1, Unit 3 - Linear Models
In the "Linear Models" unit, students learn how to recognize situations
in which key variables change at a constant rate. They learn how to express
and interpret those patterns of change in data tables, straight line
graphs, and equations in the form y = a + bx.
They identify and interpret the slopes and intercepts of linear graphs.
They learn techniques for solving linear equations and inequalities that
arise in science and business problems. They make effective use of graphing
calculators to accomplish such tasks as evaluating and solving. They
also learn to evaluate and solve linear equations using paper and pencil
Students will frequently use the language "rate of change" when talking
about the "slope" of a linear function. Also note that the letter "b" is
not reserved almost exclusively for the y-intercept as you may
have experienced in your mathematics program. The y = ax + b form
emphasizes the modeling or statistical approach to linear functions where a is
the starting value or y-intercept and b is the rate of
change. One goal is to help students develop flexible use of variables.
Thus, you will notice many different letters representing linear functions
throughout the curriculum.
Ideas from Course 1, Unit 3
- Linear data patterns: A relationship between variables x and y is
called linear if the graph of related (x, y) values is
a straight line. This graph pattern occurs when there is a constant
difference between successive y values as x values
change uniformly. That is, the ratio (change in y)/(change in x)
is constant. For example, in the next graph and accompanying table,
(5.5 - 1)/(5 - 2) = (13 - 7)/(10 - 6) = 1.5
- Equations of linear functions: Every linear relationship can
be expressed with an algebraic rule in the form y = a + bx, where a indicates
the y-intercept, (0, a) of the graph and b indicates
the slope of the graph and the rate of change in y values.
For example, the graph of y = 5 + 3x crosses
the y-axis at (0, 5)
and rises at a slope of 3 - for every increase of 1 in the value
of x, the value of y increases by 3. Because linear
functions are those with constant rates of change in the dependent
variable, they can also be thought about and represented with NOW-NEXT equations.
The linear relation with rule y =5 + 3x is
the same as that with rule NEXT = NOW + 3,
starting at 5. (See the Quick Summary of Course
1, Unit 2, and student book pages 165-169.)
- Rate of change: Appears in the table as above, on the graph
as slope, in the equation as the coefficient of x. (For example,
in y = 5 + 3x, the rate of
change is 3.)
- Modeling linear data patterns: Exact linear relationships can be
expressed with algebraic rules in the form y = a + bx.
But in many practical problems, experimental data might only suggest,
but not exactly fit,
a linear pattern. In such cases, it is often useful to summarize the
data trend by
drawing a line that matches the scatterplot of (x, y)
data pairs and to find the rule (formula) relating y- and x-coordinates
of points on that
example, the following graph shows 7 data points and the graph of y = -0.5 + x that matches the pattern in those data quite well.
- Solving a linear equation: Students have graphic, numeric,
and symbolic strategies for finding the value of x that
makes equations like a + bx = c true.
For example, to solve the equation 5 + 4x = 13,
they can scan a table or graph of y = 5 + 4x in
search of points with coordinates (x, 13). They can also
apply properties of equality to reason like this:
If 5 + 4x = 13, then 4x = 8
(subtract 5 from both sides).
If 4x = 8, then x= 2 (divide both sides
- Solving systems of linear equations: In this unit, students
solve simple systems of linear equations like y = 3x and y = 5 - 2x. That
means finding a pair of values (x, y)
that satisfy both conditions. Once again, they have at least three
strategies available for this kind of problem. They can graph the two
linear functions and look for coordinates of the intersection point.
They can scan tables of values for the two functions, looking for points
where a single x value produces the same y value
They can also use symbolic reasoning as follows:
If y = 3x and y = 5 - 2x, then 3x = 5 - 2x.
So, 5x = 5 (add 2x to both sides).
So, x = 1 (divide both sides by 5).
So, y = 3 (substitute x = 1 in either function rule).