Productivity and Graphing Linear and Quadratic Functions

In this lesson students interpret key features of graphs for both linear and quadratic functions in the context of total and marginal production. The lesson begins with a short video about a young entrepreneur who designed his own line of bowties. Students then predict the relationship between number of workers and production of bowties. Students test their predictions by participating in a production activity making paper bowties. Next, they sketch graphs of their total and marginal product, and describe the key features of their graphs. The lesson closes with students graphing two datasets and deciding which dataset most realistically describes the relationship between number of workers and production.

Approximately 90 minutes (two class periods steps 1–14 on day one, 15–20 on day two.)

• Define the marginal physical product and the law of diminishing marginal returns.
• Explain why the relationship between labor and output is quadratic (all else constant) and not linear.
• Explain the features of a linear function and a quadratic function given a table of data.
• Sketch a graph of a quadratic function and of a linear function given a description of the function.

• Activity 1: Prediction Graph, one copy for each group.
• Activity 2: Bow Tie Template, 10 copies for each group.
• Activity 3: Production Recording Sheet, one copy for each group.
  Answer Key Activity 3, one copy for the teacher
• Activity 4: Features of a Production Graph, one copy for each group.
  Answer Key Activity 4, one copy for the teacher
• Activity 5: Production Data: Linear or Quadratic, one copy for each student.
  Answer Key Activity 5, one copy for the teacher
• Stapler or roll of tape, one per group.
• Colored pens or markers, two per group.
• Scissors, one pair per group.

• “Success Stories: Mo’s Bows ” video clip from ABC’s Shark Tank Season 6, Week 20.

Figure 1: Graph of Marginal Product for a Manufacturing Firm

1. Which of the following statements best describes the graph in Figure 1 above?
   1. The domain is x ≥ 0.
   2. The graph represents a constant growth rate.
   3. The graph represents a linear function.
   4. The function is decreasing after x = 5. 
2. Which of the following statements best describes the range of the graph in Figure 1 above?
   1. The range includes negative values because adding workers eventually results in less total output.
   2. The range includes negative numbers because some workers are less productive than others.
   3. The range includes only positive numbers because you cannot make “negative” amounts of product.
   4. The range includes only positive numbers because it represents number of workers.
3. Which of the following statements best explains why a graph of marginal physical product is not typically linear?
   1. Some workers work harder than others.
   2. Not all workers work the same number of hours.
   3. The quantity of other resources, like equipment, is held constant.
   4. The quality of other resources, like equipment, often varies.
4. Jeremiah owns a bakery. He currently has three employees and is able to produce 48 dozen cupcakes per day. He would like to double his daily production so he plans on hiring three more workers. What additional data would we want in order to evaluate Jeremiah’s plan to double production?Sample answer: in order to evaluate Jeremiah’s plan to double production, we would need to know the marginal product of each additional worker. It is not enough to know total production in order to predict the effect of additional workers. While production may double, the increase in production is not linear; it is more likely quadratic. For Jeremiah this means that doubling workers from three to six may result in less than doubling of output. The reason for this is that increasing one input, in this case workers, while holding the other inputs constant, in this case ovens and other baking equipment, will eventually result in decreasing marginal product. The available space in the ovens, or number of mixers may limit production.

20. Ask students:
    1. What is the relationship between the number of workers and amount of a good or service produced? (This depends on whether or not you are considering total product or marginal product. In general we expect that as the number of workers increases, we would see the total product increase at a decreasing rate. If we consider the marginal product, we expect the marginal product to increase at a decreasing rate, eventually decrease at an increasing rate, and become negative. This is due to the law of diminishing marginal returns: as you increase one input while holding the other inputs constant, the marginal product will eventually decrease.)
    2. What is the definition of marginal product? (Marginal product is the additional quantity produced when the use of a resource increases by one unit while all other resources are held constant.)
    3. What is the law of diminishing marginal returns? (Describes a phenomenon observed in all short-run production processes, when at least one input (usually capital) is fixed. As more and more units of a variable input (usually labor) are added to the fixed input, the additional (marginal) output associated with each increase in units of the variable input will eventually decline. In other words, successive increases in a variable factor of production added to fixed factors of production will result in smaller increases in output.)

    Why is the relationship between labor and marginal output quadratic (all else constant) and not linear? (The law of diminishing marginal returns states that as one input is increased while all other inputs are held constant, the additional output per increase in units of input will eventually decrease. We can explain this by imagining more and more workers trying to use the same amount of capital resources, like scissors or table space. Eventually each worker will be less productive than the previous worker, and as they get in each other’s way, may actually result in negative marginal product.)

In this lesson students interpret key features of graphs for both linear and quadratic functions in the context of total and marginal production. The lesson begins with a short video about a young entrepreneur who designed his own line of bowties. Students then predict the relationship between number of workers and production of bowties. Students test their predictions by participating in a production activity making paper bowties. Next, they sketch graphs of their total and marginal product, and describe the key features of their graphs. The lesson closes with students graphing two datasets and deciding which dataset most realistically describes the relationship between number of workers and production.

1. Show students the “Success Stories: Mo’s Bows ” clip from ABC’s Shark Tank Season 6, Week 20.
2. Ask students:
   1. Where was Mo producing his bow ties before the factory? (In his home)
   2. Why did Mo want to move production to a manufacturer like Robert Stuart Inc.? (He wanted to increase production of his bow ties.)
   3. What do you predict is the relationship between number of workers in the manufacturing facility and the number of bowties produced? (Answers will vary, but will probably include an increase in the number of bow ties produced for every new worker. Accept all answers without judgment, but ask students to provide reasons for their predictions.)
3. Divide students into small groups (4-6 students).
4. Hand out Activity 1: Prediction Graph, one per group.
5. Tell students:
   • Create a graph that shows what you predict to be the relationship between number of workers and number of bow ties produced. Remember, the input variable is the number of workers (x-axis) and the output variable is the number of bow ties (y-axis).
   • Be sure to label your graph and be prepared to describe the features of your graph to the class. For example, describe what the graph looks like and how the shape of the graph represents the relationship between number of workers and bow ties produced.
6. After groups have completed their graphs, ask a representative from each group to describe their group’s graph to the class. (Responses will vary. As this is the activating portion of the lesson, allow students to give their responses without prompting them to use vocabulary from the standards.)
7. Tell students:
   • You are now going to have an opportunity to test your predictions about the relationship between number of workers and number of bow ties produced. Since you don’t have sewing machines in class, you will be making paper bow ties.
   • You will work in groups. Each group will represent a different bow tie manufacturer. I will give each group materials and supplies: one pair of scissors, two markers (or colored pens), tape (or stapler), Activity 2: Bow Tie Template (ten copies per group), and Activity 3: Production Recording Sheet (one copy per group).
   • During the first round, each group will select one group member to be a worker in the factory. The worker will have two and a half minutes to complete as many bow ties as possible. Each bow tie must be neat and complete in order to be counted. Remember, we would be selling these in Neiman Marcus!
   • Each group will have a manager who will count the number of bow ties produced and check for quality. (Depending on group size, this person can also record the total produced in each round or that job can be given to a different group member.) I (teacher) will be the timekeeper. Production will start when I say “go,” and stop at the end of one minute when I say “stop.”
   • In round two, each group will add a second worker. The two workers will use the same materials and supplies to create as many bow ties as they can in two and a half minutes. In round three, each group will add a third worker. In round four, each group will add a fourth worker.
8. Hand out supplies and materials to each group. Depending on how students are grouped, you will need to restrict the production space. Usually the space of one or two desks (pushed together) makes an adequate production space. The goal is to limit the space to help students discover that the size of the manufacturing facility (among other factors) affects their ability to produce. Therefore, it is important to make sure students do not work on the floor or at another desk other than the one specified as the production space.
9. Ask students to look at Activity 2 and follow along as you demonstrate how to make one bow tie.
10. Start the four 2.5-minute production rounds, making sure that production starts and stops on time and making sure that the quality control and counting of bow ties after each round is done correctly. Ask students to fill in the data for their production round on Activity 3, Columns 2 and 3. Note: the bow ties should be counted after each round and then collected so that only bow ties made in that round are counted (not cumulative production). Below is a table with sample data for the production activity:

11. Ask students what the word marginal means. (Answers will vary.) Tell students
    economists use the word marginal to mean “additional.” Specifically, when economists are thinking about production of goods or services, they are interested in “marginal physical product,” or the additional quantity produced when the use of a resource increases by one unit while all other resources are held constant.
12. Instruct students to complete Column 4 by recording the additional or marginal bow ties produced in rounds two, three, and four and Column 5 by recording the total number of bow ties produced in all rounds. See sample table.
13. Have students analyze their data. Ask them:
    1. What happened to the number of bow ties produced as you added workers? (Answers will vary. Some groups may comment that the total amount of bow ties increased after each round. Other groups might comment that the number of bow ties produced rose in each round but the additional bow ties produced fell in subsequent rounds, and so on.)
    2. What was the resource that increased by one unit in our bow tie manufacturing? (Workers.)
    3. What were the resources that were held constant? (Scissors, markers, tape/staplers, work space.)
    4. Why would an entrepreneur like Mo care about marginal physical product? (Answers will vary but may include that entrepreneurs could use the information to decide the best number of employees to hire based on a given amount of capital resources.)
    5. Why would an entrepreneur like Mo care about the total product produced? (Answers will vary but may include that entrepreneurs may want to know how much is being produced in a given day so they can compare it to the total costs of the day.)
14. Tell students:
    Working in groups, create two graphs, one graph of the relationship between total product and number of workers and a second graph of the relationship between marginal product and the number of workers. You will create these graphs on the back of Activity 3: Recording Sheet.After you complete your graphs, be prepared to describe your graphs to the class. To help you describe your graphs, answer the questions on Activity 4: Features of Graphs.
15. After students have completed Activity 4, have groups present their graphs to the class. This can be done using a document camera or by a paper pass strategy. Alternatively students can create graphs on poster paper.
16. Tell students to look at their prediction graphs and think about the similarities and differences between their prediction and the data they collected from the production activity.
17. Ask students:
    1. What features of your graphs are similar? (Answers will vary.)
    2. What features of your graphs are different? (Answers will vary.)
    3. What was accurate about your original prediction? (Answers will vary.)
    4. What parts of your prediction do you need to revise? (Answers will vary.)
    5. When you graphed your prediction graph, did you graph total production or marginal product? (Answers will vary, but it is likely that students graphed total production. If they graphed total production, make sure to compare the prediction graph to the total production graphs from the activity).
    6. Was your prediction linear or quadratic? How do you know? (Answers will vary for the first part. It is linear if there is a constant change in output per increase in unit of input; for every additional worker, for example, the number of bow ties increases by the same amount. It is exponential (quadratic in this case) if it increases at first and then decreases. While we don’t expect total production to decrease, we do expect the marginal product to decrease.)
    7. Was your marginal production graph linear or quadratic? How do you know? (Answers may vary, but graphs should be close to quadratic. Increases as a decreasing rate may have even started decreasing.)
    8. Why is the marginal product graph not linear? (Answers will vary, but students will probably comment that they only had one pair of scissors, or one stapler, or a fixed space to work in, or some people were faster workers than others, etc. Guide students in their answers toward the definition of marginal physical product and towards an understanding that even though the number of workers increased, the materials available to use were held constant. If we increase the number of workers but don’t give them additional tools, at first they will be able to be more productive, probably because they could divide the labor (i.e. one person did the cutting, a second the folding and stapling and so on), but at some point the workers start getting in each other’s way or there are not enough resources for each person to be as productive as the person before them.)
    9. Tell students that they have just demonstrated the law of diminishing marginal returns. This law describes a phenomenon observed in all short-run production processes, when at least one input (usually capital) is fixed. As more and more units of a variable input (usually labor) are added to the fixed input, the additional (marginal) output associated with each increase in units of the variable input will eventually decline. In other words, successive increases in a variable factor of production added to fixed factors of production will result in smaller increases in output.
    10. What was accurate about your original prediction? (Answers will vary.)
    11. What parts of your prediction do you need to revise? (Answers will vary, but may include what was wrong about their original predictions. Encourage students to explain why their prediction was incorrect and how what they have learned has changed their thinking.)
18. Hand out Activity 5: Production Data, Is it Linear or Quadratic?
19. Tell students to complete Activity 5 working in small groups. This activity provides students with two datasets. One is linear and one is cubic, resulting in a constant and quadratic marginal product respectively. After students calculate the marginal product, they graph the marginal product and describe the features of the graph. Then students are asked to decide which of the two graphs is a more realistic representation of production in the real world and explain their reasoning.