With these solutions you walk through various stages of formalizing. Especially students who have problems with formal and abstract mathematics, have a chance to benefit from that. At the same time they will understand more about abstract solutions.
By solving the task in different ways the students will expand their methodological repertoire. In the future they will have the opportunity to use different approaches to solve similar mathematical problems. Students also acquire skills in the areas of solving a problem and reasoning
We will try to solve the problem in four ways: visually, using table(spreadsheet), graphically and analytically (without using derivative* ), using GeoGebra or TI-Nspire™ (see Classroom activities and teacher resources):
- Visualisation of the Problem
The applet shows the rectangle limited by the wall on the left. By moving the slider, the dimensions of the rectangle change. - Table
We use the point C (base, height) and the point S(base, area). By moving the slider the coordinates of C and S fill the table. We can read the base and the height giving maximal area from the table. - Graph
The graph will show area as a function of base. It can be obtained in two ways: either directly by moving the slider (in this case the table should be disabled) or by plotting the points that correspond to the table (as we normally do in the classroom). Reading the coordinates of point S at its the highest position we get the result. - Analytical Solution
Area can be expressed as a quadratic function of the base (x) under the condition: x + 2h = 10, where h is the height of the rectangle. By calculating the coordinates of the vertex we get the result. It can be proved by writing the quadratic equation into GeoGebra's Input Bar to get the graph. If our equation is correct, the points from section 3 will lie on it. By using the command Extreme the coordinates of vertex will appear in the algebraically window [or following similar procedure with TI-Nspire™]
Are all results equal? Why the differences? Which method is the best? Which is the fastest?