Slope Changes Everything!
A critical analysis of the practical and graphical
applications of slope
Shelbi K. Cole
Title Page
Page 1
Unit Introduction
This unit focuses on the concept of slope as it relates to physical structures and to linear graphs.
Traditional textbooks often explore this concept by providing procedural methods for calculation
of slope with little emphasis on its role in the real world. In contrast, this unit asks students to
critically analyze the concept of slope to gain an understanding not only of how to measure slope
but also why it is necessary to agree on a conventional way of calculating it.
The unit emphasizes both the Curriculum of Core and the Curriculum of Practice. Meaningful
contexts are provided as students examine real-world situations dependent on standardized
methods of measuring slope. Students transition from viewing slope as a physical attribute to
slope as a means of interpreting graphs. The goal of this unit is to provide students with an
understanding of why slope is important both in the world and as a means of interpreting
graphical representations. The ability to effectively demonstrate change via graphical
representations is a highly desirable quality in many professions and is emphasized throughout
the unit.
Mathematics curricular analyses provided by the TIMSS studies have found curricula in the
United States highly repetitive calling for intended coverage across more grades per topic than
average (Valverde & Schmidt, 2000). The National Council of Teachers of Mathematics
(NCTM) has outlined the need for classrooms that are rich in student discourse, with high
expectations and learning tasks that focus on student understanding of mathematics in place of
the typical learning of procedures that has dominated classroom practice since the 1930’s
(NCTM, 2000). Still, reform visions are seldom transferred into mathematics classrooms as
whole group, non-differentiated instruction dominates current and past teaching practice (Stigler
& Hiebert, 2004). National standards are thrown by the wayside as the “dumbed down
textbooks” become the “defacto” mathematics program of a school (Reis & Renzulli, 1994;
Reys, Reys, & Chavez, 2004). Internationally, the United States ranks low in mathematics topic
difficulty and 20% of eighth graders attend schools where basic arithmetic is the most
challenging mathematics class offered (Cogan, Schmidt, & Wiley 2001).
This unit addresses concerns of researchers in the fields of both mathematics and gifted
education. The design of the unit calls for students to use prior knowledge to construct meaning
of the concept of slope. A topic that often gets brushed over in traditional algebra courses is
given due respect in this unit as the Parallel Curriculum Model framework has been used to
provide depth and complexity for mathematically talented students. Any “procedures” that might
be deemed efficient for use in solving problems related to this content must come from the
students themselves. Aligned with both NCTM and NAGC curriculum standards, the lessons in
this unit follow constructivist ideals in allowing students to develop understanding of the content
at deeper levels than are allowed by traditional instruction. This unit is designed to replace and
extend the concept of slope from one or two lessons to an entire unit. Though there are only five
Unit Introduction
Page 2
lessons, each takes considerable time to complete. They are designed in this way to enable
students to engage in tasks that do not seem so disconnected from one another as is often the case
in traditional mathematics courses. The entire unit should take approximately 3-4 weeks
depending on the flexibility in instruction and use of additional resources recommended.
Essential Questions Related to the Parallels
CORE




What distinguishes slope as a physical attribute from slope as a rate of change? To what
degree do the two overlap?
What information can one gather by glancing at a graph?
How do real world contexts affect the domain and range of functions?
How can the procedural concept of slope as rise/run be translated to a “neat” formula for
calculating the slope of a line through two points in a plane?
PRACTICE


Is it necessary to agree on a standardized means of measuring slope? Why?
What is the relationship between slope and scale and how might researchers or
companies alter one to portray information in a misleading way?
CONNECTIONS

How is the interpretation of slope as a physical attribute affected by properties of
physics?
UNIT OBJECTIVES



Students will be able to measure slope in physical structures and in linear graphs.
Students will be able to analyze rates of change in graphical representations and discuss
what this tells them about a graph.
Students will be able to explain the effects of changing the scale on a graph’s slope.
BACKGROUND INFORMATION
Prior Knowledge of Students
Students should come into this unit with prior knowledge related to the following areas:
 Plotting points using both standard and adjusted scales (i.e. one unit increments, ½ unit
increments, 5 unit increments, etc.)
 Identifying independent and dependent variables and provide examples demonstrating
how one affects the other
 Solving equations in one variable
 Substituting values into equations with multiple variables
Unit Introduction
Page 3
Resources
The first lesson requires a variety of materials including toy car ramps (2 different lengths,
stopwatches, rulers, and blocks that will be used to support the ramps. The second lesson
specifically requires arts and crafts materials that can be provided by teacher or students.
Throughout the unit teachers and students should have computer and graphing calculator access.
In addition, calculator based rangers (CBRs) are required for Lesson 4. Additional resources
recommended include calculator based laboratory systems (CBLs) and graph links. These
technological devices allow students to investigate slope as a rate of change in an authentic,
engaging manner.
Each lesson includes a section entitled “additional resources” that provides enrichment and
extension options for students and teachers. These resources can be used as individualized,
interest-based, talent development opportunities or simply as extensions to lessons which seem
to engage the entire class.
Content Framework
Macroconcept
M1 Time
M2 Change
Discipline-Specific Concepts
C1 Steepness
C2 Rate of change
C3 Formula
C4 Graphic representation
C5 Domain
C6 Range
C7 Scale
Principles and Generalizations
P1 Slope is a physical attribute that describes the change in level of a structure.
P2 Slope is a graphical feature that indicates the rate at which something changes over time.
P3 Slope is a consideration that generates safety precautions and restrictions on physical
structures.
P4 Slope is measurable.
National or State Standards
Unit Introduction
Page 4
SD 1 In grades 9 – 12 all students should analyze functions of one variable by investigating rates
of change.
SD 2 In grades 9 – 12 all students should draw reasonable conclusions about a situation being
modeled.
SD 3 In grades 9 – 12 all students should approximate and interpret rates of change from
graphical and numerical data.
Skills
S1 Measuring
S2 Examining
S3 Describing
S4 Analyzing
S5 Creating
S6 Matching
S7 Connecting
S8 Hypothesizing
S9 Plotting
S10 Interpreting
Unit Assessments
Pre-Assessment
The unit pre-assessment is a means of gathering information about students’ content knowledge
prior to beginning the unit. Students provide information to demonstrate knowledge of practical
and graphical uses of slope. Since the content of the unit differs from traditional textbook
coverage, all students should be able to participate in activities without repetition of mathematics
from previous grades. Teachers should use the pre-assessment as a tool to develop questions that
scaffold and/or extend the content as students progress through the unit. The pre-assessment
questions also lend themselves to a variety or range of acceptable solutions to get students
accustomed to justifying their responses.
The integrated nature of the unit does not lend itself to using individual pre-assessment items as
indicators that students should not participate in certain components of the unit. The preassessment should be used rather to group students according to prior-knowledge levels so that
the additional resources can be offered as extensions to groups of students that learn at
accelerated rates.
Unit Introduction
Page 5
Interim Assessments

Plotting, Interpreting, and Analyzing (Ramp 2)
o This student analysis sheet for Lesson 1 can be used as a formative assessment of
the content. Students will have had a chance to think about similar questions when
they analyzed the data for Ramp 1. This formative assessment seeks to test
students’ knowledge of domain and range as they relate to real contexts and their
abilities to critically analyze the relationship between slope and speed taking into
consideration distance as an additional factor.

A Code for Handicap Ramps Model Analysis
o Use the rubric for assessing scale models and the analysis questions as assessment
of student understanding of the concept of slope as a physical attribute. The
overall goals being assessed are students’ abilities to calculate slope and compare
it to a pre-established requirement and students’ abilities to critique the
similarities and differences amongst models.

Next Time I’ll Just Take a Shower
o Students can be assessed on the graphs drawn using different scales and the
questions used to analyze these graphs. The objective is for students to understand
that graphical representation of slope indicates a rate of change and altering a
graph’s scale may change its interpretation.
Summative Assessment
 Summative Assessment A: Provides hints to guide students who need assistance getting
started.
 Summative Assessment B: No hints are included and there is an additional analysis
question.
Student assessments should be completed in groups of 3-4 students each and levels should be
selected based on performance throughout the unit. If it is not clear which assessment should be
given, start students with Assessment B then provide hints from Assessment A as needed. This
task asks students to estimate how long it will take “the wave” to get around Yankee Stadium.
They must create a graphical representation that estimates the time vs. distance (circumference)
for any size arena and develop a general rule (can be an equation or verbalization) that can be
applied to any size stadium or arena. Finally, students will be asked to connect the concept “rate
of change” to their general rule and tell how this rate relates to the graphical representation. The
following are key ideas being examined in this assessment:

Slope as a rate of change
Unit Introduction
Page 6



Connection of the concept of slope across multiple representations
Evaluation of the predictive nature of a general rule for use in a context (i.e. How long
will the trend last? What factors might change the trend of the data?)
Restrictions on domain and range relative to the context of the task
References
Cogan, L. S., Schmidt, W. S., & Wiley, D. E. (2001). Who takes what math and in which track?
Using TIMSS to characterize U.S. students’ eighth-grade mathematics learning
opportunities. Educational Evaluation and Policy Analysis, 23(4), 323-341.
NCTM. (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.
Stigler, J. W., & Hiebert, J. (2004). Improving mathematics teaching. Educational Leadership,
61(5), 12-17.
Reis, S. M., & Renzulli, J. S. (1994). Using Curriculum Compacting to Challenge the AboveAverage. Educational Leadership, 50(2), 51-57.
Reys, B. J., Reys, R. E., & Chavez, O. (2004). Why mathematics textbooks matter. Educational
Leadership, 61(5), 61-66.
Valverde, G. A., & Schmidt, W. H. (2000). Greater expectations: Learning from other nations in
the quest for ‘world-class standards’ in US school mathematics and science. Journal of
Curriculum Studies, 32(5), 651-687.
Unit Introduction
Page 7
Slope Changes Everything: Unit Overview
Lesson
Pages
Timeframe
Description
Unit PreAssessment
9 – 10
1 class period
Students complete the pre-assessment of slope
independently
Lesson 1: A Crash
Course in Slope
and Speed
11 – 23
The classroom set-up for each day of this lesson is
provided and listed by lesson component. Students
investigate the relationship between slope and speed
using matchbox cars, ramps, and stopwatches.
Lesson 2: Slope
and Handicap
Ramps
24 – 32
This lesson will take about 5 days to complete with
optional extensions.
Initiate – 1 day
Investigate 1 & Analyze 1 – 1 day
Investigate 2 & Analyze 2 – 1 day
Evaluate & Disseminate – 1 day
Additional Resources (optional) – 2 – 3 days
This lesson will take about 5 days to complete with
optional extensions.
Initiate – 1 day
Investigate – 3 days
Analyze, Evaluate & Disseminate – 1 day
Lesson 3: The
Bathtub
33 – 41
This lesson will take about 3 ½ days to complete.
Initiate – 1 day
Investigate – 1 day
Analyze – 1 day
Evaluate & Disseminate – ½ day
This lesson is a transition from slope as a measurement
of the incline of a physical object to the graphic
representation. Students are asked to view this through
an interdisciplinary lens in creating stories about a
bathtub related to a given graph.
Lesson 4: Using
CBRs
42 – 46
This lesson will take about 3 days to complete.
Students investigate graphic slope using a combination
of motion sensor and graphing calculator technology.
Lesson 5:
47– 49
Work in progress.
Work in progress.
Summative
Assessment
50– 51
This task will take students about 3 – 5 days to
complete.
Students work in ability-level groups to complete this
task related to generalizing how long it takes “the wave”
to get around a stadium and relating this to slope as a
rate of change.
Unit Overview
The investigation in this lesson requires students to plan
and build a scale model of a handicap ramp based on
their research related to slope and other measurement
regulations.
Page 8
Name: __________________________________________
Date: ___________________
Slope Changes Everything: Unit Pre-Assessment
1. Draw a graph with an increasing trend.
2. Create a ramp that has a slope of 2/5.
3. Which graph is decreasing the fastest?
Insert graphs (a) – (d)
4. A toy car travels down the same ramp 5 times. Each time, the average speed of the car is
recorded and the height of the ramp is increased by 1 inch. The data for the first 4 trials is
given.
Insert data for trials 1 – 4.
a. Is there a pattern in the data? Explain why or why not.
b. Predict the average speed of the car for Trial 5. Explain why you predicted this value.
Unit Overview
Page 9
Insert graph
5.
a. Explain why the graph in the picture might be misleading.
b. Re-draw the graph from part (a) in a way that might be considered more “fair.”
6. A student takes a typing course to learn how to type faster. His results on 5 identical
typing exams are given below.
a. Create a scatterplot of the data in the table.
Insert table with data.
b. What was the student’s average weekly improvement in words per minute (wpm)? Show
how you determined your answer.
c. Predict the number of wpm the student will be able to type in 5 more weeks if he
continues his typing instruction.
d. Do you expect the trend to continue to increase week after week as long as the student
continues with his typing course? Explain.
Unit Overview
Page 10
Lesson 1: A Crash Course in Slope and Speed
Big Algebraic Ideas
What happens if you put your car in neutral on a hill? How fast will it roll and what does this
depend on? The relationship between slope and speed is a factor in many real-life decisions such
as designating a ski slope “Bunny Hill” or “Black Diamond.” Many steep roads have signs
warning of icy conditions or asking trucks to test their brakes. In this lesson, students investigate
the relationship between slope and speed.
Lesson Sequence
Concepts
Principles
Reflection






Skills
Standards









Guiding Questions

Materials



Mathematical





Lesson 1
C1 Steepness
C2 Rate of change
C5 Domain
C6 Range
P1 Slope is a physical attribute that describes the change in
level of a structure.
P3 Slope is a consideration that generates safety precautions
and restrictions on physical structures.
P4 Slope is measurable.
S1 Measuring
S4 Analyzing
S7 Connecting
S8 Hypothesizing
S9 Plotting
S10 Interpreting
SD 2 In grades 9 – 12 all students should draw reasonable
conclusions about a situation being modeled.
SD 3 In grades 9 – 12 all students should approximate and
interpret rates of change from graphical and numerical data.
How can slope as a physical attribute affect the outcome of an
event?
What is the relationship between slope and speed?
How and why do we measure slope?
Ramps-each group should have 3 ramps, each a different
length
Blocks (for adjusting ramp heights)
Matchbox cars
Stopwatches (hundredths of seconds)
Rulers (standard centimeter)
Slope – 1. Ground that has a natural incline
Page 11
2. Ratio of the rise and run of an incline
Language
Slope = rise
run

Additional Resources
Average Speed – Distance traveled/Elapsed time
http://www.nbc30.com/video/9583821/index.html
This link provides video footage of the scene of the July accident on
Avon Mountain. Shot from different points on the mountain, it
enables students to see sharp corners and steep inclines that may have
contributed to the tragic crashes which occurred here.
http://www.senaterepublicans.ct.gov/press/herlihy/2006/042806.html
For students interested in the political reaction to the events and efforts
to enact laws to prevent future tragedies, this article discusses a bill
passed as a result of the July Avon Mountain tragedy.
http://www.wtic.com/Avon-Mountain-Runaway-Truck-Ramp-OpensFriday/1690783
Another safety response to the Avon Mountain crashes was the
construction of a runaway truck ramp. This article offers students the
opportunity to see how a specific problem is addressed in a real world
context.
http://www.thetartan.org/2007/4/16/scitech/work
This link relates the concepts of speed and slope to Physics, discussing
how roller coasters rely on acceleration and momentum. This may
help students understand why both steepness and length of the incline
are important considerations when analyzing accidents that occur on
hills.
http://www.skisafety.com/
This website discusses some of the possible lawsuits that might be
filed from ski accidents. Students interested in exploring the
relationships between hills and safety might analyze this site deciding
which lawsuits could be “slope”-related.
Classroom setup for
each lesson
component
Initiate: Students work independently reading the article but may
discuss responses with partners before classroom discussion ensues.
Investigate1, Investigate 2: Students work in groups of 3 (4, if
necessary). Each group will need a timer, car starter, and recorder.
Lesson 1
Page 12
Analyze 1, Analyze 2: Students work independently with conference
privileges. That is, they should plot, interpret, and analyze the data on
their own but compare results and interpretations with other members
of their group. Experts in fields seldom turn in work that has not been
edited and revised by one or many colleagues.
Evaluate and disseminate: This portion of the lesson should be in the
whole class discussion format guided by the teacher. Students should
be seated and ready to share ideas and evaluate peer responses.
Lesson Sequence
Initiate the lesson
Ask students to read the newspaper article 2005 ends with
second fatal crash on Avon Mountain and answer the questions
that follow. Once students have had time to respond to
questions, discuss their answers to each eliciting mathematical
connections in responses to #2.
The following questions may help elicit mathematical
connections:
 Q1: How might changing the location of the accidents
have affected the outcomes?
o Look for students to discuss the idea of the
accident occurring on a flat road or less steep
hill. The length of the hill may have also
affected the degree of damage as length allows
a vehicle to acquire momentum and speed.
 Q2: Would you expect the same level of tragedy if
mechanical failure of a vehicle was to occur on a road
with no hill? Why or why not? (If these responses did
not come up in Q1)
o This may have been answered already, but
students should respond with “no.” Without a
hill, gravity would not cause the truck to
accelerate. If it were to hit an object in its
path, the force would not be so great.
 Q3: Based on the information in the article, how steep
do you think Avon Mountain is?
o This question is designed to get students
thinking about whether or not steepness is the
only factor involved in the accidents. Look for
students to consider the length of the hill and
Lesson 1
Reflection
The goal of this initiation is to
allow students to view slope
as an important and relevant
concept in the world. It
provides purpose for the lab
they will complete later in
Lesson 1.
Page 13
the concept of acceleration in their responses.
 Q4: Which factor do you think is more important in
determining the degree of damage (considering only
vehicles with a mechanical failure such as loss of
brakes), the length of the hill or its steepness?
o This question can remain rhetorical as
students will investigate this in the lab.
Students who have suggestions may make
them, but otherwise inform them that this is a
question they should be able to answer after
completing this lesson.
Investigate 1
Give students the lab sheet What’s the Relationship Between
Distance, Slope, and Speed? and the Ramp 1 Data Table.
Discuss the concept of slope as defined in the Mathematical
Language section. Set up a ramp on a block and demonstrate
the process of measuring the rise and run and calculating the
associated slope. Leave the example on the board for students
to refer to as they perform the lab.
Distribute the materials for this section which include the
shorter ramps (one per group), blocks (four per group), cars
(one per group) and stopwatches (one per group).
As students perform their investigations, the teacher circulates
the room answering generic questions and posing enriching
questions. Use the following questions as guides to probe
students’ thinking.
 What would the average speed of the car be if we used
no blocks? What about the height of the ramp? Slope?
 What changes did you notice in your data once you
added the second block?
 Estimate the maximum number of blocks that could be
used in this experiment. Why is there a maximum?
Analyze 1
Give students the follow-up activity Plotting Interpreting and
Analyzing (Ramp 1). Ask students to construct graphs of their
data and analyze using the questions provided.
Students may require further explanation of questions
associated with #2 if they have no prior knowledge of the
concepts domain and range. Give students a related example to
help them understand. For example, consider the independent
variable “hours worked” and the dependent variable “total
Lesson 1
Many students probably know
that changing the steepness of
a hill affects the speed of a car
in neutral. This investigation
enables students to gather data
related to that assumption and
assess the change as linear or
non-linear. This investigation
is grounded in the Practice
Parallel as students assimilate
professionals in a field by
beginning with an assumption
or hypothesis, testing that
assumption, evaluating the
assumption, then relating it to
the bigger picture.
The process of plotting,
interpreting, and analyzing are
grounded in both Core and
Practice Parallels. These skills
rely on core knowledge in
creating and interpreting a
data set. Analyzing data to
draw relevant conclusions is
similar to what professionals
Page 14
weekly paycheck.” Create a realistic hourly wage with the
student and use this to derive a reasonable domain and range.
One possibility is domain 0 ≤ h ≤ 40 and range 0 ≤ t ≤ 400 for a
person making $10 an hour.
do to support initial
assumptions.
The teacher can circulate the room, read students responses, and
generate questions related to their responses or provide
feedback designed to have students’ critique their own work.
Investigate 2
Repeat Investigate 1 with longer ramps and the original lab
sheet What’s the Relationship Between Distance, Slope, and
Speed? and the Ramp 2 Data Table.
Analyze 2
Repeat Analyze 1 using the student sheet Plotting Interpreting
and Analyzing (Ramp 2).
Evaluate and disseminate
Use this time to make comparisons of results across groups.
Initiate this discussion by relating to a topic that might be of
interest to students. An example might be to ask students why
many events require more than one referee, umpire, or judge.
As students respond to this idea, indicate that it is important to
compare data not only within a group, but externally as well to
determine consistency. Then, use the following questions to
guide the evaluation and dissemination process:
 What did each group get for the slope of Ramp 1 using
one block?
 What did each group get for the average speed of the car
on Ramp 1 using one block?
o What factors contribute to the variations in
responses to these two questions?
(Repeat these questions as needed to demonstrate variation in
group responses.)
The Practice Parallel is
emphasized in this evaluation
and dissemination as students
learn that comparisons across
groups are important. Data is
subject to error regardless of
the person in charge of
measurement and it is
important for students to
develop the habit of checking
whether the error is negligible
(human error) or significant
(measured inappropriately or
inaccurately).
 Do you think the variation amongst groups was so great
that it would have affected the interpretations and
analyses of the data?
Lesson 1
Page 15
2005 ends with second fatal crash on Avon Mountain
(http://www.zwire.com/site/news.cfm?newsid=15834161&BRD=1646&PAG=461&dept_id=11035&rfi=6)
AVON, CT - In an eerie repeat of the July crash that killed four at the base of Avon Mountain,
another woman died in an accident on the mountain the morning of Dec. 22 when the car her
brother was driving slid into a Kelly Transit bus in icy road conditions.
The driver lost control of his car as he negotiated a left-hand curve while coming down Avon Mountain
and slammed into an oncoming commuter bus owned by Kelley Transit - the same Torrington-based
company whose bus was totaled in the July accident. The horrific crash July 29, which involved 20
vehicles, put the town of Avon - in its 175th anniversary year - on the map nationwide. It also defined the
year for the region by testing the response capabilities of area emergency personnel and by making road
conditions on the mountain the impetus for new truck inspection laws and calls for the state Department
of Transportation to correct the roadway. Commuters headed through the intersection of Routes 44 and
10 at Nod Road the morning of July 29 suddenly found themselves trapped in a nightmare when a fully
loaded dump truck lost control and flew into oncoming traffic. As the truck careened through the
intersection and into traffic, 19 other vehicles - including a Kelly Transit bus - piled up as a result of the
impact. Some cars were struck and dragged along with dump truck, which eventually landed on its side.
The truck and several nearby cars burst into flames; meanwhile, the bus also ignited and had to be
extinguished before firefighters could attempt a rescue of drivers trapped in their vehicles. Close to two
dozen people were injured and four perished at the scene, including the driver of the truck owned by
Bloomfield-based American Crushing & Recycling. The crash occurred around 7:30 a.m. - at the height
of rush hour. Emergency personnel responded from several surrounding towns, including Farmington,
Simsbury, Newington, West Hartford and others. The North Central Municipal Accident Reconstruction
Squad arrived hours later to painstakingly collect evidence from the crash scene. Weeks later, David
Wilcox of Windsor and his wife Donna, the owners of American Crushing and Recycling, were charged
with insurance fraud and larceny after Arcadia Insurance accused the couple of attempting to reinstate
coverage on the truck without declaring the accident had occurred. Their cases are still pending in court.
Investigators are looking into whether the crash was caused by some type of catastrophic mechanical
failure. It was also revealed that the driver of the truck was only with the company for a few days before
the crash occurred. The fiery accident spurred Gov. M. Jodi Rell to insist on immediate inspection of all
truck companies with records of violations for shoddy maintenance practices. The crash also spurred state
legislators to pass stiffer laws for truck inspections and maintenance violations. West Hartford and Avon
police stepped up patrols on the mountain and at one point, West Hartford police arrested a truck driver
for driving without all the wheels properly engaged. West Hartford police also arrested a truck driver for
operating with faulty brakes.
The state DOT met with officials from Avon and West Hartford shortly after the crash to discuss what
could be done to improve safety on Avon Mountain. New road signs on both sides of the mountain went
in immediately and plans were discussed to add shoulders and other safety improvements. State and town
officials from both sides of the mountain met again recently but the improvements that were made didn't
save the life of Alicia Banks, 42, of Bloomfield when the car her brother was driving slammed into a
Kelly Transit Bus - the same bus company involved in the July crash - around 7 a.m. Dec. 22. Police are
attributing the Dec. 22 crash, which occurred about four-tenths of a mile from the July crash, to icy roads.
Avon police are still investigating the cause of the July 29 to determine if criminal charges will be filed.
Lesson 1 Student Sheet
Page 16
QUESTIONS
1. What are the potential causes listed for each 2005 crash on Avon Mountain?
a. July crash
b. December crash
2. Do you think location was a factor in either of these crashes? Explain why or why not.
a. July crash
b. December crash
3. What steps are being taken by authorities to prevent similar tragedies in the future?
Lesson 1 Student Sheet
Page 17
Speed Investigators: _____________________________________________________
What’s the Relationship Between Distance, Slope, and Speed?
Materials:
3 ramps (each a different length)
4 blocks (uniform in size)
Stopwatch
Definitions:
Rise x – The height of the ramp supported by x blocks.
Run x – The run of the ramp supported by x blocks.
Slope x – The slope of the ramp supported by x blocks.
Time 1, Time 2, Time 3 – Time in seconds that it takes the car to travel the distance of the ramp.
Average Time – The arithmetic mean of Time 1, Time 2, and Time 3.
Average Speed – Distance/Time (Length of Ramp/Average Time)
Independent variable – those that are deliberately manipulated to invoke a change in the
dependent variables
Dependent variable – those that are observed to change in response to the independent variables
Method:
1. Select a ramp and record the length in Table 1 in centimeters.
2. Attach the ramp so that the end of the ramp is adjacent to the top of one block.
3. Measure the rise and run (in centimeters) and record those measurements in the
corresponding boxes in the table.
4. Calculate the slope of the ramp and record this value (to the nearest tenth) in the table.
5. Place the matchbox car so that the back edge of the car aligns with the intersection of the
ramp and the block.
6. Using the stopwatch, determine how long (in seconds) it takes the car to travel the
distance of the ramp. Begin timing when the car is released (NOT PUSHED) and end
when the back edge of the car is completely off the ramp. Record this time (to the
nearest hundredth second) next to Time 1.
7. Repeat Steps 5 & 6 for Time 2 and Time 3.
8. Calculate and record the Average Time in seconds.
9. Calculate and record the Average Speed in cm/s.
Lesson 1 Student Sheet
Page 18
What’s the Relationship Between Distance, Slope, and Speed?
(Ramp 1 Data Table)
Ramp Length (cm): _______________________________
Rise 1
Run 1
Slope 1
Time 1 (s)
Time 2 (s)
Average Speed
(cm/s)
Time 3 (s)
Average Time (s)
Rise 2
Run 2
Slope 2
Time 1 (s)
Time 2 (s)
Average Speed
(cm/s)
Time 3 (s)
Average Time (s)
Rise 3
Run 3
Slope 3
Time 1 (s)
Time 2 (s)
Average Speed
(cm/s)
Time 3 (s)
Average Time (s)
Rise 4
Run 4
Slope 4
Time 1 (s)
Time 2 (s)
Average Speed
(cm/s)
Time 3 (s)
Average Time (s)
Lesson 1 Student Sheet
Page 19
Speed Investigator: ________________________________________________
Plotting, Interpreting, and Analyzing (Ramp 1)
Plotting and Interpreting the Data (Slope vs. Average Speed)
What are the independent and dependent variables?
1. On a sheet of graph paper, create a set of positive coordinate axes that will be used to plot
the slope of the ramp vs. the average speed of the car.
 What is the independent variable? _____________________________
 What is the dependent variable? ______________________________
2. For each coordinate pair (slope x, average speed y), create a point to represent this data on
the graph.
 If the domain of a graph represents the set of all values that makes sense for the
independent variable, provide a reasonable domain for this graph.
 If the range of a graph represents the set of all values that makes sense for the dependent
variable, explain how this set of values depends on the domain.
 Provide a range for this graph based on the domain you gave in part (a).
Lesson 1 Student Sheet
Page 20
Analyzing the Data
 Describe the shape of the points on your graph.
 If you were to add one more block to the height of the ramp, predict the average speed of
your car. Show your work or explain how you came up with your prediction.
 What information does slope give you that height alone does not provide?
 In the next part of this lab, you will use the same blocks but a LONGER ramp. Explain
how you expect each of the following to change:
o The slope
o The average speed of the car
o The shape of the graph
Lesson 1 Student Sheet
Page 21
What’s the Relationship Between Distance, Slope, and Speed?
(Ramp 2 Data Table)
Repeat the method directions from the lab What’s the Relationship Between Distance, Slope, and
Speed? Use a longer ramp than was used in the first collection of data.
Ramp Length (cm): _______________________________
Rise 1
Run 1
Slope 1
Time 1 (s)
Time 2 (s)
Average Speed
(cm/s)
Time 3 (s)
Average Time (s)
Rise 2
Run 2
Slope 2
Time 1 (s)
Time 2 (s)
Average Speed
(cm/s)
Time 3 (s)
Average Time (s)
Rise 3
Run 3
Slope 3
Time 1 (s)
Time 2 (s)
Average Speed
(cm/s)
Time 3 (s)
Average Time (s)
Rise 4
Run 4
Slope 4
Time 1 (s)
Time 2 (s)
Average Speed
(cm/s)
Time 3 (s)
Average Time (s)
Lesson 1 Student Sheet
Page 22
Speed Investigator: ________________________________________________
Plotting, Interpreting, and Analyzing (Ramp 2)
Plotting and Interpreting the Data
1. Repeat the steps from Plotting and Interpreting the Data to create a graph for the new data
set.
 Have the domain and range changed? Explain why or why not.
Analyzing the Data
 Were your predictions correct about the changes in slope, average speed, and shape of
graph? For each, explain how you know.
 What, if any, conclusions can you make about the effect of ramp length on the average
speed of a car? Explain.
 If you were researching the crash on Avon Mountain, what questions might you
investigate that knowledge of mathematics and Physics could help you answer?
Lesson 1 Student Sheet
Page 23
Lesson 2: Slope and Handicap Ramps
Big Algebraic Ideas
Steep inclines are sometimes unavoidable as we seek roadways that pass through hilly and
mountainous terrain. Since slope is a measurable attribute, however, safety restrictions can be
placed on man-made structures that require an incline. Structures such as handicap ramps, roofs,
and playground slides have guidelines and/or federal mandates for many measurements, one of
which is slope.
Lesson Component
Concepts
Principles
Description




Skills
Standards









Guiding Questions


Materials
Additional Resources





C1 Steepness
C2 Rate of change
P1 Slope is a physical attribute that describes the change in
level of a structure.
P3 Slope is a consideration that generates safety precautions
and restrictions on physical structures.
P4 Slope is measurable.
S1 Measuring
S2 Examining
S3 Describing
S7 Connecting
S5 Creating
S4 Analyzing
SD 2 In grades 9 – 12 all students should draw reasonable
conclusions about a situation being modeled.
SD 3 In grades 9 – 12 all students should approximate and
interpret rates of change from graphical and numerical data.
Why is the measurement of slope important to certain
constructions?
Is slope maintained from an actual structure to a scale model?
Why or why not?
Craft sticks
Toothpicks
Construction paper
Glue guns
Scissors
http://www.occupationalhazards.com/Issue/Article/43291/Preventing_Outdoor_Sam
eLevel_Slips_Trips_and_Falls.aspx
In addition to the ADA/OSHA recommendation for “maximum slope”
for handicap ramps, this site provides a “minimum slope” suggestion.
This provides an extension for students to consider.
 Is the minimum slope suggestion for safety or some other
reason? Provide support for your answer.
 Highways often post maximum and minimum speeds. Why
Lesson 2
Page 24
are both essential?
http://www.cpsc.gov/cpscpub/pubs/325.pdf
This website is the Public Playground Safety Handbook. Handicap
ramps are not the only structures with safety restrictions on their
slopes. Students can examine the guidelines for slope (or angle) of
playground slides. Students can also use this concept to examine the
relationship between angle and slope.
http://www.ramphelp.com/
Are students interested in seeing exactly how different ramps are
built? This site provides information on building handicap ramps, bike
ramps, skate ramps, and others. Students who are interested in sports
that use ramps can discuss how these slopes defy linearity. They might
develop a way to measure slope at a particular point (a topic of
calculus), rather than overall slope.
http://www.gillettenewsrecord.com/articles/2008/02/21/news/local%20news/news05
.txt
This article extends outside of mathematics into the lives of
handicapped individuals. Accessibility to certain areas can be
restricted by other variables besides ramp availability. Students
interested in making the world more accessible for individuals can
begin with this article.
http://adopt-a-ramp.org/
Classroom setup for
each lesson
component
Students and teachers who would like to take this idea to the next level
and give back to communities can visit this website for information on
volunteering to build ramps at private homes. This activity is an
opportunity for talented students to use mathematics in a meaningful
way.
Initiate: This discussion should be held in a computer lab with one or
two students per computer.
Investigate: Students work in groups of 2 or 3.
Analyze: Students work independently with conference privileges.
Evaluate and disseminate: This portion of the lesson should be in the
whole class discussion format guided by the teacher. Students should
be seated and ready to share ideas and evaluate peer responses.
Lesson Sequence
Reflection
Initiate the lesson
Give students the initiation student sheet Slope and Safety: A
Code for Handicap Ramps. The initiation task can be done most
This initiation extends the
ideas of Lesson 1 in
Lesson 2
Page 25
authentically in a computer lab as the third question asks
students to perform a research task.
Give students time to consider and respond to the first couple of
questions. Key discussion points for these questions should
focus on the following:
 The slope of a handicap ramp must be greater than zero
because it goes from the ground to a higher location. A
handicap ramp must not be too steep or people might get
injured.
 A handicap ramp might have a turn in it because a
straight ramp would be too steep. It also might have a
turn due to desired beginning and ending points.
 Other important measurements include the width of the
ramp, the height of the rails, the height of the door to
which the ramp is being built. Students should justify
their reasoning for each measurement being important.
demonstrating the importance
of slope as a measurable
quantity. It flows naturally
from the previous lesson
where students investigated
slope and speed allowing
students to make connections.
Students should investigate the ADA/OSHA standard for
handicap ramps using Internet search engines. Ask students to
record the source of their information.
 The ADA/OSHA standard for handicap ramps is 1/12.
(http://tshandiramps.com/Library/T&S%20Enterprises%2053675-1.pdf)
IDEAS FOR EXTENSION:
 Refer students to the website listed under additional
resources http://www.cpsc.gov/cpscpub/pubs/325.pdf. Ask
students to determine the recommended slope for
playground slides. Students can draw a picture with the
indicated slope and analyze why they think this
restriction is important. Do they think that all
playground slides adhere to this slope guideline? (If
there is a nearby playground, students can actually
perform the necessary measurements to determine if
playgrounds adhere to this guideline.)
 Connect the ideas to the previous lesson. How does this
relate to the ideas of slope and speed that we discovered
in Lesson 1?
o Since slope and speed are related and
increasing the slope increases the speed (when
distance is held constant), it is essential for a
maximum slope to be imposed to control the
speed of non-mechanical wheelchairs.
Investigate
Give students the Handicap Ramp Scale Model Design Task.
Lesson 2
This investigation seeks to tie
Page 26
Before beginning the task, students should perform further
in the concept of proportional
research related to handicap ramp regulations. Students can look reasoning with the current
to OSHA and ADA for further information.
study of slope.
Mathematically talented
Once the research has been done, it is necessary for students to
students are often exposed to
determine an appropriate scale factor. If necessary, remind
repetitious exercises involving
students that a scale factor is a number that each linear
proportional calculations year
dimension can be multiplied by to obtain the scale
after year. This lesson seeks to
measurement.
tie those calculations into a
meaningful context.
EXAMPLE: A measurement of 36 inches can be drawn or built
4 inches using a scale factor of 1/9.
Students and teachers should
expect models to appear
Students should choose a reasonable scale factor based on the
different as students select
largest and smallest features in the blueprint. Remind them also different scale factors for their
that a scale factor must remain constant for every feature in the designs. The differences
model.
should be addressed as
students evaluate their own
Once a scale factor has been chosen, students must develop a
work and that of other groups
way to build the handicap ramp that suits the needs of the owner recognizing that scale factor
and adheres to federal safety mandates. Provide scaffolding as
alone accounts for size
needed to struggling groups by suggesting that they build the
differences but all models
known structures on their scale models prior to deciding how
might be correct
the ramps will be built. They can then test different ramp
representations of the original
designs on the already constructed portions of the model to
structure.
check for safety violations.
Analyze
Once students have built their scale models, they can reflect on This analysis allows students
their design choices and analyze the model using the student
to independently reflect on the
sheet A Code for Handicap Ramps Model Analysis.
work they did as a group
allowing teachers to gauge
individuals’ understanding of
the content.
Evaluate and disseminate
This is a time for students to share their responses to the
Teachers can use the analysis
analysis questions. The following suggestions can guide the
questions as a formative
discussion of each question:
assessment. Sharing responses
as part of the evaluation
 1. Students should mention that the height does not
process enables students to
exceed the maximum recommended height of 30 inches
critique their own designs and
(scale conversions of this measurement will vary). Also, those of other groups. The
students should know that the minimum width for the
Practice Parallel is evident as
ramp is 36 inches and this would be converted based on students view the designs
their scale factor. Students may have included handrail
through the lens of inspectors
guidelines which recommend 34 to 38 inches above the
rather than builders.
surface of and parallel to the ramp. (If students were
Lesson 2
Page 27
extremely thorough, they may have even more guidelines
inherent in their designs.)
 2. Discussions should focus on how students are able to
tell that their ramps meet all of the established
guidelines. If asked, they should be able to explain how
they can use a ruler and the rise/run calculation to find
the slope of their ramp. Since the ramps require a turn
to end at the location designated by the homeowner
students must have at least two separate slope
calculations such that m < 1/12. This is a good time to
discuss the idea of using a scale factor in reverse to
check the accuracy of measurements.
 3. Students should calculate the slope of their ramps
using the actual and scale measurements to show
equivalence. Slight differences may occur due to
rounding and measurement error.
 4. Students have time to discuss the different scale
factors and design choices. Students should critique the
work of classmates providing positive feedback and
suggestions for improvement in design. If appropriate,
ask which scale factor provided the best representation
of the actual model.
Lesson 2
Page 28
OSHA Investigator: __________________________________________________
Slope and Safety: A Code for Handicap Ramps
1. Have you ever seen a wheelchair ramp with
a turn in it? Why do you suppose the
designer put a turn in it rather than making
it straight?
2. What factors must be considered before constructing a wheelchair ramp? What
measurements must be taken? Explain.
Lesson 2 Student Sheet
Page 29
3. Use the Internet to determine the Occupational Safety and Health Administration’s
(OSHA) and Americans with Disabilities Act’s (ADA) regulation for maximum
slope of a handicap ramp. What is the regulation?
4. The bottom of the front door of a house is 2 ½ feet off the ground. The owner wants
to build a straight handicap ramp beginning at his door.
(i)
What is the minimum ramp length the owner can use and still meet
the OSHA regulation? Show your work or explain.
(ii)
Is this a realistic ramp size for most homeowners? Explain your
thinking.
Lesson 2 Student Sheet
Page 30
Ramp Builders: ______________________________________________________________
Handicap Ramp Scale Model Design Task
Suggested Materials:
 Craft sticks
 Toothpicks
 Construction paper
 Glue guns
 Scissors
You are in charge of the design and engineering of a handicap ramp for a new home. You have
received the blueprints providing you with the dimensions of the owner’s front porch and other
relevant information. The owner has requested a ramp that begins at his porch (he has said that
he doesn’t care which part of the porch) and ends on his driveway midway between the garage
and the road.
Be sure to perform further research to locate information on height, width, and run requirements
before beginning construction (record this information and source using a MS Word or MS Excel
document and print a copy for each member of your group). Your task is to develop a threedimensional SCALE MODEL of the porch, the driveway, and the new ramp. It is not necessary
to build the garage though the model should indicate the location of the structure for proper
referencing.
Insert blueprint
Lesson 2 Student Sheet
Page 31
Model Analyst: ___________________________________________________
A Code for Handicap Ramps Model Analysis
1. What scale factor did you choose for your model? Provide the calculations you used
to determine this scale factor and a rationale for your final decision.
2. Explain how you know that the ramp you built adheres to all safety regulations
established by OSHA and/or ADA. Include regulations beyond slope.
3. Does the slope of the ramp in the model agree with the slope that the ramp would
have on the actual structure? Provide all relevant calculations and explanations
needed to support your answer.
4. Compare and contrast your model with that of other groups focusing on the
following (use additional paper if necessary):
 The origin of the ramps
 The size of the models
 The design (shape) of the ramps
Lesson 2 Student Sheet
Page 32
Lesson 3: The Bathtub
Big Algebraic Ideas
What does slope look like as a rate of change? How can a quick glance at a graph provide the
same information that might be obtained from reading an entire paragraph or page? Glancing at a
graph and its slope provides a wealth of information about the variables being measured.
Lesson Component
Description
Concepts






Principles

Skills






Standards
Guiding Questions
Materials
Additional Resources
Lesson 3
C1 Steepness
C2 Rate of change
C4 Graphic representation
C5 Domain
C6 Range
C7 Scale
P2 Slope is a graphical feature that indicates the rate at which
something changes over time.
S2 Examining
S3 Describing
S4 Analyzing
S5 Creating
S7 Connecting
S10 Interpreting
 SD 1 In grades 9 – 12 all students should analyze functions of
one variable by investigating rates of change.
 SD 2 In grades 9 – 12 all students should draw reasonable
conclusions about a situation being modeled.
 SD 3 In grades 9 – 12 all students should approximate and
interpret rates of change from graphical and numerical data.
 What are the differences between gradual and rapid rates of
change when interpreting graphs?
 How does changing the scale of a graph affect the “quick
glance” interpretation?
 Blank transparencies and markers
 Student page: The Bathtub
 Composition paper
 Timed Task Cards (Teacher should have 8-10 copies of each
card.)
http://www.hydrothermix.com/html/graph.html
This website provides temperature comparison graphs for whirlpool
tubs using different heating elements. Students can analyze the three
graphs to determine what message (if any) the vendor is attempting to
relay to the consumer. This can be used to extend the content of
Page 33
Lesson 3 to look at graphs of other continuous variables.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L388
This Illuminations lesson provides a method for developing an
equation for the height of bathwater over time as it drains. This lends
itself to some interesting extension questions for mathematically
talented students including:
 Is the graph of height of bath water over time as the tub fills up
linear? Why or why not? What other factors must be
considered to answer this question?
 Is the graph of the height of bath water over time as the tub
drains linear? Why or why not?
Students can use graphs developed using this data collection method
to analyze and re-draw the graph in this lesson more accurately.
http://math.rice.edu/~lanius/Algebra/stress.html
For students who already have a firm grasp of slope as a rate of
change, this website extends this idea to parabolic functions. Students
can be challenged to look at tangent lines to curves and interpret the
meanings of such lines in the context of given problems.
Classroom setup for
each lesson
component
Initiate: This task can be done independently or in pairs.
Investigate: This task should be done independently or with one other
person.
Analyze: Students work independently with conference privileges.
Evaluate and disseminate: This portion of the lesson should be in the
whole class discussion format guided by the teacher. Students should
be seated and ready to share ideas and evaluate peer responses.
Lesson Sequence
Initiate the lesson
Ask students to consider something that changes over time.
Then provide students with a blank transparency and marker
and ask them to draw those changes graphically, labeling axes
to clarify graphs. NOTE: If you do not have access to sufficient
materials, substitute graph paper and use chalkboard to display.
Examples can be provided for students who are unable to think
of a change on their own. Examples of things that change over
time:
Lesson 3
Reflection
This initiation relies on the
Connections Parallel, asking
students to relate the question
to something they already
know about whether it be
another academic content area
or their own lives.
Page 34




The number of leaves on New England trees
The number of cars on the highway in a given city
The number of students in the cafeteria
A person’s height from birth to age 16
Display and discuss graphs students have created emphasizing
“steepness” to show gradual or rapid ascent or decline. Also
emphasize slope discussing whether a graphical change
demonstrates an increasing or decreasing trend.
Sample questions for the first example relating the number of
leaves to the time of year might include:
 What month of the year does your graph begin with?
Why did you choose that month? Did anybody do a
similar graph but start with a different month?
 Can you explain the increases and decreases in the graph
and what time of the year is represented by those
changes? Are those changes gradual or rapid?
 Why do you suppose it is important to say that these are
“New England” trees? How would location alter the
graph? Would the graph look different in Connecticut
than in northern Vermont?
 What would a good domain and range for this graph be?
(This question may be difficult to give actual numbers,
but students should be able to define the domain and
range in terms of the independent and dependent
variables if they are continuous.)
Investigate
Give students The Bathtub task. Read the directions for the task
to see if students understand what is required.
Introduce the Timed Task Cards.
 Timed Task Cards – Students must work on the task
for the indicated number of minutes before the
teacher provides them with Timed Task Cards.
These cards are intended to scaffold material for
students who are struggling with the task. The cards
for this lesson should be used in the following ways:
o 10-minute card: This card provides a
suggestion for axes labels. If students have
already labeled their axes 10 minutes into the
task, do not provide them with this card.
o 15-minute card: This card provides a
suggestion for estimating scale and providing
units. If students have already completed
this task, do not provide them with this card.
Lesson 3
Asking students questions
about where their graphs
begin and end connects to the
concepts of domain and range
introduced earlier and
provides a nice transition into
the topic of scale.
The goal of the differentiation
in this lesson is to maintain a
high level of task demand for
students capable of
completing the work
unassisted. Mathematicians
have been known to take
hours, months, even years to
solve single problems of
mathematics and students
should be exposed to this
struggle as highlighted in the
Practice Parallel.
Page 35
o 20-minute card: This card provides tips for
interpreting the changes in the graph if
students have already begun this task, do not
provide them with this card.
Once students have created their stories, they should develop a
way to test the accuracy of the scales they provided with their
graphs.
Analyze
Ask students to consider the scale they used on the graph of The
Bathtub. In order to analyze the effects of scale on a graph, ask
students to re-draw the graph using the following scales:
1) A scale that is ½ as big as the scale they chose for the
original drawing.
2) A scale that is twice as big as the scale they chose for
the original drawing.
NOTE: In order for students to see the differences clearly, it is
important that they use the grids provided in order to maintain
the size on each graphical unit.
By developing a test of
accuracy, students are
essentially creating a way of
testing hypothesized values.
Critical analyses of graphical
results tie into the Practice
Parallel as experts must
examine the work of others in
their field with caution.
Changing scale creates more
or less dramatic appearances
in slope and can severely limit
a graph’s interpretability.
Use the student sheet Next Time I’ll Just Take a Shower for
analysis of the new graphs.
Evaluate and disseminate
Ask students to share their original bathtub stories. Students
should critique the stories of their classmates focusing on the
directions and rates of change as they align with story details.
The Connections Parallel is
emphasized as students
become mathematical story
writers and readers. In
Students should also discuss the effects of changing the scale. In addition, students share
particular, ask students to share their responses to the final
connections they have made
question to obtain a variety of different “unfair” graphs that
between the mathematical
might be used to trick consumers. Examples include:
content and the real world.
 Number of weeks on a weight loss drug and pounds lost
(unfair graph might have dependent variable axis in 1/10
increments)
 Time in weeks and value of particular stock
 Number of years on the job vs. Salary
Be sure to ask students which audience the graphs might be
intended to “trick.”
Lesson 3
Page 36
Assessment Rubric for Task: The Bathtub
Objective

Provide an
appropriate
context
0 – No evidence
that student has
met objective
1 – Student is
working toward
objective
2 – Student has
met objective
Student’s story does
not reflect the
positive and negative
changes in the graph.
Student’s story
reflects the positive
and negative
changes in the graph
but does not take into
account the rate of
change (steepness).
Student’s story
accurately
depicts the
positive and
negative changes
in the graph and
addresses rates
of change using
appropriate
details.
The axes labels do
not reflect indicated
variables.
One axis label
represents a variable
rate but the other
does not OR the
axes labels are
variable rates but do
not match the details
of the story.
The axes labels
make sense with
the story and
represent
variable rates of
change.

Provide
appropriate
axes labels

Estimate an
appropriate
scale to
match the
context of the
graph
The scale is
inappropriately
labeled or does not
provide a good
estimate for the
representative
variables.
One axis has an
appropriate scale and
matches the context
but the other does
not.
Both axes have
appropriate
scales for the
indicated
variables and
reflect the context
of the story.

Develop a
method for
testing the
feasibility of
estimated
scale
The student’s method
does not test the
variables indicated in
an adequate manner.
The student provides
an appropriate
method for
measurement but
does not relate the
method to the details
of story or graph.
The student’s
method provides
information for
testing both
variables and
linking the results
to the graph.
Lesson 3
Page 37
10 – Minute Card
Consider using time and height of
water as the variables for your
graph. Which axis should
represent time and which should
represent height of water?
15-Minute Card
Estimate how high off the floor
bath water might be. Use a ruler
to determine this height and an
appropriate unit.
What unit of time makes sense for
a person taking a bath?
20-Minute Card
Is the line on the graph going up or
down? What does this tell you
about the height of the water?
Is the change in the line rapid or
slow? What might this indicate is
occurring in your story?
Lesson 3
Page 38
The bathtub
Task: The graph above shows what is happening in “the bathtub.” Perform each of the following tasks
related to the graph:
1.
2.
3.
4.
Label the axes with variables that make sense with the graph’s title.
Estimate a scale for each axis and label the scales on the graph. Be sure to include units.
Write a story that models the changes occurring in the graph.
Describe how you would test the accuracy of the scale you provided in the bathtub activity.
Lesson 3 Student Sheet
Page 39
Re-draw the graph so that each box represents ½ and twice the unit you chose in original
scale.
Insert grids for scale change drawings
Lesson 3 Student Sheet
Page 40
Bathwater Specialist: _______________________________________________________
Next Time I’ll Just Take a Shower
1. What is the biggest difference in appearance between the graph with the original scale
and the graph with ½ that scale?
2. What is the biggest difference in appearance between the graph with the original scale
and the graph with twice that scale?
3. Explain why changing the scale might lead a person to misinterpret the overall message
of a graph.
4. Describe an independent and dependent variable combination that might lend itself to
unfair misinterpretation by a CONSUMER. Sketch fair and unfair versions of the graph.
Lesson 3 Student Sheet
Page 41
Lesson 4: Using Calculator Based Rangers (CBRs) to Graphically
Model the Speed-Slope Relationship
Big Algebraic Ideas
How does slope as a physical attribute translate to a graphical representation? Slope is the
primary indicator of a graphical trend. It is the feature that enables a person to determine
whether a trend increases or decreases and the rate at which this occurs. Of course,
altering a graph’s scale can drastically change the perceived effect of the independent
variable on the dependent variable causing misleading interpretations to be made.
Lesson Component
Concepts
Principles
Skills
Standards
Guiding Questions
Materials
Additional Resources
Description





C1 Steepness
C2 Rate of change
C4 Graphic representation
C7 Scale
P2 Slope is a graphical feature that indicates the rate at which
something changes over time.
 S2 Examining
 S3 Describing
 S4 Analyzing
 S8 Hypothesizing
 S6 Matching
 S5 Creating
 SD 1 In grades 9 – 12 all students should analyze functions of
one variable by investigating rates of change.
 SD 2 In grades 9 – 12 all students should draw reasonable
conclusions about a situation being modeled.
 SD 3 In grades 9 – 12 all students should approximate and
interpret rates of change from graphical and numerical data.
 What is the role of slope in distance vs. time graphs?
 CBR units (1:4 CBR-to-Student ratio recommended, though
lesson can be successfully implemented with a single CBR and
overhead calculator)
 Overhead calculator
 TI Graphing calculators
http://math.escweb.net/CBR/cbr.htm
This website provides information on using the programs on the
graphing calculator with CBR units. Students or teachers who are
interested in learning to use other programs in addition to the one used
in this lesson can visit this website for information.
http://www.mste.uiuc.edu/courses/ci499sp01/students/ychen17/project
Lesson 4
Page 42
336/teachplan2.html
This website provides an interdisciplinary lesson utilizing Calculator
Based Laboratory systems (CBLs) for connecting concepts of Algebra
and Physics. Teachers who find students engaged with the use of
CBRs might expand the technological repertoire of students by
introducing them to a second form of technology that communicates
its data to TI graphing calculators.
http://education.ti.com/educationportal/sites/US/productDetail/us_ti_c
onnectivity_kit.html
Since one of the many purposes of graphical data is to share with an
audience, the TI-Connectivity kit can be used to transfer calculator
based graphs to computer applications for use in presentations or lab
reports. Mathematically talented students can use this technology to
create a portfolio of graphic analyses. Classroom teachers may also
find these tools useful in having students present and analyze data.
http://eric.ed.gov/ERICDocs/data/ericdocs2sql/content_storage_01/00
00019b/80/15/1e/77.pdf
This paper “Bringing Functions and Graphs to Life with the CBL”
presented at the Carolinas Mathematics Conference in 1997 provides
research support for the use of CBLs in effectively introducing the
concept of functions to students. The author focuses on the CBL as a
tool for fostering the conceptual understanding of graphs in two
variables that is crucial to instruction of mathematically talented
students.
Classroom setup for
each lesson
component
Initiate: This discussion should be held in a computer lab with one or
two students per computer.
Investigate: Students work in groups of 2 or 3.
Analyze: Students work independently with conference privileges.
Evaluate and disseminate: This portion of the lesson should be in the
whole class discussion format guided by the teacher. Students should
be seated and ready to share ideas and evaluate peer responses.
Lesson Sequence
Initiate the lesson
Set up an object in the front of the classroom to use as a
reference point. Ask students to set up coordinate axes with
Lesson 4
Reflection
The initiation provides the
opportunity for students to
Page 43
time as the independent variable and distance as the dependent
variable. Tell students that they will be graphing the distance of
the teacher from the reference object over time.
 What units would be appropriate for each of the
variables?
o Seconds would be the most logical time unit and
feet or inches the most logical distance unit.
 What would be problematic about using minutes for the
unit of time?
o It is difficult to estimate the exact position from 0
to 1 because there are 60 seconds and the
teacher might be in constant motion.
Stand in front of the object facing the object. Ask students
where their first point will be if you begin in this position.
o The distance from the object to the teacher
should be plotted on the distance axis (the distintercept).
Tell students that you will walk forward and backward in a
straight path. They should graph the motion. Remind students
that they are graphing the distance from the object as time
passes. Students should focus on getting the general shape of
the graph rather than the exact units represented by the scale.
really think about what the
line they are drawing
represents. Mathematically
talented students
Investigate
[Section description: Students play game “match the graph”
where the CBR provides a graph and its motion sensor
picks up and graphs a person’s distance from the CBR. The
goal is for students to see how their own speed and position
affects the slope of the given graph. Students will see how
standing still creates a straight line]
Analyze
[Section description: Students will be given pictures of
different graphs and describe either how the CBR can be
used to make the graph or why it is not possible to make a
particular graph (without a time machine anyway!).
Lesson 4
Page 44
Evaluate and disseminate
[Section description: Students share ideas about
which graphs are possible and test them using a
different CBR program. Mathematically talented
students often come up with creative ways to make
graphs that seem impossible (i.e. a graph that might
require being in two places at once due to a jump).]
Lesson 4
Page 45
Insert Lesson 4 Student Sheets
Lesson 4 Student Sheet
Page 46
Lesson 5 (not yet developed): Developing a Formula for Points in
Space
[Lesson overview: Students will use their knowledge of slope and
graphs and use the investigation to help develop a formula for the line
through two given points.]
Big Algebraic Ideas
Lesson Component
Description
Concepts

Principles

Skills
Standards
Guiding Questions
Materials




Mathematical
Language

Additional Resources
Classroom setup for
each lesson
component
Initiate: Students at desks as teacher provides demonstration for
students to consider and attempt to graph.
Investigate: Students work in groups of 2 or 3.
Analyze: Students work independently with conference privileges.
Evaluate and disseminate: This portion of the lesson should be in the
whole class discussion format guided by the teacher. Students should
be seated and ready to share ideas and evaluate peer responses.
Lesson Sequence
Reflection
Initiate the lesson
Lesson 5
Page 47
Investigate
Analyze
Lesson 5
Page 48
Insert Lesson 5 Student sheets
Lesson 5 Student Sheet
Page 49
Stadium Analyst: _________________________________________________________
“Wave” Good-bye to Slope
Unit Assessment
For your final assignment on the topic of slope, you have been hired by the New York
Yankees to answer a series of questions about the fans. The Yankees are worried that the
fans are so busy doing “the wave” that they miss important moments in the game. (If you
are a fan of another team, you will have to strike a deal with that team, aka your teacher, to
alter the assignment slightly.) The tasks the Yankees have asked you to complete are listed
below.
1. Estimate how long it takes “the wave” to make its way 1 time around Yankee
Stadium. SHOW and EXPLAIN all work related to your final estimation. Use the
following suggestions to help with your estimate or develop your own plan.
a. Plan a simulation of the problem using a much smaller stadium.
b. Watch how the length of time varies with distance.
c. Create a list of materials and measurements you will need to answer the
question.
2. Create a time vs. distance graph that you could use to estimate the time it takes for
the wave to make its way once around a stadium of any size. Be sure to select
appropriate units.
HINT: Repeat the procedure you used in Question 1 for a couple of smaller
stadiums (you choose the size) to create data points.
3. Describe how a person could estimate the time it takes for the wave to make its way
around any sports arena or stadium WITHOUT using the graph.
4. Explain the “rate of change” or slope of your graph. Use the method you developed
in Lesson 5 to calculate the slope. Explain what this value means in the context of
“the wave” problem.
5. Will it take more or less time for “the wave” to travel around Giants Stadium than
Yankee Stadium? How much more or less?
Summative Assessment A
Page 50
Stadium Analyst: _________________________________________________________
“Wave” Good-bye to Slope
Unit Assessment
For your final assignment on the topic of slope, you have been hired by the New York
Yankees to answer a series of questions about the fans. The Yankees are worried that the
fans are so busy doing “the wave” that they miss important moments in the game. (If you
are a fan of another team, you will have to strike a deal with that team, aka your teacher, to
alter the assignment slightly.) The tasks the Yankees have asked you to complete are listed
below.
1. Estimate how long it takes “the wave” to make its way 1 time around Yankee
Stadium. SHOW and EXPLAIN all work related to your final estimation.
2. Create a time vs. distance graph that you could use to estimate the time it takes for
the wave to get around a stadium of any size. Be sure to select appropriate units.
3. Describe how a person could estimate the time it takes for the wave to make its way
around any sports arena or stadium WITHOUT using the graph.
4. Explain the “rate of change” or slope of your graph. Use the method you developed
in Lesson 5 to calculate the slope. Explain what this value means in the context of
“the wave” problem.
5. Will it take more or less time for “the wave” to travel around Giants Stadium than
Yankee Stadium? How much more or less?
6. The Yankees noticed that the wave went around the stadium 5 times in a row.
Estimate how many pitches were thrown during this time. SHOW and EXPLAIN all
work included in your response.
Summative Assessment B
Page 51