Traveling from
Arithmet
A number line can be a useful visual tool to
help students progress from arithmetic to
algebra in their quest to understand variables.
M
Joy W. Darley, jdarley@
georgiasouthern.edu,
teaches mathematics
content courses to preservice teachers at Georgia
Southern University in Statesboro. She is
interested in rational number instruction
and the impacts of this instruction on the
understanding of algebraic concepts.
458
any researchers and mathematics educators claim that
connections should be made
between new and prior knowledge,
because knowledge is remembered
better when it is well connected and
more readily available in new situations (Baddeley 1976; Bruner 1960;
Hiebert and Carpenter 1992; Hilgard
1957; Skemp 1976). To facilitate these
kinds of connections, the NCTM
developed Curriculum Focal Points
for Prekindergarten through Grade 8
Mathematics: “The decision to organize instruction around focal points
assumes that the learning of mathematics is cumulative, with work in the
later grades building on and deepening what students have learned in the
earlier grades, without repetitious and
Mathematics Teaching in the Middle School
●
Vol. 14, No. 8, April 2009
Copyright © 2009 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.
inefficient reteaching” (NCTM 2006,
p. 5). Historically, students have had a
difficult time transitioning from arithmetic to algebra, in part because of
the effort needed to connect the two.
When students possess a deep understanding of numbers and the connection between numbers and variables,
they should be able to make a smooth
transition from one to the other.
Research suggests that every effort
should be made to connect students’
existing fraction knowledge to a quantitative model so that fractions are
recognized as numbers before engaging in formal algebra (Bracey 1996;
Gelman, Cohen, and Hartnett 1989;
Wu 2001). Wu (2001) defines both
whole numbers and fractions as points
on the number line. In so doing, there
Photograph by Frank Fortune; all rights reserved
Joy W. Darley
etic
to
Algebra
Vol. 14, No. 8, April 2009
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Mathematics Teaching in the Middle School
459
will be a logical progression from the
teaching of whole numbers to the
teaching of fractions and ultimately to
the teaching of rational expressions.
Since our goal is for students to
better understand variables, we need
to be certain that they understand
numbers. Once the connection is made
between the two, students will be more
confident using variables. The following activities, which explicitly show
connections between numbers and
variables, use the number line as a tool
to foster such connections. Instruction
begins with using whole numbers and
continues with fractions.
Activity 1: Whole
Numbers and Integers
Use a number line that clearly defines
the unit of reference (see fig. 1). For
example, the arithmetic number line
contains 0 and 1 as the unit of reference, whereas the algebra number
line contains 0 and x as the unit of
reference. It is important for students to know that the exact distance
between “0 and 1” and “0 and x” is not
important at this time. The number 1
represents one of anything: one foot,
one candy bar length, one meter, one
pizza, or one set of objects. The variable x, however, represents an unknown quantity of anything, such as
x feet, x candy bar lengths, x meters,
x pizzas, or x sets of objects.
At this point, students can be
asked, “Where is 2 on the first number
line? Where is 3? How do you know?”
After the students are adept at using 1
as the unit of reference on the arithmetic number line, ask these questions:
“Where is 2x on the algebra number
line? Where is 3x? How do you know?
If x = 1, does this make sense? What if
x = 2? What would this look like on a
number line?” The desired connection
should resemble figure 2.
Locating 2x or 3x on the algebra
number line, given the location of 0
and x, is not a trivial task for many
460
Fig. 1 Students see a side-by-side comparison of both number lines.
Fig. 2 Students can visualize mixing whole numbers with variables.
Fig. 3 The next step involves using negative numbers.
(a)
students. After posing the previous
question to a beginning algebra class,
there was noticeable silence before the
first incorrect guess was given. Even
though students may understand that
3 • 2 is 2 + 2 + 2, understanding 3x
as 3 • x, or x + x + x, does not always
easily follow. In fact, many students
commonly confuse 3x with x3, not
recognizing that 3x represents three
xs that have been added, whereas x3
represents three xs that have been
multiplied. Another common misconception is that the value of x is always
1. Students commonly confuse the
“value” of x with the “coefficient” of
Mathematics Teaching in the Middle School
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Vol. 14, No. 8, April 2009
(b)
x. Emphasizing that x is the same as
1x, because it represents one of an unknown quantity, such as one 2 in the
former example, should eliminate this
confusion. Replacing x with several
numbers would be helpful as well.
After students are adept at locating
x, 2x, and 3x on the number line, and
perhaps 4x for a sufficient number of xs,
present the number line in figure 3a and
ask students to locate −1 and −3. Have
students describe how they determine
these locations. After they are comfortable identifying the locations with
negatives, ask the class to find –x and
–3x on the algebra number line. Suggest
Activity 2: Fractions
Just as with integers, we begin our
work with fractions on number lines
that clearly define our unit of reference (as shown in fig. 1). In this
second activity, both the measurement
and the sharing interpretations of
fractions are used. Hence, we interpret
the fraction a/b in this way:
1. a out of b equal parts in the interval [0, 1]
2. The size of the portion when a
is divided into b equal parts (Wu
2002)
To address the measurement interpretation listed in item (1) above, ask
questions such as the following:
• If Pat divided a one-mile track into
three equal parts and ran two of
them, what part of a mile did Pat
run?
• Pat ran four of these equal parts.
What part of a mile did she run?
• Where are 2/3 and 4/3 located on
the arithmetic number line? How
do you know?
Photograph by Frank Fortune; all rights reserved
specific values for x, such as 2. Invite
students to draw a number line and
identify the locations of the pertinent
values. The desired connection should
resemble figure 3b. For the purpose of
this activity, x represents a positive number. The activity should be extended,
at a later date, so that x represents a
negative number to avoid the common
misconception that a varible such as x is
always positive.
Traveling back and forth on the
number line connecting arithmetic
and algebra is a healthy practice that
both deepens conceptual understanding and helps prevent misunderstandings. Before students work with fractions and rational expressions, they
should benefit from revisiting whole
numbers and integers and explicitly
connecting this knowledge to algebra.
Before students work
with fractions, they
should revisit whole
numbers and integers.
Using the measurement interpretation, students should take the interval from 0 to 1 on the number line
and divide it into three equal parts.
Students name the second part as the
fraction 2/3 and the fourth part as
4/3. (See fig. 4.)
Once students are successful with
the fractions, ask them the following
questions:
• If Pat divided an x-mile track into
three equal parts and ran two of
them, what part of a mile did Pat
run?
• If Pat ran four of these equal parts,
what part of a mile did she run?
• Where are the rational expressions
(2/3)x and (4/3)x located? How do
you know?
• If the track was 3 miles long (x = 3),
would both expressions make sense?
• Would both expressions be true for
any value of x?
To address the algebraic version, a
student ideally uses an extension of
the same fraction interpretation using
x as the unit of reference in place of 1.
As noted earlier, some students will
have more difficulty with the algebra
and may need to revisit the arithmetic
many times before connecting with
the algebra number line.
These questions help students
understand the connection between
numbers and variables, especially
the relationship between fractions
and rational expressions. Now they
should see all of the above as simply
points on a number line. Having this
concrete representation promotes a
deeper understanding of a variable
as (merely) a symbol representing a
number. This knowledge can lessen
the fear of learning algebra.
Once students are proficient with
the measurement interpretation of
fractions, it is safe to explore the
sharing interpretation, discussed in
item 2 above. Making sense of the
sharing interpretation of fractions assumes that one can locate a given fraction on a number line. To address the
Fig. 4 A side-by-side comparison illustrates the algebraic connection.
Vol. 14, No. 8, April 2009
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Mathematics Teaching in the Middle School
461
Traveling back and forth on the number
line connecting arithmetic and algebra
is a healthy practice that both deepens
conceptual understanding and
helps prevent misunderstandings
sharing interpretation, start by asking
this question:
portion if x is divided into three equal
parts. Ask them if x/3 would make
sense for any value of x and to make a
conjecture about the location of (1/3)x
and x/3 on the number line.
Next, ask this question:
If Jack has 1 candy bar and wants
to divide it equally among three
friends, how much of a candy bar
would each friend get?
If Jack has 2 candy bars and wants
to divide them equally among
three friends, how much of a candy
bar would each friend get?
Encourage students to think about the
size of each portion if 1 is divided into
three equal parts. Be sure that students connect their responses to the
number line, such as the example in
figure 5a. If Jack has x candy bars and
wants to divide them equally among
three friends, how much would each
friend get? To get students thinking,
ask them to consider the size of each
This question involves the size of each
portion if 2 is divided into three equal
parts, as shown in figure 5b. Have
students express this question on the
number line. If Jack has 2x candy
bars and wants to divide them equally
Fig. 5 Dividing 1, 2, and 4 candy bars help students explore more variables.
Students use a number
line to visualize dividing
candy bars.
among three friends, how much
would each friend get? This question
is asking about the size of each portion if 2x is divided into three equal
parts. Students should consider if
(2x)/3 would make sense for any value
of x. Keeping in mind the number
line in figure 5b, ask students to find
locations of (2/3)x and (2x)/3 on the
number line.
To extend this sharing interpretation to improper fractions, pose this
question:
If Jack has 4 candy bars and wants
to divide them equally among
three friends, how much of a candy
bar would each friend get?
(a)
This question is asking students to
think about the size of each portion
if 4 is divided into three equal parts.
On the number line, the results
should resemble figure 5c. When
moving to variables, the question
becomes this: If Jack has 4x candy
bars and wants to divide them equally
among three friends, how much
would each friend get? (What is the
size of each portion if 4x is divided
into three equal parts?) Would (4x)/3
make sense for any value of x? What
is true about the location of (4/3)x
(b)
(c)
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Vol. 14, No. 8, April 2009
Photograph by Frank Fortune; all rights reserved
and (4x)/3 on the number line?
A conceptual knowledge of the
measurement interpretation of a
fraction not only leads to a deeper
understanding of the sharing interpretation but also lends itself to a
richer foundation for understanding fractions as ratios. For example,
a student who interprets (2/3)x as
being two-thirds of some quantity
x will be able to better visualize the
following problem:
launching pad to introduce the ratio
2 to 1 highlights the cumulative
nature of mathematics and is therefore
beneficial in building more confident
problem solvers.
The ratio of boys to girls in Mrs.
Jones’s class is 2 to 1. There are 18
students in the class. How many
are boys, and how many are girls?
1. If Jack has 1 candy bar and wants
to divide it equally among three
friends, how much of a candy bar
would each friend get?
2. If Jack has 2 candy bars and wants
to divide them equally among
three friends, how much of a candy
bar would each friend get?
In the first question, 1 explicitly refers
to 1 candy bar. Consequently, most
students both illustrate and get the
correct answer, 1/3, whether they
Fig. 6 Expressing a ratio on a number line and as parts and a whole
There are 18 students in Mrs.
Jones’s class. Two-thirds are boys.
How many are boys, and how
many are girls?
To make sense of this problem, a student may simply replace the unit
of reference x with 18 as shown in
figure 6. The thought process involved in understanding the previous
problem should enhance students’
knowledge of a similar problem involving ratios:
The Unit of Reference
Why do many students, although
they are skilled at both partitioning
intervals and sharing, continue to have
difficulties with fractions? One possible source of confusion may be the
unit of reference. The two Jack and
the Candy Bar questions specifically
address this issue:
2
(18) = 12
3
2
(18) = 12
3
22 (18) = 12 11 (18) = 6
(18) = 12
(18) = 6
33
33
1
(18) = 6
3
12 boys and 6 girls
2
1
(18) = 12
(18) = 6
3
3
Let b = # of boys
1
(18) = 6
3
2
1
(18) = 12
(18) = 6
# of boys  2
b
3
3
Let b = # of boys
 =
# of girls  1 18 − b
# of boys  2
b
b = # of boys
Let =
# of girls  1 18 − b
(
)
2 18 − b = b
(
)
(
)
2 18 − b = b
# of boys  2
b
36
6 − 2b == b
# of girls  1 18 − b
36 = 3b
2 18 − 12
b == bb
36
6 − 2b = b
36
6 − 2b = b
12 boys and 18 − 12, or 6, girls
36 = 3b
36 = 3b
12 = b
12 = b
Using the fractions 2/3 and 1/3 as a
Vol. 14, No. 8, April 2009
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Mathematics Teaching in the Middle School
463
Fig. 7 Sam’s correct number line (a) and Vanessa’s incorrect unit of reference (b) allow
the teacher to assess their understanding.
(a)
(b)
draw number lines or candy bars to
visualize the question. In this case, 1/3
implies both 1/3 candy bar and 1/3
group, or total pieces. However, in the
second question, the unit of reference
is not explicitly stated. Notice that
figure 7 shows Sam’s and Vanessa’s
work, both beginning algebra students. They each drew correct pictures
for the second question. Sam gave
the correct answer, 2/3, implying 2/3
candy bar. When Vanessa was asked
the meaning of 1/3, she answered,
“One-third candy bar.” When asked
if that answer would make sense, she
replied, “No, I meant 1/3 pieces.” If
the unit of reference was the number
of pieces in her drawing, Vanessa’s answer would be correct. However, the
question asks, “How much of a candy
bar would each get?” not “What part
of all the candy did each get?” Opportunities such as this allow rich discussion concerning the unit of reference.
For example, when students present
2/3 as an answer, teachers should ask,
“Two-thirds of what?”
464
Recommendations
Traveling back and forth from arithmetic to algebra on the number line
is a practice that should promote a
solid foundation for learning algebra.
In particular, conceptual knowledge
of numbers, especially fractions, is an
important link to a conceptual understanding of variables. We should
neither assume that our students are
aware of the “arithmetic to algebra”
connections nor assume that they
understand the underlying arithmetic concepts. Instead, we should
patiently travel the road in both
directions with them, asking many
questions along the way, to better
prepare our students for thinking
algebraically.
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