(Suitable for CAS extension)
This activity involves a curve "with history"; an excellent example of combining geometry and algebra. This function, the so-called "Witch of Agnesi" is defined by a geometric description. After implementing the construction, students are then challenged to find the equation of the constructed curve. This equation, in turn, allows further investigations and generalisations, including some from the field of analysis - an alternative to conventional curve sketching.
The first and second parts of the task are suitable for students of secondary school age with knowledge of the theorems of intersecting lines and the laws of similarity, as well as the Pythagorean theorem. Methods of differential calculus are required only for the final task.
http://compasstech.com.au/TNSINTRO/TI-NspireCD/Exemplary_Activities_PDF/Act10_WitchofAgnesi.pdf
Tuesday, December 1, 2009
Activity 10: The Witch of Agnesi
Activity 9: Playing Rugby
What is the optimal position for a rugby player attempting a conversion?
The aim of this activity is to determine this optimal position and then to study the variations in the angle q obtained depending on the position where the try was scored.
This task enables us to evaluate the advantage gained by positioning oneself as close as possible to the posts before scoring the try.
Activity 8: Random Rectangles
http://compasstech.com.au/TNSINTRO/TI-NspireCD/Exemplary_Activities_PDF/Act8_RandomRectangles.pdf
Activity 7: Algebra Tools
This activity explores the equivalence of algebraic expressions in expanded and factored form, using patterning with CAS to expose commonly held student misconceptions.
The big algebraic mathematical ideas this activity explores are equivalence and symbol sense. More precisely, the activity speaks to the following curriculum expectations: expand and simplify second-degree polynomial expressions involving one variable that consist of the product of two binomials [e.g., (2x + 3)(x + 4)] or the square of a binomial [e.g., (x + 3)2 ], using a variety of tools (e.g., algebra tiles, diagrams, computer algebra systems, paper and pencil) and strategies (e.g. patterning)
http://compasstech.com.au/TNSINTRO/TI-NspireCD/Exemplary_Activities_PDF/Act7_AlgebraTools.pdf
Activity 6: Exploring the Parabola
This activity explores the key features of the parabola, both geometrically and algebraically. A variety of interactive representations support student learning as they build their understanding of this important curve and its real world applications.
The primary objective in the study of parabolas in many high school curricula, tend to be algebraic, moving quickly to the study of the quadratic function. Key defining features of this function are geometric in nature. Students often misrepresent other curves as 'parabolic' simply because they have a similar appearance. It is therefore important for students to understand some of the properties of a parabola, features that make this curve both unique and important. This activity supports students in actively linking some of the geometric and algebraic properties of a parabola.
Activity 5: Meeting a Friend
A version of this problem was set as the final question for the 2005 New South Wales Higher School Certificate examination in Mathematics. Copyright is held by the New South Wales Board of Studies.
http://compasstech.com.au/TNSINTRO/TI-NspireCD/Exemplary_Activities_PDF/Act5_MeetaFriend.pdf
Activity 4: The Diminishing Square
Study the diagram provided. A smaller square has been constructed inside a larger square, as shown.
A point x is located on the base of the larger square. (As shown) The smaller square is constructed using similar points on each of the remaining sides of the larger square. If x is the midpoint of the base, what is the ratio between the area of the larger square and the smaller square?
Explore the relationship between the position of this point and the area of the smaller square.
http://compasstech.com.au/TNSINTRO/TI-NspireCD/Exemplary_Activities_PDF/Act4_DiminishingSquare.pdf
