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Course Profile
Principles of Mathematics, Grade 9 academic, Catholic
Unit 1
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Acknowledgements
Public District School Board
Writing Team - Mathematics- Academic
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
Easterm 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 Revisors:
Revision Date: March/April
1999.
Additional Codes:
Eastern Ontario Catholic
Curriculum Cooperative
Institute for Catholic
Education
Unit #1: Exploring Relationships
Activity 1
| Activity 2 | Activity 3
Time: 20 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, 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 3c, 4b, 5a, 7j
Strand(s): Number
Sense and Algebra, Relationships.
Overall
Expectations: NAV.01, NAV.02, NAV.04, REV.01, REV.02, REV.03.
Specific
Expectations: NA1.01, NA1.02,
NA1.03, NA1.04, NA1.05, NA1.06, NA2.01, NA2.03, NA2.04, NA2.05, NA2.06, NA4.03,
RE1.01, RE1.03, RE1.04, RE1.05, RE1.06, RE1.07, RE2.01, RE2.02, RE2.04, RE2.05,
RE2.06, RE3.02, RE3.03, RE3.04.
Activity Titles (Time + Sequence)
|
Activity 1 |
Exploring Linear Relationships Bouncing Balls |
7 hours |
|
Activity 2 |
Exploring Non-Linear Relationships Mathematical Marathon |
7 hours |
|
Activity 3 |
Exploring Motion |
6 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, graphing and algebraic 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, graphic and algebraic models for exploring linear and non- linear relationships, dependencies and constraints.
Transform/Manipulate develop numeric, graphical and algebraic 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: 7 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
patterns in the relationship being modeled.
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.
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. J
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. J
organize and analyze data, using appropriate techniques and technology. J
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 collected from experiments or from secondary sources. J
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 that 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) and 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?
(Note that 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:
* Title Page
* Materials
* Group Members and Roles
* Procedure (should be a detailed, one page description of how your group went about doing the experiment)
* Observations (the completed chart along with any other interesting observations you and your group may have noted)
* Discussion (the rebound height versus drop height graph of your groups data along with a visual line of fit through your data points)
* 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 graphing calculators should be provided in written form as well as orally.
Activity #2: Exploring Non-Linear Relationships - Mathematical Marathon
Time: 7 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:
Relationships
Overall Expectations
By the end of this course, students will:
determine relationships between two variables by collecting and analysing data. J
compare the
graphs and formulas of linear and non-linear relations. J
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. J
organize and analyse data, using appropriate techniques (e.g., making tables and graphs, calculating measures of central tendency) and technology. J
construct tables of values and graphs to represent non-linear relations derived from descriptions of realistic situations. J
construct tables of values and scatter plots for non-linearly related data collected from experiments or from secondary sources; sketch a curve of best fit. J
identify, by calculating finite differences in its table of values, whether a relation is linear or non-linear. J
determine the meaning of negative exponents and of zero as an exponent from activities involving graphing, using technology, and from activities involving patterning. J
represent very
large and very small numbers, using scientific notation. J
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 that 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 4, 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.
Students 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.
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.
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: 6 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 versus
a 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:
determine relationships between two variables by collecting and analyzing data.
compare the graphs and formulas of linear and non-linear relations. J
describe the connections between various representations of relations.
Specific Expectations
By the end of this course, students will:
demonstrate facility with critical numerical skills, including mental mathematics, estimation, operations with integers, and operations with rational numbers. J
distinguish between exact and approximate representations of the same quantity and choose appropriately between them in given situations.
collect data, using appropriate equipment and/or technology. J
organize and analyse data, using appropriate techniques and technology. J
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. J
communicate the findings of an experiment clearly and concisely, using appropriate mathematical forms and justify the conclusions reached. J
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. J
describe, in written form, a situation that would explain the events illustrated by a given graph or the relationship between two variables. J
describe the effect on the graph and the formula of a relation of varying the conditions of a situation they represent.
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, Modelling 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
Accommodation
Provide clean written instructions and steps as needed. Adjust the number of comments required to allow students to fully participate.
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