
- 14 . Criterion 1 for similarity of right-angled triangles: Problems
- 15 . Criterion 2 for similarity of right-angled triangles
- 16 . Criterion 2 for similarity of right-angled triangles: Problems
- 17 . Criterion 3 for similarity of right-angled triangles
- 18 . Criterion 3 for similarity of right-angled triangles: Problems
- 19 . Applications of similarity criteria
- 20 . Filling in sentences
lesson 12
Relations between the sides and angles of a triangle- 1 . Sine, cosine, tangent, and cotangent of acute angles
- 2 . Signs of sine, cosine, tangent, and cotangent
- 3 . Relations between sine, cosine, tangent, and cotangent
- 4 . Reduction formulas
- 5 . Simplifying expressions
- 6 . Trigonometric functions in a triangle
- 7 . Trigonometric functions in a quadrilateral
- 8 . Calculating the area of a triangle using the sine of an angle
- 9 . Law of sines
- 10 . Law of sines: Problems
- 11 . Law of sines in quadrilaterals
- 12 . Law of cosines
- 13 . Law of cosines: Problems
- 14 . Law of cosines in quadrilaterals
- 15 . Solving triangles
lesson 13
Vectorslesson 14
Circumference and circlelesson 15
Polygon inscribed in a circlelesson 16
Polygon circumscribed around a circle- 1 . Circumscribed polygons
- 2 . Circumscribed regular triangle
- 3 . Circumscribed right-angled triangle
- 4 . Circumscribed isosceles triangle
- 5 . Circumscribed scalene triangle
- 6 . Circumscribed quadrilateral
- 7 . Circumscribed square
- 8 . Circumscribed rhombus
- 9 . Circumscribed regular polygon
- 10 . Circumscribed trapezoid