Vocabulary: Thinking With Mathematical Models
Concept  Example 
Function: a relationship between 2
variables, say (x, y), so that, for any given value of x, a unique value of y can be calculated from an equation or read from a graph or a table. Linear Function: A relationship where the dependent variable changes at a constant rate in relationship to the change in the independent variable. The pattern of change can be recognized from a table or graph, and can be described using words or symbolic expressions. NonLinear Function: A relationship where the dependent variable does not change at a constant rate. This nonconstant rate will appear in the table, and will cause the graph to be a curve.

• Y = 2x and y = 5 – 0.5x are linear functions. In general y = mx + b represents a linear function. Y = 0.5x^{2} and y = 2^{x} are examples of nonlinear functions. • Students are familiar with the pattern of a constant rate of change in y shown below. (See Moving Straight Ahead)
• The two types of patterns shown below

Algebraic Expression: A combination of symbols and operations that can be evaluated if the values of the variables are given. Linear Equation: This might be an equation in 2 variables, such as y = 2x + 3, where y changes at a constant rate in relation to changes in x (see “function” above), or in one variable, such as 17 = 2x + 3 (which is just a particular case of y = 2x + 3), or 3x – 2 = 2x + 3. (See Moving Straight Ahead.) Solving Linear Equations: In the first case above, y = 2x + 3, there is an infinite number of solutions, each of which is pictured as a point on the graph of the line y = 2x + 3. In the other two cases there is just one value of x that makes the equal sign true. (See Moving Straight Ahead.) Slope: The slope of a line the ratio of vertical change to horizontal change, or (change in y) divided by (corresponding change in x). (See Moving Straight Ahead.) YIntercept:
The point where the graph of a 
• 2x + 3 is an algebraic expression. 3x – 2 is a different expression. Notice that we are not asked to FIND a value of x for either of the expressions. We may substitute ANY values of x into these expressions to evaluate the expression. For example, if we insert x = 1, we get 5 for the value of the first expression. • If we connect 2 expressions with an equal sign, we are asserting they are equal for some value(s) of the variable. For example, the linear equation 3x – 2 = 2x + 3 is only true for one value of x. One efficient way of solving this equation would be to do the same operations to both sides, using Properties of Equalities: 3x – 2 – 2x = 2x + 3 – 2x, or, x – 2 = 3. x – 2 + 2 = 3 + 2, or, x = 5 Or to use a graphical way of solving. (See Moving Straight Ahead)

Linear Inequality: A comparison between 2
linear expressions, such as 17 > 2x + 3, or 3x – 2 < 2x + 3. This time we want to find the solutions that make the inequality sign true.

The rules for solving equations (see above) apply to solving inequalities, except that when we have to multiply or divide both sides of the inequality by a negative the order is reversed. For example: • 3x – 2 < 2x + 3 3x – 2 – 2x < 2x + 3 – 2x (subtracting 2x from both sides) x – 2 < 3 x – 2 + 2 < 3 + 2 (adding 2 to both sides) x < 5. This means that any value lass than 5 is a solution for the original inequality. • 3x – 5 < 4x + 2 3x 5 4x < 4x + 2 4x (subtracting 4x from both sides) 7x 5 < 2 7x – 5 + 5 < 2 + 5 (adding 5 to both sides) 7x < 7 (dividing both sides by 7) (Notice that the inequality sign reversed when both sides were divided by 7. This happens because dividing (or multiplying) by a negative changes positives to negatives and vice versa, and we know that positive integers are in the OPPOSITE order from negative integers, That is, 3 < 2, but 3 > 2.) 
Direct variation: A relationship between 2 variables in which an increase in one variable by a particular factor creates an increase in the other variable by the same factor. A direct variation relationship, between x and y, for example, always has the form y = ax, so this is a particular case of a linear relationship. Since this can also be written as , we can say that in a direct variation relationship the ratio of the variables is always a constant.

• y = 0.6x shows a direct variation between x and y. As x increases, y also increases. Substituting 2 for x, we get 1.2 for y. If we double the value of x, to 4, we get double the value of y, 2.4. In every case the value of .
• Y = 2x + 3 would NOT be a direct 
Inverse Variation: A relationship between 2 variables in which an increase in one variable by a particular factor causes a decrease in the other variable by the same factor. An inverse variation relationship between x and y always has the format . Since this can also be written as xy = a we could say that in an inverse variation the product of the variables is always constant. The graph has a characteristic curved shape.

• shows an inverse variation between x and y. As x increases, y decreases. When x = 0.1, y = 20. If we multiply the given x value by a factor of 4, say, then the new yvalue is one fourth of what it was. . In every case xy = 2.

Mathematical Modeling: a process by which mathematical objects and operations can be used as approximations to reallife data patterns. We know that the mathematical model will not fit the real life example exactly, but there is enough of a pattern in the situation to make the model (which in this unit is a linear or nonlinear function) fit reasonably well, and make predictions reasonable.

• Suppose we collect data about the outside temperature as a plane ascends. We would see that the temperature falls as the height increases. But we would have no way of making any accurate prediction unless we look for a pattern in the data. If we place the data in a table and look at the pattern of change, we might be able to determine if the relationship is approximately linear or not. We might also make a graph and examine the shape of the graph, to determine if the relationship is linear, or if the pattern fits some other relationship we recognize. 