As part of their mathematical training, students go from using intuitive, inductive geometry, based on observation, to using deductive geometry. They discover the properties of figures by constructing and observing them. Little by little, they stop relying on measuring and start to use deduction as the basis for their reasoning. By referring to data, initial hypotheses or accepted properties, students prove statements that they believe are true (known as conjectures), which are then used to prove new ones.
In elementary school, students developed their measurement sense1 by making comparisons and estimates and taking various measurements using conventional and unconventional units of measure. They designed and built measuring instruments and used invented and conventional ones. They calculated direct and indirect measurements.2 They also located numbers on an axis and in a Cartesian plane. They constructed and compared different solids, focusing on prisms and pyramids. They learned to recognize the nets of convex polyhedrons and tested Euler’s theorem. They described circles and described and classified quadrilaterals and triangles. They observed and produced frieze patterns and tessellations, using reflections and translations. Lastly, they estimated and determined different measurements: lengths, angles, surface areas, volumes, capacities, masses, time and temperature.
In Secondary Cycle One, students construct and manipulate relations or formulas, particularly when calculating the perimeter and area of geometric figures,3 using arithmetic and algebraic concepts and processes. They learn the concept of similar figures, look for unknown figures resulting from a similarity transformation, determine arc measurements and calculate the area of segments, using the concept of proportionality. By studying lines, plane figures and solids, students identify properties and relationships between measurements. Lastly, they are introduced to deductive reason, in which they use different statements (definitions, properties, axioms, previously proven conjectures) to justify the steps in their approach or validate conjectures.
In Secondary Cycle Two, students construct and manipulate relations or formulas when calculating the area and volume of solids and determining unknown measurements in right triangles or other triangles, using metric and trigonometric relations. If necessary, they convert various units of measure. They refine their understanding of congruence or similarity, particularly by studying the conditions that allow them to conclude that triangles are congruent or similar. They analyze and optimize situations using the concept of equivalent geometric figures. The concept of vectors is introduced and builds on what students have learned about linearity in the previous cycle. In these various contexts, students use different types of reasoning, particularly deductive reasoning, to validate conjectures.
The following tables present the learning content associated with geometry. By basing themselves on the concepts and processes targeted, students develop the three competencies of the program, which in turn enable students to better integrate the mathematical concepts and processes presented.
- Spatial sense and analyzing situations involving geometric figures
- Analyzing situations involving measurements
|1.||Unlike in elementary school, in secondary school, measurement is part of geometry.|
|2.||Calculating a perimeter or area and graduating a ruler are examples of direct measurements. Reading a scale drawing, making a scale drawing, measuring an area of a figure by decomposing it, calculating the thickness of a sheet of paper based on the thickness of several sheets are examples of indirect measurements.|
|3.||In a geometric space of a given dimension (0, 1, 2 or 3), a geometric figure is a set of points representing a geometric object such as a point, line, curve, polygon or polyhedron.|