Geometric Exploration 2
Materials Needed
Access to Geometer's Sketchpad Software, computer lab, paper, pencil.
Sequence of Lesson
Description: Now have the students discuss amongst themselves in a group what the procedure is for solving the area of a polygon bounded by the equation, y=(1/2)x, the x-axis, and an arbitrary vertical line, x=a. Ask them how this might change if the slope was a different number, like six? Can they find a similar procedure for a polygon bounded by the function y=2x+3, the x-axis, y-axis and x=a? If students are having trouble, remind them that this sort of trapezoid is really just a triangle on top of a rectangle. Or they could view it as a triangle minus a smaller triangle! As a bonus question for students that get this far, ask if they can describe the general procedure if the chosen equation was had an arbitrary y-intercept and a slope greater than zero. To help the tackle this bonus question, you can suggest to the students that they could plot points on the graph that represent the area of the polygon for certain values ‘a’ and then hypothesize what the graph looks like in general. It’s important to recall that at this point, the nature of the exploration is meant to be exploratory-like, ie, more about the journey of exploration of math, than it is that they solve the equation. So the bonus question is meant to be taken with this spirit of exploration, and not about rigidly finding the right answer.
Access to Geometer's Sketchpad Software, computer lab, paper, pencil.
Sequence of Lesson
Description: Now have the students discuss amongst themselves in a group what the procedure is for solving the area of a polygon bounded by the equation, y=(1/2)x, the x-axis, and an arbitrary vertical line, x=a. Ask them how this might change if the slope was a different number, like six? Can they find a similar procedure for a polygon bounded by the function y=2x+3, the x-axis, y-axis and x=a? If students are having trouble, remind them that this sort of trapezoid is really just a triangle on top of a rectangle. Or they could view it as a triangle minus a smaller triangle! As a bonus question for students that get this far, ask if they can describe the general procedure if the chosen equation was had an arbitrary y-intercept and a slope greater than zero. To help the tackle this bonus question, you can suggest to the students that they could plot points on the graph that represent the area of the polygon for certain values ‘a’ and then hypothesize what the graph looks like in general. It’s important to recall that at this point, the nature of the exploration is meant to be exploratory-like, ie, more about the journey of exploration of math, than it is that they solve the equation. So the bonus question is meant to be taken with this spirit of exploration, and not about rigidly finding the right answer.