Views of Isometric Geometry

346 February 2012 • teaching children mathematics www.nctm.org

Two ways of drawing cubes on dot paper connect students to

life outside their classroom.

By Ryan Andrew Nivens, Tara Carver Peters, and Jesse Nivens

Copyright © 2012 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.

www.nctm.org teaching children mathematics • February 2012 347

Current mathematical standards expect students

in the upper elementary grades to learn and use

three-dimensional models and represent them in

two dimensions using isometric and orthogonal represen-

tations. We address the challenges of teaching students and

teachers alike, including those who have little or no experi-

ence with this branch of geometry.

The situationIn the early years of the NCTM Standards movement, many

articles on isometric drawings and measurements appeared

for secondary teachers (e.g., Litwiller and Duncan 1989;

Spikell 1990). The expectations emphasize that students

in the upper elementary grades begin working with three-

dimensional geometry. Specifi cally, the Geometry Standard

for grades 6–8 states that students will be able to do the

following:

Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defi ning properties…. Recognize and apply geometric ideas and relation-ships in areas outside the mathematics classroom, such as art, science, and everyday life. (NCTM 2000, p. 232)

As the recommendations of the Standards began to take

hold, curriculum writers responded, and some of the math-

ematical content at the high school level moved to the

middle school level. These changes in thinking were not

sudden; they had begun years earlier through innovative

research studies. A particularly infl uential study conducted

by Ben-Haim, Lappan, and Houang (1985) showed how

students in grades 5–8 could improve their skill in moving


from isometric views to three-dimensional models. Specifi cally, the authors concluded that “concrete experiences with cubes—building, representing in two-dimensional drawings, and reading such drawings—are helpful in improv-ing students’ performance” (p. 407). A key point here is that students’ hands-on involvement over a long period of time provides essential experience for them to understand both repre-sentations and questions about them.

Battista and Clements (1998) described the diffi culties of working in two and three dimen-sions to study volume and the related formulas, with a specific focus on elementary school students in grades 3 and 5. Before Battista and Clements conducted their interviews with stu-dents, the students in their study had received traditional instruction showing how to multiply length, width, and height to compute volume. The students demonstrated diffi culty in both predicting and computing the number of cubes within drawn and physical models alike. These diffi culties emphasize the need for students to have time with hands-on interaction when working between two and three dimensions.

In a recent assessment (see fi g. 1), 56 percent of fourth-grade students correctly answered a problem from the 2007 National Assessment

of Educational Progress (NAEP). Although this percentage may seem low, this item shows that the Standards’ recommendation to incorporate geometry in grades before high school is being implemented and that a majority of students are capable of performing on such topics. How-ever, students should experience playing with blocks at an early age, especially in a mathe-matical context. Because the blocks are stacked at a depth of one foot, a two-dimensional rep-resentation is all that this problem represents. Having only 56 percent of assessed fourth grad-ers answer correctly then seems low.

Two of the authors work in Tennessee, which was one of the fi rst states to adopt the Common Core State Standards. Tennessee standards call for students in both grade 5 and grade 8 to work with two- and three-dimensional fi gures. Specifi cally, in grade 5, students must “build, draw, and work with prisms by means of orthogonal views, projective views, and nets” (TDE 2009–2010). This requirement suggests that any students entering the middle grades should have a basis in working in two- and three-dimensional mathematics. Furthermore, the NCTM Standards recommend more than just mathematical content.

A final element of the expectations is the Communication Standard, which states that students should “use the language of mathemat-ics to express mathematical ideas precisely” (NCTM 2000, p. 268). This Standard suggests that we expect our children in the upper elementary grades to work with and achieve a competence with two- and three-dimensional representations of fi gures.

Although middle-grades teachers may not see this as a diffi cult issue, the terminology used within our state standards posed a problem for teachers and students alike. Communication is a key part of working within this area, and many times teachers and students fi nd such terms as orthogonal and isometric unusual and confus-ing. A fi rst step in working with our students is ensuring that teachers themselves know what these words mean. A further step toward excel-lence is to see where these terms and ideas originated and how they embody the world around us. Although assessments and research studies rely on abstract diagrams, a connection to art and life is a required piece of the instruc-tion provided to students, and initial activities

This item is from the 2007 NAEP Mathematics test: “Sierra built the block tower with 1-foot cubes. How many cubes did she use?” Possible multiple-choice answers were 4, 6, 8, and 10.

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should incorporate a hands-on approach. While working with prospective elementary

school teachers at East Tennessee State Univer-sity, we found that many of them were initially apprehensive about teaching these concepts and working with manipulatives. They lacked confi dence in their geometric abilities, and the vocabulary involved was their first obstacle. Furthermore, the teacher candidates had never experienced thinking this way before, so it was a true learning experience for them. After some hands-on activity and a little practice, they were excited about these concepts because they had learned a new skill and a new way of thinking about objects from both orthogonal and isomet-ric views. They gained self-confi dence in their abilities to work with geometric problems and concepts, and they are more likely to teach these geometric concepts to their students and enjoy themselves in the process. In turn, their students will be more motivated and inspired to learn geometry and mathematics. We now look at how this topic connects to life outside the classroom and present successful ways we have found to teach these concepts.

Connection to lifeOrthogonal views are used in a variety of appli-cations and are easily recognized for their use in architecture, surveying, and many forms of engineering. Blueprints of buildings are of par-ticular interest because they convert existing three-dimensional spaces into two-dimensional plans, as used in surveying, along with specify-ing new three-dimensional spaces from two-dimensional plans, as used in architecture. Another use of orthogonal views that we expect students to use is a map. Cartography converts sections of a spherical planet into usable maps by simplifying them into an orthogonal view.

An isometric view is useful in graphics because each object is represented by just one view with fi xed proportions, so no rendering is required to manipulate the object’s appearance as it recedes from or advances toward the viewer. Isometric views are common in video games and other software applications. Using an iso-metric view in a computer environment lends a greater feeling of realism, as it looks lifelike and three-dimensional while avoiding the need for three-dimensional rendering, which would require more of the computer’s processing

Defi ning the termsWolfram|Alpha offers a mathematically oriented set of defi nitions useful to teaching mathematics. NCTM’s Illuminations site presents a useful fourth defi nition. Below we present the historical back-ground of the art and design elements underlying the concept of a two-dimensional representation of a three-dimensional object.

Orthogonal views

In terms of elementary school classroom use, an orthogonal view—“having a set of mutually perpendicular axes; meeting at right angles”—can be more simply defi ned by how it is used: by depicting a three-dimensional object on a two-dimensional surface with no attempts made to show perspective. Therefore, a tower composed of three stacked cubes would be seen from the front or back as three squares, one on top of the other (see fi g. 2).

Isometric views

Showing three-dimensional objects isometri-cally—“having equal dimensions or measure-ments”—still involves depicting them on a fl at surface without perspective. However, the iso-metric form allows for showing up to three sides of each object at once. Often the grid lines in an isometric view appear as a rhombus (see fi g. 3).

Perspective views

Wolfram|Alpha describes this view as “the appearance of things relative to one another as determined by their distance from the viewer.” When creating a perspective view of a three-dimensional object, measures are taken to give the illusion that the object exists in realistic space, even though it is depicted on a two-dimensional surface. The grid lines on a perspective drawing will therefore appear wider on the side closest to the viewer and narrower in the back, with all lines becoming smaller as they get nearer to the horizon line (see fi g. 4).

Mat plan

On the NCTM Illuminations website, we found a fourth defi nition that we use. A mat plan is “a top view of a solid, with the number of cubes appear-ing in each vertical column displayed in the corre-sponding box.” Mat plans are a use of the specifi c orthogonal view from the top, with additional information given to indicate the exact height of each section. This leads some students to make errors, as we will discuss in a later section.

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Three stacked cubes depict an orthogonal view.

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Notice that in an isometric view of three cubes, three surfaces can be seen (front, side, and top), although the lines do not appear to diminish toward the horizon.

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As in the isometric view, this perspective view shows three surfaces. However, the cubes appear to shrink toward the back, which makes them more realistic in terms of what the eye is accustomed to seeing.

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resources. This view is especially useful in online games. Such games as Facebook’s popu-lar Farmville run on a server and are streamed to the user’s computer, so performance is dependent on all graphics and data being small enough to easily stream over a common broad-band Internet connection.

Finally, an understanding of two-dimen-sional representations of three-dimensional objects combined with art and literature can enrich everyone’s life. A recent trip to a local museum revealed a display of handcrafted quilts. Most of the quilts used patterning and colors to create objects of beauty, but one quilt in particular stood apart from the others. It was pieced together with a hexagonal grid and revealed an image of cubes in an isometric pat-tern. Students might explore drawing cubes on isometric paper by fi rst starting with a hexagon or rhombus.

ApplicationIn this section, we outline a how-to approach for drawing on isometric dot paper, one of the biggest challenges we have encountered with both teachers and students. We suggest two methods to share with beginners on how to draw a single cube. One method starts with a hexagon with a side length of three units, allowing for a larger initial drawing before moving to the more standard use of one-unit length cubes. First draw a hexagon, and then draw in the diagonals (see fi g. 5). This approach allows a teacher to see if students have drawn the proper fi gure by checking for six equilat-eral triangles, each with one dot in the center. Once students can draw this hexagon, they can then add half a diagonal, stopping at the center point. Using a different color makes it easier for students to see the letter Y inside the hexagon (see fi g. 6).

A second method allows students to start with a rhombus that will serve as either the top or bottom of the cube: Draw a cube as viewed from the top, construct the vertical lines, and connect them (see fi g. 7).

Regardless of which method you choose, shading is usually the part that helps bring out the three-dimensional quality of a cube representation. Once students can draw a single cube, we frequently present them with a mat plan and have each student build the cor-

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This method of drawing a hexagon uses three diagonals to create a representation of a cube on isometric dot paper.

Start by sketching a regular hexagon using isometric dot paper.

Construct (draw) the three “long diagonals” found in the hexagon.

How many “squares” do you see? You should see six. (Can you see the cube?)

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Susan Scheer of Gray, Tennessee, created the Sour Grapes quilt to illustrate Aesop’s fable “The Fox and the Grapes.”


responding fi gure out of wooden blocks (see the online appendix for additional figures). From that point, we ask students to draw inde-pendently. This trial-and-error method allows students freedom to experiment and deter-mine how the paper works with fi gures more complex than a single cube. Teachers should use this time to look for students who have not made use of the paper correctly. Students may try to treat the isometric paper as though it were square dot paper and may need someone to direct them back to making use of the hexa-gon or rhombus approach; others may have diffi culty placing their cubes together.

Students using the hexagon approach are not always immediately successful. They typically realize that something is wrong and will ask for assistance, offering the teacher a chance to direct students to help one another or to hold a whole-class discussion about what incorrectly drawn figures represent in three dimensions.

Students generally enjoy the task of reading a mat plan, building a structure with cubes, and then drawing it on paper. As they become profi cient with their drawings, you might add a fi nal component: Have them draw orthogonal views of their structures. This is a slightly easier task (although by no means easy), and students are typically quick to make the connection between the mat plan and the orthogonal views. Although the mat plan represents only the top view of the structure, students may incorrectly decide that all orthogonal views require numbers to be inserted within the boxes, usually as an indication of depth. This may seem logical, but this format is not con-ventionally used in the curriculum.

ConclusionThe Common Core State Standards for Math-ematics (CCSSI 2010) asserts that students in grade 5 should be able to understand concepts of volume measurement:

A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume and can be used to measure volume. (p. 37)

Although the language in this document is traditional, many states have adopted standards

Here is a method of drawing a hexagon with three half-diagonals to represent a cube on isometric dot paper.

Start by sketching a regular hexagon on the isometric dot paper. Construct (draw) only half of the long diagonal lines. They will resemble the letter Y or an upside down Y. Shade the top or bottom for emphasis if you like.

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These fi gures show a method of drawing a cube on isometric dot paper by starting with a rhombus.

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Start by sketching a regular rhombus on the isometric dot paper.

Next, construct (draw) three vertical lines of the same length as the rhombus side length.

Last, connect the endpoints of these vertical segments to complete the cube.

Shade the top or bottom for emphasis if you like.

containing vocabulary that initially presents a challenge. For example, when the new Stan-dards were introduced in Tennessee, many elementary school teachers did not understand the meaning of orthogonal views or isometric views and had no idea how or what to teach the students. Elementary school teachers often lack the necessary geometry background required for teaching these concepts to their students. Furthermore, many veteran elemen-tary school teachers never studied this aspect


of geometry as students. Therefore, teachers must learn the content of the new geometry standards before they can teach students.

We maintain a belief that teaching, espe-cially in the case of three-dimensional geom-etry, should be hands on. The aforemen-tioned fi ndings of researchers and our own experiences support hands-on teaching. In our work with fi fth-grade students, we sup-plied them with wooden blocks and activity sheets that contain isometric dot and regular dot paper. The key for success is found not only in using the wooden blocks but also in conducting a lesson that allows the learn-ers to grow into the lesson. Such instruction means that the teacher serves as a facilitator of learning instead of dictating procedure by offering examples. Students enjoy this type of classroom environment and remain focused on the activity. As the student work in fi gures 8–10 online shows, students make efforts to draw the fi gures, and they get better with practice. Many teachers we have worked with are amazed at just how well their students per-form when given proper instruction coupled with a hands-on learning experience with wooden blocks.

Although the activities in this article show student work based on one task, any teacher with access to an Internet connection can fi nd outstanding resources and activities on the NCTM Illuminations website. These online resources are digital, but we still maintain that teachers should not only present their students with wooden blocks with which to build their

models but also require their use. The use of tangible, physical models should necessarily precede the work that continues in a digital environment.

REFERENCESBattista, Michael, and Douglas H. Clements. 1998.

“Finding the Number of Cubes in Rectangular Cube Buildings.” Teaching Children Mathemat-ics 4 (January): 258–64.

Ben-Haim, D., Glenda Lappan, and R. T. Houang. 1985. “Visualizing Rectangular Solids Made of Small Cubes: Analyzing and Effecting Students’ Performance.” Educational Studies in Math-ematics 16 (November): 389–409.

Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics. Washington, DC: National Gov-ernors Association Center for Best Practices and the Council of Chief State School Offi cers. http://www.corestandards.org/the-standards/mathematics/grade-5/measurement-and-data/.

Litwiller, Bonnie H., and David R. Duncan. 1989. “Area Formulas on Isometric Dot Paper.” Mathematics Teacher 82 (May): 366–69.

National Council of Teachers of Mathematics (NCTM). Illuminations. “Using Cubes and Isometric Drawings: Lesson 4, Mat Plans.” http://illuminations.nctm.org/lessonDetail.aspx?ID=L611.

———. 2000. Principles and Standards for School Mathematics. Reston, VA: NCTM.

Spikell, Mark A. 1990. “Equilateral Triangles on an Isometric Grid.” Mathematics Teacher 83 (December): 740–43.


Tennessee Department of Education (TDE). Tenne-see Mathematics Standards 2009–2010 Imple-mentation for Grade 5 Mathematics, Standard 4—Geometry and Measurement, Checks for Understanding: 0506.4.3. http://www.tn.gov/education/ci/math/index.shtml.

Wolfram|Alpha™ LLC. 2011a. “Isometrical.” http://www.wolframalpha.com/input/?i=isometrical.

———. 2011b. “Orthogonal.” http://www.wolfram alpha.com/input/?i=orthogonal&t=macw01.

———. 2011c. “Perspective.” http://www.wolfram alpha.com/input/?i=perspective&a=*C .perspective-_*Word-.

Ryan Andrew Nivens, nivens@ etsu.edu, teaches elementary mathematics methods courses at East Tennessee State University in Johnson City, Tennessee. He enjoys conducting professional development for middle school mathematics teachers. Tara Carver Peters, tara [email protected], teaches secondary math at Sullivan South High School in Kingsport, Tennessee. She currently teaches AP calculus AB and BC, precalculus, and algebra 2. Jesse Nivens, [email protected], teaches design and web development at Missouri

State University in Springfield and is interested in the relationships among graphic design, mathematics, and computer science.

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