This lesson uses a discovery approach. The lesson entails students using GSP to manipulate triangles and observe the relationship between the sums of the squares of two sides and the square of the third side as one angle of the triangle is changed. The goal of the lesson if for students to observe that when one angle is equal to 90 degrees, the sum of the squares of the two adjecent sides is equal to the square of the side opposite the angle. Once students have made this observation, the instructor can explain the significance of their discovery and how it relates to the Pythagorean Theorem. This lesson would be appropriate for a seventh or eighth grade mathematics class.
Additional Learning Outcomes:
Students will need to understand the following concepts before this lesson is begun:
Square of a number
Ideally, a computer lab would that offers a computer for each individual child to use. Each computer should have GSP. If a computer lab is not available, one computer equipped with a projection device could be used to demonstrate the ideas to the students.
Briefly review the ideas of right angles, adjacent sides, opposite side, and square of a number.
Students should use GSP to create a triangle with line segments. The vertices of the triangle should be labeled.
Any type of triangle is fine. See a sample sketch below:
Students should use the tools of GSP to measure the length of each side and the measure of each angle. These measurements should be left to appear on the screen.
Students should use the calculate option of GSP to calculate the square of each side.
Students should use the calculate option of GSP to find the sum of the squares of each set of adjacent sides. For the triangle above this would be the sum of the squares of sides AB and BC, the sum of the squares of sides BC and AC, and the sum of the squares of sides AC and AB. An example of this can be seen below.
Students should now try to drag one of the vertices around. As they do this, the students should attempt to observe and note any changes in the sums of the squares and the square of the third side and the measure of the angle. Hopefully, the students will observe that one angle is equal to 90 degrees the sum of the squares of the two adjacent sides is equal to the square of the opposite side. This discovery should lead them to an understanding of the Pythagorean Theorem.
Students should use the paper and pencil to note the observations they made.
The instructor should explain that the leg opposite the right angle is known as the hypotenuse. The two sides adjacent to the right angle should be identified as the legs.
Click here to investigate this in GSP prior to attempting this lesson.
Intermath Investigations for Triangles
Teachers can assess the students be reading their written conclusions of their observations. Additional assessments could be done using teachers created worksheets with sample problems.
As an extension students could be asked to use the Pythagorean Theorem to test to see if a triangle is a right triangle. An additional extension would be to give the students triangles that are none to be right triangles with the length of one side missing. The students could be instructed to use the Pythagorean Theorem to find the length of the missing side, this could be either a leg or the hypotenuse.
The teacher could stipulate exactly which names to give to which angles. Which angle to manipulate could also be stated. The teacher could create a chart with the measure of the manipulated angle, the square of the opposite side, the sum of the squares of the adjacent sides, and the sum of the squares of the adjacent sides. To make the activity simpler, the teacher could fill in the values for the measure of the manipulated angles. This could be done to ensure that the students would fill in the chart for a 90 degree angle. The students could use the chart as a guide. The students could be told to notice what happened to the values in the chart when the angle was 90 degrees.
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