UNIT THREE
RELATIONS AND FUNCTIONS
MATH 421A
23 HOURS
Revised may 3, 00
UNIT 3: Relations and Functions
52
Previous Knowledge
With the implementation of APEF Mathematics at the Intermediate level, students should be able
to:
- Grade 7, 8 and 9 - graph linear equations using a table of values
- Grade 7, 8 and 9 - determine if an ordered pair is a solution to a linear equation
- Grade 7 - given a graph, create a story which describes the graph and label the axes
- Grade 8 and 9 - interpret both linear and non-linear graphs
Overview:
- Review the Cartesian coordinate plane
- investigate binary relations
- investigate linear relations and the line of best fit
- investigate non-linear and general relations
- study functions and their applications
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SCO: By the end of grade
10 students will be expected
to:
A2 analyze graphs or charts
of situations to identify
specific information
Elaborations: Instructional Strategies/Suggestions
Cartesian Coordinate Plane (5.1)
Students should be introduced to this plane with respect to axes,
quadrants, origin and points on the plane defined using ordered pairs.
Binary Relations
Binary Relations are relationships between two sets of data written
in a number of different ways - ordered pairs, table of values, graph,
words, equations and arrow diagrams. The domain of a relation
represents the first elements while the range represents the second
elements. The domain is usually known as the independent variable
and the range is the dependent variable.
Example:
The following represents the same relation in a number of ways:
Ordered Pairs - (-2, -4), (0, -2), (2, 0), (4, 2)
C1 gather data, plot the data
using appropriate scales,
and demonstrate an
understanding of
independent and
dependent variables and
domain and range
Table of Values -
Graph -
Words - The difference of two numbers is two.
Equation -
x-y = 2
Arrow Diagram -
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Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Cartesian Coordinate Plane (5.1)
Pencil/Paper
The Writing Coordinates activity on page 208 in
Mathpower 10 is an excellent introduction to the Cartesian
Coordinate Plane. As well the Writing Statements activity is a
good lead-in for the entire chapter
Binary Relations
Activity
Using a spring, mass hanger and slotted masses (Hooke’s
Law Apparatus ; ask your friendly neighbourhood Physics
Teacher) complete the following exercises:
a) complete the table
b) graph the data
c) determine the domain and range
d) draw an arrow diagram
e) explain the relation in words
f) write an equation
Cartesian Coordinate Plane
Mathpower 10 p. 208
Use a metre stick to measure from the table top (or the floor)
to the slotted mass holder, record this on scrap paper. Put the
.100 kg mass on the holder and re-measure from the table top
to the slotted mass holder the difference in the two
measurements is the stretch for that mass and is recorded in
the table.
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Algebra: Structure and Method,
Book 1
Binary Relations
Mathpower 10 p.212 #1-6
p.214# 1-17 odd, 17,
20, 21
Math 10
p.266-273, 312-318
Worthwhile Tasks for Instruction and/or Assessment
Binary Relations (cont’d) (5.1)
Pencil/Paper/Technology
In the L1 column are the amounts you were going over the
speed limit.The fines for speeding are shown in the L2 column
of the following table. Graph the binary relation using the TI83 and determine the domain and range.
Stat 1:Edit if L1 and L2 are not cleared then:
cursor up using >to the top of L1 press clear enter repeat
for L2
Now enter the data
now press
shown
2nd Y= Stat Plot 1: enter
Select features as
set the window dimensions as shown
now press graph
This yields a discrete graph( non-connected points).
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Suggested Resources
SCO: By the end of grade
10 students will be
expected to:
C3 solve problems using
graphing technology
C5 construct and analyze
graphs and tables
relating two variables
C10 sketch graphs from
words, tables and
collected data
C17 determine if a graph is
linear by plotting in a
given situation
C41 use interpolation,
extrapolation and the
equation to predict
and solve problems
Elaborations- Instructional Strategies/Suggestions
Linear Relations and Line of Best Fit (5.2)
Invite students to discuss their ideas on the meaning of linear.
Hopefully they will see the word line within the word.
The student is invited to explore relationships of data that are linear.
The problems that students must solve are:
a)given the equation, generate a table of values ,draw a line through
these points and determine the domain and range. See Mathpower 10
Ex. 1 p.217 & Ex. 2 p.218. Students should see that equations of degree
one are linear. Again use Pencil/Paper and the TI-83
b) given data or generate data , plot the data and get the line of best fit.
See Mathpower 10 explore exercises, p.216-7 This can be done using
Pencil/Paper and the TI-83. The first explore exercise is perfectly linear
so that there is no judgement needed to find the line of best fit. The
second explore exercise is more realistic of the type of data that is
typically collected during any experiment in science, economics,
sociology and many other fields.
Perhaps a logical order is to go from contrived problems where the
data is perfectly linear to show more realistic problems where students
actually have to make a judgement as to where to draw the line. Not
until this more realistic data is dealt with, will the idea of line of best fit
have any meaning to students.
Once the line of best fit is drawn, students will be expected to
interpolate and extrapolate(Mathpower 10 p.216) and as well determine
the domain and range.
Interpolation - finding the x or y value of an ordered pair that falls
between the given ordered pairs on a graph
Extrapolation - finding the x or y value of an ordered pair that falls
outside the given ordered pairs on a graph
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Worthwhile Tasks for Instruction and/or Assessment
Linear Relations and line of best fit (5.2)
Pencil/Paper/Technology
Determine the domain and range of the following graph.
This is a continuous graph.
(unbroken line)
Suggested Resources
Linear Relations and line of best fit
Mathpower 10 p.219#1-19 odd,
21-29,31
Note: # 28 generates a good scatter
plot. Use of the TI-83 is
recommended for #21-24.
Math 10
Answer
Pencil/Paper/Technology
In a certain experiment the following data was collected for
men. Draw a scatter plot and determine the line of best fit.
Then, using extrapolation, predict how long it would take for
an 80 year old man to run the 100m. Then, using
interpolation, predict how long it would take a 50 year old
man to run the 100m.
If you use the TI-83
Stat 1:Edit then enter the data
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p.274-299
Worthwhile Tasks for Instruction and/or Assessment
Linear Relations and Line of Best Fit (cont’d) (5.2)
to draw the scatter plot
2nd Y= 1:Plot 1
and set the preferences to
To get the line of best fit Stat < calc 4:Lin Reg 2nd L1, 2nd
L2, vars < Y vars 1: Function 1: Y1
enter
then press Graph If you then press Trace and > so that
you can cursor along till x . 80 and x . 50 then you can see
the answers of about 30s and 21s.
Computer Spreadsheet/TI-83
Mathpower 10 p.222-3 provide some examples where
spreadsheets can used.
Research Project/Presentation(oral or written)
Do research on one of the following;
< price of pizza based on number of toppings
< cost of mailing a package based on mass
< cost of renting a car based on the days of rental
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Suggested Resources
Linear Relations & Line of Best Fit
SCO: By the end of grade
10 students will be
expected to:
C36 describe real world
relationships depicted
by graphs, tables of
values and written
descriptions
Elaborations- Instructional Strategies/Suggestions
Non-Linear Relations (5.3)
Allow students to explore problems where the data doesn’t form a
linear relation.
Many examples can be found from various areas; science, business,
etc. that can allow students to discover that relations in the real world
can be non-linear as well.
Shown below is one example.
A ball is dropped from the top of a tall building. The distance fallen is
shown in the table below as a function of time.
a) plot the data and draw a smooth curve through the points
b) from the graph, determine the distance fallen after 5 seconds
c) from the graph, determine the time required to fall 100m
C37 determine the equation
for the curve of best
fit with/ and without
technology
C54 sketch graphs from
words and tables and
from real data
collected in
experiments
Before any interpolation can be done the points must be joined with a
smooth curve. If you choose to use the TI-83 these are the steps if the
data is in L1 and L2:
Stat < 5:QuadReg
2nd L1 , 2nd L2 , vars < y vars 1:Function 1:Y1
Enter
Trace > and cursor along the curve to yield
graph
It takes about 4.5s to
fall 100m.
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Worthwhile Tasks for Instruction and/or Assessment
Non-Linear Relations (5.3)
Research/Presentation
Find information on one of the following using the Internet or
other sources( print, people in community). Draw a scatter
plot and sketch a curve which best fits the data. Justify your
answer.
< population on PEI over the past 100 years in 10 year
intervals.
< average monthly temperature for a year.
< average temperature for July for the past 20 years
< tons of lobster caught per season over the past 20 years( do
this for both the North and South side fisheries)
< number of acres of potatoes grown on PEI each year over
the past 50 years
< annual number of tourists to visit PEI each year over the
past 30 years
< number of acres of woodland on PEI each year over the past
40 years
Suggested Resources
Non-Linear Relations
Mathpower 10
p.225 #1-5,8, 12a,b
13
www.gov.pe.ca/af/agweb
http://www.wri.org/wri/trends/
climate1.html
Here is a sample table from the agriculture website. Graph
columns 2, 3 and 4 vs column 1 to see if any are nonlinear. Do some research to try to explain some of the trends
are showing up on the graphs ( weather, good crops in other
provinces, countries, etc.)
Worthwhile Tasks for Instruction and/or Assessment
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Suggested Resources
Non-Linear relations (cont’d) (5.3)
Pencil/Paper/Technology/Manipulatives
Use the following diagrams to complete the given table,
then plot the data and draw a smooth curve through the
points. Predict the surface area of a cube that has a side length
of 8 cm. If the surface area of a cube is 80 cm2 , what is the
side length of the cube?
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Problem Solving Strategies
Math Power 10 p.247 #4,9,12
SCO: By the end of grade
10 students will be
expected to:
C79 represent and analyze
relationships, tables,
verbal rules, equations
and graphs
Elaborations- Instructional Strategies/Suggestions
General Relations (5.4)
Graphs < words
Challenge students to interpret real world graphical information.
Interpretation will be in communication form (oral or written). Ask
students to describe various scenarios to the other members in their
group. Some of the graphs have no labels on the axes and allow students
to interpret the situation creatively
An example is shown below:
Describe an activity that this graph could be describing.
Words < graphs
Students will also be invited to sketch graphs of information which
has been described in words.
A group of young people go to an amusement park and take a ride on a
ferris wheel. Sketch a graph representing the group’s height above the
ground from the moment they get on the ride vs time.
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Worthwhile Tasks for Instruction and/or Assessment
General Relations (5.4)
Group activity
Create a graph with the axes labelled (Similar to ex. 4, p.230
mathpower 10). and then interchange these with the other
groups. Now each group must try to develop a scenario to
match the graph.
Ranger Activity (See Appendix to get the Ranger Program)
Use the Ranger Program to allow students to match their
distance from a wall to a given graph on the TI-83.
Example: press Pgrm choose Ranger press enter and enter
Choose 2:set defaults
These settings will be fine
Real Time Yes
Time(s)
15
Display
Dist
Begin on
Enter
Smoothing None
Units
Meters
Note to teachers: Before a standard graph is shown, have
someone do a walk in front of the CBR and have a discussion
analyzing the graph. This could be done a number of times to
see various types of graphs.
To get a standard graph
now cursor up to the top of the screen and over to Main
Menu and press enter
Now press 3:Applications and 1:Meters 1:Dist match
now press enter and a graph is displayed.
Set the CBR on a table and with the TI-83 connected to the
view-screen and aim the CBR directly at a wall. Position a
student in front of the CBR, press enter and walk towards
and/or away from the CBR so that it matches the graph
shown.
To try it again, press enter 2:New Match.
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Suggested Resources
General Relations
Mathpower 10 p.229 #1-4, 8, 10-12
Math 10 Independent Study Guide
p. 63 # 6
Mathpower 10 Test Generator CD
rom
This student needs more practice.
The solid line is the CBR generated
graph; the dotted line is the student’s
attempted match. Encourage students
to discuss how the walk could be
improved on.
SCO: By the end of grade
10 students will be
expected to:
Elaborations- Instructional Strategies/Suggestions
Functions (5.5)
Invite student groups to read and discuss p.232-233. Key terms they
should become familiar with are:
< function
C44 apply function
notation and language
< dependent variable
< independent variable
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Worthwhile Tasks for Instruction and/or Assessment
Functions (5.5)
Use the vertical line test to see if the following are
functions.
Suggested Resources
Functions
Mathpower 10 p.234 #1-7, 9-16, 18,
20, 22-28
Note to teachers:
To draw a parabola that opens to the right.
In Y1 enter
In Y2 you can enter
or with the cursor beside Y2 press
! vars < yvars 1:Function 1:Y1
Answer not a function
Answer
It is a function
Journal
Use your research skills to write a rule used to determine
the cost of mailing a parcel as a function of the mass of the
parcel.
Enrichment
Simplify:
a) If f(x) = 3 ! 4x, write and simplify f(a+2)
b) If f(x) = 2x2 + x ! 4, write and simplify f(2b! 5)
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Note to teachers:
To draw a vertical line:
press 2nd draw 4:vertical
now press < or = to position the line
where you want it. Once it is where
you want, press enter
SCO: By the end of grade
10 students will be
expected to:
C44 apply function
notation and language
Elaborations- Instructional Strategies/Suggestions
Applications of Linear Functions (5.6)
Many situations in real life yield linear functions having constants of
proportionality which represent the steepness of the lines. In the next
unit we will be looking at this constant in more detail and calling it
slope.
The students will be asked to investigate situations which fall into two
categories:
C53 model real world
phenomena using
linear/quadratic
functions
C67 describe general
properties of linear
functions and
visualize their graphs
Direct variation
Where the graph of the function is a straight line passing through
the origin. (Ex. A job where one earns straight commission, distance
travelled vs time, amount earned vs hourly rate of pay, )
Partial variation
Where the graph of the function is a straight line that doesn’t pass
through the origin. (Ex. A job where one earns base salary plus
commission)
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Worthwhile Tasks for Instruction and/or Assessment
Applications of Linear Functions (5.6)
Pencil/Paper/Technology
Direct
Harry rented a carpet cleaner for $12 per day. Let d
represent the number of days the machine was rented. Let C
represent the cost.
a) make a table of values for 5 days
b) draw a graph of C vs d
c) write an equation for the function
d) how much would it cost Harry to rent the machine for 8.5
days?
Pencil/Paper/Technology
Partial
A school council wants to sell school T-shirts. There is a
charge for having a digital master of the school logo made.
From the graph below find:
a) the cost of the digital master
b) the cost of having 100 T-shirts made
c) the cost of having 200 T-shirts made
d) the cost of a T-shirt
Journal
What is your understanding of the differences between direct
and partial variation?
Journal
In your own words, explain the constant of proportionality.
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Suggested Resources
Applications ...
Mathpower 10 p.238
Direct #1-3,17,23
Partial # 16, 20, 21
Math 10
p.290-299
Extra worthwhile tasks for Page 12 General Relations:
From the following graphs, choose the graph that best describes each of the following situations. Explain
the choices that you have made.
1) John’s ability to maintain his pace running up hill.
2) the amount of daylight depends on the time of year.
3) The cost of a taxi-cab is $2 plus $1 per minute.
4) The path of a golf ball.
5) The amount of dough needed to make pizza crusts is calculated from knowing the diameter.
6) A runner’s strategy is to start quickly , to slow to an even pace and finally to sprint towards the finish.
7) The number of cigarettes smoked affects your breathing in a negative way.
Which graph describes the following:
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A wagon moves along then crashes into a wall and stops.
We
climbed a hill an sledded down it.
Ken
dra is speeding down a highway and is stopped by a
poli
ce officer. The officer gives her a ticket and she
continues on her way. Draw a graph which best describes her motion
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