Chapter 3: Linear Functions Section 3: Slope as Rate of Change
Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 3-24
Linear Functions: Slope as Rate of Change Teacher Notes
Objectives
•
To associate the slope of a straight line with a constant rate of change.
•
To calculate the rate of change from data points on a line, using the correct units.
•
To read from a linear graph: the rate of change, the scale of the axes, and the correct units.
Math Highlights
This section highlights the use of the slope formula to find the rate of change from graphs and
data. Students may need to review the slope formula; see previous sections on the calculator.
rate of change =
m
=
y
2
N
y
1
x
2
N
x
1
units
Emphasize the use of appropriate units. For example, in the
Fill the Pool
example in the
Overview
, the students see a pool filling at a constant rate of change. Two data points from the
growth graph or table shown are (4,2) and (6,3). To find the slope, students should calculate:
rate of change =
m
=
3
N
2
6
N
4
ft
hr
=
1
2
ft
hr
= .5
ft
hr
Mention to students that if the graph of a real problem were nonlinear, the calculation of the rate of
change using two data points gives the
average
rate of change over the interval chosen.
Common Student Errors
•
Neglecting to write the appropriate units.
•
Misunderstanding how to use the
position
with respect to a starting place rather than
distance. In the slope formula, y
2
−
y
1
gives the
signed distance
traveled.
•
Specifying which point is (x
1
,y
1
) and which point is (x
2
,y
2
). Show students that both
calculations give the same answer. Some students are confused when they see a division by
two negative numbers that results in a positive growth. Discuss this with the students. Watch
for an incorrect substitution where students switch the order in the numerator or
denominator. For example, if the data points (0,0) and (10,2) give the growth of a plant in cm
per day, the rate of change is calculated:
Correct:
2
N
0
10
N
0
cm
day
=
1
5
cm
day
or
0
N
2
0
N
10
cm
day
=
M
1
M
5
cm
day
Incorrect:
0
N
2
10
N
0
cm
day
•
Students need to be exposed to other variables besides
x
and
y
. For example, it is useful to
use
t
for time and
d
for distance.
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