the Focal Points
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problem solvers: problem
Ma ri e H a u k a n d B e v e rl e y K u l a
Jelly bean jumble
◗
The classic question of mixing water and
wine (Ball 1896, p. 27) is the basis of the
May problem, which can be solved without
using computation.
Problem scenario
Jon has a bag of green jelly beans, and Emily has
a bag of red jelly beans. Jon takes a handful of his
green jelly beans and puts them in Emily’s bag
of red ones. She shakes the bag to mix the jelly
beans. Then, without looking, she takes from her
bag as many jelly beans as Jon had given her and
puts them in his bag. Are there more of Emily’s
red jelly beans in Jon’s bag or more of Jon’s green
jelly beans in Emily’s bag?
More questions and a table 1 template appear
in the students’ activity sheet on p. 511.
Background
Before introducing this problem to your students, wrestle with the problem a bit yourself.
Interestingly, what at first glance may seem to
be a probability problem is nothing of the sort.
Review what the problem is asking—it is not
asking about the ratio of green to red or red to
green jelly beans in each bag—in fact, the question asks how many jelly beans of an alien color
are now in each bag. Perhaps surprisingly, this
number is the same for both Jon and Emily.
Think of it this way: If Jon puts five jelly beans
in Emily’s bag, then Emily puts five jelly beans in
Jon’s bag, the total number of jelly beans in each
bag remains consistent, and the number of jelly
beans in each bag that are not the original color
is also consistent. Why?
508
Here is one possible scenario. Let’s say that
Jon has six green jelly beans and Emily has eight
red ones. Jon takes three of his green jelly beans
and gives them to Emily. She mixes the green
jelly beans with her red ones and randomly picks
three to give to Jon. This is what could happen:
1. Emily picks three green jelly beans (highly
unlikely but possible) to give to Jon. This
leaves zero green jelly beans in Emily’s bag,
and zero red ones go into Jon’s bag.
2. If Emily picks two green jelly beans and one
red jelly bean, then one green jelly bean
remains in Emily’s bag and one red jelly bean
goes into Jon’s bag.
3. If Emily picks one green and two red jelly
beans, then two green jelly beans are in
Emily’s bag, and two red jelly beans go
into Jon’s bag.
4. If Emily picks zero green and three red jelly
beans, then three green jelly beans remain in
Emily’s bag, and three red jelly beans go into
Jon’s bag.
We can vary the number of jelly beans that each
of them has. We can vary the size of the “pick.”
The total number of jelly beans (in each bag)
that originally belonged to the other person will
be equal after the trades have been made.
Classroom setup
Introduce the problem to the students verbally
or in writing. Ask them to discuss the problem
in pairs or in small groups, to predict and record
what they think will happen, and to explain
their reasoning.
Provide each group with two paper bags and
two colors of cubes or chips to represent the jelly
beans. Invite students to act out the problem
five or six times, varying the number of cubes
they exchange.
As students explore the problem, observe
whether they begin with the same number of
cubes in each bag. If they do, ask why they chose
May 2010 • teaching children mathematics Copyright © 2010 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.
www.nctm.org
• Was your prediction correct?
• Were you surprised with the results?
• Why, or why not?
Gather students’ results into one display table
for a whole-class discussion. Some students
may have conflicting or incorrect results. You
may want to ask students to repeat some experiments for verification. During the whole-class
discussion, use such prompts as the following:
• Observe the results; what do you notice?
• Does it matter if Jon and Emily begin with the
same number of jelly beans? Explain how you
know.
• Describe the mixtures of jelly beans in
each bag after Jon and Emily have made an
exchange.
• What happens if Emily does not shake her bag
of jelly beans?
• Why do you think we get these results?
Several ways exist for you to explore the
results of exchanging a particular number of
jelly beans. Table 1 shows all possible results for
exchanging five jelly beans. If students have difficulty understanding, try using cubes to model
the process of determining all possible results
when you first exchange one jelly bean, then
two jelly beans, then three jelly beans, and so on.
Drawings and diagrams may also help students
keep track of the possible ways that the jelly
beans could be exchanged. The main challenge
of this problem is for students to understand
why the number of red jelly beans in the green
bag is always the same as the number of green
jelly beans in the red bag.
www.nctm.org
Where’s the math?
Because the suggested strategy employs a number of trials and a table of outcomes, it looks like the problem involves experimental and theoretical probability,
but it does not. Instead, students reproduce a situation and explore patterns in
their results. Students are using algebraic thinking when they explore and model
the problem with concrete materials, represent and analyze their results in a table,
and then describe the relationship between the red and green jelly beans in the
two bags. They also have the opportunity to make connections between the various representations of the problem: concrete, physical, visual, oral, and written.
As they work on the problem in pairs or small groups, students can communicate
their mathematical thinking to one another and engage in refining their conjectures
about the mixtures of jelly beans in each bag. A whole-group discussion about the
results is an occasion to continue to evaluate the conjectures of their peers and to
develop their own mathematical thinking. The first extension provides an opportunity for students to describe patterns in words, although students might also choose
to use ratios or fractions. The second extension is a chance to work on a generalization of their results.
Share your students’ work
The goal of “problem solvers” is to foster
improved communication among teachers by
posing one problem each month for grades K–6
teachers to try with students. Every teacher can
become an author.
Present this problem scenario to your class.
Note how students respond to the problem,
what problem-solving strategies they use, and
how they explain or justify their reasoning.
Send your thoughts and reflections—including
information about how you posed the problem, students’ work samples, and photographs
of your problem solvers in action—by July 1,
2010, to Julie S. Long, Department of Elementary Education, 551 Education South, Faculty
tab le 1
to do so and challenge them to investigate what
happens if they begin with different numbers of
cubes in each bag. Alternatively, if they do not
start with the same number of cubes, ask them
why and challenge them to also investigate what
happens if they begin with the same number of
each color.
You may wish to have students record their
results using the provided chart. If not, remove
the chart before you duplicate the activity sheet.
Once students have had an opportunity to
explore the problem, ask them to revisit their predictions. You might ask such questions as these:
This chart shows all possible results for exchanging five cubes
between two bags regardless of the total number of jelly
beans in either bag at the beginning.
No. of cubes
exchanged
No. of red
cubes in the
green bag
No. of green
cubes in the
red bag
Are the
numbers
the same?
5
5
5
yes
5
4
4
yes
5
3
3
yes
5
2
2
yes
5
1
1
yes
5
0
0
yes
teaching children mathematics • May 2010 509
problem solvers: problem
of Education, University of Alberta, Edmonton,
Alberta, Canada, T6G 2G5; or e-mail her at
[email protected]. Include your name,
grade level, and the school’s name, which will be
acknowledged (unless you indicate otherwise)
should your submission be selected for publication in a subsequent issue of TCM.
this section of the “problem solvers” department features a new problem for students. Readers are encouraged to submit problems to the editors to be considered for future “problem solvers” columns. Receipt of
problems will not be acknowledged; however, problems
selected for publication will be credited to the author.
Additional resources
R EF ER EN C E
Visit www.nctm.org/tcm to copy text from
the online version of the students’ activity
sheet on the next page. Paste it into your own
document and modify it for your students.
Ball, W. W. Rouse. Mathematical Recreations and
Problems of Past and Present Times. 3rd ed.
London: Macmillan, 1896.
Marie Hauk, mathaliv@interbaun, teaches mathematics
education courses to preservice teachers at the University of Alberta and facilitates professional development
sessions for practicing teachers. Beverley Kula, bkula@
ualberta.ca, is an instructor in mathematics education
at the University of Alberta. Problem solving is a primary
focus of all her courses with preservice teachers. Edited
by Julie S. Long, an assistant professor at the University
of Alberta in Edmonton, Alberta, Canada. Each month
Visit www.nctm.org/catalog for more NCTM
resources, including professional development
offerings, online resources, and other publications.
NCTM has published a collection of past “problem
solvers” columns:
Sakshaug, Lynae E., Melfried Olson, and Judith
Olson. Children Are Mathematical Problem
Solvers. Reston, VA: National Council of
Teachers of Mathematics, 2002.
NCTM’s Online Professional Development
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that’s convenient, affordable and immediately useful!
NCTM offers a variety of online professional development opportunities that make it easy for
multiple educators to participate from one site, for one low price.
Take a closer look at hot topics within math
education with these onetime 60-minute online seminars presented by
experienced mathematics educators.
These two-part workshops include
3 hours of Standards-based content and provide activities and teaching
strategies that can be immediately applied in the classroom.
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learn more and register!
to lea
510
May 2010 • teaching children mathematics ewrkshp-smnr809_433
NCTM journals
Oct.
www.nctm.org
➺ problem solvers activity sheet
Jelly bean jumble
Name_____________________________
Jon has a bag of green jelly beans, and Emily has a bag of red jelly beans. Jon takes a handful
of his green jelly beans and puts them in Emily’s bag of red ones. She shakes the bag to mix
the jelly beans. Then, without looking, she takes as many jelly beans from her bag as Jon had
given her and puts them in his bag.
1.Are there more of Emily’s red jelly beans in Jon’s bag or more of Jon’s green jelly beans in
Emily’s bag? In other words, who has more jelly beans of the other person’s color?
2.Look for a pattern in your results. Describe the pattern.
3.Why do you think this pattern works?
TABLE 1
Use this table to represent your results.
No. of cubes
exchanged
No. of red
cubes in the
green bag
No. of green
cubes in the
red bag
Are the
­numbers
the same?
4.What do you think might happen if you used three colors of jelly beans and three different
people?
5.Would the pattern also work if Emily and Jon each have two different colors of jelly beans?
For example, Jon has a bag of green and yellow jelly beans, and Emily has a bag of red and
yellow jelly beans. Explain your reasoning.
From the May 2010 issue of
problem solvers: solutions
A visit to
your school
➺ problem
The May 2009
problem involves
measuring circles,
estimating, and
making connections
to nature. When
students consider
the number of
leaves a tree can
produce each
summer, they may
realize how tedious
counting them one
by one would be.
However, students
find that math can
come to the rescue.
The May 2009
article included a
bonus problem.
Sample solutions
and a discussion of
the bonus problem
accompany the
online version of
this article at
www.nctm.org/tcm.
Throughout the past nine months, our
problem-solving excursions led us to exciting
worldwide destinations. For our final adventure, our destination is your school. Whether
Crown
it is in the city, country, mountains, or along a
shore, your school is that special place where
you and your friends go to learn. While you
solve this month’s problem using information
that you gather at your school, we are going
solve the problem using data from Pierce
Lake Elementary School in Chelsea, Michigan,
where one of our editors teaches.
As we walked from the school to the playground, we noticed several trees growing on the
Trunk
campus. Standing under one of them, we looked
up toward the sky and saw many branches reaching out away from the trunk. We focused our
attention on the tree’s leaves.
Locate a tree near your school. Examine its
Roots
branches and leaves. Focus on the size, shape,
and number of leaves. Estimate the number, and
describe how you arrive at your estimate.
Now imagine arranging all the leaves from your tree into a puzzle with no space between
the leaves. On the ground under the tree, use string or yarn to outline the circumference of
the tree’s crown. The crown is the part of the tree where the branches and leaves form (see
the sketch above). Would the leaves fill the circular area marked by the string or yarn? If so,
would they fill a single layer or many layers? Describe how you determined your answer.
Extensions
• Trace a leaf on grid or graph paper. Is it possible to use your tracing to determine how
many leaves will be inside the circle around your tree?
• Teachers at the upper grades may want to extend this introduction by discussing terms
such as chlorophyll and photosynthesis.
◗
After presenting an academic year’s
worth of math problems connected to
locations around the world, the May
2009 “problem solvers” problem asked students to consider their own backyard or
neighborhood. Children were to choose a tree
in their area and ask a question people often
think about (especially in the fall): “How
many leaves are on that tree?”
512
May 2010 • teaching children mathematics The solution this month comes from Brian
Schad of Michigan and his fifth-grade students
at Lawton Elementary School in Ann Arbor.
Trees and leaves have always been an important part of life for these Midwesterners—
furnishing both fall colors that make the season
beautiful and raking jobs after the leaves fall
from the trees. Schad’s class eagerly took on
the problem by trying to figure out how many
www.nctm.org
BRIAN SCHAD
Using string to map the
edge of the tree’s crown
and comparing the string
to the farthest reaches
of the various branches,
the Lawton students
determined that the
shape approximated a
circle, so they measured
its diameter.
leaves were on one of the trees on their school
campus. Students discussed various ways of
estimating the total leaf count, including using
the dimensions of the crown.
Schad wanted the youngsters to focus on the
crown and the related measurements. He shared
with them that some foresters have determined
that, on average, trees yield about four-anda-half layers of leaves in the circle under the
crown (see the “Where’s the math?” section of
the original problem). He then challenged his
students to use this information, some careful
measurements, and their mathematical knowhow to determine the number of leaves on their
chosen tree.
Students first investigated the area under the
tree, the shape that the crown makes, and how
to find the area of that shape. They used string
to map the edge of the crown shape, comparing
the string on the ground to the farthest reaches
of the various branches above them. The shape
approximated a circle, so the children decided
to measure its diameter. However, the tree’s
crown was not perfectly circular; the class determined that taking several measurements of the
approximate diameter and then averaging them
would be a good approach to the problem.
www.nctm.org
Students worked the string into a circular
shape so that it would be easier to determine
the area. They measured the circle and found
the diameter to be about 7 meters. Measuring
in several different directions, students determined diameters between 6 3/4 meters and
7 1/4 meters. They averaged the numbers for a
diameter of 7 meters, or 700 centimeters.
Next, students had to calculate the circle’s
area. Because they were going to use halfcentimeter­ graph paper to find the area of a
leaf, they decided to use the same units for the
radius. Schad expounds on the class discussion
about translating the units:
We changed the 7-meter diameter to centimeters and determined that the radius of a
700-centimeter diameter is 350 centimeters.
We discussed the area of a circle to be pi
times the radius squared. We determined that
the area underneath the tree was 384,650
square centimeters.
Note that the class must have used 3.14 as an
approximation of pi and that 384,650 square
centimeters for the area of the crown is also
an approximation.
teaching children mathematics • May 2010 513
problem solvers: solutions
BRIAN SCHAD
The youngsters learned
quickly that they could
get an accurate count
of the number of leaves
on a single branch but
that this method was not
the most efficient. They
searched for an alternate
method.
After calculating the size of the circle under
the tree, students had to find the number of
leaves that would fit in that circle. Selecting
various leaves, students traced them on halfcentimeter graph paper. Then they converted
their findings to square centimeters. Students
thought of the leaves on the ground as a puzzle and tried to determine how many pieces
would fill the area. Their teacher explains:
Each student traced a leaf on 1/2-cm graph
paper and tried to determine the number of
1/4-cm2 squares in their leaf. We used this
very small grid to increase our accuracy of
determining the leaf’s area. They blocked off
sections of the inside of the leaf with rectangles to make the area calculation easier.
They used the formula of length times width
to determine the area of the rectangular
shapes. Whatever irregular shapes fell along
the edge of the leaves were counted individually. Partial squares were estimated and
combined to form full squares.
Four smaller squares make up a square
centimeter, so after counting the squares,
students divided their results by four to
determine square centimeters. We discussed
this concept by drawing a square centi­
514
May 2010 • teaching children mathematics meter and showing how many of the smaller
squares (1/2 by 1/2) are inside. The leaves
were not all the same size. They averaged
43 square centimeters. Their areas varied
between 35 and 52 square centimeters.
The next task for the students was to find the
number of leaves that would cover the area of
the circle under the crown of this tree. They did
so by dividing the area of the circle representing the crown by the average area of a single
leaf. Then they multiplied this number by 4.5 to
determine the number of leaves equivalent to
four-and-a-half layers of leaves.
To calculate the number of leaves that would
cover one layer under the circular section
under the tree, each student determined how
many leaves would cover one layer under the
tree. They divided the area under the tree by the
average leaf size:
384,650/43 = 8945
They multiplied their answer by the average
number of layers under a tree and found the
number of leaves on the tree:
8945 × 4.5 = 40,252
www.nctm.org
Thank you to the fourth- and fifth-grade students from Lawton Elementary School in Ann
Arbor, Michigan, for their work on this problem. The editors would like to also acknowledge
and thank Robert Mann for his contributions
to this article.
BRIAN SCHAD
Aha!—the class determined that this
tree had about 40,000 leaves, a reasonable
approximation. Of course, the actual number
of leaves will vary with the kind of tree and
the size of its leaves. In fact, many factors can
affect the estimate.
The leaf problem begins with a contextual
setting that many of us can relate to—how
many leaves are on a tree? As these Michigan
fifth graders demonstrated, to find answers
to this simple question, we rely on many
mathematical skills and concepts: numerical
calculations, estimation, and measurement as
well as critical thinking and organized problem solving. Students used the relationship
between the size of the leaves and the circle
under the tree’s crown to determine their estimate. Their method for solving this seemingly
unwieldy problem incorporated measurement
skills, the area formula of a circle, a means for
determining the area of irregular shapes, and
multistep problem solving.
Even as we have traveled the world to discover interesting mathematics problems, the
leaf problem shows us that some of the most
intriguing and exciting problems have been
“hanging out” in our own backyards.
To find the number of leaves
that would fit in the circle
under the tree, class members
traced leaves on half-centimeter
graph paper, measured them,
and converted their findings to
square centimeters.
Students blocked off sections of
their tracings with rectangles.
They counted irregular shapes
along the edges individually,
estimating and combining
partial squares to make their
area calculations easier.
Edited by Joseph Georgeson, jgeorgeson@
usmk12. org, the middle school math department
chair and a teacher of eighth-grade students at the
University School of Milwaukee in Wisconsin, and
Sarah Bunten, [email protected],
a third-grade teacher at Pierce Lake Elementary
School in Chelsea, Michigan. Each month this section of the “problem solvers” department discusses
the classroom results of using problems presented
in previous issues of Teaching Children Mathematics. See detailed submission guidelines for all TCM
departments at www.nctm.org/tcmdepartments.
www.nctm.org
BRIAN SCHAD
Sample solutions and a discussion of the May
2009 “problem solvers” bonus problem
accompany the online version of this article at
www.nctm.org/tcm.
teaching children mathematics • May 2010 515
problem solvers: solutions
Traveling the world
bonus problem
➺ problem
2
This past year the “problem solvers” department editors have led readers on “tours” to different geographic locations to present mathematics-related problems. In August we began
by visiting Washington, D.C., and then proceeded to the Grand Hotel on Mackinac Island in
Michigan; the AquaDom in Berlin, Germany; the city of Rome to see the Presidential Flag of
Italy; the Liske Barn quilt in Sac County, Iowa; China’s Silk Trail; the public library in Fredonia,
Wisconsin; and the center of the continental United States in Lebanon, Kansas. We finished
our adventures by looking at trees near our respective schools.
If we were to physically travel to all the locations in the order that they were presented
in “problem solvers,” how long a journey would it be? If you could visit the different locations in whatever order you choose, what would be the shortest possible distance that you
cover? List the order. You must begin and end your journeys at the town where your school
is located.
◗
The first part of the problem asked
students to determine the total mileage if
they were to visit all the places in the order
that they appeared in each TCM issue, starting
and ending at their own home towns. Students
from a fourth- and fifth-grade combination
classroom who worked this activity quickly
discovered that it is not really a “problem” at
all. It is simply an exercise that involves finding a Web site to determine the total number of
miles between locations. Then students add the
distances. Using Web site data, this class from
Lawton Elementary School in Ann Arbor, Michigan, determined the total round-trip distance
from their home location to be 25,291 miles.
The actual problem solving takes place in the
second part of the problem. Students were again
to begin at their hometowns (in this case, Ann
Arbor), “visit” all the locations where problems
were presented, and return to their own towns.
However, for this part of the activity, students
could visit the locations in any order. They were
to determine which order would result in the
shortest overall distance.
We knew that the second part would be
challenging for elementary school students,
who generally lack the tools and the training at
this age that the solution requires. Such skills
May 2010 • teaching children mathematics are often not developed until students are
enrolled in high school or college classes that
deal with discrete mathematics and traveling
salesman problems (see Tannenbaum [2007]
for a description of these problems and some
common algorithms and methods for obtaining
A combined class
of fourth and fifth
graders quickly
discovered that the
bonus problem’s
first part is not
really a problem
but an exercise in
using a Web site to
determine the total
number of miles
between locations.
BRIAN SCHAD
The May 2009
“problem solvers”
department
presented students
with an opportunity
to solve a bonus
problem linking
all TCM volume 15
“problem solvers”
problems into a
single problem.
Throughout the
year, the “problem
solvers” editors
led readers on a
worldwide tour of
cities, posing math
problems related to
particular locations.
The bonus problem
asked students to
consider “visiting”
all the previous
locations. The
original, two-part
problem listing each
original location
appears above and
to the right.
www.nctm.org
problem solvers: solutions
Begin in Ann Arbor, Michigan, and then
travel to Washington, D.C.; Berlin, Germany;
Rome, Italy; Dunhuang, China; Lebanon,
Kansas; Sac County, Iowa; Port Washington,
Wisconsin; Mackinac Island, Michigan; and
end back in Ann Arbor. The total distance for
this trip is 17,406 miles.
With Sean’s method, the travel within the
United States actually occurs in two sections.
He first travels east on the way to Europe. Upon
returning from China, he starts in the westernmost U.S. location in Kansas and travels east
again. Although the plan of visiting all locations
within each continent seems sound, Sean’s solution shows that such a plan may not always yield
the minimum distance.
Sean’s value of 17,406 miles is close to the
shortest distance of 17,369 miles that his teacher
found. Both methods use the same order of
locations, except that Berlin and Rome are
reversed. Although the fourth and fifth graders
did not test every possible route and cannot
prove that they did, in fact, find the shortest
distance, the students were pleased with their
www.nctm.org
work and were excited about their final results
and analyses.
R ES OU R C E
table 1
Tannenbaum, Peter. Excursions in Modern Mathematics. 6th ed. Upper Saddle River, NJ: Pearson
Education, 2007.
Fourth graders Jungyun and Tanvi’s work for the shortest trip
includes the girls’ feelings about their results.
The resource we used for maps was www.geobites.com/. This whole problem
was fun. It was exciting to know the real distances. We did four of these
sheets—each one a different way. The lowest we got was 18,017.
Route
Miles
Ann Arbor, MI to Washington, DC
382
Washington, DC to Lebanon, KS
1092
Lebanon, KS to Sac City, IA
260
Sac City, IA to Port Washington, WI
365
Port Washington, WI to Mackinac Island, MI
232
Mackinac Island, MI to Berlin, Germany
4102
Berlin, Germany to Rome, Italy
725
Rome, Italy to Dunhuang, China
4113
Dunhuang, China to Ann Arbor, MI
6746
Total
tab le 2
solutions). However, the problem allows students
to attempt and compare various solutions while
using mathematical skills and general reasoning
abilities to investigate a real-life situation.
The Michigan students eagerly tackled the
minimum-distance problem, attempting and
comparing several routes until realizing that an
overwhelming number of possible routes exist.
Yet, they persevered and tried to focus on those
routes that seem to have the best chance—logically—of being the shortest.
Table 1 and table 2 illustrate the shortest and
longest trip, respectively, of the four routes that
fourth-grade students Jungyun and Tanvi found
and their trip plan: Visit all the locations on one
continent (North America), then all those in
Europe, and finally those in Asia; then return
home. Many of us would probably use a similar
approach when facing such travels.
Although the girls’ approach seems reasonable, another student discovered a way to shave
two thousand miles off his current distance by
thinking about it from a different viewpoint.
After working on the problem for a while, Sean
determined the following path to be the shortest
distance:
18,017
The second table shows the longest of the four routes that
the girls found and reveals their strong organizational skills.
This was the second one we made. It came out successful. This one was
going from North America to Europe to Asia. Our worst total was 19,035.
This is how we did it.
Route
Miles
Ann Arbor, MI to Berlin, Germany
4250
Berlin, Germany to Dunhuang, China
3761
Dunhuang, China to Rome, Italy
4113
Rome, Italy to Washington, DC
4545
Washington, DC to Lebanon, KS
260
Lebanon, KS to Sac City, IA
365
Sac City, IA to Port Washington, WI
232
Port Washington, WI to Mackinac Island, MI
252
Mackinac Island, MI to Ann Arbor, MI
6746
Total
19,035
teaching children mathematics • May 2010 3