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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 9 secondary school curriculum. These materials were created by writing partnerships of school boards and subject associations. The development of these resources was funded by the Ontario Ministry of Education and Training. This document reflects the views of the developers and not necessarily those of the Ministry. Permission is given to reproduce these materials for any purpose except profit. Teachers are also encouraged to amend, revise, edit, cut, paste, and otherwise adapt this material for educational purposes.

 

Any references in this document to particular commercial resources, learning materials, equipment, or technology reflect only the opinions of the writers of this sample Course Profile, and do not reflect any official endorsement by the Ministry of Education and Training or by the Partnership of School Boards that supported the production of the document.

 

©Queen’s Printer for Ontario

 

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

 

Activity 1 | Activity 2 | Activity 3

 

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.