Standards - Mathematics

• rewriting radical expressions of rational and irrational numbers can help in recognizing geometric patterns.

• When quadratic equations do not have real solutions, the number system must be extended so that solutions exist. and the extension must maintain properties of arithmetic in the real numbers.

Students understand that:

• Complex number calculations follow the same rules of arithmetic as combining real numbers and algebraic expressions.

Students understand that:

• A matrix is a tool that can help to organize, manipulate, and interpret data.

Students understand:

• Multiplying a matrix by a scalar affects every element in the matrix equally.
• Scalar multiplication is a tool that allows all elements of a matrix to be changed in a simple manner.

Students understand that:

• Matrix addition and subtraction may be performed only if the matrices have the same dimensions.
• Matrix multiplication can be performed only when the number of columns in the first matrix equal the number of rows in the second matrix.
• There are many contextual situations where arithmetic operations on matrices allow us to solve problems.

Students understand that:

• Identity properties that apply to other number systems apply to matrices.
• The multiplicative inverse property that applies to other number systems applies to matrices.
• A matrix with a determinant equal to zero does not have a multiplicative inverse analogous to zero in the real number system not having a multiplicative inverse.
• Division by zero in the real number system is undefined.

• ln(x) gives the time needed to grow x.
• E^x gives the amount of growth after the time x.

• The conjugate of a complex number differs by the sign of its imaginary part and has the same modulus.
• The modulus of a complex number corresponds to the magnitude of a vector and, therefore, is useful in the geometric representation of complex numbers.
• Mathematical convention is that radical expressions are not left in denominators to facilitate numerical approximations. therefore, since the i is equal to the square root of -1, conventional form says that i does not appear in the denominator of a fraction.
• Different forms of a complex number quotient (indicated quotient, single complex number) may be more useful for various purposes.

• A complex number (a+bi) can be graphed in a rectangular coordinate system as (a, b).
• A complex number may be represented in the plane using equivalent polar and rectangular coordinates.
• Different representations of a complex number may be more useful for various purposes.

• Different representations of mathematical concepts (e.g., algebraic and geometric representation of complex numbers) reveal different features of the concept and each may facilitate computation and sense making in different settings.
• Mathematics is a coherent whole and structure within mathematics allows for procedures from one area to be used in another (e.g., coordinate geometry and the complex plane, vectors and complex numbers, or plotting of a conjugate of a complex number in transformational geometry).

• Representing complex numbers on a rectangular coordinate system allows the use of techniques developed for real numbers to find distances and midpoints.

• The degree of a polynomial determines the number of roots, some which may be real, complex, or used more than once.
• Only real roots will be x-intercepts on a graph.

• Functions have various types of continuity and discontinuity.
• That limits can be determined numerically, graphically, and algebraically.
• converging means that a limit exists, while diverging means that a limit does not exist.

• Magnitude as the length of a vector and direction of a vector as the measure of the angle it makes with a horizontal line
• When vectors are represented in component form, ensuing computations and applications can be accomplished.

• Vectors having the same magnitude and direction are equivalent regardless of where they are in the xy-plane.
• Vectors in standard position have a terminal point that is equal to the components of the vector.

• The result of combining multiple vector forces creates a net magnitude and direction.
• Certain situations can be represented using vectors.

• The dot product of two vectors is the sum of the products of their horizontal components and their vertical components.
• If ? = ?1? + ?1? and ? = ?2? + ?2?, the dot product of ? and ? is defined by ? ? ? = ?1?2 + ?1?2.
• Alternative Formula for the Dot Product: ? ? ? = ??? ??? cos ?, where ? is the smallest non negative angle between v and w.
• Two vectors are orthogonal when the angle between them is 90^o. To show that two vectors are orthogonal, show that their dot product is zero.
• A vector may be expressed as the sum of two orthogonal vectors, called the vector components.

• There are multiple ways to find the sum and difference of a pair of vectors.
• The magnitude of the sum of two vectors will not be the same as the sum of the magnitudes unless the vectors are in the same direction.
• The vector with the larger magnitude will have the greatest effect on the result.

• Scalar multiplication results in a dilation of the original vector where a scalar greater than 1 would increase the magnitude and a scalar from 0 to 1 would decrease the magnitude.
• A negative scalar would reverse the direction of the vector.
• The absolute value of the scalar has the resulting effect on the magnitude.