NC2766 Math&Econ

Mathematics and Economics: Connections for Life © National Council on Economic Education, New York, NY 1

Lesson

The Nature of Demand

Mathematics Focus:

Algebra 1

Mathematics Prerequisites:

Prior to this lesson, students should be able to:

➤

Make a graph using a table of values

➤

Find slope and vertical intercept

➤

Write the equation of a line given

a) a point and a slope

b) two points on the line

c) a slope and one intercept

➤

Define the dependent and independent variables

Lesson Objectives:

To apply prerequisite mathematics concepts and processes to:

➤

explain why a demand curve has a negative slope

➤

explain reasons why the law of demand exists

➤

explain why quantity demanded depends on price

Overview of Mathematics and Economics:

This lesson develops the economic tool of demand. Demand is deter-

mined by the value that people attach to a product (a good or a service).

A demand curve is a graph with a negative slope that lies in the first quad-

rant. It illustrates the inverse relationship between the quantity of a prod-

uct demanded and the price that people are willing and able to pay. A

demand curve is most accurately derived from a demand schedule, which

is a table of values relating prices to quantities demanded. To make this

lesson appropriate for algebra I students, it will be restricted to the study

of linear demand curves.

A change in the quantity demanded of a product is caused by a

change in the product’s price. The demand curve shows the extent to

which a change in price causes a change in quantity demanded. A

change in quantity demanded can be viewed graphically as a movement

along the demand curve. Other factors that influence peoples’ desire to

buy the product—such as changes in income, tastes, or the prices of

substitutes or complements—cause a change in demand as opposed to

a change in quantity demanded. A change in demand leads to a new

demand curve that is parallel to the original curve. This phenomenon is

referred to as a “shift in demand.” (See Lesson 4 for further discussion

of this concept.)

The demand curve is a graphical representation of the law of demand.

The law of demand states that holding everything else constant, there is an

inverse relationship between price and quantity demanded. As prices

increase, quantity demanded decreases. Some people buy less of an item,

thus decreasing the total quantity demanded. As prices decrease, quantity

demanded increases because people are willing and able to buy more of

the item.

Mathematics and Economics Terms

➤

Demand curve

➤

Demand schedule

➤

Dependent variable

➤

Independent variable

➤

Inverse relationship

➤

Law of demand

➤

Quantity demanded

Materials

➤

Straight edge

➤

Overhead transparencies of Visuals 1.1, 1.2, 1.3, and 1.4 and Activity 1.2

➤

One copy of Activity 1.2 and 1.3 for each student

➤

Make one copy of Activity 1.1 on colored paper. Cut the activity slips

on the solid lines and place one slip for each student in the class in a

paper bag. The following instruction is important:

To ensure that a linear function is produced, leave out the slips that rep-

resent the lower prices if you have fewer students than there are slips of

paper. In addition, be sure that you use an even number of students. This

paper bag will be used in step 3.

Estimated Time

80–100 Minutes

Procedures

1. Warm-Up Activity: Have the students answer the questions in Visual

1.1. Discuss the answers with the class.

■

COMMENT When discussing the answers to the problems, emphasize

the following points:

2 Mathematics and Economics: Connections for Life © National Council on Economic Education, New York, NY

Lesson 1 The Nature of Demand

➤

In mathematics, the horizontal axis is used for the independent

variable and the vertical axis is used for the dependent variable.

➤

Since the relationship is only valid for positive numbers of lunch

charges and a positive balance remaining, the line should be graphed

in the first quadrant only.

➤

Meaningful variables are used. The variables l (quantity of lunches

charged) and r (remaining balance) are substituted for x and y, respec-

tively, in the equation

y = mx + b.

Answers to the questions include:

a) r = –2l + 50

b) independent—quantity of lunches charged; dependent—remaining

balance.

c) See Visual 1.2.

d) m = –2 and b = 50

e) Nine lunches.

2. Tell the students that today they will be looking at the market for the

latest popular CD. Ask the students, “What CD would you most like to

buy?” In this activity, they should all imagine that they are studying the

market for this CD. They will be looking at the relationship between the

number of CDs that people are willing and able to purchase and CD prices.

3. Distribute a copy of Activity 1.2 to each student. Then have each stu-

dent take a slip of paper at random from the paper bag that you have pre-

pared in Activity 1.1. Tell the students that the number on each slip rep-

resents the maximum price that each student is willing and able to pay for

the CD. Tell the students, “The number of CDs [or other product] that buy-

ers are willing and able to buy at each and every price is known as the

quantity demanded.”

4. Tell the students that they will now complete a table of values called a

demand schedule. Have the students look at their slips of paper. Ask the

students to raise their hands if they are willing and able to pay $32 for the

CD. No one should raise his or her hand, so enter a 0 for quantity demand-

ed in your table of values on the projected overhead. Have the students

put a 0 in the column “Quantity Demanded of CD” on their copy of Activ-

ity 1.2. Ask the students, “In real life, why is no one willing and able to pay

$32 for this CD?” Answers will vary but be sure that the reason that the

“price is too high” is given. Then ask, “Who is willing and able to pay $30

for the CD?” Two people should raise their hands so write “2” under

quantity demanded to correspond with the price of $30. Next ask, “Who

is willing and able to pay $28 for the CD?” Four people should raise their

hands: the people with $28 on their slips as well as the students with $30

Mathematics and Economics: Connections for Life © National Council on Economic Education, New York, NY 3

Mathematics and Economics: Connections for Life

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on their slips. Ask the students, “Why would the individuals who would

pay $30 also be willing and able to buy the CD for $28?” For each price

level, two additional students should raise their hands. Continue this exer-

cise until the table of values is completed and all hands are raised. Work

together with the class to fill in the Quantity Demanded column in the

table of values in Activity 1.2. As the students fill in the table on their indi-

vidual sheets, you should complete the table of values on the overhead

projector.

Answers are given in Visual 1.3.

Tell the students, “In economics, this table of values is called a demand

schedule.” Have the students look at the table of values and explain why

the quantity demanded decreases as the price increases. Also ask why

the quantity demanded increases as the price decreases.

■

COMMENT As the price increases, fewer people are willing and able

to purchase the item. As the price decreases, more people are willing and

able to purchase the item. For example, more people buy items when

they are on sale than at a higher regular price.

5. Ask the students to identify the dependent and independent variable

in the price/quantity demanded relationship. Remind the students that in

mathematics, the independent variable is usually placed on the horizontal

axis and the dependent variable on the vertical axis. Economists do the

reverse, placing the independent variable (price) on the vertical axis and

the dependent variable (quantity demanded) on the horizontal axis. Thus

each ordered pair is in the form:

(dependent variable, independent variable) =

(quantity demanded, price).

Students should then complete the ordered pair column of the demand

schedule.

Answers are given in Visual 1.3.

6. In Part 2 of Activity 1.2 have the students graph the line from the table

of values in Part I. Remind them to label the axes with the appropriate

variable. Tell the students that in economics, the line that they have

graphed is called a demand curve. Economists and mathematicians often

call a linear graph a curve. Have the students label this curve, “Demand.”

Use Visual 1.4 to demonstrate how the graph should appear.

■

COMMENT Tell the students that all demand curves are not linear and

even though this curve is a straight line, it is referred to as a “curve.”

7. Ask the students to record the characteristics of the graph and the rea-

son why the graph has each characteristic. You may need to use leading

4 Mathematics and Economics: Connections for Life © National Council on Economic Education, New York, NY

Lesson 1 The Nature of Demand

questions to help your students identify all of these characteristics. Dis-

cuss the students’ answers as a class.

The answers should include:

a) The graph is linear because for each $1 increase in price, the quantity-

demanded decreases by one.

b) The line has a negative slope because as the price decreases, the

quantity demanded increases. Or as the price increases, the quantity

demanded decreases.

c) The vertical intercept is $32, or the line intersects the vertical axis at

P = $32. This is because no one is willing and able to buy the CD for a

price of $32.

d) The line should not intersect the horizontal axis because an infinite

number of people would want the CD for a price of zero dollars.

e) The graph lies entirely in the first quadrant because the quantity

demanded and the price are never negative.

8. Tell the students,

“In mathematics, the slope of the line is typically viewed as

change in dependent variable

change in independent variable.

Because economists put the independent variable on the vertical axis and

the dependent variable on the horizontal axis, we must refer to slope as

change in the variable on the vertical axis

change in the variable on the horizontal axis.

Therefore, in this example, the slope is

change in price

change in quantity demanded.

The students should now calculate the slope. Remind the students that

the negative slope means that there is an inverse relationship between

price and quantity demanded. The

law of demand says, holding every-

thing else constant, when price rises, the amount that people are willing

and able to buy falls, and when price falls, the amount that people are will-

ing and able to buy rises.

■

COMMENT The slope is

Mathematics and Economics: Connections for Life © National Council on Economic Education, New York, NY 5

Mathematics and Economics: Connections for Life

◆

”

$1

________ _____

2 CDs 1 CD

9. For Part 3 of Activity 1.2, divide the students into three groups and

have each group find the equation of the demand curve using one of the

following methods (assign a different method to each group):

Group 1: Point and slope method

Group 2: Two points method

Group 3: Slope and vertical intercept method

■

COMMENT The equation of the demand curve is P = 32 – Q

. When

the students have found and compared their equations for the demand

curve, regroup the students so that in the new groupings each method is

represented. Students should share their methods with other members

of the group. Have the students record each method on their copy of

Activity 1.2. (This method is commonly called the “jigsaw method” in

cooperative learning.)

EXTENSIONS

1. Activity 1.3 can be used as an additional assignment.

■

COMMENT

Answers to Activity 1.3

1. Figure A

2. A demand schedule shows the quantity demanded of an item at dif-

ferent prices.

3. Price and quantity demanded

4. Quantity demanded is the dependent variable because it is affected by

price, which is the independent variable.

5. The

law of demand states that, holding everything else constant,

there is an inverse relationship between price and quantity demanded.

That is, as price increases, the quantity demanded decreases. As price

decreases, the quantity demanded increases.

6. P = 70



⁄6Q

or Q

= 6P + 420. Students might write the equation

either way. Economists usually write the equation as Q

= 6P + 420.

Economists refer to the equation, P = 70 –

⁄6 Q

, as the “inverse

demand curve.”

7. P = 69 –

⁄3Q

or Q

= 1.5P + 103.5. Students might write the equa-

tion either way. Economists usually write the equation as

= -1.5P + 103.5

Economists refer to the equation, P = 69 –

⁄3Q

, as the “inverse

demand curve.”

8. Same answer as problem 7.

6 Mathematics and Economics: Connections for Life © National Council on Economic Education, New York, NY

Lesson 1 The Nature of Demand

◆

Mathematics and Economics: Connections for Life © National Council on Economic Education, New York, NY 7

Mathematics and Economics: Connections for Life

9. (a)

Price in cents Quantity Demanded

(Number of People)

75¢ 1

70¢ 3

65¢ 6

60¢ 21

55¢ 33

50¢ 33

45¢ 38

40¢ 38

35¢ 46

9. (b and c)

(d) By using the ordered pairs (1,75) and (46,35) to establish our straight

line, the slope of the demand curve is



⁄9. If we assume that no stu-

dent is willing to pay more than 75 cents for a can of soft drink, then

the vertical intercept is approximately 76 cents (75

⁄9).

(e) Since quantity demanded is the dependent variable, economists write

the equation of the line of best fit for the demand curve as Q

= 

⁄8P

+ 85.5. Some students might write the equation for the “inverse

demand curve” which is P=



⁄9 Q

+ 76.

0 4 8 12162024283236404448

QUANTITY

PRICE (IN CENTS)

Straight line

approximation

8 Mathematics and Economics: Connections for Life © National Council on Economic Education, New York, NY

VISUAL 1.1 ▲ Warm-Up

A student opens a school lunch account with an opening balance of $50. Lunch

costs $2.00 a day, and the student can have the cost charged against the

account each day.

ANSWER THE FOLLOWING QUESTIONS:

a) Write the equation for the line representing the balance remaining in the school lunch

account in terms of the number of lunches charged.

b) What are your independent and dependent variables?

c) Sketch the graph for 0 to 25 lunches.

QUANTITY OF LUNCHES CHARGED

d) What is the slope of the line? (Remember: Slope is the rate of change.) What is the vertical

intercept of the line?

e) If the remaining balance is $32, how many lunches have been charged?

REMAINING BALANCE

Mathematics and Economics: Connections for Life © National Council on Economic Education, New York, NY 9

VISUAL 1.2 ▲ Schedule from Warm-Up

0 2 4 6 8 101214161820222426

QUANTITY OF LUNCHES CHARGED

REMAINING BALANCE

VISUAL 1.3 ▲ Demand Schedule

32 0 (0,32)

30 2 (2,30)

28 4 (4,28)

26 6 (6,26)

24 8 (8,24)

22 10 (10,22)

20 12 (12,20)

18 14 (14,18)

16 16 (16,16)

14 18 (18,14)

12 20 (20,12)

10 22 (22,10)

8 24 (24,8)

6 26 (26,6)

4 28 (28,4)

2 30 (30,2)

Price of CD in $

(Independent Variable)

Quantity Demanded of CD

(Dependent Variable)

(Dependent Independent

Variable, Variable)

VISUAL 1.4 ▲ Demand Curve for CDs

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

QUANTITY OF CDS DEMANDED

PRICE OF CD

Demand

ACTIVITY 1.1

Reproduce this page on colored paper, cut out the slips, and place them in a

paper bag. Students will randomly select slips for an activity to create a

simulated demand schedule.