Mathematical Models with Applications
Entrepreneurship: Keeping Score with Mathematics
Unit III: Finance
Teacher Notes, Student Activities, and Transparencies
81
Unit III: Finance
Section 1: Assumptions
Teacher Notes
Section 1: Assumptions
Teacher Notes
Context Overview:
Making assumptions is an important part of the entrepreneurial experience. In this section,
students will learn how to create an assumption statement based on the assumptions needed to
make projections for a business. The teacher will discuss various kinds of financial statements.
Each student will complete an assumption statement for Grape Expectations using realistic or
justifiable numbers as estimates.
TEKS Addressed:
Mathematical Models with Applications: The student is expected to
1A - compare and analyze various methods for solving a real–life problem;
1B - use multiple approaches (algebraic, graphical, and geometric methods) to solve
problems from a variety of disciplines;
2A - interpret information from various graphs, including line graphs, bar graphs, circle
graphs, histograms, and scatterplots to draw conclusions from the data;
2B - analyze numerical data using measures of central tendency, variability, and correlation
in order to make inferences;
3A-formulate a meaningful question, determine the data needed to answer the question,
gather the appropriate data, analyze the data, and draw reasonable conclusions.
Goal:
The goal of this section is to give students an overview of the financial statements necessary to
start and maintain a business and to provide more information on Grape Expectations, the
entrepreneurial case that will be used to demonstrate concepts.
Materials:
Grape Expectations Case (1 per student)
Assumption Statement: Grape Expectations (1 per student)
Procedures:
Show the transparency of the financial statements for a business, discuss the various statements
and introduce the following terms to the class.
Gross Profit: The gross profit is the net sales minus the cost of sales.
Income Statement: The income statement presents your venture’s performance over a specified
period of time in terms of sales, costs, and profits or losses.
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Unit III: Finance
Section 1: Assumptions
Teacher Notes
Balance Sheet: A balance sheet is a statement of the financial position of a venture at a specific
point in time.
Cash Flow Statement: The cash flow statement keeps a record of the amount of cash that goes
in and out of the business.
These are some of the financial aspects of Grape Expectations that students will investigate
before developing their own business ideas.
Student Activity—Assumption Statements
Ask the students to read the student activity case study, Grape Expectations: An Entrepreneur is
Born. They received some of this information in Unit II, but the case tells the entire story. Then
give students the Assumption Statement for Grape Expectations. Have the students answer the
questions in Part I of Student Activity: Assumption Statements. Possible answers to the questions
follow below.
Questions and Possible Answers:
1. What does it cost to make each CD (including the cover), and what must be purchased in
order to produce each CD?
It costs $2.00 to duplicate the CD; $.50 for the plastic case; $.50 for the 4-color jacket. The
total cost per CD is $3.00.
2. Throughout the year, what are the main activities that Chuck believes will aid his sales?
He believed he would sell 175 at a school concert in January and 125 at the St. Valentine’s
Day Dance in February. He figured he wouldn’t make any sales in March, April, and May
because he would be busy with spring break and exams. He planned to sell around 25 CDs at
the End–of-School-Year BBQ in June. He would be working on the farm over the summer, so
there would be no sales in July or August. In September, he figured he’d sell 175 CDs at the
Back-to-School Dance, when Grape Expectations would appear again. He believed he could
sell 100 units in October, when the band would perform at a pre-game concert and dance. In
November, he guessed he’d sell 150 because of the Thanksgiving Dance. And finally, he
figured many people would want to give copies of the CD as gifts during the holidays, so he
estimated he’d sell 300 units during that time.
2. As a guide, lead the students as they complete Part II (entering the information into the
statement). They should use the answers to Question #1 in order to complete the table. On the
next page is a copy of the completed table.
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Unit III: Finance
Section 1: Assumptions
Teacher Notes
Assumption Statement for Grape Expectations
Costs – PER UNIT
Cost to duplicate one CD
Cost for one plastic case
Cost for each 4-color CD jacket
Other costs (per unit)
Total Cost for one CD, ready to sell
Expenses
Royalties rate to be paid to artists
Cost of goods percentage
Other expenses
Monthly Activities to Sell CDs
School concert
$2.00
.50
.50
$3.00
10% As agreed upon by Chuck
Dickens with artists
20% Price, divided by total cost of
goods
Expenses not on list, but should
be
Month
January
Valentine’s Day dance
February
No sales – exams
March
No sales – back-to-school pressures
April
No sales – Final Exams prep
May
End-of-School-Year BBQ
June
No sales – no school
July
No sales – no school
August
Back-to-School Dance
September
Pre-Game Concert & Dance
October
Thanksgiving Dance
November
Holiday Dance
December
Other sales
Total CD Units Sold for the Year
# of CDs Sold
175
125
0
0
0
25
0
0
175
100
150
300
1,050
Monthly
@ $15.00 each
$2,625.00
$1,875.00
$375.00
$2,625.00
$1,500.00
$2,250.00
$4,500.00
$15,750.00
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Unit III: Finance
Section 1: Assumptions
Case Study
Case Study
Grape Expectations: An Entrepreneur Is Born
It had been an awesome night! Everyone was at last night’s school dance.
And everyone believed that they had witnessed the launch of the newest,
and perhaps greatest, rock and roll band since the Beatles. The name of
the band was Grape Expectations.
One of the students, Chuck Dickens, was in charge of media, and his job
was to record the performance for the school’s local cable channel. He did
his job very well, and it was not long before everyone at school wanted a
CD copy of Grape Expectations’ performance. At first, being the good guy
that he is, Chuck gave a few friends a free copy of the CD, thinking that
would satisfy everyone’s hunger for more of this new group. But instead,
putting a few free CD copies into circulation only fueled the fire, and the
number of students begging for a copy became a burden. Chuck decided
he’d like to try to make money from the CDs. So he began to explore the
possibility of starting up a new company that would be called Grape
Expectations.
The first thing he did was talk to his parents to get their advice and support.
He also talked to his school counselor, his math teacher, and a few other
adults at the school who had either been in business themselves or knew
something about it. His parents were very enthusiastic and offered to help
as long as Chuck’s grades did not slip.
A deal is struck! Chuck had enjoyed a long-term friendship with the band
members, and they unanimously agreed that they would love to get the
added exposure for their music. They were hopeful that it really would
launch their music careers. So Chuck and the band members drew up a
short legal agreement (Chuck’s mom is a lawyer and wrote up a legal
agreement for no charge). Chuck would record the band’s music, market
and sell it (probably on a CD), and pay the band members 10 percent of
everything he sold (gross sales). Since there were five members in the
band, 10 percent would divide evenly among the band members, with each
receiving 2 perent of the sales. It was agreed that Chuck would call his
company Grape Expectations, and that was just great with the band. The
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Unit III: Finance
Section 1: Assumptions
Case Study
copyright notice on the CD would read: © Grape Expectations, 20011. All
rights reserved.
Chuck recorded seven or eight of the band’s most popular songs onto a
master CD, which he could then copy and sell to anyone and everyone he
could. All of the music was original and had been written by the band
members together, so there was no dispute about who would receive the
bulk of the royalties. Since they recorded and sold only their own music, it
wasn’t necessary to pay royalties to other artists or other companies.
It would be easy to make lots of copies of the CD. Chuck could either take
them to the local copy shop and get them duplicated at a cost of $2.00
each, or he could purchase a CD burner and copy them himself. Chuck
decided to take the easier route and just have the copies made by the local
copy shop, as he needed them. The copy shop would require a copy of his
legal agreement with the band, but after the first set of copies were made it
would be easier to make other copies as needed.
Costs. Chuck estimated his costs to be as follows:
$2.00 to copy each CD, with the name and copyright notice on the CD
.50 for the plastic case to put the CD into
.50 for each 4-color jacket that is displayed in the plastic CD case
with the band logo, and the words to the songs
$3.00 total cost for one CD, ready to sell
Chuck figured that he could sell the CDs at $15.00 each. With a cost of
only $3.00, that would mean that his cost of goods percentage would be
reasonable, at 20 percent. (Cost of goods percentage = total cost of goods
divided by price = $3.00 / $15.00 = one-fifth = 20 percent).
Chuck figured that over the year he could sell a total of 1,050 CDs. He
believed he would sell 175 at a school concert in January and 125 at the
St. Valentine’s Day Dance in February. He figured he wouldn’t make any
sales in March, April, and May because he would be busy with spring break
and exams. He planned to sell around 25 CDs at the End-of-School-Year
A copyright notice has three elements: The copyright symbol (©), the name of the person or company claiming copyright
ownership, and the year. The words “All rights reserved.” Mean that any and all rights of ownership pertaining to the copyrighted
material belong to the copyright holder. Effectively, the band turned over their ownership rights to Grape Expectations, in
exchange for 10% royalties.
1
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Unit III: Finance
Section 1: Assumptions
Case Study
BBQ in June. He would be working on the farm over the summer, so there
would be no sales in July or August. In September, he figured he’d sell 175
CDs at the Back-to-School Dance, when Grape Expectations would appear
again. He believed he could sell 100 units in October, when the band would
perform at a pre-game concert and dance. In November, he guessed he’d
sell 150 because of the Thanksgiving Dance. And finally, he figured many
people would want to give copies of the CD as gifts during the holidays, so
he estimated he’d sell 300 units during that time. Even though there were
only about 2,000 families in the schools, he figure his estimates were
conservative since others outside of the school (families, friends, other
fans, etc.) would buy, too, especially since the school had agreed to run the
Grape Expectation’s performance on the local cable channel from time to
time at no charge to Chuck.
Since he would run the business out of his house, his expenses would be
minimal. He figured that he would pay himself 5 percent of sales and be
okay. Of course, he would also have to give 10 percent of his pay to Uncle
Sam for taxes. He had agreed to pay his artists (also called Grape
Expectations) 10 percent of gross sales (all sales). He also had asked a
graphic arts student at school to design the jacket for the CD, and Chuck
agreed to pay him $50, one time, for the art work. Chuck felt that he could
successfully market the CDs by having about 2,000 flyers printed up and
circulated at school and school activities. He had the flyers printed up in
January for $60.
The thought of preparing financial statements for his business was a bit
frightening, so his father (an accountant) provided him with some easy-touse, fill-in-the-blank forms he could use to prepare the following financial
statements.
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Unit III: Finance
Section 1: Assumptions
Assumption
Statement
Revenues
(forecasted)
Income
Statement
Cash Flow
Statement
Balance Sheet
Case Study
To detail his costs, expenses, and the
activities necessary to produce his monthly
unit sales
To predict the revenues (sales) he would
make, he needed to predict the number of
CD units he would sell and multiply that by
the price he would charge for the CD
To predict whether his business would
make a profit or lose money
To be sure that he would not run out of
cash and would have enough money to pay
all of his bills
To see whether he was “building” a
company. If his net equity (net worth) went
up, he would be making money and
creating value. If not, he should consider
doing something else because the business
probably would not stand the test of time.
Chuck was pleased to realize that he could probably make about $15,750
in sales by selling 1,050 CDs at $15.00 per CD. Not bad for a part-time
business.
The Future. Now Chuck could see the potential for even more sales if he
were successful. He could try other recording artists. He could market the
CDs outside of the school. He could try to get his CDs into other stores in
other towns, and he was really excited about the possibility of putting
information about the CD onto a website he would develop. If he did that,
he could take orders over the web and possibly deliver the music over the
web, too. Of course, these ideas presented more questions and other
issues. But Chuck was excited about the potential of his new business.
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Unit III: Finance
Section 1: Assumptions
Student Activity
Student Activity
Assumption Statements
Part I. Refer to the Grape Expectations case in order to answer the following questions.
1. What does it cost to make each CD (including the cover), and what must be
purchased in order to produce each CD?
2. Throughout the year, what are the main activities that Chuck believes will aid his
sales?
Part II. Using the answers to questions 1 and 2, complete the assumptions statement
below.
Assumption Statement: Grape Expectations
Costs – PER UNIT
Cost to duplicate one CD
Cost for one plastic case
Cost for each 4-color CD jacket
Other costs (per unit)
Total cost for one CD, ready to sell
Expenses
Royalties rate to be paid to artists
Cost of goods percentage
10%
20%
Other expenses
Monthly Activities to Sell CDs
Month
As agreed to by Chuck
Dickens and artists
Price, divided by total cost
of goods
Expenses that are not on
the list
# of
CDs
Sold
Monthly Sales
January
February
March
April
May
June
July
August
September
October
November
December
Total
Sales
for the
year
Other sales
Total CD Units Sold for the Year
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Unit III: Finance
Section 2: Projections
Assumptions
Statement
Prices
Units Sold
Transparen
Income
Statement
Income
- Expenses
=Net Profit
(Loss)
Balance
Sheet
Assets
- Liabilities
= Net Worth
(Not included in
this unit)
Note: These are
your
“assumptions”
about price,
sales, etc.
Note: These are the two basic
financial statements:
1) Income Statement
2) Balance Sheet
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
Unit III: Finance
Transparency
Section 2: Projections
Section 2: Projections
Teacher Notes
Context Overview:
In the first activity, students read about revenues (sales), and they complete a revenue table for
Grape Expectations. In the next activities, students will read more information about the
company Grape Expectations, and they will use this information to complete a one-month
projected income statement and a one-year projected income statement for Grape Expectations.
Goal:
This section should provide students with an understanding of the income flow of a business.
TEKS Addressed:
Mathematical Models with Applications: The student is expected to
1A - compare and analyze various methods for solving a real–life problem;
1B - use multiple approaches (algebraic, graphical, and geometric methods) to solve
problems from a variety of disciplines;
2A - interpret information from various graphs, including line graphs, bar graphs, circle
graphs, histograms, and scatterplots to draw conclusions from the data;
2B - analyze numerical data using measures of central tendency, variability, and correlation
in order to make inferences;
3A-formulate a meaningful question, determine the data needed to answer the question,
gather the appropriate data, analyze the data, and draw reasonable conclusions;
Materials:Projected Revenue Worksheet (1 per student)
Sales Projection Table (1 per student)
Assumptions Statement for Grape Expectations (completed in section 1)
One-Month Projected Income Statement (Worksheet (1 per student)
One-Year Projected Income Statement (1 per student)
Procedures:
Activity 1: Projected Revenue for Grape Expectations
1. Ask the students to read the information on the Projected Revenue Worksheet.
2. Help the students fill in the Projected Revenue Worksheet for Grape Expectations. A
transparency for the first three months may be used for this purpose. The completed
Projected Revenue Worksheet is on the next page.
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Unit III: Finance
Section 2: Projections
Revenues
# of Units
Sold
x Price of
CD
Gross
Sales
from CDs
Transparen
Projected Revenue Worksheet for Grape Expectations
Mar
Apr
May
June
July
Aug
Sept
Jan
Feb
175
125
0
0
0
25
0
0
175
100
$15
$15
$15
$15
$15
$15
$15
$15
$15
$15
0
0
0
$375
0
0
$2,625 $1,875
Oct
$2,625 $1,500
Student Activity 2: One-Month Projected Income Statement
1. Hand out the One-Month Projected Income Statement for Grape Expectations worksheet to each studen
2. Ask the students to read the information on the income statement (there is new information regarding so
Expectations).
3. Help the students complete the Projected Income Statement for Grape Expectations for January. Some o
to complete the income statement was given in the case.
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
Unit III: Finance
Section 2: Projections
Transparen
One-Month Projected Income Statement for Grape Expectations
January 2001
Gross Sales (# of Units Sold x Price)
____175_______ x _____$15______ =
____$2,625____
Cost of Sales (Cost per Unit x # of Units Sold)
____$3.00_____ x _____175______ =
____$525______
Gross Profits (Gross Sales – Cost of Sales)
____$2,625____ - ____$525______ =
____$2,100____
Expenses
Salaries
Payroll Tax
Royalties Paid to Band Members
Jacket Design
Advertising Flyer
Bank Fees
Supplies
Cell Phone
Car Expense
5% of Gross Sales = $131.25
10% of Salaries = $13.13
10% of Gross Sales = $262.50
One-time Fee = $50
One-time Fee = $60
____$5________
____$15_______
____$20_______
____$10_______
Total Expenses
____$566.88___
Net Profit (Gross Profit – Total Expenses)
____$2,100____ - ___$566.88____ =
___$1,533.12__
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
Unit III: Finance
Section 2: Projections
Teacher Notes
Student Activity 3: One-Year Projected Income Statement for Grape Expectations
1. Explain that a one-year income statement shows all 12 one-month income
statements on one single statement.
2. Hand out a copy of the One-Year Projected Income Statement for Grape
Expectations to each student.
3. Ask the students to complete the One-Year Projected Income Statement for Grape
Expectations. Use the transparency of the first two months to guide the
discussion. A completed copy of the statement appears on the next page.
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Unit III: Finance
Section 2: Projections
Teacher Notes
One-Year Projected Income Statement for Grape Expectations
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Gross Sales
Cost of Sales
Gross Profit
$2,625
$525
$2,100
$1,875
$375
$1,500
0
0
0
0
0
0
0
0
0
$375
$75
$300
0
0
0
0
0
0
$2,625
$525
$2,100
Expenses
Salaries and Wages
10% Payroll Tax
10% Royalties Paid to Artists
Jacket Design
Advertising Flyer
Bank Fees
Supplies
Cell Phone
Car Expense
Total Expenses
$131.25
$13.13
$262.50
$50
$60
$5
$15
$20
$10
$566.88
$93.75
$9.38
$187.5
0
0
$5
$15
$20
$10
$340.63
0
0
0
0
0
$5
$15
$20
$10
$50
0
0
0
0
0
$5
$15
$20
$10
$50
0
0
0
0
0
$5
$15
$20
$10
$50
$18.75
$1.88
$37.50
0
0
$5
$15
$20
$10
$108.13
0
0
0
0
0
$5
$15
$20
$10
$50
0
0
0
0
0
$5
$15
$20
$10
$50
$131.25
$13.13
$262.50
0
0
$5
$15
$20
$10
$456.88
($50)
($50)
($50)
$191.87
($50)
($50)
$1,643.12
Net Profit (Loss)
$1,533.12 $1,159.37
Gross sales equals the number of units sold times the price. The price is $15.00, and the number of units sol
follows: January – 175, February – 125, March – 0, April – 0, May – 0, June – 25, July – 0, August – 0, Se
100, November – 150, December – 300.
The cost of sales equals the cost per unit times the number of units sold. The cost per unit is $3.00.
The gross profit equals the gross sales minus the cost of sales.
Total expenses are the sum of all the expenses. Net profit is the gross profit minus the total expenses. A loss
parentheses.
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
Unit III: Finance
Section 2: Projections
Student Activity
Student Activity 1
Projected Revenue
Estimating or projecting sales is pretty much a guessing game if a business
is just getting started or has no history of selling its product. One of the
biggest mistakes most entrepreneurs make is to overestimate sales. They
either put too high a price on their product, or they overestimate the
number of customers they think will buy the product. Sometimes it takes
longer than anticipated for customers to become aware of a product and to
feel confident enough to buy it. Many times, the entrepreneur will simply
“want” sales of a certain number, and he or she thinks that by simply
plugging in the number on the spreadsheet, the sales will magically
happen. This can be very hazardous, especially if the entrepreneur agrees
to pay employees or agrees to other expenses before he or she actually
generates the sales needed to pay for those expenses. But it is possible to
make a realistic projection of sales by researching and considering
carefully the product’s price and desirability in the marketplace.
Price
One of the best ways to set a realistic price for a product is to take a survey
to see what potential customers would actually pay for the product. Another
way is to try to sell the product at several different prices to see which price
best suits the customers and their needs. For example, a high price might
be okay when selling directly to students who are excited about the product
and who want it right then and there. A slightly lower price might be
appropriate after a product has been on the market for a while, and an “on
sale, limited time offer” might re-ignite interest in the product.
Units Sold
The product’s price and desirability certainly impact sales—specifically, the
number of units sold. The more units you sell, the higher the sales become.
Sales (Revenue)
You can project what your sales, or revenue, will be by guessing the
number of units you think you can sell based on your customer survey and
market research and multiplying that number by the price. The result is
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Unit III: Finance
Section 2: Projections
Student Activity
your “projected sales.” The word “projected” implies that you are
estimating, or that this is your goal. Since no one knows what the future
may hold, you are saying, “I think I can sell this many units of my product . .
. during this month . . . at this price . . . or over this period of time.”
(Formula: # of units x price = projected sales).
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Unit III: Finance
Section 2: Projections
Student Activity
Projected Revenue Worksheet for Grape Expectation
Fill in the revenue table below for Grape Expectations. Refer to the Assumption Stat
Expectations in order to determine the number of units sold each month and the pric
Projected Revenue Worksheet for Grape Expectations
Revenue
# of units
sold
x price of
CD
= gross
sales
from
CDs
Jan
Feb
Mar
Apr
May
June
July
Aug
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
Sept
Oct
Unit III: Finance
Section 2: Projections
Student Activity
Student Activity 2
One-Month Projected Income Statement for Grape
Expectations
The Income Statement
The income statement covers a period of time, such as a month or
a year. If you are looking at a one-month income statement, you
will see the sales, expenses, and net profit for that entire month.
The formula for net profit is sales minus expenses. If expenses
exceed sales, the result will be a net loss.
Chuck opened an account for Grape Expectations at the local
bank. He chose an account that charged a $5 monthly fee, but it
required no minimum balance. He also decided to get a cell phone
for $20 a month, and he estimated that his car expenses would be
$10 a month.
Because he planned to make the CDs only when an order was
placed, he did not need an office or inventory. However, he
decided he might need some supplies, such as order forms and a
receipt book as well as other general office supplies. He figured
these supplies would cost him about $15 per month.
Chuck was then ready to put his business on paper! Complete the
One-Month Projected Income Statement for Grape Expectations
for the month of January.
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Unit III: Finance
Section 2: Projections
Student Activity
One-Month Projected Income Statement for Grape
Expectations
January 2001
Gross Sales (# Units Sold x Price)
______________ x ______________ = ______________
Cost of Sales (Cost per Unit x # of Units Sold)
______________ x ______________ = ______________
Gross Profits (Gross Sales – Cost of Sales)
______________ - ______________ = ______________
Expenses
Salaries
Payroll Tax
Royalties Paid to Band Members
Jacket Design
Advertising Flyer
Bank Fees
Supplies
Cell Phone
Car Expense
______________
______________
______________
______________
______________
______________
______________
______________
______________
Total Expenses
______________
Net Profit (Gross Profit – Total Expenses)
______________ - ______________ = ______________
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Unit III: Finance
Section 2: Projections
Student Activity 3
One-Year Projected Income Statement for Grape Expecta
Complete the One-Year Projected Income Statement for Grape Expectations. Refer to the One-M
as well as the Assumption Statement.
Projected Income
Statement
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Gross Sales
Cost of Sales
Gross Profit
Expenses
Salaries and Wages
10% Payroll Tax
10% Royalties Paid to Artists
Jacket Design
Advertising Flyer
Bank Fees
Supplies
Cell Phone
Car Expense
Total Expenses
Net Profit (Loss)
The gross sales are the number of units sold times the price.
The cost of sales is the cost per unit times the number of units sold.
The gross profit is the gross sales minus the cost of sales.
Total expenses are the sum of all the expenses.
Net profit is the gross profit minus the total expenses. A loss is denoted with parentheses.
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
Unit III: Finance
Section 2: Projections
Projected Revenue
Revenues
Jan
Feb
# of Units Sold
x Price of CD
Gross Sales from
CDs
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
Unit III: Finance
Section 2: Projections
Projected Income Statement
Jan
Gross Sales
Cost of Sales
Gross Profit
Expenses
Salaries and Wages
10% Payroll Tax
10% Royalties Paid to Artists
Jacket Design
Advertising Flyer
Bank Fees
Supplies
Cell Phone
Car Expense
Total Expenses
Net Profit (Loss)
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
Feb
Unit III: Finance
Section 3: The Cash Flow Statement and the Balance Sheet
Teacher Notes
Section 3: The Cash Flow Statement and the Balance Sheet
Teacher Notes
Context Overview:
In this section, students will learn about the importance of a cash flow statement. They will then
work though an example using Grape Expectations. They will read about the purpose of a
balance sheet and complete an example for Grape Expectations.
Goal:
To gain an understanding of a cash flow statement and a balance sheet, and to recognize that
monitoring cash is very important.
TEKS Addressed:
Mathematical Models with Applications: The student is expected to
1A - compare and analyze various methods for solving a real–life problem;
1B - use multiple approaches (algebraic, graphical, and geometric methods) to solve
problems from a variety of disciplines;
2A - interpret information from various graphs, including line graphs, bar graphs, circle
graphs, histograms, and scatterplots to draw conclusions from the data;
2B - analyze numerical data using measures of central tendency, variability, and correlation
in order to make inferences;
3A-formulate a meaningful question, determine the data needed to answer the question,
gather the appropriate data, analyze the data, and draw reasonable conclusions;
Materials:
Cash Flow Pre-reading (1 per student)
Cash Flow Statement (1 per student)
One-Year Projected Income Statement for Grape Expectations (completed in previous section)
Balance Sheet for Grape Expectations (1 per student)
Procedures:
Student Activity 1: Cash Flow Statement for Grape Expectations
1. Hand out the Cash Flow Statement for Grape Expectations to each student and ask them to
read the first section, which describes what a cash flow statement is. Ask the students, “Why
is a Cash Flow Statement important?”
Possible answer: If you run out of cash, you are in danger of going “out ofbBusiness!
Point out to students that running out of cash is a problem that many entrepreneurs
experience.
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
104
Unit III: Finance
Section 3: The Cash Flow Statement and the Balance Sheet
Teacher Notes
2. Help the students complete the Cash Flow Statement for Grape Expectations by filling the
columns on the transparency. A completed statement follows on the next page.
Student Activity 2: Balance Sheet for Grape Expectations
1. Ask students to use Grape Expectation’s Cash Flow Statement and oOne-Year Income
Statements to complete the Balance Sheet for Grape Expectations. A transparency of the
first few months is provided.
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
105
Unit III: Finance
Section 3: The Cash Flow Statement and the Balance Sheet
Teacher Notes
Cash Flow Statement for Grape Expectations
Cash Flow Statement
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
Dec
$2,625
$1,875
0
0
0
$375
0
0
$2,625
$1,500
$2,250
$4,500
0
0
0
0
0
0
0
0
0
0
0
0
$2,625
$1,875
0
0
0
$375
0
0
$2,625
$1,500
$2,250
$4,500
$947.50
$612.50
$50
$50
$50
$162.50
$50
$50
$837.50
$500
$725
$1,400
$103.13
0
0
0
$20.63
0
0
$144.38
$82.50
$123.75
$247.50
$715.63
$50
$50
$50
$183.13
$50
$50
$981.88
$582.50
$848.75
$1,647.50
$1,159.37
($50)
($50)
($50)
$191.87
($50)
($50)
$1,643.12
$917.50
$1,401.25
$2,852.50
$1,533.12
$2,692.49
$2,642.49
$2,592.49
$2,542.49
$2,734.36
$2,684.36
$2,634.36
$4,277.48
$5,194.98
$6,596.23
$2,692.49
$2,642.49
$2,592.49
$2,542.49
$2,734.36
$2,684.36
$2,634.36
$4,277.48
$5,194.98
$6,596.23
$9,448.73
CASH INFLOWS
Cash Sales - Collected
Other Sales
Total Cash Inflows
TOTAL CASH OUTFLOWS
New Accounts Payable
Salaries, Wages & Taxes
$144.38
Total Cash Outflows $1,091.88
CASH INFLOWS MINUS
$1,533.12
OUTFLOW
Previous Cash Balance
0
Ending Net Cash Flow $1,533.12
Cash Sales are the same as monthly sales which are determined by multiplying the number of units sold times the price.
Other Sales are the sales made to customers on a credit basis. For Grape Expectations this is always zero.
New Accounts Payable are all costs, fees, and royalties associated with the product or service. This amount will vary according to the amount of product or
service sold.
Salaries, Wages & Taxes are all salaries, commissions, wages and taxes paid to employees during the month.
Previous Cash Balance is the amount of Cash owned by the company at the end of the previous month.
Ending Net Cash Flow is the amount of Cash the company owns at the end of each month.
Example: January : Cash Sales = 175 sold x $15 each = $2625; New Accounts Payable = cost of CDs + 10% of sales to band members + Jacket + Advertising
Flyer + Bank Fee + Supplies + Cell Phone + Car Expenses = $525 + $262.50 + $50 + $60 + $5 + $15 + $20 + $10 = $947.50; Salaries, Wages & Taxes = Sally’s
commission + Taxes = $131.25 + $13.13 = $144.38; Total Cash Outflows = $947.50 + $144.38 = $1,091.88; Cash Inflows minus Outflows = $2,625 $1,091.88 = $1,533.13; Previous Cash Balance = 0; Ending Net Cash Flow = $1,533.13 – 0 = $1,533.1
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
106
Unit III: Finance
Section 3: The Cash Flow Statement and the Balance Sheet
Teacher Notes
Balance Sheet for Grape Expectations
Balance Sheet
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
Dec
Year
Current Assets
Cash $1,533.13
$2,692.50
$2,642.50
$2,592.50
$2,542.50
$2,734.37
$2,684.37
$2,634.37
$4,277.49
$5,194.99
$6,596.24
$9,448.74
$9,448.74
Other Assets
Total Assets $1,533.13
$2,692.50
$2,642.50
$2,592.50
$2,542.50
$2,734.37
$2,684.37
$2,634.37
$4,277.49
$5,194.99
$6,596.24
$9,448.74
$9,448.74
Current Liabilities
Accounts Payable
0
0
0
0
0
0
0
0
0
0
0
0
0
Other Liabilities
0
0
0
0
0
0
0
0
0
0
0
0
0
Total Liabilities
0
0
0
0
0
0
0
0
0
0
0
0
0
Total Equity
Contributed Capital
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Retained Earnings
1,533.13
2,692.50
2,642.50
2,592.50
2,542.50
2,734.37
2,684.37
2,634.37
4,277.49
5,194.99
6,596.24
9,448.74
9,448.74
Total Shareholders' Equity
1,533.13
2,692.50
2,642.50
2,592.50
2,542.50
2,734.37
2,684.37
2,634.37
4,277.49
5,194.99
6,596.24
9,448.74
9,448.74
Total Liabilities & Equity
1,533.13
2,692.50
2,642.50
2,592.50
2,542.50
2,734.37
2,684.37
2,634.37
4,277.49
5,194.99
6,596.24
9,448.74
9,448.74
YES
YES
YES
YES
YES
YES
YES
YES
YES
YES
YES
Does the Balance Sheet
balance?
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
YES
YES
107
Unit III: Finance
Section 3: The Cash Flow Statement and the Balance Sheet
Student Activity 1
Cash Flow Statement for Grape Expectations
The cash flow statement keeps track of one thing only: cash. When running a business, you n
“cash position” very carefully. Without enough cash, you might not be able to pay your bills,
inventory, or pay your expenses. If you run out of cash, you are in danger of going “out of Bu
Basically, you determine the total amount of cash taken into your business as a result of sales
amount of cash spent to run your business. (Formula: Cash In – Cash Out = Net Cash)
Cash is not the same as sales. For example, you may have sold some of your product “on cre
pays you late, or not at all, this could get you into trouble since you are continuing to spend m
create your product, but you haven’t been paid yet for the products you have sold. This probl
demise of many businesses that eventually must declare bankruptcy due to “insufficient cash
What a dilemma. You can sell your product, but you don’t have enough cash because you are
much or not being paid at all.
For Grape Expectations, we are assuming that all sales are made on a “cash basis,” meaning
collection of cash occurs within the month. Otherwise, the balance would have to include “ac
receivable”—another way of saying, “money that is owed as a result of sales, but has not bee
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
Unit III: Finance
Section 3: The Cash Flow Statement and the Balance Sheet
Use Grape Expectation’s One-Year Income Statement to complete the following Cash Flow State
Cash Flow Statement for Grape Expectations
Cash Flow Statement
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
CASH INFLOWS
Cash Sales - Collected
Other Sales
Total Cash Inflows
TOTAL CASH OUTFLOWS
New Accounts Payable
Salaries, Wages & Taxes
Total Cash Outflows
CASH INFLOWS MINUS
OUTFLOW
Previous Cash Balance
Ending Net Cash Flow
Cash Sales are the same as monthly sales, which are determined by multiplying the number of un
Other Sales are the sales made to customers on a credit basis. For Grape Expectations this is alw
New Accounts Payable are all costs, fees, and royalties associated with the product or service. Th
according to the amount of product or service sold.
Salaries, Wages & Taxes are all salaries, commissions, wages, and taxes paid to employees duri
Previous Cash Balance is the amount of Cash owned by the company at the end of the previous m
Ending Net Cash Flow is the amount of Cash the company owns at the end of each month.
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
Unit III: Finance
Section 3: The Cash Flow Statement and the Balance Sheet
Student Activity 2
Balance Sheet for Grape Expectations
Use Grape Expectations Cash Flow and one-year Income Statement in order to c
Sheet. See the notes on the next page describing each row in the table
Balance Sheet
Jan
Feb
Mar
Apr
May
June
July
Current Assets
Cash
Other Assets
Total Assets
Current Liabilities
Accounts Payable
Other Liabilities
Total Liabilities
Total Equity
Contributed Capital
Retained Earnings
Total Shareholders' Equity
Total Liabilities & Equity
Does the Balance Sheet
balance?
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
Aug
Sept
Unit III: Finance
Section 3: The Cash Flow Statement and the Balance Sheet
Cash Flow Statement for Grape Expectations
Cash Flow Statement
Jan
CASH INFLOWS
Cash Sales - Collected
Other Sales
Total Cash Inflows
TOTAL CASH OUTFLOWS
New Accounts Payable
Salaries, Wages & Taxes
Total Cash Outflows
CASH INFLOWS MINUS OUTFLOW
Previous Cash Balance
Ending Net Cash Flow
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
Feb
Unit III: Finance
Section 3: The Cash Flow Statement and the Balance Sheet
Balance Sheet for Grape Expectations
Balance Sheet
Jan
Current Assets
Cash
Other Assets
Total Assets
Current Liabilities
Accounts Payable
Other Liabilities
Total Liabilities
Total Equity
Contributed Capital
Retained Earnings
Total Shareholders' Equity
Total Liabilities & Equity
Does the Balance Sheet balance?
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
Feb
Unit III: Finance
Section 4: Mathematical Models and Projections
Teacher Notes
Section 4: Mathematical Models and Projections
Teacher Notes
Context Overview:
Students will write a mathematical function to model the expense of producing CDs. They will
also write a function to model Grape Expectations’ gross sales. They will notice that the price
and the number of units sold play a big role in determining the total profit. The students will then
perform a break-even analysis for Grape Expectations based on these models.
Goal:
To use a mathematical model to examine the break-even point for Grape Expectations.
TEKS Addressed:
1) The student uses a variety of strategies and approaches to solve both routine and non-routine
problems: The student is expected to
1B - use multiple approaches (algebraic, graphical, and geometric methods) to solve
problems from a variety of disciplines;
1C - select a method to solve a problem, defend the method, and justify the reasonableness of
the results.
Materials:
“A Mathematical Model for Grape Expectations” activity (1 copy per student)
Assumption Statement for Grape Expectations (Completed in Section 1)
Procedures:
Student Activity: A Mathematical Model for Grape Expectations
Hand out the “A Mathematical Model for Grape Expectations” activity to each student. Lead the
students through the worksheet. Questions and possible answers follow below.
Questions and possible answers:
1. A. What price is Grape Expectations charging per CD?
$15.00
B. Write a function that models revenues, y, as a function of the number of CDs sold, x. y =
$15x
2. A. For Grape Expectations, how much does it cost to make one CD?
$3.00 – $2.00 to duplicate one CD; $.50 for one plastic case; $.50 for one 4-color CD
jacket.
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
113
Unit III: Finance
Section 4: Mathematical Models and Projections
Teacher Notes
B. Write a mathematical function that models the expenses of duplicating CDs (y) as a
function of the number of CDs sold (x)
y = $3x
3. A. For Grape Expectations, what are the other expenses of producing the CDs for one year?
Salary and wages (5% of revenue), payroll tax (10% of wages), royalties (10% of
revenues), jacket design and advertising flyer = $110; all other fees (bank, supplies, etc.)
= $50 per month = $710 per year
B. Write a mathematical function that models the expenses paid in royalties to the band (y)
as a function of the number of CDs sold (x)
Royalty expenses = 10% of revenues; y = .1(15x) = 1.5x
C. Write a mathematical function that models the expense of Chuck’s salary (y) as a
function of the number of CDs sold (x).
Chuck’s salary = 5% of revenues; y = .05(15x) = .75x
D. Write a mathematical function that models the payroll tax expenses on Chuck’s salary (y)
as a function of the number of CDs sold (x).
Payroll tax = 10% of Chuck’s salary; y = .1(.05(15x)) = .075x
4. Write a mathematical function that models all the expenses of producing CDs (y) as a
function of the number of CDs sold (x).
y = 1.5x + .75x + .075x + 3x + 710; y = 5.325x + 710
5. Based on our model, how much profit would Chuck make if he sold:
A. 200 CDs in 1 year?
Expense = 5.325x + 710 = $1775; Sales = 15x = $3000; Profit = $3000 - $1775 =
$1225
B. 20 CDs in 1 year?
Expense = 5.325x + 710 = $816.50; Sales = 15x = $300; Profit = $300 - $816.50 = $516.50
• What does a negative number mean for Grape Expectations? It means the
company is losing money.
6. Based on a price of $15.00 per CD, how many CDs will Chuck have to sell in order to break
even (meaning that he does not make or lose money) for the month?
This is a break-even analysis. Students may solve by graphing y = 15x and y = 5.325x + 710
to find the intersection of the two lines. They may solve using a table on the graphing
calculator, or they may solve the system of equations for x (the number of CDs). 15x =
5.325x + 710; x = 73.38 . . . (approximately 74 CDs).
•
What is the expense of producing 74 CDs? $1104.05
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
114
Unit III: Finance
Section 4: Mathematical Models and Projections
•
•
•
•
•
•
Teacher Notes
How much revenue does Chuck make by selling 74 CDs? $1110.00
How much profit does Chuck make by selling 74 CDs? $5.95
What is the expense of producing 73 CDs? $1098.73
How much revenue does Chuck make by selling 73 CDs? $1095.00
How much profit does Chuck make by selling 73 CDs? -$3.73
Why does Chuck earn a negative profit by selling 73 CDs? It costs more to produce
the CDs than he made.
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
115
Unit III: Finance
Section 4: Mathematical Models and Projections
Student Activity
Student Activity
A Mathematical Model for Grape Expectations
Answer the following questions in order to help you make a mathematical model of
sales and expenses for Grape Expectations.
1. A. What price is Grape Expectations charging per CD?
B. Write a function that models revenues (y) as a function of the number of CDs sold
(x).
2. A. For Grape Expectations, how much does it cost to make one CD?
B. Write a mathematical function that models the expenses of duplicating CDs (y) as
a function of the number of CDs sold (x).
3. A. For Grape Expectations, what are the other expenses of producing the CDs for
one year?
B. Write a mathematical function that models the expenses paid in royalties to the
band (y) as a function of the number of CDs sold (x).
C. Write a mathematical function that models the expense of Chuck’s salary, y, as a
function of the number of CDs sold, x.
D. Write a mathematical function that models the payroll tax expenses on Chuck’s
salary (y) as a function of the number of CDs sold (x).
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
116
Unit III: Finance
Section 4: Mathematical Models and Projections
Student Activity
4. Write a mathematical function that models all the expenses of producing CDs (y) as
a function of the number of CDs sold (x).
5. Based on our model, how much profit would Chuck make if he sold:
A. 200 CDs in 1 year?
B. 20 CDs in 1 year?
6. Based on a price of $15.00 per CD, how many CDs will Chuck have to sell in order to
break even (meaning that he does not make or lose money) for the month?
Mathematical Models with Applications / Entrepreneurship: Keeping Score with Mathematics
117