A fifteen years old learner is expected to be able to:

 

 

1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

 

 

 

 

2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

 

 

 

3. Know there is a complex number i such that , and every complex number has the form a + bi with a and b real.

 

 

 

4. Use the relation and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

 

 

 

5. Find the conjugate of a complex number; use conjugates to find quotients of complex numbers.

 

 

 

6. Solve quadratic equations with real coefficients that have complex solutions.

 

 

 

7. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

 

 

 

8. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

 

 

 

9. Add, subtract, and multiply matrices of appropriate dimensions.

 

 

 

10. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

 

 

 

11. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

 

 

 

12. Know and apply the Binomial Theorem for the expansion of in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.

 

 

 

13. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

 

 

 

 

 

 

14. Solve quadratic equations in one variable

• Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
• Solve quadratic equations by inspection (e.g., for ), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

 

 

 

 

15. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle .

 

 

 

16. Represent a system of linear equations as a single matrix equation in a vector variable.

 

 

 

17. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

 

 

 

18. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

 

 

 

 

19. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

 

 

 

 

20. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

 

 

 

21. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.

 

 

 

22. Prove that all circles are similar.

 

 

 

23. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

 

 

 

24. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems