Course Profile Functions, Grade 11, University/College
Preparation, Catholic and Public
Unit
3: Financial Applications of Sequences
and Series
Time: 25 hours
Activity 3.1 | Activity 3.2 | Activity
3.3 | Activity 3.4 | Activity 3.5 | Activity
3.6 | Activity 3.7 | Activity 3.8 | Activity
3.9
Unit
Description
Students work with arithmetic and geometric sequences
and series. This knowledge serves as the basis for applications in the field of
personal finance. Students develop the formula for compound interest and solve
problems related to compound interest and annuities. As skills are developed,
students use spreadsheets to investigate the cost of borrowing when interest
rates, compound periods, lending terms, etc., are varied. The activities are
designed to reflect the type of decisions that students are likely to face in
the future. Students apply skills with linear and exponential functions.
Students are expected to summarize the results of their investigations in a
clear and concise manner.
Unit Synopsis Chart
|
Activity |
Time |
Expectations |
Assessment |
Tasks |
|
3.1
Investigating Attributes of Sequences |
150 min |
FAV.01, FA1.01,
FA1.02 |
Inquiry, Communication |
Work with
sequences |
|
3.2 Summing Up:
Arithmetic Sequences and Series |
150 min |
FAV.01, FAV.02,
FA1.04, FA1.05, FA2.04 |
Knowledge,
Application, Inquiry |
Develop formula
for arithmetic sequences and series |
|
3.3 Compound Interest:
Exploring Geometric Sequences |
150 min |
FAV.01, FAV.02,
FA1.03, FA1.04, FA2.01, FA2.02, FA2.05 |
Knowledge,
Application, Communication |
Use a financial
application to investigate geometric sequences. Develop the compound interest
formula |
|
3.4
Applications: Finding the Amount and the Present Value of a Long Term
Investment |
75 min |
FAV.02, FAV.03,
FA2.02, FA3.01, FA3.02 |
Knowledge,
Application |
Extend knowledge
of the compound interest formula and introduce financial applications of a graphing
calculator |
|
3.5 Introduction
to Geometric Series |
75 min |
FAV.01, FA1.05 |
Knowledge,
Application |
Develop the
formula for the sum of a geometric series |
|
3.6
Applications: Finding the Amount and the Present Value of an Annuity |
150 min |
FAV.02, FAV.03,
FA2.02, FA2.03, FA3.05 |
Inquiry,
Communication |
Apply geometric
sequences and series to finding the amount and present value of an annuity |
|
3.7 What Happens
When?: Changing the Time, Rate and Amount |
150 min |
FAV.03, FA3.01,
FA3.05 |
All |
Use financial
applications of a graphing calculator to investigate the effect of changing
conditions when borrowing and saving |
|
3.8 Mortgages:
How They Work |
150 min |
FAV.03, FA3.02,
FA3.03, FA3.04, FA3.05 |
Application,
Communication |
Use technology
to generate amortization tables |
|
3.9 Financial
Decision Making: A Case Study |
300 min |
All |
All |
Summative
Assessment Activity |
Note: 150 minutes has been allotted to practice essential
skills, review and to do other assessments.
Assessment &
Evaluation of Student Achievement
Some of the problems contained in the activities can
be used to assess the inquiry process. Short quizzes can be used to assess
Knowledge/Understanding and Application. Teachers might choose to give a short
quiz on Arithmetic Sequences and Series, another on Geometric Sequences and
Series and a third on the Compound Interest Formula and applications of the
formula. In addition (or alternatively) a test on Activities 1 to 5 might be
given. In Activities 6 to 8, students will apply skills from Activities 1 to 5
to investigate finance. Activity 9 is designed as a culminating assessment tool
for the entire unit, with emphasis on financial applications. Communication
skills should be assessed continuously throughout the unit. Students should use
proper mathematical form, appropriate notation and explain their answers in a
clear and concise manner. The Inquiry Rubric and the Communication Rubric
produced by OAME are resources to use to assess student achievement.
Activity 3.1: Attributes of
Sequences
Time: 150 minutes
Description
Students work with a variety of sequences. Arithmetic
and geometric sequences provide a background for future financial applications
and the Fibonacci sequence, as well as the sequence created by the Tower of
Hanoi problem are included for interest and to emphasize that there are a
variety of sequences that can be studied. Students are also introduced to
notation used for sequences to be applied later with financial applications.
Strand(s) & Learning
Expectations
Strand(s): Financial Applications of Sequences and Series
Overall
Expectations
FAV.01 - solve problems involving arithmetic and
geometric sequences and series.
Specific
Expectations
FA1.01 - write terms of a sequence, given the formula
for the nth term;
FA1.02 - determine a formula for the nth term
of a given sequence.
Prior Knowledge &
Skills
Understanding linear and exponential relations may help
in transferring knowledge to relations studied in this activity.
Planning Notes
·
Provide
the students with a copy of the student activity.
·
Use
of a graphing calculator may be helpful, but is not essential. By graphing the
relationships students can determine if the sequences are linear, quadratic,
exponential, or other.
Teaching/Learning
Strategies
Students should work individually or in pairs to work
with the sequences. Discussion of the Fibonacci sequence (introduced in
question 3) and the Tower of Hanoi problem (introduced in question 4) will
ensure that students do not assume that the only types of sequences are
arithmetic and geometric.
Student
Activity
Investigate the
following situations.
1. Fill
in the chart below. The triangle is an equilateral triangle.
|
Side Length |
1 cm |
2 cm |
3 cm |
4 cm |
5 cm |
|
Perimeter |
|
|
|
|
|
a) Predict the perimeter of the
triangle if the side length is 20 cm.
b) Identify the type of
relationship between the perimeter and the side length.
c) Create
a formula that can be used to find the perimeter of an equilateral triangle
with
side length n.
d)
Verify
that your formula works.
2. The cells in a culture divide every hour.
a) If there are 500 cells now,
how many cells will there be in 1 hour?
b) Create a table showing the
number of cells at the end of each hour for 5 hours.
c) What type of relationship
exists between the number of cells and the time?
d) Predict the number of cells
after 8 hours.
e) Create an equation that
relates the number of cells to the time.
f) Verify that your formula
works.
3. You have 12 sticks of different whole number
lengths in a bag. What must their lengths be if, when you pull out any 3 of
them, you cannot make a triangle? What is the shortest length of the longest
stick?
4. Tameez has a board with 3 pegs. The first peg
has 5 flat blocks, all of different sizes, arranged by size with the smallest
on top and the largest on the bottom. He wants to move these blocks to the
third peg in a minimum number of moves. He has set up rules for himself which
include:
i) only
1 block can be moved at a time.
ii) A
larger block cannot be put on a smaller one.
a) What
is the least number of moves Tameez can take to move the blocks? Explain your reasoning.
b) Predict
how many moves would be required if there were 10 blocks on the first peg.
c) What
is the general relationship between the number of blocks and the number of
moves required?
5. In each of these examples a pattern can be
found. Try to find the pattern and determine the next two terms for each of the
following.
a) 1, 3, 5, 7, __, __
b) 12, 24, 48, __, __
c) 
, 
, 
, __, __
These are examples of sequences of numbers. A sequence is a set of
numbers, shapes, letters, etc. that are in a distinct or recognizable pattern.
6. In the sequence 3, 6, 9, 12, 15, 18… the
fourth term is 12. We write this as t4 = 12.
State the values of t2 and t6. The terms of
this sequence represent the perimeters found in question 1 above. We found that
P = 3n where n represents the side length. For the sequence 3, 6, 9, 12, …
tn = 3n. Find t30 and t50
Teacher
Facilitation
The sequence developed in question 3 of the student
activity is the Fibonacci Sequence. Two of the follow-up questions also result
in the generation of this very interesting sequence. After completing the
student activity present the students with some examples where, given a general
term, they must find the first few terms (see follow-up questions). Students
should also be presented with examples where, given the general term and the
value of the term, they can determine where the term is located in the
sequence.
Follow-up
Questions
1. You have 7 steps to climb. You can go up 1
step or 2 steps at a time. In how many different ways can you climb the steps.
Use the chart below to help organize your work.
|
Number of
steps |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
|
Number of
possible ways |
|
|
|
|
|
|
|
2. Using nickels and dimes only, in how many
different ways can you make up various sums of money?
(i.e., $0.05, $0.10, $0.15, $0.20, $0.25, $0.30); etc.
3. Given the general term, state the first 4
terms of each sequence:
a) tn = 3n +
1 b) tn = 2n
4. Find tn for each of the
sequences in question in the previous activity.
Assessment &
Evaluation of Student Achievement
Students’ ability to read and interpret directions
could be informally assessed as they work. Students could be asked to write a
journal entry describing the results of their investigation. Student work on
questions 3 and 4 could be collected and assessed using a rubric which focuses
on inquiry. Where verification or explanation is required, communication could
be assessed.
Accommodations
Use concrete materials for students to better
understand the activity.
Activity 3.2: Summing Up:
Arithmetic Sequences and Series
Time: 150 minutes
Description
Students investigate, through financial applications
and other contexts, arithmetic sequences and series and develop the formulas
for the general term of an arithmetic series and the formula for the sum of the
terms of an arithmetic sequence. During the investigation, students make a
connection between arithmetic sequences and linear growth.
Strand(s) & Learning
Expectations
Overall
Expectations
FAV.01 - solve problems involving arithmetic and
geometric sequences and series;
FAV.02 - solve problems involving compound interest
and annuities.
Specific
Expectations
FA1.04 - determine the value of any term in an
arithmetic or a geometric sequence, using the formula for the nth term of the
sequence;
FA1.05 - determine the sum of the terms of an
arithmetic or a geometric series using appropriate formulas and techniques;
FA2.04 - demonstrate an understanding of the
relationships between simple interest, arithmetic sequences and linear growth.
Prior Knowledge &
Skills
·
first
and second differences to identify whether a relationship is linear or
quadratic
·
notation
associated with sequences and series
·
use
of a graphing calculator
Planning Notes
·
A
brief review of simple interest may be necessary before beginning this
activity.
·
Part
A: Students work independently on the investigation.
·
Part
B: Prepare an overhead with the problem only. Have the students work in groups
of four or fewer to investigate the problem.
·
It
would be helpful if students had access to graphing calculators.
Teaching/Learning
Strategies
Part
A: Teachers may need to emphasize the relationship between arithmetic sequences
and linear growth. Ensure that students have obtained the correct formula for
the general term of an arithmetic sequence.
Part B: Encourage students to develop a strategy to
find the number of handshakes accurately. After the students have worked in
groups, discuss the various strategies. Be sure to discuss the strategy of
introducing the people to the room one at a time in order to develop the
concept of an arithmetic series.
Student Activity
Part A
Carolyn received a gift of $1000 from her grandmother
when she graduated from Grade 8. The money is to be invested to help Carolyn
pay for university when she graduates from high school. She buys an investment
that pays 4% per year simple interest. Complete the chart below to determine
the amount that Carolyn will have at the end of each year for the next five
years.
|
End of Year |
Annual Interest |
Total Amount |
|
1 |
40 |
1040 |
|
2 |
40 |
1080 |
|
3 |
|
|
|
4 |
|
|
|
5 |
|
|
1. Predict the amount that Carolyn would have if
she kept the investment for ten years. On what are you basing your prediction?
2. What is the type of relationship between the
number of years she has the investment and the total amount? Hint: Examine
the first differences.
3. Determine the equation that best models the
relationship between the year and the total amount.
4. Use the equation to predict the year that
Carolyn will double her investment.
5. The total amounts that she has at the end of
each year can be written as a sequence similar to the sequence 7, 10, 13, 16, 19, 22, …
This sequence can be
written 7, 7 + 3, 7 + 3 + 3, 7 + 3 + 3 + 3, 7 + 3 + 3 + 3 + 3, … or using the
notation developed in Activity 3.1.
t1
= 7 + 0(3)
t2
= 7 + 1(3)
t3 = 7 + 2(3)
t4
= 7 + 3(3)
t20
= 7 + _(3)
tn
= 7 + ( )(3)
Simplify the last
term of this sequence. This is called the general or “nth” term of the
sequence.
6. Write the total amounts that you found in the
chart above as the terms of a sequence.
7. Find the “nth” term using the method
used in question 5. Compare this to the equation found in question 3.
8. Use
this method to find tn for each of the following sequences.
a) -10,
-6, -2, 2, 6, …
b) 12,
8, 4, 0, -4, …
c) 
, 1, 
, 2, …
The sequences in this section are called arithmetic.
Describe the properties of any arithmetic sequence.
9. Give an example of an arithmetic sequence.
Explain why you think it is arithmetic.
10. In general, every arithmetic sequence has a
first term, a, and the difference between successive terms is a
constant, d. Complete the missing information below.
t1
= a
t2
= a + d
t3
= a + 2d
t4
= a + 3d
t25
= a + ( )d
tn
= a + ( )d
The last term is the general term of any arithmetic
sequence.
Part B
1. How Many Handshakes?
a) Determine
the number of handshakes required in a room that contains 5 people if each
person shakes hands with every other person once?
b) If
one more person enters the room, how many more handshakes will occur? What is
the new total number of handshakes? Predict the number of handshakes that will
occur if there are 10 people in the room.
You may find the following chart helpful to
organize your work
|
Number of
People |
Number of Handshakes |
Total |
|
1 |
0 |
0 |
|
2 |
0+1 |
1 |
|
3 |
0+1+2 |
3 |
|
4 |
|
|
|
5 |
|
|
c) Identify
the type of relationship that exists between the number of people in the room
and the total number of handshakes.
d) Find
an equation that best models this relationship. The number of handshakes if
there are 5 people in the room can be found by adding the following 0+1+2+3+4.
The numbers 0,1,2,3,4 are the terms of an arithmetic sequence. Explain why.
When the terms of an arithmetic sequence are added, the result is called an
arithmetic series.
e) The
number of handshakes required if there are 10 people in the room can be found
be adding the terms of the arithmetic series 0+1+2+3+4+5+6+7+8+9.
S10 = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7
+ 8 + 9
S10 = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2
+ 1 + 0
Adding these two rows, we get
2S10
= 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9
2S10
= 10(9)
S10
= 
= 45
f) Use
a similar process to find the total number of handshakes required if there were
20 people in the room, 50 people in the room. Find the total number of
handshakes required if there are n people in the room.
2. The same method can be used to find the sum
of the terms of any arithmetic series.
a) Use this method to find the
sum of the terms of the following arithmetic series:
i) Find
the sum of the first 9 terms of the series 9 + 11 + 13 + 15 + 17 + 19 +…
ii) Find
the sum of the first 10 terms of the series 10 + 12 + 14 + 16 + 18 + 20+…
b) Note that
the sum of the first 5 terms can be found by the following S5
= 
Create a formula to
find the sum of the first 8 terms.
c) Create a formula to find the sum of the first
n terms.
d) Recall
that t1 = a, and tn = a + (n-1)d
for an arithmetic sequence. Verify that the following formula represents the
sum of n terms of an arithmetic series.
S = 
e) Verify that S10 in the previous example is 45 by letting a =
0, and d = 1.
3. Friedrich Gauss was a famous Mathematician.
When he was 10 years old his teacher gave him the problem of adding the whole
numbers from 1 to 100. The problem was meant to keep him busy for several
minutes, but he immediately said that the sum was 50(101). Explain how he might
have used the Sn formula to get his result.
Teacher
Facilitation
Students need to develop skills with arithmetic
sequences and series. At the end of each part of the activity the concepts
should be summarized and additional problems assigned from the students’ text.
Extension
S10 = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9
can also be written 
. This notation is called Sigma or Summation Notation.
1. Write the series represented by
a) 
b) 
2. Write the following series using summation
notation
a) 1 + 3 + 5 + 7 + 9 + 11 b) 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18
Assessment &
Evaluation of Student Achievement
Learning skills, such as teamwork, could be informally
assessed while students work in small groups. After students have had an
opportunity to consolidate skills on arithmetic sequences and series, teachers
could give a short quiz to assess Knowledge/Understanding. Teachers might
collect solutions to questions 1 and 3 and assess Thinking/Inquiry and Problem
Solving. Questions 2a and 2e could be used to assess Application.
Accommodations
·
Handouts
should be provided for students who have difficulty reading from the overhead screen.
·
Partner
student within teams to provide support for students who require peer
assistance
Activity 3.3: Compound Interest:
Exploring Geometric Sequences
Time: 150 minutes
Description
Students develop the concept of a geometric sequence
through the use of compound interest. They then develop the formula for the
general term of a geometric sequence using patterns discovered. The formula
will be applied to develop the compound interest formula.
Strand(s) & Learning
Expectations
Strand(s): Financial Applications of Sequences and Series
Overall
Expectations
FAV.01 - solve problems involving arithmetic and
geometric sequences and series;
FAV.02 - solve problems involving compound interest
and annuities.
Specific
Expectations
FA1.03 - identify sequences as arithmetic, geometric
or neither;
FA1.04 - determine the value of any term in an
arithmetic or a geometric sequence, using the formula for the nth term of the
sequence;
FA2.01 - derive the formulas for compound interest and
present value, the amount of an ordinary annuity, and the present value of an
ordinary annuity, using the formulas for the nth term of a geometric
sequence and the sum of the first n terms of a geometric series;
FA2.02 - solve problems involving compound interest
and present value;
FA2.05 - demonstrate an understanding of the
relationships between compound interest, geometric sequences, and exponential
growth.
Prior Knowledge &
Skills
·
simple
interest formula
·
use
of a graphing calculator and/or spreadsheet software
Planning Notes
·
Students
need access to a graphing calculator or spreadsheet software.
·
Students
can work individually or in pairs on the activity.
·
Spreadsheets
might be used to generate tables.
·
The
rule of 72 might be used where doubling applies.
Teaching/Learning
Strategies
When students are completing the chart in the
activity, suggest that they look for efficient ways of calculating the new
principal. This will lead to the development of the geometric sequence. After
students have completed Part A, the teacher should assign some follow-up
questions from the text, or other resources. Before beginning Part B, discuss
the vocabulary related to compound interest. Define conversion or compound
period and discuss the meaning of statements like “interest is compounded
semi-annually”. Students may need some help developing the sequences in Part B.
Make sure that they have the correct meanings for the variables in the compound
interest formula.
Student Activity
Part A
In
Activity 3.2, Part A, Carolyn placed the money her grandmother had given her in
an investment that paid 4% per year simple interest. Understanding simple
interest is very important to understanding how interest is actually
calculated. It is very rare that an investment like the one in Activity 3.2
would be used. Usually interest is calculated on a regular basis and deposited
in the account. The next time that interest is calculated the principal has
increased. This is called compound interest because interest is paid on
the interest that has been deposited. Complete the chart below to determine the
amount that Carolyn will have at the end of four years if interest is 4% per
year, compounded annually, and the interest earned is deposited at the end of
each year.
|
Time (years) |
Principal at the Beginning of the Year |
Interest for the year |
New Principal at the End of the Year |
|
Now |
- |
- |
$1000.00 |
|
1 |
$1000.00 |
1000(0.04)(1) = 40 |
1040.00 |
|
2 |
1040.00 |
1040(0.04)(1) = 41.60 |
1081.60 |
|
3 |
|
|
|
|
4 |
|
|
|
1. Compare the amount she has at the end of four
years to the amount you found in Activity 3.2, Part A.
2. Determine the type of relationship that
exists between the number of years that have elapsed and the total amount of
money that Carolyn has accumulated.
3. Use a graphing calculator to find an equation
that best models this relationship.
4. Predict the amount that Carolyn will have at
the end of 10 years.
5. Determine approximately how long it will take
for Carolyn to double her money. Can you find more than one method of doing
this?
Consider the sequence below:
1000,
1000(1.04) 1000(1.04)2, 1000(1.04)3, 1000(1.04)4
6. Calculate the value of the 5 terms of this
sequence.
7. Compare the values to the “New Principal” in
the chart above. What do you notice? Explain.
8. Use the pattern in the original sequence to
calculate the amount that Carolyn will have at the end of 7 years.
9. Use the pattern to develop a formula for tn.
Compare this to the regression equation found above. Explain what you found.
The sequence above is called a geometric sequence. Describe the
properties of a geometric sequence.
10. Give an example of a sequence that is neither
arithmetic nor geometric. Explain why they are not.
11. The geometric sequence 3, 6, 12, 24, … can be written in the form
3,3(2), 3(2)2, 3(2)3,… Find the value of the 5th term of
the sequence. Find an expression for tn .
12. Any geometric sequence can be written in the
form a, ar, ar2, ar3, ar4, … a is the
first term and r is the ratio of successive terms. Find an expression
for tn.
Part B
In the first part of this activity you developed the
formula for the nth term of a geometric sequence by looking at a problem
involving compound interest. You discovered that the first term of the sequence
1000, 1000(1.04), 1000(1.04)2,
1000(1.04)3,
1000(1.04)4, is the principal deposited and the remaining
terms represent the amount of money that has accumulated at the end of each
year for 4 years.
Write a sequence that represents the amount of money
that will accumulate if $5000 is invested at 6% per year, compounded
semi-annually. Determine tn. In this case, what does n represent?
Write a sequence that represents the amount of money
that will accumulate if $10 000 is invested at 8% per year, compounded
quarterly. Determine tn. What
does n represent?
Write a
sequence that represents the amount of money that will accumulate if the
principal is P and the interest rate per year is r. Interest is compounded
monthly. Determine tn. What
does n represent?
The compound interest formula is A = P(1 + I)n. Explain
what A, P, I, and n represent.
Follow
up Questions
At the end of each part of the activity, assign
related questions. Some additional examples may be necessary. At the end of
Part A, present the students with several different sequences (many examples
will be available in the texts) and have them identify the type of sequence.
Discuss how the formula for compound interest can be rearranged to find the
present value of an investment or loan. Assign related questions for the students
to develop skills.
Assessment &
Evaluation of Student Achievement
Communication skills might be assessed after having
the students write a journal entry summarizing their findings. Informal
assessment of Learning Skills should be ongoing. Teachers could assign and
collect solutions to a problem that requires the use of the compound interest
formula to determine whether students are able to apply their knowledge to
problem solving. Solutions to question 5 could be completed in a variety of
ways. Thinking/Inquiry and Problem Solving might be assessed here.
Activity 3.4: Applications: Finding the Amount and the
Present Value of a Long Term
Investment
Time: 75 minutes
Description
Students extend their knowledge of compound interest
and present value in order to make informed decisions about investments.
Students learn to use the Financial Applications feature on a graphing
calculator as well as the spreadsheet capabilities of software programs.
Strand(s) & Learning
Expectations
Strand(s): Financial Applications of Sequences and Series
Overall
Expectations
FAV.02 - solve problems involving compound interest
and annuities;
FAV.03 - solve problems involving financial decision
making, using spreadsheets or other appropriate technology.
Specific
Expectations
FA2.02 - solve problems involving compound interest
and present value;
FA3.01 - analyse the effects of changing the
conditions in long-term savings plans;
FA3.02 - describe the manner in which interest is
calculated on a mortgage (i.e., compounded semi-annually but calculated
monthly) and compare this with the method of interest compounded monthly and
calculated monthly.
Prior Knowledge &
Skills
Understanding of the compound interest formula
Planning Notes
·
Students
need access to a graphing calculator or spreadsheet software.
·
Prepare
a reference sheet that can be distributed to the students outlining keystrokes
on the graphing calculator.
·
In
the example provided, the steps required on a TI-83+ graphing calculator are
outlined. If other technology is used, the instructions will require
adjustment.
Teaching/Learning
Strategies
Students require the use of a graphing calculator to
complete this activity as it is written. Teachers may wish to lead the students
using the overhead calculator, if one is available, so that the students can
see the screen that they should have on their own calculator.
Student
Activity
In the previous activity you calculated the amount
that Carolyn would accumulate if $1000 was deposited for 4 years at 4%/a
compounded annually. In this activity you will learn how to use the “Finance”
feature of the TI83+ graphing calculator to evaluate this amount and
investigate other related problems.
1. Press [MODE], cursor down to FLOAT 0, 1, 2,
etc. Select [2]. The calculator will now automatically round all values to two
decimal places. All values will be rounded to the nearest cent.
2. Press [APPS], [ENTER] to select Finance,
[ENTER] to select TVM Solver. On the screen you should see
N=
(number of interest payments)
I%=
(annual interest rate)
PV=
(present value of the investment or loan)
PMT=
(periodic payment)
FV=
(future value of the investment or loan)
P/Y=
(payments per year)
C/Y=
(compound periods per year)
PMT:
END BEGIN
Press 4 [ENTER] to store 4 years. Press 4
[ENTER] to store 4%. Press –1000 [ENTER] to indicate a cash outflow of $1000.
Press 0 [ENTER] to show a payment of 0 (Periodic payments will be considered in
future sections. In this activity this value will always be 0.) Press 0 [ENTER] to show a future value of 0.
Press 1 [ENTER] to show 1 payment per year. Press 1 [ENTER] to show 1 compound
period per year. Select PMT: END. This indicates the payments are due at the
end of each period. Cursor up to FV. Press [ALPHA] [Solve]. The value of the
investment at the end of 4 years will appear.
3. Carolyn discovers she can invest the $1000 at
4%/a compounded semi-annually. Adjust the values to calculate how much she will
have at the end of 4 years if she chooses this option.
Cursor to N=. Press 8 [ENTER] to indicate 8 interest payments. Cursor to
FV=. Press 0 [ENTER]. Cursor to P/Y=. Press 2 [ENTER] to indicate 2 payments
per year. Cursor to C/Y=. Press 2 [ENTER] to indicate 2 compound period per
year. Cursor to FV=. Press [ALPHA] [Solve]. The value of the investment after 4
years will appear.
4. Find the amount that will accumulate if
Carolyn finds an investment with interest at 4%/a compounded monthly.
Compare the following investments.
$8000 is invested at 5.25%/a compounded
semi-annually for 5 years.
$8000 is invested at 5%/a compounded monthly
for 5 years.
Which is the better investment? Why?
5. Carolyn will need about $15 000 for her first
year of university. (This includes tuition, books, residence and spending
money.) How much would she have to
invest now at 4%/a compounded monthly to ensure she has enough 4 years from
now?
Go to the Financial Applications program on the
TI-83+ graphing calculator. Enter the following: N=48, I=4%, PV=0, PMT=0,
FV=15000, P/Y=12, C/Y=12. Cursor to PV and press [ALPHA] [Solve]. Explain what
each of the values entered means.
Follow
up Questions
Provide the students with opportunities to practise
skills with the finance program on the calculator. There will be many examples
available in the students’ text. Students should be encouraged to check their
results (for a few examples) by using the compound interest formula.
Assessment &
Evaluation of Student Achievement
The focus of assessment should be on the consolidation
of skills needed in problem solving. Students should be able to solve problems
with and without the use of technology. Unless the teacher specifically
requests the use of the formula, either type of solution should be acceptable.
Resources
TI-83+ Graphing Calculator Guidebook, Chapter
14 – Applications, Section 14-4
Activity 3.5: Introduction to
Geometric Series
Time: 75 minutes
Description
Students investigate a geometric series using an
activity requiring financial decision making. The formula for the sum of a
geometric series is developed.
Strand(s) & Learning
Expectations
Overall
Expectations
FAV.01 - solve problems involving arithmetic and
geometric sequences and series.
Specific
Expectations
FA1.05 - determine the sum of the terms of an
arithmetic or a geometric series, using appropriate formulas and techniques.
Prior Knowledge &
Skills
General term of a geometric sequence
Planning Notes
Access to a graphing calculator is helpful, but not
essential.
Teaching/Learning
Strategies
Make an overhead that states the problem only. Have
the students investigate the problem. The students could make a table of values
for each situation and use a graphing calculator to find an equation that
models the total amount he is paid. Before completing the activity, discuss the
fact that the amounts Sasha receives each day, in the second scenario, are the
terms of a geometric series.
Student
Activity
Sasha’s neighbours are planning on taking a vacation
for two weeks (14 days). They have asked him to look after their cat and to
water their plants. The neighbours have offered to pay him $5 per day or $0.01
the first day, $0.02 the second day, $0.04 the third day, $0.08 the fourth day,
etc. Which method of payment should Sasha choose?
1. Sasha
notices that the amounts he will earn each day if he chooses the second plan
are the terms of the geometric sequence
0.01, 0.01(2)2, 0.01(2)3,…,0.01(2)13
He uses the following method to calculate the
amount that he will have at the end of two weeks.
S14 = 0.01 +
0.01(2) + 0.01(2)2 + 0.01(2)3 + … + 0.01(2)13
2S14 = 0.01(2) + 0.01(2)2 +
0.01(2)3 + … + 0.01(2)13 + 0.01(2)14
2S14
– S14 = 0.01(2)14 – 0.01
S14 = 
= $163.83
2. Use this method to develop a formula to find
the amount that Sasha will earn if he works any number of days.
3. If the neighbours are away for 7 days which
method of payment should Sasha choose?
4. The sum of the terms of a geometric sequence
is called a geometric series. Any geometric series can be written using the
following form Sn = a + ar
+ ar2 + ar3 … + arn-1
Use the method that Sasha used to develop the formula
for the sum of the terms of a geometric series.
Teacher
Facilitation
Ensure that the students have developed the correct
formula for the sum of the geometric series. Assign additional questions using
the formula. Make sure that some of the questions are related to financial
applications.
Follow
up Questions
1. Find the amount that Sasha would earn if his
neighbours were away for a month. Discuss whether this method of payment is
reasonable.
2. Write the geometric series using summation
notation.
Assessment &
Evaluation of Student Achievement
A short quiz could be used to assess knowledge and
understanding of expectations related to geometric sequences and series.
Communication could be assessed using question 1 from the activity.
Accommodations
Hand outs should be provided for students who have
difficulty reading off of the overhead
Activity 3.6: Applications:
Finding the Amount and Present Value of an Annuity
Time: 150 minutes
Description
Using the example of a student who is saving money
earned from a paper route, students investigate the amount of an annuity to
determine interest, number of payments and the total amount. In the second part
of the activity students calculate the present value of an annuity given the
scenario of a hockey player who would like to set up a scholarship fund.
Strand(s) & Learning
Expectations
Overall
Expectations
FAV.02 - solve
problems involving compound interest and annuities;
FAV.03 - solve problems involving financial decision
making using spreadsheets or other appropriate technology.
Specific
Expectations
FA2.02 - solve problems involving compound interest
and present value;
FA2.03 - solve problems involving the amount and the
present value of an ordinary annuity;
FA3.05 - communicate the solutions to problems and the
findings of investigations with clarity and justification.
Prior Knowledge &
Skills
·
Working
knowledge of geometric series
·
Solving
equations
Planning Notes
·
Students
will need access to a graphing calculator and/or spreadsheet software.
·
Before
beginning the activity discuss situations where students encounter periodic
payments, both in light of amount accumulated and the amount needed to
accumulate a specified sum (for example savings plans, scholarship funds, car
loans)
·
Students
need to be shown how to set up the first line diagram. Terms of the series
should be left in unsimplified form (i.e., 
).
Teaching/Learning
Strategies
The night before the activity students could be
directed to investigate in the media any situations where periodic payments are
used. This could be used to initiate discussion before the student activity
begins.
Student
Activity 3.6.1: How Much Do We Have?
Time: 75
minutes
Part A
Jeremy is in Grade 8. He has a paper route and wants
to save for his college education. He determines that he has $100 per month to
put into an account at 6%/a compounded monthly. How much will he have at the
end of five years for his college education?
1. Set up a line diagram showing the amount paid
each month and the amount accumulated for each payment at the end of the time
period.
2. Show that the amounts form a geometric
series.
3. Calculate the sum of this geometric series
using 
Part B
Based on an average cost per year to attend a
university or college of your choice, how much should be invested each month to
obtain the goal of that amount? Assume 6%/a compounded monthly for 5 years.
Teacher Facilitation
Part A: Make sure the students’ line diagrams show
several payments and the value of these payments at the end of the annuity.
This will be important for the next activity when the amount invested at the
end of each interval will not be known. The generality of determining amounts
using the 
formula is important.
Part B: If the students have not had the time to
investigate actual amounts, suggest they use $10 000 per year. The teacher
might suggest looking at other scenarios. For example, the student may be able
to obtain a better interest rate.
Follow-up Questions
1. Ask if the amount accumulated for Jeremy’s
college education is realistic. Investigate the costs of going to university or
college for one year at various educational institutions.
2. Have the students fill in a table such as the
following.
|
Amount |
Payment |
Number of Payments |
Conversion Period |
Interest Rate pa |
|
? |
$300 |
20 |
Semi-annually |
8% |
|
? |
$500 |
24 |
Quarterly |
12% |
|
? |
$100 |
36 |
Monthly |
12% |
|
? |
$4000 |
10 |
Annually |
8% |
|
$10 000 |
? |
40 |
Semi-annually |
8% |
|
$3000 |
? |
24 |
Quarterly |
12% |
|
$5000 |
? |
18 |
Monthly |
12% |
|
$12 000 |
? |
15 |
Annually |
6% |
Student
Activity 3.6.2: How Much Do We Need?
Time: 75 minutes
A retired hockey star wants to set up a scholarship
fund to assist an underprivileged child who would like to go to a
post-secondary institution. He wants to ensure that the student will have $6000
per year for five years. How much should he give to the institution, now, to
ensure that this can happen, if the institution is able to invest the money at
10%/a compounded annually.
1. Set up a line diagram showing the present
value of each of the $6000 payments.
2. Explain why the amounts form a geometric
series.
3. Calculate the sum of the geometric series
using the formula
Teacher
Facilitation
Remind the students that they will need to find the
present value of an amount when calculating the present value of each $6000
payment.
Follow-up
Questions
Set up a table similar to the one in Activity 3.6.1 to
find the present value of an annuity given the same kind of conditions
illustrated in the table. Students should also complete questions that involve
knowing the present value of the annuity and find the periodic payment.
Assessment &
Evaluation of Student Achievement
Student work could be collected and assessed using an
inquiry rubric. Students’ solutions should be organized and written using
proper mathematical form. This is especially important when setting up the line
diagram. Teachers could informally (or formally) assess progress and provide
feedback to students to assist them to improve their communication skills.
Activity 3.7: What Happens When?:
Changing the Time, Rate and Amount
Time: 150 minutes
Description
Students will solve problems involving periodic
payments. They will investigate the results of changing different parameters.
Strand(s) & Learning
Expectations
Overall
Expectations
FAV.03 - solve problems involving financial decision
making using spreadsheets or other appropriate technology.
Specific
Expectations
FA3.01 - analyse the effects of changing the
conditions in long-term savings plans;
FA3.05 - communicate the solutions to problems and the
findings of investigations with clarity and justification.
Prior Knowledge &
Skills
·
Formula
for the sum of a geometric series
·
Solve
a quadratic equation
Planning Notes
Students will require access to a graphing calculator
or spreadsheet software.
Teaching/Learning
Strategies
Due to the number of parameters involved in these
problems, it would be appropriate to complete a number of examples with the
students. It may be helpful to use the overhead view screen projector with the
graphing calculator, if it is available. The intent of this activity is to
provide the students with an efficient method of investigating the result of
changing some of the conditions when investing or borrowing. Students should be
aware of the mathematics involved from the previous activity, but should make
use of available technology to perform the calculations.
Student Activity
Example
1
Grant and Kera are both 75 years old. Kera is very
money conscious. She was 20 years old when she began investing $1000 a year into
an RRSP paying an average of 6%/a compounded annually. Grant, on the other
hand, did not start to invest until age 50. He made an annual deposit of $3000
beginning at
age 50. The average interest rate he received on his investment was 8%/a
compounded annually. What amount does each have today? What should Grant have
invested each year in order to have the same amount as Kera at age 75? If Grant
could only afford to invest $3000 per month, what average rate of interest
would result in his saving the same amount as Kera?
Solution
To find the amount that Grant has, use the TI-83+
graphing calculator. Select [APPS], [1],[1] to choose the financial
applications.
Enter the following
N=25, I=8%, PV=0, PMT=-3000 (this indicates a cash
outflow of $3000 per year), FV=0, P/Y=1, C/Y=1, Select PMT:END. Cursor to FV,
then push [ALPHA], [solve].
Now try the rest of the problem using the calculator.
Example
2
Stacey buys a $1000 RRSP today. After one year she adds
$2500. By the end of the second year, the money has grown to $3851 as it has
earned interest over time. What rate of interest does the money earn?
1. Apply the compound interest formula to both
the $1000 and $2500 investments. Assume that interest is compounded annually at
the rate, i, to grow to $3851 to get
1000(1 + i)2 + 2500(1 + i) = 3851
2. If this equation is written 1000x2
+ 2500x – 3851 = 0, what does x represent.
3. Use the quadratic formula and your calculator
to solve for x and i.
4. There are two solutions to the quadratic
equation. Which one would you choose? Why?
Teacher
Facilitation
There are many types of variables that can be adjusted
in problems related to finance. Teachers should spend time discussing other
scenarios and assign similar problems from other resources. In example 2, the
interest rate for both investments is compounded annually, consider what will
happen if the interest rate were compounded more frequently for one of the two.
Follow
up Questions
1. Maria received $50 on her 16th
birthday, and $70 on her 17th birthday, both of which she
immediately invested in the bank with interest compounded annually. On her 18th
birthday, she had $134.97 in her account. Draw a time line and calculate the
annual interest rate.
2. Raul’s grandparents invested $1000 in a GIC
for him on his 15th birthday and $2000 on his sixteenth birthday.
During each of the two years the money earned 8.5% compounded annually.
a) Draw
a time line to show the situation.
b) How
much was in the fund on his 17th birthday?
Assessment &
Evaluation of Student Achievement
Provide the students with two or three problems.
Student ability to read, interpret, and solve the problems could be assessed.
The students might be asked to hand in their solutions. Since a range of
performances is likely, a rubric could be used to assess Application to Problem
Solving and Communication.
Resources
TI-83+ Graphing Calculator Guidebook – Chapter
14.
Activity 3.8: Mortgages: How They
Work
Time: 150 minutes
Description
In this activity students will investigate various
scenarios related to mortgages. They will compare methods of interest
calculation, generate amortization tables for mortgages and analyse the effects
of changing conditions of a mortgage.
Strand(s) & Learning
Expectations
Overall
Expectations
FAV.03 - solve problems involving financial decision
making, using spreadsheets or other appropriate technology.
Specific
Expectations
FA3.02 - describe the manner in which interest is
calculated on a mortgage (i.e., compounded semi-annually but calculated
monthly) and compare this with the method of interest compounded monthly and
calculated monthly;
FA3.03 - generate amortization tables for mortgages,
using spreadsheets or other appropriate software;
FA3.04 - analyse the effects of changing the
conditions of a mortgage (e.g., the effect on the length of time needed to pay
off the mortgage of changing the payment frequency or the interest rate);
FA3.05 - communicate the solutions to problems and the
findings of investigations with clarity and justification.
Prior Knowledge &
Skills
Students should have acquired an understanding of
annuities.
Planning Notes
Access to the TI-83+ calculators is required. If
different technology is used, the instructions in this activity will have to be
modified.
Teaching/Learning
Strategies
Students could be asked to visit a local financial
institution to determine the method of interest calculation on loans and
mortgages. An alternative would be to invite a guest speaker from a local
financial institution to discuss these issues with the class. Before beginning
this activity, teachers should introduce students to some of the language
associated with mortgages. It is important for students to be comfortable using
the financial applications of a graphing calculator or spreadsheet software.
Teachers may choose to have the students use a computer spreadsheet to generate
the amortization schedules.
Student
Activity
Mortgages are a special type of annuity. They are also
a fact of life for most Canadian families. When applying for a mortgage through
a financial institution, it is important to know how much your family can
afford to pay on a monthly basis, the term of the amortization, and the
interest rate charged by the financial institution. The TI-83+ calculator has a
software package that will help determine the value of some of these
parameters.
Ron and Lynda are purchasing a new home. They will
need to borrow $90 000 for the home. The amortization period, which best fits
their financial situation is 25 years. Discuss why this might be so. The
current interest rate is 8.7%/a. Interest is compounded semi-annually and
calculated monthly. The couple wants to know how much it will cost them each
month to pay off the loan. Assume that they will make one payment per month.
To solve this problem, select [APPS], [1], [1] (this
selects the TVM solver). Enter, N=25x12, I=8.7, PV=90 000, PMT=0, FV=0, P/Y=12,
C/Y=2, select PMT:END. Cursor to PMT, press [ALPHA], [solve]. This finds the
payment of $727.52. (It will appear as a negative because this is a cash
outflow.)
1. Create a schedule of payments for the first 5
years of the mortgage. Include a graph showing the declining balance at the end
of each year.
3. Create a new schedule of payments for 5
years.
Teacher
Facilitation
Teachers may wish to prepare a handout, which covers
the menu and the steps required to compute the amortization schedule on the
TI-83+ calculator.
Assessment &
Evaluation of Student Achievement
Learning skills and the ability to read and interpret
problems might be informally assessed. Students could be asked to write a
report on information obtained from visiting financial institutions or from
listening to a guest speaker. This report could be assessed for Communication
skills. The amortization tables could be handed in and assessed for Application
and Knowledge/Understanding.
Activity 3.9: Financial Decision
Making: A Case Study
Time: 300 minutes
Description
Students are presented with a realistic situation
involving financial decision making. They analyse the situation and suggest
financial decisions to be made. Students are required to write a report
justifying their decisions using the mathematics learned in this unit.
Strand(s) & Learning
Expectations
Overall
Expectations
FAV.01 - solve problems involving arithmetic and
geometric sequences and series;
FAV.02 - solve problems involving compound interest
and annuities;
FAV.03 - solve problems involving financial decision
making, using spreadsheets or other appropriate technology.
Specific
Expectations
It is expected that as many specific expectations as
possible in the Financial Applications of Sequences and Series strand will be
addressed in this summative assessment activity.
Planning Notes
Teachers may choose to review some of the major
concepts discussed in the previous activities (e.g., computation of interest,
annuities).
Teaching/Learning
Strategies
Conduct a group discussion to enable students to
generate an expense report for a typical month in the life of a family of four
(two adults, two children in elementary school). Include in the report all
expenses related to the management of a home. Emphasize in the student activity
that a full and detailed report on the
Student Activity
Part 1
The
After 5 years, the mortgage must be renewed with an
interest rate of 8.5%/a compounded semi-annually and calculated monthly. How
will this effect their monthly expenses?
Part 2
Write a short report about long-term financial and
educational planning. What have you learned? How has this information changed
the way you view a mortgage? How could this influence your financial choices?
You may wish to comment on the feelings that other people might have, or what
they might know or not know about this matter. You may want to describe your
own personal long-term plans in this area.
Assessment &
Evaluation of Student Achievement
All categories of the Achievement Chart should be
assessed in this summative assessment activity. Since a range of performances
is likely, a rubric might be used. Teachers might choose to use a
pencil-and-paper test to assess Knowledge/Understanding of expectations in the
unit that are not specifically addressed in this activity.
Suggested
Rubric to be used with Activity 3.9
|
Categories |
Level 1 |
Level 2 |
Level 3 |
Level 4 |
|
Thinking/Inquiry
and Problem Solving - analyses the
effects of changing variables in financial applications |
- takes into
account only a limited number of factors in analysing the problem - is unable to
devise a reasonable financial plan |
- takes into
account some of the factors in analysing the problem - offers a
solution which is somewhat justified by the facts given |
- takes into
account most of the factors in analysing the problem - makes a
reasonable suggestion based on the facts given |
- takes into
account all, or almost all, of the factors in analysing the problem - identifies
more than one option based on the facts given |
|
Communication - communicates
with clarity and justification |
- communicates
with limited clarity and limited justification |
- communicates
with some clarity and some justification |
- communicates with
general clarity and justification |
- communicates
with a high degree of clarity and justification |
Note: A student whose achievement is below level 1
(50%) has not met the expectations for this assignment of activity
Accommodations
Allow extra time for those students who need it.
Resources
Banks and trust companies
The personal financial planner at a local financial
institution
Course
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