## Amazing Profit

- Lesson

Students use equations to determine eBay profit on new technology. EBay is an online auction agency. For a limited time after a “new” product’s street release date, it is possible to track the profit that sellers make for auctioning them on eBay. Students use previous data of selling prices to derive a linear equation for the “closing bid price” on a product.

This lesson is designed for students who already know how to determine linear equations from a coordinate graph. Its purpose is to illustrate an application of understanding relationships and trends.

To begin the lesson, ask students questions such as:

- What is eBay?
- What is an auction?
- What is a bid?
- What is supply and demand?
- What is the relationship between supply and demand?
- Is there any way you can think of that relationship that may help people make money on eBay?

Allow students to discuss their thoughts and share any personal experiences they may have. The starter questions are a lead-in for the activity sheet. This will allow you to assess students’ current experience and understanding of supply and demand. Clarify any misconceptions of these issues.

Display the Amazing Profit Overhead. Hide the equations on the bottom, and give students time to figure out the equations. Review the answers at the bottom and provided additional examples of problems relating to linear equations if necessary.

The Amazing Profit Activity Sheet explores approximate data from 2006, as collected by a middle school teacher. It has been manipulated to create a linear relationship. You may wish to discuss this with students after the activity. During the lesson, focus on interpreting data.

Read through the introduction together. Then, allow students to complete the activity individually or in pairs, helping them as needed.

### Answers

**1.** PS3 Seller’s Profit: $4,900, $4,400, $3,650, $2,900, $1,900, $1,150

**2.**

**3. ***y* = –250*x* + 5500

**4. ***y* = –250*x* + 4900

**5. ***x*-values represent days the PS3 is on sale on eBay. Negative *x*-values are not important, because they represent time before the PS3 went on sale.

**6.*** y*-values represent money paid for the PS3. Negative *y*-values for profit represent a loss for the seller. Negative *y*-values for the Mean Closing Bid line would represent a seller paying a buyer to take the unit off their hands.

**7.**The lines are parallel, because the profit line will always be $600 less than the closing bid. $600 is the price of the original unit.

**8.**The lines have negative slope. That means the longer PS3s are selling, the less profit the sellers will get, and the less money the buyers will bid for PS3s.

**9.** Predictions should be between the 19th and 20th day.

**10**

a.If the sellers buy PS3s for $600, the lowest price they could set for a profit would be $601.b.Substituting $601 foryin the Mean Closing Bid equation from Question 3 gives a result of approximately 19.6 days.

c.The Mean Closing Bid line is just above thex-axis because there is almost no profit.

**11.** Student answers will vary. Some student responses may be: “Most people who wanted a PS3 could have gotten one,” “Most people can’t afford the extremely high price so they would wait to get one at a store,” “More PS3s make it to the stores so less people want to get one on eBay.”

- Markers or colored pencils
- Rulers
- Amazing Profit Overhead
- Amazing Profit Activity Sheet

**Assessment Options**

1. Other products have faced similar phenomena on eBay. Complete the activity sheet with these data points:

Day Mean Closing Bid on eBay Profit on a $400 Xbox 360 0 $2,000 4 $1,600 7 $1,300 12 $800

Day Mean Closing Bid on eBay Profit on a $250 Wii 0 $1,000 3 $850 6 $600 10 $400 2. Have students answer the initial questions in a journal, and informally check it to assess students' current experience and understanding of supply and demand.

**Extensions**

1. Use the data table below to draw a line of best fit. The students will find that the data does not precisely lie upon a single line, but there is a line that passes very closely to each of the data points. Discuss line of best fit and other types of regression.

Day Mean Closing Bid on eBay Profit on a $200 iPhone 0 $1,280 2 $1,160 5 $1,100 9 $900 2. Have students research other eBay trends and identify another similar phenomena that occurred and provide evidence as a comparison to the other examples.

**Questions for Students**

1. Can you think of a situation where there might be a positive slope for profit on items selling on eBay?

[Answers will vary.]

2. What other kinds of items might sell for high prices in the future?

[Answers will vary.]

3. How fast do you think the lines of profit for other items would come down?

[Answers will vary.]

4. Do you think the profit and closing bids will always have an approximately linear relationship?

[Answers will vary.]

**Teacher Reflection**

- What do you see as the next steps for strengthening your students’ mastery with interpreting graphs?
- Were there any questions a majority of your students struggled with overall? in a certain class?
- Was students’ level of enthusiasm/involvement high or low? What types of products do you think could be used to spark greater enthusiasm in the future?
- Did you challenge the achievers? How?
- Was your lesson appropriately adapted for the diverse learner?
- Was your lesson developmentally appropriate? If not, what was inappropriate? What would you do to change it?
- How did your lesson address auditory learning styles?
- What were some of the ways that the students illustrated that they were actively engaged in the learning process?
- What worked with classroom behavior management? What didn't work? How would you change what didn’t work?

### Learning Objectives

Students will:

- Determine ta linear equation for selling price of and profit from “new” technology on eBay.
- Interpret a graph of the constructed equations.
- Analyze relationships and trends from multiple representations of data.

### NCTM Standards and Expectations

- Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.

- Use graphs to analyze the nature of changes in quantities in linear relationships.

- Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.

- Make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit.

### Common Core State Standards – Mathematics

Grade 6, Expression/Equation

- CCSS.Math.Content.6.EE.A.3

Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

Grade 6, Expression/Equation

- CCSS.Math.Content.6.EE.B.6

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

Grade 6, Expression/Equation

- CCSS.Math.Content.6.EE.C.9

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

Grade 7, Expression/Equation

- CCSS.Math.Content.7.EE.A.2

Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that ''increase by 5%'' is the same as ''multiply by 1.05.''

Grade 8, Expression/Equation

- CCSS.Math.Content.8.EE.B.5

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

Grade 8, Expression/Equation

- CCSS.Math.Content.8.EE.B.6

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Grade 8, Functions

- CCSS.Math.Content.8.F.B.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Grade 8, Functions

- CCSS.Math.Content.8.F.A.3

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s^2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP1

Make sense of problems and persevere in solving them.

- CCSS.Math.Practice.MP4

Model with mathematics.

- CCSS.Math.Practice.MP5

Use appropriate tools strategically.

- CCSS.Math.Practice.MP7

Look for and make use of structure.