investigations
Christine Cullen and Joseph Gaymore
Ocean Quest
T
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 is—
multidimensional in content;
open ended, with several acceptable solutions;
an exploration requiring a full period or longer
to complete;
centered on a theme or event; and
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 “Investigations” presents a scripted
sequence and set of directions for a mathematical
exploration for the purpose of communicating what
happened in this particular classroom, Principles
Christine Cullen, [email protected], has taught for six years in Pinellas County, Florida, the
past three years at Campbell Park Elementary. She is currently working on her master’s
degree with a mathematics-science emphasis. Along with other teachers at Campbell Park
Elementary, she has developed an integrated curriculum with a marine science theme. Joseph
Gaymore, [email protected], previously served as a math coach and classroom teacher at
Lealman Avenue Elementary, St. Petersburg, Florida. He is now a mathematics staff developer
for Pinellas County Schools. His focus this year is on increasing the level of conversation about
mathematics in the classroom so that all students can benefit.
Edited by Sue McMillen, [email protected], and Jodelle S. W. Magner, magnerjs@
buffalostate.edu, who both teach mathematics and mathematics education classes at Buffalo State College in Buffalo, New York 14222. “Investigations” highlights classroom-tested
multilesson unit activities that develop conceptual understanding of mathematics topics. 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 submit manuscripts appropriate for this department by accessing tcm.msubmit.net. Manuscripts are limited to a maximum of twelve double-spaced typed pages and two
reproducible pages of activities.
344
and Standards for School Mathematics (NCTM
2000) encourages teachers and students to explore
multiple approaches and representations when
engaging in mathematical activities. Each investigation will come alive through students’ problemsolving decisions and strategies in readers’ own
classrooms. As a result of their exploration, students incorporate their reasoning and proof skills
as they evaluate their strategies. The use of multiple
approaches creates the richness that is so engaging
in an investigation. It also helps students find new
ways of looking at problems and understand different ways of thinking about them.
The Ocean Quest investigation focuses on developing algebraic thinking and communication in
the classroom. Specifically, students represent and
analyze mathematical situations using algebraic
symbols and then look for and apply relationships
between the quantities to arrive at a logical answer
(NCTM 2000). Students work through the process
of solving systems of equations using a table with
pictures. Through classroom discourse, students
move away from random guess and check toward
more logical algebraic thinking.
The Investigation
Learning goals, rationale, and
pedagogical context
Traditionally, arithmetic was the only mathematics taught in the primary grades. Formal algebra
instruction was postponed until adolescence
because young children were assumed to not yet
have the developmental capacity to think abstractly.
Nevertheless, arithmetic is algebra to the extent that
it provides opportunities for making and expressing generalizations. Children too young to master
formal symbolic algebra are capable of identifying
patterns, reasoning with variables, and identifying possible values for an unknown. It is essential
for students to learn to represent and manipulate
unknowns in symbolic form. Introducing algebra
at a younger age may help students understand the
Teaching Children Mathematics / February 2008
Copyright © 2008 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.
concept of variables. Through classroom discussion, children not only understand more about variables but also can see other approaches to solving
equations using variables. Algebra is for all ages.
The earlier the intervention, the more algebraic
thinking develops.
This investigation provides children with an
opportunity to reason logically and determine values for combinations of different objects. It is an
adaptation of Challenge #30, “Smiles,” located on
the NCTM Web site www.figurethis.org/ (see fig. 1).
The Ocean Quest investigation involves similar
activities; however, it has been modified to make a
connection to the school’s marine science theme.
Campbell Park Elementary Marine Science
Center is a unique elementary school located in
St. Petersburg, Florida. The school comprises a
diverse population of approximately 550 students.
One major focus of the school is to integrate a
stimulating marine science theme into the general
curriculum. To enhance instruction, the school
collaborates with the University of South Florida
Marine Center, St. Petersburg Pier Aquarium,
Florida Marine Research Institute (FMRI), Eckerd
College, MOTE Marine, and the Florida Aquarium.
Ocean Quest was taught to a fourth-grade class by
classroom teacher Christine Cullen and a district
staff developer, Joseph Gaymore.
One of the investigation’s goals is to further
develop students’ understanding of logical reasoning. Promoting and valuing all classroom discourse
can enhance the development of logical reasoning,
which can be developed indirectly within the context of classroom discussion (Chapin, O’Connor,
and Anderson 2003). By encouraging students to
share their thinking process with their classmates,
the gap can be lessened between those students who
rely completely on random guess-and-check and
those who are comfortable using logical reasoning.
Through classroom discourse that makes student
thinking public, students can learn that there is
more than one way to solve a problem.
Objectives of the investigation
Students will—
develop their ability to reason with and represent
with variables;
move away from random guess-and-check to
a more logical approach for finding values for
variables in a system of equations; and
understand various approaches to solving the
same problem.
Teaching Children Mathematics / February 2008
Figure 1
Which is worth more? (www.figurethis.org)
Materials
Lesson 1 (1–2 days)
Each student needs the following materials:
“How Much Is Each Symbol Worth?” (see
fig. 2)
“How Much Does Each Fish Cost?” (see
fig. 3a)
“How Much Does Each Whale Weigh in Tons?”
(see fig. 4a)
Each teacher and each pair of students need the following materials:
pattern blocks (rhombus, triangle, and hexagon)
Lesson 2 (1–2 days)
The teacher presents the visual aid “What Is the
Length of Each Shark in Feet?” (see fig. 5) to the
entire class. Suggested formats include, but are not
limited to, recreating the grid on a poster or an overhead transparency. In addition, each student needs
the following material:
“What Is the Length of Each Shark in Feet?”
(See activity sheet with no symbols.)
345
Figure 2
The students were introduced to a system of equations.
How Much Is Each Symbol Worth?
32
29
25
24
33
40
37
Previous knowledge
The students in this investigation have previously
solved for the values of shapes representing numbers in expressions and equations. They have also
been exposed to the use of variables in expressions,
equations, and word problems.
Lesson One
Introducing systems of
equations
The purpose of the initial activity was to introduce
the children to systems of equations in a
nonthreatening manner. We began by distributing rhombus, triangle, and hexagon pattern blocks to each pair of
students and also placing the blocks
on a document projector. The hexagon was assigned a value of five, and a
total value of twenty was given for the sum of all
346
the shapes. As expected, when asked to determine
possible values of the rhombus and the triangle,
students responded with a wide range of answers.
Next, we assigned a value of six to the triangle
and asked students to determine the possible value
of the rhombus. After about ninety seconds, Lisa
responded with “nine.” When Tiffany was asked if
she agreed or disagreed, she agreed and explained,
“All of the shapes equal twenty. [The rhombus] has
to equal nine, because five plus six equals eleven,
and eleven plus nine equals twenty.” We restated
Tiffany’s reasoning to allow others who initially
did not understand to hear it again. Throughout
subsequent class discussions, teachers often restated
students’ responses in this manner.
Now that the students had solved for an unknown
variable in a single equation, a system of equations
was introduced. Students were given the grid (see
fig. 2) and asked to determine the value of each of the
following symbols: a cloud, a flower, and a sun. The
row and column sums were shown to the right of each
row and below each column respectively, but none of
the individual values was given. Thus, the grid represents a system of equations: seven equations in three
unknowns. Students in grades 3–5 should begin to
understand that different models for the same situation could give the same results (NCTM 2000). You
may want to remind students that each picture has a
unique value and that the same picture always has
the same value in a given grid.
We were curious to see how children approached
this problem: guess-and-check, logical reasoning, or
a combination of both. Initially, many of the students
simply guessed values for the symbols regardless
of whether they resulted in the correct sums. When
they were asked to share their values, Zanquesha answered, “The sun is equal to seven.” Oscar
responded, “I disagree because the answer is eight.
If the three suns in the bottom row are seven, then all
three [together] will be twenty-one, not twenty-four.
It needs to be twenty-four, and eight three times is
twenty-four.” Tina was asked to restate Oscar’s explanation and then indicated that she agreed “because
eight plus eight plus eight are the only numbers that
will get twenty-four.”
Only a few students were able to determine that
the bottom row containing the same three symbols
(suns) might be the easiest place to start. Zanquesha
suggested that the bottom row would be the easiest
place to begin because the numbers all have to be
the same when the symbols are the same, and “it is
easier to find the answer when all the numbers are
the same. Three suns equal twenty-four. In the other
Teaching Children Mathematics / February 2008
rows, the symbols are different. You can’t figure
out how many each is worth. When they are all the
same, it is like division; they all are an equal part.”
Students were invited to add their own thoughts
to what others were saying. Doing so helped
several children understand that many different
approaches could lead to an answer. For example,
when asked to add to Zanquesha’s comment, Chase
said, “I think she means twenty-four divided by
three equals eight, so each sun has to be worth
eight points.” When asked to provide an equation
to represent this idea, Tina stated that three times
eight equals twenty-four, and Sammy offered,
“Eight plus eight plus eight equals twenty-four.” At
this point, we suggested that the students write the
numeral 8 in each grid section containing a sun.
Now that the students agreed the sun had a value
of eight, they were asked to determine the value of
the cloud by first thinking alone, then sharing their
thinking with a partner, and finally sharing with
the class. Chase looked at the top row and suggested, “If you subtract eight from thirty-two, you
get twenty-four. There are two clouds, so twelve
plus twelve gives you twenty-four.” Sammy shared
that he divided twenty-four by two to get twelve.
TaVasha said she multiplied twelve times two to
get twenty-four. The students recorded the numeral
12 in the sections containing the clouds. Then they
solved for the value of the flowers. Once all three
symbols had values, we modeled how to check each
of the row and column sums.
The second activity, “How Much Does Each Fish
Cost?” (see fig. 3a), used different types of fish that
the students were familiar with or would become
familiar with through the school’s marine science
integration: a beta fish, a goldfish, or a clown fish.
As in the first activity, one symbol, the beta fish,
was used three times in a single row. We asked the
students to find the cost of each beta. Working with
a partner, most of the children determined rather
quickly that the value of the beta fish was $2. This
time, we encouraged students to determine on their
own which symbol to solve for next. (One pair’s
solution is shown in fig. 3b). In the ensuing classroom
discussion, we noticed students taking more of an
interest in adding to one another’s ideas. They were
also becoming more confident about solving systems
of equations.
The third activity for this first lesson, “The Weight
of Whales,” showed a blue whale, a gray whale, and
a right whale (see fig. 4a). Students had a little more
difficulty with this problem because the sums were
larger. As in the previous activities, one symbol, the
Teaching Children Mathematics / February 2008
Figure 3
(a) The second activity used symbols familiar to the students.
How Much Does each fish cost?
SUM
$6
$12
$19
$16
SUM
$13
$19
$21
Goldfish
Beta
Clown Fish
(b) One student-pair’s solution to the activity “How Much Does Each Fish
Cost?”
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Figure 4
(a) Students had more difficulty when the activity dealt with larger sums.
How Much Does Each whale weigh in tons?
SUM
330
170
200
185
SUM
215
Right Whale
375
Gray Whale
I went with row 2. How I figured it out was to
subtract the weight of the right whale from 170,
which gave me 60. Since there are two gray
whales in that row, I divided 60 in half and got
30, so [each] gray whale weighs 30 tons.
295
Blue Whale
(b) Working in pairs, students displayed various approaches to the same
problem.
right whale, was used three times in a single row. We
prompted each pair of students by asking which row
or column would be a good place to start. After a minute or so, Alex replied, “The first row.” When asked to
explain why he agreed or disagreed with Alex, Chase
agreed: “It is easier to find what each [right] whale is
worth. If all three [right] whales together are worth
330, they have to be the same number.” We now felt
comfortable that all of our students had a solid, systematic way to find a starting point.
Working in pairs, the students finished solving the
system of equations. Two different approaches are
shown in the student solutions in figure 4b. The students shared their solutions during a whole-class discussion. For example, Tina’s explanation for finding
the value for the weight of the gray whale follows:
By making the discussion public, students now saw
that different ways exist to solve this problem correctly and that the values do not have to be found in a
certain order. Many students wanted to discuss their
own solutions and were excited to see that they had
correct answers, too.
Lesson Two
Abstract variables
After two days of activities using only pictures as the
variables, the activity “What Is the Length of Each
Shark in Feet?” allowed for the more abstract representation of a variable by a letter. “Children develop
notions of the idea and usefulness of variables, which
they may express with a box, letter, or other symbol
to signify the idea of a variable as a place holder”
(Cuevas and Yeatts 2001, p. 3). Children need exposure to variables at a young age to help them develop
a clear concept of what a variable is and to understand
it in the abstract later. Cuevas and Yeatts note that
“from the lower elementary grades, students need to
have experiences with the uses of variables to develop
facility with them in different contexts” (2001, p. 3).
Our lesson began with students sharing their concepts of a variable. Alena offered, “It’s a letter used in
a problem when you don’t know what [the number]
is.” Sammy elaborated, “It is the unknown number.”
The students apparently understood what a variable is,
348
Teaching Children Mathematics / February 2008
but could they use one correctly in a problem-solving
activity? We hoped that after experiences with using
pictures, students were ready to move on to the more
abstract use of a letter in place of the picture.
We placed the grid with shark symbols (see fig. 5)
on the document projector, and the students identified the symbols as a hammerhead shark, a tiger
shark, and a bull shark. Each student had the activity
sheet (with an empty grid) to use for finding shark
lengths. Students were quick to respond after being
asked, “What variables can we use to represent each
shark?” The consensus was h for hammerhead, t for
tiger shark, and b for bull shark. They quickly wrote
the variables in the corresponding places in their
blank grids, demonstrating their understanding of
which picture each variable represented.
The students began to solve for h, t, and b.
Although many of them obtained correct answers,
some students experienced difficulty in explaining
how they arrived at their answers. Most students
immediately figured out the value of h by using
the bottom row with three hammerhead sharks. We
anticipated that the students would then go to row
1 or column 2; however, this was not the case. We
noticed some students attempting to solve row 3, but
it was impossible because they had determined the
value for only one variable, h. We observed others
trying to solve row 2, but only h was known, and row
2 had two bs and one t. Only a few students were
working on row 1, which was solvable because h was
known and the row had only one other unknown, b.
Although we fully expected the students to have
an easier time after finding the value of h, this was a
perfect opportunity to stop for a whole-class discussion on why row 2 or row 3 might not be the best
place to go next. We reminded the students that they
knew the value of h was 10 and asked them how that
fact could help them solve for the sharks in row 2.
One student replied that b equaled 20 and t equaled
1. When asked if those were the only possible values
for b and t, another student replied that her answers
were 15 for b and 6 for t. After being asked how they
could be absolutely sure of the values of b and t, the
students studied the grid. Tina raised her hand:
Actually, row 1 would be easier than row 3. In row
1, we have two hs, which will give us 20, since one
h equals 10. And that leaves 14 feet for b. We can’t
figure out row 3 [yet] because there is one of each
shark. And all we know is that h equals 10; b and t
can be different numbers. Now that we have row 1
where b = 14, we can figure out what row 3 is and
[what] the tiger shark (t) [is].
Teaching Children Mathematics / February 2008
Figure 5
Overhead visual aid (with symbols)
What is the length of each shark in feet?
SUM
34
35
31
30
SUM
52
37
41
Tiger Shark
Bull Shark
Hammerhead Shark
After Tina’s explanation, many aha’s occurred
in the class. Students who had been unsure about
how to proceed were able to
restate Tina’s reasoning and
now understood. Her contribution helped make the learning
public. Seeking additional contributions, we asked if anyone figured it
out a different way. Sammy raised his
hand: “I tried column 2 next. If h equals ten, then
in column 2, there are three hs, which equal thirty
feet, and only a t left. If the column equals thirtyseven, then thirty-seven minus thirty equals seven,
so the tiger shark is seven feet.” Students were now
confident enough on systems of equations to create
their own.
Lesson Three
Creating their own systems
In the third lesson, students were asked to create a
grid system of equations for a friend to solve. Most
349
Figure 6
Students learned to create their own
systems of equations.
(a) A student-created grid shadowing the
format of those used in class.
students kept their grids simple (see fig. 6a), using
common shapes with easy numbers. Some of the
more advanced students tried to make a grid that
was not impossible to solve but would require some
deep thinking for the student who was trying to
solve it. One such attempt is shown in figure 6b
(the first row sum should be 240, not 140). We
were impressed that not one student just plugged in
numbers. They first created their three symbols on
the back of their papers, assigned a value for each,
and then placed each symbol in the grid. We were
reminded again that not only were the students
learning how to solve systems of equations, but
they were also learning to create their own system
of equations.
Reflections
(b) A student-created grid uses a column—instead of a row—with all the same objects. The
first row has an error; the sum should be 240.
This was a great activity to use with the students
because of the advanced level of classroom discussion and learning. Students were motivated to participate. Moreover, the feeling of accomplishment
also prompted more students to think about each
problem in a systematic way. In the end, many students were so enthusiastic that they started creating
their own puzzles.
Although we will definitely use this project
again, we will make a couple of changes—for
example, tweak the activities to be a bit more challenging and allow the students to verbally grapple
longer with the problems before we intervene.
Overall, we were able to understand the students’
thinking processes through the use of classroom
discussion. Students observed a variety of ways to
solve the problems. This discussion of emerging
strategies helped students make sense of the mathematics and promoted their use of logical reasoning over random guess-and-check. We believe our
students are now ready to tackle more challenging
systems of equations.
References
Chapin, Suzanne H., Catherine O’Connor, and Nancy
Anderson. Classroom Discussion: Using Math Talk
to Help Students Learn, Grades 1–6. Sausalito, CA:
Math Solutions, 2003.
Cuevas, Gil J., and Karol Yeatts. Navigating through Algebra in Grades 3–5. Reston, VA: National Council of
Teachers of Mathematics, 2001.
National Council of Teachers of Mathematics (NCTM).
“Figure This: Math Challenges for Families, 2004.”
www.figurethis.org.
——— . Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000.
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Teaching Children Mathematics / February 2008
Activity Sheet. Ocean Quest
WHAT IS THE LENGTH OF EACH SHARK IN FEET?
Use a variable to represent the different types of sharks
Solve to find the length of each type of shark in feet.
SUM
34
35
31
30
SUM
Teaching Children Mathematics / February 2008
351