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 encouraged to amend, revise, edit, cut, paste, and otherwise adapt this
material for educational purposes.
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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
Acknowledgements
Public District School Board Writing Teams -
Mathematics
Course Profile Writing Team
Myrna
Ingalls, Lead Writer, York Region District School Board
Shirley
Dalrymple, York Region District School Board
Carolyn
Gallagher, Kawartha Pine Ridge District School Board
Mary
Howe, Ontario Association for Mathematics Education
Irene
McEvoy, Peel District School Board
Lionel
LaCroix, Peel District School Board
Christine
Surtamm, Peel District School Board
Reviewers
Bill Clarke, Mark Pankratz, Kelly Searle,
Ottawa Carleton DSB: Angela Con, Kawartha Pine Ridge DSB; Donna Del Re, Peel
DSB; Sandra Emms Jones, Waterloo Region DSB; Gary Flewelling, Ontario
Mathematics Co-ordinators Association;
Ron Lewis, Rainbow DSB; Bob McRoberts, York Region DSB
Lead Board
Peel
District School Board
Allan
Smith, Project Manager
Partner Boards
Kawartha Pine Ridge District School Board,
Ottawa Carleton District School Board, Rainbow District School Board, Waterloo
Region District School Board, York Region District School Board
Associations
Ontario Association for Mathematics Education
(OAME)
Ontario
Mathematics Co-ordinators Association (OMCA)
Unit 1:
Constructing Graphical Models Through Investigation
Time:
35 hours
Unit
Description
Students will gather, analyze, manipulate, and
display data from primary and secondary sources to model and communicate
results about both linear and non‑linear situations. Many contextual
problems will be studied to ensure that students gain depth of understanding
through meeting the same specific expectations in different contexts. Students
will conduct investigations to verify or refute their own conjectures, using
lines or curves of best fit, tables and pattern descriptions. They will
communicate their findings and describe trends. A rich contextual foundation for
subsequent algebraic studies will be built in this unit. Several different
types of technologies will be introduced for gathering, analyzing and
displaying data.
Strands and Expectations
Some specific expectations from the Number Sense
and Algebra, and Measurement and Geometry Strands have been combined with
overall expectations REV.01, REV.02, REV.03 from the Relationships Strand.
Weaving together the expectations of the strands in this way will help students
make connections.
Relationships Strand Specific Expectations: RE1.01, 02, 03, 04, 05, 06, 07; RE2.01, 02, 03, 04, 05, 06; RE3.01, 02, 03, 04
Number Sense and Algebra Specific Expectations: NA 1.01, 03, 04; NA2.01, 02, 03, 04, 05, 06; NA 3.06; NA 4.01, 02, 03
Measurement and Geometry Specific Expectations: MG1.01,04; MG 2.01, 02, 03, 04
Activity Titles
What follows is a suggested sequence, with
timing, for teaching Unit 1. This profile develops only the activities that
depart from traditional pencil and paper skill development. These activities
are designed to help students make sense of mathematics by working through
concrete experiences to develop their understanding of various mathematics
concepts. The need for remediation and further development of skills will arise
from the activities.* Up to 960 of the 2100 minutes have been allotted
for work, as needed, on Cartesian graphing, integers, percents, exponents,
basic algebra, graphing calculator and spreadsheet skills, to name a few. As
the new elementary curriculum becomes fully implemented, the use of this time
will change.
|
Activity 1: |
What's My Style?
Gather, Organize & Display Learning Styles Data (1 variable) |
75 minutes |
|
Activity 2: |
What's Our Class
Profile? Gather, Organize & Display 1 Variable Data |
75 minutes |
|
Activity 3: |
Is There a
Relationship Here? Searching for Two Variable Relationships |
150 minutes |
|
Activity 4: |
What Type of
Relationship Is This? Interpret and Analyze Two Variable Data |
75 minutes |
*Time for: activity completion, other
activities that address the same expectations, building graphing calculator
skills, other Skill Building, quizzing (distribute as needed throughout
activities) 200 minutes
Activity 5: |
A Cagey Problem! Discovering
Linear & Quadratic Relationships between Geometric Measures |
90 minutes |
|
Activity 6: |
A Design Problem.
Factors Affecting Steepness of a Linear Graph |
150 minutes |
|
Activity 7: |
Fold It! Discovering an Exponential Relationship
Practising and
Extending Exponent Skills |
75 minutes |
*Practise and extend exponent skills 225
minutes
|
Activity 8: |
Walk This Way!
Describing Constant & Non‑Constant Speeds in Distance/Time
Relationships |
75 minutes |
|
Activity 9: |
Tell Me a Story:
Interpreting and Analyzing Graphs in Contexts |
75 minutes |
|
Activity 10: |
What Does It Mean?
Steepness as Rate of Change |
75 minutes |
*Time for: activity completion, other activities
that address the same expectations, building graphing calculator skills, other
Skill Building, quizzing (distribute as needed throughout activities) 160
minutes
*Sample Assessment activities (practise performance
tasks form EQAO, OAME, NCTM) or further skill development 150 minutes
*Time to Consolidate Skills 225
minutes
|
Activity 11: |
Testing Roofs: A
Summative Assessment Activity |
150 minutes |
|
Activity 12: |
Testing Skills: Sample
Questions for a Summative Pencil and Paper Test |
75 minutes |
Prior Knowledge Required
See Appendix.
Unit Planning Notes
•
The first
activity is intended to help the teacher discover the range of learning styles in
the class, and to help students better understand themselves as learners.
Teachers might use this information to balance and adjust the types of learning
and assessment activities that they use.
•
Introduce linear
and non‑linear relationships using concrete materials, and use words,
units, and representative letters from a wide variety of contexts, with no x's
and y's until Unit 2.
•
Practise using
spreadsheets to record data and draw graphs, graphing calculators to enter data
into lists and display graphs using the appropriate lists, Calculator Based
Laboratory (CBL) and Calculator Based Ranger (CBR) equipment to gather and
display data.
Note:
Only activity 8
requires the use of graphing calculators and CBL or CBR equipment. It is
expected that this activity will be used some time after September, giving
teachers time to become comfortable with the introductory applications of
technology that this activity requires. Should it happen that graphing
technology is not available to the teacher during Unit 1, it is recommended
that the use of activities 8 & 9 be postponed until later in the course.
Together, these activities present a powerful way for students to develop a
concrete understanding of rate of change and spacial sense involving
interpretation of graphs.
There are many more
opportunities for the use of technology than those suggested in this profile,
and it is possible to find many activities that require no technology. This
profile shows many more sample activities requiring no technology than
requiring it. As teachers get more of the equipment in their schools and become
more comfortable with its use, they may gather more activities that benefit
from the use of technology.
•
Gather examples
of other activities, in different contexts that address the same expectations
as each activity outlined. You may wish to use these as followups, warmups, or
substitutes. New textbooks and publications from OAME/OMCA and others will
contain appropriate examples.
•
Develop a plan
and gather materials appropriate to diagnose and develop prior learning skills
as they are identified throughout the unit. Some students will need remedial
work to perform at the expected standard. 16 hours out of 35 hours have
been earmarked in Unit 1 for skill development. It is to be used throughout the
unit as the need for skills arises within the contexts of the activities.
•
Involve students
whose prior learning skills are not in need of remediation, in extending
questions and alternate extending activities.
•
The relative need
for remedial vs extending experiences may vary according to the group of
students in a class.
•
The concrete
experiences presented in unit 1 are important preparation for the abstract
algebraic treatment of linear relations that will occur in unit 2. Both units
give students opportunities to develop higher level thinking and communication
skills, an expectation prominent in the curriculum policy document.
Teaching/Learning Strategies
There are many types of skills required of
students as they engage in the activities of this Unit. In the teacher
facilitation section of each activity, skills of the following types have been
identified, in bold, as they are needed: numeric skills, measurement
skills, communication skills, analytic skills, use of technology skills,
and collaborative skills. A teacher may find that direct instruction is
needed for any of the skills identified. It is intended that repeated, short,
direct instruction in these skills, within a wide variety of contexts, will
improve student skills in an incremental and natural way.
Teachers will be working diagnostically with
students to determine which students do and which do not need remedial support
on items identified in the Prior Learning section of each activity. Direct
teaching of necessary skills for one group can happen at the same time that
extending and enriching activities are used by another group of students.
Flexibility of timing and structure are needed so that all students are engaged
in meaningful tasks. Time has been allotted for this in the *asterisked time
periods.
Students will often be working in pairs or
small groups, but growing independence is also a goal.
Activities that are suggested as teaching tools
could be used as assessment tools, and vice versa, since assessment activities
should be learning activities.
Assessment/Evaluation
Techniques
When students do open‑ended, multi‑dimensional
work that requires them to perform in a situation which calls for mathematics, it
is not useful to score their work on the basis of right or wrong, alone.
Inviting students to show what they know and explain their reasoning means
assessing a piece of work, or an entire performance, as opposed to a correct or
incorrect solution. Teachers will need to look at the strengths and weaknesses
of the whole piece of work or an entire performance, as it pertains to the
specific Expectations conveyed in the purpose of the task. Work can be scored
holistically, with consistent standards, using a rubric. Rubrics are required
when there is a range of student responses possible and when there is a need
for teachers to be much more precise about criteria for assessment. Several
examples of assessment activities and their scoring rubrics have been included
in this Profile. Rubrics will be the most effective means of measuring student
performance on the Thinking/Inquiry/Problem Solving and Communication and
Applications in unfamiliar settings categories of the Achievement Chart.
Most traditional pencil and paper tests do not
offer students opportunities to demonstrate Level 4 performances. This Profile
includes sample questions for pencil and paper tests that do allow students to
demonstrate Level 4 work.
Resources
Heid, M., Algebra in a Technological World.
Addenda Series. 1995 NCTM
Asp, G. et. al., Graphic Algebra : Explorations with a Graphing Calculator. Key Curriculum Press
Meridian Creative Group, CBL Explorations in Algebra
Murdock, J. and Kamischke, E., Advanced Algebra Through Data Exploration. Key Curriculum Press
MCTM/SIMMS Integrated Mathematics A Modeling Approach Using Technology. Simon & Schuster Custom Publishing, 401 Linfield Hall, Bozeman, MT 59717‑2810
Specht, Jim, More Than Graphs: Activities for TI Graphics Calculators. 1996 Key Curriculum Press
Coxford, A. et. al., Contemporary Mathematics in Context. 1997 Everyday Learning Corp., P.O. Box 812960, Chicago, Il 60681 (ISBN 1‑57039‑475‑X)
Texas Instruments Real-world Math with the CBL System; 25 Activities Using the CBL and TI-82
Texas Instruments Explorations-Modeling Motions: High School Math Activities with the CBR, 1997
MARS
- Mathematics Assessment Resource Service.
Appendix ‑
Prior Mathematics Knowledge That Students Bring to Unit 1
Patterning and Algebra
•
find patterns and
describe them using words and algebraic expressions
•
write an
algebraic expression for the nth term in a numeric sequence
•
complete a table of
values and write words to explain the pattern (from grade 7)
•
use variables to
write equations and algebraic expressions from patterns and complex statements
Data Management
•
collect primary
data using both a whole population (census) and a sample of classmates
•
assess bias in
data collection methods
•
manipulate and
present data using spreadsheets or search databases for information and use the
quantitative data to solve problems
•
construct
frequency tables, stem-and-leaf plots, line graphs, comparative bar graphs,
circle graphs, and histograms, with and without the use of technology, and use
information to solve problems (e.g. extrapolate..., predict...)
•
read and report
information about data presented on the graphs listed above
•
understand the
difference between a bar graph and a histogram
•
know that a
pattern on a graph may indicate a trend
•
from grade 8
geography: construct a variety of graphs, charts, diagrams, and models to
organize information (e.g. graphs that demonstrate correlations between two
population characteristics, such as literacy and birth rates)
•
understand and
apply the concept of the best measure of central tendency and determine the
effect on a measure of central tendency of adding or removing a value
•
make inferences
and convincing arguments that are based on data analysis
Number Sense and Numeration
•
compare and order
fractions, decimals and integers
•
arithmetic
operations including order of operations involving fractions, decimals and
integers
•
solve and explain
multi‑step problems involving simple fractions, decimals, integers,
percents, ratios and unit rates
•
express repeated
multiplication as powers
•
express whole
numbers in expanded form using powers and scientific notation
Unit 1 ‑ Constructing Graphical Models Through
Investigation
Activity 1 ‑ What's My Style?
Time: 75 minutes
Description
In this introductory activity students will
gather information about their own learning styles using a learning styles
inventory (a website reference is provided)
and organize their results using graphs. The data from this activity will be
used for analysis in a number of activities which follow. In addition,
information on students' learning styles will give mathematics teachers
pedagogically useful information about their grade nines.
Strand(s) and Expectations
Strands: Relationships; Number Sense and Algebra
Expectations: This activity reviews expectations RE1.0, RE1.06 and NA1.03 in the context of one variable data in preparation for the work with two variable data that follows.
Planning Notes
The activity described here makes use of the
learning styles inventory “Assessing Your Learning Style”, which can be found
at <. This activity can be adapted for use with
another learning styles inventory, if the teacher so chooses. Copies of the
learning styles inventory are needed for each student in the class. A class
list chart with columns to record each student's scores in each of the two
learning styles categories is needed. A blank copy of this chart could be
prepared on an acetate for use with the overhead projector, or in the form of a
wall chart to be posted in class.
The student's scores would be
compiled and computed by the student as follows, in this sample chart:
|
Students |
List
A scores |
List
B scores |
|
Andrew |
14 |
9 |
|
Thansha |
6 |
12 |
|
Iris |
7 |
15 |
|
Class total scores |
27/60 |
36/63 |
|
Class total scores as percentages |
45% |
57% |
A second chart to
tally students’ preferred learning styles should also be prepared ahead of
time. A sample is provided.
|
|
Tally |
Total Number of Students |
Percentage of Students |
|
stronger auditory |
|
|
|
|
stronger visual |
|
|
|
|
similar strengths |
|
|
|
|
class totals |
|
|
|
The teacher should become familiar with the
learning styles inventory ahead of time in preparation for any questions which
may arise.
Prior Knowledge Required
Percent calculations, construction of circle
and bar graphs.
Teaching/Learning Strategies
Student
Activity: Students will complete the learning styles inventory and tally their
scores for each of the different learning styles. These results will be copied
onto the class list. Once the class results are compiled, the students will
discuss ways to organize the learning styles data for the entire class (e.g.
bar graph, circle graph) and together graph them.
Teacher Facilitation:
• introduce learning styles and how information about
learning styles can be used by students and their teacher.
C
introduce
the activity and inform the class that this information will be used in subsequent
activities. Also provide numeric
skills instructions for students to evaluate and score their own learning
styles inventory results and assign this task to the students. (5 minutes),
•
circulate
around the classroom, assisting students in completing their learning styles
inventories as needed (20 minutes),
•
compile
the learning styles scores on the class list chart (10 minutes)
•
review/demonstrate
numeric skills of percent calculation, and graphical communication
skills of how to construct bar and circle
graphs by constructing graphs of the entire class’s results as follows: (20 minutes)
1)find the class's total scores in each
category
2) calculate each total as a
percentage of the maximum number of responses possible in each category
3) tally the numbers of
students whose learning style preferences lie in each of the three categories
(auditory - higher on list A, visual - higher on list B, tactile-kinesthetic -
approximately the same on lists A and B)
4) calculate each score in 3)
as a percentage of the entire class in preparation for constructing a circle
graph
5 )point out the difference
between the class total scores as percentages and the most prominent styles as
percentages. Relate these to mean and mode and point out that in this case mean
takes into account all of the data whereas mode takes into account only part of
the data.
6) construct a bar graph to
show the relationship between the response rates (as percentages) for the list
A and B items within the class. Then construct a circle graph to show the
distribution of students amongst the
three learning styles categories.
• assign students the following task for homework (5
minutes) Add whatever skills questions that seem appropriate. Emphasize
to students that they should demonstrate their numeric, communication, and
analytic skills in completing homework.
Student Activity:
1) Construct a bar graph to compare your response rates for the list A and B items and answer the following question: “How does your learning style profile fit that of the class as a whole?”
2)
Form a hypothesis about
the similarity of your own learning style profile with that of a family member
or friend, and test your hypothesis by administering the learning styles
inventory to that person so that the results can be compared.
Assessment/Evaluation
Techniques
None.
Accommodations
The teacher should be prepared to assist
individual students who have difficulty performing these tasks. Parts of the inventory
may be completed as a whole class exercise to get the students started and
individuals who have difficulty with the text may be grouped with other
students for assistance. If a student has difficulty with a question it may be
left blank.
Resources
The teacher may wish to do some research on
learning styles and how to incorporate this into their teaching. The website
cited above is a good start. Another useful reference is Teaching and
Learning Stlyes: Celebrating Differences, OSSTF, 1986.
Activity 2 - What's
our class profile?
Time:
75 minutes
Description
In this activity, students will continue to gather information about
themselves and the class. Following a brief review of graphing, the students will
graph one item from the class data and draw an inference about the class from
this.
Strand(s)
and Expectations
Strand: Relationships
Specific expectations: The activity relates to expectations
RE1.04, RE1.05, and RE1.06, in the context of one variable data, in preparation
for the work with two variable data that follows.
Planning
Notes
Needed are copies of the personal traits questionnaire found at the end of this activity, and a class list chart with ten columns, to record students' responses for each of the eight questions from the questionnaire. Part A of the questionnaire will be completed during activity 2 and part B, during activity 3. As in the first activity, this chart could be prepared on an acetate, or in the form of a wall chart. Prepare examples of a comparative bar graph, and a histogram, and calculations for measures of central tendency. The teacher should assist in making available the materials students will need to present their group graph to the class (e.g., chart paper, markers, overhead acetates, compasses, protractors, and calculators). Prepare a homework task where students will interpret information from various types of graphs.
Prior
Knowledge Required
Using mode, organizing and graphing data.
Teaching/Learning Strategies
Student Activity: Discuss responses from the previous
day's homework. Students will complete part A of the personal traits
questionnaire and have their answers recorded on the class list by the teacher.
Working in groups, students will graph class data and look for trends.
Teacher Facilitation:
• Lead
the class discussion of the previous day's homework on learning styles, helping
students with their communication skills. Conclude this by having students
relate the different learning styles to specific behavior patterns that are
familiar to them. (e.g., an auditory learner may prefer radio to TV) (10
minutes)
• Get
the students started on part A of the personal traits questionnaire.
• Record
students' answers for the personal traits questionnaire as soon as they have
completed it (15 minutes).
• Provide
a review of the communication skills, of the use and construction of
graphs, and the numeric skills of calculation of mean, median, and mode
from the raw data, and the analytic skill of the appropriate use of each
of these measures. (This should be a quick review of grade 8, not a lesson.)
(10 minutes)
Brainstorm ways in which to
demonstrate collaborative skills of working effectively in a group. The
following guidelines are suggested:
1) assign specific roles to each member of the
group. For the current activity these could be coordinator, person who
calculates, graphics designer, and writer. (Assign roles to special needs
students first to encourage their participation)
2) each person contributes to their group's
work
3) listen carefully to what other group members
say and ask questions when needed
4) help and encourage other members of your
group
5) keep working until everyone in your group
understands your results and can explain them fully.
• Organize
the class into ten groups and assign each group the task of graphing the class
results for one of the questionnaire items. Direct students to use numeric
skills to calculate an appropriate measure of central tendency for their
data, and to use communication and analytic skills to prepare a
description of any trends or suggestions about the class indicated by their
graph and to justify their conclusions.
• Circulate
within the class and help individual students as needed. (15 minutes)
• Have
the groups use their communication skills to present their completed
graph to the class and share their conclusions. (15 minutes)
• Have
each group use their analytic skills to
complete a single evaluation of their teamwork
• Introduce the homework (5 minutes).
Assessment/Evaluation
Techniques
Observations of student's team work and organizational skills could be recorded at this time. Students could be asked to reflect on their group’s work using the following questions.
Group Evaluation
1. What made your group effective as a team?
2. What could you do differently to work more effectively as a team in the future?
3. Assess your group’s efforts under the following criteria. Place a checkmark in the appropriate space for each.
Rarely Sometimes Usually Always
a) Every
person contributed to the group’s work. ____ ____ ____ ____
b) Every
person listened carefully to other group members. ____ ____ ____ ____
c) We
asked questions when needed. ____ ____ ____ ____
d) We
helped and encouraged each other. ____ ____ ____ ____
e) We
kept working until everybody in the group understood ____ ____ ____ ____
the results and could explain them fully.
Accommodations
Assist individuals as needed and provide examples that students can
use as a reference. The inventory can be read to students and responses can be
scribed.
Resources
There are many examples of the different types of graphs in current textbooks, magazines, and newspapers for students to find and bring in.
Appendix
Personal Traits Questionnaire
Part A: Answer the following questions in the spaces provided.
1. What is your gender, male or female? __________
1. What is your age in months? Round down to the nearest whole month. __________
1. On average, how much time in hours do you spend watching TV per day? __________
1. How many children (including yourself) live at your home? __________
1. On average, how many hours to you sleep each night? ___________
1. How much time in minutes, on average, did you spend on your homework each school night during your previous semester at school? __________
1. On average, how much time in minutes do you spend listening to music each day? __________
1. On average, how many minutes do you spend reading per week for your own enjoyment? __________
1. How much time in hours, on average, do you spend per week doing physical activities such as exercise, active games, sports, etc. Round off to the nearest half hour. __________
1. On average, how many hours do you spend per week doing chores or other household tasks? __________
Part B: You will complete this
section with the help of a partner.
1. Hand span is the distance from the tip of your thumb to the tip of your baby finger when your hand is stretched out as far as possible. What is your span of your right hand in centimetres? __________
1. Forearm length is measured from the crease on the inside of your elbow to the crease on the inside of your wrist. What is length of your left forearm in centimetres? __________
1. Arm span is the distance from finger tips to finger tips when your arms are stretched out horizontally as far as possible. What is your arm span in centimetres? __________
1. What is the length of your right foot in centimetres? __________
1. Stride length is the distance from the tip of your toe on one foot to the tip of your toe on the other when you take a normal walking step. What is your stride length in centimetres? __________
1. What is your height in centimetres? _________
Activity 3: Is There a
Relationship Here? Searching for Two Variable Relationships
Time: 150 minutes
Description
This activity introduces the students to
two variable relationships and the possibility that some sets of data will not yield
a relationship. Students will use data collected in the previous activity and
newly collected data to form an hypothesis, find relationships between
variables, make scatter plots, choose appropriate axes and draw a line or curve
of best fit. The graphing calculator projection panel can be used as a
demonstration to display data and underline the need for decisions concerning
scale.
Strand: Relationships
Specific Expectations: RE1.01, RE1.04, RE1.05, RE1.06,
RE2.02, RE2.0, RE2.05
Planning Notes
The teacher will supply metre sticks, measuring tapes and the activity outline, "Is There a Relationship Here?" Each student should have a copy of the Personal Traits Questionnaire for the previous activity.
Emphasis will be placed on students exploring and constructing relationships between sets of data. The data from activities 1 and 2 should be displayed on a bulletin board or wall or organized using a spreadsheet and made available to the students. The class set of data must not identify individual students by name to avoid any discomfort that may stem from a discussion of physical characteristics.
It is appropriate to introduce the notion that there are formal, mathematical ways to find a line of best fit. This would be an excellent springboard for discussion about criteria for such a line and the need to communicate these criteria clearly. Teachers may wish to use Fit-ness from the Harvard Balanced Assessment before or during this activity to introduce students to the criteria for choosing a line of best fit.
The view screen for the graphing calculator is a useful device to
illustrate the scatter plot of the data and generate discussion around the
‘viewing window' using technology and by hand with the class. Teachers will
instruct students on how to use the list, stat plot and window functions on a
graphing calculator.
Prior Knowledge
Required
Patterning and Algebra Grade 7 and 8: Recognize patterns and use them to make predictions.
Data Management and Probability Grade 7 and 8: Manipulate and present data using spreadsheets; identify and describe trends in graphs, using informal language to identify growth, clustering, and simple attributes; know that a pattern on a graph may indicate a trend.
Grade 8 Geography: Constructs a variety
of graphs, charts, diagrams and models to organize information (including
correlations between two population characteristics).
Teaching/Learning
Strategies
Student Activity: Students will work in pairs to
measure their hand spans, forearm lengths, arm spans, foot lengths, stride
lengths and heights, to complete part B of the personal traits questionnaire.
Students should record their data using the data base or wall chart started
earlier.
Teacher Facilitation: The teacher will group the students
in same gender pairs and distribute the materials for this measurement
activity. While students are working, the teacher will circulate to assist
students with the measurement and collaborative skills needed to complete the task.
Student Activity: Students will work with a partner to
carry out the following, “Is there a Relationship Here?” Upon completion of the
activity, students will present their findings to the class, reporting on
questions 1, 3, and 5.
Is There a Relationship Here?
A relationship between two sets of data can help you to make predictions that lead to informed decisions about important matters such as setting up a small business or the effectiveness of a new medicine. You will be looking for connections between two of the data sets that you have collected. For example, can foot size be used to estimate a person's height? Is the amount of time that you spend watching TV related to the amount of time that you spend on homework?
1. What question about the sets of data that have been collected would you like to answer? Write this question down and then form a hypothesis. The hypothesis is your best guess, it answers the question , "What do you think the relationship will be?"
2. Create a scatter plot of the relationship in #1. Answer these questions:
a. What are your variables?
b. On which axis will you place each variable?
c. What scale will you use for each axis?
3. Does your scatter plot show a definite relationship between the data sets? If so, describe the relationship. Consider these questions:
a. Does the scatter plot reveal an obvious pattern?
b. Do the points seem to describe a straight line or a curve?
c. Does the value of one variable increase or decrease as the value of the other increases?
4. Make a list of the criteria for drawing a line of best fit that were identified by the class. Using the list of criteria, draw a line or curve of best fit through the points on your scatter plot.
5. Describe the relationship between your data sets. Consider these questions:
a. What are the variables that are represented in this relationship?
b. Is this relationship linear or non linear?
c. Does the value of one variable increase or decrease as the value of the other increases?
d. Is the rate of increase or decrease low? high?
e. Does the relationship support your hypothesis? If not, create a new hypothesis. Explain your reasoning.
f. Can you think of an equation that would describe this data?
6. a. Plot your data using the list
and stat plot functions of your graphing calculator.
b. Set the window and explain your
choices.
c.
Describe and explain differences
between your hand scatter plot and the calculator’s image.
Teacher Facilitation: The teacher will set the stage for this
activity by describing the following scenario:
Marketing analysts study data
gathered from a population to predict social trends so that they can decide
whether or not a product or service will be in demand in a certain area. It is
often necessary to know if there is a relationship between two variables or
not.
Brainstorm
with the whole class to develop analysis skills. Ask students to
identify sets of data that may be related (e.g., relationship between hand span
and foot size, amount of time spent watching TV and score for a specific
learning style, arm span vs height). Choose one example, formulate a hypothesis
and discuss how you would prove or disprove it using the class data. Discuss
ways to represent the data visually. Review the graphical communication
skills involved in plotting points on a grid to create a scatter plot. The
class should also consider the decisions to be made about scale and whether or
not it is appropriate to use a break in an axis. An arbitrary decision
regarding the placement of variables on the axes must also be made by the
students since dependence and independence will not be obvious at this stage.
The goal of this activity is to
identify both linear and non linear patterns, and to realize that some sets of
data are not related in an obvious way. The students' communications and
analysis skills will dictate how much guidance is needed in formulating an
hypothesis. The teacher will direct students as needed, to use a scatter plot.
During the activity, the teacher
will circulate and prompt. When most students have worked on question 3, the
teacher will lead a whole class discussion to further develop analysis
skills. Students will be asked to
list criteria for a line of best fit (e.g., distance from the points,
number of points above and below the line, number of points on the line,
whether or not the origin is on the line, ignoring outliers). The students
should not draw a line of best fit before the class discussion that identifies
the criteria for this task. Students can judge each others work to determine the
line of best fit. Some students may extend their communication skills by
writing an equation to describe their line, but this is not necessary at this
point.
As students use their communication
skills to present their findings, the teacher will ask probing questions as
needed to ensure that the key concepts described in the activity description
emerge, using analysis skills (e.g., How does the trend shown in your
graph compare to your hypothesis?).
Assessment/Evaluation
Techniques
The teacher should make observations regarding the students' independence and teamwork skills using the rubric in Appendix 1. Observational assessment can also be used together data on students’ knowledge, and problem solving skills. This would be the ideal time to provide formative assessment feedback to students regarding their communication skills using “Rubric for Assessing Student Presentations”.
Appendix
Rubric for Assessing Student Presentations
|
Formulate
a hypothesis associated with a relationship between two variables RE 1.01 |
C iidentifies the variables, with help C Fformulates a hypotheses that does not describe the relationship |
C Iidentifies the variables, with help C Fformulates a hypothesis that describes the relationship, with some help |
C Iidentifies the variables C Fformulates a hypothesis that describes the relationship |
C Iidentifies the variables C Fformulates with confidence a hypothesis that describes the relationship |
|
Describe
trends and relationships observed in data RE 1.05 |
C Mmakes correct inferences from data, with much teacher support C Eexplanations of the differences between the inferences and the hypothesis unclear and incomplete C Hhas difficulty discussing any relationship that exists in the data |
C Mmakes correct inferences from data, with some prompting C Eexplanations of te differences between the inferences and the hypothesis unclear C Ddiscusses only one relationship that exists in the data with minor errors |
C Mmakes correct inferences from data with minor errors C Eexplains the differences between the inferences and the data C Ddiscusses only one relationship that exists in the data |
C Mmakes correct inferences from data C Cclearly explains the differences between the inferences and the hypothesis C Ddiscusses other relationships that might exist in the data |
|
Communicate
the findings of an experiment RE 1.06 |
C Ccommunicates unclearly C Uuses little or no justification |
C Ccommunicates results with some inappropriate forms C Uuses faulty logic to justify conclusions |
C Ccommunicates clearly using appropriate forms C Jjustifies relationship with respect to the class |
C Ccommunicates and justifies clearly and concisely C Ggeneralizes relationship beyond the context of the classroom |
C
Activity 4:
Interpreting and Analyzing Two Variable Data
Time: 75 minutes
Description
This activity builds on the students' ability to identify linear and
non linear relationships using tables and graphs based on primary and secondary
data. Data will be selected by the teacher to allow for the use of finite
differences (particularly the first differences) in the classification of
relationships as linear or non linear. Dependence and independence, and the
four quadrants of the Cartesian plane will be introduced in context as needed.
Strands: Number Sense and Algebra,
Relationships, Analytic Geometry
Specific Expectations: NA1.01, RE1.04, RE2.01B, RE2.02B,
RE2.04B, RE2.05B, RE3.03B, AG1.01
Planning
Notes
A variety of data tables that represent linear (with positive and
negative slopes) and non linear relationships must be prepared for this
activity. Current (secondary) data could be gathered from the Internet (for
example, at www.statcan.ca ), but keep in mind that this activity requires data
with a readily apparent relationship between the variables. Data can be
generated easily using simple formulas from science (distance vs time, speed vs
time, volume of a gas vs temperature, voltage vs current, mass vs volume,
conversion between Fahrenheit and Celsius temperature scales) and other
everyday applications (loan balance, bank account balance, sales tax gas consumption,
perimeter vs area, metric conversions).
The calculation of first differences will require the subtraction of 2nd minus 1st, 3rd minus 2nd, etc, values of the dependent variable. The subtraction of integers should be reviewed here to facilitate finding finite differences when negative integers are used. A number of the tables should contain data with negative values and/or a negative slope.
Linear and non linear graphs are needed for the second half of this activity. The graphs should include grid lines or dots and the scales must be easy to read.
The concepts of dependence and independence may be introduced when the students are choosing a variable to start constructing their tables of values. A class discussion about whether or not it matters which variable is chosen in order to read the value of the other variable, and when it matters in the design of an experiment could clarify these issues for the students (e.g., If a person was traveling at 100 km/h on a highway, the distance traveled would depend on the time traveled).
Prior Learning
Number Sense and Numeration Grade 8: Add and subtract integers
Patterning and Algebra Grade 7 and 8: Recognize patterns and use them to make predictions
Data Management and Probability Grade 7 and 8: Identify and describe trends in graphs, using
informal language to identify growth, clustering, and simple attributes; know
that a pattern on a graph may indicate a trend.
Teaching/Learning
Strategies
Student Activity: Students will carry out the following activity in small, heterogeneous ability groups.
Interpreting and Analyzing Two Variable Data
1. A
stone is dropped off the top of a tall building. Two sets of
data were recorded and are shown below.
i)
|
Time (s) |
Speed (m/s) |
First Difference in the speed (new value - previous value) |
|
0 |
0 |
<10-0 = < < < < |
|
1 |
10 |
|
|
2 |
20 |
|
|
3 |
30 |
|
|
4 |
40 |
|
|
5 |
50 |
|
ii)
|
Time (s) |
Distance (m) |
First Difference in the distance fallen |
|
0 |
0 |
< < < < < |
|
1 |
5 |
|
|
2 |
20 |
|
|
3 |
45 |
|
|
4 |
80 |
|
|
5 |
125 |
|
a) Using the entries in the first two columns of each table,
graph each relationship on a separate grid. When choosing your axes, use time
as the independent variable.
b) What is the speed of the stone and the distance fallen when
the time is 2.5 seconds? Explain your reasoning, including the assumptions you
made.
c)
When
would the speed of the stone be 80 m/s? Explain your reasoning.
d)
When
would the distance travelled by the stone be 245 m? Explain your reasoning.
e)
Which
relation is linear?
f)
For
the linear relation, form an equation that models the data.
g)
Complete
the third column for each table. Contrast the differences formed in the two
tables.
h)
How
does the type of differences in the table relate to the shape of the graph?
i)
How
does the constant first difference for table i) relate to the graph and to the
equation for this
relationship?
2. The
air temperature on a cold, clear night is measured at the beginning of each
hour.
|
Time (hours) |
Temperature ( °C) |
First Difference in the temperature |
|
0 |
‑2 |
< < < < |
|
1 |
‑5 |
|
|
2 |
‑8 |
|
|
3 |
‑11 |
|
|
4 |
‑14 |
|
a) Calculate
the first difference for the temperature.
b)
Do
you think that this data describes a straight line or a curve? Explain your
reasoning.
c)
Create
a scatter plot of the data and draw a line or curve of best fit. Does the graph
support your answer to part b?
d)
Find
an equation to model the data.
e)
When
would the temperature be ‑20 °C? Explain how you arrived at your
answer.
f)
What
would be the temperature if the time were 2.5 hours? Explain two different ways
to answer
this question.
3. A
cold drink is left out in the sun on a warm day. The graph of its temperature
vs time follows.
a) Is
this graph linear or non-linear?
b) What
pattern do you think will exist in the first difference for this relationship?
Explain your reasoning.
c) Use
the table to record the values of time and temperature for five points on the
graph and calculate the first difference for temperature.
|
Time (minutes) |
Temperature (EC) |
First Difference |
|
|
|
< < < < |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
d) Do the first differences support your
answer to part b? Explain why or why not.
4. Tara
wishes to keep track of the number of kilometres that she travels on a basketball
team road trip. She forgets to begin making observations until she is 50 km
from home. She begins timing the trip (time equals zero hours) at the 50 km
point. The graph below provides distance vs time data for Tara's trip.
a)