# Change and Invariance

**Systems of Two Linear Equations**

Functions of two variables. Examples of situations modeled with linear functions of two variables. Students’ difficulties in translating word problems into algebra. Representations of functions of two variables with tables, algebraic expressions, and graphs. Representation of linear relations among the variables with lines on the plane. Solving 2 linear equations of two variables; interpretation of the number of solutions.

**An Example-Based Introduction to the Idea of Limit**

Decimals with an infinite number of digits as limits of sequences. Measuring with increased accuracy. The physical limitations when using mathematical models in the real world. The idea of limit and of vertical and horizontal asymptotes . Applications to arithmetic operations and the middle school classroom (dividing by zero, dividing by large numbers, cutting in half repeatedly). Approximating solutions to equations.

**Slope and Rate of Change**

Slopes as indicators of the rate of change of a function generalizations of the rates introduced through a fraction (and the corresponding rational number). Average rate of change of a function over an interval and its geometric representation as slope of a secant. Instantaneous rate of change introduced as the limit of the average rate of change and its geometric counterpart as slope of a tangent line. Comparison of linear functions to non-linear functions, whose rate of change varies in different intervals.

**The Slope Function**

Introduction of the derivative as the function slope at the point or rate of change at the point. Comparison of derivatives for different types of functions (constants, linear, quadratic, exponentials, 1/x). Reconstruction of a function given its derivative. Applications to issues relevant to middle school students like interpretation of sentences as “the speed of an automobile is increasing slower and slower”. Representations through graphs, tables of values, and algebraic expressions.