Download Unit One Radicals 15 Hours Math 521B

UNIT ONE
RADICALS
15 HOURS
MATH 521B
Revised Nov 9, 00
19
SCO: By the end of grade
11 students will be
expected to:
A3 demonstrate an understanding of the role of
irrational numbers in
applications
Elaboration - Instructional Strategies/Suggestions
What is a radical?
Invite a short discussion by student groups on the term “radical”. Ideas
they come up with could be listed on the board. These might include:
< a radical is a compact way of writing an irrational number similar to
scientific notation being a compact and convenient way of writing
very small or very large numbers.
< parts of a radical are
! radical symbol
! radicand
! index (degree of the root)
radical symbol
A4 approximate square
roots
9
Index º
B9 use the calculator
correctly and
efficiently for various
computations
3
34 » radicand
Estimation of Radicals
Encourage student groups to use guess and check to estimate the values
of various degree roots and check their accuracy using a scientific
calculator.
Simplifying Radicals
a) entire to mixed
Challenge students to reduce the size of the radicand as much as
possible and still have an equivalent radical expression. The idea is to
find two factors of the radicand, one of which is the highest possible
perfect square (cube, etc.).
48 = 16 × 3 = 4 3
Ex.
3
48 = 3 8 × 3 6 = 2 3 6
For even degree roots of variable radicands the concept of principal
root must be considered.
Example: the square root of 16 can be either ± 4
The symbol
ˆ
means principal square root (or positive root)
16 = 4 and not ! 4
20
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
What is a radical?
Journal
Write a short explanation of the advantages of writing
numbers in radical form.
What is a radical?
Estimation of Radicals
Estimation/Technology
Use guess and check to approximate, to the nearest hundredth,
the value of the following. Then use a calculator to check the
accuracy of your guess.
Estimation of Radicals
Math Power 10 p.14 # 3-17 odd
Algebra, Structure and Method
Book 2 p.268 # 33-38
a)
b)
3
c)
4
d)
5
40
9
20
200
Technology
Use decimal approximations to arrange the following from
smallest to largest.
15 , 2 2 ,
5 ,4,
3
1)
128
2)
68x
3
128
5
160
16x
4)
5)
3
Math Power 10 p.15 # 31-33
50 , 2.3
Simplifying Radicals
a) entire to mixed
Pencil/Paper
Write each of the following in simplest form.
3)
Problem Solving
Proof by contradiction p.39 #2
Simplifying Radicals
a) entire to mixed
Math Power 10 p.19 # 1-15 odd
p.20 # 59
Algebra, Structure and Method
Book 2 p.268 # 19,20,31,39-42
3
Applications
Math Power 10 p.20 # 56,57
4
21
SCO: By the end of grade
11 students will be
expected to:
Elaboration - Instructional Strategies/Suggestions
Simplifying Radicals (cont’d)
Only when the radicand is a variable, must we consider the concept of a
principal root.
For example, if the question is: Simplify:
x
2
then we must ensure
that the answer is positive. We do this by using absolute value symbols.
A8 demonstrate an understanding of and apply
properties to
operations involving
square roots
x2 = x
b) mixed to entire
Challenge students to write a mixed radical as an entire radical.
Note to Teachers: As a rationale, look at arranging in order from
smallest to largest:
3
2
, 2 3 , 3 3 , 5 2
Operations with Radicals
a) add and subtract
Challenge students to attempt problems like
8 + 2 . Have the
students use a calculator to get the approximate values of
and 2 ,
then add these values. Now invite students to do the problem without
evaluating the square roots first. Students might try adding the radicands
8
to get 10 .
Through discussion and discovery the rule below should be developed:
Adding or subtracting radical expressions is equivalent to combining
variable expressions with like terms.
Example:
2x + 3x = x + x + x + x + x = 5x
2 5+3 5= 5+ 5+ 5+ 5+ 5=5 5
22
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
b) mixed to entire
Math Power 10 p.19 # 25-30,31-33
b) mixed to entire
Pencil/Paper
Write each of the following as entire radicals
1) 4 3
2) 33 2
3) 2 4 3
Pencil/Paper
Without using a calculator, arrange the following from
smallest to largest:
a) 3 5 , 2 11 , 4 3 , 5 2
b) 63 2
, 2 10 , 34 3 , 7 2
Operations with Radicals
a) add or subtract
Math Power 10 p.23 # 15-28
p.24 # 67
Operations with Radicals
a) add or subtract
Pencil/Paper
Simplify:
Algebra, Structure & Method Book 2
p.272 # 1-12
32 + 8 + 3 50
2) 34 32 − 4 162 + 4 512
1)
3)
3
81 +
3
−24 +
3
192
Journal
Explain the rules regarding adding or subtracting radical
expressions.
23
SCO: By the end of grade
11 students will be
expected to:
Elaboration - Instructional Strategies/Suggestions
Operations with Radicals
b) Multiply and divide
Challenge student groups to carry out investigations about multiplying
and dividing radical expressions like:
2× 8
A8 demonstrate an understanding of and apply
properties to
operations involving
square roots
20
5
or
Students could use a calculator to evaluate the individual radical
expressions before doing the multiplying or dividing. They could
develop notions about the rules for solving these types of problems if
the multiplying or dividing is done first.
2 × 8 = 16 = 4
20
5
= 4 =2
The students should appreciate the reversibility of the operations and
when it is advantageous to do a particular operation first.
Note to Teachers; Any radical expressions can be multiplied if
changed to exponential form. (See p.11)
24
Worthwhile Tasks for Instruction and/or Assessment
Operations with Radicals
b) Multiply and divide
Choose sparingly from:
Math Power 10 p.19 #34-45
p.23 # 29-42
p.24 # 65
Operations with Radicals
b) Multiply and divide
Pencil/Paper
Simplify:
1) 2 3 × 3 5
2)
4 6
2 3
Applications
Math Power 10 p.26-7 # 1-4 green
p.38 # 93
3) 4 3 (5 2 − 6 5 )
4) ( 2 3 +
Suggested Resources
5 )( 7 − 6 2 )
5) 3 2 ( 2 3 12 − 3 36 )
Pencil/Paper
Determine the area of this rectangle.
Algebra, Structure & Method Book 2
p.268 # 13-18,25-28
p.272 # 19-24,31,32
p.276 # 1-6,9-12
15-18,21-26
28-30
4− 2
2+ 2
Pencil/Paper
Find the distance from A to B in simplest radical form.
2 3
4 2
2 3
25
SCO: By the end of grade
11 students will be expected
to:
Elaboration - Instructional Strategies/Suggestions
Rationalizing denominators
Initiate a discussion on what rational and non-rational denominators
would look like. Hopefully students could predict that a non-rational
denominator is in an irrational form (for our purposes that means a
radical). Challenge student groups to develop a method for making the
denominator rational.
A8 demonstrate an understanding of and apply
properties to
operations involving
square roots
For a rational expression with a monomial radical denominator,
multiply by the radical part of the denominator over itself.
B2 develop algorithms and
perform operations on
irrational numbers
4
4
2
4 2
2 2
=
×
=
=
6
3
3 2
3 2
2
Ex.
For a rational expression with a binomial radical denominator,
multiply by the conjugate of the denominator over itself.
3
Ex.
5 +1
=
3
5 +1
5 −1
×
5 −1
=
3 5−3
4
For rational expressions with higher degree monomial radical
denominators, multiply by the correct form of (1).
Ex.
3
1
=
2
3
1
×
2
26
3
3
4
=
4
3
4
2
Worthwhile Tasks for Instruction and/or Assessment
Rationalizing denominators
Choose sparingly from:
Math Power 10 p.20 # 46-54
p.23 # 43-52
p.25 # 17-22
Rationalizing denominators
Pencil/Paper
Rationalize the following denominators:
1)
2)
2 5
6 3
3)
3− 6
2
3
2 +5 3
Applications
Math Power 10 p.24 # 68-70
3
4)
5
23 3
Pencil/Paper
If a rectangle has an area of 6 cm2 and a width of
cm, what is the length in simplest radical form?
Suggested Resources
5− 2
Pencil/Paper
Calculate the perimeter of each figure in simplest radical
form. Then write the ratio of the perimeter of the larger figure
to that of the smaller figure in simplest radical form
(remember to rationalize the denominator).
Journal
Write a short series of instructions to explain how radical
denominators are rationalized.
27
Algebra, Structure & Method Book 2
p.267-8 # 9-12,22,29,
30,32
p.272 # 13-18
SCO: By the end of grade
11 students will be
expected to:
B9 use the calculator
correctly and
efficiently for various
computations
Elaboration - Instructional Strategies/Suggestions
Rational exponents
Invite students to discuss and give examples of expressions with
rational exponents. Challenge students to write the following in order
from smallest to largest.
271/3, 271, 274/3, 271/2, 272, 270, 272/3 , 27!1/3
Because all have the same base, they should be able to order them
simply by looking at the relative sizes of the exponents. If, however, the
bases were not all the same they would have to be able to evaluate each
expression.
On the TI-83:
Students should also understand that raising a base to a power of ½ is
equivalent to taking the square root of the base. Allow student groups to
try various operations on their calculator to evaluate 271/2. Hopefully
they will on their own find that 271/ 2 = 27 .
If they look at 271/3 on their calculator hopefully it can be equated with
cube rooting. Looking at 272/3 and using power law, this can be thought
1/3 2
of as (27 ) or
e27 j
2
3
m
an
=
n
a m = (n a ) m
ˆ the operations done to the 27 are cube rooting and squaring. The
order that these are done in are reversible.
In general, for rational exponents:
a) the denominator of the exponent is the index of the radical
b) the numerator of the exponent is the power to which the answer
from part (a) is raised
Students should see the connection between radical expressions and
rational exponents.
Most radicals represent irrational numbers but many times it is more
convenient and compact to do operations like +, !, × and ÷ with the
irrational numbers in radical form.
Extension:
Perhaps a mention of the historical use of logarithm tables could occur
here. For example, a problem like 73/5 is easily solved today with a
scientific calculator. Not too many years ago this type of problem would
be solved using logarithms. A logarithm is a specialized form of an
exponent.
28
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Rational exponents
Math Power 10 p.37
Rational exponents
Discussion
Arrange the following in order from smallest to largest:
271/3 , 271 , 274/3 , 271/2 , 272 , 270 , 272/3
Algebra, Structure & Method Book 2
p.457 # 1-8
p.458 # 1-19 odd
p.458 # 29-35 odd
Pencil/Paper
Write in simplest radical form (simplify further, if possible):
a) 84/3
e)
b) 4!3/2
f)
c) 6251/4
g)
6
83
8
94
10
505
Problem solving
Guess and Check p.11 #3,5,6,9
d) 95/2
Pencil/Paper
Write in exponential form and simplify if possible:
x ×3 x ×6 x
a)
b)
4
x ×6 x ÷3 x
Pencil/Paper
Express in simplest radical form (Hint: it may be necessary to
temporarily convert to exponential form)
27 × 6 27
10
32 ÷
8
4
29
SCO: By the end of grade
11 students will be
expected to:
Elaboration - Instructional Strategies/Suggestions
Radical equations (5.7)
Invite students to discuss strategies for solving equations. Hopefully
they will see that the same method is always used, ie. whatever
operations have been performed on the variable must be undone.
C12 solve linear, simple
radical, exponential
and absolute value
equations or linear
inequalities
For the problem:
2 x +1 + 4 = 0
the steps are:
1) Isolate the radical term
2 x + 1 = −2
x + 1 = −1
x +1 = 1
2) Square both sides
x = −2
3) Solve for the variable
4) Verify by substituting into the original equation. Here we see the
answer does not verify; it is an extraneous root. The left and right sides
are not equivalent.
5) State the conclusion
There are no real roots for this problem.
Note to the Teacher: Steps 4 and 5 can be eliminated for odd-degree
radical equations.
Worthwhile Tasks for Instruction and/or Assessment
30
Suggested Resources
Radical equations (5.7)
Radical equations
Guess and Check
Solve and verify:
Math Power 11 p.323 # 23-36
Algebra, Structure & Method Book 2
p.280 # 1-10
x =3
1)
2)
x −6=0
3)
x +1 = 5
4) 2 x − 3 = 4
x + 2 = 0
5)
Pencil/Paper
Solve algebraically, verify your answer:
1)
3 x − 1 = 20
2)
4− x =7
3)
x+2 +9=4
4)
3
x +1 = 2
5)
4
x−2 =2
Written Assignment
Create a problem where there is a radical equation to solve.
Research
Search various fields of study (ex. Physics, Chemistry,
Economics, Business, etc.) to find two formulas containing a
radical. Give a short description of their use in those fields.
Research
Write a short paper on the life and contributions to
mathematics of Julius Wilhelm Richard Dedekind.
31
SCO: By the end of grade
11 students will be
expected to:
C12 solve linear, simple
radical, exponential
and absolute value
equations or linear
inequalities
Elaboration - Instructional Strategies/Suggestions
Equations with Rational Exponents
Invite students to discuss possible methods of solving problems like:
x2/3 = 16
One possible method might be by guess and check. Perhaps someone
might say; “ let’s read the operations done to the “x” and undo them.
The operations are:
1) cube rooting
2) squaring
We must square root and cube to solve for x. If we employ power law
and raise both sides to the reciprocal power that is in the problem ...
(x2/3)3/2 = 163/2
x=
e16 j
3
= 64
Exponential Equations
The more difficult problems require logarithms but the simpler ones can
be done fairly easily.
Allow students to try to solve problems like 3 x!2 = 81 by any method;
perhaps guess and check. Some students might notice that the bases can
be made the same thus allowing them to be disregarded.
Ex.
3 x!2 = 81
3x ! 2 = 3 4
The only way the left side equals the right side is if the exponents are
equivalent. Looking at only the exponents:
x!2=4
x=6
Worthwhile Tasks for Instruction and/or Assessment
32
Suggested Resources
Equations with Rational Exponents
Equations with Rational Exponents
Group Activity
Solve each of the following:
Algebra, Structure & Method Book 2
p.457 # 23-30
p.458 # 43-50
1) (2 + x)½ = 3
2) (3x ! 1)2/3 = 4
3) (5 + y)!1/3 = ½
Exponential Equations
Exponential Equations
Pencil/Paper
Solve each of the following:
Algebra, Structure & Method Book 2
p.461 # 9-12 oral
p.462 # 19-29
1) 2x!1 = 32
2) 41!2x = 128
Application
A bacteria culture starts with 2,000 bacteria. After 5 hours the
estimated number of bacteria is 64,000. What is the time
required for the population to double for this culture?
Solution
N(t) = N(0) 2t/d
64,000 = 2,000 × 25/d
32 = 25/d
25 = 25/d
5 = 5/d
d = 1 hour
Project
Look for at least two examples where formulas have the
variable in the exponent.
33