# MSP:MiddleSchoolPortal/Geometry in 3-d

### From Middle School Portal

### Geometry in 3-D Introduction

It’s a three-dimensional world out there! And while dealing with that in a mathematics class can complicate instruction, it can also enhance and enrich middle school geometry. The online resources featured in Geometry in 3-D actively engage students in exploring a variety of geometric shapes, at times through lessons that involve building models or creating paper nets that fold into three-dimensional shapes; at other times, through technology that allows students to rotate and zoom in on figures, noting their attributes and complexity.

## Contents

Other lessons offer problems on surface area and volume, a part of every middle school curriculum. The problems, each with a different twist on the subject, challenge students to reconsider their understanding of how to measure solids. Activities for developing spatial sense, another primary objective in teaching geometry, are also featured. Finally, there are online galleries of geometric solids, included for the rare opportunity they offer to show your students the beauty in mathematics.

In Background Information, you will find workshop sessions developed for teachers and other materials that may interest you as a professional. Each resource deals specifically with three-dimensional geometry topics that align with the geometry and measurement standards recommended by NCTM.

We hope Geometry in 3-D proves a valuable resource as you plan activities that move your students from two-dimensional to three-dimensional math!

### Background Information for Teachers

If the mathematics of three dimensions feels less certain to you than that of two dimensions, you are not alone! These professional resources support the teaching of solid geometry in both the areas of shape and measurement. They include two workshop sessions that review the material, while allowing for hands-on experimentation; a book chapter that gives the underpinnings and relevance of the whole study of solids; and a set of activities addressed to teachers for their classrooms. We hope you will enjoy revisiting and reinforcing your knowledge of this practical, and sometimes beautiful, topic of geometry.

**Solids**
Through these activities, developed for teachers, you can explore three-dimensional figures by building and manipulating the Platonic solids, exploring their properties and some of the geometric relationships between them. Other investigations focus on nets and cross-sections of solids. This is one session in the free online course Learning Math: Geometry.

**Measurement Relationships**
This online workshop for teachers begins with an examination of the relationships between area and perimeter but goes on to the proportional relationship between surface area and volume and some of its applications. For instance, you will construct boxes and use graphs as you try to find the dimensions of the rectangular prism that holds the maximum volume. Video segments of teachers working on this activity lead to further discussion in this session from Learning Math: Measurement.

**Shape**
"Shape is a vital, growing, and fascinating theme in mathematics with deep ties to classical geometry but goes far beyond in content, meaning, and method," writes the author of "Shape," a chapter in On the Shoulders of Giants: New Approaches to Numeracy (National Academies Press, 1990). The chapter focuses on the significance of shape as a concept, its place in mathematical history, the tools used to study it, and its connections to art and other fields. Throughout, the language is accessible, and the author continually highlights the relevance of the topic to the school curriculum. Excellent professional background reading! On the Shoulders of Giants is available free online.

**Classroom Polyhedral Activities**
These are behind-the-scenes lesson ideas for teachers. George W. Hart, polyhedral master, gives ideas and instructions on how to construct polyhedral models from paper, soda straws, wood, and the Zometool kit. Although Hart does not give step-by-step directions here, he does make his ideas clear and shows a picture of each model.

### 3-Dimensional Shapes

To know a geometric shape, learners need to play with it—hold it, see it from all angles, even create it. These resources offer activities on making actual models that students can hold, but also virtual activities in which they can view, move, and rotate figures too complex to make in the classroom. Not every shape is here—only polyhedra, the objects most encountered in school mathematics. But there’s a wide range of materials here, arranged in order from activities for the youngest of the middle grades through materials more appropriate for high school students. We hope your students enjoy getting to know 3-D shapes.

**2D to 3D Morphing**
As students follow the directions on the printable pages, they construct a pyramid, a cube, and an octahedron. They can see the flat two-dimensional polygons rise up to form three-dimensional polyhedra. Each page is decorated with colorful images of the Cyberchase team so that one image appears on each face of the constructed three-dimensional objects.

**Scaling the Pyramids**
These several activities engage students through their fascination with the sheer size of the Great Pyramid. In one hands-on activity, students use a template to construct a scale model of the Great Pyramid. In another, students are given the actual dimensions for two other pyramids and challenged to create their own models.

**Geometric Solids and Their Properties**
A five-part lesson plan has students investigate several polyhedra through an applet. Students can revolve each shape, color each face, and mark each edge or vertex. They can even see the figure without the faces colored in—a skeletal view of the "bones" forming the shape. The lesson leads to Euler’s formula connecting the number of edges, vertices, and faces, and ends with creating nets to form polyhedra. An excellent introduction to three-dimensional figures!

**Platonic Solids (Grades 6-8)**
Students examine in detail the five Platonic solids—their shapes, vertices, edges, and regular polygonal faces. With the virtual manipulative, they can rotate each solid, viewing it from every angle, change its size, then use the transparent mode to see only the skeletal structure of the polyhedron. The site is part of the National Library of Virtual Manipulatives.

**Studying Polyhedra**
What is a polyhedron? This lesson defines the word. Students explore online the five regular polyhedra, called the Platonic solids, to find how many faces and vertices each has, and what polygons make up the faces. An excellent applet! From this page, click on Polyhedra in the Classroom. Here you have middle school classroom activities to pursue with a computer. Developed by a teacher; the lessons use interactive applets and other activities to investigate polyhedra. Activities extend to paper nets to print out and fold, studies of buckyballs and crystals, and cube-coloring problems.

**Slicing Solids (Grades 6-8)**
So what happens when a plane intersects a Platonic solid? This virtual manipulative opens two windows on the same screen: one showing exactly where the intersection occurred and the other showing the cross-section of the solid created in the collision. Students decide which solid to view, and where the plane will slice it. The site is part of the National Library of Virtual Manipulatives.

**Polyhedral Solids**
The site opens with: "The study of polyhedra is one of those special areas of Mathematics which allow the amateur and expert to work with an equal delight." Each page here gives a crisp, succinct definition of a type of polyhedra: Platonic, Archimedean, stellated, prisms and antiprisms—and even more! Each set is carefully illustrated with a view to clear instruction.

**Virtual Polyhedra: Encyclopedia of Polyhedra**
The ultimate site for those who want to play with every kind of polyhedron! You may simply let your class view and manipulate the virtual objects, or study the polyhedra more in depth through the background explanations and the exercises offered. An incredible collection of material!

### Measuring a Solid

Teaching measurement is always a challenge; in three dimensions, the challenge increases. Many students never really understand volume or surface area, although they can memorize the formulas and even apply them on tests. These resources have been selected with an eye to helping students enter into the concepts of volume and surface area through practical problems, hands-on experiences, and applets they can manipulate to actually see how these measurements are affected by change in a figure’s dimensions.

**Keeping Cool: When Should You Buy Block Ice or Crushed Ice?**
Which would melt faster: a large block of ice or the same block cut into three cubes? The prime consideration is surface area. A complete solution demonstrates how to calculate the surface area of the cubes as well as the large block of ice. Related problems involve finding surface area and volume for irregular shapes and examining the relationship between surface area and volume in various situations.

**Cordwood**
This problem scenario, set in Alaska, asks students to find the volume of a shed, applying the standard formula, and then to determine the number of cords of wood needed to fill it. Finally, they must calculate the cost of the wood. The Village Math site features more than 25 math lessons that involve application-oriented problems relevant to life in Alaska today.

**How High? Geometry (Grades 6-8)**
Using an online simulation, students investigate conservation of volume by pouring a liquid from one container to a container of the same shape, but of a larger size. Students choose from four shapes: rectangular prism, cylinder, cone, and pyramid. The smaller version of the selected shape is shown partially filled with liquid; the base dimensions of both containers are also given. Using this information, students use a slider to predict how high the liquid will rise when poured into the larger container. On "pouring" the liquid, students can compare their prediction with the results. Multiple problems are available for each of the shapes.

**Popcorn: If You Like Popcorn, Which One Would You Buy?**
Students are directed to use popcorn to compare the volumes of tall and short cylinders formed with 8-by-11-inch sheets of paper. A simple but visual and motivating way of comparing volume to height in cylinders! The solution offered explains clearly all the math underlying the problem.

**Surface Area and Volume**
This applet enables students to form and rotate both rectangular and triangular prisms. They can set the dimensions (width, depth, and height), observing how each change in dimension affects the shape of the prism as well as its volume and surface area. This is a quick way to collect data for a discussion of the relationship between surface area and volume. Users can rotate the figure and call for its frontal, side, or back view—very interesting with a triangular prism!

**Pyramid Applet**
This applet allows students to set the width, height and length of a pyramid. They then see the initial cutout (the net) and watch it fold into the pyramid specified. For better viewing, the pyramid can be rotated. At this point, the surface area and the volume are shown. No activities accompany the applet, except for the challenge to try to minimize the surface area while maximizing the volume. From the Office for Mathematics, Science and Technology Education (MSTE), a division of the College of Education at the University of Illinois at Urbana-Champaign.

**Three Dimensional Box Applet: Working with Volume**
With this applet, students create boxes online in order to explore the relationship between volume and surface area. The screen first shows a rectangular piece of graph paper. Students “cut” four squares of a size determined by the student from the corners of the rectangle. The cut surface then folds to form a box whose dimensions, surface area, and volume are displayed onscreen. Since various sizes of graph paper can be selected, data can quickly be collected and the relationship between volume and surface area explored.

### Spatial Sense

Your students already have well-developed spatial sense, as you have seen in their sports and dancing, but in the realm of mathematics, that spatial sense may seem nonexistent. All the activities in this unit help to develop a degree of comfort in the 3-D math zone, but the exercises below aim particularly to increase learners’ ability to move between the two-dimensional and three-dimensional worlds of school mathematics.

**Space Blocks: Algebra (Grades 6-8)**
Working in three dimensions requires and develops spatial sense. In these activities, students use an applet to model shapes with cubes, constructing a figure, for example, that has eight cubes and the largest possible surface area. In other problems, they connect a colored cube to the one pattern that can be folded into that cube.

**Decorating Boxes: How Do You Print a Ribbon on a Box?**
Here is the problem: four boxes are shown opened flat. Each box pattern shows a ribbon crossing several sides of the box. Students are asked to decide which of the box patterns can be folded into a box that has a ribbon running continuously around it. The essential problem is to visualize the three-dimensional object that can be made from a two-dimensional pattern.

**Using Cubes and Isometric Drawings**
Students first explore the online isometric drawing tool, which allows them to draw and manipulate figures built with “blocks” on isometric paper. In further lessons in the unit, they construct three-dimensional figures using a front-right-top view and, later, only a mat plan. The preliminary work needed to know the drawing tool is well worth the exercise in spatial sense and reasoning.

### Art in 3-D

*Thus, the chief reason for studying regular polyhedra is still the same as in the time of the Pythagoreans, namely, that their symmetrical shapes appeal to one's artistic sense.*- —H.M.S. Coxeter

H.M.S. Coxeter, an eminent geometer, became a mathematician because of his love of symmetry. Your students may also become intrigued with geometric shapes simply because of their beauty, order, and structure. These resources offer visual, sometimes interactively animated, representations of three-dimensional objects. You may find that they open a new avenue to mathematics for your students.

**Polyhedron Models Custom Built**
A ton of textual information on what polyhedra are, their history, classification, and structure, but graced with clear, colorful images! These pages are for that student who falls in love with 3-D geometric art, or the class that wants to really know how polyhedra were named, what makes one different from another, or simply wants to see a good picture of almost any type of symmetric figure in solid geometry.

**Polyhedra Collection**
Galleries of polyhedra! All types are shown here using static images and three-D objects that can be rotated, so as to see the polyhedron from all sides. Great for showing the connection of mathematics to art and for visualizing the abstract.

**Polyhedra and Art**
Here you will find images and accompanying text on polyhedra as they appeared in art from pre-Renaissance times to the 20th century. For example, geometric drawings from Leonardo da Vinci, Johannes Kepler, and M.C. Escher, among others, are not only portrayed in detail but also explained in terms of art history and mathematical structure.

**Math That Makes You Go Wow - A Multi-Disciplinary Exploration of Non-Orientable Surfaces**
This unique site was created by teachers as a supplement in a late middle school or early high school math course. The developers designed the site “to convey the beauty and fascination of such mathematical objects as the sphere, the torus, the Klein bottle, the Mobius band, the real projective plane and beyond (3-manifolds), etc.” It includes illustrations that may be rotated on-screen, sections on the history and philosophy of these unusual solids, and the connection of these figures to other areas.

### SMARTR: Virtual Learning Experiences for Students

Visit our student site **SMARTR** to find related virtual learning experiences for your students! The **SMARTR** learning experiences were designed both for and by middle school aged students. Students from around the country participated in every stage of SMARTR’s development and each of the learning experiences includes multimedia content including videos, simulations, games and virtual activities. Visit the virtual learning experience on **3D Geometry.**

### Careers

**The FunWorks**
Visit the FunWorks STEM career website to learn more about a variety of math-related careers (click on the Math link at the bottom of the home page).

### NCTM Standards

In its discussion of the Geometry Standard, Principles and Standards for School Mathematics takes care to include three-dimensional shapes as appropriate for study from preK on:

*"Through the study of geometry, students will learn about geometric shapes and structures and how to analyze their characteristics and relationships. Spatial visualization—building and manipulating mental representations of two-dimensional and three-dimensional objects and perceiving an object from a different perspective-is an important aspect of geometric thinking"* (NCTM, p. 41).

In particular, middle grades students should investigate a variety of geometric shapes, becoming comfortable with the characteristics of three-dimensional solids. They are encouraged to develop their visualization skills through hands-on experiences, such as building concrete models and exploring with software that can show a geometric solid from every angle. It is expected, as well, that students at the middle school level "recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life" (p. 232).

The Measurement Standard also applies to the study of three-dimensional shapes since volume and surface area are necessarily tied to these figures. For some solids—prisms, pyramids, and cylinders—middle school students should be guided to develop the formulas for these measurements (p. 244).

The resources offered here address each of these standards to some degree. They can help your students examine the geometric shapes of the 3-D world, develop their understanding of measurement and their powers of visualization, and come to see the artistry of mathematics in polyhedra.

## Author and Copyright

Terese Herrera taught math several years at middle and high school levels, then earned a Ph.D. in mathematics education. She is a resource specialist for the Middle School Portal 2: Math & Science Pathways project.

Please email any comments to msp@msteacher.org.

Connect with colleagues at our social network for middle school math and science teachers at http://msteacher2.org.

Copyright September 2006 - The Ohio State University. This material is based upon work supported by the National Science Foundation under Grant No. 0424671 and since September 1, 2009 Grant No. 0840824. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.