Download Unit Five Rational Expressions 18 Hours Math 521B

UNIT FIVE
RATIONAL EXPRESSIONS
18 HOURS
MATH 521B
Revised Nov24, 00
107
SCO: By the end of
grade 11 students will be
expected to:
B1 model (with concrete
materials and pictorial
representations) and
express the
relationship between
arithmetic operations
and operations on
algebraic expressions
and equations
Elaborations - Instructional Strategies/Suggestions
Note to Teachers: Students may need a day or two of review on factoring
What is a Rational Expression?
Encourage students to create a definition for a rational number.
Common fraction definition
A rational number is a number of the form
a
where a, b 0 I and b… 0.
b
Decimal fraction definition
A rational number is any terminating or non-terminating repeating
decimal fraction.
Ex.
1
= 0.5
2
2
= 0. 6
3
1
= 0.142857
7
A rational algebraic expression is a fraction where the numerator and/or
denominator are polynomials.
Simplifying Rational Expressions
Challenge students to simplify expressions like
1)
8
12
2)
15 x 3 y 4
20 x 2 y 7
3)
x 2 + 3x + 2
x2 − x − 6
After a short discussion with students on how a rational expression is
reduced, students should be able to develop rules for arithmetic or
algebraic rational expressions.
Find the largest possible factor common to the numerator and
denominator. For algebraic expressions this will involve:
< factor the numerator and denominator separately
< state the restrictions
< cancel out the common factors
x 2 + 3 x + 2 ( x + 1)( x + 2 ) x + 1
Ex:
=
=
, x ≠ −2,3
x 2 − x − 6 ( x − 3)( x + 2 ) x − 3
Rational expressions give people who use statistics flexibility in using
formulas in their fields.
108
Worthwhile Tasks for Instruction and/or Assessment
What is a Rational Expression?
Group Activity
Generate a list of as many different appearing rational
numbers as possible.
Discussion
For what values are the following rational expressions
undefined:
x
x+4
3x − 1
2)
2x − 5
4x
x + 5x + 6
x +1
4) 2
x − 3x + 10
1)
3)
2
Suggested resources
What is a Rational Expression?
Math Power 10 p.158 # 51
Algebra, Structure & Method Book 2
p.229 # 21-28
See Worksheet at the end of the unit
on restriction implications.
Problem Solving
p.146 #1-3
Journal
Evaluate each rational expression for x = 3 and notice a
pattern:
1)
4
x−3
2)
2
x −9
2
3)
1
x + 4x + 3
2
Simplifying Rational Expressions
Performance
a) For pattern 1 express the number of stars in the nth diagram
in terms of “n”.
b) Repeat for pattern 2.
c) write rational expression that represents the ratio of pattern
1 to pattern 2. Simplify the expression.
d) for what values is the rational expression undefined?
Pattern 1
continued on p. 5
109
Simplifying Rational Expressions
Choose sparingly from
Math power 10 p.158 #19-47 odd
p.159 # 53,54,56
Algebra, Structure & Method Book 2
p.228 #1-15 odd
29-35 odd
SCO: By the end of grade
11 students will be
expected to:
B1 model (with concrete
materials and pictorial
representations) and
express the
relationship between
arithmetic operations
and operations on
algebraic expressions
and equations
B8 understand the
relationships that exist
between operations on
integers, fractions and
decimals, and
operations on
algebraic expressions
Elaborations - Instructional Strategies/Suggestions
Multiplying or Dividing Rational Expressions
Challenge student groups to simplify rational arithmetic and algebraic
expressions. They should appreciate that the process is the same for
both. Students must state any restrictions on the variable for rational
algebraic expressions.
Ex.
1)
18 14
×
21 24
2)
10 15
÷
21 28
3)
x2 + x − 6
x−3
×
2
x + 2 x − 15 x − 2
a2 − 4
2a − 4
4)
÷ 2
a + 3 a + 2a − 3
B37 demonstrate number
and operation sense in
solving inequality
relationships and
operating on rational
expressions
Worthwhile Tasks for Instruction and/or Assessment
110
Suggested resources
Simplifying Rational Expressions
continued from p. 3
Simplifying Rational Expressions
Journal
Why must restrictions be considered for rational expressions?
Multiplying or Dividing Rational
Expressions
Math Power 10
Multiplying or Dividing Rational Expressions
p.163 #17-27 odd
#37-47 odd
p.164 #53-59
Algebra, Structure & Method Book 2
p.234 # 3-21 odd
Group Activity
Write two rational expressions with binomial denominators
that have a product of
8
x − x−6
2
111
SCO: By the end of grade
11 students will be
expected to:
B1 model (with concrete
materials and pictorial
representations) and
express the
relationship between
arithmetic operations
and operations on
algebraic expressions
and equations
B8 understand the
relationships that exist
between operations on
integers, fractions and
decimals, and
operations on
algebraic expressions
Elaborations - Instructional Strategies/Suggestions
+, ! Rational Expressions
Use and investigate a range of examples to allow student groups to
generate algorithms for adding and subtracting rational expressions.
Begin with examples that have numeric denominators, progressing to
examples containing algebraic denominators. Hopefully, student groups
will be able to generate the algorithms and see the common method no
matter what the denominator.
Good examples to challenge the groups with might be:
a)
5 1 3
+ −
6 2 8
b)
3
1
5
− 2 + 3
5a a
2a
c)
x
x−4
−
x + 3 x +1
x 2 − 9 x + 20 x 2 − 9 x + 18
d) 2
−
x − 7 x + 12
x2 − 9
B37 demonstrate number
and operation sense in
solving inequality
relationships and
operating on rational
expressions
112
Worthwhile Tasks for Instruction and/or Assessment
Suggested resources
Multiplying or Dividing Rational
Expressions
Multiplying or Dividing Rational Expressions
Pencil/Paper/Discussion
Write, but don’t simplify, expressions for ªABC and ªDBC.
Then write and simplify an expression for the ratio of the
area of ªABC to the area of ªDBC.
+, ! Rational Expressions
Pencil/Paper
Find the perimeter of this triangle:
Performance
Write two rational expressions with binomial denominators
that have a sum of
4x − 5
.
( x + 1)( x − 2)
+, ! Rational Expressions
Choose sparing from:
Math Power 10
p.167 #19,21,23
p.168 #29-31
p.172 #27-31 odd
#43-53 odd
# 64,66
p.173 # 67
Algebra, Structure & Method Book 2
p.237 #1,3,7,8,27,
#33-36
Problem Solving
p.203 #3,6,7
113
SCO: By the end of grade
11 students will be expected
to:
Elaborations - Instructional Strategies/Suggestions
Rational Equations
Allow time for student groups to investigate a range of rational
equations including:
A6 apply properties of
numbers when operating
upon and expressing
equations
< equations with numerical denominators
< equations with algebraic denominators
< applications
< geometric applications
D4 apply the Pythagorean
Theorem
Remind groups that restrictions to the denominator are important here
also.
D5 apply and determine
formulae for perimeter,
area, surface area and
volume
114
Worthwhile Tasks for Instruction and/or Assessment
Suggested resources
Rational Equations
Rational Equations
Research/Presentation
Find the length of the bases and the altitude of the trapezoid if
it has an area of 2 m2.
Algebra, Structure & Method Book 2
p.245 #1,3,11,17,19
p.246 #5,6,9
p.249 #13-23 odd
p.250 #5,6,9,12
Pencil/Paper
Find the dimensions of the rectangle below which has a
shaded area of 64 cm2.
Pencil/Paper
Solve each of the following:
a)
x
x − 2
x +1
−
=
−1
8
3
6
b)
x2
2x
1
=
+
2
15
10
c)
y
2
10
−
= 2
y − 2
y + 3
y + y − 6
115
Rational Restrictions Worksheet
Pencil/Paper/Technology
Graph each of the following and show the discontinuity (applicable restrictions):
x2 − 1
a) y =
x +1
x2 − 2x + 1
b) y =
x −1
c) y =
x2 − 1
x −1
d) y =
x2 − 4
x + 2
x2 + x − 6
e) y =
x − 2
x2 + 2x − 3
f) y =
x + 3
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Note to Teacher:
TI-83 Solution
If you press zoom 4: decimal it will show the hole in the graph. You may
have to set the window dimensions to multiples of the basic to get some of
the problems to fit on the TI-83 screen.
basic dimensions
the dimensions I used
press graph
if you press trace and cursor along the graph to where x = ! 1, you will see no function value
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