MATHEMATICS 20-1
Systems of Equations and Inequalities
High School collaborative venture with
Edmonton Christian, Harry Ainlay, J. Percy Page, Jasper
Place, Millwoods Christian, Ross Sheppard and W. P.
Wagner, M. E LaZerte, McNally, Queen Elizabeth,
Strathcona and Victoria
Edm Christian High: Aaron Trimble
Harry Ainlay: Ben Luchkow
Harry Ainlay: Darwin Holt
Harry Ainlay: Lareina Rezewski
Harry Ainlay: Mike Shrimpton
J. Percy Page: Debbie Younger
Jasper Place: Matt Kates
Jasper Place: Sue Dvorack
Millwoods Christian: Patrick Ypma
Ross Sheppard: Patricia Elder
Ross Sheppard: Dean Walls
W. P. Wagner: Amber Steinhauer
M. E. LaZerte: Teena Woudstra
Queen Elizabeth: David Underwood
Strathcona: Christian Digout
Victoria: Steven Dyck
McNally: Neil Peterson
Facilitator: John Scammell (Consulting Services)
Editor: Jim Reed (Contracted)
2010 - 2011
Mathematics 20-1
Systems of Equations and Inequalities
Page 2 of 38
TABLE OF CONTENTS
STAGE 1
DESIRED RESULTS
PAGE
Big Idea
4
Enduring Understandings
4
Essential Questions
4
Knowledge
5
Skills
6
STAGE 2
ASSESSMENT EVIDENCE
Transfer Task (on a separate page which could be photocopied & handed out to
students)
Investing in the Future by Looking in the Past
Teacher Notes for Transfer Task and Rubric
Transfer Task and Rubric
Rubric
Possible Solution
9
10
14
17
STAGE 3 LEARNING PLANS
Lesson #1
Solving Systems Graphically
21
Lesson #2
Solving Systems Algebraically
25
Lesson #3
Linear Inequalities Two Variables
28
Lesson #4
Quadratic Inequalities One Variable
33
Lesson #5 Quadratic Inequalities Two Variables
36
Mathematics 20-1
Systems of Equations and Inequalities
Page 3 of 38
Mathematics 20-1
Systems of Equations and Inequalities
STAGE 1
Desired Results
Big Idea:
The solution to a problem may not be a single value, but a range of values. Using test
points and intervals is foundational to further study in mathematics.
Implementation note:
Post the BIG IDEA in a prominent
place in your classroom and refer to
it often.
Enduring Understandings:
Students will understand …







A system involving quadratic equations can have no solution one solution, two
solutions or infinitely many solutions.
Quadratic inequalities involve a solution with a range of values.
An inequality in one variable has a range of x-values for its solution. (linear
axis)
An inequality in two variables has a range of coordinates for its solution.
(shaded region)
Solutions can be found graphically or algebraically.
The solution to an inequality does not include the line (or curve) if an equal sign
is not included.
When graphing inequalities, the line (or curve) should be broken if an equal
sign is not included and solid if the equal sign is included.
Essential Questions:




Where are inequalities used in real life?
Does a system involving a quadratic always have a solution?
When is it appropriate to have a range of values as a solution?
Will the methods we learned in Math 10C work for systems in involving a
quadratic equation?
Implementation note:
Ask students to consider one of the
essential questions every lesson or two.
Has their thinking changed or evolved?
Mathematics 20-1
Systems of Equations and Inequalities
Page 4 of 38
Knowledge:
Enduring
Understanding
List enduring
understandings (the fewer
the better)
Specific
Outcomes
List the reference #
from the Alberta
Program of Studies
Description of
Knowledge
The paraphrased outcome that the group is targeting
Students will know …
Students will understand…
*RF 6


A system involving
quadratic equations
can have no solution
one solution, two
solutions or infinitely
many solutions.

the points of intersection are the solutions of a system
of linear-quadratic or quadratic-quadratic equations
a system of linear-quadratic or quadratic-quadratic
equations may have zero, one, two or an infinite number
of solutions
Students will know …
Students will understand…
RF 7, 8


Quadratic inequalities
involve a solution with
a range of values.

Students will know …
Students will understand…

An inequality in one
variable has a range of
x-values for its
solution. (linear axis)
test points can be used to determine the solution region
that satisfies an inequality
a solid line should be used when the boundary line is
included in the inequality. A broken line should be used
when the boundary line is not included in the inequality
RF 8


test points can be used to determine the solution region
that satisfies an inequality
a solid point indicates that it is part of the solution,
whereas an open dot is not part of the solution
Students will know …
Students will understand…
RF 7


An inequality in two
variables has a range
of coordinates for its
solution. (shaded
region)

test points can be used to determine the solution region
that satisfies an inequality
a solid line should be used when the boundary line is
included in the inequality. A broken line should be used
when the boundary line is not included in the inequality
Students will know …
Students will understand…
RF 7

Solutions can be found
graphically or
algebraically.
Mathematics 20-1


the graph of a linear or quadratic inequality can be
sketched with or without technology
linear or quadratic inequalities can be used to solve
some problems
Systems of Equations and Inequalities
Page 5 of 38
Students will know …
Students will understand…
RF 7


The solution to an
inequality does not
include the line (or
curve) if an equal sign
is not included.

test points can be used to determine the solution region
that satisfies an inequality
a solid line should be used when the boundary line is
included in the inequality. A broken line should be used
when the boundary line is not included in the inequality
Students will know …
Students will understand…
RF 7


When graphing
inequalities, the line
(or curve) should be
broken if an equal sign
is not included and
solid if the equal sign
is included.
8888
I*RF =
a solid line should be used when the boundary line is
included in the inequality. A broken line should be used
when the boundary line is not included in the inequality
Relations and Functions
Skills:
Enduring
Understanding
List enduring
understandings (the
fewer the better)
Specific
Outcomes
List the reference
# from the
Alberta Program
of Studies
Description of
Skills
The paraphrased outcome that the group is targeting
Students will be able to…
Students will understand…
*RF 6


A system involving
quadratic equations
can have no solution
one solution, two
solutions or infinitely
many solutions.
explain, using examples, why a system of linearquadratic or quadratic-quadratic equations may have
zero, one, two or an infinite number of solutions
Students will be able to…
Students will understand…
RF 7, 8

Quadratic inequalities
involve a solution with
a range of values.
Mathematics 20-1
 explain, using examples, how test points can be used to
determine the solution region that satisfies an inequality
 sketch, with or without technology, the graph of a linear or
quadratic inequality
 determine the solution of a quadratic inequality in one
variable, using strategies such as case analysis,
graphing, roots and test points, or sign analysis; and
explain the strategy used
 interpret the solution to a problem that involves a
Systems of Equations and Inequalities
Page 6 of 38
quadratic inequality in one variable
Students will be able to…
Students will understand…
RF 8

 determine the solution of a quadratic inequality in one
variable, using strategies such as case analysis,
graphing, roots and test points, or sign analysis; and
explain the strategy used
 represent and solve a problem that involves a quadratic
inequality in one variable
 interpret the solution to a problem that involves a
quadratic inequality in one variable
An inequality in one
variable has a range of
x-values for its
solution. (linear axis)
Students will be able to…
Students will understand…
RF 7

 explain, using examples, how test points can be used to
determine the solution region that satisfies an inequality
 explain, using examples, when a solid or broken line
should be used in the solution for an inequality
 sketch, with or without technology, the graph of a linear or
quadratic inequality
 solve a problem that involves a linear or quadratic
inequality
An inequality in two
variables has a range
of coordinates for its
solution. (shaded
region)
Students will be able to…
Students will understand…
RF 6, 7, 8

Solutions can be found
graphically or
algebraically.
Mathematics 20-1
 relate a system of linear-quadratic or quadratic-quadratic
equations to the context of a given problem
 determine and verify the solution of a system of linearquadratic or quadratic-quadratic equations graphically,
with technology
 determine and verify the solution of a system of linearquadratic or quadratic-quadratic equations algebraically
 explain the meaning of the points of intersection of a
system of linear-quadratic or quadratic-quadratic
equations
 explain, using examples, why a system of linear-quadratic
or quadratic-quadratic equations may have zero, one, two
or an infinite number of solutions
 solve a problem that involves a system of linear-quadratic
or quadratic-quadratic equations, and explain the strategy
used
 explain, using examples, how test points can be used to
determine the solution region that satisfies an inequality.
 explain, using examples, when a solid or broken line
should be used in the solution for an inequality
 sketch, with or without technology, the graph of a linear or
quadratic inequality
 solve a problem that involves a linear or quadratic
inequality
 determine the solution of a quadratic inequality in one
variable, using strategies such as case analysis,
graphing, roots and test points, or sign analysis; and
explain the strategy used
Systems of Equations and Inequalities
Page 7 of 38
Students will be able to…
Students will understand…
RF 7


The solution to an
inequality does not
include the line (or
curve) if an equal sign
is not included.
explain, using examples, when a solid or broken line
should be used in the solution for an inequality
Students will be able to…
Students will understand…
RF 7

When graphing
inequalities, the line
(or curve) should be
broken if an equal sign
is not included and
solid if the equal sign
is included.

explain, using examples, when a solid or broken line
should be used in the solution for an inequality
* RF = Relations and Functions
Implementation note:
Teachers need to continually ask
themselves, if their students are
acquiring the knowledge and skills
needed for the unit.
Mathematics 20-1
Systems of Equations and Inequalities
Page 8 of 38
STAGE 2
Assessment Evidence
sired Results Desired Results
Results Desired Results
Investing in the Future by Looking in the Past
Teacher Notes
There is one transfer task to evaluate student understanding of the concepts relating
to systems of equations and inequalities. A photocopy-ready version of the transfer
task is included in this section.
Implementation note:
Students must be given the transfer task & rubric at
the beginning of the unit. They need to know how
they will be assessed and what they are working
toward.
Each student will:
 Analyze the given data
 Create equations that match the data by performing a regression on each set of
data using technology
 Create graphs to compare the values of each investment
 Interpret the graphs and make decisions based on the interpretation
Implementation note:
Teachers need to consider what performances and
products will reveal evidence of understanding?
What other evidence will be collected to reflect
the desired results?
Mathematics 20-1
Systems of Equations and Inequalities
Page 9 of 38
Investing in the Future by Looking at the Past - Assessment Task
A year ago, after a few months of hard work, you and your friends
found yourselves with some money to invest. You decided to invest
a portion of this money in the stock market and found a broker to
manage your investments. You chose companies that were familiar
to you and hoped for the best. The following table shows the amount
of money each person invested in stocks at the end of last summer
and the value of that investment after a year.
Name
Ali
Ben
Khalid
Dimitro
Pauline
Sam
Amount of money
invested
$800
$1100
$900
$1000
$900
$1000
Value of investment
after 12 months
$464
$956
$1212
$1144
$588
$1144
Investing in the Future by Looking at the Past
Though some people in the group had some good fortune, overall the
group lost money. You’ve decided to analyze last year’s market activity to
see how you could’ve better managed your investments. Below is a table
showing the activity of the six investments.
Company
invested
in (and name
of investor)
Coach (Ali)
Month
Share Value
(unit price)
0
3
6
9
12
23.45
36.02
38.57
31.10
13.60
34.12
34.12
34.12
34.12
34.12
800.00
1229.00
1316.00
1061.00
464.00
Google (Ben)
0
3
683.23
468.01
1.61
1100.00
6
9
12
381.37
423.29
593.79
1.61
1.61
1.61
1.61
753.50
614.00
681.50
956.00
0
46.56
19.33
900.00
978.00
1056.00
1134.00
1212.00
1000.00
1198.00
1288.00
1270.00
1144.00
McDonald's
(Khalid)
Nike (Dimitro)
Number of
Shares
3
50.59
6
9
12
54.63
58.67
62.70
19.33
19.33
19.33
19.33
0
3
6
9
12
44.07
52.80
56.77
55.97
50.42
22.69
22.69
22.69
22.69
22.69
Market
Value of
Investment
Investing in the Future by Looking at the Past
Analysis
An excellent way to assess the changes in each investment is through
graphical representation. Using the values given in the table above,
perform a regression to determine the quadratic equation of each
investment except McDonald’s, for which you will find a linear equation.
Each of these equations should describe the market value of the investment
as a function of time. Round all values to the nearest tenth, where
necessary.
Once you have determined the equation for each of the investments, you
will be creating graphs to compare the value of different investments at any
given time. For each of the following pairs of investments, respond to the
indicated questions.
Company
invested
in (and
name of
investor)
Royal
Bank
(Pauline)
Telus
(Sam)
Month
Share Value
(unit price)
0
29.20
3
6
9
12
46.82
51.01
41.76
19.08
0
39.10
3
6
9
12
34.00
32.63
36.03
42.61
Number of
Shares
Market Value
of
Investment
30.82
30.82
30.82
30.82
30.82
900.00
1443.00
1572.00
1287.00
588.00
25.58
25.58
25.58
25.58
25.58
1000.00
874.00
856.00
946.00
1144.00
Investing in the Future by Looking at the Past
Situation #1 – Dimitro vs Pauline
1. When is their investments equal in value?
2. When is Dimitro’s investment greater in value than Pauline’s investment?
3. After one year, how much greater was Dimitro’s investment in value than
Pauline’s?
4. If the trend continues for both investments, will the market value of each
investment ever be equal again?
Situation #2 – Khalid vs Ali
1.
2.
3.
4.
When is their investments equal in value?
When is Ali’s investment greater in value than Khalid’s investment?
After one year, how much greater was Khalid’s investment in value than Ali’s?
If the trend continues for both investments, will the market value of each
investment ever be equal again?
Situation #3 – Khalid vs Sam
1. When is their investments equal in value?
2. When is Sam’s investment greater in value than Khalid’s investment?
3. After one year, how much greater was Khalid’s investment in value than
Sam’s?
4. If the trend continues for both investments, will the market value of each
investment ever be equal again?
Situation #4 – Ben vs Sam
1.
2.
3.
4.
When is their investments equal in value?
When is Sam’s investment greater in value than Ben’s investment?
After one year, how much greater was Sam’s investment in value than Ben’s?
If the trend continues for both investments, will the market value of each
investment ever be equal again?
Extension
1. If you could go back and choose one of these investments, which would you
choose?
2. If you were allowed to sell your shares one time during the year and buy into a
different stock, which stocks would you choose and when would you trade?
How much profit would you make that year?
3. If you were allowed to stocks as many times as you would like, which choices
would you make? Discuss and justify your investment decisions
Assessment
Mathematics 20-1
Systems of Equations and Inequalities
Investing in Stocks Rubric 1
Component
Description of Requirements
- Analyzes information and performs the appropriate
regression.
Mathematical
- Creates appropriate equations based on the data
Content
with and/or without technology
- Creates accurate graphs
Presentation
- Communicates findings using graphs
of Data
- Includes mathematical vocabulary, notation and
symbolism
Interpretation
- Explains significance of key findings in the graphs
of Data
- Shows clear understanding of graphical data by
accurately answering the assigned questions
Assessment
IN 1 2 3 4
IN 1 2 3 4
IN 1 2 3 4
IN 1 2 3 4
IN 1 2 3 4
Investing in Stocks Rubric 2
Level
Criteria
Math
Content
Comparison
1
Excellent
4
All required
elements are
present and
correct
Math
Content
Comparison
2
All required
elements are
present and
correct
Math
Content
Comparison
3
All required
elements are
present and
correct
Math
Content
Comparison
4
All required
elements are
present and
correct
Graphs
Presentation
of data is
clear,
precise and
accurate
Provides
insightful
explanations
Explains
Choices
Proficient
3
All required
elements are
present but
may contain
minor errors
Adequate
2
Some
required
elements are
missing, or
contain
major errors
All required
Some
elements are
required
present but
elements are
may contain
missing, or
minor errors
contain
major errors
All required
Some
elements are
required
present but
elements are
may contain
missing, or
minor errors
contain
major errors
All required
Some
elements are
required
present but
elements are
may contain
missing, or
minor errors
contain
major errors
Presentation of Presentation
data is
of data is
complete and simplistic
unambiguous and
plausible
Provides
Provides
logical
explanations
explanations
that are
complete but
vague
Limited
1
Most
required
elements are
missing or
incorrect
Insufficient
Blank
No score is
awarded as
there is no
evidence
given
Most
required
elements are
missing or
incorrect
No score is
awarded as
there is no
evidence
given
Most
required
elements are
missing or
incorrect
No score is
awarded as
there is no
evidence
given
Most
required
elements are
missing or
incorrect
No score is
awarded as
there is no
evidence
given
Presentation
of data is
vague and
inaccurate
Presentation
of data is
incomprehensible
Provides
explanations
that are
incomplete
or
confusing.
No
explanation is
provided
When work is judged to be limited or insufficient, the teacher makes decisions
about appropriate intervention to help the student improve
Mathematics 20-1
Systems of Equations and Inequalities
Page 15 of 38
Glossary
boundary line – A line on the coordinate plane that separates the plane into two
regions
compound inequality – A compound statement expressing two inequalities at the
same time
inequality – A mathematical statement comparing expressions that may not be equal.
These can be written using the symbols less than ( ), greater than ( ), less than or
equal to ( £ ), greater than or equal to ( ³ ), and not equal to ( ¹ ). [Math 20-1 (McGrawHill, page 588)]
model - Method of simulating real-life situations with mathematical equations to
forecast their future behavior (source)
solution region – All the points in the Cartesian plane that satisfy an inequality. Also
known as the solution set. [Math 20-1 (McGraw-Hill, page 590)]
system of equations – A group of equations that are to be considered at the same
time
systems of linear-quadratic equations – A linear equation and a quadratic equation
involving the same variables. A graph of the system involves a line and a parabola.
[Math 20-1 (McGraw-Hill, page 591)]
systems of quadratic-quadratic equations – Two quadratic equations involving the
same variables. A graph of the system involves two parabolas. [Math 20-1 (McGrawHill, page 591)]
test point – A point not on the boundary of the graph of an inequality that is
representative of all the points in a region. A point that is used to determine whether
the points in a region satisfy the inequality. [Math 20-1 (McGraw-Hill, page 591)]
verify - Verifying a solution ensures the solution satisfies any equation or inequality by
using substitution
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Mathematics 20-1
Systems of Equations and Inequalities
Page 16 of 38
Possible Solution to Transfer Task Title
Nike vs Royal Bank
Royal Bank (Pauline): y = -23x 2 + 250x + 900
Nike (Dimitro): y = -6x 2 + 84x + 1000
-6x 2 + 84x +1000 = -23x 2 + 250x + 900
17 x2 166 x  100  0
x =
(
2 (17)
166 ± 27556 - 4 ×17 ×100
x =
)
166 ± 20756
34
x = 0.65, 9.12
The two plans are equal at 0.65 months and 9.12 months.
The market value of Dimitro’s investment is greater than Pauline’s from 0.65 months
to 9.12 months. 0.65 < M < 9.12
(
)
After one year:
The market value of Pauline’s investment in Royal Bank is $588.
The market value of Dimitro’s investment in Nike is $1144.
Dimitro’s investment is worth $556 more than Pauline’s.
If the trend continues for both investments, Pauline’s and Dimitro’s investments will
never be equal again.
Mathematics 20-1
Systems of Equations and Inequalities
Page 17 of 38
McDonalds vs Coach
McDonalds (Khalid): y = 26x + 900
Coach (Ali): y = -19x 2 + 200x + 800
26x + 900 = -19x 2 + 200x + 800
19x 2 -174x +100 = 0
x =
(
2 (19)
174 ± 30276 - 4 ×19 ×100
x =
)
174 ± 22676
38
x = 0.62, 8.54
The two plans are equal at 0.62 months and 8.54 months.
The market value of Ali’s investment is greater than Khalid’s from 0.62 months to 8.54
months. 0.62 < M < 8.54
(
)
After one year:
The market value of Ali’s investment in Coach is $464.
The market value of Khalid’s investment in McDonalds is $1212.
Khalid’s investment is worth $748 more than Ali’s.
If the trend continues for both investments, Ali’s and Khalid’s investments will never be
equal again.
Mathematics 20-1
Systems of Equations and Inequalities
Page 18 of 38
Google vs Telus
Google
Telus
11.5x 2 -150x +1100
6x 2 - 60x +1000
Solve for points where there meet:
11.5x 2 -150x +1100 = 6x 2 - 60x +1000
5.5x 2 - 90x +100 = 0
a = 5.5, b = -90, c = 100
90 ± (-90)2 - 4(5.5)(100)
-b ± b 2 - 4ac
90 ± 5900
®
®
2a
2(5.5)
11
x = 15.16, 1.20
x = 1.20 ® 6x 2 - 60x +1000 ® 6(1.20)2 - 60(1.20) +1000 = 288.64
They are equal at 1.20 months with a value of $288.64
If the trends continue they will meet again at 15.16 months and their value would be:
x = 15.16 ® 6x 2 - 60x +1000 ® 6(15.16)2 - 60(15.16) +1000 = 1469.35
They are again equal at 15.16 months with a value of $1469.35
In the first year Google is worth less then Telus from 1.20 months till the end of the
year.
{1.20 < m £ 12}
Mathematics 20-1
Systems of Equations and Inequalities
Page 19 of 38
McDonalds vs Telus
McDonalds
Telus
26x + 900
6x 2 - 60x +1000
Solve for points where there meet:
26x + 900 = 6x 2 - 60x +1000
6x 2 - 86x +100 = 0
a = 6, b = -86, c = 100
86 ± (-86)2 - 4(6)(100)
-b ± b 2 - 4ac
86 ± 4996
®
®
2a
2(6)
12
x = 13.06, 1.28
x = 1.28 ® 26x+900 ® 26(1.28) + 900 = 933.28
They are equal at 1.28 months with a value of $933.28
If the trends continue they will meet again at 13.06 months and their value would be:
x = 13.06 ® 26x+900 ® 26(13.06) + 900 = 1239.56
They are again equal at 13.06 months with a value of $1239.56
In the first year McDonalds is worth more then Telus from 1.28 months till the end of
the year.
{1.28 < m £ 12}
Mathematics 20-1
Systems of Equations and Inequalities
Page 20 of 38
STAGE 3
Learning Plans
Lesson 1
Solving Systems Graphically
STAGE 1
BIG IDEA: The solution to a problem may not be a single value, but a range of values. Using test points
and intervals is foundational to further study in mathematics.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …
 Does a system involving a quadratic always
have a solution?
 When is it appropriate to have a range of
values as a solution?
 Will the methods we learned in Math 10C work
for systems in involving a quadratic equation?

A system involving quadratic equations can
have no solution one solution, two solutions or
infinitely many solutions.
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 the points of intersection are the solutions of a
system of linear-quadratic or quadraticquadratic equations
 a system of linear-quadratic or quadraticquadratic equations may have zero, one, two
or an infinite number of solutions
 relate a system of linear-quadratic or quadraticquadratic equations to the context of a given
problem
 determine and verify the solution of a system of
linear-quadratic or quadratic-quadratic
equations graphically, with technology
 explain the meaning of the points of
intersection of a system of linear-quadratic or
quadratic-quadratic equations
 explain, using examples, why a system of
linear-quadratic or quadratic-quadratic
equations may have zero, one, two or an
infinite number of solutions
 solve a problem that involves a system of
linear-quadratic or quadratic-quadratic
equations, and explain the strategy used
Implementation note:
Each lesson is a conceptual unit and is not intended to
be taught on a one lesson per block basis. Each
represents a concept to be covered and can take
anywhere from part of a class to several classes to
complete.
Mathematics 20-1
Systems of Equations and Inequalities
Page 21 of 38
Lesson Summary

Students will solve linear-quadratic and quadratic-quadratic systems
graphically.
Lesson Plan
Hook
Scenario on page 432 in the McGraw Hill Textbook
Lesson Goal
Solve linear-quadratic and quadratic-quadratic systems
Activate Prior Knowledge
Review the three possible solutions of a linear-linear system of equations.
Solve the following system graphically and algebraically. Determine the numbers of
solutions in each case.
1. x + y = -5
y = -x + 5
2. x + y = -5
y = -x + 3
3. x + y = -5
3x + 2y = 4
Lesson
Students will sketch a scenario, which would involve: no solution, one solution, or
many solutions, between a linear-quadratic system.
Repeat for a quadratic-quadratic system.
Discuss each scenario with technology (students need to generate equations that go
with the scenarios created above).
Mathematics 20-1
Systems of Equations and Inequalities
Page 22 of 38
Going Beyond
Discuss coincident parabolas and systems of other functions (i.e. rational, radical,
absolute value, etc.)
Graph systems of equations without technology.
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 8.1)
Pre –Calculus 11 (Pearson: 5.4)
Supporting
http://members.shaw.ca/jreed/math20-1/program5.htm (Ron Blond’s applet)
Assessment
Exit slip- students will determine their own linear-quadratic system and sketch their
solution and explain their solution.
Glossary
boundary line – A line on the coordinate plane that separates the plane into two
regions
compound inequality – A compound statement expressing two inequalities at the
same time
model - Method of simulating real-life situations with mathematical equations to
forecast their future behavior (source)
system of equations – A group of equations that are to be considered at the same
time
systems of linear-quadratic equations – A linear equation and a quadratic equation
involving the same variables. A graph of the system involves a line and a parabola.
[Math 20-1 (McGraw-Hill, page 591)]
Mathematics 20-1
Systems of Equations and Inequalities
Page 23 of 38
systems of quadratic-quadratic equations – Two quadratic equations involving the
same variables. A graph of the system involves two parabolas. [Math 20-1 (McGrawHill, page 591)]
verify - Verifying a solution ensures the solution satisfies any equation or inequality by
using substitution
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Other
Mathematics 20-1
Systems of Equations and Inequalities
Page 24 of 38
Lesson 2
Solving Systems Algebraically
STAGE 1
BIG IDEA: The solution to a problem may not be a single value, but a range of values. Using test points
and intervals is foundational to further study in mathematics.
ENDURING UNDERSTANDINGS:
Students will understand …


A system involving quadratic equations can
have no solution one solution, two solutions or
infinitely many solutions.
http://members.shaw.ca/jreed/math201/program5.htmSolutions can be found
graphically or algebraically.
ESSENTIAL QUESTIONS:
 When is it appropriate to have a range of
values as a solution?
 Will the methods we learned in Math 10C work
for systems in involving a quadratic equation?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 the points of intersection are the solutions of a
system of linear-quadratic or quadraticquadratic equations
 a system of linear-quadratic or quadraticquadratic equations may have zero, one, two
or an infinite number of solutions
 relate a system of linear-quadratic or quadraticquadratic equations to the context of a given
problem
 determine and verify the solution of a system of
linear-quadratic or quadratic-quadratic
equations graphically, with technology
 determine and verify the solution of a system of
linear-quadratic or quadratic-quadratic
equations algebraically
 explain the meaning of the points of
intersection of a system of linear-quadratic or
quadratic-quadratic equations
 explain, using examples, why a system of
linear-quadratic or quadratic-quadratic
equations may have zero, one, two or an
infinite number of solutions
 solve a problem that involves a system of
linear-quadratic or quadratic-quadratic
equations, and explain the strategy used
Mathematics 20-1
Systems of Equations and Inequalities
Page 25 of 38
Lesson Summary

Students will solve linear-quadratic and quadratic-quadratic systems
algebraically and through problem solving.
Hook – Show video clips of basketball players doing an “alley-oop”.
http://www.youtube.com/watch?v=C3buFTUNNec
http://www.youtube.com/watch?v=qQl9ES9vdAo&feature=fvsr
Discuss the relevance of the intersection point of the quadratic equations representing
the trajectory of the ball and the player doing the “dunk”.
Lesson Goal
In this lesson students will model, solve algebraically, verify and interpret solutions of
a system of linear-quadratics and quadratic-quadratic equations.
Activate Prior Knowledge
1. Solve and verify the following linear system algebraically (using both substitution &
elimination).
2x – 3y = -2
4x + y = 24
2. The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a
certain day, 2200 people enter the fair and $5050 is collected. How many children
and how many adults attended?
Lesson
Review the Quadratic Formula and how it is used to find roots. Emphasize that the
roots are the solution to the system.
Solve and verify linear-quadratic system and a word problem by:
 substitution
 elimination.
Explain the meaning of the solution. If the quadratic formula is used to solve,
emphasize the meaning of the roots of the resultant equation that is created from
the substitution/elimination.
Solve and verify quadratic-quadratic system and a word problem by
 substitution
 elimination
Mathematics 20-1
Systems of Equations and Inequalities
Page 26 of 38
Explain the meaning of the solution. If the quadratic formula is used to solve,
emphasize the meaning of the roots of the resultant equation that is created from
the substitution/elimination.
Describe a real life application that would represent this system.
Going Beyond
Solving other systems algebraically involving different types of functions (cubic,
polynomial, radical, absolute value etc.)
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 8.2)
YouTube videos: http://www.youtube.com/watch?v=C3buFTUNNec
http://www.youtube.com/watch?v=qQl9ES9vdAo&feature=fvsr
Supporting


Provide all equations in the modelling situations.
Solve graphically first and explain the solutions in the context of the question.


Assessment
Exit Slip
Think Pair Share
Glossary
model - Method of simulating real-life situations with mathematical equations to
forecast their future behavior (source)
system of equations – A group of equations that are to be considered at the same
time
verify - Verifying a solution ensures the solution satisfies any equation or inequality by
using substitution
Other
Mathematics 20-1
Systems of Equations and Inequalities
Page 27 of 38
Lesson 3
Linear Inequalities Two Variables
STAGE 1
BIG IDEA: The solution to a problem may not be a single value, but a range of values. Using test points
and intervals is foundational to further study in mathematics.
ENDURING UNDERSTANDINGS:
Students will understand …





An inequality in one variable has a range of xvalues for its solution. (linear axis)
An inequality in two variables has a range of
coordinates for its solution. (shaded region)
Solutions can be found graphically or
algebraically.
When graphing inequalities, the line (or curve)
should be broken if an equal sign is not
included and solid if the equal sign is included.
The solution to an inequality does not include
the line (or curve) if an equal sign is not
included.
ESSENTIAL QUESTIONS:
 Where are inequalities used in real life?
 When is it appropriate to have a range of
values as a solution?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 test points can be used to determine the
solution region that satisfies an inequality
 a solid line should be used when the boundary
line is included in the inequality. A broken line
should be used when the boundary line is not
included in the inequality.
 the graph of a linear or quadratic inequality can
be sketched with or without technology
 linear or quadratic inequalities can be used to
solve some problems
 explain, using examples, how test points can
be used to determine the solution region that
satisfies an inequality
 explain, using examples, when a solid or
broken line should be used in the solution for
an inequality
 sketch, with or without technology, the graph of
a linear or quadratic inequality
 solve a problem that involves a linear or
quadratic inequality
Lesson Summary

Students will learn to solve linear inequalities.
Mathematics 20-1
Systems of Equations and Inequalities
Page 28 of 38
Lesson Plan
Hook
Go to a shopping website of interest to purchase two items of your choice. Determine
how many of each item can be purchased for at most $500.
Website possibilities:
American Eagle: http://shopping.yahoo.com/browse/clothing-accessoriesshoes/;_ylt=Aix3vYSHYnjSxVJ8xTbzpY4EgFoB
Sportcheck: http://www.sportchek.ca/home/index.jsp
Amazon: http://www.amazon.ca/
1. How many of each of the two items can you afford to buy? Discuss. You have
begun the process of solving a linear inequality.
2. How could this be represented algebraically?
3. Discuss how this might be represented graphically. (This meant for discussion only
at this point)
4. Discuss the relevance of the domain and range in this scenario ( x ³ 0, y ³ 0 ).
Lesson Goal
Students will solve linear inequalities algebraically and graphically.
Activate Prior Knowledge
Review rearranging equations, graphing linear equations from slope-intercept form
and general form, multiplying or dividing by a negative number causes the inequality
sign to change direction. Review discrete vs. continuous data.
Lesson
Student lead lesson:
Sketch the following on a horizontal number line:
x= 2
x≥ 2
x≤ 2
Sketch the following on a vertical number line:
y= 2
y≥ 2
y≤ 2
Mathematics 20-1
Systems of Equations and Inequalities
Page 29 of 38
Sketch the following on a coordinate plane. Show proof of why your answer is
correct.
y= x
y≥ x
y≤ x
Sketch the following on a coordinate plane. Show proof of why your answer is
correct.
y = 2x + 1
y ≥ 2x + 1
y ≤ 2x + 1
What is different about the last equation and how does it affect your graph. In the
discussion about “proof of how we know the answer is correct” and test point
needs to be addressed.
Teacher - Led lesson:
Have students graph a line. Example: y =
3
x + 2.
4
Label the line “C”, the region above the line “A”, and below the line, label it “B”.
The line separates the plane into 3 regions, A, B, and C.
1. Which inequality below best describes region A? Discuss.
3
y > x +2
4
3
y < x +2
4
3
2. a. Sketch y ³ x + 2 . What would it look like? Which regions from example one
4
would be included in my graph? Discuss. Use shading. (Teacher note: Since the
line is included, the solution would include both region A and the line C)
3
x + 2 . What would it look like? Which regions from example one
4
would be included in my graph? Discuss. Use shading. (Teacher note: Since
it includes only region B, the line C would be represented as a broken line.)
b. Sketch y <
In general, if an equal sign is included, the line will be solid to show that every point
on the line is included in the solution. If the equal sign is not included, the line is
shown as a broken line as it is not part of the solution.
Teacher may want to check students’
prior knowledge of solving a linear
inequality in one variable. Ex. -2x < 10
Mathematics 20-1
Systems of Equations and Inequalities
Page 30 of 38
3. Sketch 2x - 3y < 6.
(Note to teacher: use a table of values, x and y intercepts, or rearrange for y)
Step 1: Sketch the line 2x - 3y = 6, but should it be solid or broken? (broken)
Step 2: Which region should be shaded?
Method 1: If you chose to rearrange for y, you would shade above if there is a
____ sign and below if there is a _____ sign.
Method 2: Try the test-point method. Pick any point that is not on the line. (Hint:
0, 0 is a good one to use). Substitute the values into the original equation. If the
resulting statement is true, then shade the region containing that point. If it is false,
shade the region not containing that point.
Hence, the region that should be shaded is above the line.
Method 3: Using the GDC, graph the line by rearranging for y, and shade above
the line. (Teacher: demonstrate this)
4. Apply knowledge to revisit the questions posed in the hook. Determine how many
of each item can be purchased for at most $500.
Reiterate the discussion of continuous vs. discrete data.
Going Beyond
Could do a system of linear inequalities for fun. (See Math 20 Applied)
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 9.1)
Supporting
Assessment
Exit Slip.
Solve graphically without the use of technology:
3x - 5y > 15
Mathematics 20-1
Systems of Equations and Inequalities
Page 31 of 38
Glossary
boundary line – A line on the coordinate plane that separates the plane into two
regions
compound inequality – A compound statement expressing two inequalities at the
same time
inequality – A mathematical statement comparing expressions that may not be equal.
These can be written using the symbols less than (<), greater than (>), less than or
equal to (≤), greater than or equal to (≥), and not equal to (≠). [Math 20-1 (McGrawHill, page 588)]
solution region – All the points in the Cartesian plane that satisfy an inequality. Also
known as the solution set. [Math 20-1 (McGraw-Hill, page 590)]
system of equations – A group of equations that are to be considered at the same
time
test point – A point not on the boundary of the graph of an inequality that is
representative of all the points in a region. A point that is used to determine whether
the points in a region satisfy the inequality. [Math 20-1 (McGraw-Hill, page 591)]
verify - Verifying a solution ensures the solution satisfies any equation or inequality by
using substitution
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Other
Mathematics 20-1
Systems of Equations and Inequalities
Page 32 of 38
Lesson 4
Quadratic Inequalities One Variable
STAGE 1
BIG IDEA: The solution to a problem may not be a single value, but a range of values. Using test points
and intervals is foundational to further study in mathematics.
ENDURING UNDERSTANDINGS:
Students will understand …





Quadratic inequalities involve a solution with a
range of values.
An inequality in one variable has a range of xvalues for its solution. (linear axis)
Solutions can be found graphically or
algebraically.
When graphing inequalities, the line (or curve)
should be broken if an equal sign is not
included and solid if the equal sign is included.
The solution to an inequality does not include
the line (or curve) if an equal sign is not
included.
ESSENTIAL QUESTIONS:
 Where are inequalities used in real life?
 When is it appropriate to have a range of
values as a solution?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 test points can be used to determine the
solution region that satisfies an inequality
 a solid line should be used when the boundary
line is included in the inequality. A broken line
should be used when the boundary line is not
included in the inequality.
 the graph of a linear or quadratic inequality can
be sketched with or without technology
 linear or quadratic inequalities can be used to
solve some problems
 explain, using examples, how test points can
be used to determine the solution region that
satisfies an inequality
 explain, using examples, when a solid or
broken line should be used in the solution for
an inequality
 sketch, with or without technology, the graph of
a linear or quadratic inequality
 solve a problem that involves a linear or
quadratic inequality
 determine the solution of a quadratic inequality
in one variable, using strategies such as case
analysis, graphing, roots and test points, or
sign analysis; and explain the strategy used
 represent and solve a problem that involves a
quadratic inequality in one variable
 interpret the solution to a problem that involves
a quadratic inequality in one variable
Lesson Summary

Students will learn to solve quadratic inequalities in two variables.
Mathematics 20-1
Systems of Equations and Inequalities
Page 33 of 38
Lesson Plan
Lesson Goal
Students will be able to solve quadratic inequalities in two variables.
Activate Prior Knowledge
Have students sketch the following and indicate a test point to verify solution.
y≥x+1
y < -2x + 1
Review quadratic function graphing including the vertex and direction of opening.
Note: This is assuming that the Quadratic Function unit has been completed.
y = x2 - 9
Lesson
Sketch the following inequalities:
y ≥ x2
y ≥ -x2
y < x2
y < -x2
Use a test point to verify solution and explain why you would shade above or below
the graph.
Try the following inequalities:
y ≥ (x - 2)2
y ≤ (x + 3)2 - 2
y < (x - 4)2 + 2
When given in the ax2 + bx + c = 0 form, have students use the quadratic formula to
find the roots.
Example:
Solve 2x2 - 7x > 12 (see example 3 on p. 482 of McGraw Hill)
Going Beyond
Given the following functions:
y = x2 – 1
y = -x2
Sketch the graph of –x2 ≥ x2 – 1.
Mathematics 20-1
Systems of Equations and Inequalities
Page 34 of 38
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 9.1)
Supporting
Ron Blond’s Applet: http://members.shaw.ca/jreed/math20-1/program5.htm
Assessment
Exit Slip: Solve the following inequality both graphically and algebraically.
2x2 + 12x - 11 > x2 + 2x + 13
Glossary
boundary line – A line on the coordinate plane that separates the plane into two
regions
compound inequality – A compound statement expressing two inequalities at the
same time
inequality – A mathematical statement comparing expressions that may not be equal.
These can be written using the symbols less than (<), greater than (>), less than or
equal to (≤), greater than or equal to (≥), and not equal to (≠). [Math 20-1 (McGrawHill, page 588)]
solution region – All the points in the Cartesian plane that satisfy an inequality. Also
known as the solution set. [Math 20-1 (McGraw-Hill, page 590)]
test point – A point not on the boundary of the graph of an inequality that is
representative of all the points in a region. A point that is used to determine whether
the points in a region satisfy the inequality. [Math 20-1 (McGraw-Hill, page 591)]
verify - Verifying a solution ensures the solution satisfies any equation or inequality by
using substitution
Other
Mathematics 20-1
Systems of Equations and Inequalities
Page 35 of 38
Lesson 5
Quadratic Inequalities Two Variables
STAGE 1
BIG IDEA: The solution to a problem may not be a single value, but a range of values. Using test points
and intervals is foundational to further study in mathematics.
ENDURING UNDERSTANDINGS:
Students will understand …






Quadratic inequalities involve a solution with a
range of values.
An inequality in one variable has a range of xvalues for its solution. (linear axis)
An inequality in two variables has a range of
coordinates for its solution. (shaded region)
Solutions can be found graphically or
algebraically.
When graphing inequalities, the line (or curve)
should be broken if an equal sign is not
included and solid if the equal sign is included.
The solution to an inequality does not include
the line (or curve) if an equal sign is not
included.
ESSENTIAL QUESTIONS:
 Where are inequalities used in real life?
 When is it appropriate to have a range of
values as a solution?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 test points can be used to determine the
solution region that satisfies an inequality
 a solid line should be used when the boundary
line is included in the inequality. A broken line
should be used when the boundary line is not
included in the inequality.
 the graph of a linear or quadratic inequality can
be sketched with or without technology
 linear or quadratic inequalities can be used to
solve some problems
 explain, using examples, how test points can
be used to determine the solution region that
satisfies an inequality
 explain, using examples, when a solid or
broken line should be used in the solution for
an inequality
 sketch, with or without technology, the graph of
a linear or quadratic inequality
 solve a problem that involves a linear or
quadratic inequality
 determine the solution of a quadratic inequality
in one variable, using strategies such as case
analysis, graphing, roots and test points, or
sign analysis; and explain the strategy used
 represent and solve a problem that involves a
quadratic inequality in one variable
 interpret the solution to a problem that involves
a quadratic inequality in one variable
Mathematics 20-1
Systems of Equations and Inequalities
Page 36 of 38
Lesson Summary

Students will learn how to solve quadratic inequalities in two variables and
apply to word problems.
Lesson Plan
Lesson Goal
 Students will solve quadratic inequalities in two variables and apply to word
problems.
Activate Prior Knowledge
Express the following on a number line
x=0
x+1>0
x>0
2x – 1 ≤ 0
x≤0
Review the use of set notation to express solution.
Review the sketching of quadratic function with a focus on the vertex and direction of
openings.
Lesson
Have students graph, shade and state the solution space in set notation for the
following:
y ≤ -3(x - 4)2 + 2
y ≥ 2(x + 3)2 - 5
y < x2 – 3x - 6
y < 2x2 + 3x + 5
Example Word Problem:
Do an example word problem, such as McGraw Hill p.498 #10
Going Beyond
Mathematics 20-1
Systems of Equations and Inequalities
Page 37 of 38
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 9.3)
Supporting
Ron Blond’s Applet: http://members.shaw.ca/jreed/math20-1/program5.htm
Assessment
Exit Slip: Graph the following without the use of technology.
y < -2(x – 1)2 - 5
Glossary
boundary line – A line on the coordinate plane that separates the plane into two
regions
compound inequality – A compound statement expressing two inequalities at the
same time
inequality – A mathematical statement comparing expressions that may not be equal.
These can be written using the symbols less than (<), greater than (>), less than or
equal to (≤), greater than or equal to (≥), and not equal to (≠). [Math 20-1 (McGrawHill, page 588)]
solution region – All the points in the Cartesian plane that satisfy an inequality. Also
known as the solution set. [Math 20-1 (McGraw-Hill, page 590)]
test point – A point not on the boundary of the graph of an inequality that is
representative of all the points in a region. A point that is used to determine whether
the points in a region satisfy the inequality. [Math 20-1 (McGraw-Hill, page 591)]
verify - Verifying a solution ensures the solution satisfies any equation or inequality by
using substitution
Other
Mathematics 20-1
Systems of Equations and Inequalities
Page 38 of 38