A2Ch0806 Natural Logarithms
Goal
p1
To evaluate natural logarithmic expressions.
To solve equations using natural logarithms.
14.0 Students understand and use the properties of exponents to simplify logarithmic
numeric expressions and to identify their approximate values.
15.0 Students determine whether a specific algebraic statement involving logarithmic
or exponential functions is sometimes true, always true, or never true.
13.0 Students translate between logarithms in any base.
Ca. State
Standard
Natural Logarithmic Functions
Natural Logarithmic Functions
The natural number e ! 2.71828. The function y = e x
has an inverse, the Natural Logarithmic Function.
Definition
Natural Logarithmic Function
If y = e , then log e y = x, which is
commonly written as ln y = x.
x
y = e and ln y = x are inverse functions.
x
cam09a2_te_1101.qxd
Try : Write each as a single natural logarithm.
Simplifying Natural Logarithms
04/25/07
06:05 pm
1
Page 611
A 5 ln 2 ! ln 4
2. Teach
Generating a Sequence
EXAMPLE
Write 2 ln 12 ! ln 9 as a single natural logarithm.
2 ln 12 ! ln 9 = ln 12 2 ! ln 9 Power Prop
a. To create one side of the Koch snowflake, replace each
Draw the first four figures of the pattern.
with
.
Guided Instruction
Activity
Teaching Tip
Help students understand that
each time a new person is added
to the group, the number of new
calls will equal the number of
people who were already in the
group.
b. Write the number of segments in each figure above as a sequence.
When you apply the
construction from Example 1
to an equilateral triangle,
you form the Koch
snowflake.
1, 4, 16, 64, c
c. Predict the next term of the sequence. Explain your choice.
1
Each term is 4 times the preceding term. The next term is 64 ? 4, or 256.
There will be 256 segments in the next figure in the pattern.
E
D
E
D
E
D
C
B
D
D
E
E
Test-Taking Tip
10 ft
When you have to
repeat a step several
times, be careful to
use the correct number
of steps.
8.5 ft
04/25/07
6.1 ft
( )
1 a. Start with a square with
sides 1 unit long. On the right
side, add on a square of the same
size. Continue adding one square
at a time in this way. Draw the
first four figures of the pattern.
5.2 ft
Original height of ball: 10 ft S
After 1st bounce: 85% of 10 = 0.85(10) = 8.5 S
06:05 pm
After 3rd bounce: 0.85(7.225) < 6.141 S
After 4th bounce: 0.85(6.141) < 5.220 S
Page 611
Write 12 ln 7 + 12 ln y as a single natural logarithm.
2 Suppose you drop a ball from
a height of 100 cm. It bounces
Instruction
backGuided
to 80% of its
previous
height.
How high will it go after
Activity
its fifth bounce? about 32.8 cm
.
You can use a variable, such as a, with positive integer subscripts to represent the
terms in a sequence.
2nd term 3rd term c
1st term
T
a1
T
a2
n - 1 term
nth term
n + 1 term
c
T
an - 1
T
an
T
an + 1
c
c
1, 4, 16, 64, c
Lesson 11-1 Mathematical Patterns
611
c. Predict the next term of the sequence. Explain your choice.
L4
1
2
B
A
B
E
E
D
E
D
C
C
D
D
E
E
Test-Taking Tip
When you have to
repeat a step several
times, be careful to
use the correct number
of steps.
cam09a2_te_1101.qxd
04/25/07
Solving Natural Logs
Real-World
EXAMPLE
10 ft
= ln 7y
Additional Examples
6.1 ft
5.2 ft
Original height of ball: 10 ft S
Page 611
After 3rd bounce: 0.85(7.225) < 6.141 S
After 4th bounce: 0.85(6.141) < 5.220 S
with
3
. Suppose you drop a ball from
2
2nd term
3rd term
c
n - 1 term
nth term
n + 1 term
c
T
a1
T
a2
T
a3
c
T
an - 1
T
an
T
an + 1
c
c. Predict the next term of the sequence. Explain your choice.
1
Learners EL
Each term is 4 timesEnglish
the preceding
term. The next term is 64 ? 4, or 256.
Write a3 on the board. Point out that the 3 is called a
segments
the next
pattern.
subscriptin
because
the figure
numberinisthe
written
below the
Sometimes you can find the next term in a sequence by using a pattern from the
terms that come before it.
2
A
B
B
C
C
E
E
D
D
E
E
Real-World
Test-Taking Tip
10 ft
When you have to
repeat a step several
times, be careful to
use the correct number
of steps.
8.5 ft
Original height of ball: 10 ft S
4 + ln ( x + 3) = 13
Connection
to Topology
2
4
5
A
A
B
B
E
D
E
D
C
C
D
D
E
E
Test-Taking Tip
2nd term
T
a2
3rd term
T
a3
c
c
n - 1 term
T
an - 1
nth term
T
an
Lesson 11-1
L4
English Learners
Have students compare the recursive formula in
Example 3 with the composition of a function with
itself in Lesson 7-6.
8.5 ft
n + 1 term
T
an + 1
Mathematical Patterns
Additional Examples
6.1 ft
5.2 ft
Original height of ball: 10 ft S
e
1st term
T
a1
2nd term
T
a2
3rd term
T
a3
c
c
n - 1 term
T
an - 1
Advanced Learners
English Learners
L4
n + 1 term
T
an + 1
Mathematical Patterns
06:05 pm
611
1
4 + ln ( x + 3) = 13
2
x " ±90.017 ! 3 Calculator, e9 / 2 = e9
x " +87.017, ! 93.017
Page 611
2. Teach
Solve ln (12x + 5 ) ! ln 3 = 7
ln (12x + 5 ) ! ln 3 = 7
ln 12x + 5 = 7
3
EXAMPLE
Generating a Sequence
c. Predict the next term of the sequence. Explain your choice.
.
Guided Instruction
)
1
(
)
Sometimes you can find the next term in a sequence by using a pattern from the
terms that come before it.
2
Simplify
4
5
A
A
B
B
E
D
E
D
C
C
D
D
E
E
EXAMPLE
Real-World
ln 2 x+15
Connection
3
Test-Taking Tip
10 ft
When you have to
repeat a step several
times, be careful to
use the correct number
of steps.
8.5 ft
EXAMPLE
Connection
to Topology
You may wish to tell students that
the Koch snowflake is an example
of a fractal. Fractals have been
the subject of much mathematical
research since the mid-twentieth
century.
e
= e7
12x + 5 = e7
3
7
x = 3e ! 5
12
x " 3289.899 + 5
12
" 274.575
CD, Online, or Transparencies
Additional Examples
Multiple Choice Suppose you drop a handball from a height of 10 ft. After the ball
hits the floor, it rebounds to 85% of its previous height. About how high will the
ball rebound after its fourth bounce?
E
D
C
B
E
D
C
B
A
3
C
B
A
2
C
B
A
1
Sqr Root both sides
Help students understand that
each time a new person is added
to the group, the number of new
calls will equal the number of
people who were already in the
group.
Each term is 4 times the preceding term. The next term is 64 ? 4, or 256.
There will be 256 segments in the next figure in the pattern.
1 Describe the pattern formed. Find the next three terms.
a. 27, 34, 41, 48, c
b. 243, 81, 27, 9, c
Divide by 3; 3, 1, 13 .
Add 7; 55, 62, 69.
Exponentiate, e
e
(
1, 4, 16, 64, c
Subtract 4
x + 3 = ± e9
with
b. Write the number of segments in each figure above as a sequence.
When you apply the
construction from Example 1
to an equilateral triangle,
you form the Koch
snowflake.
9
Solve for x
Activity
Teaching Tip
Solve 4 + ln ( x + 3) = 13
( x + 3)2 = e9
( x + 3)2 = ±
9
2
611
a. To create one side of the Koch snowflake, replace each
Draw the first four figures of the pattern.
2
eln( x+3) = e9
Simplify
611
x=e !3
EL
Write a3 on the board. Point out that the 3 is called a
subscript because the number is written below the
variable. Note that sub means underneath. Ask: What
other words begin with the prefix sub? samples:
A 2nd look
c
c
Write a3 on the board. Point out that the 3 is called a
subscript because the number is written below the
variable. Note that sub means underneath. Ask: What
other words begin with the prefix sub? samples:
subway, subterranean
04/25/07
Exponentiate using e
9
x + 3 = e2
nth term
T
an
Lesson 11-1
Have students compare the recursive formula in
Example 3 with the composition of a function with
itself in Lesson 7-6.
9
2
2 Suppose you drop a ball from
a height of 100 cm. It bounces
back to 80% of its previous
height. How high will it go after
its fifth bounce? about 32.8 cm
You can use a variable, such as a, with positive integer subscripts to represent the
terms in a sequence.
611
2
=e
2 a. About how high will the ball rebound after the seventh bounce?
b. After what bounce will the rebound height be less than 2 ft?
a. 3.2 ft
b. 10th bounce
EL
2
Divide by 2
b. Write the number of 1-unit
segments in each figure above as
a sequence. 4, 7, 10, 13, . . .
c. Predict the next term of the
sequence. Explain your choice.
16; Each term is 3 more than the
preceding term.
ln( x+3)
c
ln ( x + 3) = 9
Power rule
1 a. Start with a square with
sides 1 unit long. On the right
side, add on a square of the same
size. Continue adding one square
at a time in this way. Draw the
first four figures of the pattern.
After 1st bounce: 85% of 10 = 0.85(10) = 8.5 S
c
subway, subterranean
Subtract 4
CD, Online, or Transparencies
Connection
The ball will rebound about 5.2 ft after the fourth bounce.
The correct choice is D.
cam09a2_te_1101.qxd
Advanced Learners
Real-World
After 2nd bounce: 0.85(8.5) = 7.225 S
2 Suppose you drop a ball from
a height of 100 cm. It bounces
back to 80% of its previous
height. How high will it go after
its fifth bounce? about 32.8 cm
You can use a variable, such as a, with positive integer subscripts to represent the
terms in a sequence.
Connection
to Topology
EXAMPLE
You may wish to tell students that
the Koch snowflake is an example
of a fractal. Fractals have been
the subject of much mathematical
research since the mid-twentieth
century.
2
After 3rd bounce: 0.85(7.225) < 6.141 S
After 4th bounce: 0.85(6.141) < 5.220 S
b. Write the number of 1-unit
segments in each figure above as
a sequence. 4, 7, 10, 13, . . .
c. Predict the next term of the
sequence. Explain your choice.
16; Each term is 3 more than the
preceding term.
The ball will rebound about 5.2 ft after the fourth bounce.
The correct choice is D.
1st term
T
a1
EXAMPLE
10 ft
When you have to
repeat a step several
times, be careful to
use the correct number
of steps.
After 2nd bounce: 0.85(8.5) = 7.225 S
2 a. About how high will the ball rebound after the seventh bounce?
b. After what bounce will the rebound height be less than 2 ft?
a. 3.2 ft
b. 10th bounce
1
ln ( x + 3) = 9
2 ln ( x + 3) = 9
ln ( x + 3) = 92
Multiple Choice Suppose you drop a handball from a height of 10 ft. After the ball
hits the floor, it rebounds to 85% of its previous height. About how high will the
ball rebound after its fourth bounce?
E
D
C
B
E
D
C
B
A
3
C
B
A
2
C
B
A
1
After 1st bounce: 85% of 10 = 0.85(10) = 8.5 S
After 3rd bounce: 0.85(7.225) < 6.141 S
After 4th bounce: 0.85(6.141) < 5.220 S
Help students understand that
each time a new person is added
to the group, the number of new
calls will equal the number of
people who were already in the
group.
Each term is 4 times the preceding term. The next term is 64 ? 4, or 256.
There will be 256 segments in the next figure in the pattern.
1 a. Start with a square with
sides 1 unit long. On the right
side, add on a square of the same
size. Continue adding one square
at a time in this way. Draw the
first four figures of the pattern.
5.2 ft
2
Guided Instruction
2
Sometimes you can find the next term in a sequence by using a pattern from the
terms that come before it.
Additional Examples
6.1 ft
.
1, 4, 16, 64, c
CD, Online, or Transparencies
Connection
Multiple Choice Suppose you drop a handball from a height of 10 ft. After the ball
hits the floor, it rebounds to 85% of its previous height. About how high will the
ball rebound after its fourth bounce?
E
D
C
B
E
D
C
B
A
D
C
B
A
D
C
B
A
5
EXAMPLE
EXAMPLE
with
1 Describe the pattern formed. Find the next three terms.
a. 27, 34, 41, 48, c
b. 243, 81, 27, 9, c
Divide by 3; 3, 1, 13 .
Add 7; 55, 62, 69.
You may wish to tell students that
the Koch snowflake is an example
of a fractal. Fractals have been
the subject611
of much mathematical
research since the mid-twentieth
century.
variable. Note that sub means underneath. Ask: What
2. Teach
Generating a Sequence
c. Predict the next term of the sequence. Explain your choice.
Help students understand that
each time a new person is added
to the group, the number of new
calls will equal the number of
people who were already in the
group.
b. Write the number of segments in each figure above as a sequence.
Lesson 11-1 Mathematical Patterns
611
1, 4, 16, 64, c
Solve 4 + ln ( x + 3) = 13
b. Write the number of segments in each figure above as a sequence.
When you apply the
construction from Example 1
to an equilateral triangle,
you form the Koch
snowflake.
Check :
Teaching Tip
Page 611
EXAMPLE
Activity
Teaching Tip
Guided Instruction
a height of 100 cm. It bounces
back to 80% of its previous
height. How highActivity
will it go after
its fifth bounce? about 32.8 cm
3
other words begin with the prefix sub? samples:
1 Describe the pattern formed.
Find the next three terms.
subway, subterranean
a. 27, 34, 41, 48, c
b. 243, 81, 27, 9, c
Divide by 3; 3, 1, 13 .
Add 7; 55, 62, 69.
A
06:05 pm
Check
2. Teach
ln ( 2x ! 4 ) = 6
Notice the exponent
ln ( 2 x ! 4 3
3ln ( 2x ! 4 ) = 6
Power rule ) = 6
ln ( 2 ( 5.69
3
5) !
ln ( 2x ! 4 ) = 2
Divide
by 43 ) = 6
ln (11.39 ! 3
ln( 2 x!4 )
2
4 ) using
e
=e
Exponentiate,
=6e
ln
( 7.39 ) 3 = 6
2x ! 4 = e2
Simplify
ln 403.583
2
=6
x= e +4
Solve for x
6.0003 " 6
2
x " 7.3891 + 4
Use a calculator
2
" 5.695
Simplify
1st term
Have students compare the recursive formula in
willwith
be 256
Example 3 with the composition of There
a function
itself in Lesson 7-6.
4
04/25/07
a. To create one side of the Koch snowflake, replace each
Draw the first four figures of the pattern.
b. Write the number of 1-unit
segments in each figure above as
a sequence. 4, 7, 10, 13, . . .
c. Predict the next term of the
sequence. Explain your choice.
16; Each term is 3 more than the
preceding term.
Solve ln ( 2x ! 4 ) = 6
When you apply the
construction from Example 1
to an equilateral triangle,
you form the Koch
Advanced Learners L4
snowflake.
3
cam09a2_te_1101.qxd
1
Solving Natural Logarithms
The ball will rebound about 5.2 ft after the fourth bounce.
correct choice is D.
Generating a Sequence
1 TheEXAMPLE
You can use a variable, such as a, with positive integer subscripts to represent the
terms in a sequence.
2
Simplify
1 a. Start with a square with
sides 1 unit long. On the right
side, add on a square of the same
size. Continue adding one square
at a time in this way. Draw the
first four figures of the pattern.
After 2nd bounce: 0.85(8.5) = 7.225 S
2 a.
how high
theof
ball
rebound
the seventh
bounce?
a. About
To create
onewill
side
the
Koch after
snowflake,
replace
each
b. After
what
will the
rebound
height
be less than 2 ft?
Draw
thebounce
first four
figures
of the
pattern.
a. 3.2 ft
b. 10th bounce
1
Product Prop.
CD, Online, or Transparencies
Connection
8.5 ft
1
After 1st bounce: 85% of 10 = 0.85(10) = 8.5 S
06:05 pm
D 3( ln 5 + ln x ) – ( ln 25 + 5 ln x )
ln 52
x
1
Connection
to Topology
= ln ( 7y ) 2
Multiple Choice Suppose you drop a handball from a height of 10 ft. After the ball
hits the floor, it rebounds to 85% of its previous height. About how high will the
ball rebound after its fourth bounce?
E
D
C
B
A
5
D
C
B
A
4
C
B
A
3
C
B
A
2
EXAMPLE
You may wish to tell students that
the Koch snowflake is an example
of a fractal. Fractals have been
the subject of much mathematical
research since the mid-twentieth
century.
611
Sometimes you can find the next term in a sequence by using a pattern from the
terms that come before it.
1
ln 4 3x
Teaching Tip
Help students understand that
each time a new person is added
to the group, the number of new
calls will equal the number of
people who were already in the
group.
7 + 12 ln y = ( ln 7 + ln y ) 2 Power Prop
T
a3
1 ln
2
b. Write the number of segments in each figure above as a sequence.
Each term is 4 times the preceding term. The next term is 64 ? 4, or 256.
There will be 256 segments
in theLearners
next figure EL
in the pattern.
English
Have students compare the recursive formula in
Write a3 on the board. Point out that the 3 is called a
1 Describe
Example 3 with the composition
of a function
with formed.
the pattern
Find the
next three
terms. is written below the
subscript
because
the number
itself in Lesson 7-6.
variable. Note that
sub means
Ask: What
a. 27, 34, 41, 48, c
b. 243,
81, 27, underneath.
9, c
other words begin with
theby
prefix
Divide
3; 3,sub?
1, 13 samples:
.
Add 7; 55, 62, 69.
subway, subterranean
Advanced Learners
C 1 ln 3 + 1 ln x
4
4
Simplify
2. Teach
EXAMPLE
Generating
a Sequence
a.
About how high
will the ball rebound
after the seventh bounce?
After
what
bounce
willthe
theKoch
rebound
height be
less than
2 ft?
a. b.To
create
one
side of
snowflake,
replace
each
with
a. 3.2the
ft first four figures of
b.the
10th
bounce
Draw
pattern.
When you apply the
construction from Example 1
to an equilateral triangle,
you form the Koch
snowflake.
Simplify
b. Write the number of 1-unit
segments in each figure above as
a sequence. 4, 7, 10, 13, . . .
c. Predict the next term of the
sequence. Explain your choice.
16; Each term is 3 more than the
preceding term.
The ball will rebound about 5.2 ft after the fourth bounce.
The correct choice is D.
12
ln x 3 y
Additional Examples
After 2nd bounce: 0.85(8.5) = 7.225 S
cam09a2_te_1101.qxd
3ln x + ln y
B
Quotient Prop
CD, Online, or Transparencies
Connection
Multiple Choice Suppose you drop a handball from a height of 10 ft. After the ball
hits the floor, it rebounds to 85% of its previous height. About how high will the
ball rebound after its fourth bounce?
E
D
C
B
A
5
C
B
A
4
C
B
A
3
C
B
A
2
C
B
A
1
Real-World
EXAMPLE
( )
You may wish to tell students that
the Koch snowflake is an example
of a fractal. Fractals have been
the subject of much mathematical
research since the mid-twentieth
century.
Sometimes you can find the next term in a sequence by using a pattern from the
terms that come before it.
2
Connection
to Topology
EXAMPLE
2
= ln 12
9
= ln 144
9
= ln 16
1 Describe the pattern formed. Find the next three terms.
a. 27, 34, 41, 48, c
b. 243, 81, 27, 9, c
Divide by 3; 3, 1, 13 .
Add 7; 55, 62, 69.
ln 8
6.1 ft
5.2 ft
Original height of ball: 10 ft S
1 a. Start with a square with
sides 1 unit long. On the right
side, add on a square of the same
size. Continue adding one square
at a time in this way. Draw the
first four figures of the pattern.
After 1st bounce: 85% of 10 = 0.85(10) = 8.5 S
Quotient rule
Exponentiate, e
Simplify
After 2nd bounce: 0.85(8.5) = 7.225 S
Simplify
After 3rd bounce: 0.85(7.225) < 6.141 S
After 4th bounce: 0.85(6.141) < 5.220 S
x = ± e9 ! 3
Solve for x
x " ± 8130.084 ! 3 Simplify
x " ±90.017 ! 3
Simplify
x " +87.017, ! 93.017
b. Write the number of 1-unit
segments in each figure above as
a sequence. 4, 7, 10, 13, . . .
c. Predict the next term of the
sequence. Explain your choice.
16; Each term is 3 more than the
preceding term.
The ball will rebound about 5.2 ft after the fourth bounce.
The correct choice is D.
2 a. About how high will the ball rebound after the seventh bounce?
b. After what bounce will the rebound height be less than 2 ft?
a. 3.2 ft
b. 10th bounce
You can use a variable, such as a, with positive integer subscripts to represent the
terms in a sequence.
1st term
T
a1
2nd term 3rd term c
T
T
a2
a3
c
n - 1 term
T
an - 1
nth term
T
an
n + 1 term
T
an + 1
Lesson 11-1 Mathematical Patterns
Advanced Learners
L4
Have students compare the recursive formula in
Example 3 with the composition of a function with
itself in Lesson 7-6.
2 Suppose you drop a ball from
a height of 100 cm. It bounces
back to 80% of its previous
height. How high will it go after
its fifth bounce? about 32.8 cm
c
Calculator
c
611
English Learners EL
Write a3 on the board. Point out that the 3 is called a
subscript because the number is written below the
variable. Note that sub means underneath. Ask: What
other words begin with the prefix sub? samples:
subway, subterranean
Solve for x
611
Simplify
A2Ch0806 Natural Logarithms
p2
Try : Solve
ln x = 0.1
A
1.105
B
ln ( 3x ! 9 ) = 21
C
D
ln x + 2 = 12
3
4 ln 5x = 8
±1.478
E
2 ln ( x ! 7 ) = 10
+155.413,
!141.413
439, 605, 247.8
( )
488, 262.4
cam09a2_te_1101.qxd
cam09a2_te_1101.qxd
04/25/07
06:05 pm
1
1
with
.
1, 4, 16, 64, c
c. Predict the next term of the sequence. Explain your choice.
1
Each term is 4 times the preceding term. The next term is 64 ? 4, or 256.
There will be 256 segments in the next figure in the pattern.
4
5
A
A
B
B
B
E
E
E
D
C
Test-Taking Tip
E
D
C
Real-World
EXAMPLE
CD, Online, or Transparencies
Connection
Additional Examples
Multiple Choice Suppose you drop a handball from a height of 10 ft. After the ball
hits the floor, it rebounds to 85% of its previous height. About how high will the
ball rebound after its fourth bounce?
E
D
C
10 ft
When you have to
repeat a step several
times, be careful to
use the correct number
of steps.
8.5 ft
Connection
to Topology
You may wish to tell students that
the Koch snowflake is an example
of a fractal. Fractals have been
the subject of much mathematical
research since the mid-twentieth
century.
Sometimes you can find the next term in a sequence by using a pattern from the
terms that come before it.
2
E
D
C
B
A
D
C
B
A
D
C
B
A
EXAMPLE
6.1 ft
1 a. Start with a square with
sides 1 unit long. On the right
side, add on a square of the same
size. Continue adding one square
at a time in this way. Draw the
first four figures of the pattern.
5.2 ft
Original height of ball: 10 ft S
After 1st bounce: 85% of 10 = 0.85(10) = 8.5 S
After 2nd bounce: 0.85(8.5) = 7.225 S
After 3rd bounce: 0.85(7.225) < 6.141 S
After 4th bounce: 0.85(6.141) < 5.220 S
b. Write the number of 1-unit
segments in each figure above as
a sequence. 4, 7, 10, 13, . . .
c. Predict the next term of the
sequence. Explain your choice.
16; Each term is 3 more than the
preceding term.
The ball will rebound about 5.2 ft after the fourth bounce.
The correct choice is D.
2 a. About how high will the ball rebound after the seventh bounce?
b. After what bounce will the rebound height be less than 2 ft?
a. 3.2 ft
b. 10th bounce
2 Suppose you drop a ball from
a height of 100 cm. It bounces
back to 80% of its previous
height. How high will it go after
its fifth bounce? about 32.8 cm
You can use a variable, such as a, with positive integer subscripts to represent the
terms in a sequence.
cam09a2_te_1101.qxd
term 3rd term c
T
T
a2
a3
c
06:05 1st
pm term
Page 6112nd
04/25/07
T
a1
1
n - 1 term
T
an - 1
Generating a Sequence
EXAMPLE
Advanced Learners
nth term
T
an
n + 1 term
T
an + 1
Have students compare the recursive formula in
Example 3 with the composition of a function with
itself in Lesson 7-6.
1
Each term is 4 times the preceding term. The next term is 64 ? 4, or 256.
There will be 256 segments in the next figure in the pattern.
2
A
A
B
B
C
C
E
E
D
D
E
E
Real-World
EXAMPLE
Test-Taking Tip
10 ft
When you have to
repeat a step several
times, be careful to
use the correct number
of steps.
EXAMPLE
Additional Examples
8.5 ft
6.1 ft
5.2 ft
Original height of ball: 10 ft S
1 a. Start with a square with
sides 1 unit long. On the right
side, add on a square of the same
size. Continue adding one square
at a time in this way. Draw the
first four figures of the pattern.
After 1st bounce: 85% of 10 = 0.85(10) = 8.5 S
After 3rd bounce: 0.85(7.225) < 6.141 S
After 4th bounce: 0.85(6.141) < 5.220 S
b. Write the number of 1-unit
segments in each figure above as
a sequence. 4, 7, 10, 13, . . .
c. Predict the next term of the
sequence. Explain your choice.
16; Each term is 3 more than the
preceding term.
The ball will rebound about 5.2 ft after the fourth bounce.
The correct choice is D.
2 a. About how high will the ball rebound after the seventh bounce?
b. After what bounce will the rebound height be less than 2 ft?
a. 3.2 ft
b. 10th bounce
1st term
2nd term 3rd term c
T
a1
T
a2
T
a3
c
n - 1 term
nth term
n + 1 term
c
T
an - 1
T
an
T
an + 1
c
Lesson 11-1 Mathematical Patterns
611
Write a3 on the board. Point out that the 3 is called a
subscript because the number is written below the
variable. Note that sub means underneath. Ask: What
other words begin with the prefix sub? samples:
subway, subterranean
06:05 pm
1
Word Problems
611
Page 611
EXAMPLE
with
.
1
Each term is 4 times the preceding term. The next term is 64 ? 4, or 256.
There will be 256 segments in the next figure in the pattern.
B
A
5
A
A
B
B
C
C
E
D
D
E
E
EXAMPLE
Real-World
Test-Taking Tip
10 ft
When you have to
repeat a step several
times, be careful to
use the correct number
of steps.
EXAMPLE
8.5 ft
Additional Examples
6.1 ft
5.2 ft
Original height of ball: 10 ft S
1 a. Start with a square with
sides 1 unit long. On the right
side, add on a square of the same
size. Continue adding one square
at a time in this way. Draw the
first four figures of the pattern.
After 1st bounce: 85% of 10 = 0.85(10) = 8.5 S
After 2nd bounce: 0.85(8.5) = 7.225 S
After 3rd bounce: 0.85(7.225) < 6.141 S
After 4th bounce: 0.85(6.141) < 5.220 S
b. Write the number of 1-unit
segments in each figure above as
a sequence. 4, 7, 10, 13, . . .
c. Predict the next term of the
sequence. Explain your choice.
16; Each term is 3 more than the
preceding term.
The ball will rebound about 5.2 ft after the fourth bounce.
The correct choice is D.
2 a. About how high will the ball rebound after the seventh bounce?
b. After what bounce will the rebound height be less than 2 ft?
a. 3.2 ft
b. 10th bounce
You can use a variable, such as a, with positive integer subscripts to represent the
terms in a sequence.
1st term
T
a1
2nd term 3rd term c
T
T
a2
a3
c
n - 1 term
T
an - 1
nth term
T
an
n + 1 term
T
an + 1
Lesson 11-1 Mathematical Patterns
Advanced Learners
E
Test-Taking Tip
L4
Have students compare the recursive formula in
Example 3 with the composition of a function with
itself in Lesson 7-6.
2 Suppose you drop a ball from
a height of 100 cm. It bounces
back to 80% of its previous
height. How high will it go after
its fifth bounce? about 32.8 cm
c
c
611
English Learners EL
Write a3 on the board. Point out that the 3 is called a
subscript because the number is written below the
variable. Note that sub means underneath. Ask: What
other words begin with the prefix sub? samples:
subway, subterranean
611
1
Real-World
Connection
e
5x
2
10 ft
8.5 ft
6.1 ft
EXAMPLE
Connection
to Topology
You may wish to tell students that
the Koch snowflake is an example
of a fractal. Fractals have been
the subject of much mathematical
research since the mid-twentieth
century.
= 1.4
= 0.7
5.2 ft
5x
2
Original height of ball: 10 ft S
1 a. Start with a square with
sides 1 unit long. On the right
side, add on a square of the same
size. Continue adding one square
at a time in this way. Draw the
first four figures of the pattern.
ln e = ln 0.7
5x = ln 0.7
2
5x ! ".357
2
x ! ".143
After 1st bounce: 85% of 10 = 0.85(10) = 8.5 S
After 2nd bounce: 0.85(8.5) = 7.225 S
After 3rd bounce: 0.85(7.225) < 6.141 S
After 4th bounce: 0.85(6.141) < 5.220 S
b. Write the number of 1-unit
segments in each figure above as
a sequence. 4, 7, 10, 13, . . .
c. Predict the next term of the
sequence. Explain your choice.
16; Each term is 3 more than the
preceding term.
2 a. About how high will the ball rebound after the seventh bounce?
b. After what bounce will the rebound height be less than 2 ft?
a. 3.2 ft
b. 10th bounce
You can use a variable, such as a, with positive integer subscripts to represent the
terms in a sequence.
1st term
T
a1
Solve for x
2nd term 3rd term c
T
T
a2
a3
c
n - 1 term
T
an - 1
nth term
T
an
n + 1 term
T
an + 1
Lesson 11-1 Mathematical Patterns
2 Suppose you drop a ball from
a height of 100 cm. It bounces
back to 80% of its previous
height. How high will it go after
its fifth bounce? about 32.8 cm
c
Solve for x
Use a calculator
Simplify
Simplify
Use a calculator
Solve for x
611
Write a3 on the board. Point out that the 3 is called a
subscript because the number is written below the
variable. Note that sub means underneath. Ask: What
other words begin with the prefix sub? samples:
subway, subterranean
Simplify
Take the natural log.
c
Try : Solve
Take the natural log.
Divide by 2
English Learners EL
L4
Have students compare the recursive formula in
Example 3 with the composition of a function with
itself in Lesson 7-6.
Add 4
Substract 5.7
CD, Online, or Transparencies
Additional Examples
The ball will rebound about 5.2 ft after the fourth bounce.
The correct choice is D.
Connection
to Topology
You may wish to tell students that
the Koch snowflake is an example
of a fractal. Fractals have been
the subject of much mathematical
research since the mid-twentieth
century.
CD, Online, or Transparencies
Connection
Multiple Choice Suppose you drop a handball from a height of 10 ft. After the ball
hits the floor, it rebounds to 85% of its previous height. About how high will the
ball rebound after its fourth bounce?
E
D
C
B
E
A = Pert
Interest formula.
0.06t
254.25 = 200e
Sub. 254.25 ( A ) , 200 (P ) , & 0.06 (r )
0.06t
1.271 = e
Divide each side by 200.
ln 1.271 = ln e0.06t Take the natural log. of each side.
ln 1.271 = 0.06t
Simplify.
ln 1.271
=t
Solve for t.
0.06
4!t
Use a calculator.
The money has been invested for about 4 years.
Sometimes you can find the next term in a sequence by using a pattern from the
terms that come before it.
4
D
Help students understand that
each time a new person is added
to the group, the number of new
calls will equal the number of
people who were already in the
group.
c. Predict the next term of the sequence. Explain your choice.
2
3
D
Activity
Teaching Tip
1, 4, 16, 64, c
E
D
C
E
D
When you have to
repeat a step several
times, be careful to
use the correct number
of steps.
Guided Instruction
b. Write the number of segments in each figure above as a sequence.
E
D
C
B
A
D
C
B
A
E
D
C
2. Teach
Generating a Sequence
1 Describe the pattern formed. Find the next three terms.
a. 27, 34, 41, 48, c
b. 243, 81, 27, 9, c
Divide by 3; 3, 1, 13 .
Add 7; 55, 62, 69.
2
E
D
C
An initial investment of $200 is now valued at $254.25.
The interest rate is 6%, compounded continuously.
How long has the money been invested?
a. To create one side of the Koch snowflake, replace each
Draw the first four figures of the pattern.
When you apply the
construction from Example 1
to an equilateral triangle,
you form the Koch
snowflake.
1
B
English Learners EL
L4
Have students compare the recursive formula in
Example 3 with the composition of a function with
itself in Lesson 7-6.
04/25/07
B
E
D
C
B
2 Suppose you drop a ball from
a height of 100 cm. It bounces
back to 80% of its previous
height. How high will it go after
its fifth bounce? about 32.8 cm
You can use a variable, such as a, with positive integer subscripts to represent the
terms in a sequence.
cam09a2_te_1101.qxd
A
611
Connection
to Topology
After 2nd bounce: 0.85(8.5) = 7.225 S
Advanced Learners
A
Advanced Learners
CD, Online, or Transparencies
Connection
Multiple Choice Suppose you drop a handball from a height of 10 ft. After the ball
hits the floor, it rebounds to 85% of its previous height. About how high will the
ball rebound after its fourth bounce?
E
D
C
B
E
D
C
B
A
5
D
C
B
A
4
D
C
B
A
3
5
EXAMPLE
c
You may wish to tell students that
the Koch snowflake is an example
of a fractal. Fractals have been
the subject of much mathematical
research since the mid-twentieth
century.
Sometimes you can find the next term in a sequence by using a pattern from the
terms that come before it.
2
4
C
B
A
3
C
B
A
2
C
B
A
1
Use a calculator
Guided Instruction
1 Describe the pattern formed. Find the next three terms.
a. 27, 34, 41, 48, c
b. 243, 81, 27, 9, c
Divide by 3; 3, 1, 13 .
Add 7; 55, 62, 69.
1
2e
Activity
Teaching Tip
c. Predict the next term of the sequence. Explain your choice.
5x
2
Multiple Choice Suppose you drop a handball from a height of 10 ft. After the ball
hits the floor, it rebounds to 85% of its previous height. About how high will the
ball rebound after its fourth bounce?
c
group.
Help students understand that
each time a new person is added
to the group, the number of new
calls will equal the number of
people who were already in the
group.
+ 5.7 = 7.1
Sometimes you can find the next term in a sequence by using a pattern from the
terms that come before it.
Help
students
understand
Write a3 on the board. Point out that
the
3 is called
a that
each time
a new
person is added
subscript because the number is written
below
the
to the group, the number of new
variable. Note that sub means underneath.
Ask: What
calls will equal the number of
samples:
the prefix sub?
people
who were already in the
subway, subterranean
1, 4, 16, 64, c
Guided Instruction
Each term is 4 times the preceding term. The next term is 64 ? 4, or 256.
There will be 256 segments in the next figure in the pattern.
2
b. Write the number of segments in eachother
figure above
as abegin
sequence.
words
with
When you apply the
construction from Example 1
to an equilateral triangle,
you form the Koch
snowflake.
.
c. Predict the next term of the sequence. Explain your choice.
Simplify
2. Teach 611
.
5x
2
1 Describe the pattern formed. Find the next three terms.
a. 27, 34, 41, 48, c
b. 243, 81, 27, 9, c
Divide by 3; 3, 1, 13 .
Add 7; 55, 62, 69.
Take the natural log.
Solve e 3x!1 ! 4 = 9
e3x!1 ! 4 = 9
e3x!1 = 13
ln e3x!1 = ln13
3x ! 1 = ln 13
ln 13 + 1
x=
3
2.565
+1
x"
3
x " 1.188
with
English Learners EL
L4
Divide by 4
2e
1, 4, 16, 64, c
Subtract 1.2
Lesson 11-1 Mathematical Patterns
a. To create one side of the Koch snowflake, replace each
Draw the first four figures of the pattern.
with
b. Write the number of segments in each figure above as a sequence.
When you apply the
construction from Example 1
to an equilateral triangle,
you form the Koch
snowflake.
Help students understand that
each time a new person is added
to the group, the number of new
calls will equal the number of
people who were already in the
group.
1 Describe the pattern formed. Find the next three terms.
a. 27, 34, 41, 48, c
b. 243, 81, 27, 9, c
Divide by 3; 3, 1, 13 .
Add 7; 55, 62, 69.
3
5x
Guided Instruction
b. Write the number of segments in each figure above as a sequence.
2. Teach
Generating a Sequence
Activity
Teaching Tip
Solve 4e + 1.2 = 14
4e 3x + 1.2 = 14
4e 3x = 12.8
e3x = 3.2
ln e3x = ln 3.2
3x = ln 3.2
3x ! 1.163
x ! 0.388
3x
EXAMPLE
Solve 2e 2 + 5.7 = 7.1
Activity
Teaching Tip
When you apply the
construction from Example 1
to an equilateral triangle,
you form the Koch
snowflake.
Page 611
a. To create one side of the Koch snowflake, replace each
Draw the first four figures of the pattern.
2. Teach
Generating a Sequence
EXAMPLE
a. To create one side of the Koch snowflake, replace each
Draw the first four figures of the pattern.
2
06:05 pm
Solving Exponential Equations
Solving Exponential Equations
1
04/25/07
Page 611
A e
x+1
611
= 30
2.401
2x
B e 3 + 7.2 = 0.1
1.605
C 4 ! 2e x = !23
2.603
D 23 e4 x ! 13 = 4
0.468
E 1.2e!5 x + 2.6 = 3
!0.220