Conceptualizing Ratios
with Look-Alike Polygons
N A N C Y K. D W Y E R, B E T T Y J. C A U S E Y - L E E,
[email protected], is a mathematics supervisor with the Detroit Public Schools in
Detroit, Michigan. They are both interested in professional development and preservice teacher education.
NEKEYA IRBY , [email protected], is a mathematics
teacher at Gompers School, a National Exemplary School,
in Detroit, Michigan. She is interested in integrating technology into the middle school mathematics curriculum.
The authors wish to thank the anonymous MTMS reviewers for their many helpful suggestions and comments.
426
l
N E K E Y A M. I R B Y
N THE MIDDLE GRADES, STUDENTS BUILD ON,
and extend their understanding of, fractions. They
begin to use fractions as ratios and expressions of
part-whole and other relationships. Using fractions
as ratios requires interpreting fractions differently than
when they represent part of a region or set of objects.
For example, the first term of a ratio may tell the number of objects in one set, and the second term may tell
the number of objects in a completely different set.
Figure 1 shows examples of several different kinds of
ratios (Kennedy and Tipps 1997).
When using fractions to symbolize part-whole relationships, we commonly solve problems using all
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
Copyright © 2003 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.
PHOTOGRAPHS BY NEKEYA M. IRBY; ALL RIGHTS RESERVED
NANCY DWYER, [email protected], teaches at the
University of Detroit Mercy and BETTY CAUSEY-LEE,
AND
• Between objects in two groups: In a classroom in which each child has 5 pencils, the
ratio of children to pencils is 1 to 5. This ratio
may be represented by the expression 1:5 or
by the common fraction 1/5.
• Between a subset of a set and the whole
set: If a set of 10 books contains 3 red books,
the ratio of red books to books is 3 to 10,
3:10, or 3/10.
• Between the sizes of two sets: If a room
that is 30 feet in length is compared with a
hallway that is 100 feet in length, then the
ratio of room length to hall length is 30 to 100,
30:100, or 30/100, or 3 to 10, 3:10, or 3/10.
• Between objects and their cost: If two
pencils cost 7¢, the ratio between pencils and
cost is 2 to 7, 2:7, or 2/7.
• Between the chance of one event’s occurrence and the occurrence of all possible events: If a 6-sided number cube is
tossed, the probability that a 6 will show is 1
to 6, 1:6, or 1/6.
Fig. 1 Different types of ratio comparisons
four operations; however, when fractions are used
as ratios to express relationships, they cannot be
added, subtracted, multiplied, or divided. Great
care must be exercised when introducing fractions
as ways of indicating ratios to ensure that students
clearly understand their use and meaning.
Students can develop in-depth understanding of
fractions as ratios through experiences with a variety of concrete models. The classroom activity with
similar polygons described in this article illustrates
how a carefully chosen investigation can help students move flexibly from concrete objects to various ratio representations of abstract ideas.
1. All the rectangles drawn from the same diagonal will be similar.
2. Diagonals that are close together will also
have ratios that are close.
3. On 1-centimeter graph paper, draw a diagonal for each set of similar rectangles. Draw
rectangles where the diagonal crosses the
grid; these points will form the rectangles’
upper-right vertexes. The rectangles shown
have a 3:4 ratio of width to length.
(a)
Similar rectangles
1. Use centimeter graph paper.
2. Draw a diagonal.
3. The bottom left point will be a vertex of every
triangle.
4. Mark off equal increments on the horizontal.
5. Find the points where the diagonal crosses
the intersecting lines on the grid. Join these
points to the points marked off on the horizontal to make the third side of each new triangle.
Preparing the Materials
PREPARE AND DUPLICATE A COLLECTION OF SIMilar polygons for children to cut out and sort. Rec-
tangles are easy to make, but triangles and other
polygons can also be used. Figures 2a and 2b
show an easy way to draw similar rectangles and triangles (Van de Walle 1994). Other similar polygons
can be made using this method. Make three or four
sets of similar shapes with about three or four sizes
of each shape. The following sets make a good mixture of rectangles to sort:
•
•
•
•
3/4: 3 × 4, 6 × 8, 9 × 12, 12 × 16
1/2: 1 × 2, 2 × 4, 3 × 6, 4 × 8, 5 × 10, 6 × 12
2/5: 2 × 5, 4 × 10, 6 × 15, 8 × 20
5/6: 5 × 6, 10 × 12, 15 × 18
(b)
Similar triangles
Fig. 2 Instructions for drawing similar figures
Trace the rectangles on plain paper, or use centimeter graph paper to facilitate measuring, and duplicate the pages for students to cut out.
V O L . 8 , N O . 8 . APRIL 2003
427
PHOTOGRAPH BY NEKEYA M. IRBY; ALL RIGHTS RESERVED
Conducting the Activity
WE WORKED WITH A CLASS OF FIFTH- AND SIXTH-
grade students to explore ratio concepts using sets of
similar rectangles. After the students cut out the rectangles, they sorted them to find which ones were
similar, or seemed to have the same shape. Then, we
asked the students to justify their classifications in
some way. We encouraged them to describe, in as
many ways as possible, why they placed rectangles
in the same or different sets. Sets of rectangles with
more obvious differences, such as those with ratios
that were farther apart, were easier to distinguish.
Some students stacked the rectangles, in much the
same way as they were originally drawn on graph
paper, and decided that a set is similar because all
the rectangles get bigger in the same way. Salud described the 2 to 5 set as having “very long lengths
and very skinny widths.” Joseph discovered that the
rectangles in the 5:6 set “kind of look like squares.”
Including a set of squares with the sets of rectangles
is a good way to reinforce the differences between
length and width in the rectangles. The squares increase by the same amount on every side. The increase in the length of the rectangles, however, is different from the increase in the width.
Next, we asked students to measure the sides of
the polygons to see if doing so would help them explain why some rectangles seemed to belong together and others did not. They used metric rulers
because measuring in centimeters gives more accurate and usable data than measuring in inches does.
After all the rectangles were measured, the students worked in pairs to find ways to enter their
data into tables. Because many different tables are
possible, this exercise encouraged students to think
about how to organize their data in a meaningful
presentation. The pairs then shared their tables
with the class. Students might be asked to copy
their work onto transparencies for this purpose.
Figure 3 shows three student tables.
Width of rectangle
1
2
3
4
Length of rectangle
2
4
6
8
Length of rectangle
2
4
6
8
Width of rectangle
5
10
15
20
Width
3
6
9
12
Length
4
8
12
16
The teacher’s role in this part of the investigation
was to prompt students to raise and discuss questions,
look for patterns, and express relationships in words
and with mathematical symbols. After entering data
into a table, Brandon noticed that in one set, the lengths
were all multiples of 5 units, whereas in another set, the
lengths were all multiples of 3 units. Others noticed that
although two sets had widths that were multiples of 3,
only one of these sets had lengths that were twice as
long as the widths. Some students discovered a multiplication pattern when they extended their tables. They
found that they could multiply both the length and
width of the smallest rectangle in a set by a given number to get the width and length of another rectangle in
the set. We encouraged the students to write sentences
to describe all the relationships they discovered.
Representing Relationships Algebraically
STUDENTS CAN ALSO WRITE ALGEBRAIC REPRE-
sentations for their verbal descriptions. If the rectangles are numbered from smallest to largest and
n is used to represent the order of a rectangle in the
list, then, for the rectangles of ratio 3 to 4, the width
is always 3 × n and the length is always 4 × n. This
method can be used to generate a list of possible
members of each set, including members that may
be missing from the sets of paper rectangles.
At this stage, students can be shown how to use
fraction representations to express ratio comparisons. Teachers should emphasize the properties
being compared instead of the terms numerator and
denominator. For example, we can write the comparison of 3-centimeter width to 4-centimeter length as
3/4. Students must understand that the fraction notation 3/4 represents a comparison. The part-whole
meaning that they have always associated with 3/4,
such as three parts of a whole that is divided into four
equal parts, does not make sense in this context.
Exploring Equivalent Ratios
WE ASKED THE STUDENTS TO WRITE RATIOS
Fig. 3 Three sample tables of side-length data
428
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
comparing width to length for each rectangle. They
=
15
18
15
20
=
=
20
24
18
24
=
not similar
6
Nonsimilar rectangles do not have
equivalent-fraction
ratios.
3
4
≠
3
6
3
Fig. 4 Similar rectangles and equivalent fractions
6
3
To make another 3:4
rectangle—
• add 3 to the width, and
• add 4 to the length.
4
8
5
}
12
12
=
similar
3
2
If I add 2 centimeters to
the length and the width, I
do not get a 3:4 rectangle.
}
10
9
4
}
6
=
8
=
⋅
⋅
3 2 6
=
4 2 8
}
5
6
6
}
4
=
Similar rectangles
have equivalentfraction ratios.
2
}
3
8
}}
noticed that in each set of similar rectangles, the ratios they had written were all equivalent ratios.
Some students observed that the reason for this occurrence was that making the sides of one rectangle twice as long as those of a second rectangle is
the same as multiplying both numbers of the ratio
by 2. Others noticed that all the ratios in a similar
set can be renamed to the same ratio (see fig. 4).
We then asked the students what would happen to
the ratios if they compared length to width. They
discovered that the ratios were still equivalent.
Students at this stage are making a transition
from additive to multiplicative reasoning. Darnel
thought that the 3:4 ratio was the same as the 5:6
ratio because 2 was added to each of the numbers 3
and 4 to get 5 and 6. To correct this misconception,
we had the students visually compare the 3:4 and
5:6 sets to search for look-alike rectangles between
the two sets. After this comparison, the students
concluded that the two sets were different. Joseph
described the 5:6 rectangles as looking almost like
squares. Darnel pointed out that the shorter sides
in the 3:4 set were not as long as the shorter sides
in the 5:6 set. To help further in correcting the additive misconception, we asked the students to trace
the 3 × 4 rectangle on a sheet of 1-centimeter graph
paper, shown as a shaded rectangle in figure 5a.
By adding 2 centimeters to the width and 2 centimeters to the length, students made a new 5 × 6 rectangle (see fig. 5a). Next, we asked students to draw a
6 × 8 rectangle on the same grid. When the diagonals of the three rectangles were drawn, the students discovered that the 3 × 4 and 6 × 8 rectangles
shared the same diagonal, but the 5 × 6 rectangle
had a different diagonal (see fig. 5b).
Students can be further convinced that the additive relationship does not apply by scaling the ratios. Multiply the ratios 3:4 and 5:6 by scale factors
until either two widths or two lengths (one from
each set) are the same, as follows:
6
(a)
Experimenting with addition to make new rectangles
The 3:4 and the 6:8
rectangles have the
same diagonal.
25
30
Liana found that when comparing the equal widths
of 15 centimeters, for example, the rectangle from
the 3:4 set must have a length of 20 centimeters
and the rectangle from the 5:6 set must have a
length of 18 centimeters. Chico made a similar discovery about different widths when comparing rectangles from two sets in which both rectangles had
lengths of 12 centimeters.
The 5:6 rectangle does
not have the same
diagonal as the 3:4 or
the 6:8 rectangle.
(b)
Comparing diagonals to check similarity
Fig. 5 Correcting an additive misconception about ratios
V O L . 8 , N O . 8 . APRIL 2003
429
Using Proportional Reasoning
to Make Predictions
STUDENTS CAN BE ENCOURAGED TO BEGIN TO USE
proportional reasoning by applying what they have
learned to make predictions about similar rectangles.
In this class, we began by asking, “What width must a
rectangle have if its length is 20 centimeters and it belongs to the 3:4 set of similar rectangles?” To answer
this question, the students must determine the scaling
factor. Darnel, the same student who earlier thought
that 2 centimeters could be added to each side of a
rectangle to get another similar rectangle, was later
able to easily answer this scaling question. Questions
of this type enable students to use proportional reasoning to solve for unknowns in a proportion without
using cross-multiplication. Presenting proportions as
equivalent ratios in fraction form emphasizes the relationships in the ratios without introducing the crossmultiplication procedure, as shown below:
3
3 × 5 15
=
⇒
=
4 20
4 × 5 20
In this activity, students first explored the similarity relationship by visually sorting rectangles.
Then, they described the observed relationships
orally and in written summaries. Next, they examined proportions as equivalent ratios in fraction
form. Finally, they were ready to represent these
similarity relationships by graphing their data.
Representing the Data Graphically
WE HAD THE STUDENTS GRAPH THE LENGTH-
versus-width or width-versus-length measurements
for each set of similar polygons and extend a line
through the data points, as shown in figure 6. They
length
I can find other 3:4 rectangles on my graph. First, I
draw a horizontal line from
length 10 centimeters to the
diagonal. Then, I draw a vertical line from the diagonal
down to the width. The new
rectangle is 10 centimeters
by 7.5 centimeters.
width
The 10-by-7.5-centimeter rectangle is exactly 4 centimeters
longer and 3 centimeters wider than another new rectangle
that is 6 centimeters by 4.5 centimeters.
Fig. 6 Using a graph to find new 3:4 rectangles
430
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
then used their graphs to make predictions about the
width of a rectangle in a particular set when given the
length. For example, Tanisha found that a rectangle
in the 3 to 4 set with a length of 24 centimeters would
have to have a width of 18 centimeters.
Encouraging Mathematical Reasoning with
Good Questions
ASKING GOOD QUESTIONS THAT ENCOURAGE
students to think about the relationships represented in the graph is important. Students will also
learn important questions to ask themselves when
modeling any data graphically. Good questions
might include the following:
•
•
•
•
What are the variables?
How are they related?
How can you describe the relationship in words?
As the length increases, how does the width
change?
• How can one of the larger rectangles be obtained
from the smallest rectangle in the set?
Each rectangle in the 3 to 4 set of similar rectangles is always 3 × n centimeters larger in width and
4 × n centimeters larger in length than the next
smaller rectangle in the set. For example, n can be a
positive integer, such as
3
6 9 12
⇒ ,
,
, ...,
4
8 12 16
where n = 2, 3, 4, . . . , or n can be a decimal, such as
3
1.5 4.5 7.5 10.5
⇒
,
,
,
, ...,
4
2 6 10 14
where n = 0.5, 1.5, 2.5, 3.5, . . . . This relationship
holds true for every rectangle in the set. We asked
the students to further test this discovery by using
other points on their diagonal lines to make and
measure rectangles with side measures that were
not even multiples of 3 centimeters and 4 centimeters. For example, Chico found that when he drew a
line on the graph to the point representing 10 centimeters in length, he found a corresponding width
of 7.5 centimeters. The resulting rectangle was exactly 3 centimeters wider and 4 centimeters longer
than a 4.5-by-6-centimeter rectangle, which also corresponded to a point on the graph. This relationship
in the graph is called the rate of change. Pick any
point on the diagonal representing the width and
length of a rectangle in the 3:4 set. Adding 3 centimeters to the width and 4 centimeters to the length
always results in another rectangle in the 3:4 ratio
set. After several trials, the students discovered
a similar relationship with subtraction. Subtracting
3 centimeters and 4 centimeters from the width and
length, respectively, of any rectangle in the 3:4 set
always results in another rectangle in the 3 to 4 set.
Extensions
MORE ADVANCED STUDENTS CAN INVESTIGATE
the effect of scale factor on perimeter and area.
They will discover that multiplying both the width
and length by 2, for example, produces a rectangle
with twice the perimeter but four times the area of
the original.
An exploration with similar triangles naturally
leads to side-altitude comparisons, as well as sideside ratios. In forming these ratios, care must be
taken to compare corresponding sides. When the
sides of one triangle are multiplied by a given number to obtain a second similar triangle, the length of
the altitude will increase by the same factor.
Final Thoughts
PRINCIPLES AND STANDARDS FOR SCHOOL MATH-
ematics (NCTM 2000) states, “The ways in which
mathematical ideas are represented is fundamental
to how people can understand and use those ideas”
(p. 67). Data can be represented in several different
ways, and each representation tells us something
that is not obvious from other representations.
Investigating side-length ratios of similar polygons
can help students both verbalize and conceptualize
ratio relationships in several different representations.
The activity described in this article offers a fun and
rewarding mathematical opportunity to help students
move from additive to multiplicative reasoning and
leads to discussions of scaling and proportions. Proportional reasoning is developed as students use
knowledge of equivalent ratios to make predictions
and solve for unknowns, without introducing the
process of cross-multiplication. Finally, graphing their
data leads students quite naturally to a discussion of
the idea of rate of change in a linear relationship.
References
Kennedy, Leonard M., and Steve Tipps. Guiding Children’s Learning of Mathematics. Belmont, Calif.:
Wadsworth Publishing Co., 1997.
National Council of Teachers of Mathematics (NCTM).
Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000.
Van de Walle, John A. Elementary School Mathematics:
Teaching Developmentally. White Plains, N.Y.: Longman Publishing Co., 1994. V O L . 8 , N O . 8 . APRIL 2003
431