MATHEMATICS 20-1
Rational Expressions and Equations
High School collaborative venture with
M. E LaZerte, McNally, Queen Elizabeth, Ross Sheppard,
Strathcona and Victoria
M. E. LaZerte: Teena Woudstra
Queen Elizabeth: David Underwood
Ross Sheppard: Dean Walls
Strathcona: Christian Digout
Victoria: Steven Dyck
McNally: Neil Peterson
Facilitator: John Scammell (Consulting Services)
Editor: Jim Reed (Contracted)
2010 - 2011
Mathematics 20-1
Rational Expressions and Equations
Page 2 of 33
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)
Rational Reno’s
Teacher Notes for Transfer Task
Transfer Task
Rubric
Possible Solution
7
10
12
13
STAGE 3 LEARNING PLANS
Lesson #1
Simplifying Rational Expressions
14
Lesson #2
Multiplying Rational Expressions
18
Lesson #3
Dividing Rational Expressions
22
Lesson #4
Adding and Subtracting Rational Expressions
26
Lesson #5
Solving Rational Equations
30
Mathematics 20-1
Rational Expressions and Equations
Page 3 of 33
Mathematics 20-1
Rational Expressions and Equations
STAGE 1
Desired Results
Big Idea:
The study of rational expressions is foundational to further levels of mathematics.
Rational expressions, which are fractions involving variables, have applications in
fields like physics, chemistry, biochemistry, circuitry, economics and calculus.
Implementation note:
Post the BIG IDEA in a prominent
place in your classroom and refer to
it often.
Enduring Understandings:
Students will understand …


The operations on rational expressions are similar to fractional operations.
Rational expressions and equations may have restrictions.
Essential Questions:




What real life situations/careers require rational expressions?
Why is dividing by zero undefined?
If a fraction can be represented pictorially can a rational expression also be
represented pictorially?

X
Implementation note:
Ask students to consider one of the
essential questions every lesson or two.
Has their thinking changed or evolved?
If I can graph a vertical line, why is
Mathematics 20-1
5
undefined?
0
Rational Expressions and Equations
Page 4 of 33
Knowledge:
Enduring
Understanding
List enduring
understandings (the
fewer the better)
Specific
Outcomes
List the reference
# from the
Alberta Program
of Studies
Students will understand…

Students will know …
*AN.4, AN.5, AN.6
The operations on
rational expressions
are similar to fractional
operations.




Students will understand…

Description of
Knowledge
The paraphrased outcome that the group is
targeting
when to find a common denominator
factoring a denominator leads to a common
denominator
that division is multiplication of the reciprocal
that common factors between the numerator
and denominator reduce to one
Students will know …
*AN.4, AN.5, AN.6
Rational expressions
and equations may
have restrictions.



there may undefined values when there is a
variable in the denominator
restrictions are found on original expressions
a solution may not be a restriction
8888
*AN = Algebra and Number
Mathematics 20-1
Rational Expressions and Equations
Page 5 of 33
Skills:
Enduring
Understanding
Specific
Outcomes
List the reference #
from the Alberta
Program of Studies
Description of Skills
Students will be able to…
Students will understand…
*AN.4, AN.5, AN.6




The operations on
rational expressions
are similar to fractional
operations.

+, -, x, ÷ rational expressions
write equivalent forms of rational expressions
solve a rational equation, limited to numerators
involving monomials, binomials and trinomials
model a situation using a rational equation
Students will be able to…
Students will understand…
*AN.4, AN.5, AN.6

Rational expressions
and equatons may
have restrictions.


identify NPV
verify a solution
*AN = Algebra and Number
Implementation note:
Teachers need to continually ask
themselves, if their students are
acquiring the knowledge and skills
needed for the unit.
Mathematics 20-1
Rational Expressions and Equations
Page 6 of 33
STAGE 2
1
Assessment Evidence
Desired Results Desired Results
Rational Reno’s
Teacher Notes
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.
Consider
The LearnAlberta Graphing Tool can be used to generate interactive
resources. See Applet Notes and Graphing Tool Quick Start for
additional details.
TT Applet with Monomial Denominator
TT Applet with Binomial Denominator – pictured above
TT Applet with Trinomial Denominator
 files were added to the EPSB Understanding by Design share site
Applet Notes:
 You will need to install the Graphing Tool from Learnalberta:
http://www.learnalberta.ca/content/megt/graphing_tool.html
 To view the downloaded applet, click and drag the applet file onto the
application icon.
 Click the first (red) display screen.
Mathematics 20-1
Rational Expressions and Equations
Page 7 of 33
Graphing Tool Quick Start
 You will need to download/install the Graphing Tool from
Learnalberta:
http://www.learnalberta.ca/content/megt/graphing_tool.html
 Launch Graphing Tool.
 Click on the red Graphing Tool launch screen. It will be replaced with a grey
screen.
 Select "Create a Text Entry Function" from the Functions/Relations menu.
 To recreate the Binomial Denominator applet, click the fraction option then
entered ax in the numerator and x – b in the denominator. Click [ OK ].
 Select "Setup to Draw Graph(s) from Graph(s) menu.
 Drag the "f" button to "Graph One:" and select a color. Click [ OK ].
 When parameters a, b, c, are included, display one or more using the
Parameter menu.
Hook
Show video from Dan Meyer on bean counting.
source: http://vimeo.com/22156608




Stop video after the second person completes the task and have students
discuss how long they think the combined effort of the two people would take. It
would serve to reason that the combined time would be faster than the fastest
individual’s time. Then show remainder of the video. Use this reasoning to help
students reason out the first problem.
Teachers may want to incorporate an activity like the bean cup activity. The
challenge would be to maintain a consistent rate by both participants.
Here is a YouTube video similar to question 1, showing all the steps for solving
the problem. http://www.youtube.com/watch?v=3Gv8UNd4xJ8
Additional resources to strengthen student’s skills required for solving rational
equations.
http://www.mrperezonlinemathtutor.com/A2/8_1_Rational_Equations.html
http://staff.argyll.epsb.ca/jreed/
Mathematics 20-1
Rational Expressions and Equations
Page 8 of 33
Teacher Notes for Rational Reno’s Transfer Task
Glossary
equation - A statement of equality between two expressions
expression – A general term that ultimately represents a number. An expression can
consist of numbers, variables and operations on these.
non-permissible value – Any value for a variable that makes an expression
undefined [Math 20-1 (McGraw-Hill Ryerson: page 589)]
operation – Associates two or more members of a set with one of the members of
the set. The basic operations in mathematics are addition, subtraction, multiplication,
division and exponentiation.
rational equation – An equation formed by setting a rational expression equal to 0
rational expression – An algebraic fraction with a numerator and denominator that
are polynomials [Math 20-1 (McGraw-Hill Ryerson: page 590)]
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.
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
Rational Expressions and Equations
Page 9 of 33
Graphic 1
Graphic 2
Graphic 3
Glossary
equation - A statement of equality between two expressions
expression – A general term that ultimately represents a number. An expression can
consist of numbers, variables and operations on these.
non-permissible value – Any value for a variable that makes an expression
undefined [Math 20-1 (McGraw-Hill Ryerson: page 589)]
operation – Associates two or more members of a set with one of the members of the
set. The basic operations in mathematics are addition, subtraction, multiplication,
division and exponentiation.
rational equation – An equation formed by setting a rational expression equal to 0
rational expression – An algebraic fraction with a numerator and denominator that
are polynomials [Math 20-1 (McGraw-Hill Ryerson: page 590)]
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.
Assessment
Mathematics 20-1
Rational Reno’s
Rubric
Level
Excellent
4
Proficient
3
Adequate
2
Limited
1
Performing
calculations
for Question
1
Checking
solution(s)
against
NPV’S and
domain
Correctly
solving the
rational
equation
Setting up
a logical
rational
equation
Setting up
fractions to
represent
data
Performing
calculations
for question
2
Checking
solution(s)
against
NPV’S and
domain
Correctly
solving the
rational
equation
Setting up
a logical
rational
equation
Setting up
fractions to
represent
data
Insufficient or
Blank
Criteria
No score is
awarded
because there is
no evidence of
student
performance.
No data is
presented.
No explanation
is provided.
No score is
awarded
because there is
no evidence of
student
performance.
No data is
presented.
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.
Possible Solution to Rational Reno’s
1. Dave’s Rate:
1
6
1
9
Time together: x
Your Rate:
Equation:
x x
+ =1
6 9
LCD = 18
No NPV'S
3x 2x 18
+
=
18 18 18
5x = 18
x = 3.6
2. Let x represent the time required for Samantha to complete the job
working alone. Then, (x + 3) is the time required for you to complete the
job working alone.
1
1
1
+
=
x x +3 2
LCD is 2x x + 3
(
)
NPV'S are 0 and - 3
(
2 x +3
(
)
2x x + 3
)
+
(
(
)
)
x x +3
2x
=
2x x + 3
2x x + 3
(
)
2x + 6 + 2x = x 2 + 3x
0= x2 -x -6
(
)(
0 = x -3 x +2
)
x = 3, - 2


When x = -2 the equation checks, but it does not make physical
sense to have a negative time.
When x = 3 , the equation checks.
If x = 3 and x + 3 = 6
Samatha can do the job in 3 hours and you can do the job in 6 hours.
Mathematics 20-1
Rational Expressions and Equations
Page 13 of 33
STAGE 3
Learning Plans
Lesson 1
Simplifying Rational Expressions
STAGE 1
BIG IDEA: The study of rational expressions is foundational to further levels of mathematics. Rational
expressions, which are fractions involving variables, have applications in fields like physics, chemistry,
biochemistry, circuitry, economics and calculus.
ENDURING UNDERSTANDINGS:
Students will understand …

ESSENTIAL QUESTIONS:

Rational expressions and equations may have
restrictions.



What real life situations/careers require
rational expressions?
Why is dividing by zero undefined?
If a fraction can be represented pictorially can
a rational expression also be represented
pictorially?
If I can graph a vertical line, why is
5
0
undefined?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …






that common factors between the numerator
and denominator reduce to one
there may undefined values when there is a
variable in the denominator
restrictions are found on original expressions
a solution may not be a restriction
Mathematics 20-1
write equivalent forms of rational expressions
identify NPV
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.
Rational Expressions and Equations
Page 14 of 33
Lesson Summary



Students will compare the process of simplifying a fraction to the process of
simplifying a rational expression.
Students will factor polynomial expressions in the numerator and denominator
of a rational expression in order to simplify.
Students will discuss the causes of non-permissible values and determine their
values in rational expressions by factoring the denominator where necessary.
Lesson Plan
Prior to this lesson, students should have reviewed basic operations on fractions and
factoring polynomials.
Relating equivalent forms of rational expressions to equivalent fractions
1. Introductory activity (activate prior knowledge)
Put students in groups of 4 and give students fractions such as
3
and ask them to
4
3
and encourage students to come up with fractions that
4
they think will be different from other students.
 Have one student in each group write all their fractions on the board
 Look over the answers with the students and pick out interesting ones.
Hopefully some student will put in a variable, which you can use as a lead into
rational expressions.
3x
 If no students do that, you can propose an equivalent fraction (ie.
) with
4x
variables and ask students if they think it is equivalent and have a discussion.
 This leads to the discussion about “Is there any value that x can not be?”
give 3 equivalent fractions to
2. Transition to lesson’s main concept (equivalent rational expressions)
At this point you can suggest 2x and ask students to write an equivalent expression to
3x
using binomials.
4x
3 x +1

is an equivalent form involving fractions with binomials. Again discuss
4 x +1
(
(
)
)
“Is there any value that x can not be?”
Mathematics 20-1
Rational Expressions and Equations
Page 15 of 33

Discuss the result (as students may say the answer is 0 versus 1) when
x +1
simplifying
. If there is confusion, suggest to students to use
x +1
(
(
)
)
multiplication to check the quotient.
(
) (
) (
) (
)
0 x +1 ¹ x +1 , 1 x +1 = x +1


Put the following four on the board and ask students if they are equivalent.
2 x +1
2x + 2
x 2 + 3x + 2
( )
2 ( x - 2)
2x - 4
x2 -4
Ask students how we can tell
(
(
)
)
x +1
2x + 2
x 2 + 3x + 2
and
are
equivalent
to
.
2x - 4
x2 -4
x -2
Possible answers are:
o Factor the expressions and simplify like a fraction
o Substitute a number into both to verify you get the same number
 After trying a few numbers, you could suggest 2 and discuss nonpermissible values with students.
 Do a number of examples so students get the idea that only the
number 2 can’t be used to verify.
3. What is a non-permissible value?
You can discuss with students the difference between

Talk about

For

0
2
and .
2
0
6
 3 because we can fit three 2’s into 6.
2
0
= 0 , because 2 fits into 0, zero times.
2
2
For = undefined, because 0 fits into 2, an infinite number of times.
0
For which values are the following rational expressions undefined (for what values is
the denominator equal to zero?
Now students could be given some rational expressions and asked to simplify them
and discuss NPVs:
(m + 1) m - 4
n5
12x 3 y
12x 2
x 2 + 13x + 42
(
6x
)
(m + 2)(m + 5)
n 2  n  20
x 2 - 36
36x 2 y 2
It may be worthwhile to show students that the simplification of rational expressions
follows the same steps as prime factorization.
ex. The steps to simplify
Mathematics 20-1
are the same as
Rational Expressions and Equations
Page 16 of 33
2x 2 + 12x + 16 2( x 2 + 6x + 8) 2( x + 2)( x + 4) 2( x + 2)
=
=
=
3( x - 4)( x + 4) 3( x - 4)
3x 2 - 48
3( x 2 - 16)
In both of these examples, show how factors that are common to the numerator and
denominator can be removed as a common factor. It is important however that NPVs
are always identified before simplifying.
Going Beyond
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 6.1)
Supporting
Assessment
Glossary
equation - A statement of equality between two expressions
expression – A general term that ultimately represents a number. An expression can
consist of numbers, variables and operations on these.
non-permissible value – Any value for a variable that makes an expression
undefined [Math 20-1 (McGraw-Hill Ryerson: page 589)]
rational expression – An algebraic fraction with a numerator and denominator that
are polynomials [Math 20-1 (McGraw-Hill Ryerson: page 590)]
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
Rational Expressions and Equations
Page 17 of 33
Lesson 2
Multiplying Rational Expressions
STAGE 1
BIG IDEA: The study of rational expressions is foundational to further levels of mathematics. Rational
expressions, which are fractions involving variables, have applications in fields like physics, chemistry,
biochemistry, circuitry, economics and calculus.
ENDURING UNDERSTANDINGS:
Students will understand …


ESSENTIAL QUESTIONS:

The operations on rational expressions are
similar to fractional operations.
Rational expressions and equations may have
restrictions.



What real life situations/careers require
rational expressions?
Why is dividing by zero undefined?
If a fraction can be represented pictorially can
a rational expression also be represented
pictorially?
If I can graph a vertical line, why is
5
0
undefined?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …









when to find a common denominator
factoring a denominator leads to a common
denominator
that common factors between the numerator
and denominator reduce to one
there may undefined values when there is a
variable in the denominator
restrictions are found on original expressions
a solution may not be a restriction
Mathematics 20-1
+, -, x, ÷ rational expressions
write equivalent forms of rational expressions
identify NPV
Rational Expressions and Equations
Page 18 of 33
Lesson Summary

Students will learn that rational expressions are a special type of fraction and
therefore the operation of multiplying is the same.
Lesson Plan
9
of all money for prizes and the jackpot winner
14
Hook: A certain lottery pays out
2
of the payout. For one particular lottery, there are 3 winners so each winner
5
1
gets of the jackpot payout. What fraction of the total money collected does each
3
jackpot winner get?
gets
Get a couple of students to put their answers on the board. Talk about how you could
9 2 1 3×3 2 1 3
· · =
· · =
simplify
before multiplying compared to multiplying
14 5 3 2 × 7 5 3 35
9 2 1 18
3
· · =
first and then simplifying to
.
14 5 3 210
35
Lesson:
Since the operations on rational expressions are the same as regular fractions, give
students the following examples to try with a partner. Tell students to come up with a
strategy for multiplying any rational expression.
2x + 10
7x
·
21x
3x + 15
x -2 x -3
x + 4 x +1
2.
·
x -2 x +4
x +2 x -3
1.
(
(
)(
)(
) (
) (
)(
)(
)
)
Students may have different strategies such as:
1. Multiply all terms together first, students should see that they will get a 4th degree
polynomial on the top and bottom and then are not able to simplify.
2. Cancelling factors that are common in the numerator and denominator.
( x - 2) ( x - 3 ) · ( x + 4) ( x + 1) = ( x + 1)
( x - 2) ( x + 4 ) ( x + 2) ( x - 3) ( x + 2)
Mathematics 20-1
Rational Expressions and Equations
Page 19 of 33
At this point mention that anytime we have variables in the denominator, we have to
list all the non-permissible values before simplifying (cancelling factors).
A variety of examples should be given.
Ask students to develop a way to simplify the following:
x 2 - 5x + 6 x 2 + 5 x + 4
×
x 2 + 2x - 8 x 2 - x - 6
Many students will probably see from the previous example that they run into the
problem of not being able to simplify once they have multiplied and will not be able to
factor the quartic polynomials.
Going Beyond
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 6.2)
Supporting
Assessment
Glossary
equation - A statement of equality between two expressions
expression – A general term that ultimately represents a number. An expression can
consist of numbers, variables and operations on these.
non-permissible value – Any value for a variable that makes an expression
undefined [Math 20-1 (McGraw-Hill Ryerson: page 589)]
Mathematics 20-1
Rational Expressions and Equations
Page 20 of 33
operation – Associates two or more members of a set with one of the members of the
set. The basic operations in mathematics are addition, subtraction, multiplication,
division and exponentiation.
rational equation – An equation formed by setting a rational expression equal to 0
rational expression – An algebraic fraction with a numerator and denominator that
are polynomials [Math 20-1 (McGraw-Hill Ryerson: page 590)]
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
Rational Expressions and Equations
Page 21 of 33
Lesson 3
Dividing Rational Expressions
STAGE 1
BIG IDEA: The study of rational expressions is foundational to further levels of mathematics. Rational
expressions, which are fractions involving variables, have applications in fields like physics, chemistry,
biochemistry, circuitry, economics and calculus.
ENDURING UNDERSTANDINGS:
Students will understand …


ESSENTIAL QUESTIONS:

The operations on rational expressions are
similar to fractional operations.
Rational expressions and equations may have
restrictions.



What real life situations/careers require
rational expressions?
Why is dividing by zero undefined?
If a fraction can be represented pictorially can
a rational expression also be represented
pictorially?
If I can graph a vertical line, why is
5
0
undefined?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …










when to find a common denominator
factoring a denominator leads to a common
denominator
that division is multiplication of the reciprocal
that common factors between the numerator
and denominator reduce to one
there may undefined values when there is a
variable in the denominator
restrictions are found on original expressions
a solution may not be a restriction
Mathematics 20-1
+, -, x, ÷ rational expressions
write equivalent forms of rational expressions
identify NPV
Rational Expressions and Equations
Page 22 of 33
Lesson Summary

Students will learn that rational expressions are a special type of fraction and
therefore the operation of dividing is the same.
Lesson Plan
Activate Prior Knowledge
Spend some time reviewing dividing fractions
1
 10 ¸ 2 = 10 ·
2
The same concepts also apply to rational functions
Lesson:
Let students try the first example:
5x 3x
5x 4
¸
®
·
3
4
3 3x
The equation started with no non-permissible values, but if there are any we would
state them.
When we perform the reciprocal of the second term we need to consider the nonpermissible value since there is now an unknown in the denominator. x ¹ 0
Follow the same process as outlined in multiplying rational expressions:
5x 4
20
·
®
, x ¹0
3 3x
9
We need to remember that even though the final answer has no variable the nonpermissible value still exists.
Since we have already covered multiplication of rational functions you may now give
the students more difficult questions and ask them to follow the steps:
1. Factor and check for NPV’s
2. Reciprocate and state new NPV’s
3. Simplify
Mathematics 20-1
Rational Expressions and Equations
Page 23 of 33
Example
x 2 + 5x + 4 x 2 - 1
¸
2x 2 - 8 x + 8 4 x - 8
Step 1
( x + 1)( x + 4) ( x -1)( x + 1)
¸
,x ¹2
2( x - 2)( x - 2)
4( x - 2)
Step 2
( x + 1)( x + 4)
4( x - 2)
·
, x ¹ ±1, 2
2( x - 2)( x - 2) ( x - 1)( x + 1)
Step 3
( x + 1)( x + 4)
4( x - 2)
4( x + 4)
2( x + 4)
·
®
®
, x ¹ ±1, 2
2( x - 2)( x - 2) ( x - 1)( x + 1)
2( x - 2)( x -1) ( x - 2)( x - 1)
Another possible activity:
Have the students develop their own division of rational expression question and trade
with a partner.
Going Beyond
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 6.2)
Supporting
Assessment
Glossary
equation - A statement of equality between two expressions
expression – A general term that ultimately represents a number. An expression can
consist of numbers, variables and operations on these.
Mathematics 20-1
Rational Expressions and Equations
Page 24 of 33
non-permissible value – Any value for a variable that makes an expression
undefined [Math 20-1 (McGraw-Hill Ryerson: page 589)]
operation – Associates two or more members of a set with one of the members of the
set. The basic operations in mathematics are addition, subtraction, multiplication,
division and exponentiation.
rational expression – An algebraic fraction with a numerator and denominator that
are polynomials [Math 20-1 (McGraw-Hill Ryerson: page 590)]
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
Rational Expressions and Equations
Page 25 of 33
Lesson 4
Adding and Subtracting Rational Expressions
STAGE 1
BIG IDEA: The study of rational expressions is foundational to further levels of mathematics. Rational
expressions, which are fractions involving variables, have applications in fields like physics, chemistry,
biochemistry, circuitry, economics and calculus.
ENDURING UNDERSTANDINGS:
Students will understand …


ESSENTIAL QUESTIONS:

The operations on rational expressions are
similar to fractional operations.
Rational expressions and equations may have
restrictions.



What real life situations/careers require
rational expressions?
Why is dividing by zero undefined?
If a fraction can be represented pictorially can
a rational expression also be represented
pictorially?
If I can graph a vertical line, why is
5
0
undefined?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …








when to find a common denominator
factoring a denominator leads to a common
denominator
that common factors between the numerator
and denominator reduce to one
there may undefined values when there is a
variable in the denominator
restrictions are found on original expressions
+, – rational expressions
write equivalent forms of rational expressions
identify NPV
Lesson Summary

Students will learn that rational expressions are a special type of fraction and
therefore the operations of addition and subtraction are the same.
Mathematics 20-1
Rational Expressions and Equations
Page 26 of 33
Lesson Plan
Lesson Goal

Students will be able to perform the operations of addition and subtraction on
rational expressions.
Activate Prior Knowledge
Review adding and subtracting fractions:
2 3
+ =
3 4
7 5
- =
4 6
For both of these questions students will have to find a lowest common denominator.
Have a couple of students put their answers on the board and in particular, show how
they found the lowest common denominator.
It would make sense for students to find the lowest common denominator using prime
factorization, as this is the process required for finding the lowest common
denominator for adding and subtracting rational expressions.
Since students may be able to find the lowest common denominator of 12 for both
5
1
+
without prime factorization, a harder example such as:
, where most students
96 72
will not be able to find the lowest common denominator of 216 mentally.
Lesson
Non-permissible values can be integrated throughout this lesson.
Now introduce adding and subtracting rational expressions:
Have students try the following in groups of 2 or 3 and have each group put their
answer on the board:
2
3
2 3
2
3
2
3
+
=
+
=
+
=
+
=
3x 4x
3 4x
3 x+2
x x+2

Have students develop strategies for simplifying the above rational
expressions.
Mathematics 20-1
Rational Expressions and Equations
Page 27 of 33
Move onto a similar problem such as:



1
+
( x + 1) ( x + 2) (
5
x + 2 x -1
)(
)
After the last question, students have hopefully picked out that the
denominators need to be the same (and that they need to have the same
factors).
æ 5
1ö
If they did the previous example, ç
+ ÷ , using prime factorization, they
è 96 72 ø
should have seen that you do not need to repeat a factor that is in both
denominators.
Discuss student answers, some students will find a common denominator of
x + 2 x + 1 x -1 and x + 2 x + 1 x + 2 x -1 .
(
)(
)(
)
(
)(
)(
)(
)
o As long as students completed the question correctly, you can simplify the
answer to show they are equivalent.
At this point have students list the important steps for adding and subtracting rational
expressions.
 You may have to guide students.
Continue with a variety of examples such as:
8x
14x 2 y
, where the numerator and denominators are monomials.
+
4xy 2 7x 3 y 3
3
x +1
+ 2
, where students can see the benefit of trying to simplify first before
x + 3 x - 3x - 4
3
1
¹
finding a common denominator and have a discussion of why
. Students
x + 3 x +1
can verify this with a variety of numbers.
3
2
, where there is a common factor in the denominator
x 2 - x - 6 x 2 - 2x - 3
2x + 1
x +1
, where you can factor and simplify the numerators and
2
2x - 3 x - 2 2 x - x - 6
denominators of the subtrahend and/or minuend before finding the common
denominator.
2
x2 -x -2
x2 -x -6
, where you can factor numerators and denominators and
x 2 + 6x + 5 x 2 + 5x + 6
simplify before finding the common denominator.
Mathematics 20-1
Rational Expressions and Equations
Page 28 of 33
Going Beyond
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 6.3)
Supporting
Assessment
Glossary
expression – A general term that ultimately represents a number. An expression can
consist of numbers, variables and operations on these.
non-permissible value – Any value for a variable that makes an expression
undefined [Math 20-1 (McGraw-Hill Ryerson: page 589)]
operation – Associates two or more members of a set with one of the members of the
set. The basic operations in mathematics are addition, subtraction, multiplication,
division and exponentiation.
rational expression – An algebraic fraction with a numerator and denominator that
are polynomials [Math 20-1 (McGraw-Hill Ryerson: page 590)]
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
Rational Expressions and Equations
Page 29 of 33
Lesson 5
Solving Rational Equations
STAGE 1
BIG IDEA: The study of rational expressions is foundational to further levels of mathematics. Rational
expressions, which are fractions involving variables, have applications in fields like physics, chemistry,
biochemistry, circuitry, economics and calculus.
ENDURING UNDERSTANDINGS:
Students will understand …


ESSENTIAL QUESTIONS:

The operations on rational expressions are
similar to fractional operations.
Rational expressions and equations may have
restrictions.



What real life situations/careers require
rational expressions?
Why is dividing by zero undefined?
If a fraction can be represented pictorially can
a rational expression also be represented
pictorially?
If I can graph a vertical line, why is
5
0
undefined?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …










when to find a common denominator
factoring a denominator leads to a common
denominator
that division is multiplication of the reciprocal
that common factors between the numerator
and denominator reduce to one
there may undefined values when there is a
variable in the denominator
restrictions are found on original expressions
a solution may not be a restriction
Mathematics 20-1



+, -, x, ÷ rational expressions
write equivalent forms of rational expressions
solve a rational equation, limited to
numerators involving monomials, binomials
and trinomials
model a situation using a rational equation.
identify NPV
verify a solution
Rational Expressions and Equations
Page 30 of 33
Lesson Summary




Students will learn to building rational equations based on word problems
Students will factor the denominators of a rational equation in order to identify
the least common denominator.
Students will learn to multiply a rational equation through by its least common
denominator in order to create a polynomial equation, which they will solve.
Students will verify their answer(s) for NPVs.
Lesson Plan
Hook
Consider using Dan Meyer’s Bean Counting lesson/video found here:
http://blog.mrmeyer.com/?p=9608 (this blog also offers insight into how to introduce
and discuss this problem).
Lesson Goal
Students will learn to construct a rational equation to model a real-life situation
(generally involving rates). Students will learn to solve rational equations and verify
their answer(s) in case of against non-permissible values (NPVs).
Activate Prior Knowledge
Students should be reminded of NPVs (and how to identify them) and factoring.
Lesson
Begin with an equation involving fractions and discuss the steps necessary to solve.
ex.
1 x 1
+ =
5 4 3
One possible strategy would be to identify the least common multiple of the
denominator and multiply both sides of the equation through by this LCM. This will
allow us to transform a rational equation into a polynomial one.
Mathematics 20-1
Rational Expressions and Equations
Page 31 of 33
Though the LCM of the above example is 60, let’s write 3 x 4 x 5.
é 1ù
éxù
é 1ù
(3 ´ 4 ´ 5) ´ ê ú + (3 ´ 4 ´ 5) ´ ê ú = (3 ´ 4 ´ 5) ´ ê ú
ë5û
ë4û
ë3û
(3 ´ 4) ´ éë1ùû + (3 ´ 5) ´ éë x ùû = (4 ´ 5) ´ éë1ùû
12 + 15 x = 20
15 x = 8
x =
8
15
Let’s now see how this applies to a rational equation.
x - 2 x -1 x + 2
+
=
x
x +1
2x
LCM = (2x)(x + 1)
éx - 2 ù
é x - 1ù
é x + 2ù
(2x)(x + 1) ê
+ (2x)(x + 1) ê
= (2x)(x + 1) ê
ú
ú
ú
ë x û
ë x + 1û
ë 2x û
(2)(x + 1)(x - 2) + (2x)(x - 1) = (x + 1)(x + 2)
(2x 2 - 2x - 4) + (2x 2 - 2x) = x 2 + 3x + 2
4x 2 - 4x - 4 = x 2 + 3x + 2
3x 2 - 7x - 6 = 0
(3x + 2)(x - 3) = 0
2
or 3
3
These two solutions should be verified to ensure both that they are correct
and that they are elements of the equation's domain (no NPVs).
x=-
Continue with more examples, as needed. It would likely help to show an example
where the LCM of the denominators is less obvious.
2x + 9
x
12
+
=
x + 7x + 12 x + 3 3x + 12
2
Once students understand how to solve a rational equation, move on to word
problems (such as distance-rate-time problems) and discuss strategies for
constructing a table and, from it, a rational equation.
Going Beyond
Give students an example with an irrational solution that will require the use of the
quadratic formula.
Mathematics 20-1
Rational Expressions and Equations
Page 32 of 33
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 6.4, pages 341-351)
Supporting
Assessment
Glossary
equation - A statement of equality between two expressions
expression – A general term that ultimately represents a number. An expression can
consist of numbers, variables and operations on these.
non-permissible value – Any value for a variable that makes an expression
undefined [Math 20-1 (McGraw-Hill Ryerson: page 589)]
operation – Associates two or more members of a set with one of the members of the
set. The basic operations in mathematics are addition, subtraction, multiplication,
division and exponentiation.
rational equation – An equation formed by setting a rational expression equal to 0
rational expression – An algebraic fraction with a numerator and denominator that
are polynomials [Math 20-1 (McGraw-Hill Ryerson: page 590)]
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
Rational Expressions and Equations
Page 33 of 33