## Supply and Demand

• Lesson
9-12
2

In this grades 9–12 activity, students write and solve a system of linear equations in a real-world setting. Students should be familiar with finding linear equations from 2 points or from the slope and y-intercept. Graphing calculators are not necessary for this activity, but could be used to extend the ideas found on the second activity sheet. Parts of this lesson plan were adapted from the October 1991 edition of Mathematics Teacher.

This activity focuses on analyzing supply-and-demand problems from business by solving systems of equations and finding the equations for lines.

### Prerequisites

It is assumed that students are familiar with:

• How to find a linear equation from a graph from 2 points or from the slope and the y‑intercept
• How to solve systems of 2 equations in 2 unknowns.

### Definitions

You may wish to acquaint students with the following concepts from economics at the beginning of the activity session.

Supply

• Supply can be defined as the quantity of the item that is in stock.
• Price is a factor in the supply of an item. When the price of an item is low, more people buy the item, and the supply (stock on the shelves) decreases. When the price of an item is high, the supply remains high because fewer people buy the item, leaving more inventory on the shelves.
Demand
• Demand can be defined as the quantity of merchandise the consuming public wishes to buy.
• Price affects demand. A lower price tends to increase the demand because people may feel that the item is a bargain, while a higher price tends to decrease the demand.
Supply vs. Demand
• When supply is greater than demand, the merchant suffers. The merchant has a stockpile of merchandise that is not making any money for the business.
• When demand is greater than supply, the merchant also suffers. Customers are coming to buy an item that may be sold out, and an opportunity to make a sale is missed.
Equilibrium
• The merchant is best served when supply and demand are in equilibrium. Equilibrium occurs when the price is set so that the supply and demand equal each other. The item completely sells out, but no one who wants to purchase the item goes home without it.

### Graphing Supply and Demand Against the Price

Once students have an idea of how the price of an item can affect sales, they have a context in which to understand the graphs offered on the following activity sheets.

Activity: Class Fundraiser

Having a transparency of Activity Sheet: Senior Class Buttons makes it easier for you to guide students into a discussion about supply and demand.

Students may not initially recognize what to do with 3 columns of data.

Question 1: Coach students to use the price column for x‑values and the 'supply' column for y‑values.

Question 2: Coach students to use the price column for x‑values and the demand column for y‑values.

Allow sufficient time for students to plot points and complete the graphs. If possible, check students' graphs. Students can work individually or in pairs to answer Questions 3–8. Point out that they have used 3 different ways to display and interpret the information given in this button example. The data were presented to them in a table. They used the data to construct a graph that allowed them to understand more about the problem. Then they wrote and solved a system of equations that represented the same information in yet another form.

Activity: Game Cartridges and Silver Dollars

The 2 scenarios on this activity sheet can be assigned for homework or used as additional class activities.

After working through the button example from Activity Sheet 1 and the game cartridge example on Activity Sheet 2, students may get too comfortable with values that predict a straight line. Point out to students that this result rarely occurs in real-world situations.

The second problem on Activity Sheet 2, about silver dollars,  is a good example of realistic data. Even though the supply-and-demand functions are not linear and no formulas are given to represent the functions, students can draw conclusions when they study and interpret the graphs. Point out to students that many times a line or a formula is used that is a good approximation based on the given data.

Assessment Options

1. Provide additional questions involving systems of equation questions.
2. Ask student groups to develop their own scenarios for a system of equations. You can provide the equations, or ask the groups to find their own.
3. Have students propose an item for sale. Survey students to see how many would purchase the item if it were offered at several prices and then use the data to determine the best price and the number of items that should be made.

Extensions

1. Introduce students to systems of inequalities
2. Have students work with systems of equations with 3 variables and 3 unknowns
3. Show students how to use a calculator as a tool to solve the system

Questions for Students

1. Explain the different ways the data from each problem can be represented.
2. Which representation do you prefer? Why?
3. Which representation is best for getting accurate values? Explain why.
4. Explain how it is possible to look at 3 categories of data on a 2 dimensional graph.
5. Which representation is best for seeing the trends of the data? Explain why.

Teacher Reflection

• Which method(s) did students prefer?
• How well were students able to communicate the “why” for their preference?
• Would it suffice for students to use their method of choice on assessments? Why or why no

### Learning Objectives

By the end of this lesson, students will:
• Explain factors that affect supply and demand.
• Find linear equations for given sets of supply and demand data.
• Find the equilibrium point for a system of supply and demand equations.
• Translate between table, graph, and equation representations for supply and demand data.

### NCTM Standards and Expectations

• Write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases.
• Use symbolic algebra to represent and explain mathematical relationships.
• Draw reasonable conclusions about a situation being modeled.