Grade 4: Mathematics and Economics DECISION‐MAKING UNIT MN Academic Standards in Social Studies (Economics): 4.2.1.1.1 “Apply a reasoned decision‐making process to make a choice.” 4.2.3.5.1 “Describe a market as any place or manner in which buyers and sellers interact to make exchanges; describe prices as payments of money for items exchanged in markets.” 1 2 Sessions 1 and 2 PACED Decision‐making Process/Grid This lesson introduces the basic five steps of a reasoned decision‐making process known as PACED. Students vote on possible field trip locations and use the results to create fractions. MN Academic Standards in Mathematics: 4.1.2.3 “Use fraction models to add and subtract fractions with like denominators in real‐world and mathematical situations. Develop a rule for addition and subtraction of fractions with like denominators.” 4.4.1.1 “Use tables, bar graphs, timelines and Venn diagrams to display data sets. The data may include fractions or decimals. Understand that spreadsheet tables and graphs can be used to display data.” Materials  “Field Trip Options” visual  “PACED Decision‐making Steps” visual  “PACED Decision‐making Grid” visual Session 1 a. Begin the session by explaining that individuals must make personal decisions every day. Ask the students what decisions they made before coming to school today. (What to have for breakfast, what to bring for lunch, what to wear, etc.) Were these decisions difficult or easy? (Answers will vary.) b. Tell them that groups must also make choices. Families decide where to go on vacation or which movie to see. Schools decide on playground equipment or which sports to offer. Ask students for additional examples. c. Present the following scenario: Let’s pretend our class is going on a field trip. The school will pay for the trip and gave us a list of possible places from which to choose. Our task is to decide where we want to go. We can choose one of the following locations: zoo, planetarium, science museum, fire station and police station, or the children’s theater. d. Display the “Field Trip Options” visual. Ask students to discuss what they know about each location. (Answers will vary but may include they have been to some of the sites with family. Ask them to explain what they liked or did not like about the location. They may have wanted to visit some of them but haven’t had the chance. Ask them to tell you why they are interested in going to a particular location.) 3 e. Ask: What are some things we need to consider to help us make our decision? (Examples: activity level‐‐will we be in one place the entire time or will we be able to move around; variety‐‐ are there lots of things to see or people to interact with; educational value‐‐does it relate to something we studied in class; distance from school‐‐will travel require too much of the time allotted for the field trip, and others suggested by the students.) f. Ask the students to think about which option they prefer and then vote for their favorite. Survey each student individually and place a tally mark on the visual to record each student’s vote. g. Discuss the responses using the following questions: Which option received the most votes? Which option received the second‐most votes, or came in second? Third? Fourth? Fifth? h. Explain to the students that they will use the voting results to create fractions that represent the opinions of the class about where to go on the field trip. Remind the students that a fraction is a part of a whole and that each fraction has a numerator, or the number above the line, that represents the number of equal parts of a whole being considered, and a denominator, or the number below the line, which represents the total number of equal parts needed to make up the whole. The total number of votes for each option will be the numerator. The total number of students in the class who voted will be the denominator of each fraction. h. Ask the students to figure out the fraction that represents the number of students who voted for the top choice. Record this on the “Field Trip Options” visual. (The number of tally marks beside the top choice will be the numerator and the total number of students in the class will be the denominator.) Continue by asking students to determine the fractions that represent each of the other options and record these on the visual. Session 2 a. Display “Field Trip Options” visual with tally marks and fractions from previous day. Remind the students that the class was discussing where to go on a field trip. b. Tell them that today they are going to learn how to make choices using a special guide. Display “PACED Decision‐making Steps” visual. Explain that the word PACED is an easy way to remember the steps in this method of decision‐making because each letter of the word stands for a step in the process: P is the PROBLEM or the reason why you must make a decision. A stands for ALTERNATIVES or the possible options you have to choose from. C stands for CRITERIA or things you can use to compare the desirability of the options. E stands for EVALUATION or judging how well each option meets each criterion. D stands for the final DECISION you make. 4 c. Display “PACED Decision‐making Grid” visual. Ask: What is our problem? Write this on the appropriate line of the visual. (The class must decide where to go on a field trip. This is a problem because they would probably like to go on several of the field trips but can only go on one.) What are our alternatives? (All of the options the school gave us for the trip: zoo, planetarium, science museum, fire station and police station, and the children’s theater). Tell the students the class will only consider the top 3 choices as determined by the vote in the previous session. Write these in the cells of the first column titled “Alternatives.” d. Review the discussion from the previous session about considerations or criteria that would be useful to compare the various alternatives. (See examples in Session 1, step c.) e. Guide the students to select three most important criteria to them and write these in the cells of the row just below “Criteria.” (For example: variety, educational value, travel time) f. Have the students working together discuss and evaluate the selected alternatives using the criteria they selected. Beginning with the first alternative, discuss whether or not it meets the first criteria using a “+” (positive) sign if it meets the criteria or a “‐” (negative) sign if it does not. (Suppose the first alternative is the zoo and the first criteria is activity level, since students will be moving from animal to animal most of the time, it would get a “+” sign in the cell. Suppose the second criteria is educational value and the class has just finished a unit about animals, it would again receive a “+.” However, if the third criterion is travel time and the zoo is a long distance from the school it would receive a “‐” sign in the third cell.) g. Continue evaluating each alternative in this manner. Once the grid is completed, have students count the number of “+” signs and the number of “‐” signs received by each alternative. Have students determine the “net value” which equals the number of positive signs minus the number of negative signs and write this in the last column under “Net Value.” (Using the example in f. above, the net value for the zoo would be “+ 2.”) h. Tell the students that the alternative that received the greatest net value would be the logical choice for the class trip. If two or more alternatives receive the same positive net value, then the students could put the choice to another class vote. 5 Field Trip Options Votes Fraction Zoo Planetarium Science Museum Fire and Police Stations Children’s Theater 6 PACED DECISION‐MAKING STEPS 1. P = Problem you are facing 2. A = Alternatives or options that are possible 3. C = Criteria or ways that options can be compared 4. E = Evaluation or judging the alternatives 5. D = Decision you make after your evaluation 7 PACED Decision‐Making Grid Problem: ____________________________________________ Criteria Net Value Alternatives 8 Sessions 3, 4 and 5 A Day at the Amusement Park This lesson takes students through the five Steps of the PACED process with respect to deciding what to do at an amusement park (Session 3 looks at Steps 1 and 2, Session 4 looks at Steps 3 and 4, and Session 5 looks at Step 5). Students create their own PACED decision‐making grid using addition and multiplication to make a choice. MN Academic Standards in Mathematics: 4.1.1.5 “Solve multi‐step real‐world and mathematical problems requiring the use of addition, subtraction and multiplication of multi‐digit whole numbers. Use various strategies, including the relationship between operations, the use of technology, and the context of the problem to assess the reasonableness of results.” 4.2.1.1 “Create and use input‐output rules involving addition, subtraction, multiplication and division to solve problems in various contexts. Record the inputs and outputs in a chart or table.” 4.4.1.1 “Use tables, bar graphs, timelines and Venn diagrams to display data sets. The data may include fractions or decimals. Understand that spreadsheet tables and graphs can be used to display data.” Materials  A copy of “A Day at the Amusement Park” for each student  A copy of “Decision‐making Grid” for each student Session 3 The Problem and the Alternatives a. Describe the following to students: Suppose you are at an amusement park with the following attractions and ticket prices: Rollercoaster (4 tickets), Bumper Cars (3 tickets), Water Slide (2 tickets), and Ring Toss (1 ticket). You have 6 tickets to spend. b. Ask: Why is this a problem? (You probably don’t have enough tickets to do everything you would like to do and so you have to make a choice.) Economists call this particular problem “scarcity.” Scarcity is not having enough of something to satisfy all of your wants. In this case, it is a scarcity of tickets. Ask: What other things are scarce in your life? (money to buy all the things you would like, time to do all the things you want, space in your backpack for all your book, supplies, lunch, etc.) c. Continue: So having only 6 tickets at the amusement park means you must make a choice. What does the PACED process tell you the next step should be? (List all the possible alternatives or options). 9 d. Distribute a copy of “A Day at the Amusement Park” to each student. e. Have students identify all the possible ways to spend 6 tickets and list them in Column A of the handout using “R” for Rollercoaster, “B” for Bumper cars, “W” for Water Slide, and “T” for Ring Toss. (There are nine possibilities: RW, RTT, BB, BWT, BTTTT, WWW, WWTT, WTTTT, TTTTTT.) f. Have students verify that each of these is possible by showing through addition (in Column B) and multiplication (in Column C) strategies that each alternative costs exactly six tickets. (For example, the cost of RTT can be expressed as 4 + 1 + 1 =6 or 4 + (2 x 1) = 6 tickets.) Session 4 The Criteria and the Evaluation a. Distribute a copy of “Decision‐making Grid” to each student. Have students write “ALTS” for alternatives in the box above the first column of the grid and then transfer the list of alternatives from the previous session’s worksheet down the first column of their grid. b. Ask students to give reasons why they would rank or rate one alternative as better than another. Note that the alternatives are not the individual attractions, but different combinations of the attractions. (Examples: number of different attractions in the alternative, control over what happens with each attraction in the alternative, shortness of wait time, ability to win prizes, amount of fun the attractions in the alternative are, and others suggested by the students.) c. Explain that these are called criteria and provide ways to compare the various alternatives. Have students write “Criteria” in the uppermost box of the grid. d. Discuss the criteria given in step b. above. Have the class select four to list as the criteria in the top row of their grid below “Criteria.” e. Have each student working individually determine how important each of the four criterion is to them personally by weighting each of them from 0 to 3 with 0 representing “not important at all”, 1 “somewhat important”, 2 “important”, and 3 “very important.” Have them record these “weights” in the cells below each criterion. (Their answers will vary, but must be 0, 1, 2, or 3 for each criterion.) f. Have students working individually rank how well they believe each listed alternative satisfies or meets each criterion using the following scale: 3 for “very well”, 2 for “well”, 1 for “not very well” and 0 for “not at all.” Have them record these values in the lower half of each cross‐
hatched cell of the grid. Have students report, discuss, and, if necessary, adjust their evaluation. (Their answers will vary, but must be 0, 1, 2, or 3.) 10 Session 5 The Decision a. Have each student multiply the evaluation in the lower half of each cell by the criterion weight for that column and record the answer in the upper half of each cross‐hatched cell. b. Have each student add up all the upper‐half cell values for each alternative and record their answer in the last column. The alternative with the highest total would likely be the best choice for them. c. Have students report their choices and discuss why they may be different. (Different weights for the criterion and different evaluations of the alternatives.) d. Conclude by explaining that the PACED decision‐making process and grid is not about determining the one choice that everyone should make, but is instead about making the best choice for one person based on what is possible (the alternatives) and what is important to them (the criteria). Session 6 Catch‐up, debrief, and review sessions in DECISION‐MAKING UNIT as necessary. 11 A Day at the Amusement Park
You are at an amusement park with the following attractions and ticket prices:
Roller Coaster 4 Tickets Bumper Cars 3 Tickets Water Slide 2 Tickets Ring Toss 1 Ticket You only have 6 tickets to spend.
What are all the possible ways to spend all of these 6 tickets? Complete Column A
in the table below using R for roller coaster, B for bumper cars, W for water slide
and T for ring toss.
Complete Column B and Column C to verify that each combination is possible by
showing through addition and multiplication strategies that each alternative costs
exactly six tickets.
Column A Combinations Column B
Addition Column C Multiplication 12 Decision‐making Grid 13 14 Grade 4: Mathematics and Economics PERSONAL FINANCE/BUDGETING UNIT MN K‐12 Academic Standards in Social Studies (Economics): 4.2.1.1.1 “Apply a reasoned decision‐making process to make a choice.” 15 16 Session 1 How Big is the Schoolyard? This lesson uses manipulatives to determine areas for various recreational uses of schoolyard land. MN K‐12 Academic Standards in Mathematics: 4.1.1.2 “Use an understanding of place value to multiply a number by 10, 100 and 1000.” 4.3.2.3 “Understand that the area of a two‐dimensional figure can be found by counting the total number of same size square units that cover a shape without gaps or overlaps. Justify why length and width are multiplied to find the area of a rectangle by breaking the rectangle into one unit by one unit squares and viewing these as grouped into rows and columns.” 4.3.2.4 “Find the areas of geometric figures and real‐world objects that can be divided into rectangular shapes. Use square units to label area measurements.” Materials  A copy of each of the following for each group (preferably on different‐colored paper): “Schoolyard Scarcity” “Schoolyard Scarcity: Fields” “Schoolyard Scarcity: Hockey Rinks” “Schoolyard Scarcity: Playground Areas” “Schoolyard Scarcity: Courts”  Scissors a. Distribute a copy “Schoolyard Scarcity” and scissors to groups of 2‐5 students. Explain that the school is trying to decide how to use the limited recreational space it has around its parking lot. There are four possible uses: fields (which could be used for baseball or softball), hockey rinks (which could be used as soccer fields in the summer), playground areas, and courts (which could be used for basketball or tennis). b. Distribute one sheet of each of these possibilities (“Schoolyard Scarcity: Fields,” “Schoolyard Scarcity: Hockey Rinks,” “Schoolyard Scarcity: Playground Areas,” and “Schoolyard Scarcity: Courts) to each group and have them cut out the individual items. (For example, there are four fields on the “Schoolyard Scarcity: Fields” sheet.) c. Tell the class: Suppose you know that the width and length of each Playground Area is 30 meters. Ask: What would the area of each Playground Area be? (900 meters) 17 d. Given this information, have each group use their Playground Area cut‐outs to determine each of the following:  The area of a Hockey Rink. (Two Playground Areas cover it exactly, so 900 + 900 = 1800 square meters, or 900 x 2 = 1800 square meters.)  The area of a Field. (Four Playground Areas cover it exactly, so 900 + 900 + 900 + 900 = 3600 square meters, or 900 x 4 = 3600 square meters.)  The area of a Court. (It takes two Courts to exactly cover one Playground Area so, a Court is one‐half the size of a Playground Area, or 900 x ½ = 450 square meters.)  The area of the Schoolyard (not including the Parking Lot). (It takes 12 Playground Areas to exactly cover the Schoolyard, so 12 x 900 = 10,800 square meters; alternatively, since it takes three Fields to exactly cover the Schoolyard and since we know from above that one Field is 3600 square feet, then 3 x 3600 = 10,800 square meters.) e. Have groups report other methods for determining the area of the Schoolyard using the various cut‐outs they have. (Since the entire Schoolyard could be covered by one Field, two Hockey Rinks, and four Playground Areas, the area would be (1 x 3600) + (2 x 1800) + (4 x 900) = 3600 + 3600 + 3600 = 10,800 square meters.) f. Explain to the class that 30 meters is approximately 100 feet because 1 meter is approximately 3.28 feet, so 30 meters = 3.28 x 30 = 98.4 feet ≈ 100 feet. Given this, have each group estimate the above areas in square feet.  The area of a Playground Area: 100 x 100 = 10,000 square feet  The area of a Hockey Rink: 2 Playground Areas = 2 x 10,000 = 20,000 square feet  The area of a Field: 4 Playground Areas = 4 x 10,000 = 40,000 square feet  The area of a Court: ½ Playground Area = ½ x 10,000 = 5,000 square feet  The area of the Schoolyard: 12 Playground Areas = 12 x 10,000 = 120,000 square feet 18 Schoolyard Scarcity Schoolyard
19 Parking Lot Schoolyard Scarcity: Fields 20 Schoolyard Scarcity: Hockey Rinks 21 Schoolyard Scarcity: Playground Areas 22 Schoolyard Scarcity: Courts 23 24 Sessions 2 and 3 Schoolyard Choices This lesson introduces the concept of opportunity cost and then looks at the possible ways (alternatives) to use the Schoolyard area. MN Academic Standards in Mathematics: 4.1.2.1 “Represent equivalent fractions using fraction models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives. Use the models to determine equivalent fractions.” Materials  Same as in Session 1 (except no further need for scissors)  Tape or glue (in Session 3) Session 2 a. Using the same groups and materials as in Session 1, have groups discover possible alternative ways to use the schoolyard by placing various combinations of the uses on the schoolyard diagram. (For example, if F is for Field, H is for Hockey Rink, C is for Courts, and P is for Playground Areas, then there are many alternatives including FFF, FFHH, FFHPP, FFHPPPCC, and FHHPPPCC or alternatives such as 12 Playground Areas and 24 Courts.) b. Ask: If you decide to have a Field, what must you give up in terms of: Hockey Rinks? (2 Hockey Rinks) Playground Areas? (4 Playground Areas) Courts? (8 Courts) Ask: If you decide to have a Court, what must you give up in terms of: Fields? (1/8th of a Field) Hockey Rinks? (1/4th of a Hockey Rink) Playground Areas? (1/2 of a Playground Area) c. Explain that when choosing one alternative, you usually must give up the opportunity to have something else. The next best alternative not chosen is called the opportunity cost of the choice. It can be thought of as the “opportunity lost.” d. Have students provide simple examples of the opportunity cost of choices they have made. (Examples: the opportunity cost of buying a $40 video game would be the next most valuable goods you could have purchased with the $40, perhaps a ticket to a concert; the opportunity cost of playing an hour with your friend would be the next best way you could have spent your time, perhaps reading a book; the opportunity cost of eating several slices of pizza would be the next best thing you could have eaten, perhaps hot dogs.) 25 Session 3 a. Using the same groups and materials as in Session 1, have each group discuss some possible criterion for ranking the various recreational uses of the schoolyard. Have them report these to the class for discussion. (Some possibilities are: number of kids that could use it; number of kids that would want to use it; amount of use over the year round; number of different uses; degree to which the use develops good physical skills; age‐appropriateness of the use; etc.) b. Have each group evaluate the possible alternatives with respect to the criteria they feel are most important and decide on how the schoolyard should be used. c. Have them glue or tape their chosen uses onto the “Schoolyard Scarcity” handout. d. Ask each group to present its results and discuss differences. (Groups may weigh each criterion differently or evaluate how well a given alternative satisfies a given criterion differently. Groups may internally disagree as to what is best since this is not an individual choice, but a group, or social, choice.) 26 Session 4 Deciding How to Spend Money This lesson uses the same proportional relationships that existed in the schoolyard problem and uses it to show students the decision to use or budget their money is very similar to the decision on how to use the schoolyard. MN Academic Standards in Mathematics: 4.1.2.1 “Represent equivalent fractions using fraction models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives. Use the models to determine equivalent fractions.” Materials:  “Scarcity of Money” visual a. Display the “Scarcity of Money” visual. b. Describe the following situation: Suppose you had $24 to spend and four different ways you can spend it: 3D movie tickets for $8 each, T‐shirts with a cool logo for $4 each, bottles of lemonade for $2 each, or $1 for $1 worth of savings towards buying a bicycle, a coat, or other item. You can buy as much of any of these items as you want, but you cannot spend more than $24. c. What is the problem here? (You have lots of goods you may want, but only have $24 to spend.) How is this problem like the schoolyard problem? (In the case of the schoolyard there was a limited amount of space but lots of different uses that were desired. In this case there is a limited amount of money but lots of different items that are desired.) d. Have students determine the fraction of their money ($24) represented by each good’s price. (3D movie ticket: $8/$24 = 1/3; T‐shirt: $4/$24 = 1/6; Lemonade: $2/$24 = 1/12; and Saving $1/$24 = 1/24.) e. Have students determine the fraction of the schoolyard space represented by each schoolyard use. (Field, 1/3; Hockey Rink, 1/6; Playground Area, 1/12; Court, 1/24.) f. Ask: What do you notice about these fractions? (A 3D movie ticket takes the same fraction or proportion of the income that a Field takes of the schoolyard space. Similarly, a T‐shirt takes the same fraction or proportion of the income that a Hockey Rink takes of the schoolyard space, etc.) g. Ask: What does this imply about the possible alternatives uses of the $24? (They would be the same as for the schoolyard problem with 3D movie replacing Field, T‐shirt replacing Hockey Rink, Lemonade replacing Playground Area; and Saving replacing Court.) 27 h. Explain that this problem and the possible alternatives (ways to spend the money) are thus very similar to the schoolyard problem. Ask: How is this problem different from the schoolyard problem? (The criteria used to judge the alternatives would be different and this would be an individual choice as opposed to a group or social choice.). i. Explain that budgeting a limited amount of money is much like deciding how to use limited schoolyard space. So, the PACED process could be used here as well. j. Have students think about what criteria they would use in the budgeting problem and use this to evaluate the possible alternatives and make a decision on how to spend the $24. g. Have students report on their criteria and choices. Session 5 Catch‐up, debrief, and review sessions in PERSONAL FINANCE/BUDGETING UNIT as necessary. 28 Scarcity of Money $8 3D movie ticket
$4 T‐shirt with cool logo
$2 Bottle of lemonade $1 $1 savings How would you spend $24?
29 30 Grade 4: Mathematics and Economics BUSINESS/PRODUCTION UNIT MN Academic Standards in Social Studies (Economics): 4.2.3.3.1 “Define the productivity of a resource and describe ways to increase it.” 31 32 Sessions 1 ‐ 5 The Packet Factory This lesson has students play the role of workers in packet factories, create production data based on their experience, measure productivity and analyze production. MN Academic Standards in Mathematics: 4.1.1.5 Solve multi‐step real‐world and mathematical problems requiring the use of addition, subtraction and multiplication of multi‐digit whole numbers. Use various strategies, including the relationship between operations, the use of technology, and the context of the problem to assess the reasonableness of results.” 4.2.1.1 “Create and use input‐output rules involving addition, subtraction, multiplication and division to solve problems in various contexts. Record the inputs and outputs in a chart or table.” 4.2.2.1 “Understand how to interpret number sentences involving multiplication, division and unknowns. Use real‐world situations involving multiplication or division to represent number sentences.” 4.4.1.1 “Use tables, bar graphs, timelines and Venn diagrams to display data sets. The data may include fractions or decimals. Understand that spreadsheet tables and graphs can be used to display data.” Materials:  One copy of “Production Data” for each student (plus a visual if desired)  Resources needed per packet factory: Cleared‐off table or desk One fully‐loaded stapler 10 copies of each of the four, different pattern sheets cut apart (40 total sheets) Session 1 Productivity a. Explain that the production of goods and services requires the use of natural resources (raw materials‐‐wood, minerals, water, etc.‐‐and energy‐‐oil, coal, and natural gas), human resources (workers), and capital resources (tools, equipment, machines, and buildings). Have students provide examples of resources needed to:  Grow corn. (Land/soil, seeds, water, farmer, tractor, fuel, combine, etc.)  Make a car. (Iron ore, copper, rubber, autoworkers, designers, assembly line, factory, electricity, etc.)  Provide education. (Crayons, pencils, teachers, principal, desks, chairs, bulletin boards, power for lights and heating, etc.) 33 b. Define the productivity of a resource as the amount of output it can produce divided by the number of resource inputs used during a specific time period. For example, if 36 chairs can be produced by 4 workers in a day, then the productivity of each worker is 36/4 or 9 chairs per worker per day. (Note that this is just the average or mean output produced by each unit of resource during the time period.) c. Have students calculate the productivity of each of the following resources:  5 barbers give 20 haircuts in an hour. (20/5 = 4 haircuts per barber per hour)  10 acres grow 1000 tons of potatoes a year. (1000/10 = 100 tons per acre per year)  6 machines stamp out 240 widgets in a day. (240/6 = 40 widgets per machine per day) d. Have each student create a new productivity example such as those in c. Explain that each example should include three things: (1) the amount of a resource used, (2) the amount of output produced by those resources, and (3) the amount of time it took the resources to produce the output. e. Have selected students read their examples. Have the class identify the resource used, the output produced, the time or length of the production period and then use this information to determine the productivity of the resource for that time period. (Some examples may not be correct…have students help adjust them so that they involve a resource and what that resource produces. Also, remind students as necessary that productivity is determined by dividing the amount of output produced by the amount of resources used in a given time period.) Session 2 The Packet Factory a. Tell students that the class is going to produce packets of production questions. Each packet consists of four different questions on different‐patterned pieces of paper that are stapled together in the top left‐hand corner. Demonstrate the production of a packet by first laying the four patterned pieces of paper on a desk (show the students the different patterns: triangles, circles, rectangles, and stars), then collate them, straighten them, and staple them at a 45 degree angle in the top left‐hand corner. Show students your finished packet, noting that the packet has exactly four pages and that they are all different. b. Ask: What resources are used in the production of this packet and what type of resource are they—natural (raw material), human, or capital? (Students usually quickly identify the paper as a raw material used, the teacher as a worker, and the stapler as a capital resource, but there is also the desk (another capital resource) and the staple that ends up as part of the packet (another raw material). They might also mention the space it takes to do everything. This would be part of the school building which would be a capital resource.) 34 c. Ask: What is the mathematical relationship between the number of pieces of paper used and the number of packets produced? (It would be given by the rule that the number of pieces of paper is four times the number of packets.) d. Explain that the class is now going to investigate the relationship between the number of workers and the number of packets produced. e. Ask: How many packets would be produced if there were no workers? (0, zero) f. Ask: How do you think the number of packets would change as the number of workers is increased? (It would also likely increase.) g. Have each student write down how many packets they believe could be produced by one, two, and three workers if they only had one desk, one stapler, and 45 seconds to work each time. Have students report their estimates. (Answers will vary, but it is likely they will believe that as the number of workers increases, the number of packets produced will increase proportionately, that is, if there are twice as many workers, twice as many packets will be produced.) Tell them the actual amounts will be determined in the next session. Session 3 The Production Simulation (Set‐up: Create factories—each factory should include a table/desk just far enough away from the wall that students can stand between the table/desk and the wall. On each table place four stacks of the different‐patterned paper (40 pieces in each stack) and a fully‐loaded stapler. Create at least two factories, but no more than the class‐size divided by five. Note that the more factories created, the more students that can directly participate, however, the greater the amount of resources required.) a. Distribute a copy of “Production Data” to each student. Explain that this worksheet will be used to record the number of packets produced by each factory with a given number of workers, the total production of packets by all factories, and the average production of each factory for each number of workers. Select five students to be workers in each factory that you have created. Have all students record a “0” in the first line of Column A and “0’s” under each factory in Column B based on their answer to this question in Session 2 (if there are zero workers, then zero packets would be produced). b. Assign one student at each factory the role of factory manager. The manager will control the number of workers at his/her factory according to the teacher’s instructions, make sure that packets are produced as shown earlier, and record his/her factory’s output after each round. The remaining four students will be workers at the factory. They will be added one at a time (one in the first round, two in the second round, three in the third round, and four in the last round). Finally, assign each factory a number from 1 up to 5. 35 c. Have one worker from each factory “enter” their factory. Explain that they are to produce as many packets as they can from when the teacher says “Go” to when the teacher says “Stop.” The time period will be 45 seconds. Tell managers to make sure their workers do not start early or work after time has been called. Remind students that they must produce quality packets or no potential customers would buy them. d. Say “Go” and allow 45 seconds for the students to work. After 45 seconds, say “Stop,” and have each manager count and record the results of his/her factory on the “Production Data” sheet (record a “1” in Column A and the output of his/her factory under Column B). e. Have each manager add another worker to his/her factory and repeat the above process recording after each production round. Continue until production has been completed with four workers in each factory. Session 4 Data Construction a. Have all factory managers report their results and have all students record this information on their “Production Data” sheet so that everyone has the output of their factory each round as well as the output of the other factories from each round. b. Have students use data from Columns A and B on the “Production Data” worksheet to complete Column C. (This is just the sum of the packets produced by all factories in that round.. c. Have students complete Column D by dividing the total production in each round (Column C) by the total number of workers used by all factories in that round (the number of workers at each factory times the number of factories). (For example, if there are 3 factories and when they each had 2 workers they produced 7, 9, and 8 packets respectively, then the total production would be 30 packets and the total number of workers would be 6, 2 workers x 3 factories, so in Column D the “average productivity” would be 5 = 30/6). d. Have students fill in the bottom row (“Total Production by Each Factory”) by summing the production of each factory over the four rounds. e. Have students determine the total production by all factories during all rounds by summing all the numbers in the bottom row of the table or by summing all the numbers in Column C. Have them place this total in the bottom cell of Column C. 36 Session 5 Data Analysis a. Have students take out their “Production Data” sheets completed in Session 4 (this may also be shown as a visual). b. Distribute a packet produced in Session 3 to each student. Explain that each page of the packet contains a question related to their “Production Data” sheet. Have students working individually or in pairs (or in their factory groups) answer each of the questions based on their observation/analysis of the data on the sheet. Tell them to write their answers on the back of each page and provide a reason why they think what they are reporting happened (these can be turned in for assessment if desired). c. Allow students 10 minutes to answer the four questions. d. Discuss the questions and their answers as follows: Triangles: Did all factories produce the same number of packets when they had the same number of workers? (Looking across at least one row where there is the same amount of workers, it is very likely that all the factories did not produce the same number of packets even though they had exactly the same amount of workers (and other resources). This may be the result of several different things: difference in the skills of the workers, difference in the motivation of the workers, problems during the production round (such as a jammed stapler), better organization of the production process itself, etc.) Circles: As more workers were added to each factory, what happened to the total number of packets produced? (Looking at Column C, the total number of packets produced should be rising even though it is possible for an individual factory to have experienced a problem so that its production stayed the same or even dropped in one production round. With more resources, there should be more output.) Rectangles: As more workers were added to each factory what happened to the average number of packets produced? (The average is the number of packets produced per worker (defined earlier as “productivity”). Looking at Column D, the average number of packets is likely falling as the number of workers is increased. This is related to what economists call “diminishing returns.” This is due in large part to the fact that while the number of workers was increased in each round, the amount of the capital resources (stapler, desk, and space) was not. Thus, each time a worker was added, the amount of capital available per worker was falling causing their productivity to fall.) Stars: Construct two math sentences to determine the number the total number of packets produced by all factories? (1. Sum the total amount produced by all factories in each round‐‐the sum of the numbers in Column C. 2. Sum the total amount produced by each factory in all rounds‐‐the sum of the numbers in the bottom row of the table.) 37 e. Have students compare and discuss their predictions from Session 2, step g with the actual data. (It is often surprising that the number of packets does not rise proportionately with the number of workers. Again, this is due to diminishing returns.) f. Catch‐up, debrief, and review BUSINESS/PRODUCTION UNIT as necessary. 38 Packet Factory: Triangles Did all factories produce the same number of packets when they had the same number of workers? Did all factories produce the same number of packets when they had the same number of workers? 39 Did all factories produce the same number of packets when they had the same number of workers? Did all factories produce the same number of packets when they had the same number of workers? Packet Factory: Circles As more workers were added to each factory, what happened to the total number of packets produced? As more workers were added to each factory, what happened to the total number of packets produced? As more workers were added to each factory, what happened to the total number of packets produced? 40 As more workers were added to each factory, what happened to the total number of packets produced? Packet Factory: Rectangles As more workers were added to each factory, what happened to the average number of packets produced per worker? As more workers were added to each factory, what happened to the average number of packets produced per worker? 41 As more workers were added to each factory, what happened to the average number of packets produced per worker? As more workers were added to each factory, what happened to the average number of packets produced per worker? Packet Factory: Stars Construct two math sentences to determine the total number of packets produced by all factories. Construct two math sentences to determine the total number of packets produced by all factories. Construct two math sentences to determine the total number of packets produced by all factories. Construct two math sentences to determine the total number of packets produced by all factories. 42 Production Data Column A Column B Number of Packets Produced Number Factory Factory Factory Factory Factory of #1 #2 #3 #4 #5 Workers Total Production by Each Factory Column C Total Number of Packets Column D Average
Number of Packets 43