INVESTIGATIONS
Elizabeth Mailley and Patricia S. Moyer
The Mathematical Candy
Store: Weight Matters
his department features children’s hands-on
and minds-on explorations in mathematics
and presents teachers with open-ended investigations to enhance mathematics instruction.
These tasks invoke problem solving and reasoning,
require communication skills, and connect various
mathematical concepts and principles. The ideas
presented here have been tested in classroom
settings.
A mathematical investigation—
T
• has multidimensional content;
• is open-ended and has several acceptable
solutions;
• is an exploration requiring a full class period or
longer to complete;
• is centered on a theme or event; and
• is often embedded in a focus or driving question.
In addition, a mathematical investigation
involves processes that include—
•
•
•
•
researching outside sources;
collecting data;
collaborating with peers; and
using multiple strategies to reach conclusions.
Although this department presents a scripted
sequence and set of directions for an investigation,
NCTM’s Standards (2000) encourage teachers and
students to explore multiple approaches and representations when engaging in mathematical activities. The activities in this investigation explore the
Elizabeth Mailley, [email protected], teaches first grade at Westlawn Elementary School in
Falls Church, Virginia. She is interested in using open-ended explorations and discussions to
help children better understand mathematical concepts. Patricia S. Moyer, [email protected],
teaches mathematics education courses and is the director of the Mathematics Education
Center in the Graduate School of Education at George Mason University in Fairfax, Virginia.
Her research focuses on uses of representations in mathematics and mathematics teacher
development.
Edited by Patricia S. Moyer, [email protected], George Mason University, Fairfax, VA
22030. This section is designed for teachers who wish to give students new insights into familiar topics in grades K–6. This material can be reproduced by classroom teachers for use with
their own students without requesting permission from the National Council of Teachers of
Mathematics. Readers are encouraged to send manuscripts appropriate for this section to
“Investigations,” TCM, 1906 Association Dr., Reston, VA 20191-1502.
388
concept of weight measurement. The purpose of
this investigation is to recognize the attributes of
weight, use nonstandard units to compare and
order objects by their weight, and use repetition of
a single unit to represent something larger than the
unit (NCTM 2000). These objectives represent several expectations of the Measurement Standard for
grades pre-K–2.
Introduction
Measurement incorporates several areas of mathematics, including geometry and number, and facilitates other areas of the elementary curriculum
such as science. Measurement is an important life
skill. Teachers can develop investigations that
apply to children’s everyday experiences. Weight
measurement, in particular, lends itself to hands-on
activities and student collaboration. For example,
students can gather around a balance scale and take
turns adding objects of varying weights to the
scale, counting aloud, checking for equilibrium,
and recording results.
In the four-day lesson described in this investigation, children explore weight measurement,
equivalence, and counting, and use the processes of
problem solving and communication to simulate
activities in a candy store. The candy-store theme
serves as a framework for children to practice
using balance scales as tools for standard and nonstandard measurement experiences.
Teaching Children Mathematics / April 2004
Copyright © 2004 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Materials
Day one
• a small plastic bag containing one piece of several different kinds of candy, such as an M&M,
gum, a gummy bear, a Life Saver, a chocolate
kiss, a Red Hot, a jelly bean, and a Sweet Tart;
• a bag of centimeter cubes (from a base-ten
block set);
• a balance scale; and
• four copies of the “Task Sheet: Day One” worksheet.
Day two
Each group of four student investigators will
need—
• seven paper cups, each containing a small
amount of a different type of candy, such as
M&M’s, gum, gummy bears, Life Savers, Red
Hots, jelly beans, and Sweet Tarts;
• a small plastic bag containing five chocolate
kisses;
• a balance scale; and
• four copies of the “Task Sheet: Day Two” worksheet.
Days three to four
Each group of four student investigators will
need—
• eight paper cups, each containing a small
amount of a different type of candy, such as
M&M’s, gum, gummy bears, Life Savers,
chocolate kisses, Red Hots, jelly beans, and
Sweet Tarts;
• eight empty plastic bags;
• a balance scale;
• markers or crayons; and
• four large pieces of construction paper.
Day One: Investigating
Weight and Order
Before the lesson, purchase several different kinds
of candy, such as M&M’s, gum, gummy bears,
Life Savers, chocolate kisses, Red Hots, jelly
beans, and Sweet Tarts. Students will use these
candies during each day of the investigation. On
the first day, prepare small plastic bags containing
Teaching Children Mathematics / April 2004
Photograph by Elizabeth Mailley; all rights reserved
Each group of four student investigators will
need—
one piece of each type of candy. Give one bag to
each group of four students. Also give each group
a bag of centimeter cubes, a balance scale, and four
task sheets.
At the beginning of the investigation, tell students that they will be simulating a “Mathematical
Candy Store” for the next few days. Explain that
the owner of the Mathematical Candy Store needs
their help. A computer error has erased the weights
of all the candies in the store. Students must help
the store’s owner by weighing each type of candy
and placing the candies back in their proper order,
from lightest to heaviest. This task is important
because the candy store charges more for heavier
bags of candy.
Direct students’ attention to the bags of candy at
their tables. Ask them if they have ever considered
the weight of the candy they have eaten. Discuss
why the weights of the candies might be important.
Ask the students if they have ever purchased candy
and seen the salesclerk weigh the candy to determine the cost. Encourage students to predict which
type of candy weighs the most and which weighs
the least. Show students how to place a piece of
gum on one side of the balance scale and centimeter cubes on the other side of the scale. Demonstrate how to add centimeter cubes one at a time
until the balance scale is “even.” Instruct students
389
Photograph by Elizabeth Mailley; all rights reserved
to look for the center line on the balance scale to
ensure that it lines up with the base, which indicates that the items on each side of the balance
scale are equal in weight.
After this guided introduction, students work
together to weigh each piece of candy and record
their results. Have each group weigh one piece of
gum. Check to be sure that students know how to
weigh the candy and read the scale accurately. Students should record the weight in centimeter cubes,
along with the weights of the other candies, on
their Day One task sheets. Next, students compare
the weights of the candies, order them from lightest to heaviest, and record this information on their
task sheets.
After students complete the investigation, meet as
a group to discuss their results. You should ask questions that encourage students to make comparisons,
such as “How many centimeter cubes equal the
weight of a Life Saver? How many centimeter cubes
equal the weight of one piece of gum? Which of
these two candies weighs more, and how do you
know?” As an extension to the activities on day one,
you may choose to have students suggest prices for
bags of the candies, using the weights as a guide.
Day Two: Focusing
on Equivalence
On the second day, students explore different types
of candy to find amounts that are equivalent in
weight. Before the lesson, place five chocolate kisses
in a small plastic bag for each group, and place a
390
small amount of each of the other candies in separate
paper cups. Give each group of students one bag of
the chocolate kisses, seven paper cups filled with
each of the different candies, a balance scale, and
four task sheets. Students will use the five chocolate
kisses as a referent for comparison with each of the
other candies. The objective of this lesson is to find
out how many of the other candies are equal in
weight to the five chocolate kisses. For example, it
may take thirty M&M’s to equal the weight of five
chocolate kisses. When preparing for this lesson, be
sure to put an amount of candy in each cup that
weighs more than the five chocolate kisses. This will
enable students to explore the weight relationships
and to focus on the concept of equivalence, and you
will not give away the answer to the students.
Tell students that the owner of the “Mathematical Candy Store” has returned to request their help
once again. It seems that having different prices for
different bags of candy confuses some customers.
The candy store owner has decided to make each
bag of candy the same weight so that each bag can
be sold at the same price. The owner has given the
students a bag of chocolate kisses to use for comparison. Discuss with students how to make the
bags of candy the same weight. Encourage them to
think about how the numbers of candies must
change so that each bag is equal in weight to the
five chocolate kisses. Should the numbers of candies increase, decrease, or stay the same? What
have the students learned from day one’s exploration that will help with today’s investigation?
Demonstrate how to make equal weights on the
balance scale by putting five chocolate kisses on
one side of the scale and M&M’s on the other side.
Students can make predictions about how many
M&M’s will equal the weight of the five chocolate
kisses. Add M&M’s to the scale until it shows that
the two sides are equal. After finding an amount of
M&M’s equal to the weight of the chocolate kisses,
show the students how to record this information
on their task sheet for day two.
After the introduction and demonstration, have
each group use its own scale to compare the weights
of the M&M’s and the chocolate kisses. Check that
students are using the scale properly to find equivalent amounts. As students begin to compare the other
candies, remind them that their task is to find equivalent weights by making all the candy amounts
weigh the same as the five chocolate kisses. Students
should count the number of each candy that equals
the five chocolate kisses and record this number on
the task sheet. After students find equivalent weights
Teaching Children Mathematics / April 2004
for all the candies, tell them to use the numbers 1
through 8 to order the candies from least to greatest,
based on the number they need to balance the chocolate kisses. Then meet as a class to compare and discuss results among the groups. How did students find
equal weights for all the candies? Did all the groups
find the same number of gummy bears equivalent to
the chocolate kisses? If not, why do they suppose a
difference exists?
To extend the lesson, discuss the relationship
between the order of the candy weights from day
one and the candy amounts equal to the chocolate
kisses from day two. Ask questions to determine if
the children recognize that as the weight of an individual candy increases, the amount of that candy
needed to balance the chocolate kisses decreases.
Days Three and Four:
Creating a Candy-Store
Advertisement
On the third and fourth days of the investigation,
students will use the candies to determine how
many candies should be in each bag for sale, how
much to charge for a bag of candy, and how to
advertise the candy for their own candy store. Give
each group eight paper cups, each filled with different types of candy; eight empty plastic bags; a
balance scale; markers or crayons; and four large
pieces of construction paper.
Tell the students that the candy store owner has
returned again. He is so impressed by the students’
knowledge and mathematical ability that he has
given them the job of deciding how they would set
up a new branch of the candy store. The students
must work in groups to decide whether they would
prefer to sell candy bags of equal weight that have
the same price, or bags of different weights with different prices. Have the students review their results
and findings from the activities on days one and two
of the investigation. How will students make bags of
candy of different weights? How will they make
bags of equivalent weights? How much will they
charge for the candies and why? Use the balance
scale, gummy bears, and jelly beans to review how
to make bags of candy with equal weights.
Each student will create an advertisement for a
candy store and decide whether the bags will be of
equal weight or equal number, or use another method
of his or her choosing. Show students an example of
an advertisement for a candy store on construction
paper. The example should show the name of the
candy store (for example, “Julie’s Candy Store”),
Teaching Children Mathematics / April 2004
pictures of candies such as M&M’s, and information
about the price and the number of candies (for example, 10 M&M’s for $1.00). Designing the candystore advertisement requires students to use problemsolving skills and to apply the skills of weighing and
ordering that they used on days one and two.
Once students have agreed on how they will
price the candy items in their store, they should
select two or three types of candy to sell. Students
should make small bags of the selected candies,
weigh and price the bags, and make a note of the
prices and the number of candies for their candy
advertisement posters. After students have gathered all the information for their candy-store advertisements, they can design their posters. Remind
them to name the candy store, draw pictures of the
candies they selected, write the number of candies
in each bag, and write a price for each bag. Discuss
the importance of clearly listing prices and
amounts as well as using eye-catching colors and
producing neat work. To extend the activities on
day three, the students can present their advertisements to the class on day four and justify their
decisions about weight and price.
Concluding Comments
In students’ everyday experiences, they encounter
situations in which the weight of an item influences its price. Engaging in investigations such as
this allows students to become familiar with weight
measurement and ordering the weights of objects.
In this investigation, students compared the
weights of objects, explored how the amount of an
item influences its weight, and found equivalent
weights. They had the opportunity to make connections across several areas of mathematics,
including number, measurement, and problem
solving, as well as to collaborate and make decisions that relate to the world outside the classroom.
These early experiences can strengthen students’
understanding of the application and processes of
measurement while enabling them to use numbersense skills in a meaningful context.
Reference
National Council of Teachers of Mathematics (NCTM).
Principles and Standards for School Mathematics.
Reston, Va.: NCTM, 2000.
The authors wish to thank first-grade teacher Julie
Ross and her class of young mathematicians at Westlawn Elementary School in Falls Church, Virginia. ▲
391
Task Sheet: Day One
Investigating Weight and Order
Name: __________________________
Use a balance scale and centimeter cubes to determine the weight of each candy.
Record each weight below. Use the numbers 1–8 to order the candies from the lightest
to the heaviest.
Type of Candy
Weight in Centimeter
Cubes
Order from Lightest to
Heaviest
Extension: How much would you charge for a bag of each type of candy? Write the
amount next to each type of candy.
From Teaching Children Mathematics, April 2004
Task Sheet: Day Two
Focusing on Equivalence
Name: __________________________
Use a balance scale to determine the number of each type of candy that equals the
weight of five chocolate kisses. Record the number amounts in the table below. Use the
numbers 1–8 to order the candies from the least to the greatest number.
Type of Candy
Number of Candies
Equal to Five
Chocolate Kisses
Order from Least to
Greatest Amount
Extension: What do you notice about the relationship between the order of the candy
weights from day one and the order of the candy amounts from day two?
From Teaching Children Mathematics, April 2004