In the Curriculum Mathematics ach year, students in my seventh- and eighth-grade Emath classes plan and orga- nize a schoolwide sale. This activity brings to life mathematical We’re concepts learned in the classroom. By transferring textbook math- ematics to a real-world situation, my students learn to value the math- ematics being studied and are able to understand how it can affect them in the future. To conduct this lesson, we use several technologies, includ- ing the TI-83 Plus to fi nd the best selling price for and for create charts and graphs to support decision making, and Microsoft Math Excel and Word to organize data.

Seeding the Activity Throughout the year, I focus on help- ing students understand why we learn mathematics. I am always striving to answer the familiar question posed each year, “When am I ever going to use this?” If I am to teach my students to value and appreciate mathematics I must help them see its relevance in their lives. With this in mind and with the understanding that children’s learning is built from prior learning, I developed the project We’re Poppin’ for Math as a culminating activity for my unit on Solving Systems of Linear By Brenda Levert Equations. The activity requires the Subject: Algebra, real-world students to transfer their knowledge connections of solving systems of linear equations to orchestrate a successful popcorn Audience: Teachers, teacher sale. We begin the unit after my educators eighth-grade algebra students have successfully completed the unit on Grade Level: 7–8 (Ages 12–13) analyzing linear equations. We spend Technology: Graphing calculators, a few days investigating and planning spreadsheets, word processing the popcorn sale. The actual sale takes software place in one day. To capture my students’ attention, Standards: NETS•S 3–6; NETS•T II, I begin the lesson by posing the ques- III (http://www.iste.org/standards). tion, “How many of you sold lemon- NCTM Grades 6–8 Algebra, Data ade as a child?” Often many of them Analysis and Probability, Problem raise their hands and discuss their ex- Solving, and Communication (http:// periences. I then ask them about their standards.nctm.org/document). profi t on those sales. We discuss the

20 Learning & Leading with Technology Volume 31 Number 4 Copyright © 2003, ISTE (International Society for Technology in Education), 1.800.336.5191 (U.S. & Canada) or 1.541.302.3777 (Int’l), [email protected], www.iste.org. All rights reserved. Mathematics terms cost and revenue. The students this means in relation to the project popcorn oil cost? Where can we get admit they never worried about their and sale of popcorn. I relay to the a popcorn machine? What other cost because they would simply take students that fi nding the best selling items are needed. what they needed from their pantries price is critical to having a successful This is the point at which I get at home. We agree that although it business. Students begin to realize very excited about the lesson. It is did not cost them anything, it cost that if they sold the popcorn for 25¢ at this point that I can tell they have someone. They usually laugh and a bag, considering their cost, they successfully transferred textbook say, “Yeah, our parents.” would have to sell 100 bags to break knowledge to a real-world situation. even. I pose the question “If we break Investigating the Problem even after selling 100 bags, how many Involving Others I begin by having students work in bags must we sell to realize a profi t?” We then call on the expertise of small groups to investigate a problem Students answer “101.” Familiar re- the other mathematics classes in the called Popcorn that I modifi ed from sponses are, “That is too much work school. The seventh graders are eager the Dale Seymour Publications book for such a small profi t,” or, “We only to participate. We explain what we Graphing Power. I stress that this ac- have about 400 students in our whole need and ask the seventh graders to tivity serves as a model to help guide school. There’s no way we can sell survey the school, compile the data, and direct them in their upcoming that much popcorn.” and give us a report of their investiga- assignment to plan and execute a We use the table on the graphing tion. The seventh graders survey the schoolwide popcorn sale. The modi- calculator to fi nd our maximum prof- school, asking the students if they fi ed problem reads: it. Students get excited and begin to would be willing to pay 50¢ for a interject their ideas. Someone always bag and which type of popcorn they Your class decides to sell bags suggests, “We could raise the price.” would prefer—butter, plain, caramel, of popcorn as a fundraiser. That suggestion leads to, “How or cheese. The cost to rent the popcorn much?” Students discuss what they Using Microsoft Excel, the seventh machine is $15, and it costs feel would be the best price. I steer graders organize the data to report 10¢ a bag to produce the pop- the discussion to help the students back to the eighth grade Algebra corn. The class wants to sell the decide to survey the school. We also 1 students. This activity helps the popcorn for 25¢ per bag. Using agree that we must fi nd our cost for seventh graders extend their under- this information, calculate the this project. How much does a one- standing of analyzing data by creating cost, revenue, and profi ts for pound bag of popcorn kernels cost? charts and graphs. Also, the spread- 0, 1, 10, 20, and 130 bags of How much does a gallon of sheet activity helps students work popcorn. The students consider the prob- Popcorn Sale 2003 lem and fi nd the values using basic Part I computations. Your class decides to sell bags of popcorn as a schoolwide fundraiser. The cost of the Then, the activity asks students to materials is $25.23. In addition, you will need to pay 2¢ for each bag used. Determine derive functions to represent the cost, a selling price for a bag of popcorn. Justify your rationale for determining your selling price. revenue, and profi t. Students answer 1. Write a function for determining cost. Cost = ————————————— questions that help them understand 2. Write a function for determining the revenue. Revenue = ————————— what each function means. For exam- 3. Write a function for determining the profi t. Profi t = ——————————— ple, the activity asks students to enter 4. Enter your cost and revenue functions at Y1 and Y2, respectively. Graph the functions simultaneously. What does the point of intersection tell you? the cost function into their graphing 5. What is the least number of bags of popcorn you must sell to make a profi t? calculator and generate a graph. Then How did you determine this value? the activity asks, “What does the x- 6. Enter your profi t function as Y3. Graph the function. If your class wants to make $100 axis represent?” and “What does the profi t, how many bags of popcorn do you need to sell? How did you arrive at this value? y-axis represent?” They then enter the 7. Can you have a negative profi t? What does this mean? other functions into their calculators Part II and explore the results. This helps Create a Word document to describe and analyze your problem. Include screens from your students connect multiple representa- calculator to verify your fi ndings. Use verbal, tabular, graphical, and symbolic representations tions of their data. to explain your data. Students soon discover the break- even point, and we discuss what Students use a real problem to guide their investigation of linear equations.

December/January 2003–04 Learning & Leading with Technology 21 Copyright © 2003, ISTE (International Society for Technology in Education), 1.800.336.5191 (U.S. & Canada) or 1.541.302.3777 (Int’l), [email protected], www.iste.org. All rights reserved. Mathematics fl uently with decimals, fractions, and percents. The seventh graders report My class is having a popcorn sale. We went to each class and asked the students, out the information to the Algebra 1 of cheese, butter, caramel, or regular, which class, and this starts the ball rolling. they liked the best. The most popular fl avor Interestingly enough, the seventh was butter (2) 44.2%, followed by cheese graders usually uncover the same fi nd- (1) 25.7%, then caramel (3) 20.4%, and then plain (4) 9.7%. ings year to year—buttered popcorn —Josh is the most popular.

Determining the Costs Once students collect and analyze The seventh graders present their survey data to the eighth graders. all the data, they are ready to orches- trate their sale. They feel confi dent During this project we have decided that our selling price should be 50¢. Together with in their knowledge of solving systems the cost of our materials and the added fee of 2¢ per bag, we have come to the conclusion that, we will need to sell about 53 bags to be at a breaking point. To reach a profi t of $100, of linear equations and can connect we will need to sell about 261 bags of popcorn. their knowledge and studies to a real- During this project we have decided that algebra is going to be a useful skill to know life situation. This type of activity in the business world. We enjoyed fi guring out the wonderful uses of algebra and hope to motivates the students to learn the do more of these projects in the future. This was so much fun! Now I can’t wait to sell the concepts being taught. Every year, popcorn that we worked so hard to fi gure out the price for. I defi nitely recommend doing this with your class in the future. students’ eagerness to participate is evidenced by their comments and questions. The last time I did this activity, one topic of discussion was centered on lowering the cost. A student brought in price quotes from a local movie theater and shared the prices of their popcorn. She also had a sample of the popcorn bag along with the price per bag. Another stu- dent said she would try to get her This is a picture of our table for Cost (Y1) This is a continuation of the table. The Y3 and our Revenue (Y2); X represents the represents the Profi t. If we sell 261 bags we mother’s employer to donate supplies. number of bags sold. will make $100.05! These ideas, along with other strategic thinking, proved that students knew if they could lower their cost, they could ensure a higher profi t. The class worked in small groups to collect and analyze data to de- termine the cost. We decided to purchase a large bag of kernels, 600 paper bags, and two gallons of butter- fl avored popcorn oil and to use the school’s Parent Teacher Association popcorn machine. (The school do- nated the butter-fl avored salt.) This is a graph of the information located Here are our linear equations. We derived We calculated our initial cost (for on the table. Each line represents the these equations by placing our information kernels and oil) as $25.23 and used equations in Y1, Y2, and Y3. This is the in algebraic form. it as the y-intercept in the slope inter- BREAK EVEN POINT! cept form, y = mx + b, because these goods were perishable. We decided By Michael, Heather, Will, and Renee any bags left over could be used for Eighth graders use the data gathered by the seventh graders and their technology tools to assess next year, so we represented this in- a price for the bags of popcorn. This report addresses part II on the assignment worksheet.

22 Learning & Leading with Technology Volume 31 Number 4 Copyright © 2003, ISTE (International Society for Technology in Education), 1.800.336.5191 (U.S. & Canada) or 1.541.302.3777 (Int’l), [email protected], www.iste.org. All rights reserved. Mathematics formation as our rate of change, the to explain their data. As stated in were correct. Students demonstrated slope of the line, or m. We calculated the NCTM’s Principles and Standards their ability to solve systems of equa- each bag would cost 2¢ (or .02 dol- For School Mathematics, using various tions using graphical, tabular, and lars). The students derived the cost forms of representation is “central to symbolic representations. Most im- equation y = .02x + 25.23. We used the study of mathematics. Students portant, I captured the interest the graphing calculator to explore can develop and deepen their under- of my middle school students. I various selling prices to investigate standing of mathematical concepts believe this activity will help my stu- and determine the best selling price. and relations as they create, compare, dents appreciate and value learning The graphing calculator produced and use various representations.” mathematics. Also, it helps them rec- graphical and tabular representations Using TI-Connect software and ognize technology as a valuable tool. and performed complex calculations graph link cables, students transfer- Students took ownership in this proj- so the students could focus on us- red information from their graphing ect and expressed this was a fun and ing functions to model patterns of calculators to the computer. When innovative way to “do math.” quantitative change. For example, the students’ reports refl ected that This activity’s success can be attrib- when the students used 25¢ for their they had thoroughly investigated the uted to the technological tools that selling price and viewed the table problem and had valid information generated visual images of mathemat- and graphs of their functions, they to support their decision, we set a ical concepts, facilitated organizing quickly realized they would need to date for the popcorn sale. and analyzing data, and computed sell a tremendous amount of popcorn effi ciently and accurately. As stated to break even, let alone to see a profi t. Selling the Popcorn in NCTM’s Principles and Standards Students were able to avoid perform- We sent a letter to the parents two for School Mathematics, “with calcu- ing unnecessary calculations and weeks before the sale, explaining the lators and computers, students can painlessly substitute other values project to solicit preorders. examine more examples or represen- for the cost. We took preorders, and students tational forms than are feasible by In this situation, technology recorded them according to the hand, so they can make and explore proved to be a powerful tool to purchasers’ lunch schedules. A week conjectures easily.” develop problem-solving skills before the sale, I sent a reminder note Also, the use of technology en- by allowing students the ability to home, and we collected more money gaged the students by providing feed- explore and investigate solutions. and recorded more orders. back necessary to create and validate Students derived several functions On the day of the sale, the stu- decisions. This empowered the stu- to represent revenue and continued dents set up and popped popcorn. dents and gave them confi dence their investigation on their graphing We fi lled preorders and boxed them in their mathematical abilities. calculators. By graphing the cost in paper boxes according to their Finally, this project encouraged and revenue functions simultane- lunch schedule. We wrote the teach- students to go beyond the textbook ously, we explored various break- er’s names on the box and waited applications and extend their learning even points. Students used the table for them to come into the cafeteria to real-world applications through the function to show the data in another for lunch. In most cases, all the stu- use of technology. form. Through their investigation, dents in the class purchased a bag of students decided 50¢ per bag of pop- popcorn. If not all students in a class Brenda Levert teaches Grade corn was the most reasonable price. had paid, we just included a couple 7 and 8 math at the Academy for Academics and Arts in The students wrote the cost func- of extra bags. We really did not want Huntsville, Alabama. She was tion y = .50x. They used the table to leave anyone out. Our presale was a recipient of the 2000 Toyota to discover they would make $100 such a success we did not compromise TIME Grants for Teachers to if they sold 261 bags of popcorn. This our profi ts by giving away bags of purchase technology for her investigation helped them justify their popcorn to these students. We no- classroom. She is also a National Board Certifi ed Teacher in early adolescence math. conclusion that a popcorn sale would ticed that if a couple of students had be profi table. They used screenshots not prepaid, their teacher or a parent Margaret L. Niess, L&L’s from their TI-83 Plus to create a sent in extra money to cover this cost. mathematics editor, is a profes- Word document summarizing their sor and director of science and mathematics teacher prepara- decision. They were instructed to Extending the Learning tion at Oregon State Univer- include in their document graphical, When all was done, students reported sity. For more, see http:// tabular, and symbolic representations a profi t of $108. Their predictions oregonstate.edu/~niessm/.

December/January 2003–04 Learning & Leading with Technology 23 Copyright © 2003, ISTE (International Society for Technology in Education), 1.800.336.5191 (U.S. & Canada) or 1.541.302.3777 (Int’l), [email protected], www.iste.org. All rights reserved.