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Reprinted from Barbara Moses, ed., Algebraic Thinking, Grades K–12: Readings from NCTM’s School-Based Journals and Other Publications (Reston, Va.: National Council of Teachers of Mathematics, 2000), pp. 194–96. Originally appeared in Mathematics Teaching in the Middle School 2 (February 1997): 290–92. © 1997 by NCTM. Building Equations Using M&M’s Victoria Borlaug T his activity uses M&M’s and is designed to be used in an algebra class after instruction in solving equations. It actively involves students in identifying the variable, formulating an equation, and then solving the equation. To begin, the teacher and each student are given one bag of M&M’s. The1.69-ounce, or 47.9 gram, bag works well. A more economical plan is to purchase a large bag of M&M’s and distribute some candy to each individual. The instructor starts the activity by counting the number of each color of his or her M&M’s. The teacher’s results are written on the chalkboard for everyone to see (see fig. 1). Then students are instructed to count their candies but to keep their numbers secret. Each student then receives a set of questions (fig. 2). The instructor selects a student, say, Sabrina, and asks her to fill in the first question appropriately on the basis of the M&M’s data: I would have to add (or eat) ______ red candies to have the same number of red candies as the teacher. How many red candies do I have? Suppose, for example, that Sabrina had 11 red candies and the teacher had 9 red candies. Then Sabrina would say, “I would have to eat two red canVictoria Borlaug, [email protected], teaches at Walters State Community College, Morristown, TN 37813-6899. She is actively involved with teacher training through conference presentations and leadership workshops. 194 Fill in the blanks with the teacher’s M&M’s candy data. Color Number Brown ________ Yellow ________ Red ________ Green ________ Orange ________ Tan/blue ________ Total ________ Fig. 1. Students’ tally M&M’s dies to have the same number of red candies as the teacher. How many red candies do I have?” The other students in the class will work to determine the answer to Sabrina’s question. Students can easily solve this first problem without using algebra. However, the questions on the handout become progressively more difficult. At some point the students may decide for themselves that algebraic solutions are desirable. Or when enough students begin to have trouble answering the questions, the teacher can point out that algebraic techniques are a valuable tool to help them find solutions. Before tackling the more challenging questions, the class may want to go back and practice using algebraic equations to solve the easier questions they had done earlier. When using algebraic techniques, the students define the variable, build an equation based on the word problem, and then solve the equation. One algebraic solution to the previous example would be as follows: Let x represent the number of Sabrina’s red candies x–2=9 x–2+2=9+2 x = 11 Answer: Sabrina has 11 red candies. 1. I would have to add (or eat) ______ red candies to have the same number of red candies as the teacher. How many red candies do I have? 2. If I doubled the number of tan candies I have, then I would have ______ tan candies. How many tan candies do I have? 3. If I tripled the number of yellow candies I have, I would have ______ more yellow candies than the teacher. How many yellow candies do I have? 4. If I added 15 brown candies to my bag, the teacher would have to add ______ brown candies to his or her bag for us to have the same number of brown candies. How many brown candies do I have? 5. If I ate 3 of my orange candies, then put my orange candies together with the teacher’s orange candies, we would have ______ orange candies. How many orange candies did I start with originally in my bag? 6. Suppose another student had a bag of M&M’s exactly like mine so we each started with the same number of each color candy. If we combined our candy, then I ate 5 of our red candies, we would have ______ red candies left. How many red candies did I start with originally in my bag? 7. My brown, yellow, and green candies total ______. I have ______ more (or fewer) brown candies than yellow candies. I have ______ fewer (or more) green candies than yellow candies. How many brown candies do I have? How many yellow? How many green? 8. I have a total of ______ candies in my bag. I have more ______ (or fewer) brown candies than orange candies. If I eat all my brown and orange candies, I will have ______ candies left. How many brown candies did I eat? How many orange candies did I eat? Fig. 2. Students’ M&M’s questionnaire Sabrina would then verify whether the answer to her question was correct. Many strategies become apparent for incorporating this activity into the classroom after the instructor and the class have worked through some examples. One way is to have students work in small groups to build equations and find solutions. Then the class could compare solution strategies and answers. Sometimes groups will define the variable differently, resulting in different equations, yet they will have the same answers. For example, consider the following question posed by a student: I have a total of 61 candies in my bag. I have 9 more brown candies than orange candies. If I eat all my brown and orange candies, I will have 32 candies left. How many brown candies did I eat? How many orange candies did I eat? The variable x could represent the number of orange or of brown candies. The respective equations and solutions follow. Note that both have the same answer. Solution 1: Let x represent the number of orange candies and let x + 9 represent the number of brown candies: 61 – x – (x + 9) = 32, 61 – x – x – 9 = 32, 52 – 2x = 32, 52 – 52 – 2x = 32 – 52, –2x = –20, –2x = –20 ; –2 –2 x = 10 orange candies, so x + 9 = 19 brown candies. Answer: 10 orange and 19 brown candies were eaten. Solution 2: Let x represent the number of brown candies and let x – 9 represent the number of orange candies: 61 – x – (x – 9) 61 – x – x + 9 70 – 2x 70 – 70 – 2x –2x –2x –2 x = = = = = = 32, 32, 32, 32 – 70, –38, –38 ; –2 = 19 brown candies, so x – 9 = 10 orange candies. Answer: 19 brown and 10 orange candies were eaten. 195 Comparing the two equations from solutions 1 and 2 can lead to discussion and insight into different ways to approach the same problem. After completing the handout shown in figure 2, students may enjoy creating questions of their own, which may include more M&M’s questions or relate to other real-life situations. The instructor may review the students’ questions for clarity, and appropriate ones may be used as questions in a class contest. Students often experience a sense of 196 pride when the class works on a question that they have created. Students enjoy this activity. It gives them a chance to represent concrete situations as equations and practice the algebraic techniques involved with solving those equations. An added bonus is that the students enjoy eating their M&M’s at the end of the lesson. As always, it is a good idea for teachers to have alternative eating treats to give to those students who cannot eat M&M’s.