- 1.
- Appreciation of mathematical patterns:
M 147 presents several mathematical patterns:

- (a)
- elementary proof patterns;
- (b)
- an almost-axiomatic pattern of trigonometry;
- (c)
- elementary duality patterns such as function inverses and the great geometry-algebra pattern;
- (d)
- elaboration of the Pythagorean pattern to circular trigonometry.

- 2.
- Awareness of applications:
M 147 presents the following applications while laying groundwork for study of further applications in subsequent courses:

- (a)
- compound interest and exponential growth and decay;
- (b)
- trigonometric applications such as surveying and circular or oscillatory motion;

- 3.
- Mastery of some mathematical tools:
- (a)
- geometric effects of algebraic transformations;
- (b)
- algebraic effects of geometric transformations;
- (c)
- algebraic address of exponential and trigonometric phenomena;
- (d)
- use of ``appropriate technology'' to investigate problems.

- 4.
- Mathematics as a language:
- (a)
- algebraic language and its geometric consequences;
- (b)
- geometric language and its algebraic consequences;
- (c)
- grammar of communication with computers and calculators.

M 147 relates to the General Learning Outcomes of the University Core-Curriculum Philosophy as follows:

- 1.
**Critical Thinking/Problem-Solving Skills**This course promotes critical thinking and problem-solving skills on a very narrow class of abstract problems. It is not a ``problems'' course: for each mathematical topic covered, the course presents an array of problems which can be addressed successfully using the insights and techniques appropriate to the topic.Periodically, the students must have some problems presented free of any particular context, in hopes that they break down the ``topic-to-problem'' structure of the course's presentation to arrive at a successful ``problem-to-topic'' solution.

- 2.
**Communication Skills**An ongoing battle is to encourage the student to document problem solutions with appropriate prose, mnemonic variable names and labeled diagrams.This eventually aids the student in communicating problem descriptions and solutions to others as well as in communicating about problems with the student's own self.

Student who have attained these communication skills and habits exhibit the ability to solve non-routine problems, problems not immediately transparent, problems whose solution may require time spent several different days.

- 3.
**Cultural Perspective**Persons lacking the minimal algebra and geometry perspectives of M 147 are at a disadvantage if life ever*requires*coping with scientific exposition or argument. That is, such persons are walled off from any appreciation of the basis of the scientific part of our culture.- 4.
**Breadth of Knowledge and Intellectual Perspective**M 147 aims to promote a sensitivity toward numeric inputs and quantitative relationships in general. Thus, M 147 is at the launch point of an acculturation effort which can lead to membership in the western scientific-technical subculture.

- 1.
- Be able to use the concepts of function, relation, and graphs.
- 2.
- Be able to use the algebraic and geometric
*language*of mathematics correctly and effectively. - 3.
- Have skill with manipulative algebra
and the basic properties of elementary
polynomial, rational, and transcendental functions.
- 4.
- Particular trigonometric knowledge and skills:
- (a)
- a working knowedge of right-triangle trigonometry and the trigonometric functions in this setting: , , , , , and .
- (b)
- a working knowedge of the unit-circle
- i.
- circle geometry: tangent lines, arc length, radian measure.
- ii.
- trigonometry and the circular functions in this setting: , , , , , and . This includes the famous ``by-heart'' values of the functions: at , , , , , and their radian-measure versions.

- (c)
- a working knowledge of trigonometric graphs: sinusoids, , , and the inverse-trigonometric graphs.
- (d)
- a working knowledge of the basic identities: the Pythagorean identities, the sum formulas and the multiple-angle formulas.
- (e)
- skill at writing proofs of trigonometric or logarithmic identities.
- (f)
- skill at solving trigonometric equations and the use of graphs or symmetry to find all their solutions.

- 5.
- Have skill in translating problems to relevant prose, graphical, diagrammatic, or algebraic form.
- 6.
- Exhibit a working understanding of reflection, symmetry, and translation of graphs.
- 7.
- Be able to solve ``routine'' problems efficiently and have at-least-occasional success with more challenging problems.
- 8.
- Be able to cope with the problems inherent in solving equations for polynomials with degree greater than 2 and elementary transcendental equations.
- 9.
- Be able to avoid calculator-generated gaffes. Although
M 147 does
*not*teach calculator or computer skills as an alternative to algebraic techniques, it does bear some responsibility to study use of powerful graphing and algebraic calculators and attendant pitfalls.