Business mbf3c power of compound interest

Document Sample

```					                                                                                                        Comment [AI]: Minor formatting
Contextualized Learning Activities (CLAs)                                     changes.

For the “other required credits” in the bundle of credits, students in a Specialist High Skills Major
program must complete learning activities that are contextualized to the knowledge and skills
relevant to the economic sector of the SHSM. Contextualized learning activities (CLAs) address
curriculum expectations in these courses.

This CLA has been created by teachers for teachers.
It has not undergone an approval process by the Ministry of Education.

Contact Information

Board                      Rainbow District School Board

Development date           January 2008

Contact person             Keri Endanawas
Sheila Giroux
Program Coordinator – Numeracy
Phone                      705-522-2320
705-523-3308
Fax                        705-523-3314

E-mail                     endanak@rainbowschools.ca
girouxs@rainbowschools.ca

Major

Course code and course      MBF3C – Foundations for College Mathematics
title

Name of contextualized      The Power of Compound Interest
learning
activity/activities

Brief description of        In these activities, the students will explore different investment
contextualized learning     options in simple interest and in compound interest problems.
activity/activities         Students will identify the main features of each and recognize trends.
The activity culminates in exploring the benefits associated with two
alternatives for funding a scholarship.

Duration                    6 hours
Overall expectations        B1 Compare simple and compound interest, relate compound
interest to exponential growth, and solve problems involving
compound interest

B2 compare services available from financial institutions, and solve
problems involving the cost of making purchases on credit

Specific expectations       B1.1 determine, through investigation using technology, the
compound interest for a given investment, using repeated
calculations of simple interest, and compare, using a table of values
and graphs, the simple and compound interest earned for a given
principal (i.e., investment) and a fixed interest rate over time

B1.2 determine, through investigation (e.g., using spreadsheets and
graphs), and describe the relationship between compound interest
and exponential growth

B1.3 solve problems, using a scientific calculator, that involve the
calculation of the amount, A (also referred to as future value, FV),
and the principal P (also referred to as present value, PV) using the
`     an
compound interest formula in the form A  P 1  i [or
`    an
FV  PV 1  i      ]

B1.4 calculate the total interest earned on an investment or paid on
a loan by determining the difference between the amount and the
principal [e.g., using I  A @P (or I  FV @PV )]

B1.5 solve problems, using a TVM Solver on a graphing calculator or
on a website, that involve the calculation of the interest rate per
compounding period, i, or the number of compounding periods, n, in
`    an               `     an
the compound interest formula A  P 1  i [or FV  PV 1  i ]

B2.2 gather and interpret information about investment alternatives
(e.g., stocks, mutual funds, real estate, GICs, savings accounts), and
compare the alternatives by considering the risk and the rate of
return

Essential Skills and work   Numeracy – Money Math
habits                      Thinking – Job Task Planning and Organizing
Oral Communication
expectations (if
applicable)

Instructional/Assessment Strategies
Teacher’s notes
There are many considerations in the development of a positive learning environment

Processes
The seven mathematical processes can be referred to as the ‘actions of math.’ In the revised
curriculum, these process expectations have been highlighted in their importance since they
support the acquisition and use of mathematical knowledge and skills. They can be mapped to
three categories of the Achievement Chart – Thinking, Communication and Application. The
fourth category, Knowledge and Understanding, connects to the overall and specific
expectations of the course, which can be referred to as the ‘mathematical concepts’.
Students apply the mathematical processes as they learn the content for the program.

The combination of the mathematical processes and expectations are embedded in the
achievement chart as the following:

Knowledge and Understanding
Concept Understanding         Procedural Fluency

Thinking
Problem Solving             Reflecting            Reasoning and Proving
Application
Selecting Tools and Computational Strategies                 Connecting

Communication
Communicating         Representing

Literacy Strategies
Mathematics is the most difficult content area material to read because there are more
concepts per word, per sentence, and per paragraph than in any other subject; the
mixture of words, numerals, letters, symbols, and graphics requires the reader to shift
from one type of vocabulary to another.
Leading Math Success, Report of the Expert Panel
for Mathematical Literacy Gr. 7 – 12

Improved student achievement demands an emphasis on developing literacy competencies
linked to mathematics learning. To consolidate understanding, learners need opportunities to
share their understanding both in oral as well as written form. Weakness in reading or writing
skills provides barriers to success in problem solving. This resource explicitly embeds at least
one literacy strategy in every lesson.

Starting points for teachers:
 Use strategies to develop vocabulary and comprehension skills, including
 word walls
 Frayer model
 concept circles
 Use strategies relating to the organization of information
 “inking your thinking” – having students write down their thoughts
 concept maps
 anticipation guides
 Use strategies to help students understand features of textbooks and graphics
 highlight key words
 think aloud
More details and strategies can be found in Think Literacy: Cross-CurricularApproaches,

Context
Over the past several years, there has been remarkable growth in the need to understand and
apply basic principles of money management and investment. Technology has been at the
front of this surge. Technology allows a larger proportion of the population to access the tools
required to make sound financial decisions with respect to their business and personal
investments. Entrepreneurs sometimes make investments in local education as part of their
social responsibility. They will fund awards, bursaries and scholarships at local educational
institutions including secondary schools. In doing so, entrepreneurs often need to examine
alternatives in consideration with their cash flow, and assets to allow the maximum return to
the school with minimum cost. To do so, entrepreneurs need a sound understanding of simple
and/or compound interest.
Strategies

Lesson 1 - Simple Interest
 This lesson will be review of previous material for many students.
 Allow students to work in small groups to solve the examples rather than demonstrate
the calculations for students.
 Provide graph paper, placemat paper, chart paper with markers for each group.
 Be aware of students who continue to struggle working with percents and provide
support when required.
 Pearson Math 11 (2008) p. 212-216. Additional practice may be assigned for the text
as determined by student’s needs.

Lesson 2 - Compound vs. Simple Interest
 Be certain that students are linking the two ideas together.
 A computer lab is required for this activity but students should work in pairs.
 Students need to be able to compare and contrast the two types of interest.
 Pearson Math 11 (2008) p 217-222
 MBF3C Support Materials Unit 8 www.oame.on.ca

Lesson 3 - The Compound Interest Formula
 The teacher should ensure that students are connecting the ideas of all three lessons
so far in this unit.
 Pearson Math 11 (2008) p 223-234
 MBF3C Support Materials Unit 8 www.oame.on.ca

Lesson 4 - The TVM Solver
 Students use technology in graphing calculators to further explore compound interest,
particularly when needing to solve for i or n in the compound interest formula.
 As a homework activity, students will discover the Rule of 72 (the product of the annual
interest rate and the number of years for the investment to double is 72).
 Emphasis on exponential growth and the effect of different compounding periods on the
accumulated amount continues.
 MBF3C Support Materials Unit 8 www.oame.on.ca
 Taking Stock in Your Future www.investorED.ca

Lesson 5 - Philanthropy
 Students use the compound interest formula in conjunction with technology to explore
the present value of an investment given the accumulated amount.
 Students will explore two different payout options, make conjectures and make a final
argument in writing.
 This is the summative evaluation of the unit.
 MBF3C Support Materials Unit 8 www.oame.on.ca
 Taking Stock in Your Future www.investorED.ca

Assessment and Evaluation of Student Achievement

1. Paper/Pencil Quiz (L1)                    Diagnostic

2. Peer Assessment (L2)                      Formative

3. Journal (L2)                              Formative

4. Checklist (L3)                            Formative

5. Journal (L3)                              Formative

6. Gallery Walk (L3)                         Formative

7. Oral Discussion (L4)                      Formative

8. Philanthropy Assignment (L5)              Summative

Assessment tools

Achievement Chart
Rubric

To elevate student interest, the teacher may pull from current events to teach the elements of
this unit. These can be incorporated as mathematical examples, investigations or to
encourage dialogue and discussion of current investment trends and options.
Resources

Authentic workplace materials

Human resources

Print

Pearson Math 11 (2008)

Video

Software

Websites

www.investorED.ca
www.oame.on.ca
Other

Graphing Calculator

Accommodations

    Through groups set up and identified by the teacher, an entry point to problems
for all students can be ensured.
    The teacher circulates during small group discussion, making notes, and can step
in to provide additional support and or review for students in need.
    The use of graphing calculators, spreadsheets, scientific calculators in addition to
tasks requiring little to no technology allows the teacher to differentiate meeting
the needs of all learners. In the final task (Day 5), students may wish to use the
    Embedding literacy strategies into each day’s activities supports all Mathematics
Language Learners in the classroom. The teacher should refer to the word walls
whenever possible throughout each day’s lesson to support students’ acquisition
of precise mathematical vocabulary.
    As always, any modifications and/or accommodations outlined in a student’s IEP
Simple Interest

SHSM Business CLA Day 1                                                                                     MBF3C
Description                                                                                   Materials
Students review simple interest and working with percents in preparation for further work     Placemat paper
with compound interest.                                                                       Chart paper
Markers
Graph paper
Calculators

Assessment
Opportunities
Minds On…   Whole Class – Small Group
In groups of 3 or 4, have students discuss when they have ever loaned money or                Make sure that a
made a financial investment.                                                                  wide variety of
 How were the funds re-paid? How much was invested, how much was paid                   responses are
out at the end?                                                                     presented and
 What are some advantages to lending money? Are there any                               accepted. Provide
 Would you ever lend money to someone with whom you did not have a                      when required.
close personal relationship? Why or why not?
Groups record their findings on chart paper and share with class. Teacher facilitates
discussion making reference to “Interest”, “Investing”, and “Lending Institutions”.

Action!     Whole Class – Small Group
When working
When banks and businesses lend money to individuals and companies, interest is                through examples,
charged. The bank or business is investing in the person borrowing the money.                 students should
There are different ways of determining the interest charged.                                 work in their small
groups on placemat
Example: You loan \$250 to your brother in January so that he can buy an 8GB mp3               paper. The teacher
Player. He plans to repay you in two months. You agree to lend the money but ask              should consolidate
him to repay \$260 for your trouble.                                                           with the whole class
a) What interest, in dollars, did you charge?                                            after each example.
b) What percentage of the principal (\$250) is the interest?
c)
Solution:
a) \$10 per 2 months
\$10
fffffffffffffff
\$250                                                                               Definition should be
 0.04                                                                             student’s glossary
 4%                                                                               of terms and the
word should be
Simple Interest: interest earned only on the principal, using the formula I  Prt ,           added to the
where I is the simple interest, P is the principal, r is the annual interest rate as a        classroom word
decimal, t is the time in years.                                                              wall.

Note that the interest rate in the example was not an annual rate, it was the rate for
2  1
fffffff
two months or        fff of a year.
12 6
Example: Your brother does not repay you until the following January. Assume you
continue to charge him \$10 for each two months that he owes you the money.
a) Make a table of values to record the Accumulated Amount now owing every
two months.
c) How much interest did you charge over the course of a year?
d) What is the annual interest rate?

Solution:
Month              Total Owing (\$)
March                         260
May                           270
July                          280
Sept                          290
Nov                           300
a)     Dec                           310

Total Ow ing (\$)

320
310
300
290
280
270
260
250
240
230
March   May      July      Sept      Nov       Jan

b)
I  A @P
c)     Over the course of 1 year,     \$310 @\$250 . So \$60 interest was
 \$60
charged over the course of 1 year.
60
fffffffffff
d)      250 Therefore, 24% interest was charged over the course of 1 year.
 0.24
 24%
Note that interest is charged 6 times over the course of the year, 4% is
charged every 2 months and 6 B 4%  24% .

Simple interest is often used for short team investments. When the interest rate is an
annual rate, time in months or days must be written as a fraction of 1 year.

Example: You invest \$4000.00 at your financial institution. How much interest do you
earn for each of the following situations:
a) 2.25%/a simple interest for 60 days;
b) 2.25%/a simple interest for 5 months;
c) 3%/a simple interest for 5 months?
Solution:
60
a)   P  \$4000.00; i  0.0225; t  fffffffffff
365
I  Prt
60
 \$4000.00 B 0.0225 B fffffffffff
365
 \$14.79
5
fffffff
b) P  \$4000.00; i  0.0225; t 
12
I  Prt
f      g
`           5
a fffffff
 \$4000.00 0.0225
12
\$37.50
5
c)    P  \$4000.00; i  0.03; t  fffffff
12
I  Prt
5
 \$4000.00 B 0.03 B fffffff
12
 \$50
Note that more interest is earned the longer the investment time and the higher the
interest rate.

Curriculum Document/Quiz/Marking Scheme Assess students’ ability to apply their
understanding of Simple Interest. BLM1

Consolidate    Note that the simple interest graph you created showed a linear relationship. When
Debrief        we looked at the total amount owed, what was the y-intercept? (Ans: Principal or
initial amount owing) If we label each month where interest was charged as 1, 2, 3,…
what is the slope of the line? (Ans: the product of the two month interest rate and the
Principal)

Generate an expression for determining the interest earned on a \$1200 investment
that earns 6.5% simple interest. The time is variable.
a) What would the graph of the equation look like?
b) What is the y-intercept? the slope?
c) What do the y-intercept and the slope represent in this situation?

Solution: P  \$1200; i  0.065; t is a variable
I  Prt
 \$1200 B 0.065 B t                                                          Note that the y-
I  \$78t                                                                     intercept here is
a)The graph of this equation would be linear.                                        different from the
b)The y-intercept is 0 and the slope is 78.                                          graph of “Total
c)The y-intercept represents the amount of interest earned at the beginning of       Amount Owing”.
the investment and 78 is the product of the Principal and the annual interest      Explain the reason
rate.                                                                              for this difference.
Home Activity or Further Classroom Consolidation
Concept Practice        a)     In your own words, describe Simple Interest.
b)     How is Simple Interest calculated?
Differentiated                                                                                               Student responses
Exploration   c)   Create an “interesting” Simple Interest example to be solved by a classmate   will vary.
tomorrow. You must do the solution to your problem as homework.
Keep It Simple!

Name:
Date:

Josie holds the following bonds in her investment portfolio. Each bond earns simple interest at the given rate.
 A \$1200 Ontario Savings Bond earning 6.7% per year
 A \$5000 Canada Savings Bond earning 5.9% per year
 A \$5000 corporate bond earning 8.25% per year

Each year, Josie must report the interest earned on these bonds on her tax return. She has the Ontario Savings Bond for
the entire year, the Canada Savings Bond for 10 months and the corporate bond for 95 days.

a)    What is the total interest she must report on her tax return? Show all of your calculations. (9 marks)
b)    Graph the line showing the total interest earned on the Ontario Savings Bond for each of the twelve months in the
year. Determine the y-intercept and the slope and state their meaning in the context of the question. (5 marks)

Solution:

a) For the Ontario Savings Bond (2 marks):
P  \$1200; i  0.067; t  1
I  Prt
 \$1200 B 0.067 B 1
 \$80.40

For the Canada Savings Bond (3 marks):
10
P  \$500; i  0.059; t  fffffff
12
I  Prt
10
 \$500 B 0.059 B fffffff
12
 \$24.58

For the Corporate Savings Bond (3 marks):
95
P  \$5000; i  0.0825; t  fffffffffff
365
I  Prt
95
 \$5000 B 0.0825 B fffffffffff
365
 \$107.36

On her tax return, Josie must report \$80.40  \$24.58  107.36  \$212.34 (1 mark)

b)
Month            1      2         3         4         5         6         7          8         9         10       11      12
Number
Interest         6.70   13.40     20.10     26.80     33.50     40.20     46.90      53.60     60.30     67.00    73.70   80.40

Earned (\$)

Total Interest Earned

100
Interest Earned (\$)

80
60
40
20
0
1   2   3     4    5     6      7      8      9      10     11     12
Month Number

The y-intercept is 0 and this is the total interest earned during the year on January 1.
rise          \$80.40 @\$6.70
The slope is ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ; this represents to total interest earned on the investment each
`                        a
run 12 @1 months
 \$6.70 per month
month.
Compound vs. Simple Interest

SHSM Business CLA Day 2                                                                                   MBF3C
Description                                                                                 Materials
A first look at compound interest. Students will use the Simple Interest formula            Computer lab with
repeatedly along with a spreadsheet to calculate the Total Interest earned with simple      enough stations for
compounding periods.                                                                        students to work in
pairs
Computer projector
Assessment
Opportunities
Minds On…   Whole Class – Pairs
In pairs student solve each other’s problems that were created as part of the
homework. If there is a pair without problems to be solved, encourage them to create
one problem together and solve it as a pair.

Administer Quiz (BLM1.1) if not completed yesterday and take up solutions making
noteworthy points.

Think-Pair-Share
Students discuss
You are a millionaire at age 60!!! Your grandparents invested \$2500 when you are            the reasonableness
born in a savings bond.                                                                     of this statement in
pairs. Each pair will
share their
perspective and
rationale with the
class.
Action!     Whole Class – Small Groups
Build on mp3 player question from yesterday….

Example: Suppose in March when your brother cannot repay you for the mp3 player
loan, you decide that you don’t want to be without the money for much longer. You           Students work on
tell him that either he has to pay the interest every two months (\$10) or you will now      this in pairs. The
consider the Principal amount to be \$250 + \$10 = \$260 and charge the 4% interest for        teacher takes up
the next two months. He cannot pay the \$10 so you charge interest on \$260. This             the solution using a
continues for the entire year.                                                              computer projection
of one pair’s work.
a)     Make a spreadsheet showing the Compounding Period, Total Principal and          It is important to
Interest Charged and Total Interest Charged.                                    circulate so that
b)     Graph Total Interest against Compounding Period using the spreadsheet           students have a
functions.                                                                      good understanding
of the calculations
Solution:                                                                                   to be carried out.
The teacher may
wish to show a table
students how each
entry in the second
row is obtained.
d
rio
Pe

ed
l

arg
ing

ipa

est
nd

Ch
nc

er
ou

Pr i

Int
st
mp

ere
tal

tal
Co

To

To
Int
1   \$250          \$10              \$10
2   \$260       \$10.40           \$20.40
3   \$270       \$10.82           \$31.22
4   \$281       \$11.25           \$42.46
5   \$292       \$11.70           \$54.16
a)                                     6   \$304       \$12.17           \$66.33

Total Interest

\$70

\$60
Total Interest Charged (\$)

\$50

\$40

\$30

\$20
\$10                                                  word wall and
glossary.
\$0
1      2           3          4        5   6
Encourage students
Com pounding Period
to use a table of
b)                                                                                        differences to
predict the shape of
Compounding Period: The time interval at which interest is added to the principal.             the graph.
Teacher debriefs
Small Group Discussion:                                                                        the discussion and
 Why does the interest charged increase every two months?                                 shows a projection
 At the end of the year, how much additional interest is charged by not                   of a graph with the
paying the \$10 out each time?                                                         linear and
 What are the advantages of paying the interest as it accrues? Why might                  exponential graphs
someone not choose to do this?                                                        on the same chart.
 What shape is the resulting graph? Compare this to yesterday’s graph.

 The interest charged increases because each time the interest is rolled into
the Principal amount and essentially he has “borrowed” more money.
 The additional interest charged is \$66.33 - \$60.00 = \$6.33.
 If you pay the interest as it accrues then you save money and end up paying
less total interest at the end. Someone may choose not to do this because
they don’t have enough money. You might want to do this if it is an                   Add to word wall
investment rather than a loan as you will earn more total interest over the           and glossary.
life of the investment.
 The resulting graph is exponential. Yesterday’s was linear. Exponential
curves grow more rapidly than the linear graphs…bad for a loan but good
for an investment.

The kind of interest that works this way is called Compound Interest.                      Students work in
pairs at computer
Have students work in their groups to create a word-wall card for Compound Interest.       stations.

Example:
a) Suppose you invested \$1500 in a Canada Savings Bond for seven years.
The simple interest rate is 6.35%/a. Make a spreadsheet showing the total
interest earned at the end of each year. Calculate the first differences.
b) Suppose you invested \$1500 in a Canada Savings Bond for seven years.
The compound interest rate is 6.35%/a. Make a spreadsheet showing the
total interest earned at the end of each year. Calculate the first differences.   Note that the First
c) Plot Total Interest against Year for each situation on the same chart.              Differences for
Simple Interest are
Solution:                                                                                  constant – the
Year                       Total Int  First Diff                               graph is linear.
1       \$95.25
2     \$190.50      \$95.25
3     \$285.75      \$95.25                               The First
4     \$381.00      \$95.25                               Differences for
5     \$476.25      \$95.25                               Compound Interest
6     \$571.50      \$95.25                               are not constant.
The total interest
a)                            7     \$666.75      \$95.25
changes by an
Year                       Total Int  First Diff                               increasing amount
1      \$95.25                                           each year – the
2    \$196.55 \$101.30                                    graph is not linear.
3    \$304.28 \$107.73                                    Ask students to
4    \$418.85 \$114.57                                    calculate further
5    \$540.70 \$121.85                                    differences to
6    \$670.28 \$129.58                                    identify the shape
7    \$808.10 \$137.81                                    as exponential.
b)
Interest

\$800.00
Total Interest

\$600.00
Simple
\$400.00
Compound
\$200.00

\$0.00
1   2   3    4      5   6
Year

c)

Consolidate              The points for simple interest lie on a straight line.
Debrief
      The points for compound interest lie on a curve.
      Simple interest illustrates linear growth.
      Compound interest illustrates exponential growth.
      In the first year of the example, the simple interest and the compound
interest are the same.
    Money grows more rapidly when interest is compounded.

Revisit the Minds On questions – A millionaire at age 60!!                          Pairs revisit their
 If interest is earned monthly, how many times will the bond earn interest     responses to the
over the lifespan?                                                       Minds On question
 Estimate the interest rate that would result in this Accumulated Amount.      posed at the
Explore on your calculator or using a graph.                             beginning of class.
 Is this situation realistic?

Home Activity or Further Classroom Consolidation

Peer Assessment of Part a) of Homework in small groups (3 – 4)

Concept Practice
a)   Explain the difference between simple interest and compound interest.
Include tables and graphs in your explanation.

Reflective Journal is collected for Part B and teacher provides written feedback.

    A simple interest bond that earns 3.7% per year for three years
Reflection
    A compound interest bond that earns 3.6% per year for three years

Solution:
a) See Consolidate and Debrief.
b) Suppose \$1000 is invested.
I  Prt
A simple interest bond earns    \$1000 B 0.037 B 3
 \$111.00
A compound interest bond earns…
I  Prt
…in year 1  \$1000 B 0.36 B 1
 \$36.00
I  Prt
…in year 2  \$1036 B 0.36 B 1
 \$37.30
I  Prt
…in year 3  \$1073.30 B 0.036 B 1
 \$38.64
…total interest earned over 3 years:
\$36.00  \$37.30  \$38.64  \$111.94
The compound interest bond earns more.
Peer Assessment – Different Interests

Indicate Y/N in the left hand column beside each item. Staple this page to the back of your classmate’s
work. Remember to be positive in your written feedback. This activity provides you and your classmates
an opportunity to improve your work and understanding.

Is simple interest well defined? (refer to the word-wall card)

Is compound interest well defined? (refer to the word-wall card)

Based on the written description, can you tell what the difference is between simple interest and
compound interest?

Does the description use correct mathematical vocabulary (e.g. interest, principal, present value,
compound, amount, compounding)?

Are there tables included to show the difference between simple interest and compound interest?

Are there graphs included to show the difference between simple interest and compound interest?

Are there any recommendations you could provide to your peer? Remember to use positive
wording.

Reviewer’s signature: _________________________________
The Compound Interest Formula

SHSM Business CLA Day 3                                                                                       MBF3C
Description                                                                                     Materials
Chart paper
`    an
Students are introduced to the compound interest formula, A  P 1  i             , through      Markers
Tape
investigation.
Blank word wall
cards
Assessment
Opportunities
Minds On…   In Pairs
Thinking back to the mp3 player example (first example) from yesterday, encourage               Teacher circulates
students to express the entries under Total Principal (the Accumulated Amount) as a             and prompts
product of initial Principal (\$250) and some other value. Encourage students to think           students with
about the operation they are performing each time to come up with the revised                   appropriate
expression.                                                                                     questions as
`       a                                                                  required.
\$260  \$250 1.04
`       a        `       a`        a       `       a2                  Teacher uses
\$270.40  \$260 1.04  \$250 1.04 1.04  \$250 1.04                                               checklist to assess
`        a       `        a3                                       student
\$281.22  \$270.40 1.04  \$250 1.04                                                             understanding of
`       a4                                                             the introduction to
\$292.46  \$250 1.04                                                                            compound interest
from the previous
Have students conjecture a formula based on the values of the principal, P, the                 day.
interest rate, i, and the number of times interest is charged, n to calculate the
Accumulated Amount of the loan, A.

Selecting Tools and Computational Strategies/Observation/ Checklist: Observe
students, watching their calculator use and patterning skills.
Action!     Whole Class – In Pairs
The compound interest formula allows us to quickly make compound interest
calculations without the need for a chart.
`       an
A P 1 i
A is the total Accumulated Amount of investment (or future value)
P is the Principal
i is the interest rate as a decimal
n is the number of compounding periods
Example: Suppose you invest \$1000 at 6.37% compounded annually for 15 years.
a) Calculate the Accumulated Amount at maturity.
b) Calculate the interest earned.

Solution:
a) A = ?
P = \$1000
i = 0.0637
n = 15
`       an
A P 1 i
`                a15
 \$1000 1  0.0637
`          a15
 \$1000 1.0637
 \$2525.15
The Accumulated Amount of the investment is \$2525.15
b)   To calculate the interest earned, calculate the difference between the
Accumulated Amount at maturity and the principal invested.
I  A @P
 \$2525.15 @\$1000.00
 \$1525.15
The interest earned is \$1525.15

Interest can be compounded at different intervals each year. Compounding could be:
 Annually                                                                        Through mediating
 Semi-annually                                                                   respectful class
 Quarterly                                                                       discussion, the
 Bi-weekly                                                                       teacher can ensure
 Semi-monthly                                                                    that each word wall
 Weekly                                                                          card is complete
 Daily                                                                           and accurate.
Assign each group of four one word. As a group, they create a word-wall card for
each compounding period. A reporter from each group will present and provide an
overview of the group’s word wall card to the whole class before posting it in the
classroom.

When interest is not compounded annually then in the formula, i is the interest rate
per compounding period and n is the number of times interest is compounded.

To calculate i, divide the annual interest rate by the number of times interest is
earned/charged per year.

Example: Calculate the Accumulated Amount of an investment of \$600 that is             i is calculated by
invested for 7 years at 5.5%/a compounded quarterly.                                   dividing the annual
interest rate by the
Solution: A = ?                                                                        number of
P = \$600                                                                     compounding
i  0.055 D 2                                                                   periods in one year.
 0.0275                                                                        n is calculated by
n  7B 4                                                                        multiplying the
`
 28an                                                                          number of years
A P 1 i                                                                             and the number of
`               a28                                                         compounding
 \$600 1  0.0275                                                                     periods in one year.
 \$1282.46
The Accumulated Amount at maturity is \$1282.46

Example: You are saving for a spring break vacation in two years’ time. Your bank
will pay you 3.6%/a compounded monthly. How much do you need to invest now to
have \$2500 to pay for your vacation?
0.036
fffffffffffffffff
Solution: A  \$2500; P  ?; i                      0.003; n  2 B 12  24
`   an
12
A P 1 i
`              a24
\$2500  P 1  0.003
`        a24
\$2500  P 1.003
\$2500
ffffffffffffffffffffff
24
P
1.003
\$2326.58  P
You have to invest \$2326.58.

Consolidate   Small Groups                                                                            Students work in
Debrief       If you have a choice between two investments with the same interest rate except one     small groups and
offers simple interest and the alternative offers compound interest, which should you   record their
paper. Once all
responses have
been posted, the
class goes on a
gallery walk, each
individual records
the different
responses and
these are discussed
in the small groups.
Home Activity or Further Classroom Consolidation
Application       Journal Entry:
x                                          Allow students to
You have learned that the equation  y  ab describes exponential growth. Explain        discuss the
x
how the compound interest formula is related to y  ab . Include an example in          question in small
begin their
independent
responses.
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
Na
me

Ca
n   use
ex
po
ne
nt
b
Assessment Checklist

utt
Un                                 on
de
rs t
Calculator use and patterning

an
ds
r      ep
ea
te d
ds                                       lt
1    to
ipli
ca
inte                                       ti   on
res
t   ra t
Ide                               e
nti
fie
s th
eP
r in
cip
a   l
Ex
pr e
s   ses
in     ter
est
as
Re                              a        de
cog                                     c im
niz                                             an
e     s th
ep
att
er n
The TVM Solver

SHSM Business CLA Day 4                                                                                 MBF3C
Description                                                                               Materials
Students will use the TVM (Time Value Money) Solver to calculate different amounts in     TI-83Plus or TI-
compound interest problems.                                                               84Plus calculators
Assessment
Opportunities
Minds On…   Small Groups
With the assistance of the graphing calculator (not the TVM Solver application),
students work on the following problem…

\$500 is invested at 4%/a compounded quarterly. How long would it take to grow to
\$600.

Solution 1: Using a table.
d
Perio
ding
po un

i pal

unt
es t
Prin c
Com

Am o
Inter

0       \$500.00                0.01      \$500.00
1       \$500.00                0.01      \$505.00
2       \$500.00                0.01      \$510.05
3       \$500.00                0.01      \$515.15
4       \$500.00                0.01      \$520.30
5       \$500.00                0.01      \$525.51
6       \$500.00                0.01      \$530.76
7       \$500.00                0.01      \$536.07
8       \$500.00                0.01      \$541.43
9       \$500.00                0.01      \$546.84
10       \$500.00                0.01      \$552.31
11       \$500.00                0.01      \$557.83
12       \$500.00                0.01      \$563.41
13       \$500.00                0.01      \$569.05
14       \$500.00                0.01      \$574.74
15       \$500.00                0.01      \$580.48
16       \$500.00                0.01      \$586.29
17       \$500.00                0.01      \$592.15
18       \$500.00                0.01      \$598.07
19       \$500.00                0.01      \$604.05

Therefore, it will take 19 compounding periods or 19/4=4.75 years (4years, 9months).
To grow to \$600.00.

Solution 2: Using a graph and the trace function on the graphing calculator.
0.04
A  \$600; P  \$500; i  fffffffffffff 0.01; n  ?
4
`     an
A P 1 i
`     an
A  500 1.01
`   ax
Graph: y  500 1.01

It will take 19 compounding periods or 19/4 = 4.75 years.

Action!   Teacher-led Discussion

There is a built-in function on the graphing calculator, TVM Solver, that allows us to   Use the overhead
easily determine any of the variables.                                                   projection screen to
demonstrate use of
On the TI-83Plus enter the key sequence [APPS] [1] [1]                                   the TVM Solver

Define the screen variables:

N is the number of years
I% is the interest rate as a percent (not a decimal)
PV is the Present Value or Principal
PMT is the payment amount
FV is the Future Value or Accumulated Amount
P/Y is the number of payments per year                                                   N = 5 (years)
C/Y is the number of compounding periods per year                                        I% = ?
PV = 5300
Example: You invest \$5300 in a fund where the interest is compounded semi-               PMT = 0 (we are
annually. At the end of 5 years your investment is worth \$6500. What was the             not making regular
interest rate of your investment?                                                        payments to the
investment)
Solution:                                                                                FV = -6500
(negative because
we may withdraw
from the account)
P/Y = 1 (no regular
payments)
C/Y = 2
(compounded semi-
annually)

After entering all of
the information
place the cursor
The interest rate was 4.12%.                                                             beside I% and key
in [ALPHA]
[ENTER] to solve.

Students work on
this problem in
pairs.

Remind students
that more frequent
compounding
results in a greater
amount of interest
Example: For how many years did you invest \$5300 at 4.12%/a compounded                   earned.
monthly, if the value now is \$6511?

Solution:

The money was invested for 5 years.

Consolidate    Small Groups
Debrief

What difference would consumers notice in a loan that carries a 5%/a interest rate
compounded annually versus the same interest rate compounded monthly? Include
an example and calculations to support your explanation.

The teacher may want to carry this out as a placemat activity. Each group divides
chart paper into the same number of sections as the number of members of the group.      Small Group
Each person works independently in their chart paper area. A rectangle in the centre     Discussions:
is reserved for the group’s final thoughts. One member from each placemat group will     Placemat.
share the group’s response with the whole class.
Think Literacy:
The teacher assesses for understanding by listening.                                     Cross-
CurricularApproach
es, Mathematics,
66-71

Home Activity or Further Classroom Consolidation
Concept Practice   1.   You need \$4500 to buy a car. You find a savings investment that pays 3.75%
compounded monthly. How much would you need to invest now if you wait 2

2.   Approximately how long would it take for a \$2500 investment to double if it earns
9.5%/a compounded
a)   annually?
b)   quarterly?
c)   monthly?
d)   daily?

3.   How long would it take for a \$1000 investment to double is interest is
compounded annually at
a) 4%/a?
b) 6%/a?
c) 8%/a?
d) 12%/a?
Do you notice any pattern in your answers that you could use to predict the
doubling time for the same investment at 9%/a?

Assign other practice/reflection questions to suit the needs of the students.

Solutions:

1.    N = 24
I% = 3.75%
PV = ?
PMT = 0
FV = 4500
P/Y = 1
C/Y = 12

PV = \$1832.13 I would need to invest \$1832.13 to have enough money to
purchase the car in two years.

2.   N=?
I% = 9.5
PV = 2500
PMT = 0
FV = -5000
P/Y = 1
a) C/Y = 1
N = 7.64
It will take 7.64 years (or 7 years, 8 months) for the investment to double.
b) C/Y = 4
N = 7.38
It will take 7.38 years (or 7 years, 5 months) for the investment to double.
c) C/Y = 12
N = 7.32
It will take 7.32 years ( or 7 years, 4 months) for the investment to double.
d) C/Y = 365
N = 7.29
It will take 7.29 years (or 7 years, 3.5 months) for the investment to double.

3.   N=?
I% = variable
PV = 1000
PMT = 0
FV = -2000
P/Y = 1
C/Y = 1
a) I% = 4
N = 18 It will take 18 years for the investment to double.
b) I% = 6
N = 12 It will take 12 years for the investment to double.
c) I% = 8
N = 9 It will take 9 years for the investment to double.
d) I% = 12
N = 6 It will take 6 years for the investment to double.

The product of the interest rate and the number of years to double is always 72.
This means that for an investment at 9%, it will take 72 D 9  8 years for the
investment to double in value.
Group Assessment

Group Number ________

Strengths:





Misconceptions:

#

#
Philanthropy

SHSM Business CLA Day 5                                                                                       MBF3C
Description                                                                                     Materials
Computer lab with
Students explore the Present Value of money by considering different options for              access to 1
providing a scholarship at a local secondary school. Students will work in pairs to explore   computer per pair.
the activity thus providing support and entry points to all students. Each student will
submit an individual response to the activity to be graded as a summative task. Students
may decide to write a complete report to the company’s board of directors, make an oral
presentation, or make an electronic presentation to the board of directors.

Assessment
Opportunities
Minds On…     Think-Pair-Share
The Scholarship?                                                                              Students work in
pairs and make an
The company you work for would like to provide a graduation scholarship at a local            initial prediction
secondary school. They would like the award to be given each year for the next ten            providing their
(10) years valued at \$500.00. Currently they have \$5000.00 ready to be put aside for          reasoning. This
the award.                                                                                    part of the activity
 Should the finance department issue a cheque to the school for \$5000.00                should be done
 Provide advice in a written or oral format as part of the opening piece to             spreadsheet or a

Action!       In Pairs

     If the company decides to issue one payment for all ten years to the school,
 If the school invests this money at current interest rates (use the web
to locate these and provide sources), determine the future value of the
investment assuming the interest is compounded monthly. Note: the
first \$500 is issued immediately, the second is invested for 12 months,
the third for 24 months, etc.
 Given the total future value of this initial investment, if the school
wanted to award the entire future value, how much should each award
be worth? Note that changing the award amount will change how
much is invested and thus change the total interest.
     Suppose the company issues \$500 per year to the school and places the
remaining amount in a GIC (use the web to determine current rates and
provide sources). How much does the company need to put aside today?
That is, they pay out \$500 now and invest enough to be worth \$500 in 12
months and \$500 in 24 months and so on…

Consolidate   Individually
Debrief
     Given this information, make a complete presentation to the board of
directors providing details of your analysis and a final, justified
recommendation.
Home Activity or Further Classroom Consolidation

Complete activity for submission and grading.

Curriculum Document/Achievement Chart/Evaluation Rubric
January 2008                           Specialist High Skills Major (SHSM)                                           31
Evaluation Rubric – Philanthropy

Criteria                     Level 1                  Level 2                  Level 3                  Level 4
- Solve problems       The student solves       The student solves       The student solves       The student uses
related to             simple compound          more complex             complex compound         compound interest to
compound interest      interest problems to     compound interest        interest problems to     explore possibilities
determine the            problems that may        accurately determine     beyond the lump sum
amount given the         require                  the present value of     and payment plan
present value, the       rearrangement of the     the payment plan         options presented.
interest rate and the    compound interest        option.                  (e.g. explore the
compounding              formula. (e.g. solve                              possibility of
period.                  for present value                                 accepting a lump sum
given accumulated                                 to invest and
amount, interest rate                             withdraw amounts at
and compounding                                   various intervals of
period).                                          time).
- Compare services     The student makes        The student makes        The student              The student explores
available from         limited mention of       some reasonable          incorporates the         investment
financial              the different            mention of the           difference between       opportunities not
institutions.          investments              different investments    investing in a GIC       mentioned here as
presented here.          presented and            compounded monthly       part of their final
incorporates them        and investing at prime   recommendation.
loosely into their       compounded annually                 OR
recommendation.          into their               The student makes a
recommendation.          good argument for or
against one method
of investment in their
final recommendation.
- Apply reasoning      The student uses         The student uses         The student uses         The student makes
skills to make         limited                  mathematics to justify   mathematical             thorough use of
mathematical           mathematical             conclusion and           reasoning to             mathematical
conjectures, and       reasoning to             employs some             formulate any            concepts presented in
justify conclusions.   formulate                mathematical             conjectures and then     this unit to formulate
conjectures.             reasoning to             carries out supporting   conjectures and to
Conjectures seem         formulate conjectures.   work to justify          support their position.
to be mostly                                      conclusions.             The student then
instinctive with                                                           proceeds to use the
limited                                                                    tools and concepts
consideration given                                                        mentioned to verify
to the concepts                                                            (or deny) the
explored in this unit.                                                     conjecture and make
a final conclusion.
- Communicate          The student makes        The student makes        The student uses         The student makes
visually and in        limited use of           some use of              precise mathematical     thorough use of all of
writing.               mathematical             mathematical             language, charts,        the mathematical
language, charts,        language, charts,        graphs, and tables of    communication tools
graphs, and tables       graphs, and tables to    values to clearly        mentioned and
in their work. The       communicate their        communicate the          incorporates them
mathematical             message.                 message in their         seamlessly into their
reasoning is not                                  conjectures and final    work.
clearly                                           recommendation.
communicated to
the audience.

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 7 posted: 4/20/2014 language: Unknown pages: 31