Download Building Equations Using MandMs

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.
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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
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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.