Knowledge

Abilities

• Length of a circle.
• Perimeter and surface of a rectangle.
• Surface of a right-angled triangle.
• Volume of a right-angled parallelepiped starting from a paving.

Geometrical and numerical work can constitute a privileged ground to approach reasoning on circumscribed deductive small islands, especially in connection with lengths and surfaces.

• Decimal writing and operations +, -, ×.
• Division by a whole number:
  • quotient and remainder in Euclidean division,
  • approximate division.
• Truncation and round fractional
• Writing of the quotient of two whole numbers, simplifications.

Pupils are familiar with integer numbers but not with decimal numbers. It is necessary to work on the significance of decimal writing and to link this with work on operations and the multiplication and division by 0.1; 0.01 or 0.001. Writing fractions appeared only in very simple examples in primary school. It is now necessary to link this with decimal writing.

Multiplication of two decimal numbers.

Literal algebraic calculation

• Substitution of numerical values for letters in a formula.

Numerical functions

• Changes of surface units and of length units.
• Application of a rate of percentage.
• Study of examples involving or not proportionality.

To know how to approach multiplicative situations whose treatment makes it possible to use or to highlight the properties of linearity or the presence of a proportionality factor.

Representation and organization of data

• Examples to read and draw up tables and graphs.

Configurations, constructions and transformations.

• Ring.
• Particular triangles, triangles.
• Rectangle, rhombus. 
• Transformation of figures by axial symmetry.
• Right-angled parallelepiped.

It is necessary to develop training which binds pupils to the demonstration. Preparation of deductive reasoning in particular by taking into account information in diverse forms.

Locations, distances and angles.

• Positive X-coordinates on a graduated line.
• Location by relative integers, on a graduated line (abscissa) and in the plane (with coordinates).

Knowledge

Abilities

• Measurements of time.
• Sum of the angles of a triangle.
• Area of a parallelogram, a triangle, a disc.
• Side surface and volume of a right prism, a cross-section cylinder.
• Quotients of usual magnitudes.
• Volume of a pyramid.
• Volume and side surface of a circular cone.

Numbers and numerical calculation

• Sequence of calculations, operational priorities.
• Product of two fractions.
• Comparison, sum and difference of two fractions with equal or multiple denominators.
• Comparison, sum and difference of relative numbers in decimal writing.
• Operations (+, -, ×, :) on relative numbers in decimal or fractional form (not necessarily simplified).
• Powers of relative numbers and exponent.
• Scientific notation of the numbers.
• Keys Ö and cos of a calculator, opposite.

In the third year, control of the four operations with decimal numbers, and  with fractional numbers.

Multiples and dividers, criteria of divisibility. The control of the procedures is acquired through activity, especially the resolution of problems. To distinguish the nature of numbers: exact numbers, values displayed on a computer screen, values approached with a given precision. To practise the various operations: without calculator, mentally and with a machine. Mental calculation allows consolidation of knowledge, for instance multiplication tables, as well as control of use of a computer by determining the order of magnitude of a result.

Literal calculation

• Equalities:
  • k (a + b) = ka + kb
  • k (a - b) = ka - kb
• Test of an equality or an inequality by substitution of numerical values for one or more variables.
• Expansion of expressions.
• Effect of the addition and the multiplication on the order.
• Simple linear equations with one unknown.

In the second year, substitution of numbers for letters makes it possible to carry out numerical calculations, to include/understand and control the writing rules of literal algebraic expressions.

Resolution of equations and inequalities, conservation of equalities, inequalities. The distributivity of the multiplication compared to the addition is not limited to numerical examples, it leads to a first contact with the concept of identity. Development of expressions, concepts of numerical functions

Mean velocity

• Calculations with percentages.
• Changes of units for quotients of usual magnitudes. 
• Applications of proportionality.

Examples, counterexamples, particular cases in opposition to the general case.

Initiation in reasoning using counter-examples.

Representation and organization of data

• Classes, organisation of a statistical distribution.
• Frequencies.
• Bar charts, pie-charts.
• Cumulative distributions. Frequencies.
• Averages.
• Starting the use of spreadsheet-graphics packages.

Step consisting in synthesizing in numerical or graphic form the information collected on all the elements of a population. Study of statistical statements in the form of tables or graphs while being interested in the relevance of the choice of the classes and the mode of chart selected. Distinction between the case where one has data on the whole population and the case where the data relate to a grouping of the population in class intervals. Utilisation of spreadsheets graphics packages making it possible to undertake in experiments the search for a distribution in classes adapted to a problem arising by visualizing the various paces of the associated diagrams.

Configurations, constructions and transformations. 

• Construction of triangles (with instruments and/or geometrical software).
• Centroid of a triangle. Parallelogram. 
• Transformation of figures by central symmetry.
• Right prisms, cylinders of cross-section.
• Triangle: Theorems relating to the midpoints of two sides.
• Triangles determined by two parallel straight lines cutting two secants: proportionality of lengths.
• Orthocenter of a triangle.
• Right-angled triangle and its circumscribed circle.
• Transformation of figures by translation.
• Pyramid, circular cone.

In solid geometry, construction of solids starting from development of these solids. In plane configurations and geometrical transformations: it aims to obtain on geometrical objects a look which comes from evolution of points of view, confrontation of configurations represented by geometrical figures. Using geometrical transformations is a step to be got for all technical or scientific uses. Gradual passage from a vision of geometrical figures to one of the whole plane. Translation is appropriate to mark such an evolution. Conservation of alignments, distances and angles by a translation.

It is the composition of different translations which will make useful the introduction of the vectors.

In Thales triangle situation, all the useful results of proportionalities are presented starting from the situation obtained by having two secants “cut” by two parallels. 

Locations, distances and angles.

• Location on a graduated line.
• Distance between two points.
• Location in the plane (coordinates).
• Triangular Inequality.
• Graphic relation of proportionality: representation. 
• Pythagoras' theorem and its reciprocal.
• Distance from a point on a line;
• Tangency to a circle.
• Cosine of an acute angle.

Triangular inequality appears from the second year. The construction of triangles leads naturally to it. It means simply to know, for example, that it is useless to try to build a triangle whose side lengths are respectively 10 cm, 15 cm, 26cm , that a triangle whose sides lengths are 11 cm, 15 cm and 26 cm will be flattened.

Knowledge

Abilities

Magnitudes and measurements

• Usual magnitudes and compound magnitudes.
• Surface area of the sphere, volume of the sphere

In a situation using magnitudes, one of the magnitude being a function of the other one. To represent graphically the situation in an exact way if that is possible, in an approximate way if not . Read and interpret such a representation.

To know and use the fact that in an enlargement or a reduction of  ratio k: 

• the surface area of a surface is multiplied by k²
• the volume of a solid is multiplied by k³

Numbers and numerical calculation

• Calculations involving radicals.
• Simple examples of algorithms (successive differences, Euclide...).
• Numerical applications with a calculator.
• Irreducible fractions.

To determine if two natural integers have no common divisor, but 1.

To simplify a fraction to make it irreducible.

To know that if a indicates a positive number Öa is the positive number whose square is a. To use equalities with numerical examples where a is a positive number: (Öa)²= a = Öa² . With numerical examples to determine the number x such as x² = a where a indicates a positive number.

To use the equalities::

Ö(ab) = ÖaÖb and Ö(a/b) =Öa/Öb

with numerical examples, where a and b are two positive numbers

Literal algebraic calculation

Factorization (identities).

Problems that can be reduced to first degree equations. Inequalities.

Linear systems with 2 unknowns.

To factorize expressions such as
(x+1)(x+2)-5(x+2); (2x+1)+(2x+1)(x+3)

Knowing the equalities:

• (a + b) (a - b) = a² - b²
• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²

and using them on numerical and simple literal expressions such as:

• 101² = (100+1)² = 100² + 200 + 1
• (x+5)² - 4 = (x+5+2) (x+5-2)

To use the fact that the relative numbers ab and bc are in the same order as b and c if c is strictly positive, in the opposite order if c is strictly negative. To solve an equation such as: ax + b = cx + d . To determine an equation and to solve a problem leading to an equation, an inequation or a system of two simple equations.

Numerical functions

• Linear functions and affine functions.

To know the notation x®ax for a fixed numerical value of a.

To determine the algebraic expression of a linear function starting from the data of a number and its image.

To represent a linear function graphically.

On the graph of a linear function, to find the image of a given number, and to read the number from the image of this number.

To know the notation x®ax+b for fixed numerical values of a and b.

To determine an affine function defined by data of two numbers and their images. To represent an affine function graphically.

On the graph of an affine function, to find the image of a given number and to read the number from the image of this number.

Representation and organisation of data

• Representation and organisation of data
• Characteristics of position of a statistical series. To approach characteristics of dispersion of a statistical series.
• Initiation in the use of spreadsheets graphics packages in statistics.

A statistical series being given (in the form of list or table, or by a chart), to propose a median value of this series and to give its significance.

A statistical series being given, to determine its extent or that one of a given part of this series.

Configurations, constructions and transformations. 

• Solid geometry.
• Sphere.
• Problems of plane sections of solids. Properties of Thalès.
• Vectors and translation.
• Vector equalities.
• Composition of 2 translations.
• Composition of two central symmetries rotation, angle, regular polygon, images of figure by a rotation, regular polygon and inscribed angle.

To know that the section of a sphere by a plane is a circle. To know how to place the center of this circle and how to calculate its radius knowing the radius of the sphere and the distance of the plane from the centre of the sphere. To know the nature of the sections of the cube, the right-angled parallelepiped by a plane parallel to a face, or an edge. To know the nature of the sections of the cylinder of cross-section by a plane parallel or perpendicular to its axis. To represent and determine the sections of a circular cone and a pyramid by a plane parallel to the base. To know and use in the right-angled triangle the relations between the cosine, the sine or the tangent of an acute angle and the lengths of two sides of the triangle.

To use the vectorial equality :

vect (AB) + vect(BC) = vect(AC)

and to connect it to two successive translations.

To build a representation of the vector sum using a parallelogram.

To know and to use the vector writing.

To express that the translation which transforms A into B transforms also C into D.

To link this vector writing to a flattened parallelogram ABCD. 

To know the image by a given rotation of a point, a circle, a line, a segment, a ray. To build an equilateral triangle, a square, a regular hexagon knowing its center and a top.

To know and use in a given situation the two following theorems “ d and d’ are two secant lines in A. B and M are two points of d , distinct from A, C and N are two points of d’, distinct from A. If lines (BC) and (MN) are parallel then

AM/AB = AN/AC = MN/BC

Are d and d' and two secant lines in A. Are B and M two points of d, distinct from A. Are C and N two points of d ', distinct from A. If

AM/AB = AN/AC

and the points A, B, M and points A, C, N are in the same order, then lines (BC) and (MN) are parallel.

To compare an inscribed angle and the angle with the center intercepting the same arc.

Locations, distances and angles.

Right angled triangle, trigonometric relations, and distance between two points in an orthonormal coordinate system of the plane.

Coordinates of a vector in the plane in an orthonormal coordinate system.

Sum of two coordinates of two vectors in the plane in an orthonormal coordinate system.