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Course
Profile Foundations of
Mathematics, Grade 9 applied, Catholic
Unit 1
Course Profiles are
professional development materials designed to help teachers implement the new Grade
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Acknowledgments
Public District School
Board Writing Team - Mathematics- Applied
Lead Board
Ottawa-Carleton
Catholic School Board
Sandra
Bender, Manager
Department:
Mathematics
Course Developer(s):
Arlene Corrigan, Renfrew County Catholic District School
Board
Dominique Levac,
Catholic District School Board of Eastern Ontario
Maureen Vincentine, Algonquin-Lakeshore Catholic School
Board
Linda Sloan, Ottawa
Carleton Catholic School Board
Carolyn Boyer, Ottawa Carleton Catholic School Board
Tom Steinke, Ottawa
Carleton Catholic School Board
Len St. Clair, Catholic District School Board of Eastern
Ontario
Nora Buckley,
Algonquin-Lakeshore Catholic School Board
Sue Trew, Dufferin-Peel Catholic District School Board
Brian McCudden, Toronto Catholic District School Board
Margaret Sinclair, Toronto Catholic District School Board
David Kurzinger, Toronto Catholic District School Board
Paul Costa, Toronto Catholic District School Board
Development Date:
February/March 1999
Course Revisor(s):
Revision Date:
March/April 1999.
Additional Codes:
Eastern Ontario
Catholic Curriculum Cooperative
Institute for Catholic
Education
Unit
#1: Exploring Relationships
Time: 25
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 begin to explore both
linear and non-linear relationships arising from meaningful problems. Students
will develop numeric and graphic and 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 3c, 4b, 5a, 7j
Strand(s): Number
Sense and Algebra, Relationships
Overall
Expectations: NAV.01, NAV.02, REV.01, REV.02, REV.03.
Specific Expectations: NA1.01, NA1.02, NA1.03, NA1.04, NA1.05, NA1.06, NA2.04, NA2.05, RE1.01, RE1.02, RE1.03, RE1.04, RE1.05, RE1.06, RE1.07, RE2.01, RE2.02, RE2.04, RE2.05, RE3.01, RE3.02, RE3.03.
Activity Titles (Time and Sequence)
|
Activity 1 |
Exploring Linear
Relationships - Bouncing Balls |
8 hours |
|
Activity 2 |
Exploring Non-Linear
Relationships - Mathematical Marathon |
9 hours |
|
Activity 3 |
Exploring Motion |
8 hours |
Unit
Planning Notes
In this unit, students
will be actively gathering and analyzing data. Manipulatives are required for activities 1 and 2 (balls, metre sticks,
compasses, rulers,...). Graphing calculators and motion detectors are necessary
for Activity 3, which involves a comparison of liner and non-linear
relationships between distance and time. For schools in which this technology
is not yet readily available, Activity 3 might be postponed until a later time
in the course.
|
Look for text boxes
like this one for points at which skill development can be done as needed in
the context of the activity |
Sufficient time has
been allotted for each activity to include time that is available for skill
development.
Prior
Knowledge Required
Students should have some
facility with numeric and graphing skills. When direct instruction is required,
this should occur as needed within the context of 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 and non-linear relationships.
Explore/Investigate- through hands-on
investigations of linear and non-linear relationships.
Model/Formulate- develop numeric and
graphic models for exploring linear and non-linear relationships, dependencies
and constraints.
Transform/Manipulate- develop numeric and
graphical 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 and make inferences to extend their
learning.
Communicate- individually and in
groups, orally and in writing, communicate the findings of their investigations
by defending their numeric and graphic mathematical models and explaining their
reasoning.
Assessment/Evaluation
performance tasks
paper and pencil tasks (e.g.,
quizzes, worksheets, small assignments)
written reports
oral presentations
observation
Resources
Graphing Calculators (e.g., TI82/83/83Plus)
Motion Sensors (e.g., Calculator-Based Ranger)
Spreadsheet (e.g., Quattro Pro or Excel)
Internet
Manipulatives (e.g., balls, metre sticks, compasses,...)
Atlas
Student Textbook
Activity #1: Exploring
Linear Relationships - Bouncing Balls
Time: 8
hours
Description
In this activity
students will explore the relationship between the drop and rebound height of a
ball. They will represent the data numerically and graphically. They will
analyze the data to determine any pattern in the relationship being modeled.
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 self-directed responsible life long
learner who applies effective communication, decision-making, problem-solving and
resource management skills
a collaborative contributor who works
effectively as an interdependent team member
Strands: Relationships
Overall Expectations
By
the end of this course, students will:
determine relationships between two
variables by collecting and analyzing data.<
describe the connections between various
representations of relations.
Specific Expectations
By the end of this course, students will:
pose problems, identify variables, and
formulate hypotheses associated with relationships;
collect data, using appropriate equipment
and/or technology<
organize and analyze 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
hypotheses
construct tables of values and scatter plots
for linearly related data involving direct variation collected from experiments or from secondary sources<
Planning
Notes
Prior to beginning the
activity:
place the students in groups of 4
provide each group with materials: (a metre
stick, masking tape, 2.5 m of blank cash register tape, a ball - tennis or
rubber, copies of student handout, paper for recording purposes)
Prior
Knowledge Required
ratio
(proportional reasoning)
representing data in charts
graphing ordered pairs
choosing appropriate scales
measurement skills
Teaching/Learning
Strategies
Getting Ready
Students
will be placed in groups of four.
Teacher will demonstrate a sample ball
bounce so students are clear of what drop height and rebound height are.
Explain that to do the experiment
efficiently and accurately, each member of the group must choose and perform a
specific task. Each student should record the names of their group members
along with the task that each member is to perform.
Each group will need the following
materials:
metre
stick, masking tape (several strips), 2.5 m of blank cash register tape paper,
a ball, 4 copies of the Student Handout (Appendix A), several blank sheets of
paper to record the experiences of the group.
Before the students begin, it is crucial
that they know that this is not just a fill in the missing numbers activity.
Like with any experiment, they must carefully observe and record the procedures
and data. This will be crucial when they write their report.
Beginning the Activity
Students can work in the class or hall.
Ask probing questions to each group as you
circulate through the hallway:
-
Where does the ball dropper hold the ball relative to the marked height?
-
Where does the ball bounce height recorder mark the rebound height of the
ball?
-
Ball height can be marked at the bottom, top, or middle of the ball! It is
therefore important that the drop height and rebound height are marked in a
consistent fashion.
-
Are all the group members contributing to the best of their ability?
-
What is the role of each group member?
As the groups complete their experiments
have them share the group results and observations.
Have the students begin to do a rough copy of their rebound height versus drop height graph. The groups will undoubtedly
call you over to ensure they are setting up their axes and graphing their data points properly.
|
This is an
appropriate time to ensure that the students graphing abilities are
adequate. Direct instruction may be required. |
Ball Bounce Report
Each student is now responsible for
preparing a Ball Bounce Report. The
report can be very similar to a
science lab report, which includes:
Φ Title Page
Φ Materials
Φ Group Members and Roles
Φ Procedure
(should be a detailed, one page description of how your group went about doing the experiment)
Φ Observations (the completed chart along with any other interesting observations you
and your group may have noted)
Φ Discussion (the rebound height versus drop height graph of your groups data along
with a visual line of fit through
your data points)
Φ Conclusion (describe in your own words, the relationship between the drop height
and rebound heights of each of the
three balls)
Assess the Ball Bounce Report using the rubric (Appendix B). The key parts of
the report are the procedure, graphs and conclusions. Be sure to post the Math
Reports around your room to celebrate the work of your students.
Follow-Up
Have students hypothesize what a table of
values and graph for a ball which rebounds three quarters its drop height would
look like.
What would a graph for a superball look
like?
Could you predict how high a ball would rebound
if it were dropped from the top of the C.N. Tower?
Analyzing Data Using Technology
|
At this point you
may wish to show students how to input data into lists, setup a scatter plot
and perform a linear regression on a graphing calculator (TI82/83/83Plus). |
Have your students input their data from
the bouncing ball experiment into lists in a graphing calculator.
Have your students construct a scatter plot
using the graphing calculator.
Have your students perform a linear
regression.
Allow the students to critique the
resulting line of best fit generated by the calculator.
The line will probably not pass through the
origin. It will make sense to students that if you don't drop a ball, it won't
bounce! This should allow students to select the origin as a carefully selected
point through which a line of best fit should pass.
In groups allow students to share
strategies to select a second point through which a line of best fit might
pass.
|
You may wish to
provide scatter plots for your students, where students carefully select two
points through which they can draw a line of best fit. They should defend
their choice of points based on the context from which the data points were
derived. |
Assessment/Evaluation
1. Observational rubric
for group data collection (Appendix C)
2. Rubric for the
individual written report (Appendix B)
Resources
1. manipulatives (e.g., bouncing balls of various sizes, metre sticks, ...)
2. class set of graphing calculators (e.g., TI82/83/83Plus)
3. http://www.ti.com/calcs/doc
4. Textbook
Accommodations
1. Students should be
given the option of doing an oral presentation in place of, or to complement a
written report.
2. When assigning roles
to members, be sure to assign a role to students that is not an area of
limitation.
3. Steps and
procedures for using graph calculators should be provided in written form as
well as orally.
Activity
#2: Exploring Non-Linear Relationships - Mathematical Marathon
Time: 9
hours
Description
In the spirit of the
Terry Fox Marathon can we create a fund-raiser to raise billions of dollars to
help the plight of the homeless in Canada and the U.S.? If we do a marathon
along the border of Canada and the U.S. how much can we expect an individual
participant to raise?
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
Overall Expectations
By
the end of this course, students will:
determine relationships between two
variables by collecting and analysing data.<
compare the graphs of linear and non-linear
relations.<
demonstrate understanding of the three
basic exponent rules and apply them to simplify expressions.
Specific Expectations
By
the end of this course, students will:
collect data, using appropriate equipment
and/or technology<
organize and analyse data, using
appropriate techniques and technology<
construct tables of values and graphs to
represent non-linear relations derived from descriptions
of realistic situations;<
identify, by calculating finite differences
in its table of values, whether a relation is linear or non-linear.<
determine the meaning of negative exponents
and of zero as an exponent from activities involving
graphing, using technology, and from activities involving patterning.<
represent very large and very small
numbers, using scientific notation.<
enter
and interpret exponential notation on a scientific calculator, as necessary in
calculations involving very large and very small numbers.
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.
Planning
Notes
Use the Mandelbrot's story "The Length of
the British Coastline" as an introduction. (Appendix D)
Each student requires a map of North
America with the Canada/U.S. border clearly defined and a pair of compasses.
Students work in pairs.
Spreadsheets/charting software, or graphing
calculators will be helpful.
Prior
Knowledge Required
measurement
skills
organizing data in charts
graphing ordered pairs
Teaching/Learning
Strategies
"The length depends on the
step size !"
Begin with a brief whole class discussion of
what information is required to answer the question posed. When the question of
border length emerges, introduce the Mandelbrot story, "How long is the
British coastline". (Appendix D)
In pairs, students use the map of the
Canada/US border and instructions for "How to do a Structured Walk"
(Appendix D) to collect and record measurements in columns with headings,
"step size", "number of steps", "remaining distance".
Each pair should do at least 6 structured walks (3 each) using a range of step
sizes from 0.4 cm to 3 cm.
Students calculate distance estimates
(using a formula) for each step size, and plot the ordered pairs (step size,
distance), using a spreadsheet or graphing calculator if available.
|
You may wish to
ensure all students are able to substitute into a formula so they are able to
calculate the perimeter. |
All students make a paper and pencil version
of the plot and describe it in words. They will notice that the points do not lie approximately on a straight
line.
The teacher will ensure that students make
the connection between this and the non-linear nature of the plot. Students use
the regression capabilities of a calculator to investigate possible curves of
best fit, and make the connection with the exponential model.
|
Lead into a
discussion about the meaning of negative exponents in this context. |
In groups of four, students have discussions
to consider the bigger problems: "How long is the border really?" and
"How much money could one person raise?"
|
Ensure students are
able to make the conversion of scale from cm to km, and can use scientific
notation to represent the larger distances. |
Students need to consider factors such as
method of travel along the border over land/water, distance covered in a day,
costs incurred per day/period.
Report
Students write a report which includes:
- an explanation of the problem in their
own words;
- a chart and graph of the data with
a discussion of results
- their estimate of the amount to be
raised with a complete justification including
assumptions and calculations.
Students will
now consolidate and enhance their understanding of the three basic exponent
rules by completing assignments from the textbook. Include questions with the
exponent rule for the power of a power.
This would also
be a good time to enter and interpret exponential notation on a scientific
calculator, since some distances will be quite large. Again, use textbook
assignments to involve applications with very small numbers.
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. Students write a
brief paragraph, describing how they decided that the relationship between
estimate of distance vs. step size is non-linear, followed by a reflection of
their ideas, discoveries and concerns/difficulties that arose from the
activity. This can be assessed for clarity in communicating mathematical ideas.
3. Teacher evaluates
final written report (Appendix B).
Resources
1. World atlas
2. Lewis, Ron.
"Fractals in Your Future", (http://www.eureka.ca/resources/fiyf/chapter1.html)
3. Benoit Mandelbrot
website
4. Spreadsheet
(computer lab) and/or Graphing Calculators (class set)
5. Compasses, ruler,
graph paper
Accommodations
1. Students should be given the option of doing an
oral report on tape in place of, or to complement
a written report.
2. The pacing of the activity and complexity of
the procedures can be adjusted as required.
Activity
#3: Exploring Motion
Time: 8
hours
Description
In this activity,
students will explore the concept of rate (relationship between distance and
time) by moving in front of a motion sensor. They will develop a sense of what
type of motion leads to a linear relation or non-linear relation. The
instantaneous graphic representation provided by the technology is a powerful
tool that allows all students to develop a graphical model from their own
motion. This activity is ideal in forming students understanding of predicting
the graphical outcome of an event.
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 self-directed responsible life long
learner who applies effective communication, decision- making, problem-solving and resource management skills
a collaborative contributor who works
effectively as an interdependent team member
Strands: Relationships,
Number Sense and Algebra
Overall Expectations
By
the end of this course, students will:
consolidate numerical skills by using them
in a variety of contexts throughout the course<
determine relationships between two
variables by collecting and analyzing data
compare the graphs of linear and non-linear
relations<
describe the connections between various
representations of relations
Specific Expectations
By
the end of this course, students will:
determine strategies for mental mathematics
and estimation and apply these strategies throughout
the course;<
demonstrate facility in operations with
integers, as necessary to support other topics of the course<
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, explain the
differences between the inferences and the
hypotheses<
communicate the findings of an experiment
clearly and concisely, using appropriate mathematical
forms and justify the conclusions reached<
construct tables of values and graphs to
represent non-linear relations derived from descriptions
of realistic situations
demonstrate an understanding that straight lines
represent linear relations and curves represent
non-linear relations;<
identify the slope of a linear relation as
representing a constant rate of change;<
Planning
Notes
Equipment required:
a class set of graphing calculators
one motion sensor for each group of
students
one projection unit and compatible graphing calculator
Prior
Knowledge Required
Collecting,
organizing and analyzing data.
Recognizing relationships as being linear or non-linear numerically and graphically.
Teaching/Learning
Strategies
Students observe a teacher and/or student
walking demonstration using a motion sensor, graphing calculator and projection
unit. You may wish to show students how you would walk in front of a motion
sensor to generate the letter "V".
Students, in small groups (ideally in
pairs), explore different types of motion by walking back and forth in front of
their motion sensors. You may wish to challenge the students to generate the
first letter of their first name by walking in front of the motion sensor.
Using technology, students will create several distance/time graphs.
|
Have the students
explore where the graphing calculator has stored the data represented on the scatter
plot. When the students have located the lists, have them determine what each
list represents. Students can also trace along the scatter plot and discuss
the significance of the ordered pairs in the context of their motion. |
Students observe a teacher and/or student
walking demonstration of linear and non-linear motion.
Students, in small groups (ideally in
pairs), practise and explore linear and non-linear motion using their motion
sensors. Using technology, students will create several distance/time graphs of
linear motion and several distance/time graphs of non-linear motion. Students
should record their graphs with a description of their motion.
Students can design their walk to create a
graph that is a) a straight line with a positive slope; b) a straight line with
a negative slope; c) several lines with a combination of positive and negative
slopes.
Have students walk at different speeds and in different directions so that they not only investigate positive and negative slopes, but different ratios as well. (Refer to "Explorations, Modelling Motions: High School Activities with the CBRTM."
Discuss with students the meaning of positive and negative integers in this context.
Oral Presentation
Students make oral presentations of their
group results.
Report
Each student submits a written report of
their graphs and a description of their motion that gave rise to their graphs.
Paper and Pencil Assessment Tasks
Students observe a given motion and then predict
and defend the nature of the resulting distance/time graph.
Students to describe the type of motion
that would result from a given distance/time graph.
Performance Assessment Task
Using the "Distance Match"
application on the Ranger program, students walk in front of a motion sensor so
as to imitate a given distance/time graph.
Assessment/Evaluation
1. Teacher Observation (Appendix C)
2. Oral Presentation.
3. Written Report (Appendix B)
4. Performance Assessment Task.
5. Paper and Pencil Tasks.
Resources
1. "Getting Started with CBR",
Texas Instruments
2. "Explorations,
Modeling Motion: High School Activities with the CBR", Texas Instruments
3. http://www.ti.com/calc/docs
4. "Life by the Numbers, Video
Number 7,
PBS 1998
Accommodations
1. Provide clean
written instructions and steps as needed. Adjust the number of comments
required to allow students to fully participate.
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