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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.

Non-Linear Function: A relationship where the
dependent variable does not change at a constant
rate. This non-constant rate will appear in the
table, and will cause the graph to be a curve.








• Y = 2x and y = 5 – 0.5x are linear
In general y = mx + b
represents a linear function. Y = 0.5x2 and
y = 2x are examples of non-linear

• 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
are non-linear. They are investigated
further in Frogs, Fleas and Painted Cubes
and in Growing, Growing, Growing.


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

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.)

Y-Intercept: The point where the graph of a
function crosses the y-axis. Since the point is on
the y-axis the coordinates are (0, something). So
we could say the y-intercept is the value of the
dependent variable when x is 0. (See Moving
Straight Ahead.)

• 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

• 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
variation. When x = 10, y = 23. If we
double x, say x = 20, then the
corresponding y value is not doubled.
There is no constant value for .


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


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 y-value 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 real-life 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 non-linear function) fit
reasonably well, and make predictions



• 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.