The proofs presented will always be adapted to your grade level.
Students in Secondary Cycle 1 should be familiar with the definitions and properties of triangles, quadrilaterals and special straight lines. In addition, they must understand the classification of angles and the relationships between angles, in particular the theorems associated with parallel lines that are intersected by a secant.
Students in Secondary Cycle 2 will need to know the concepts mentioned above as well as the Pythagorean theorem, the minimum conditions for congruent and similar triangles, and the metric relations in right triangles. You may also be asked to construct proofs using analytic geometry.
Of course, these lists are not complete. Students may also be asked to formulate a conjecture or to prove a theorem that is not in the field of geometry. This means students need to be able to draw on prior knowledge, in particular with regard to the definition of even and odd numbers.
Proofs help to develop a student's critical thinking skills and mathematical rigour.
See the 2 videos below for further explanations.
The 1st video is for all secondary school students.
The 2nd video is aimed more specifically at Secondary 4 students, as it covers concepts only seen at this level.
To prove a conjecture is false, all that is needed is one counterexample.
When you're asked to formulate a conjecture, you must provide a minimum of 3 different examples. For instance, start with a simple example that satisfies the criteria of the question. Then, for the subsequent examples, work with more general figures or try using negative numbers.
After completing each example, try to establish a link between the elements you began with and those you obtained as a result of your calculations and observations. In the end, you should be left with just one conclusion, which is a generalization of all your observations. This last conclusion is your conjecture, which you can express in your own words. There's more than one way to write a conjecture. What's important is that it be clear and precise, and that nothing is left out.
To prove that a theorem is true, a structured approach is required, consisting of the following elements:
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Starting hypotheses: everything given in the statement or clearly identified in the accompanying figure.
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The conclusion: what we're trying to prove.
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The proof: a list of statements and justifications.
Every statement made must be justified, either using the starting hypotheses, previous statements, or with other definitions, properties or theorems.
Here is a table of the useful definitions, properties and theorems presented in the 1st video:
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Definitions |
Triangles
Quadrilaterals
Special Lines
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|---|---|
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Properties |
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Theorems |
Angles
Minimum Conditions for Congruent Triangles
Minimum Conditions for Similar Triangles
Metric Relations in a Right Triangle
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Proofs of theorems in mathematical branches other than geometry are performed in the same way.
Proofs in analytic geometry also use the same structure, but instead of using theorems, the justifications are based on algebraic calculations using formulas from analytic geometry, such as the distance between 2 points or the slope of a line.
Finally, even if some problems don't explicitly ask for a proof, it's still important to justify each statement. For example, even if it's not mentioned in the question, it's always necessary to prove that figures are similar before establishing a proportion between the corresponding sides.