PLEASE NOTE:
This document may contain a number of charts and graphics that
could be problematic for your computer configuration. It is recommended that you use the "pdf" version for
printing this document and this file for working with or adapting the Course
Profile to meet your instructional needs.
Course Profile
Foundations of Mathematics, Grade 9 applied, Catholic
Unit #2: Modeling Linear
Relationships
Activity 1 | Activity 2 | Activity 3 | Activity
4
Time: 35 Hours
Unit
Developer(s)
Arlene Corrigan, Dominique
Levac Maureen Vincentine, Linda Sloan, Carolyn Boyer , Tom Steinke, Len St.
Clair, Nora Buckley, Sue Trew, Brian McCudden, Margaret Sinclair, David
Kurzinger, Paul Costa
Development
Date: February/March,
1999.
Unit Description
In this unit, students and teachers will explore numerical,
graphical and algebraic models (tables, graphs and equations) of linear
relationships arising from meaningful problems. Students will develop numeric,
graphic and algebraic skills as needed in the context of the activity. Various
forms of assessment are built into all the activities.
Strand(s) & Expectations
Ontario
Catholic School Graduate Expectations: CGE 2b, 3c, 3e, 4f, 5a, 5g.
Strands: Number Sense and Algebra, Relationships, Analytic Geometry.
Overall
Expectations: NAV.01, NAV.02, NAV.03, NAV.04, REV.01, REV.02, REV.03,
AGV.01, AGV.02, AGV.03.
Specific
Expectations: NA1.01, NA1.02, NA1.03, NA1.04, NA1.05, NA1.06, NA2.04,
NA2.05, NA3.01, NA3.02, NA3.03, NA3.05, NA4.01, NA4.02, NA4.03, RE1.01, RE1.02,
RE1.03, RE1.04, RE1.05, RE1.06, RE1.07, RE2.01, RE2.02, RE2.03, RE3.01, RE3.02,
RE3.04, AG1.01, AG1.02, AG1.03, AG2.01, AG2.02, AG2.03, AG2.04, AG3.01, AG3.02,
AG3.03, AG3.04, AG3.05.
Activity Titles (Time and Sequence)
|
Activity 1 |
Modeling Motion - Walking the Line |
8 hours |
|
Activity 2 |
Modeling Linear Relationships - The Help Line |
9 hours |
|
Activity 3 |
Modeling Linear Relationships - Environmental
Issues |
9 hours |
|
Activity 4 |
Modeling Linear Relationships - Bouncing Balls |
6 hours |
Unit
Planning Notes
The activities in this unit embed the use of
technology, in particular graphing calculators. For schools in which this
technology is not yet readily available, teachers may adapt the activities.
With the exception of the first, all activities can be accomplished without the
use of graphing calculators, if some changes are made. Bear in mind that,
without the use of graphing calculators, some of the activities are likely to
take more class time.
The use of graphing calculators is essential for
Activity 1. The key ideas introduced in this activity include the use of finite
differences to describe a constant rate of change, and the introduction of the
significance of m and bin the equation y=mx+b. These concepts do arise in later activities, and teachers
are advised to make adjustments, as necessary. Activities 2 and 3 involve the
use of a computer lab for spreadsheet and Internet activities. Manipulatives
must be available for Activity 4. The student textbook should be used for
numeric, graphic, and algebraic skill development at appropriate points in the
activities.
Sufficient time has been allotted for each activity to
ensure time is available for skill development.
|
Look for text boxes like this one for points at
which skill development can be done as needed in the context of the activity. |
Prior
Knowledge Required
Students should be able to comfortably model linear
and non-linear relationships numerically and graphically. Should direct
instruction be required, this should occur as needed within the context of the
activities as opposed to before the activities. All students should be able to
engage fully in all of the activities.
Teaching/Learning
Strategies
Students will:
Hypothesize -
formulate hypotheses associated with linear relationships.
Explore/Investigate- through hands-on investigations
of linear relationships.
Model/Formulate- develop numeric, graphic,
algebraic and geometric models for exploring linear relationships, dependencies
and constraints.
Transform/Manipulate- develop numeric,
graphical, algebraic and geometric skills as needed in the context of their
investigations to allow them to move within and between representations.
Infer/Conclude
-
re-evaluate their hypotheses in light of their learning make inferences to
extend their learning.
Communicate- individually and in
groups, orally and in writing, communicate the findings of their
investigations, defending their numeric, graphic, algebraic and geometric
mathematical models and explaining their reasoning.
Assessment/Evaluation
performance tasks
paper and pencil tasks
written reports
oral presentations
observation
Resources
Graphing Calculators (e.g., TI82/83/83Plus)
Graphing Software (e.g., Graphmatica or Zap-A-Graph)
Motion Sensors (e.g.,
Calculator-Based Ranger)
Spreadsheet (e.g.,
Quattro Pro or Excel)
Internet
Manipulatives (e.g.,
balls, metre sticks, compasses,...)
Dynamic Geometry Software (e.g., Geometer's SketchPad, Cabri, TI92, Java SketchPad)
Student Textbook
Activity
#1: Modeling Motion - Walking the Line
Time: 8 hours
Description
In this activity, concepts of slope and y-intercept
will be addressed formally by linking the characteristics of a linear graph,
and its algebraic representation y = mx +
b, to the motion of the students. Students will explore the concept of rate
(relationship between distance and time) by moving in front of a motion sensor (e.g., CBR).
Strand(s)
and Expectations
Ontario Catholic School Graduate Expectations:
The graduate is expected to
be:
a reflective and creative thinker who thinks reflectively and creatively
to evaluate situations and solve
problems
a collaborative contributor who achieves excellence,
originality, and integrity in ones own work
and supports these qualities in the work of others
a collaborative contributor who works effectively as an
interdependent team member
Strands: Number Sense and Algebra, Relationships, Analytic Geometry
Overall Expectations
At the end of Grade 9,
students will:
determine, through investigation, the relationships between the
form of an equation and the shape
of its graph with respect to linearity an non-linearity;<
determine, through investigation, the properties of the slope
and y-intercept of a linear relation;<
Specific Expectations
Students will:
collect data, using appropriate equipment and/or
technology;<
organize and analyse data, using
appropriate techniques and technology;
communicate findings of an experiment
clearly and concisely, using appropriate mathematical
forms;<
identify the geometric significance of m and b in the equation y = mx + b through investigation<
demonstrate facility in operations with
percent, ratio, rate and rational numbers, as necessary to support other topics
of the course (e.g. analytical geometry, measurement)
Planning
Notes
Equipment required:
a class set of graphing calculators
one motion sensor
(e.g., CBR) for each group of students
one projection unit with compatible graphing
calculator
Prior
Knowledge Required
collecting, organizing and analyzing data using
technology
modeling relations numerically and graphically
concept of rate (relationship between distance and
time)
Teaching/Learning
Strategies
1. Students observe a teacher and/or student
walking demonstration of a constant rate of change using a motion sensor (e.g., CBR), graphing calculator and
projection unit. (Note that it may take practice to model a constant walking
speed.).
2. Students, in small groups (ideally in pairs),
explore constant rates of change by walking back and forth in front of a motion
sensor. Using technology, they will create and record distance/time graphs for
various constant rates of walking using different starting points.
|
Have students explore finite differences using the
data stored in the lists. This can serve as a numeric link to the concept of
a constant rate of change. This may be an opportunity to consolidate
students' skills in performing operations with rational numbers. |
3. Have students hypothesize about the type of
motion that would lead to a horizontal line (x
= a) and the type of motion that would lead to a vertical line (y = b). Allow the students to test
their hypotheses using the motion sensors.
4. Students prepare a written report describing
how a constant rate of change is represented by the slope of a linear relation.
Students must be able to defend their results and predictions.
|
The geometric significance of m and b in the equation
y = mx + b will also be
investigated in the context of walking in front of a motion sensor. |
Report
Students present their results in the form of a
written report. The report should show the students ability to communicate
their ideas clearly and concisely.
Paper and Pencil Task
Students match
linear graphs with their algebraic representation in the form y = mx + b.
Performance Assessment Task
Students model, by walking in front of a motion
sensor, a linear relation expressed algebraically in the form y = mx + b.
Assessment/Evaluation
1. Observe students for learning and for
evidence of their problem solving and inquiry skills as they proceed through
the activity (Appendix C)
2. Written Report (Appendix B)
3. Performance Assessment Task
4. Paper and Pencil Task
Resources
1. Getting Started with CBR, Texas
Instruments
2. Explorations, Modeling Motion: High School
Activities with the CBR, Texas Instruments
3. www.ti.com/calc/docs
4. Life
by the Numbers", Video Number 7, PBS, 1998
Activity
#2: Modeling
Linear Relationships - The Help Line
Time:
9 hours
Description
The students of your school plan to sponsor a school
in the Dominican Republic by sending computers along with some students who
will provide instruction while they experience the culture. A car wash (in the school
parking lot) will be held, as one of the fundraising activities, to pay for the
air fares and to purchase the computers. How much money is likely to be raised
by the car wash?
The goal is to raise $50 000 in total. If this goal
is reached, how many students and how much equipment are you likely to be able
to send?
Strand(s)
and Expectations
Ontario Catholic School Graduate Expectations:
The graduate is expected to
be:
a responsible citizen who witnesses
Catholic social teaching by promoting equality, democracy, and solidarity for a just, peaceful and
compassionate society
Strands: Number Sense and Algebra, Relationships, Analytic Geometry
Overall Expectations:
At the end of Grade 9,
students will:
consolidate numerical skills by using them
in a variety of contexts throughout the course;<
manipulate first-degree polynomial
expressions to solve first-degree equations;<
solve problems, using the strategy of
algebraic modelling;<
describe the connections between various
representations of relations;
demonstrate understanding of the three
basic exponent rules and apply them to simplify expressions.
Specific Expectations:
Students will:
manipulate
first-degree polynomial expressions to solve first-degree equations;<
use algebraic modelling as one of several
problem-solving strategies in various topics of the course;
communicate solutions to problems in
approximate mathematical forms and justify the reasoning
used in solving the problems.<
construct tables of values, graphs, and formulas
to represent the linear relations derived from descriptions
of realistic situations;<
describe the effect on the graph and the
formula of a relation of varying the conditions of a situation they represent.<
identify the equation of a line in any of
the forms y = mx + b as a standard
form for the equation of a straight
line, including the special cases x = a, y = b;<
graph lines by hand, using a variety of
techniques;
graph lines, using graphing calculators or
graphing software;
determine the equation of a line, given the
slope and y-intercept, the slope and
point on the line, and two
points on the line;
demonstrate facility in operations with
integers, as necessary to support other topics of the course (e.g.,
polynomials, equations, analytic geometry);
determine, from the examination of
patterns, the exponent rules for multiplying and dividing monomials and the
exponent rule for the power of a power, and apply these rules in expressions
involving one and two variables;
add and subtract polynomials, and multiply
a polynomial by a monomial;
expand and simplify polynomial expressions
involving one variable.
Planning
Notes
This task will require the use of a spreadsheet.
Book a computer lab and ensure you have a
spreadsheet application.
This task provides an excellent opportunity
for students to explore the capabilities of a spreadsheet program for tables
and graphs.
When chart is set up for the spreadsheet,
select new window and arrange windows vertically, side-by-side, to manipulate
tables and graphs simultaneously (refer to software help for spreadsheet
instructions).
Prior
Knowledge Required
Some facility with spreadsheet
applications.
Facility modeling linear relationships
numerically, graphically.
Some facility modeling linear relationships
algebraically.
Teaching/Learning Strategies
Part 1: y = mx
Begin with a simple investigation of how the amount raised
depends on the number of cars washed. Students generate table with headings
"number of cars washed" ,"$ amount raised". Each students
decides how much to charge for a car wash and how many cars they can reasonably
expect to wash in a day.
|
Teacher ensures students enter the correct formula
for the second column. |
Students look at each other's plots
and discuss what is the same, what is different, what makes the difference.
They need to make the connection that the more they charge per car , the
steeper the plot.
Suppose y represents the amount of money raised in dollars and x represents the number of cars. Write
the equation describing y in terms of x for their graph. Students share the
results by writing them on the board.
|
General
equation y = mx is developed in the
context of this activity. |
At this point students start a new
spreadsheet and graph y = mx for
various m-values beginning with m = 1. Whole class discussion of whether
a line is appropriate for the car wash data. Ask students to come up with an
example where a continuous line is appropriate.
|
Explore situations
which give rise to data that could reasonably be modeled with a continuous
versus a discrete graphical model. |
Suppose you raise $48 washing 8 cars.
Enter these values in your spreadsheet to plot a point to correspond to this
situation. Then change m so that the
line y = mx goes through this point.
|
Teacher provides several more points for students
to repeat/practise this process. |
Students tabulate y = mx for these different m-values in the spreadsheet and look for
patterns in the tables. Teacher ensures students make the connections between
and among the finite differences, the m-value,
and the slope of the line.
Ask students what m represents in the
context of the problem. Make connection that slope is a rate.
Part 2: y
= mx + b
Revisit the initial problem and
introduce the fixed cost of supplies for the car wash. Adjust the formula and
re-evaluate the amount raised. Plot the pairs relating number of cars washed to
amount raised. Compare graphs to the original graphs and discuss what is the
same, what is different, and what makes them different. (This is an opportunity
to introduce the words direct or partial variations).
On a new spreadsheet, introduce b into the equation y = x + b or y = mx + b, m = 1, b = 0, and graph it for various b-values.
|
As an exercise, have students change b, so that the line passes through
previously selected points. Have students change m or b, so that the
line passes through previously selected points. |
Teachers should be prepared to
discuss why there are different lines that may pass through any one point.
Revisiting the initial problem and starting
a new spreadsheet, students should re-plot their data for x-values of 3 rather than 1. Have students calculate the finite
differences in the next column over. Class discussion should raise points that
differences are all the same, all are 3 times the m value, and why 3 times?
(i.e. make the connection with size of increment). Have students change the x-values to 5, 10, 12, 16, 22, and 30,
observing the changes to the finite differences note any patterns that occur.
Challenge the students with data from
a previous car wash event to see if they can determine the slope from the x and y-values. Then write a formula to find the slope from a list of x and y-values. Include in your discussion whether it matters which pairs
of x and y-values are used. Help students convert their spreadsheet formula
into the mathematical slope formula.
In answering the question: How many
students and how much equipment are you likely to send?, students need to
research air fares and price of computers. Investigating the possible
combinations of the number of computers to send and the number of students to
accompany them, students can answer questions like: "If you wanted to send
15 computers at $2000 each, how many students would be able to accompany them
at a cost of $1250 each? If you only send 8 students, how many computers can
you send?"
Part 3: Assume students have reached the $50 000 goal
What are other possible combinations
that enable you to spend the entire $50 000?
Letting x represent the number of computers and y represent the number of students, write an equation that connects x and y. Have students start a new spreadsheet, entering the x and y-values they have found, and graph the line using the ordered
pairs. Whole class discussion would include questions like: What is the slope
of the line? What is the y-intercept?
What is the equation in the form y = mx +
b? We also have 2000x +1250y = 50 000. Are these the same? How can
you show this?
|
Teacher should ensure that students can manipulate
one (y = mx + b) into the other |
In discussing how useful the line is
in this problem, students should consider questions like do all solutions to
the problem lie on the line? Do all points on the line represent solutions to
the problem? Students should follow up their research by writing a report to
answer the question: How many students and how much equipment are you likely
to send?, giving supporting arguments.
Part 4: Possible Extension
Now would be a good time for the
teacher to diagnose students' ability to work with integers and remediate as
necessary. This could then be extended to lessons on manipulating polynomial
expressions, supported by textbook resources. When multiplying and dividing
monomials, highlight the exponent rules covered in Activity One. Include the
exponent rule for the power of a power.
Assessment/Evaluation
Observation of students working with
spreadsheet and solving problems. (Appendix C)
At the end of the second day a
written reflection on the characteristics of the graph with equation y = mx + b.
Cumulative quiz to assess acquisition
of basic skills in calculating slopes, and rearranging equations.
Written submission with individual answers
to question posed, in the description, with justifications.
Resources
1. Spreadsheet software
2. Textbooks
3. Newspapers and advertising flyers
4. Internet
Accommodations
1. Students can be given options to a written report.
2. A symbolic algebraic manipulator (e.g., TI89, TI92, Maple) could be used as a compensatory tool for
any students for whom algebraic manipulation is a learning obstacle.
Activity
#3: Modeling Linear Relationships - Environmental Issues
Time: 9 hours
Description
The students will collect data on the use of energy
specifically electricity. They will use the properties of linear relations
and their numeric, graphic and algebraic skills to examine trends and solve
problems related to environmental data. This activity gives students and
teacher opportunities to examine our lifestyle and understand our
responsibilities of Christian stewardship. Students use the analysis of data to
consolidate numeric skills.
Strand(s)
and Expectations
Ontario Catholic School Graduate Expectations:
The graduate is expected to
be:
a reflective and creative thinker who examines, evaluates and
applies knowledge of interdependent
systems (physical, political, ethical, socio-economic and ecological) for the development of a just and compassionate
society.
a responsible citizen who respects the environment and uses
resources wisely.
a reflective and creative thinker who thinks reflectively and
creatively to evaluate situations and
solve problems.
Strands: Number Sense and Algebra, Relationships, Analytic Geometry.
Overall Expectations:
At the end of Grade 9,
students will:
determine relationships between two
variables by collecting and analysing data;
describe the connections between various
representations of relations;<
determine, through investigation, the
properties of the slope and y-intercept
of a linear relation;<
consolidate numerical skills by using them
in a variety of contexts throughout the course.<
Specific Expectations
Students will:
pose problems, identify variables, and
formulate hypotheses associated with relationships
collect data, using appropriate equipment
and/or technology;<
organize and analyse data, using
appropriate techniques and technology;
describe trends and relationships observed
in data, make inferences from data, compare the inferences with hypotheses about the data, and explain the
differences between the inferences and
the hypotheses<
Construct tables of values, graphs, and
formulas to represent the linear relations derived from descriptions of realistic situations;
determine the equation of a line of best
fit for a scatter plot, using an informal process;<
determine values of a linear relation by
using the formula of the relation and by interpolating
or extrapolating from the graph of the relation;<
identify the geometric significance of m and b in the equation y = mx + b through investigation;<
determine the equation of a line, given the
slope and y-intercept, the slope and
point on the line, and two
points on the line;<
demonstrate facility in operations with
percent, ratio, rate and rational numbers, necessary
to support other topics of the course;<
Planning
Students will require access to the Internet or
other source of up-to-date data.
Book a computer lab with Internet access.
This activity does not require graphing software, however spreadsheets or graphing calculators could be used to advantage.
Prior
Knowledge Required
Plotting data using scaled axes.
Fitting a visual line of best fit.
Finding slope between two points.
Writing equations of lines given slope and
intercept.
Teaching/Learning
Strategies
Collecting and Displaying Data
From Environment Canada and statistics sites for
world data, or from an encyclopedia, students collect the following data for at
least 15 countries over approximately 20 years.
Statistics on electricity consumption by
country
Population
Average monthly temperatures
Plot total electricity used in Canada against years.
The teacher should emphasize the importance of choosing a suitable scale since
the issue of scale will be an important one in graphing any real life
statistics.
Ask students to consider why the amount of
electricity used has increased. Can it be something other than each person
using more? Students should notice that there has been an increase in
population. How could we remove the population effect from the graph to see if
each person is using more?
Calculate per capita electricity use for the given
years.
Discuss students predictions of a possible
relationship between the variables of per capita use and time (measured over 20
years).
Plot electricity used per capita in Canada
against time. Assign a particular year to be 0 and then use an increment of 1
for each year thereafter.
Analyzing the Data
In pairs, students fit a visual line of best fit
using the convention: a line that passes through as many points as possible and
has approximately half the remaining data above the line and half below.
Calculate the slope by selecting two points on the
line, find the intercept and develop an equation for the line of best fit.
|
At this time teachers may want to give extra practice finding an equation of a line using the slope and intercept form. |
Discuss the meanings
of m, and b in relation to the particular data.
Slope is the rate of change of electricity use per
year.
The intercept will be the amount used per capita in
the starting year.
Calculate interpolated and extrapolated values by
substituting into the equation.
|
Teachers may want to take time here to ensure that
students have adequate practice at substituting in simple algebraic equations.
A context should be provided, so that students can determine if their answer
is reasonable. Example: Let y = 20x + 12. If x = 25, find the value of y. |
Students analyze the predicted values and discuss
whether they are realistic. For example, can per capita use of electricity
continue to grow infinitely at a constant rate? What limitations exist?
In this context compare the domain and range of the
plotted data to that of the line of best fit.
Also at this time emphasize the distinction between
discrete data as this plot displays (the only values known are at particular
points) and the continuous values shown by the line of best fit.
In groups of approximately four, students
repeat the process above for two other countries and then present their graphs,
defend the calculations of m, b, and
the equation and give their interpretation of slope and intercept.
Reporting
Based on their results and the results of
the other groups students write a report to include:
- a description of
the trends noticed,
- a comparison of
Canadas place in the world regarding consumption of electricity and
recommendations for the future.
Students investigate alternate sources of
energy. At present much electricity production is reliant on coal or nuclear
plants. New technology is emerging that should bring the advantages of
electricity to everyone while protecting the environment.
In pairs, students use the Internet or recent
articles to gather the following information about a renewable source of energy
such as wind or solar power.
Consider such things as:
the advantages of one source of power over
another.
the initial cost for a Canadian house that
has this type of power source.
the quantity of additional back up energy
required to supplement the energy provided by
traditional sources.
Students share their findings and discuss the
advantages of renewable energy from the position of our responsibility to the
worlds people and to our planet.
Consolidating Numerical Skills
Students continue the analysis of present
energy production and the options of renewable energy with opportunities to
consolidate numerical skills.
|
Consolidate
numeric skills: |
|
Scientific notation (e.g., in calculations
of pollutants they deal with very large and very small numbers). |
|
Rate (e.g., of pollutants in quantities
of air and water, of per capita use of water, oil, natural gas). |
|
Ratio (e.g., comparison between
quantities of emissions produced by different forms of energy). |
Assessment/Evaluation
1. Student collecting, plotting and analyzing of
data will be assessed with an observation checklist. (Appendix C)
2. For the group assignment on two other countries:
Plots, lines of best fit, slopes, intercepts, equations and interpretation of
these will be handed in and assessed for completion and accuracy.
3. Written report on the results of the country
comparisons to be handed in and assessed for communication and application of
procedures.
4. A set of questions on finding unit rates,
calculating slopes, finding an equation based on slope and intercept or two
points, and finding values by substituting in simple algebraic formulas will be
marked for knowledge and understanding.
Resources
1. http://www.energy.ca/ELECTRIC.html
2. http://www.statcan.ca
Accommodation
Reports can vary by mode of presentation and by
level of complexity.
Activity
#4: Modeling Linear Relationships - Bouncing Balls
Time: 6 hours
Description
Students will explore the linear relationship
between a balls drop height and rebound height in a technology rich
environment. Students, in unit one, will have developed a sense of the
difficulty of accurately collecting data by hand for this activity. Here, they
will use graphing calculators and motion sensors to collect the data accurately
and efficiently. They will develop and explore numeric, graphic, algebraic, and
geometric
models of
the resulting linear relationship.
Strand(s)
and Expectations
Ontario Catholic School Graduate Expectations:
The graduate is expected to
be:
an effective communicator who presents
information and ideas clearly and honestly and with sensitivity to others;
a reflective, creative and holistic thinker
who demonstrates flexibility and adaptability.
Strands: Number Sense and Algebra,
Relationships, Analytic Geometry, Measurement and Geometry
Overall Expectations
At the end of Grade 9,
students will:
solve problems, using the strategy of
algebraic modelling;<
determine relationships between two
variables by collecting and analysing data;
determine, through investigation, the
properties of the slope and y-intercept
of a linear relation;<
formulate conjectures and generalizations
about geometric relationships involving two- dimensional
figures, through investigations facilitated by dynamic geometry software, where
appropriate.<
Specific Expectations
Students will:
use algebraic modelling as one of several
problem-solving strategies in various topics of the course (e.g. relations, measurement, direct and partial
variation, Pythagorean theorem, percent);<
compare algebraic modelling with other
strategies used for solving the same problem;
communicate solutions to problems in
approximate mathematical forms (e.g., written explanations,
formulas, charts, tables, graphs) and justify the reasoning used in solving the
problems;<
collect data, using appropriate equipment
and/or technology (e.g., measuring tools, graphing calculators, scientific probes, the Internet; Sample problem: Drop a ball from varying
heights, measuring the rebound
height each time.);<
organize and analyse data, using
appropriate techniques (e.g., making tables and graphs, calculating measures of central tendency) and technology (e.g.,
graphing calculators, statistical
software, spreadsheets);
construct tables of values and scatter
plots for linearly related data collected from experiments
(e.g., the rebound height of a ball versus the height from which it was dropped);<
determine the equation of a line of best
fit for a scatter plot, using an informal process (e.g., a process of trial and error on a
graphing calculator; calculation of the equation of the line joining two carefully chosen points of the
scatter plot).<
Planning
Notes
The teacher will have to have several balls and a
class set of graphing calculators and motion sensors. The teacher should book time
in the computer lab for the dynamic geometry exploration.
Prior
Knowledge Required
Lists, scatter plots, linear regressions on
a graphing calculator
Visual line of best fit
Teaching/Learning
Strategies
Collecting
Data With Technology
Teacher demonstrates ball bounce using a
graphing calculator, motion sensor and the Ball Bounce application on the
Ranger program.
Students collect data from ball bounce
experiment in pairs with graphing calculator and motion sensor.
Analyzing the
Data With Technology
Teacher demonstrates the Trace, List,
Scatter Plot, Regression analysis sequence on the graphing calculator. The
process involves tracing along the graph to read, interpret, and record the
peak heights as drop height and rebound height (see Unit 1, Activity 1). The
drop heights and rebound heights will be input into lists. Set up a scatter
plot. Perform a linear regression on the data.
Students, in pairs, use the Trace, List,
Scatter Plot, Regression sequence to analyze their data with the graphing
calculator.
Teacher demonstrates the Least Squares Fit
on Dynamic Geometry Software.
Students, in pairs, critique the algebraic
model given by the graphing calculator and refine their algebraic model by
exploring other, potentially better lines of fit, with the Least Squares Fit
geometric model using Dynamic Geometry Software.
|
Discuss different points one could choose through
which a line of best fit might be drawn.
The justification for the selection of points must be firmly grounded
in the context of the situation which gave rise to the data. Different scenarios should be explored. |
Report
Each student prepares a written report
which they defend their algebraic
model of a line of best fit.
Assessment/Evaluation
1. Observation Rubric (Appendix C)
2. Paper and Pencil Task
3. Written Report (Appendix B)
Resources
1. "TI Explorations: Modeling with CBR", Texas Instruments
2. http://www.ti.com/calc/docs
Accommodations
Students for whom the display on the graphing
calculator is too small, should be provided with a viewscreen and a fluorescent
palette or they should work on a computer using a spreadsheet application.
Appendix
A: STUDENT HANDOUT - BALL BOUNCE
Materials:
Φ Assorted balls (tennis, golf, table tennis,
crazy...)
Φ Masking tape
Φ Metre stick
Φ Roll of blank cash register tape (or other
paper to tape on wall)
Problem:
What is the relationship between
the height that a ball is dropped and the height that the ball bounces?
Directions:
In groups of four, choose a role for each
group member
(material
collector, ball dropper, ball bounce height recorder, recorder).
Drop the ball from the first specified
height, recording the bounce height.
Drop the ball from the same height two times
to get an average drop height.
Repeat the process for the other specified
drop heights.
Be sure to keep a detailed record of how
your group goes about carrying out this investigation.
Data:
|
Drop Height (cm) |
Ball Type __________________ Bounce Height (cm) |
|
200 |
|
|
175 |
|
|
150 |
|
|
125 |
|
|
100 |
|
|
75 |
|
|
50 |
|
|
25 |
|
Analyzing Your
Data & Presenting Your Findings
Each member of the group is required to prepare a BALL BOUNCE REPORT that should include:
list of group members and their roles;
detailed description of how your group
carried out this investigation;
complete data chart (above);
graph of the bounce height versus the drop
height for each of the balls
(the vertical axis is the bounce
height and the horizontal axis is the drop height);
visual
line of fit through the data points for each ball;
description in words of the relationship
between the drop height and bounce height for each of the balls.
Appendix
B: SAMPLE RUBRIC - WRITTEN REPORT
(Note: This Rubric is designed specifically for Unit
1, Activity 2. It can be used as a model for other activities)
|
|
|
Level 1 |
Level 2 |
Level 3 |
Level 4 |
|
Model/ Formulate |
Measure-ment |
Incomplete and/or step sizes not appropriate |
Measurements complete but some step sizes not appropriately chosen. |
Measurements complete with appropriate choice of step sizes |
Student shows deeper understanding of problem by using creative means to generate additional measurements |
|
Transform/ Manipulate |
Calculating Distance Chart/ Plotting Scale Conversion |
Many errors in distance calculations Many errors in scales, labels, titles and/or many points plotted incorrectly. Many errors in use of scale and/or use of scientific notation |
Some errors in distance calculations Some errors in scales, labels, titles and/or some points plotted incorrectly. Many errors in use of scale and/or use of scientific notation |
Almost all distances calculated correctly Charts correctly titled. Appropriate scales and labels on graphs. Points accurately plotted. Uses map scale to convert distances with minor errors. |
All distances calculated correctly Uses map scale to convert distances accurately |
|
Infer/ Conclude |
Analysis of Finite Differences Regression Analysis Deciding on distance along the border, and calculation of costs etc. |
Finite difference calculations missing or contain many errors. Understanding of difference between graphs of linear and non-linear relationships not demonstrated. Analysis not done Calculations incomplete |
Some finite differences not calculated correctly. Infers correctly that relationship is non-linear. Recognizes that linear model does not fit but does not identify exponential model as correct. Cost calculations contain errors and/or no justification for quantities used and/or no explanation of assumptions made. |
Calculates finite differences correctly and infers correctly that relationship is non-linear. Correctly identifies the exponential model as the closest fit for the data Calculations complete and correct. Student clearly states all assumptions made and justifies all quantities used. |
As level 3 but also looks for patterns in the differences calculated. As level 3 but also discusses the limitations of the exponential model for this problem Student extends problem in some way. e.g. Compares several schedules to maximize profits from marathon, or includes a detailed discussion of the notion of an infinite border length. |
|
Communi-cate |
Communi- cation |
Poor use of mathematical language and/or arguments weak. |
Does not use complete sentences and/or mathematical language. Arguments may not be fully developed or may be unclear |
Communicates clearly and effectively using mathematical language. Develops arguments fully, clearly stating all assumptions and considerations involved. |
Communicates persuasively and effectively using mathematical language. Develops arguments fully, clearly stating all assumptions and considerations involved. |
Appendix C: SAMPLE OBSERVATIONAL
RUBRIC
|
Criteria |
Level 1 |
Level 2 |
Level 3 |
Level 4 |
|
Engages in
task |
Begins task but with
need of considerable
prompting |
Begins task with some
prompting |
Begins task without
need of prompting |
Begins task promptly
and encourages
others to
begin |
|
Applies appropriate strategies |
Requires consistent support
and prompting
to pursue
alternate strategies |
Pursues alternate strategies
with frequent
assistance and limited
prompting |
Pursues alternate strategies
with only limited assistance |
Actively
pursues alternate strategies independently |
|
Uses resource
materials effectively and independently |
Does not refer to notes,
text or other resources before seeking assistance |
Rarely refers to notes,
text or other resources before seeking assistance |
Frequently refers to
notes, text or other
resources before
seeking assistance |
Consistently
refers to notes, text or other resources before seeking assistance |
|
Works effectively
with others in the group |
Assumes
passive role and contributes infrequently and often in a limited way |
Assumes
passive role and contributes usually only
when prompted |
Assumes
active role and contributes freely to the group |
Assumes
leadership role and tries to encourage all to contribute |
|
Contributes effectively to the work of
the group |
Contribution
is limited and only when prompted |
Contribution
is infrequent but often will volunteer ideas |
Contribution
is frequent and does not require prompting to share ideas |
Contribution
is frequent and builds on
others ideas |
|
Uses the materials and resources effectively |
Requires
frequent support in finding and
using the materials. |
Will
often need support in finding or
in using the materials |
Needs
only limited and infrequent support in finding or in using the materials |
Needs little or no support
to find and use materials, and consistently assists others to find and use
materials |
|
Is an active problem solver |
Will explore very few possibilities and often
stops before it is solved Rarely seeks alternate
solutions |
Will explore some possibilities
but may stop before problem is solved Sometimes seeks alternate
solutions |
Will explore a few possibilities
until the problem is solved Often
seeks alternate solutions |
Will
explore many possibilities until the problem is solved Will
seek alternate solutions |
Choose the criteria that you can effectively and efficiently
assess in the time you have available. Do not assess all criteria at one time.
All criteria can be assessed as students work in groups, but some criteria can
also be assessed as students work independently.
Appendix D: Supporting Materials
for Unit 1, Activity 2
1. Mandelbrot Story
Benoit Mandelbrot, a famous mathematician (who is
still living!) was commissioned to find the length of the British coastline.
After months of working on this problem, he came back with his response. His
response was not particularly well received by those who had paid him to give
them an "answer". His "answer" was: "It
depends!!!"
This activity is designed to allow students to
develop, through a hands-on investigation, an appreciation for Mandelbrot's
puzzling answer. The "dependency" is a rich avenue for exploration
and discussion. The mathematical models students will construct to represent
this scenario generate non-linear relationships (exponential). The
"dependency" arises from the step size one chooses.
2. Structured Walk
(a) Set your compass to your chosen step size (e.g.
5 cm) (L).
(b) "Walk" along the borderline with your
compass, counting the number of "steps" (n).
(c) Measure the remainder with your ruler (r).
(d) Record your L, n and r values in a chart (see
below) and calculate the length (perimeter P) using the formula P = nL + r.
(e) Repeat steps (a) - (d) for your next chosen step
size (e.g. 4 cm).
|
Step Size (L in cm) |
Number of Steps (n) |
Remainder (r in cm) |
Perimeter/Length (in cm) |
|
5 |
|
|
|
|
4 |
|
|
|
|
3 ... |
|
|
|
Continue to Unit 3 | Back to Unit 1 | Back to Course
Profiles main menu