Extending Calculus Reform to Prealgebra
Paper Presented at The Sixth Conference On the Teaching of Mathematics,
on June 20, 1997
by
Umesh P. Nagarkatte, Medgar Evers College, CUNY, Brooklyn, NY 11225
email:umnme@cunyvm.cuny.edu
Abstract: The paper discusses construction of modeling type problems on topics in Prealgebra  percent, signed numbers, equations, ratio and proportion that connect systematically elementary and advanced concepts extending "Calculus Reform" to Prealgebra. The approach has been classtested for three semesters starting Spring 1996 by more than 400 students each semester.
Introduction: During the intersession of January 1996, I took upon myself to develop lab problems for Prealgebra  the first remedial course  offered by our department every semester with an enrollment of about 400 students. I am a member of the Basic Skills Committee of the Mathematics Department of my college, Medgar Evers College, Brooklyn of the City University of New York. We wanted problems satisfying the following conditions.
The problems that:
Logistics of the Design of Problems: In our committee we first laid down the sequence of the topics to be taught that implement, in general, the AMATYC standards. The list of topics is given in Appendix 2. Basically, we wanted to introduce Whole Numbers, and Graphs, Decimals and two types of Percent Problems, which are applications of decimals, in the beginning of the course. These topics were to be followed by Order of Operations, Geometrical formulas on perimeters and areas as applications of substitution, Metric system, Commutative, Associative and Distributive laws, Combining like terms, Integers, Equations and Problems on percent as applications of equations, Numerical and Algebraic fractions, Fractional Equations. Building formulas from a numerical pattern was also an important goal.
The 4hour Prealgebra course met usually three times a week, one of which was to be the lab hour. We wanted five problems per lab hour. I felt that for the students to be able to solve the problems in the lab, they should have practice of that type of problems in their two class periods with the instructor also. So ten problems per week for classwork had to be created. In this manner it was necessary to have available a minimum of 195 problems for a 13week semester. In January 1996 and the Spring 1996 semester I developed more than 200 problems for the Prealgebra course. As I started developing the problems in the spirit of reformed calculus or item No. 5, I realized that they easily implemented the other four conditions. All Prealgebra  MTH 002  instructors and students in the college have used these problems for the last three semesters. They have responded to the problem sets enthusiastically. We have included these problems in the upcoming revision of the textbook Prealgebra, which Dr. Berenbom and I developed, to be custom published for our College. The first edition of the book was published by Hartcourt Brace in 1991.
Important Consequences:
Bibliography that helped me create the problem sets is enclosed.
In the following I would like to present a selection of the problems based on different concepts involved in the course.
Note: In all problems indicate the answer using appropriate units and not just a number.
Week 1
Concepts and skills: Place Value System, Operations on Whole Numbers, Bar and Line Graphs
1. A chef, who weighed 197 pounds, went on a diet. The first week the chef lost six pounds. During the five subsequent weeks, he lost two pounds each week.
Week 
1 
2 
3 
4 
5 
6 
Weight 
197 
If he continued dieting and lost weight at the same rate as the previous week, what will be his weight at the end of seventh week?
2. A pizza delivery van has to cover the distance from the store to a customer's house with a choice of several streets. Time, in minutes, needed to cover each street is shown.
3. A retailer wants to make an $87 profit on a TV that he paid $214 for.
# TV's 
1 
2 
3 
4 
5 
6 
8 
10 
12 
20 
Profit ($) 
87 
4. Mary is a salesperson at Great Discounts Auto Store. The following table summarizes her sales for the last four weeks.
Week 
Sales in $ 
1 
$4,050 
2 
1,075 
3 
900 
4 
2,425 
Week 
Sales in $ 
Weekly Change 
1 
$1,050 
xxxxxxxx 
2 
1,075 
$, increase/decrease 
3 
1900 
, increase/decrease 
4 
2,425 
, increase/decrease 
5. A Beverage and Snacks Company has income from sales from 19911996 as in the following table. (Additional goal of the problem: Using appropriate scales)
Year 
1991 
1992 
1993 
1994 
1995 
1996 
$inMillions 
17500 
19300 
21900 
25000 
28700 
29400 
Year 
1991 
1992 
1993 
1994 
1995 
1996 
$ in Millions 
17500 
19300 
21900 
25000 
28700 
29400 
Yearly Increase 
xxxxxx 
6. A $20,000 investment will grow, depending on your tax bracket as in the following table. Tax Deferred means tax will be paid after 5 years.
Years 
0 
1 
2 
3 
4 
5 
20% tax bracket 
20,000 
21,100 
22,300 
23,500 
24,800 
26,200 
Tax Deferred 
20,000 
21,400 
22,900 
24,500 
26,200 
28,100 
Years 
0 
1 
2 
3 
4 
5 
20% tax bracket 
20,000 
21,100 
22,300 
23,500 
24,800 
26,200 
Yearly Increase 
xxxxxx 
_____ 
_____ 
______ 
______ 
______ 
Years 
0 
1 
2 
3 
4 
5 
Tax Deferred 
20,000 
21,400 
22,900 
24,500 
26,200 
28,100 
Yearly Increase 
xxxxxx 
_____ 
______ 
_______ 
______ 
_____ 
7. Complete the following square using numbers 2 through 10, only once, so that total horizontally, vertically and diagonally is 18. Some boxes have already been filled to get you started.


9 



10 
Week 2
Topics and Skills: Decimals, Operations on Decimals. Rounding and Estimation
1. Pam drove her car at 55 mph.
(a) In 4 hours, how much distance does she cover?
(b) Fill in the blanks in the following table. (Note that Distance = Speed x Time)
Time (hours) 
0 
1 
2 
4 
6 
8 
Distance (miles) 
(c) Draw a line graph using the table in (b).
2. The Brown family drives a car for a distance of 300 miles from Albany to Buffalo at a speed of 60 mph with a onehalf hour lunch break in Syracuse which is 150 miles away from Albany.
(a) Complete the given table.
Time (hours) 
0 
1 
2 
3 
3.5 

Distance (miles) 
(b) Draw a line graph of distance against time.
(c) What was the average speed for the entire trip?
3. A cab driver drives 800 miles per 6day week.
Day 
0 
1 
2 
3 
4 
5 
6 
Distance(miles) 
4. An investment of $100 doubles every 4 years.
(a) How much will be the balance at the end of 16 years?
(b) Fill in the following table.
Years 
0 
4 
8 
12 
16 
Balance ($) 
(c) Draw a line graph of balance against years.
5. A parking garage charges $6 for the first hour. The hourly rate for parking after the first hour is $3, with the maximum rate of $20. Partial hours are rounded to the next hour.
(a) If a car is parked for 2 1/2 hours, what will be the parking charges?
(b) If a car is parked for 3 hours 40 minutes, what are the parking charges?
(c) Complete the following table.
Hours 
½ 
1 
1 ½ 
2 
2 ½ 
3 
3 1/2 
4 
4 1/2 
5 
6 
Parking charges 
(d) Draw a graph of parking charges for the first hour. Draw a graph for the second hour. Repeat the process for 6 hours. Describe in your words how the graph appears.
6. A video store has a starting expense of $5,000.
(a) If the store rents $3 per tape, how many tapes should be rented to break even?
(b) If a profit of $3,000 is made, how many tapes were rented?
7. Every week Joe earns $15 per hour for the first forty hours, and $23 per hour overtime.
Hours 
10 
20 
30 
40 
45 
50 
60 
Wages ($) 
8. A plant 10 inches long grows 1.5 inches every week until it reaches 16 inches.
Week 
0 (start) 
1 
2 
4 
6 
8 
12 
Height (inches) 
10 
9. Match with each description in (a) through (d), the appropriate graph in Ga  Gd and then fill in the blanks in (a) through (d).
Week 3
Concepts and Skills: Changing fractions to decimals, decimals to fractions. Changing percent to decimals and fractions, Changing fractions or decimals to percent. Finding percent of a number
1. Complete the following table based on data from the census of a village.
Ages 
017 
1829 
Over 30 
Total 
# males 
320 
500 
1800 

# females 
210 
750 
2000 

Total 

Fraction representing male population 

% of male population 
2. The following chart is from an annual report of a corporation.
Using the above chart answer the following questions.
(a) What was the increase in sales from 19901991. Find the percent increase. (Hint: To find % increase for 1991. find the fraction: Increase from 19901991/Sales in 1990. Express your answer as a percent.)
(b) Complete the following table.
Year 
1990 
1991 
1992 
1993 
1994 
Sales 

Yearly Increase 
xxxxxxx 

% Increase 
xxxxxxx 
3. During a semester Malcolm scores 38/40, 45/50, 32/40, and 36/40 in his 4 exams. What was his percent score in each exam? What was his average for the semester?
4. A population survey of 900 people about the job performance of the President was taken. The results are given in the following table, with some information missing. Fill in the missing information, and answer the questions (a) through (c).
Male 
Female 
Total 

Approve 
285 
680 

Disapprove 
120 

Total 
900 
(a) What percent of male population approve the job performance?
(b) What percent of female population disapprove the job performance?
(c) What percent of population approve the job performance?
5. A beverage company's profits in December 31, 1994 and December 31, 1995 are given in the following table. Complete the following table. This table will be part of the annual report for the company for its share holders. (Dollars in millions.)
December 31, 1995 
December 31, 1994 
Percent Change (+/) 

Beverages 
1,567 
1,150 

Snack foods 
1,235 
1,025 

Restaurants 
640 
724 

Total 
6. One semester Cathy takes three 4credit and two 3credit courses. She gets 2 A's in a 4credit courses, 1 A in a 3credit course. She gets a B in the remaining courses. The grade of an A is worth 4 points and a B is worth 3 points.
A 
B 

# 4credit courses 

Points per course 

# Credits per course 

Total points in 4credit courses 
A 
B 

# 3credit courses 

Points per course 

# Credits per course 

Total points in 3credit courses 
7. Potato and other snack chip consumption (in "Lbs per Capita" meaning pounds per person) in the countries around the world is given in the following table.
Country 
US 
Canada 
United Kingdom 
Australia 
Mexico 
Other 
Chip Consumption 
18 
9 
8.5 
6 
3 
.2 
(a) What percent of Australia's chip consumption is US chip consumption?
(b) What percent of all the chip consumption around the world is US chip consumption?
8. A snack company's profits are given in the following Piechart. If the company's profits were $125,000,000 last year, how much profit was made from Potato and Tortilla chips?
9. The following chart indicates the next monthly payment on a loan of $1,000 at different interest rates per year (APR).
Annual Rate (APR) (%) 
6 
9 
12 
18 
21 
24 
Monthly rate (APR/12)(%) 
0.5 

Monthly payment ($) 
5 
10. A salesperson gets a salary of $500 per week and 3% commission on all the sales she makes. If she sold $52,000 amount of merchandise in one week, what was her total income that week?
Week 4
Concepts and Skills: Exponents, Order of Operations, Variable Expressions, Perimeter, Area, Volume
1. In a business investment doubles every six years.
(a) Complete the following table and draw a smooth line graph of balance against time in years.
Time (in years) 
0 
6 
12 
18 
24 
30 
Balance ($) 
1,000 
2,000 
(b) Divide the balance by the initial investment and enter into the blanks titled "Ratio". Write the answer in exponential form and also as a number.
Time (in years) 
0 
6 
12 
18 
24 
30 
Balance ($) 
1,000 
2,000 

Ratio 
1 
(c) Draw a graph of Ratio against time.
2. (a) A photograph is surrounded by a 2"wide frame. What is the area of the glass to frame the photograph?
(b) Complete the following table.
Size of picture (in inches) 
4" W x 6" L 
4" W x 6" L 
6" W x 8" L 
6" W x 8" L 
9" W x 11" L 
9" W x 11" L 
width of frame 
1" 
2" 
1" 
2" 
1" 
2" 
Area of glass 
3. The formula for converting kgs into pounds (lb) is given by: 1 kg. = 2.2 lb
Complete the following table and plot a line graph of lb. against kg.
kg 
1 
2 
6 
8 
lb. 
4. The formula F = 1.8 C + 32 relates to converting Celsius to Fahrenheit where C is temperature in degrees Celsius and F is the temperature in degrees Fahrenheit.
(a) If the boiling point of water is 100o C, what is it in degrees Fahrenheit?
(b) Complete the following table.
Celsius 
10 
20 
60 
80 
100 
Fahrenheit 
(c) Draw a graph of Fahrenheit against Celsius temperature.
5. The growth of bacteria in a culture is given by the formula: N = A 2t, where t is the elapsed time in hours, N is the weight of bacteria, in grams. Complete the following table and draw a graph of weight against time, given A = 100 g
time 
1 
2 
6 
8 
Weight 
6. A formula for simple interest is given by I = Prt, where I is the interest, P is the principal (initial investment or loan), r is the rate of interest per period t is the time expressed as the number of periods. Complete the following table with the interest rate of 8.2%.
P 
1,000 
2,000 
5,000 
8,000 
t 
1 
5 
2.4 
6.2 
I = Prt 
7. Simone invests $10,000 in a growth stock and $5,000 in an equity index stock. The growth stock went up by 35.75% and equity index stock went up by 25.28%. Complete the following table. How much is Simone's investment now?
Growth 
Equity index 

Initial Investment 
10,000 
5,000 
Rate 

Increase ($) 

Current value ($) 
8. 2 gallons of 40% acid solution is mixed with 3.5 gallons of 20% acid solution. What is the concentration of the mixture? Use the following table to answer the question.
40% solution 
20% solution 
Mixture ( _____ %) 

Volume 

Amount of acid (gallons) 
9. A 16quart radiator has been filled with 30% antifreeze.
Amount drained (qt.) 
Pure Antifreeze in (30%) solution 
Pure Antifreeze (100 %) 
Pure antifreeze in Solution 
% solution in the radiator 
1

16 x .3  .3 qt = _________ 
1 qt 

2

16 x .3  ___ = _________ 


4

16 x .3  ___ = ________ 

6

16 x .3  ___ = _________ 
Week 5
Concepts and Skills: Circles, Signed numbers, addition, and multiplication of signed numbers. Introduction to graphing using positive and negative numbers.
1. The circumference of a circle is given by the formula: C = 2p r where p » 3.14, r = radius of the circle.
(a) If Jason wants to make circles of radii of 5, 10, 20, 40 inches from wire, how much wire should he get? First complete the table in (b).
(b) Complete the following table.
Radius (in.) 
5 
10 
20 
40 
Circumference (in) 
(c) Draw a graph of the circumference against the radius using the table in (a).
2. The area of a circle is given by the formula: A = p r2 where p » 3.14, r = radius of the circle.
(a) Complete the following table.
Radius (cm.) 
5 
10 
20 
30 
Area (cm2) 
(b) Draw a smooth line graph of the area against the radius using the above table.
3. The distance of a falling object from the ground is given by the formula:
d = .5gt2 + 512, where g = 32, t is measured in seconds.
(a) Complete the following table
t (sec) 
0 
1 
2 
2.5 
5 
6 
d (feet) 



(b) Interpret the result at t = 6 seconds.
(c) Draw a smooth graph of distance against time using the table in (a).
4. The formula F = 1.8 C + 32 relates to converting Celsius to Fahrenheit where C is temperature in degrees Celsius and F is the temperature in degrees Fahrenheit.
(a) If the freezing point of water is 0o C, what is it in degrees Fahrenheit?
(b) Complete the following table.
Celsius 
10 
20 
60 
40 
0 
10 
20 
100 
Fahrenheit 
(c) Draw a graph of Fahrenheit against Celsius temperature.
5. In a country there is a flat tax of 17% over $30,000. If a family makes less than $30,000 it gets a subsidy of 17% of the difference between the income and $30,000. The formula is given by A = 17%(I  30000), where A is the amount of tax or subsidy, I is the income. Complete the following table.
Income ($) 
40,000 
30,000 
20,000 
10,000 
5,000 
Amount ($) 
6. (a) Complete the following table which gives the absolute values (distances from zero) of numbers.
Number 
20 
10 
5 
4 
0 
4 
5 
10 
Absolute Value 
(b) Draw a graph of the absolute value against number from the table.
Week 6
Concepts and Skills: Subtraction of signed numbers, The Commutative and Associative Laws, The Distributive Law, Combining like terms
1. To test the effectiveness of tutoring on a group of students, a test was administered before tutoring and after tutoring. The following table represents difference of students' scores from the passing mark. Complete the third row of the table showing by how much each student progressed.
Before tutoring 
15 
20 
20 
5 
0 
5 
10 
10 
After tutoring 
5 
5 
10 
25 
5 
21 
20 
20 
Change 
2. If an item is not sold during the first week, a store reduces it by 10% per week. After the item reduces to 60% of the original marked price it is removed from the store.
(a) Complete the following table.
Week 
1 
2 
3 
4 
5 
6 
Price ($) 
250 
(b) Write a formula for the price, using the unknown P for price in terms of weeks elapsed.
(c) Draw a graph of Price against Weeks.
3. A car is traveling at 65 mph from Washington, DC to Pittsburgh, PA, a distance of 300 miles.
Time (hour) 

Distance (miles) 
4. A submarine is coming to the surface at a rate of 5 m per minute. Complete the table in (a)
(a) Find the distance from the surface of water.
t (min) 
0 
1 
2 
3 
5 
6 
d (meters) 
50 



(b) Write a formula of d in terms of t.
5. Jerome drives a motorcycle at 50 mph south on I95 from Wilmington, DE. Rimi drives a car at 60 mph on the same route after 1/2 hour.
Time (hours) 
0 
.5 
1 
2 
3 
4 
5 
Jerome's distance (miles) 

Rimi's distance (miles) 
6. Find the sum of each row, column and diagonal in the following table.
a + x 
a  (x + y) 
a + y 
a  (x  y) 
a 
a + (x  y) 
a  y 
a + (x + y) 
a  x 
7. A roundtrip from Newark to Miami Beach is $175 and cost of a hotel room per day is $55.
# Days 
1 
2 
3 
4 
5 
6 
Total cost ($) 
8. Frank invests x dollars in a social choice stock and y dollars in a global stock. During the year, the social choice stock went up by 30% and global stock went up by 25%.
Social choice 
global 

Initial Investment 
x 
y 
Rate 

Increase ($) 

Current value ($) 
Week 7
Concepts and Skills: Solutions of equations, Solving equations by addition, division and both techniques, Word Problems
1. The formula F = 1.8 C + 32 relates to converting Celsius to Fahrenheit where C is temperature in degrees Celsius and F is the temperature in degrees Fahrenheit.
(a) Complete the given table.
City 
New York 
Chicago 
San Francisco 
Los Angeles 
Temperature (o F) 
23 
15 
57 
72 
Temperature (o C) 
(b) Solve the above formula for C.
2. A $20,000 investment grew for the last four years as given in the table. Find the yearly increase and complete the table if the investment grows at the same annual percentage rate (APR).
Years 
0 
1 
2 
3 
4 
5 
Yearend Balance 
20,000 
24,000 
28,800 
34,560 
41472 

Yearly Increase 
xxxxxx 
_____ 
_____ 
______ 
______ 
______ 
Yearly % increase 
3. If the cab fare is $25, and Mary has 2 tendollar, 4 fivedollar and 8 onedollar bills. Complete the following table that indicates the bills she can combine to make the fare $25.00.
tendollar bill 
fivedollar bill 
onedollar bill 
Total Value 
2 
0 
25 

1 
7 
25 

0 
x 
25 

0 
x 
25 

x 
5 
25 
4. Pam starts driving her car at 65 mph from Chicago to Cincinnati, a distance of 295 miles.
(a) Fill in the blanks in the following table. (Note that Distance = Speed x Time)
Time (hours) 
0 
1 
2 
t 
Distance (miles) 
0 
(b) Draw a line graph of the table in (a).
(c) Write an equation of distance in terms of time t. Use the equation to find the time needed to cover 200 miles. Verify this fact from the graph in (b).
5. A salesperson makes $275 and 10% of all sales per week..
Sales ($) 
0 
1000 
2000 
x 
Weekly Salary ($) 
275 
Week 8
Concepts and Skills: Solutions of equations, Solving equations by addition, division and both techniques, Word Problems
1. A population survey or 900 people about the job performance of the President was taken. The results are given in the following table, with some information missing.
Male 
Female 
Total 

Approve 
240 

Disapprove 
120 

Total 
900 
2. Every week Joe earns $15 per hour for the first forty hours, and $23 per hour overtime.
Hours 
10 
20 
40 
45 
60 
x 
Wages ($) 
Week 
0 
1 
2 
4 
5 
x 
Height (Inches) 
10 
4. The perimeter of a rectangle is given by the formula: P = 2L + 2W. A rectangle has perimeter = 184 cm; W is its width and the length is 8 cm longer than the width.
5. Measurements of a tree are taken at different heights from the ground and is listed in the following table. By assuming the tree to be a stack of cylindrical slabs of height 2 feet, find the volume of the tree as follows.
Find the radius at the given heights.
Height from the ground (feet) 
0 
2 
4 
6 
8 
10 
Circumference (inches) 
35 
28 
21 
14 
10 
7 
Radius (inches) 
Radius (inches) 

Radius (feet) 

Area of crosssection (squarefeet) 

Volume of slab of ht. 2ft. (cubic feet) 
Total volume = ____ cubic feet.
6. The line graph below lists the number of vehicles using natural gas. What was the (percent) rate of increase for the year 199596. If the rate of increase is the same, estimate the number of vehicles in 1997. (Source: The Natural Gas Vehicle Coalition, Energy Department)
7. A credit card company charges 1.5125% monthly finance charge for any unpaid balance. If the finance charge was $80, how much was unpaid balance?
8. The average price of a home in a city went down to $85,000 by 8% during the year 1995. What was the average price at the end of 1994?
9. Area of a rectangle is given by the formula: A = LW, where L is the length and W is the width. If the area of plywood board is 36 m^{2}, and the width is 4 m, find the length of the board.
Week 9
Concepts and Skills: Fractions (Rational Numbers), Rational Expressions, Reducing, Mixed Numbers and Improper fractions, Operations on Fractions and Rational Expressions, Operations on Mixed Numbers
X 
Y 
3 
2 
6 

9 

12 
(a) What fraction is boiling water?
(b) How many gallons of tap water is used?
Team 1 
Team 2 

Won 
16 
20 
Lost 
20 
24 
Tapes 
Profit ($) 
0 
2500 
1 
3  2500 
1000 
3*1000  2500 = 
x 
Year 
1990 
1991 
1992 

# of years 
0 
1 
2 
x 
Value of the car 
16950 
Week 10
Concepts and skills: Fractional Equations, Cross Multiplication, Ratio and Proportion, Word Problems on Fractions
(a) Complete the following payment schedule for the Johnson family.
Payment # 
Unpaid Balance 
Finance Charge 
Amount of Balance paid 
Installment 
1 

2 

3 

4 
(b) How much more does the Johnson family pay on the easy payment plan than it would have if it paid the entire $3500 at once?
Social choice 
global 
Total 

Initial Investment ($) 
x 
15,000 

Rate (%) 

Increase ($) 
4,100 
This ratio is called P/E ratio. Which of the above chains has a higher P/E ratio? If you are investing in one of the two chains which chain would you prefer?
Month 
12/94 
1/95 
2/95 
3/95 
4/95 
5/95 
6/95 
7/95 
8/95 
9/95 
10/95 
11/95 
12/95 
Approx. Rate 

Change over the previous month. 

Finance charge for $500. 
One staircase with each step 1 ft wide and 9 inch high has been selected.
Run a sale 
Advertise More 
Door Prizes 
Increased Profits 
Increased Profits / Expenses 
200,000 
xxxxxxxxxxxx 
10,000 
$1,000,000 

xxxxxxxxx 
500,000 
10,000 
$1,000,000 

200,000 
500,000 
xxxxxxxxxx 
$2,500,000 
The table as given above is used in the "Design of Experiments", which is a part of the subject of Statistics.
Week 11
Concepts and skills: Pythagorean Theorem, Denominate Numbers
x 
1 
4 
9 
16 
25 
36 
49 
64 
81 
100 
y = Ö x 
2 
3 
5^{2 } = (5 + 2)(5  2) + 2^{2} 7^{2} = (7 + 3)(7  3) + 3^{2} 46^{2} = (46 + 4)(46  4) + 4^{2} 
79^{2 } = (79 + 1)(  ) + ^{2} 93^{2} = (93 + )(93  ) + 7 ^{2} 115^{2} = (115 + _)(  5_ ) + ^{2} 
a 
5 
7 
9 
13 

b 
24 
60 
84 

c 
13 
41 
61 
They fix a marker in the ocean and at a distance of two kilometers on either side of the mountain and they start digging at 45^{0} maintaining the same angle. How long is the tunnel?
(Comment: Based on Heron’s story. Source: Morris Kline, Mathematics and the Physical World, Page 95.)
Week 12
Concepts and skills: Word problems involving fractions, equations, formulas, fractional equations, Ratio and Proportion, Denominate numbers
Date computed 
7/1/1995 
8/1/1995 
1/1/1996 
6/30/1996 
$ in Millions 
28700 
29400 
(a) Complete the following table.
Water 
Ammonia 
Total Volume 

Original volume 
8 quarts 

Original percentage 
% 
30% 
100% 
Volume of ammonia solution to be added 
1 quart 

percentage 
% 
% 
100% 
Volume after adding ammonia solution i.e. New volume 
9 quarts 

New percentage 
% 
% 
100% 
Step 1: Create scaled down copies of L_{0} each L_{1} having length 1/3 of L_{0}. Call the new figure L_{1}.
Step 2: Create scaled down copies of L_{1} each having length 1/3 of a segment in L_{1} What is the length of each segment?
(b) If this construction is continued, and L_{3} is created, predict the length of L_{3}. Is it greater than L_{2}?
Notice that the length of L_{3} is the greatest of the three sets. One can continue these steps. The curve we obtain is called the Koch Curve. Such curves can be found in nature like a jagged edge of an ocean beach. The length of such a curve becomes infinite.
(Source: COMAP, Lexington, MA 02173, Consortium Newsletter Number 45, Spring 1993, p. 6)
(a) Complete the following table
Miles 
0 
1/5 
2/5 
4/5 
1 
2 
3 

Fare 
(b) Draw a line graph using the table in (a). A part of the graph is already drawn .
(c) If the cab fare for a trip was $12.50, how many miles did the cab cover? (correct to 1/5 mile.)
(a) Complete the table.
Shirts 
Trousers 
Ties 
Socks 
Total 

# items 
2 
2 
4 
5 pairs 
xxxx 
Price tag per item 
23 
29 
7.50 
3.50 
xxxx 
% discount 
30% 
40% 
25% 
25% 
xxxx 
Savings 

Sale price 
(Source: WAL*MART circular)
(Note: Such a graph is called a demand curve in economics.)
Week 13
Review
1. A chef went on a diet. The first week the chef lost six pounds. During the five subsequent weeks, he lost two pounds each week. The chef weighed 250 pounds the day he went on a diet.
Week 
1 
2 
3 
4 
5 
6 
Total Pounds Lost 

Chef’s weight 
2. A company's net sales from Jan. 1, 1990  Dec. 31, 1994 are described by the following chart.
Using the chart answer the following questions.
3. Match with each table in (a) through (d), the appropriate graph in Ga  Gd.
(a)
x 
0 
2 
3 

x 
y 
0 
4 
6 
8 
2x 
(b)
x 
0 
1 

4 
x 
y 
0 
1 
9 
16 
x^{2} 
(c)
x 
0 
1 
2 
3 
x 
y 
1 
3 
12 
24 
3(2)^{x} 
(d)
x 
0 
2 
3 
4 
x 
y 
2 
0 
1 
2 
x  2 
4. Ruben invests $5,000 in two types of stock. He invests $2,000 in a growth stock and the rest in technology stock. The growth stock went up by 18% and technology stock went down by 5%. How much is Ruben's investment now? (Hint: Completing the following table will be helpful.
Growth 
Technology 

Initial Investment 
2,000 


Rate 

Change ($) 
Total Investment ¯ 

Current value ($) 
5. A 1lb bottle of mixed nuts contains 1/2 lb. peanuts, 1/3 lb. cashews, and remaining part other nuts. At the same rate, how many pounds of nuts other than peanuts and cashews are contained in a case of 24 such bottles.
6. 1 gallon = 3.78 L, how many mL is 1 quart? (Use: 1 L = 1000 mL)
7. A salesperson makes $475 and 10% commission for sales over $500. The following table describes her salary. Find her salary if she sells merchandise worth x dollars.
Sales ($) 
0  500 
1000 
3000 
x 
Salary ($) 
475 
475 + .10(1000500) 
475 + .10(3000  500) 
8. The following graph describes time and the corresponding distance from the ground of an orange tossed in the air.
The next two questions are based on the chart given which describes the net sales of a company.
9. What was the average increase per year from 1990  1994?
Bibliography
In addition, several Annual Reports of Corporations, Newspaper cuttings and Circulars were used.
Cooperative Learning:
Appendix 1 AMATYC STANDARDS
Revised Final Draft  February 1995 (Page 18)
GUIDELINES FOR CONTENT 

Increased Attention 
Decreased Attention 
pattern recognition, drawing inferences* 
rote application of formulas 
number sense, mental arithmetic, and estimation* 
arithmetic drill exercises, routine operations in whole and real numbers 
connections between mathematics and other disciplines 
presentation of mathematics as an abstract entity 
integration of topics throughout the curriculum* 
algebra, trigonometry, analytic geometry, etc., as separate courses 
discovery of geometrical relationships through the use of models, technology, and manipulatives* 
establishing geometric relationships solely through formal proofs 
visual representation of probability as area under a curve and via probability trees; of timelines for annuities and interest; of logic and electrical circuits 
rote memorization and use of probability formulas 
integration of the concept of function across topics within and among courses

separate and unconnected units on linear, quadratic, polynomial, radical, exponential, and logarithmic functions 
analysis of the general behavior of a variety of functions in order to check the reasonableness of graphs produced by graphing utilities 
paperandpencil evaluation of functions and handdrawn graphs based on plotting points 
connection of functional behavior (such as where a function increases, decreases, achieves a maximum and/or minimum, or changes concavity) to the situation modeled by the function 
emphasis on the manipulation of complicated radical expressions, factoring, rational expressions, logarithms, and exponents with little obvious relevance to functional behavior 
connections among a problem situation, its model as a function in symbolic form, and the graph of that function* 
"cookbook" problem solving without connections 
modeling problems with experimental and theoretical probability, including estimations based on simulations* 
theoretical development of probability theorems 
* Points applicable to Prealgebra
Revised Final Draft February 1995 (Page 19)
GUIDELINES FOR CONTENT (continued) 

Increased Attention 
Decreased Attention 
collection of real data for use in both descriptive and inferential statistical techniques* 
analysis of contrived or trivial data 
exploratory graphical analysis as part of inferential procedures* 
"cookbook" approaches to applying statistical computations and tests which fail to focus on the logic behind the processes 
use of curve fittingto model real data, including transformation of data when needed 
reliance on outofcontext functions that are overly simplistic 
discussion of the fact that nonzero correlation does not imply one variable causes another 
blind acceptance of r 
use of statistical software and graphing calculators 
paperandpencil calculations and four function calculators 
problems related to the ordinary lives of students, e.g., financing items that students can afford and statistics related to sports participated in by females as well as by males* 
problems unrelated to the daily lives of most students, e.g., investments of large sums of money in savings or statistics related to sports only played by males 
matrices to organize and analyze information from a wide variety of settings 
requiring a system of equations to be solved by three methods 
graph theory and algorithms as a means of solving problems* 
algebraically derived exact answers 
* Points applicable to Prealgebra
Revised Final Draft  February 1995 (Page 20)
GUIDELINES FOR PEDAGOGY 

Increased Use 
Decreased Use 
active involvement of students 
passive listening 
technology to aid in concept development 
paperandpencil drill 
problem solving and multistep problems 
onestep single answer problems 
mathematical reasoning 
memorization of facts and procedures 
conceptual understanding 
rote manipulation 
realistic problems encountered by adults 
contrived exercises 
an integrated curriculum with ideas developed in context 
isolated topic approach 
multiple approaches to problem solving 
requiring a particular method for solving a problem 
diverse and frequent assessment both in class and outside of class 
tests and a final exam as the sole assessment 
openended problems 
problems with only one possible answer 
oral and written communication to explain solutions 
requiring only short, numerical answers, or multiple choice responses 
variety of teaching strategies 
lecturing 
Summary
These standards provide a new vision for introductory college mathematics—a vision whereby students develop intellectually by learning central mathematical concepts in settings that employ a rich variety of instructional strategies. To provide a more concrete illustration of these standards the Appendix contains a set of problems that brings them to life.
Appendix 2 Detailed Syllabus for MTH 002  PREALGEBRA
Week 
Topic/s 
1 
Whole Numbers: The Place Value System of Whole Numbers, Arithmetic of Whole Numbers, Distributive Property. Bar and Line graphs Problems for Classwork 1, Lab 1 
2 
Fractions (Brief Review); Decimals: Place Value System, Comparison of Decimals, Rounding and Estimation, Addition and Subtraction of Decimals Multiplication and Division of Decimals, Linear Measurement Problems for Classwork 2, Lab 2 
3 
Definition of Percent, Changing Percent to Decimals or Fractions, Changing a Fraction or Decimal to a Percent Finding a given percent of a given number, and finding percent when a part and whole are given, without using equations. Selected Applications of Percent, Word Problems Problems for Classwork 3, Lab 3 
4 
Exponents, Order of Operations, Variable Expressions, Geometric Formulas, Perimeter, Area, Volume Problems for Classwork 4, Lab 4 Test 1 
5 
Introduction to Signed Numbers, Addition, Subtraction and Multiplication of Signed Numbers Introduction to graphing  Instructor’s Notes Problems for Classwork 5, Lab 5 
6 
Division of Signed Numbers, Combining Like Terms, Laws of Exponents Problems for Classwork 6, Lab 6 
7 
Addition Property of Equality, Multiplication Property of Equality, Equations combining both properties, Formulas and Word Problems Problems for Classwork 7, Lab 7 Test 2 
8 
Word Problems using equations, Transposition (Optional), Word Problems on Percent using equations, Interest formula Problems for Classwork 8, Lab 8 
9, 10 
Fractions (Rational Numbers), Graphing Rational Numbers, Rational Expressions, Reducing, Mixed Numbers and Improper Fractions, Operations on Fractions and Rational Expressions, Comparison of Fractions, Operations on Mixed Numbers Problems for Classwork 9, Lab 9, Classwork 10, Lab 10 Test 3 
11 
Fractional Equations, Cross Multiplication, Ratio and Rates, Proportion, Miscellaneous Applications of Percent  Mixture Problems Problems for Classwork 11, Lab 11 
12 
Square roots, Pythagorean Theorem, Denominate Numbers Problems for Classwork 12, Lab 12 Test 4 
13 
Word Problems  Review Problems for Classwork 13, Lab 13 
Final Examination  comprehensive 