Walking Rates and Linear Relationships
Goals
• In this Problem, you will investigate these
independent and dependent variables for linear
relationships that are represented in tables
• Construct tables, graphs, and equations to
represent linear patterns of change
Students explore the walking rates of three
students and look at the walking rate and its
effect on various representations—table, graph,
and equation.
• Which axis should represent time? Distance?
Suggested Questions Ask:
Alternate Launch
Pose the following situation:
Hillary and Bill used graphs to represent the
relationship between distance they walked and
time:
• What effect does a walking rate have on the
• As you change your walking rate to a faster
rate, what effect does this have on the original
equation? (The coefficient of t will increase.)
If you have graphs and tables from the last
Problem, put them up and ask:
• Suppose you made a table, graph, or an
equation of your distance over time at your
walking rate. As you change your walking rate
to a faster rate, what effect does this change
have on the table and graph that represented
your original equation? (Students may make
conjectures at this time. The Problem will
help them develop an awareness of the
effects of the walking rate on tables, graphs,
and equations.)
Hillary’s Graph
Distance
relationship between time and distance
walked? (If you increase the rate at which
you walk, you can cover more distance in a
given time. This means that if you walk faster,
it takes less time to cover a certain distance.
Conversely, if you walk for a longer period of
time at a given rate, you will cover more
distance.)
1
[The horizontal axis usually represents time
(the independent variable) and the vertical
axis represents distance. Note: You might
have the class do Questions A and B that
involve the three students. Discuss these
questions and then assign Question C as a
follow-up to the discussion of A and B.]
Launch 1.2
You could collect some brochures for local
walkathons to post in your room.
Introduce the three students and their walking
rates.
I N V E S T I G AT I O N
questions by looking at the walking rates of
three students. You are asked to make a table
and a graph and to write an equation for each
of the walkers. Examine these to see how
changing the rate affects each form of
representation of the data.
• Describe the patterns of change between the
Bill’s Graph
Distance
1.2
Time
Time
• Tell a story of a walkathon that could be
represented by Hillary’s and Bill’s graphs.
Suggested Question
• Do either of the graphs represent a linear
relationship? Some students will argue that
Bill’s graph is linear in pieces. This is okay as
long as you emphasize that the total graph is not
linear. This graph is called piece-wise linear.
Students can work in groups of two to four.
Investigation 1
Walking Rates
19
Explore 1.2
Students should each make their own tables and
graphs, but they can discuss the questions in pairs.
This will give you an opportunity to examine
students’ ability to make tables and graphs. If a
student is having trouble getting started, you can
pair the student temporarily with another student
who seems to be on target to get started.
One of the issues that Question C is raising is
“Is every relationship linear?” If students are
struggling to see the patterns in the table, you
could suggest that they graph the data to see if it
forms a straight line.
Check to see if students are picking up on the
patterns in Elizabeth’s table. It is linear but the
independent variable is increasing by 2 seconds
each time. For each 2-second increase, distance is
increasing by 3 meters. This data represents a
linear relationship, but the walking rate is
3 meters every 2 seconds or 1.5 meters per
second—not 3 meters per second.
Summarize 1.2
Suggested Question If you assign Questions A
and B first, ask:
• How does the constant walking rate show up
in the table, graph, and equation? (It affects
the steepness of the line. It is the coefficient
of t in the equation and, in the table it, is the
constant change in distance as time changes
by 1 second. Recognizing the constant rate of
change in tables is a bit more complex. See
the following discussion as a model of how to
help students see the patterns in a table.)
Using a Table
For each unit change in time, there is a constant
change in distance. This constant rate should be
explored in the different representations. It may be
difficult for students to recognize the effect of
changing the rate of walking in a table. You can use
the ideas from the Comparing and Scaling unit.
Suggested Questions Ask:
• Trish walks at 2 meters per second. As the
number of seconds she walks increases from
1 second to 2 seconds, how does the distance
change? (The distance changes 2 meters from
20
Moving Straight Ahead
1 to 2 seconds, 2 meters from 2 to 3 seconds,
etc. Students might think of this as a rate of
increase of 2 meters for every second, or they
may reason proportionally that, if the time
doubled, then the distance should also
double, or they may think that, if the distance
value was twice the time value to start with, it
should continue to be twice the time value.
The latter two ways to think of the situation
will only work if (0, 0) is a point on the graph
or in the table.)
• If the number of seconds increases by 1, does
the distance always change by 2 units? (Yes.
Ask the class to show this on the table.
Walking Rates
Distance (meters)
Time
(seconds)
Alana
Gilberto
0
0
0
0
1
1
2
2.5
2
2
4
5
3
3
6
7.5
Leanne
For Gilberto, the entries in the table go up by
2 meters for each one-second increase. This is
what 2 meters per second means.)
• In the table for Gilberto, if the number of
seconds increases by 2 seconds, what is the
change in the distance? (4 meters. Have the
class illustrate this answer on the table.
Students can see that a change from 1 sec to
3 sec produces a change of 4 meters in the
distance. Note: Students tend to set their
tables up in units of one second because that
is what makes sense to them. This means that
they need to produce many pairs to get a
sense of what is happening over a reasonable
span of time. You may want to discuss the
advantages and disadvantages of using other
intervals such as 2 seconds or 5 seconds, etc.)
Using a Graph
It should be fairly easy to recognize the effect of
changing the rate of walking on a graph—it
changes the ‘steepness’ of the line. Some students
say that the line “goes up faster.” These intuitive
ideas are the goal at this time.
Suggested Questions Ask questions like these to
push the students’ thinking:
• What would a graph look like if someone
walked 3 meters per second? 2 meters per sec?
1
1
1 2 meters per second? 2 meters per second?
• How would each of these change the table?
• How would each of these change the
equation?
per
per
per
per
per
per
2.5 meters
5 meters
7.5 meters
10 meters
12.5 meters
15 meters
Show the vertical and horizontal changes on
the graph by drawing in dotted lines.
Using an Equation
Students may be able to recognize the effect of
changing the rate of walking on the equation
d = rate 3 time or in symbol form d = rt. The
rate (r) is the coefficient of t; it is the number by
which time is multiplied. The word coefficient is
defined in the next investigation. At this point,
students will say it is the rate at which people
walk, which is the number that “time is multiplied
by to get the distance.”
Students may want to see the graphs of these
equations on a graphing calculator. If you do this,
then ask them to answer questions similar to
those above, using the graphing calculator.
Question C
Assign (if you did not do so before) Question C.
Be sure to discuss it and emphasize how constant
rate of change between time and distance is
represented. The table of Elizabeth’s time and
distance represents a linear relationship. But
students need to note that the time is not
increasing by 1. Go back to the unit rate idea
discussed in the preceding paragraph.
other pairs of data for the table. (This will give
you a chance to see and hear how students
are thinking about rate: are they simply
adding the same to each distance value to
complete an additive pattern, or are they
paying attention to both variables? A rate
can be thought of as a single numerical
quantity, especially if you are comparing
rates. But a rate is also an internal
comparison between two quantities.)
• For those situations in Question C that are
linear, compare the walking rates with the
original three students. Who is the fastest?
Slowest? Which graph is the steepest?
Pick one or two of the situations in Question C:
1
1 second
2 seconds
3 seconds
4 seconds
5 seconds
6 seconds
• Assuming the rate remains constant, find two
I N V E S T I G AT I O N
Pick two points on one of the graphs and ask
how the time and distance changes from one point
to the next. This is an opportunity to connect back
to unit rates in Comparing and Scaling. For
example, for Leanne, in the points (3, 7.5) and
(6, 15), time increases by 3 and distance increases
by 7.5. To show that this is the same rate as
1 second per 2.5 meters, you could write:
Suggested Questions Ask:
• What are the variables?
• Describe the patterns of change between the
two variables.
• Describe what is happening in each situation.
For all of the 4 situations in Question C, ask:
• After 5 seconds who is ahead? To which
situation does the point (2, 3) belong? Explain.
Check for Understanding
If you need more examples of linear relationships
in which the independent variable, x, is changing
in intervals other than one, you can use the
following tables. One of the misconceptions that
students have is that, in a table, each value for x
must be exactly 1 greater than the previous.
Suggested Question
• Does either table of data represent a linear
relationship? Explain.
x
y
x
y
5
18
0
0
10
33
3
12
15
48
5
20
20
63
8
32
10
40
(In the first example, you can see that
33 - 18 = 15, 48 - 33 = 15, and 63 - 48 = 15.
This means that for each change of 5 in x, there is
a change of 15 in y. For a change of 1 in x, there
Investigation 1
Walking Rates
21
must be a corresponding change of 15 4 5 = 3 in
y. For the second table, changes in x are not
evenly spaced, so we have to be even more
careful. Students can apply the reasoning that
they developed in Comparing and Scaling to find
the constant rate; we see that 12 - 0 = 12. So for
a change of 3 in x we have a change of 12 in y.
This would mean a change in x of 1 should give a
change in y of 4 if the relationship is linear. This
pattern of change shows that the table represents
a linear relationship with a change of 1 in x
related to a change of 4 in y. For example, from
(8, 32) to (10, 40) in the table is a change of 2 in x.
So the change in y should be 8 and 40 - 32 = 8.
If students are dubious about either table
representing a linear relationship have them
graph the coordinate pairs and then examine the
tables again, looking for points they see on the
line but not in the table.)
Having students interpolate pairs of data for
the table, or extend the table, or produce pairs of
data given an x-value or a y-value will allow you
to assess how they are thinking of rate at this
time.
You can make some quick sketches of graphs
on the overhead and ask if they are linear: for
example, a sketch of a piece-wise linear graph (see
the preceding alternative launch for an example),
some sketches of linear graphs, and some graphs
with curves in them.
Suggested Questions Conclude the discussion by
asking:
• Was every relationship in this investigation
linear?
• How can we tell if a graph is not linear?
• How can we tell if a table does not represent a
linear relationship?
• How can we tell if an equation represents a
linear relationship?
22
Moving Straight Ahead
At a Glance
1.2
Walking Rates and Linear
Relationships
PACING 1 day
Mathematical Goals
• Describe the patterns of change between the independent and dependent variables for linear
relationships that are represented in tables
• Construct tables, graphs, and equations to represent linear patterns of change
Launch
Materials
Introduce the three students and their walking rates. Ask:
• As you change your walking rate to a faster rate, what effect does this
•
Transparency 1.2
have on the original equation?
If you have graphs and tables from the last Problem, put them up and ask:
• As you change your walking rate to a faster rate, what effect does this
change have on the table and graph that showed the results of your
original equation?
• Which axis should represent time? Distance?
Explore
Check to see if students are picking up on the patterns in Elizabeth’s table.
Summarize
Materials
If you assign Questions A and B first, ask:
•
• How does the constant walking rate show up in the table, graph and
Student notebooks
equation?
It may be difficult for students to recognize the effect of changing the
rate of walking on a table. Consider asking:
• What would a graph look like if someone walked 3 meters per second?
1
1
2 meters per second? 1 2 meters per second? 2 meter per second?
• How would each of these change the table? The equation?
Pick one or two of the situations in Question C:
• What are the variables?
• Describe the patterns of change between the two variables.
For all of the 4 situations in Question C, ask:
• After 5 seconds, who is ahead? To which situation does (2, 3) belong?
Explain.
ACE Assignment Guide
for Problem 1.2
Core 3, 4, 5
Other Applications 5, Connections 19–22, Extensions
Adapted For suggestions about adapting
ACE Exercise 4 and other ACE exercises, see
the CMP Special Needs Handbook.
Connecting to Prior Units 19: Accentuate the
Negative; 20–22: Comparing and Scaling
30; unassigned choices from previous exercises
Investigation 1
Walking Rates
23
Answers to Problem 1.2
A. 1. The greater the walking rate, the faster the
distance values increase. The walking rate is
the amount that the distance (m) for each
person is changing by as time increases by
1 s. For example, Alana’s walking rate is
1 m/s and, for each change by 1 s, the
distance increases by 1 m.
Walking Rates
Distance (meters)
Time
(seconds)
Alana
Gilberto
Leanne
0
0
0
0
1
1
2
2.5
2
2
4
5
3
3
6
7.5
4
4
8
5
5
10
12.5
6
6
12
15
7
7
14
17.5
8
8
16
20
9
9
18
22.5
10
10
20
25
10
2. The higher the rate, the steeper the graph.
Distance (meters)
Walking Rates
24
21
18
15
12
9
6
3
0
d
Leanne
Gilberto
Alana
t
0 1 2 3 4 5 6 7 8 9 10
Time (seconds)
3. Alana: d = t, Gilberto: d = 2t, Leanne:
d = 2.5t; the walking rate is represented as
the number that comes before the t. It is
the number you multiply time by to get
distance. (Note: Students will discuss later
how there is actually a 1 in front of the t in
Alana’s equation.)
24
Moving Straight Ahead
B. 1. Alana by l m; Gilberto by 2 m; Leanne by
2.5 m; this change is represented in the
table because if time increases by 1 s, the
distance values increase by 1 m, 2 m, and
2.5 m. This change is represented in the
graph because if you start on the line for
Alana’s graph and go over 1 to the right
(1 second later) and up 1 (1 meter further),
you land back on the line for Alana. If you
start on the line for Gilberto’s graph and go
over 1 to the right and up 2, you are on
Gilberto’s line. Also, if you start on the line
for Leanne’s graph and go over 1 to the
right and up 2.5, you will land on the graph
for Leanne.
2. If t increases by 5 seconds, the change for
each student is Alana by 5 m; Gilberto by
10 m; Leanne by 12.5 m. This change is
represented in the table because if time
increases by 5 s, the distance values
increase by 5 m, 10 m and 12.5 m. This
change is represented in the graph because
if you start on the line for Alana’s graph
and go over 5 to the right (5 seconds later)
and up 5 (5 meters further), you land back
on the line for Alana. If you start on the
line for Gilberto’s graph and go over 5 to
the right and up 10, you are on Gilberto’s
line. Also, if you start on the line for
Leanne’s graph and go over 5 to the right
and up 12.5, you will land on the graph
for Leanne.
3. Rates per minute: Alana: 60 m/min,
Gilberto: 120 m/min, Leanne: 150 m/min;
(walking rate per second 3 60 seconds)
Rates for one hour at a steady pace:
Alana: 3,600 m/hr; Gilberto: 7,200 m/hr;
Leanne: 9,000 m/hr.
C. Elizabeth’s and Billie’s are linear; they show
a constant walking rate. Elizabeth walks
3 meters per 2 seconds, which you can see by
the changes in the table values; and Billie
walks 2.25 m/s, which you can see in the
equation as the number you multiply t, the
time, by to find d, the distance.