ARIWASABI/THINKSTOCK
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This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Revisit
Pattern
Blocks
to Develop Rational Number Sense
A classic manipulative, used since the 1960s,
continues to offer opportunities for intriguing
problem solving involving proportions.
p
Pattern blocks are inexpensive
wooden, foam, or plastic manipulatives developed in the 1960s to help
students build an understanding of
shapes, proportions, equivalence, and
fractions (EDC 1968). The colorful
collection of basic shapes in classic
pattern block kits affords opportunities for amazing puzzle-like problem-solving tasks and for exploring
underlying mathematical relationships. These visuals can help build
students’ conceptual and procedural
understandings of rational numbers
(Reimer and Moyer-Packenham
2005). To illustrate the accessibility
and challenges of problem solving
with pattern blocks at the middle
school level, we present increasingly
challenging problems. They progress
from constructing shapes with given
proportions to creating arrangements
with given percentage, fractional, and
decimal quantities to building seven
Vol. 19, No. 6, February 2014
●
Joe Champion
and
Ann Wheeler
completely simplified representations
of any given rational number.
PatteRn BlocKs 101
The six standard pattern blocks include a yellow hexagon, a red trapezoid, a blue rhombus, a green triangle,
a tan rhombus, and an orange square
(see fig. 1). With the exception of one
of the trapezoidal bases, all the sides
of the shapes are the same length
(typically 1 inch). Moreover, the areas
MatheMatics teaching in the Middle school
337
of the hexagon, trapezoid, blue rhombus, and triangle are related through
their area proportions as:
Fig. 1 These six basic pattern block shapes begin the mathematical conversation.
2
3
hexagon : trapeziod : rhombus : triangle
= 1 :2 :3 :6
The areas of the tan rhombus and
orange square are in irrational proportion to the other four shapes and will
be mostly omitted from our problemsolving tasks (see Ellington and
Whitenack 2010 for the exact areas of
the six shapes).
A useful way to connect pattern
blocks to rational numbers is to pick
any nonzero rational value to represent the area of one shape (Lanius
1997). Once such a reference value
has been set, the value of any other
collection of pattern blocks can be
determined by the proportional
relationships among their respective
areas. Readers are invited to explore
this question:
Suppose it costs $300 to make
the hexagon out of platinum.
About how much will a
platinum triangle cost?
As an example, suppose we assume
that the hexagon represents 2 square
units of area. In that case, the trapezoid must be 1 square unit; the rhombus, 2/3 square unit; and the triangle,
1/3 square unit. Similarly, if we begin
with the smallest shape and suppose
that the triangle has a value of 6, the
following values hold: The rhombus
has a value of 12; the trapezoid, 18;
and the hexagon, 36. Once students
complete a few examples, they may
generalize their work (see table 1).
Although the proportional relationships among pattern block
shapes may seem straightforward to a
teacher, students can struggle with the
ideas of how the shapes relate to one
another. Ellington and Whitenack
(2010) tell the story of Ms. Sneider,
338
Yellow
hexagon
Red
trapezoid
Blue
rhombus
Green
2
triangle
3
Tan
rhombus
Orange
1square
32
3
2
table 1 Proportional values2of pattern block shapes are1 shown in tabular form.
3
3
3
3
21
hexagon
trapezoid
Rhombus
triangle
2
3
2
13
2
1
3
1
3
3
3
2
23
1
3
18
3
2
3
36
3
24
x
2
1
2
12
1
12
1
1
3
2
1
3
x
2
2
8
2
x
2
x
1
x
2
3
2
x
who was helping fifth-grade
students
3
x pattern
understand fractions by using
2
6
3
x
2
21
12
x
2
3
4x
2
x
x
2
6x
x3
x
x
1
2
1
lesson to2the idea of sharing, 3
students
+ =1
6
2x each
3
6
quickly understood
how giving
of six children
1 pattern block6
blocks. Students often could
3 represent
x shape
x
1 be2fair. 1
1 1
would not
Her
inventive
pattern blocks using variousx equiva6+ =1
3+ = 1
3 a real-world
6
strategy 2
related
concept
lences, such as using 3 blue6xrhombi
21 22
1
+ =1
that
students
could
easily
understand
to represent 1 yellow hexagon,
but
6
3
6
x
12 2
1
to help solve
your
could not explain their part-to-whole
+ 1 =1
1 2
1
1 1the problem. If 1
1
6+ =1
2 + 3 + 6= 1
+ =1
classes also
with learning
the
reasoning. To combat student
2 confu3
6
2 struggle
2
3
1
2
1
1 3
1 3
=1
proportions among the shapes, +you = 1
sion, Sneider created a “funky+ cookie”
2 3
6
1 to
2 create
1 your own
12 funky
12
might want
design consisting of all six 1
pattern
1 1 1
1
1 + 1 =1
1 + 1 =1
2 + 3 + 6= 1
2+ 2+ + + +
+ =1
cookie creation.
blocks (see fig. 2).
2 1
2
3 3 3
61 61 61 6 6
When asked how much1of+ the
+ + =1
=1
2 2
whole “cookie” the yellow hexagon
cReatiVe
PatteRn BlocK
13 13 13
1 1
1 1“1/6.”
1
1 + 1 = 1 1 1 11 + 1 + 1 = 1
comprised, a student answered,
2 + 2 + + + + 3=+13 = 3
+ + = 1 aRRangeMents
3 1
3 the
3
6 to
6 engaging
6 6 students
6 63
2 1 1
The student thought that because
In addition
in 1
1
1
1 61
+ + =1
+ + + + +
yellow hexagon represented3 1 of
proportional
when16sort3 the
36
6 1
6 1
6 1
6
1 1reasoning
1
1
= 1 students
1 1 1 1ing 1the different
1 + 1 + 1shapes,
shapes, the yellow hexagon1represent1 +1 +1 + 1+ +
6+ 6+ 6= 6 6
+ + + + + = 13 + 3 = 3
ed 1/6 of the whole arrangement.
6 1
6 The
6 1
6 1
6also1
6learn3 about
numbers
6 rational
2
61 61 61 2
1
1
+ + + + =1
+ =
student’s interpretation of 6
the+ shapes
and analyzing creative
6 6 6 6by designing
6
6 12
1 1 1 1 1 113 1
did not consider the different
pattern block
+ 1 + 1 + + Arrange1 relative
1 1
1 + 1 arrangements.
1 =+ 11 = 1
6 + problem-solving
6 + 6 = 6 6 63tasks
+ =
+ 6= 2
sizes of the pattern blocks, 3underscorment-based
6 1
2
6 6
6 2
61 61 1
3 1
1 1
ing the importance of “equal +parts”
can range in complexity, depending
=
+ + =
6 of
6 1
6 12
1 1 1understanding
1
1
when interpreting fractions3as 6
parts2
on your students’
+ + =
1 1 1 1
1 +1 = 1
3 + and
6 = overall
2
6 sense.
6 6 2
+ + her=
of a whole. When Sneider shifted
proportions
number
MatheMatics teaching in the Middle school
●
6 1
6 1
6 1
2
1
+ + =
6 62014
6 2
Vol. 19, No. 6, February
1 1 1
+ =
6 6 3
6
6
3
1 1 1 1
+ + =
6 6 6 2
1 1 1
+ =
6 1
6 13
1
+ =
6 6 3
Fig. 2 A funky cookie representation is
Fig. 3 These arrangements demonstrate 50 percent hexagons, 1/4 trapezoids, and
made using one of each pattern block
piece.
0.25 triangles by area.
Readers are invited to answer the following question:
DON BAYLEY/THINKSTOCK
Can you arrange 6 pattern blocks
so that trapezoids form 1/2 of the
combined area?
Arrangement tasks can help build
students’ skills in converting among
proportions, percentages, decimals,
and fractions. For example, we might
ask students to create a picture with 50
percent of the area in hexagons,
1/4 trapezoids, and 0.25 triangles.
Students would need to understand
how to interpret the different representations of rational numbers, such as
converting 1/4 and 0.25 to percentages
to see how the areas of shapes in the
picture should consist of 50 percent
hexagons, 25 percent trapezoids, and
25 percent triangles.
Figure 3 shows two of the many
possible solutions that can be built.
Teachers should emphasize the importance of inventing creative ways to
solve the problems. The relative numbers of shapes in any solution would be
proportional to the counts in figure 3,
such as 5 hexagons, 5 trapezoids, and
15 triangles. Thus, the counts of pattern blocks in solutions to the problem
should follow the proportions
hexagons : trapezoids : triangles =
1 : 1 : 3.
To create more challenging examples, assign different fractionpercentage-decimal amounts of each
shape or include more shapes, such as
constructing an arrangement that is
1/4 triangles, 60 percent hexagons,
and 0.15 trapezoids by area. Using
this example, students may want
to convert all values to fractions
(1/4 triangles, 3/5 hexagons,
3/20 trapezoids) or all values to percentages (25 percent triangles,
60 percent hexagons, and 15 percent
trapezoids) to easily see the relationships among the three different pattern
blocks. We encourage students to be
creative and depict complex examples,
such as the dancing pair in figure 4.
Arrangement-based problem-solving tasks
can range in complexity, depending on your
students’ understanding of proportions
and overall number sense.
Show this arrangement to the class
and explore the idea of using a smaller
or larger number of pattern blocks to
create a correct representation of the
given values. During the discussions,
students may realize that they only
need one dancing person to discover
the equivalent proportions of
trapezoids : hexagons : triangles = 1 : 2 : 5.
Fig. 4 This dancing-pair arrangement is composed of 1/4 triangles, 60 percent hexagons,
0.15 trapezoids by area.
Fig. 5 Colorful pattern block designs allow students to create and explore arrangements.
Source: Used with permission, MathToybox.com.
340
MatheMatics teaching in the Middle school
●
Vol. 19, No. 6, February 2014
Because students demonstrate
varying skill levels in art and mathematics, it is helpful to assign more
challenging arrangement problems
as group tasks. Some groups may
always build the most basic examples,
whereas others may spend a significant portion of the class on one intricate town or flower garden design.
During one assignment in our classes
for prospective middle school teachers, most groups proudly displayed
their pictures consisting of fewer
than ten pattern blocks. However, one
group used almost all their shapes to
construct a detailed rose that covered
the entire table. This group’s artistic
talents led to an authentic reason to
discuss rational number sense and the
use of multiplication and/or division
to find various equivalent proportions.
An excellent collection of arrangement
problem-solving tasks (e.g., build a
trapezoid that is 1/2 red and 1/2 blue)
is available through the website of
the New South Wales Department of
Education and Communities (2007).
Many electronic resources are
available for exploring pattern blocks
using computers, including applets
at the National Library of Virtual
Manipulatives (1999) and NCTM’s
Illuminations websites. In our review
of resources, we found that the Pattern Blocks tool at MathToybox.com
(Booth 2012) is the best implement
for students to use when creating and
exploring pattern block arrangements.
Students can save, print, and e-mail
their creations, as well as browse a gallery of others’ work, to use as building
blocks for their own work. The online
tool can be operated with an interactive whiteboard. Teachers can find
many example arrangements to use
for in-class activities and assignments.
See figure 5 for two such examples.
PatteRn BlocK Math: eQUal
aReas and siMPliFications
Students working with pattern blocks
often discover that there is more than
one way to express a given area. For
example, the area of a trapezoid is the
same as the combination of a rhombus
and a triangle. In cases where a set of
pattern blocks can be represented by
fewer shapes with the same combined
area, we say that the set can be simplified. Figure 6 illustrates how the set
made of 2 rhombi + 1 trapezoid
can be exchanged for 1 hexagon +
1 triangle. Because the area represented by 1 hexagon and 1 triangle cannot
be expressed with fewer blocks, we
call this resulting set completely
simplified.
Next, we can shift the focus of the
pattern block problems to a task in
which students find ways to completely simplify an arrangement of pattern
block shapes by substitution. Readers
are invited to answer:
Fig. 6 Two rhombi and a trapezoid can be replaced by a hexagon and a triangle to
obtain the same shape, or pattern block reduction.
1 trapezoid + 2 rhombi (3 shapes) reduce to 1 hexagon + 1 triangle (2 shapes)
2 rhombi. Students may naturally
start to wonder about how many different simplifications they can find,
which can be difficult to determine.
Students may feel a combination of
excitement and frustration as they
look for new ways to simplify pattern
blocks.
To help students find all the ways
to simplify pattern blocks, try an example that illustrates multiple simplifications. Suppose we begin with
22 pattern blocks in the form of
4 hexagons + 3 trapezoids
+ 10 rhombi + 5 triangles
Determine how you can completely simplify each of the
following sets of pattern blocks:
(a) 2 triangles
(b) 3 triangles
(c) 5 triangles
(d) 10 triangles
Through discovery learning, students
can see how different pattern block
combinations will lead to just a few
completely simplified sets. The exploration above illustrates four examples:
2 triangles are equivalent to 1 rhombus; 3 triangles, to 1 trapezoid; 5 triangles, to 1 trapezoid and 1 rhombus;
and 10 triangles, to 1 hexagon and
Vol. 19, No. 6, February 2014
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MatheMatics teaching in the Middle school
341
Because students demonstrate varying
skill levels in art and mathematics, it is
helpful to assign more challenging
arrangement problems as group tasks.
These can be simplified through the
steps shown in figure 7. This example
provides three new simplifications:
• 2 trapezoids simplify to 1 hexagon;
• 3 rhombi simplify to 1 hexagon;
and
• 1 rhombus + 1 triangle simplify to
1 trapezoid.
Our list of simplifications has now
doubled in size.
From our work, we see certain
rules emerge. We find that a hexagon is equivalent to 2 trapezoids, 3
rhombi, or 6 triangles. Similarly, a
trapezoid is equivalent to 3 triangles
or 1 rhombus + 1 triangle. Since
the triangle is the smallest shape, it
cannot be broken down. We have
exhausted the number of completely
simplified sets that involve a single
shape, but are there any simplifications that consist of multiple shapes?
Through more investigation, we can
see that we will only add one more
grouping to our list, the 1 hexagon +
1 triangle that is equal to the 1 trapezoid + 2 rhombi pairing mentioned in
the opening discussion of this section.
Table 2 illustrates the exhaustive list.
Armed with the complete list of
simplifications, a class can explore
the relationships among the seven
representations, given a specified value
for one shape. This will allow students
to directly connect the proportional
values among the pattern block shapes
to fractions corresponding to completely simplified sets. Table 3 summarizes how the seven pattern block
simplifications can be justified by fraction statements once the value of the
hexagon is set to 1.
An interesting extension can also
promote students’ understanding of
rational numbers: The completely
simplified sets provide a way to represent any fraction in seven essentially
different ways using pattern blocks. In
fact, given any nonzero rational number written as ab in simplest form,
students can use the seven simplified
sets to represent a/b by choosing the
value of the hexagon as one number
in the list
6a 3a 2a 3a 6a
, , , ,
.
b 4 b 3b 5b 11b
For example, suppose a student
wanted to represent the value 3/8
using 1 hexagon and 1 trapezoid.
He or she could use a table, apply trial
and error, or write an algebraic equation to find that the third fraction in
the list sets the value of a hexagon to
1/4 and a trapezoid to 1/8, resulting
in the desired total of 3/8.
the end oR a neW
Beginning?
Through problem-solving explorations, we can build students’ rational
number sense using four standard pattern blocks. Who knows the possibilities that will be unlocked by including
additional types of shapes among
the pattern blocks (e.g., those in the
Deci-Blocks™ system)? We will leave
those investigations for an extra-credit
assignment.
ReFeRences
Booth, Ginger. 2012. “Mandalar Pattern
Blocks.” http://mathtoybox.com
/mandalar/readers/
Education Development Center (EDC).
1968. Elementary Science Study.
Chicago: McGraw-Hill.
Ellington, Aimee J., and Joy W. Whitenack. 2010. “Fractions and the
Funky Cookie.” Teaching Children
Fig. 7 These steps walk through simplifying a complex combination of polygons.
4 hexagons + 3 trapezoids + 10 rhombi + 5 triangles
= 4 hexagons + (1 hexagon + 1 trapezoid) + (3 hexagons + 1 rhombus) + (1 trapezoid + 1 rhombus)
= 8 hexagons + 2 trapezoids + 2 rhombi
= 8 hexagons + 1 hexagon + 2 rhombi
= 9 hexagons + 2 rhombi
342
MatheMatics teaching in the Middle school
●
Vol. 19, No. 6, February 2014
Mathematics 16 (May): 532-39.
King, C. 2002. “Geometry Blocks.”
MathPlayground. http://www
.mathplayground.com/pattern
blocks.html
Lanius, Cynthia. 1997. “No Matter What
Shape Your Fractions Are In.” http://
math.rice.edu/~lanius/Patterns/
National Council of Teachers of Mathematics (NCTM). Illuminations.
http://illuminations.nctm.org
National Library of Virtual Manipulatives
(NLVM). 1999. http://nlvm.usu.edu/
en/nav/frames_asid_169_g_1_t_2
.html?open=activities
NSW Department of Education and
Communities. 2007. “Mathematics:
What Are Pattern Blocks?” http://
www.curriculumsupport.education
.nsw.gov.au/secondary/mathematics/
years7_10/teaching/frac.htm. State of
New South Wales through the Department of Education and Communities.
Reimer, Kelly, and Patricia S.
Moyer-Packenham. 2005. “Third
Graders Learn about Fractions
Using Virtual Manipulatives: A
Classroom Study.” Journal of Computers
in Mathematics and Science Teaching
24 (1): 5-25.
Any thoughts on this article? Send an
e-mail to [email protected].—Ed.
Joe Champion,
joe.champion@gmail
.com, teaches mathematics education at Boise
State University, Idaho.
He likes to build collaborative math websites and
explore connections between advanced math and
school math. Ann Wheeler, awheeler2@
twu.edu, is a former secondary mathematics teacher and current mathematics
education professor at Texas Woman’s
University in Denton. Champion and
Wheeler would love to hear your ideas on
this article.
3
12 3
3
331
1
13
3
3
33 1
132
233
3
32
2
31
Table 2 The seven completely simplified sets of pattern blocks are2
13listed.
2
12
2
Original Sets
Simplification
2
1
12
1 trapezoid + 2 rhombi
1 hexagon +
x
2 1 triangle
2x 1
2 trapezoids
1 hexagon 1x22
22
x
3 rhombi
1 hexagon x 2
x
2
6 triangles
1 hexagon 2xx x
3
2
32x
1 rhombus + 1 triangle
1 trapezoid
x
x3
3 triangles
1 trapezoid
3
x
3x x
2 triangles
1 rhombus xx36
63
x
x6
6
1
x1 + 2 the
Table 3 For the seven completely simplified sets of pattern blocks,
=1
61 setting
x6+2 2 =3 1 1 6
hexagon equal to 1 unit numerically justifies the relationships.
1 2 61
1 2 262 1+33 = 1
+ 1
=1 2
16
Original Sets
Simplification
2 3 Justification
+16 2
=11 1
1
1
2
3
11 2+
1+2 2 ==611
1
+
=
2
3
6
1
1
+
=
1
1 trapezoid + 2 rhombi 1 hexagon + 1 triangle
1 1 222 +232 = 16
+ 1
=1 1
2 2 1
+1 1
=11 1
2
11 32
+1+1 3=1+13 = 1
2 trapezoids
1 hexagon
12++ 312+= 3
11= 1
1 1 323
1 +23 + 3 = 1
+ +
=1 1
1 1
3 3 1
3
+1 1
+1 1
=11 1 1 1
3 rhombi
1 hexagon
3
3
3
11 6 +
1+1 6 +
1+1 6 1
=+ 1 1+ 1+ = 1
++ 13
++ 13
+= 11+6 1+6 1=6 1
3
1
1 1 636
1 +636
1 +636
1 +66
1 +66 +66 = 1
+ +
=1 1
6 triangles
1 hexagon
1 +
1 +
1 +
1 1
6 6 6
+1 6
+1 6
+1 6
+ 1
+ =1
1
1
1
1
+
=
61 6
6 6 6 61
16+3+116=6+1162+16 +16 +16 = 1
1+ 1+ 1+ + + = 1
1 rhombus + 1 triangle
1 trapezoid
6 6 6 6
1 1 3631 +666 =2
+ 1
= 1 12
3 6 1
21 1
+
=1 11 1
3
2+ 1=
11 66
1
+1+1 6 =
++ 16+= 112
=6 12
3
3 triangles
1 trapezoid
1
1 1 636
1 +6661 +626 =22
+ +
= 1 1
1 1
6 6 6
21 =1
+1 1
+
1
+ 6=1 2 1
61 6
2 triangles
1 rhombus
16+6+116=6+1163=12
1+ 1+ 1=
6 2
1 1 6661 +666 =3
+ 1
= 1 13
6 6 1
3
+ 1
= 1
6 6
3
16 +16 =13
+ =
6 6 3
Great Kids + Great Adults = Great Results
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Vol. 19, No. 6, February 2014
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Mathematics Teaching in the Middle School
343