Guided Geometric Exploration
Materials Needed
Access to Geometer's Sketchpad Software, computer lab, paper, pencil.
Sequence of Lesson
Description: Discuss the generalized methods that they have found for solving the area of traingles. This discussion should lead us to this general conclusion:The area bounded by y=mx, the x-axis and x=a is (m/2)a^2 (Write this on the board and draw an example of this type of triangle.)
Then have discuss methods for solving rectangles. This discussion should lead us to the general conclusion.
The area bounded by y=b, the y-axis the x-axis and x=a is ba. (Write this on the board and draw an example of this type of rectangle).
Now after students have been given time to mull this over with discussion, ask how these equations might be used to determine the area bounded by y=mx+b, the y-axis, the x-axis and x=a is ba. This may stimulate conversation. Some useful questions to ask if it doesn’t are: What are some ways we said you could look at trapezoids? How would we find the area of a trapezoid, if we knew the area of the triangles and rectangles that composed it? This hints at one intuitive way of finding our conclusion, but not the only one.
However, the discussion proceeds, eventually solve the general form of this equation, ie., that the shape of the polygon bounded by y=mx+b the x-axis, y-axis, y=mx+b, and x=a is (m/2)(a^2)+(b)(a).
Notice that polygons underneath the x-axis are considered to have negative area using this equation! Discuss the class in why this is the case, and for what situations it would be good (or not) for this to be the case, based on what the graphs were being used for.
Now, at this point, come up with a linear equation, such as y=(4)x+9 and put this equation up on a screen that all student can see. Be sure to have appropriate scaling so the whole graph can be seen. Now assign two different ‘a’ values to each group, such as 1 and 3.5. (Note that you may want to choose your ‘a’ values so that the calculations are fairly simplistic.) Now give the students a few minutes to calculate the area bounded by the y-axis, x-axis, x=a, and your equation. Once they are mostly finished, ask them what area the groups got for their respective ‘a’ values. Plot these on your classroom screen on the same graph as your function. After all points have been put up, ask your graph what that shape looks like. Make sure that they notice that it is a parabola. Ask your class what relation they think would plot all those points. That’s right! The same equation we just came up with: (m/2)(a^2)+(b)(a) with each m=4 and b=9 in the example given above.
Access to Geometer's Sketchpad Software, computer lab, paper, pencil.
Sequence of Lesson
Description: Discuss the generalized methods that they have found for solving the area of traingles. This discussion should lead us to this general conclusion:The area bounded by y=mx, the x-axis and x=a is (m/2)a^2 (Write this on the board and draw an example of this type of triangle.)
Then have discuss methods for solving rectangles. This discussion should lead us to the general conclusion.
The area bounded by y=b, the y-axis the x-axis and x=a is ba. (Write this on the board and draw an example of this type of rectangle).
Now after students have been given time to mull this over with discussion, ask how these equations might be used to determine the area bounded by y=mx+b, the y-axis, the x-axis and x=a is ba. This may stimulate conversation. Some useful questions to ask if it doesn’t are: What are some ways we said you could look at trapezoids? How would we find the area of a trapezoid, if we knew the area of the triangles and rectangles that composed it? This hints at one intuitive way of finding our conclusion, but not the only one.
However, the discussion proceeds, eventually solve the general form of this equation, ie., that the shape of the polygon bounded by y=mx+b the x-axis, y-axis, y=mx+b, and x=a is (m/2)(a^2)+(b)(a).
Notice that polygons underneath the x-axis are considered to have negative area using this equation! Discuss the class in why this is the case, and for what situations it would be good (or not) for this to be the case, based on what the graphs were being used for.
Now, at this point, come up with a linear equation, such as y=(4)x+9 and put this equation up on a screen that all student can see. Be sure to have appropriate scaling so the whole graph can be seen. Now assign two different ‘a’ values to each group, such as 1 and 3.5. (Note that you may want to choose your ‘a’ values so that the calculations are fairly simplistic.) Now give the students a few minutes to calculate the area bounded by the y-axis, x-axis, x=a, and your equation. Once they are mostly finished, ask them what area the groups got for their respective ‘a’ values. Plot these on your classroom screen on the same graph as your function. After all points have been put up, ask your graph what that shape looks like. Make sure that they notice that it is a parabola. Ask your class what relation they think would plot all those points. That’s right! The same equation we just came up with: (m/2)(a^2)+(b)(a) with each m=4 and b=9 in the example given above.