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Technology and other electronic media have
become popular with students of all ages. Game
programs, virtual-reality helmets, and online environments are just some of the electronic tools
that fascinate school-age students. Yet numerous engaging applications of technology for
learning school mathematics exist. This article
describes how to use virtual pattern blocks,
a virtual graphing tool, and a spreadsheet to
explore the concept of function in algebra in
grades 6 through 8. By using these electronic
tools, students have the opportunity to generate
and compare various representations of a function. The technology enables students to focus
on the mathematics of the tasks by eliminating the distraction of tedious computations.
Exploring Patterns and
Functions
Students in the middle grades study more abstract
concepts and move away from the concrete
tasks of elementary school. Sometimes they
need extra help and scaffolding to make this
transition. The use of contextualized problems
with multiple representations can help students
connect these concrete and abstract mathematical ideas. Middle school students sometimes
see using concrete materials as less sophisticated than using a technological tool. Therefore,
using a virtual mathematics applet may be more
appealing to students, yet still will provide the
necessary scaffolding and support that moves
students into abstract algebraic concepts.
Task 1 allows students to create visual models
of square frames that are used to generate data
representing both linear and quadratic relationships. In Task 2 students examine the data they
have recorded and look for patterns in the number of tiles that form the frame, the number of
interior tiles, the total number of tiles, and the
perimeter. In Task 3, students have the opportunity to verify the rules they created in Task 2 by
using the spreadsheet. Task 4 is designed to allow
students to explore a graphical representation of
their functions. And in Task 5 students analyze
the graph and solidify and extend their understanding through articulation of their thinking.
Important goals of the NCTM Standards for
middle school students are to investigate, make
conjectures, verify those conjectures, and generalize concepts (NCTM 2000). These investigations
should focus on linear and non-linear relationships that develop students’ understandings
of characteristics of each relationship and that
allow students to distinguish between the two
types of relationships. Students can explore
these relationships in a variety of ways, using
tables, graphs, and symbols so that they then
can compare the different representations. The
ability to move flexibly among various representations increases students’ mathematical power.
© 2003 National
Council of Teachers
of Mathematics
The following investigation includes five different
tasks that focus on helping students in grades
6–8 develop a comprehensive view of functions.
These tasks have the following objectives:
•
Incorporate the use of technology (virtual
pattern blocks, spreadsheets, and a graphing tool) into the familiar frames problem
•
Explore the algebraic, graphic, and
numeric representations of functions
•
Introduce students to concepts of
functions and function notation
Task 1: Generating, Gathering
and Recording Data
Task 2: Look for Patterns
and Create a Formula
The first task allows students to create visual
models of square frames that are used to generate data representing both linear and quadratic
relationships. Students begin the task by creating
a 2 x 2 frame, using squares from the virtual pattern blocks. Virtual pattern blocks can be found
at several locations on the Web, including the
NCTM Web site (http://my.nctm.org/eresources/
repository/6165/applet/shapetool/index.html ), the
National Library of Virtual Manipulatives (www.
matti.usu.edu/nlvm), and the Arcytech site (www.
arcytech.org). Using the square blocks on one of
these sites, students create a 3 x 3, 4 x 4, and
5 x 5 (and so on) frame using the virtual squares
(see fig. 1). The created shapes are called frames
because the blocks used to create the frames
are on only the outer edge of the shape; that is,
there is empty space in the interior of the frame.
For example, a 3 x 3 frame would have three
blocks on each side of the frame with an open
space the size of one block in the middle; therefore, the frame would contain eight blocks. A 4
x 4 frame would have four blocks on each side
of the frame with an open space the size of four
blocks in the middle; therefore, the frame would
contain twelve frame tiles. Interior tiles are created by using a square tile that is the same size
as those in the frame, but are a different color.
In this task, students examine the data they have
recorded and look for patterns in each of the
categories: Frame Tiles, Interior Tiles, Total Tiles,
and Perimeter. Students can begin by describing what happens from one stage to the next in
each sequence. For example, in the Frame Tiles
sequence, students should notice that numbers
increase by four in each stage. The Perimeter
sequence also grows by a common difference of
four. In the other two sequences, no common difference exists. Students should discuss these patterns and look for other characteristics or ways to
describe the number sequences. To help students
with this task, ask questions such as “What is
happening to the numbers (increasing, decreasing)? By how much are they increasing? Is there
anything the numbers have in common? How do
these numbers relate to the length of each frame?”
After creating the frames, students should
record the following data about each frame
using a standard spreadsheet program:
© 2003 National
Council of Teachers
of Mathematics
•
Frame dimensions: (for example,
2 x 2, 3 x 3, 4 x 4)
•
Side length: the length of one side of
the frame (for example, two would be
the side length for a 2 x 2 frame)
•
Frame tiles: the number of tiles that form the
frame of the square and whose edges lie on
the perimeter of the frame (for example, a
3 x 3 frame would have eight frame tiles)
•
Interior tiles: the tiles that would fit on
the interior of the square that forms
the frame (for example a 3 x 3 frame
would have one interior tile)
•
Total tiles: the sum of the frame tiles
and the interior tiles (for example a 3 x
3 frame would have nine total tiles)
•
Perimeter– the distance around the outside
of the frame (for example a 3 x 3 frame
would have a perimeter of twelve units).
After students have the opportunity to explore
the iterative pattern, they can consider what
would happen if they needed to generate data
for a 50 x 50 square or a 100 x 100 square.
Clearly, continuing the pattern is one way to find
the information, but it is not the most efficient
way. Through discussion, encourage students to
define an explicit rule for each sequence, using
the variable n to represent the length of the
side of the frame. By creating this rule, students
can generate data for frames of any length.
Task 3: Verify Your Formulas
Task 4: Explore the Function
In this task, students have the opportunity to verify the rules they created in Task 2
by using the spreadsheet. Students should
create four new columns on their spreadsheet with the following headings:
This task is designed to allow students to explore
a graphical representation of their functions.
Students can enter data as a set of ordered pairs
and as a function, and view both representations
on the same graph. For example, to view the
Frame Tiles function graphically, students might
first enter their data as a set of ordered pairs in
which x represents the length of the side of the
frame and y represents the number of frame tiles
in that frame. The entry for the third frame, 4 x
4, would be (4, 12). Clicking on PLOT creates
a line graph of this information. Students can
toggle between a connected graph and scatterplot view. Next, students can enter their rule
in the function box (4*x – 4). Clicking on PLOT
shows both graphs concurrently. Students can
compare the two graphs and begin to get a
sense of how to illustrate the data (see fig. 4).
•
Formula (Frame), in which they will
test their rule describing the pattern
for the frame tiles sequence
•
Formula (Interior), in which they will
test their rule describing the pattern
for the interior tiles sequence
•
Formula (Total), in which they will
test their rule describing the pattern
for the total tiles sequence
•
Formula (Perimeter), in which they will
test their rule describing the pattern for
the perimeter sequence (see fig. 3)
Using the spreadsheet’s formula capabilities,
students can enter the formula once under each
column heading and cut and paste it to the
remaining cells in the column to quickly generate
data. For example, the formula for Frame Tiles,
using n as the length of the side of the frame, is
4n – 4. In the first cell under the new heading titled
“Formula (Frame),” students click on the formula
button and enter 4*B2 – 4 (where B2 is the cell in
which the length of the 2 x 2 square, 2, is entered).
The number 4 should appear, which shows the
number of tiles in the frame. Selecting the cell in
which the formula was entered, copying it, and
then pasting it into the remaining cells in the column will generate results using each of the side
lengths—3, 4, 5, 6, 7, and 8—for n. These numbers should match those that the student entered
under Frame Tiles, verifying the student’s rule. If
the numbers do not match, the rule is incorrect
and the student can analyze the incorrect results
to determine adjustments to make the rule correct.
© 2003 National
Council of Teachers
of Mathematics
Task 5: Analyze the Graph
Once students have the opportunity
to gather data, look for patterns, create and
test formulas, and explore graphs, they need
opportunities to solidify and extend their understanding through articulation of their thinking.
This can occur through written and oral communication in small groups as well as wholeclass settings. Teachers may focus on a variety
of topics and questions during these discussions. Depending on the level of the class,
these discussions can lead to exploring such
concepts as linear versus quadratic functions,
function notation, and domain and range.
Conclusion
This investigation allows students to look for patterns in data they have generated in the context
of a problem, a fundamental skill in the development of algebraic thinking. Through these tasks,
students can see how developing an algebraic
expression or rule that describes the pattern
allows them to generate data quickly. By creating
their own numbers, students are able to observe
the behavior of the pattern as it emerges—how
the numbers change and how the relationship
develops between the input and output variables.
As a result, students are able to use an arithmetic
or procedural approach, in which operations are
carried out on numbers, rather than a structural
approach, in which operations are carried out on
algebraic expressions. Research on the development of beginning algebraic thinking indicates
that students find the former type of approach
more meaningful than the latter (Kieran 1992).
Exploring the functions generated by this problem through several different views—numeric,
algebraic, and graphical—helps students begin
to develop their understanding of variables.
Research indicates that exposure to a curriculum
that focuses on the use of multiple representations helps students develop algebraic reasoning skills (Moseley and Brenner 1997). Students
using multiple representations display more
flexible usage and a deeper understanding of
variables when solving problems than do students exposed to a more traditional curriculum.
© 2003 National
Council of Teachers
of Mathematics
In addition, this investigation allows students the
opportunity to explore the difference between linear and quadratic functions graphically while they
relate their explorations to the situation and data
that the graph describes. For example, students
can compare the difference in the consecutive
entries for the frame tiles to the differences in the
consecutive entries for the interior and total tiles.
They will discover that the frame tiles display a
constant difference, whereas the differences for
the other two are not constant. Through discussion, students can compare this finding to the
graphical representations and explore the relationship between the rate of change—whether it is
constant or not—and the behavior of the graph.
Finally, students can begin to develop an understanding of domain and range by comparing
the graph of the data set and the graph of the
function. For example, in looking at the graph of
both the data set and function for Frame Tiles,
students can explore the possible values for x
and y represented in each, discussing the differences and what each represents. In the data
set, values for x, the domain of the function, are
limited to the set {2, 3, 4, 5, 6, 7, 8} because they
represent the side lengths of the frames actually built. A similar result occurs for the range. In
the graph of the function, the domain and range
are both continuous, representing the result for
frames with a side length of any real number.
Introducing students to the terms domain and
range in this context establishes a more meaningful learning experience by allowing them to
connect these concepts to a familiar situation.
student activities
Task 1 Generating, Gathering,
and Recording Data
Task 2: Look for Patterns and
Create a Formula
•
Open the virtual pattern blocks at http://
matti.usu.edu/nlvm/nav/frames_asid_
170_g_2_t_2.html?open=activities.
•
•
Using the virtual squares, create a 2 x
2, 3 x 3, 4 x 4, and 5 x 5 (and so on)
squares with a frame of one color and
an interior of another color (see fig. 1).
Frame Tiles_______________________________
_________________________________________
_________________________________________
•
Open a spreadsheet and record the following
information in the first cell of each column: In
column A, “Frame Dimensions”; in column B,
“Side Length”; C, “Frame Tiles”; D, “Interior
Tiles”; E, “Total Tiles”; and F, “Perimeter.”
Interior Tiles ______________________________
_________________________________________
_________________________________________
•
•
© 2003 National
Council of Teachers
of Mathematics
Total Tiles ________________________________
_________________________________________
_________________________________________
Under the Frame Dimensions column,
enter 2 x 2, 3 x 3, 4 x4, 5 x 5, 6 x
6, 7 x 7, and 8 x 8 (see fig. 2).
Using your virtual squares, record the side
length, the number of frame tiles, the number
of interior tiles, the total number of tiles,
and the perimeter in the spreadsheet.
Study the data to find a pattern for the
sequence of numbers in each category.
Describe your pattern for each category.
Perimeter _____________________________
______________________________________
•
Using your pattern, fill in the spreadsheet
for a 10 x 10 square. Verify your pattern by
using the virtual manipulatives to generate
a 10 x 10 square. Were you right? _______
_____________________________________
•
Once you have determined each pattern,
express it by using n to represent the number
length of a side of the square frame; for
example, for the 4 x 4 square, n would be 4.
In the row following the 8 x 8 square frame
on the spreadsheet, enter n x n under the
Frame Dimensions column and type in a
formula under each of the categories.
student activities
Task 3: Verify Your Formulas
Task 4: Explore the Function
•
•
Open the Graphit! graphing tool (www.shodor.
org/interactivate/activities/graphit/index.html).
•
Enter the data for the number of frame tiles
in each frame as ordered pairs such that
x represents the length of the side of the
square and y represents the number of frame
tiles in that frame. For example, the entry
for the third frame, 4 x 4, would be (4, 12).
•
In the function box, enter the formula for
the number of frame tiles in each frame.
•
Click on PLOT. The graph will show
the data points in red and the function
graph in blue. What do you notice?
•
Toggle between the Scatter and Connected
options under Plot Type to get a different view
of the relationship between the graphs of the
data points and the graph of the function.
In columns G, H, I, and J on your spreadsheet, type in the following: Formula
(Frame), Formula (Interior), Formula
(Total), and Formula (Perimeter).
•
Use the spreadsheet’s formula tool to enter
the formula for the frame tiles in cell G2,
under the Formula (Frame) column. Copy
and paste this formula into cells G3–G8.
•
Repeat the procedure for the remaining columns: Formula (Interior), Formula
(Total), and Formula (Perimeter).
•
Check your results with your original data.
What do you notice? What adjustments do
you need to make to the formulas, if any?
Repeat these procedures for the
Interior Tiles, Total, and Perimeter.
© 2003 National
Council of Teachers
of Mathematics
student activities
Task 5: Analyze the Graph
•
What do you notice about the graphs?
________________________________________
________________________________________
•
How are the graphs of the three functions
similar?
________________________________________
________________________________________
•
How are they different?
________________________________________
________________________________________
•
Do you see anything in the formulas that
might explain this?
________________________________________
________________________________________
Figure 1: 2 _ 2, 3 _ 3, and 4 _ 4 Frames.
© 2003 National
Council of Teachers
of Mathematics
Figure 2: Spreadsheet
•
Do you see anything in the data that might
explain this?
________________________________________
________________________________________
•
Why does the function graph appear to be
continuous but the data points do not?
________________________________________
________________________________________
Figure 3: Formula Columns
Figure 4: Function and Data Graph
references
Kieran, Carolyn. “The Learning and
Teaching of School Algebra.” Handbook of
Research on Mathematics Teaching and
Learning, edited by Douglas A. Grouws,
pp. 390–419. Reston, Va.: National Council
of Teachers of Mathematics, 1992.
Moseley, Bryan, and Mary E. Brenner. Using
Multiple Representations for Conceptual
Change in Pre-Algebra: A Comparison Of
Variable Usage with Graphic and TextBased Problems. Washington, D.C. Office of
Educational Research and Improvement,1997.
National Council of Teachers of Mathematics
(NCTM).. Principles and Standards for School
Mathematics. Reston, Va. NCTM, 2000.
© 2003 National
Council of Teachers
of Mathematics