• Algebra Skills Recommendations
• Trigonometry Skills Recommendations
• Geometry Skills Recommendations
• Summaries of Recommended Readings

XI. Appendices

Specific Content Recommendations

Algebra Skills Recommendations

The following skills/facts should be contained in the curriculum of the algebra sequence and should be included in practice work given to the students:

1. Addition of fractions
   
   
1. 
2. 
   
3. Long division of polynomials
4. Addition of polynomials
   
   
2. Linear Functions and Straight Lines
   
   
1. Definition of function
2. Determination of the equation of a line given two distinct points. Equation should be checked using the two given points.
3. Applications of b.
   
   
1. Conversion of degrees Celsius to degrees Fahrenheit.
2. Conversion of miles per hour to feet per second.
   
   
4. Unit conversion
5. Writing the equation of a line in slope-intercept and in point-slope form.
6. Slopes and equations of perpendicular and parallel lines.
7. Slope of a line as a constant rate with units.
8. Solution of systems of linear equations and verification of the solution .
9. Properties of < and .
10. Solution of the following linear equation and linear inequalities.
    
    
1. 
2. 
3. 
4. 
5. 
   
   
11. Solution of a linear equality by solving and graphing the line .
12. Domains of functions of the form , , , .
13. Linear word problems.
14. Application to slope to increasing, decreasing, and horizontal lines.
    
    
3. Quadratics
   
   
1. Identification of the variable for a quadratic of a single variable
   
   
1. 
2. 
   
   
2. Identification of coefficients
   
   
1. 
2. 
   
   
3. Solutions to by the quadratic equation and verification by hand of these solutions.
4. Graph of .
5. Solutions of the following inequalities:
   
   
1. 
2. 
3. 
4. 
   
   
6. The discriminant and its applications to previous 3 problem types.
7. Completing the square to find the absolute extrema.
8. Find the midpoint of the zeros and use it to find the absolute extrema.
9. Word problems which involve finding absolute extrema.
10. Application of the discriminant to determine whether a quadratic is factorable or prime over the reals.
11. Application of the discriminant to factoring when the discriminant is the square of a rational number.
12. Factoring as , where and .
    
    
4. Functions
   
   
1. Definition of function
2. Notations for functions
   
   
1. 
2. 
3. 
   
   
3. Composition of functions: if and , then .
4. The difference quotient: .
5. Definition of the polynomial
6. Definition and construction of rational functions
7. Definition of and , where a>0, a
8. Definition of trigonometric functions
9. Domains of polynomials, rational functions, ,, and where p (x) is a polynomial
10. Graphs of lines, quadratics, rational functions, and algebraic functions
11. Solving inequalities from graphs
12. Finding domains from graphs
13. The following facts about the real numbers:
    
    
1. for all provided and
2. for all provided and
3. has no solutions if
4. for all in the domain of
5. for all in the domain of
6. has no solutions
   
   
5. Solving and Checking Equations
   
   
1. Find the domain of the rational function
2. Solve the equation by solving and checking these candidates for membership in the domain.
3. Solve the equation by solving and checking these candidates for membership in. the domain
4. Solve by solving .
5. Mathematical identities
   
   
1. provided that and
2. only when
3. 
4. for all in the domain of
5. for all in the domain of
6. provided that and
7. provided that and
   
   
6. Reduction of difficulty (on appropriate domains)
   
   
1. 
2. 
3. 
   
   
7. Solve the following equations and check by hand
   
   
1. 
2. 
3. 
4. 
5. (no solutions!)
   
   
6. Factoring polynomials over the Real numbers
   
   
1. is a prime as a polynomial for all constants
2. is prime as a polynomial
3. is prime as a polynomial if
4. is prime a perfect square if
5. factors as a product of polynomials if is the square of a rational number
6. factors using the quadratic formula
7. always factors into a product of linear polynomials and prime quadratic polynomials
8. Factor a polynomial using the remainder theorem and long division
9. If with , then
10. Factorizations of special polynomials:
    
    
1. 
2. 
3. 
   
   
11. If and , then
    
    
7. Word (Modeling) Problems
   
   
1. Unit conversions using the following facts
   
   
1. 1 mile = 5280 feet
2. 1 mile = 1760 yards
3. 1 yard =3 feet
4. and
5. 1 inch=2.54 centimeters
6. 1 pound453.5 grams
7. 60 mph=33 feet per second
8. 1 kilometer .61 miles
9. 2 radians =
   
   
2. Changing units by mimicking multiplication and division
3. Finding absolute extrema of quadratics
4. Linear rate problems
5. Simple and compound interest
6. Continuous growth and/or decay
   
   
8. Solving Procedures
   
   
1. Polynomials by the following procedure:
   
   
1. 
2. 
3. Factor if possible to
   
   
2. Rational functions by the following procedures:
   
   
1. 
2. 
3. 
4. 
5. Exclude values of the independent variable which make the denominator zero
6. Solve by factoring
7. Check candidates for solutions in the original equation, be aware of step v
   
   
3. Algebraic equations by the following procedures
   
   
1. by solving for the most difficult operation using +,-,×,÷.
2. by applying inverse of the most difficult operation to both sides of the equation.
3. by rewriting left side as and following above procedures.
   
   
4. Logarithmic equations of the form by the following procedures
   
   
1. 
2. 
3. Use the properties of logarithms:
4. Use the properties of logarithms to compress:
5. Apply inverse to both sides:
6. Solve
   
   
9. Given a graph of a function , determine the following:
   
   
1. The domain of
2. Solve
3. Solve
4. Solve
5. Solve
6. Solve
7. Determine the domains of and
   
   
10. Miscellaneous
    
    
1. Midpoint formula
2. Distance formula

Trigonometry Skills Recommendations

The following skills/facts should be contained in the trigonometry curriculum and should be included in practice work given to the students:

1. The definitions of the six trigonometric functions, using both right triangles and the unit circle.
2. Basic knowledge of the number such as:
   
   
1. The circumference of the circle divided by the diameter is .
2. The number is not 3.1415 nor is it , but is approximated by these.
   
   
3. The definition of angular measurements both in radians and degrees.
4. Convert angular measurement between radians and degrees:
   
   
1. A strong understanding of the angular measurement in radians.
2. The length of the arc subtended by an angle on the unit circle.
3. 1 radian57 degrees
   
   
5. Numerical approximations (without a calculator)
   
   
1. 
2. 
   
   
6. Values of the trigonometric functions for the standard angles which are multiples of 30 degrees and 45 degrees, without using a calculator.
7. The period of each of the trigonometric functions.
8. Given a zero of the function or , with , and the period, produce all the zeros of the function without using a calculator.
9. Solve the equations and for zeros in a given interval.
10. Domains for all six trigonometric functions.
11. Solve and on an interval.
12. Understand that has no solutions; similarly for .
13. Know the graphs of the six basic trigonometric functions.
14. Solve the following inequalities from the graph of the corresponding function:
    
    
1. ,,, and
2. ,,, and
3. ,,, and
   
   
15. Be able to quickly state and easily use the following identities:
    
    
1. Unit Circle Identities
   
   
1. 
2. 
3. 
   
   
2. Basic Division Identities
   
   
1. 
2. 
3. 
4. 
   
   
3. Even and Odd Identities
   
   
1. 
2. 
   
   
4. Euler Identities
   
   
1. 
2. 
   
   
5. Half-angle and Double-angle Identities
   
   
1. 
2. 
   
   
6. Miscellaneous Identities
   
   
1. 
2. 
   
   
7. Law of Cosines: ; It should be understood that this is a generalization of the Pythagorean Theorem.
8. Law of Sines
   
   
16. Ability to use trigonometric identities to produce the sine and cosine of 15°, 22.5° and 7°.
17. Use right triangle geometry and trigonometry to solve word problems.
18. State the meaning of , , and .
19. State the difference between and .
20. State the restrictions on the graphs of and which allow the definitions of and . (Note: The restrictions for may differ from one text to the next.)
21. Using right triangles and an expression for an inverse trigonometric function, write an algebraic expression for the basic trigonometric functions; for example, find the values for the six trigonometric functions given .
22. Use a calculator effectively, especially for applications involving non-standard angles. Operations should be performed with the calculator in both the radian and the degree mode.

Geometry Skills Recommendations

The following skills/facts should be contained in the curriculum geometry classes and should be included in practice work given to the students:

1. Lines and Planes
   
   
1. Measure of distance on a number line – using absolute value
2. Definitions of line and line segment
3. Concept of line and plane as sets of points
4. Perpendicular lines
5. Distance in plane – distance formula
6. Parallel lines in a plane
7. Transversals of parallel lines
   
   
2. Angles
   
   
1. Distinction between the angle and its measurement
2. How to measure angles
3. Properties of right angles and straight lines
4. Congruence of angles
5. Complementary and Supplementary angles
6. Angle bisector
7. Identification of certain pairs of angles cut by transversals
   
   
3. Triangles
   
   
1. Congruent triangles (sas, asa, sss)
2. Similar triangles and Proportions
3. Properties of right, isosceles, 30-60-90, and 45-45-90 triangles
4. Sum of the measures of the angles of a triangle
5. Altitude
6. Area and perimeter
7. Pythagorean Theorem
8. Area of a triangle
   
   
4. Polygons
   
   
1. Quadrilaterals (rectangle, parallelogram, and trapezoid)
2. Perimeter and area
   
   
1. Trapezoid
2. Parallelogram
3. Rectangle
4. Square
   
   
3. Similarity, proportions, and congruence for regular polygons
   
   
5. Circles
   
   
1. Measure of central angles
2. Arcs
3. Congruence of central angles in terms of arcs
4. Inscribed angles (right and straight)
5. Tangents
6. Diameter and radius
7. Circumference
8. Chords
9. as a number
10. as the measure of an angle
11. Sector of a circle
12. Area of a circle
13. Area of a sector
    
    
6. Solid Geometry
   
   
1. Distance between two points in three-space
2. Identify spheres, right circular cylinders, cones, pyramids, prisms, and parallelepipeds
3. Formulas for volumes
   
   
1. Sphere
2. Rectangular prism
3. Right circular cone
   
   
4. Formulas for lateral area/surface area
   
   
1. Sphere
2. Right circular cone
3. Right circular cylinder
   
   
7. Logic and Proof
   
   
1. Understanding of an axiomatic system
2. Truth values
   
   
1. Truth tables for "or", "and", and "not"
2. Implication
3. Converse, contrapositive, and inverse
4. Use of quantifiers, including "if", "for every", "for all", "for each", and "whenever"
   
   
3. A concept of proof as a justifiable sequence of steps from hypothesis to conclusion
   
   
1. Proof of false implication by counterexample
2. Direct proofs
3. Proof by contradiction
4. Vacuous proof
5. The logic and use of DeMorgan’s Laws

Summaries of Recommended Readings

These documents add perspective to this report and represent the national understanding of the issues in mathematics education as well as issues within the states:

• South Carolina Mathematics Framework, Columbia, South Carolina: South Carolina Department of Education, 1993.
  

  This document presents a statewide consensus of what we expect students to know and be able to do in mathematics and the changes necessary in the education system to support what teachers and students do in the classroom. The Framework is not a detailed program or a curriculum guide; it is intended to be used by policymakers, instructional leaders, teachers, and communities as a broad instructional design for the continuous improvement of mathematics education. Its chapters address the teaching and learning of mathematics; the K-12 mathematics curriculum; instructional materials; assessment; professional development of teachers of mathematics; and essential support systems.

• Precollege Preparation for College Mathematics: A Survey of South Carolina Faculty, by J. Christopher Tisdale, III, Danny W. Turner, and Gary T. Brooks, Department of Mathematics, Winthrop University, January, 1998.
  

  Do students enter South Carolina higher education institutions with the appropriate background to be successful in college mathematics? This report presents the opinion of 66 full-time faculty with recent experience teaching students in their initial college mathematics course.

  Survey results identify a high percentage of students who are under-prepared for this initial mathematics experience. The deficiencies are less severe for those aspiring to careers in technical/scientific areas. Weak-problem-solving skills and negative attitudes about learning mathematics are especially prevalent in students in non-mathematical majors.

  Other significant conclusions:
  
  

  • College faculty support the use of calculators in their classes, but are somewhat unsure if their use is detrimental during the precollege experience.
  • Students receiving credit for calculus taken in high school perform acceptably in the more advanced calculus course in college.
  • A high percentage of students are found to be deficient in the study, listening, note-taking, and test-taking skills necessary for successful performance in college.
    
    
• What Matters In College? , by Alexander W. Astin, San Francisco: Jossey-Bass, 1993; and What Matters in College?: Four Critical Years Revisited, by Alexander W. Astin, San Francisco: Jossey-Bass Publishers, 1997.
  

  The first book is the single most frequently cited work in higher education literature. This up-date provides recent information, through the tracking of 25,000 students through four years of college, to determine which aspects of the college experience provide the most impact on the students’ learning and maturation. Although the center chapters are technical in style, chapters 1 and 12 are of importance to students, parents, teachers, and guidance counselors. In fact, the conclusion could also be applied to high school teaching if properly reformulated. The primary conclusions are:
  
  

  • The number one factor influencing student development is the peer group.
  • The second most important influence is concerned faculty who are involved with the students.
  • The third most important factor is that the students experience a core curriculum (both with respect to their major and with respect to their overall college experience.)
    
    
• Counting on You: Actions Supporting Mathematics Teaching Standards, Mathematics Sciences Education Board, Washington, D.C.: National Academy Press, 1991.
  

  This brief document describes why significant change in mathematics education is necessary, what steps have been taken thus far to bring about such change nationwide, and how demanding the challenges are that teachers face in carrying out the task. It ends by describing specific actions that various members of the public can take to support the efforts of the mathematics teachers to meet the high standards they have set for their profession.
  
  

• Everybody Counts, National Research Council, Washington, D.C.: National Academy Press, 1989.
• A Challenge of Numbers, National Research Council, Washington, D.C.: National Academy Press, 1990.
• Reshaping School Mathematics, Mathematical Sciences Education Board, Washington, D.C.: National Academy Press, 1990.
• Assessment Standards for School Mathematics, National Council of Teachers on Mathematics, Reston, VA : 1995
• Mathematics and Science Achievement in Secondary School: IEA’s Third International Mathematics and Science Study (TIMMS), U. S. Department of Education, Washington, D. C.: U. S. Government Printing Office, 1998.

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