What is a CONJECTURE?
A conjecture is a mathematical statement that has not yet been rigorously proved. Conjectures can be formed when one notices a pattern that holds true for many cases.
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However, just because a pattern holds true for many cases does not mean that the pattern will hold true for all cases.
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A conjecture is an important step in problem solving; it is not just a tool for professional mathematicians. In everyday problem solving, it is very rare that a problem's solution is immediately apparent.
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In primary/elementary education, students can form a conjecture as a generalisation tool once they witness a pattern.
The problem solving process involves analysing the problem structure, examining cases, developing a conjecture about the solution, and then confirming that conjecture through proof.
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See an example of a lesson plan utilising the concept of CONJECTURE.
0. The GOAL
What we would like students to learn during this learning
The goal of the lesson is for students to discover how to calculate the area of a triangle. Grade 5 or 6.
Prior knowledge (front loading): Calculation of area of a rectangle, multiplications, geometrical terminology (triangle, base, height, right angle, perpendicular angle, right-angled triangle, scalene, isosceles, acute, obtuse etc
Equipment: Grid paper (1cm)/ workbook, ruler
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Learning Intention: Students understand that the area of triangles is one-half of the product of its base and height
Success Criteria: Students can find the area of triangles conceptionally
1. Revision
Students discuss the area of following rectangles. (revision)
Pose: What is the area of this rectangle?
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Pose: How did you find the area? (open-ended)
Pose: Is there a 'good' way to find the area of a rectangle? (open-ended)
Students independently calculate the area of the following rectangle.
Pose: What is the area of this rectangle?
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Pose: Does the 'good method' work on this rectangle? Why? Why not? (open-ended)
Area of a rectangle
.jpeg)
2. Investigation
Area of a Triangle
Students copy this triangle in their workbook.
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Pose: What is the area of this triangle? How did you find it? (open-ended)
Students may count the whole squares then add halves, quarters.
Students may guess the area.
Students likely point out that the area of this triangle is one-half of a rectangle around the shape.
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Introduce the student who drew the diagram above.
Ask the student explain his/her thinking to the class.
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Explain to students that the area of this triangle is exactly one-half of the rectangle around it.
So the area of the triangle = 6
3. Exploration
Another triangle
Ask students to draw the triangle below with a ruler.
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Pose: What is the area of this triangle?
Pose: How can we find the exact area? (open-ended)
By drawing a rectangle around the triangle, students can investigate the area.
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An extra line (dotted) may help
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Pose: We drew a different rectangle but the area was the same. Why? (open-ended)
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4. Further
Exploration
More triangles
Ask students to draw the triangle below with a ruler.
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Pose: What is your prediction of the area of this triangle? Reason? (open-ended)
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Ask students to check the area.
As a whole class, investigate the area of this triangle. Discuss with students how this could be again the same area (one-half of the rectangle around it)
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Pose: What part of the triangle is the same in each triangle? (open-ended)
Teacher note: base and height of each triangle are consistent. Focus should be on triangles, NOT the rectangle around
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Pose: Can you see a pattern? (open-ended)
Pose: Can you make a conjecture (rule)? (open-ended)
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As a class, compose a conjecture (a rule that can be obtained by gathering data)
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Possible conjecture:
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The area of a triangle is the product of base and height then halve it.
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The area of a triangle is one-half of the product of its base and height
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When you multiply the base and the height of a triangle then halve it, you get the area.
Name the conjecture with a student's name
(whoever contributed most, let's say - Mia Conjecture)
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Once you compose a conjecture, always ask:
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Pose: Would this work ALL THE TIME? (open-ended)
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Teacher note: To challenge (deny) the conjecture, students need to find a triangle that 'doesn't work'. - see logical discussion page
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Pose: Can you find a triangle that conjecture doesn't work? (open-ended)
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Students independently explore a variety of triangles changing the size of base/height.
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As class, discuss the outcome of students' investigation.
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Pose: Does Mia Conjecture work all the time?
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5. Challenge
More triangles
EXTRA:
Pose: What happens if we move the apex of the triangle even further away from the base?
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Ask students to draw the triangle below with a ruler.
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This triangle also has the base length of 4 and height of 3.
Pose: What is the area of this triangle?
Pose: Do you think the area is now larger, smaller or the same?
Pose: How can we find the area?
Teacher Note:
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So the conjecture still stands.
