Transformations and Equations
Transformations of the Plane
Functions of two variables in general building on the examples of addition, multiplication and division already introduced. Translations, dilations and reflections on the plane and comparison with similar functions on the line. Compositions and inverses of these functions. Emphasis on the comparison with arithmetic operations, properties of arithmetic operations, order of operations, parentheses.
Transformations on the Graph of Functions
Translations, dilations and reflections acting on the graphs of functions. Algebraic representation of transformations for the graph of a function. Interpretation of changes in the data modeled by a function in terms of transformations to the graph. Applications to science and everyday life. Solution of linear equations using transformations pointing out the characteristics of transformations that preserve and do not preserve solutions and their relationship with the algebraic manipulation of equations. The teaching of solutions of linear equations.
Children’s understanding of equations and expression. Geometric and algebraic representation of equations and their solutions. Parabolas and their equations under transformations and interpretation of the situations they model. The quadratic formula and its interpretation through transformations. Geometric understanding of the number of solutions to an equation.
Divisibility for Integers and Polynomials
Recall of the concept of divisibility for integers. Unique factorization for integers as product of primes. The Euclidean algorithm for the greatest common divisor. Review of basic facts about polynomials. Divisibility for polynomials, unique factorization. The relations between roots and factoring for polynomials. The parallel between the properties of integer and those of polynomials and applications to teaching on these two topics. The number of solutions of a polynomial equation.