Graphic Organizers for Teaching Advanced-Level Mathematics:
Understanding Negative Integer Exponents & Solving Linear Equations
Category: Mathematics
Grade Level: Secondary Grades
1. What is the purpose of using Graphic
Organizers in advanced-level mathematics?
Graphic organizers are visual-spatial displays
used to organize knowledge and represent
relationships among pieces of information. Due
to their visual-spatial nature, they require less
reliance on language skills. While other
techniques used in mathematics instruction,
such as the use of manipulatives and graphs,
also rely little on language skills, their
applicability to advanced-level mathematics (ie.
algebra) has been questioned (Ives & Hoy,
2003). The effectiveness of graphic organizers
in improving reading comprehension has
therefore prompted the use of graphic organizers
in teaching advanced concepts in mathematics.
2. With whom can they be used?
In advanced mathematics, graphic organizers
can be used by teachers to complement regular
classroom instruction. Graphic organizers might
be particularly useful for students with learning
disabilities who have weaker language skills and
stronger spatial and nonverbal reasoning skills.
3. How does one adapt Graphic Organizers
to mathematics instruction?
Many different types of graphic organizers have
been developed for reading comprehension and
other language instruction. Consequently, when
developing graphic organizers to use in
mathematics, one can look to literature on
reading comprehension and other literature on
strategies for creating graphic organizers (eg.
Winn, 1991). However, there are three
important points to remember when applying
graphic organizers to advanced mathematics:
I) In mathematics, the content of graphic
organizers should no longer be verbal elements,
such as words, phrases, and sentences. Rather,
content should be mathematical in nature,
consisting of numbers, other types of symbols,
expressions and equations.
II) Advanced mathematics skills emphasize
understanding concepts, patterns, and processes,
as opposed to memorization of numbers,
expressions, and equations. Consequently, the
goal of using graphic organizers in higher-level
mathematics should be to help students
recognize and understand relationships between
elements. Therefore, within graphic organizers,
the way in which elements are spatially
arranged should reinforce these relationships.
III) Graphic organizers should be used to
complement, not substitute for regular
classroom instruction. Teachers should
explicitly teach concepts while relating them to
graphic organizers.
4. What are some examples of Graphic
Organizers that can be used in higher-level
mathematics?
Suggestions for graphic organizer use in the
areas of negative integer exponents and solving
linear equations are presented below.
Understanding Negative Integer Exponents
The goal of this graphic organizer is to present
negative integer exponents as a meaningful
concept which naturally builds on the students’
previous knowledge of positive exponents. The
graphic organizer is used to draw attention to
the relationship between positive integer
exponents and negative integer exponents. The
teacher starts out by engaging students in an
interactive discussion of information related to
positive integer exponents. Such information is
visually laid out in a column of increasing
exponents and multiplying values, as shown in
Figure 1. The teacher encourages the students to
point out patterns in this column. For example,
as one goes down the column, exponents
decrease by one and values are divided by 2.
Negative integer exponents are then introduced
in a second parallel column, as shown in Figure
2, and students are encouraged to apply the rules
used with positive integer exponents (ie. as
exponents decrease by 1, divide the previous
value by 2). The teacher then encourages
students to look for more patterns in the
organizer. It becomes apparent that exponents in
the right-hand column are the opposite of
exponents in the left-hand column and that
values in each column are reciprocals of each
other. With this technique, concepts are
communicated by the relative positioning of
elements within the organizer. [For more details
and specific scripts regarding this method, see
Ives & Hoy (2003).]
Solving Systems of Three Linear Equations
with Three Variables
The frame suggested for this graphic organizer
is shown in Figure 3. The frame is divided into
cells. Again, the teacher engages the students in
an interactive discussion as they are guided
through the various steps to solving the
equations while using the graphic organizer.
The equations are first placed in the cell in the
top left corner of the graphic organizer (see
Figure 4). They are then worked through from
cell to cell in a clockwise direction. Equations
are combined in the top row and solved in the
bottom row. The Roman numerals on top of
each column correspond to the number of
variables within the equations being worked.
Thus the Roman numerals allude to one of the
goals of solving linear equations – combining
equations so that they contain fewer variables.
Figure 5 represents the completed graphic
organizer. The relative positions between
elements, as well as the relative positions
between elements and the frame, help to
emphasize relationships and concepts important
in understanding how to solve linear equations.
[For more details and specific scripts regarding
this method, see Ives & Hoy (2003)].
5. In what types of settings should Graphic
Organizers be used?
Teachers can use graphic organizers to facilitate
classroom learning. Once understood, they can
be applied independently by students.
6. To what extent has research shown
Graphic Organizers to be useful in higherlevel mathematics?
The effectiveness of graphic organizers in
teaching higher-level mathematics has been
verified informally in classrooms. Systematic
research on the graphic organizer for solving
linear equations (described above) is currently
underway to determine whether its use can lead
to improvement on mathematics achievement
measures for students with learning disabilities
(Ives & Hoy, 2003). Initial evidence from this
study suggests that students find the graphic
organizers useful.
References
1. Ives, B. & Hoy, C. (2003). Graphic
Organizers Applied to Higher-Level Secondary
Mathematics. Learning Disabilities Research &
Practice, 18(1), 36-51.
2. Kim, A., Vaughn, S. Wanzek, J. & Wei, S.
(2004). Graphic organizers and their effects on
reading comprehension of students with LD.
Journal of Learning Disabilities, 37, 105-118.
3. McEwan, S. & Myers, J. (2002). Graphic
organizers: Visual tools for learning. Orbit,
32(4), 30-34.
4. Winn, W. (1991). Learning from maps and
diagrams. Educational Psychology Review, 3,
211-247.
Websites:
Information on Types of Graphic Organizers
http://www.sdcoe.k12.ca.us/score/actbank/torganiz.htm
http://www.writedesignonline.com/organizers/
More Information on Graphic Organizers Can Be Found
In Recommendations For “Writing/Spelling” On This
Website:
http://www.oise.utoronto.ca/depts/hdap/report_writer/Spe
ll.htm
Reviewed by: Carly Guberman
26 = 64
III
II
I
25 = 32
24 = 16
23 = 8
22 = 4
21 = 2
Figure 1. Left column of a graphic organizer for
teaching negative integer exponents (Ives &
Hoy, 2003).
Figure 3. Blank graphic organizer for solving
systems of linear equations in three variables
(Ives & Hoy, 2003).
III
2x +4y+2z=16
II
I
-2x-3y+z=-5
6
-6
2 = 64
2 = 1/64
5
2x+2y-3z=-3
-5
2 = 32
2 = 1/32
24 = 16
2-4 = 1/16
23 = 8
2-3 = 1/8
22 = 4
2-2 = 1/4
21 = 2
2-1 = ½
20 = 1
Figure 2. Completed graphic organizer for
teaching negative integer exponents (Ives &
Hoy, 2003).
Figure 4. Graphic organizer for solving systems
of linear equations in three variables as it may
appear after the original equations have been
entered (Ives & Hoy, 2003).
III
2x +4y+2z=16
II
I
y+3z=11
-2x-3y+z=-5
z=3
-y-2z=-8
2x+2y-3z=-3
2x+4(2)+2(3)=16
2x+14=16
2x=2
x=1
y+3(3)=11
y+9=11
y=2
z=3
Figure 5. A completed graphic organizer for
solving systems of linear equations in three
variables (Ives & Hoy, 2003).