# Math Art Books

## A Comparative Analysis

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In my search for books similar to mine, I selected and analyzed eighteen titles. Of these eighteen, I discovered three sufficiently similar to my book to be considered "main competitors." The other fifteen books are better labeled "related works."

The main competitors and related works are the following:

I will first explain the related books so that it is easier to understand why only three books are categorized as main competitors. To better explain why the related works are not main competitors, I divided them into five sub-categories: worksheet books, art-focused books, books written for children, game books, and read-aloud books.

The fifteen related books are similar to my book only because they 1) are supplemental, and 2) attempt to blend math and art in some way.

Each of the three worksheet books contains a series of reproducible worksheets. Essentially, each worksheet contains a series of math facts plus a hidden picture or incomplete design. After solving each math fact, students follow a key in order to determine what line needs to be drawn or what region needs to be colored. A design or image slowly appears as more math facts are solved.

In comparison to my book, such books form a connection between math and art that is artificial. The math work required of students is not really related to the design or image being created. Instead, the art portion of each worksheet is basically just a reward for solving math facts. Furthermore, there is a limit to the amount of student excitement that can be generated from a series of worksheets.

The two art-focused books use math to teach art rather than art to teach math. Neither book is organized around math concepts. Instead, each chapter or activity begins with a topic or question related to visual art, which is then connected to a hodgepodge of math concepts. Thus, unlike my book, it would be difficult to use one of these activities to teach a specific math topic.

The eight books written for children have an ambitious goal. Each one attempts to simplify mathematical language and directions so as to allow students to conduct activities with minimal help from the teacher. Unfortunately, many of the activities, although quite innovative, would be very difficult for an average elementary-aged student to independently carry-out from start to finish. Furthermore, although many lessons incorporate art, most are akin to science projects: definitions are given at the beginning, mathematical “experiments” are conducted (materials are manipulated, measured, weighed, etc.), a chart is filled out, and ultimately a discovery is made or a pattern is observed.

These children’s books would be more useful if, like my book, they were written for teachers as lesson plans. Also, the books neglect to suggest grade levels for each activity. For instance, Groovy Geometry includes a simple activity that requires students to categorize various triangles. Then, a few pages later a much more advanced activity asks students to prove the Pythagorean Theorem. Nowhere in the book are grade recommendations ever made for these two very different lessons.

I only found one game book that attempts to incorporate art into the instruction of math. This book requires students to create many of the game boards required to play math games. However, the book is primarily focused on exposing children to games and puzzles that were “invented hundreds, even thousands of years ago,” and that come “from all parts of the world.” In other words, it is much more focused on games and world culture than math or art. Unlike my book, the amount of math taught by each game is minimal. Whereas each one of my lessons focuses on a specific math topic, the lessons in this book—which possess titles such as “Igba-Ita from Nigeria,” “Toma-Toda from Mexico,” and the “Ishango Bone from Congo”—only touch upon vague mathematical themes. Since its lessons are organized around topics such as “Games of Chance,” “Puzzles without Numbers,” and “Repeating Patterns,” teachers would find it very difficult to use the book to teach specific math concepts.

The read-aloud book that I analyzed is one in a series of similar books by the same author. Of the books in this series, Math-terpieces makes the most direct connection between math and art. The book uses illustrations and poetic riddles to encourage children to solve math problems by seeking patterns, symmetries, and convenient groupings. As a 32-page read-aloud, it obviously cannot cover as many specific math topics as my book. Instead, its stated teaching goals are broad—it aims to teach children to “think about numbers in pieces” so that arithmetic is “visual and less abstract.”

The three books that are arguably main competitors are similar to my book in that they 1) are supplemental, 2) focus more on math instruction than art, 3) consist of lesson-plans written for teachers, and 4) mainly present activities that require students to construct a visual and tangible representation of a mathematical concept.

Math, Manipulatives & Magic Wands is similar to my book because of its “project-oriented” approach. However, in contrast to my book, this book has poor lesson organization, less instructional depth, imprecise learning objectives, confusing written directions, and no assessments.

For instance, imagine a teacher wishing to use this book to teach a supplemental lesson on angles. She may wish to consider this book’s “Hefty Plate Math” lesson, which requires students to write angle measurements in intervals of 10° around the perimeter of a plastic plate. Unfortunately, the only way to find the Hefty Plate Math lesson is to skim through the entire book, since no table of contents or index matches specific lessons to specific math concepts.

Even if this lesson were easier to find, it would still be difficult to teach. Like almost all other lessons in this book, Hefty Plate Math is divided into five sections: Materials, Extensions, Procedures, “Magic Touch,” and NCTM Standards. Rather than offer clear and precise learning objectives, the Extensions section—which, considering its name, is peculiarly placed at the beginning of each lesson, right after Materials—is a list of disconnected and undeveloped suggestions of how the activity might be used for math instruction. The “extensions” for Hefty Plate Math are:

• Measurements of an Angle
• Compare and make models of halves, thirds, and fourths
• Show different times using the plates as a clock face
• Make a 2-3-4 color spinner to discuss probability
• Use plates for sorting

Rather than elaborate on one of these extensions, the lesson’s next section (Procedures) is a very confusing and diagram-less explanation of how to construct the project:

TWO PLATES
1. Cut from one edge of each plate to the center on both plates.
2. Slide one plate through the other at the slit until the centers meet. Turn one plate to show both colors.

THREE PLATES

1. Cut from one edge of each plate to the center on both plates.
2. Stack two plates together and line up their slits. Slide the stacked plates through the other plate until the centers meet. Turn two plates to show all three colors.

FOUR PLATES
1. Cut from one edge of each plate to the center on both plates.
2. Stack two plates together on each side and line up their slits. Slide the plates together through the other plates at the slits until the centers meet. Turn three plates to show all colors and create a circle graph.

Following these confusing directions is a section called Magic Touch, where the authors aim to share ways of making projects “extra special.” However, this section in each lesson seems no different than the preceding Extensions section. It consists of cursory ideas of how to use the activity for instruction. For example, here is the confusing Magic Touch section of Hefty Plate Math:

Teach younger students the basics of circle graphing by giving them 36 objects that can be sorted into four categories. Have them turn the plate 10 degrees or one thumbprint for each of the 36 objects. Example: If they have four blue objects, turn the plate to show four blue thumbprints. Ask them to write sentences to compare the groups on their graph.

Finally, the last section—in an apparent attempt to bolster the book’s instructional credibility—claims to align each lesson with the National Council of Teachers of Mathematics (NCTM) Standards. Here are the standards listed for Hefty Plate Math:

DATA AND PROBABILITY
Standard - The students will select and use appropriate statistical methods to analyze data

Standard - The students will develop and evaluate inferences and predictions that are based on data.

Standard - The students will understand and apply basic concepts of probability.

Unfortunately, these correlations only increase confusion. They are exceedingly vague and only deal with the topic of data and probability. What happened to the other topics mentioned earlier in the lesson, namely “measurements of an angle,” “models of halves, thirds, and fourths,” and “clock faces?”

Essentially, this overall incoherence—present in almost all the lessons in Math, Manipulatives & Magic Wands—severely hinders a teacher’s ability to use the book for effective math instruction. Indeed, in some of its lessons, academics are wholly absent. The lesson “Googol Necklace” asks students to write all the digits that make up a Googol (a 1 followed by 100 zeros) on a long strip of adding machine tape. What could possibly be the academic benefit of such a time-consuming activity?

In contrast, the lessons in my book are coherent, focused, organized, and educational. For instance, a teacher wishing to teach a supplemental lesson on angles would not need to skim through my book. Instead, the book’s table of contents conveniently states that the “Angles” lesson “teaches students to estimate and compare angle measurements.” This is also stated in the lesson’s introductory paragraph.

The lesson follows through on this promise. Students are required to construct two items: 1) an adjustable “Paper Protractor” that can be used to estimate the size of any angle, and 2) an “Angle Comparison Strip,” which allows students to easily compare thirteen angle measurements by placing them in a side-by-side sequence. Indeed, not only are the lesson’s assembly instructions not confusing and more thorough than those in Hefty Plate Math, the lesson also specifically explains how the teacher can use the project to facilitate student learning. Here is an excerpt:

Introduction to the Rotating “Paper Protractor”

Introduce the first project to students by simply asking, “What are angles?” Good responses to elicit are “angles are two connected lines,” or “angles are corners.” Draw some examples on the board and show students how to use a paper protractor (constructed beforehand by the teacher) to estimate angle sizes. For instance:

Also, teach students that the size of an angle depends on how opened or closed it is, not how long its lines are.

Next, using the paper protractor (or the teacher’s previously constructed “Angle Comparison Strip”), the teacher should focus student attention on landmark angles, namely 60, 90, 180, and 360 degrees. Point out to students that the lines of a 90 degree angle can also be called “perpendicular” lines. Also, show them that a 180 degree angle is when two lines open up to the point of forming a straight line. A 360 degree angle is an angle that has opened up so much that it has closed back on itself, making it look a lot like a 0 degree angle. To really help students remember these landmarks, the teacher or a student volunteer could perform a 180 and/or 360 degree “spin jump” (they will love it).

Constructing the Rotating “Paper Protractor"

Students begin making their adjustable paper protractor by coloring their 30 degree sheet (page 119) with a single color crayon. Then they follow three steps to cut it out. First, they cut around the circle. Second, they cut down the dotted line leading to the middle. Finally, they carefully cut out the very small black dot in the circle’s center.

After coloring and cutting the 30 degree sheet, students cutout two circle sheets (page 120). Students should cut along the dotted line on only one of these circle sheets. The three pieces, unconnected, should look like this:

Notice that the 30 degree circle is slightly smaller than the other two circles. This is because the smaller circle will need to easily rotate when it is eventually inserted between the two larger circles.

When a student finishes cutting-out all three pieces, the teacher should assist in putting them together. Connect the two plain white circles first, using four staples and a brass tack.

This creates a “container” designed to hold the 30 degree circle. Students insert one of the flaps of the 30 degree circle into the slit located on the container. The students use a turning motion to slide the 30 degree sheet between the container sheets.

None of the lessons in Math, Manipulatives & Magic Wands are as detailed, organized, or educationally focused as this lesson. Furthermore, because my book is committed to helping teachers teach specific rather than vague math concepts, it often forewarns them of common student misconceptions in a way that Math, Manipulatives & Magic Wands cannot. This allows teachers to turn potential pitfalls into valuable “teaching moments.” Here is another excerpt from my Angles lesson:

After drawing the angles in pencil, students can use a single crayon to color the interior of each angle. It is extremely common for students to make the mistake of coloring the wrong side of angles greater than 180 degrees.

However, this mistake is instructive, since it causes students to think about each angle’s reflex angle. Therefore, the teacher probably should not warn students beforehand. Just make sure to have backup angle sheets available.

Please view any of my book’s other lessons for similar evidence that my book is more focused, organized, educational, and coherent than Math, Manipulatives & Magic Wands.

The second competitor, Hands-On Math Projects with Real-Life Applications, is a 270-page book for teachers of grades three through five. The lessons are divided into six categories: Math and Science, Math and Social Studies, Math and Language Arts, Math and Art, Math and Recreation, and Math and Life Skills. The Math and Art section contains only seven lessons. The first three of these lessons—which require students to create a math cartoon, math t-shirt, and math billboard—possess vague learning objectives.

For instance, one of the first lessons, titled “Painting a T-Shirt with a Math Slogan,” states that it covers the following skills:

1. Various math skills depending on the T-shirt slogans
2. Synthesizing information related to math
Another of the book’s first three lessons, titled “Creating a Billboard That Advertises Math,” is similarly vague regarding the skills taught:

1. Various math skills depending on the content of students’ billboards
3. Using art to convey ideas about math
4. Measuring lengths
5. Using technology (if computers are used)

Essentially, rather than teach specific math concepts, the first three “Math and Art” lessons in Hands-On Math Projects with Real-Life Applications only ask students to creatively express concepts they already know.

The other four “Math and Art” lessons in this book are similar and of comparable quality to my book’s lessons because they have precise learning goals and easy-to-follow directions. However, with only four strong lessons in its “Math and Art” chapter (compared to my book’s 27 lessons), Hands-On Math Projects with Real-Life Applications is severely limited in its capacity to teach math through art.

As evidence that my lessons are as strong as these four lessons in Hands-On Math Projects with Real-Life Applications, below is a side-by-side comparison of a lesson from each book. Both lessons are similar in their learning objectives and in their directions provided to the teacher (albeit only my lesson’s directions include diagrams).

As evidence that my lessons are as strong as these four lessons in Hands-On Math Projects with Real-Life Applications, below is a side-by-side comparison of a lesson from each book. Both lessons are similar in their learning objectives and in their directions provided to the teacher (albeit only my lesson’s directions include diagrams).

Prisms and Pyramids

Materials
• Pyramid sheets, pages 133-134 (1 set per student)
• Prism sheets, page 135-137 (1 set per student)
• Scissors (1 per student)
• Scotch tape dispenser (for the teacher)
• The completed project prepared by the teacher before the lesson
Introduction

This lesson teaches students how to identify differences between two types of three-dimensional shapes: prisms and pyramids. It also teaches students to identify and count each shape’s edges, vertexes, and faces. Students will create a total of three prisms and three pyramids.

The teacher should begin by showing students a number of pre-constructed prisms and pyramids. Call on some students to describe what they notice to be the difference between the two types of three-dimensional shapes. Most students will quickly recognize that prisms have a flat top, while pyramids have a pointed top. The teacher should make sure that students also realize that the sides of prisms are rectangles (or parallelograms), while the sides of pyramids are triangles.

The teacher should also explain to students the difference between an edge, a face, and a vertex. The diagram below, drawn by the teacher and displayed at the front of the room, will help students understand these words.

The Project

Students use pages 133-134 to make their three pyramids. They start by cutting along the solid lines and folding along the dotted lines.

Students will need two half-inch pieces of tape for each pyramid—one to hold the walls together and the other to hold the base in place. The teacher should provide these pieces of tape, sticking them along the edge of each student’s desk. (The teacher may need the help of a student or assisting adult in order to distribute all the tape needed for this project.) When taping two edges together, students should first stick the tape on one edge. Then, making sure the walls are just barely touching (not overlapping), they fold the tape over so that it sticks to the other wall.

Next, students create three prisms using pages 135-137. Just like with the pyramids, students should cut along the solid lines and fold along the dotted lines. Students will need three half-inch pieces of tape for each prism—one to hold the walls together and one to hold each base in place.

Once students finish making all of their pyramids and prisms, ask them to practice counting each ones’ number of faces, edges, and vertexes.

Assessment: page 180
Making a Three-Dimensional Paper Sculpture from a Two-Dimensional Net

Most students are familiar with working with two-dimensional figures such as squares, rectangles, and triangles, which are usually described in terms of length and width or base and height. We live in a three-dimensional world, however, and the properties of three-dimensional figures include the added dimension of depth. Because students most often work with two-dimensional figures (at least in school), the concept, and certainly the visualization, of three-dimensional figures can cause confusion. In this project, students will make three-dimensional paper sculptures. As they complete their sculptures, they will become more familiar with the properties of three-dimensional figures.

Goal

Working individually, students will use nets to make a paper sculpture. The nets will be used to form cubes, tetrahedrons, and octahedrons. Suggested time: two to three class periods.

Skills Covered

1. Making a three-dimensional figure from a two-dimensional net
2. Identifying the terms (ace, edge, and vertex
3. Using visualization, spatial reasoning, and geometric modeling
Special Materials and Equipment

One 1-inch-by-3-inch-by-5-inch piece of Styrofoam for each student to use as the base of a sculpture, pipe cleaners (one for each figure in the sculpture each student will make), scissors, glue, markers, crayons, and colored pencils.

Development

• Before starting this project, make a sculpture of your own using a net. Show this sculpture to your students as an example. Also decide in advance which nets you will instruct your students to use. You may prefer that they work with only one, or you may allow them to work with two or all three. Another option is to allow your students to choose which nets they will construct. Younger students should probably be limited to one. Note that the cube is the easiest to construct and the octahedron is the most challenging.
• Begin this project by discussing with your students the difference between two-dimensional and three-dimensional figures. Show examples of some two dimensional figures, such as squares and rectangles, on the board or a screen. Point out the dimensions, for example, length and width. Next show your students your example of a three-dimensional figure. Explain that it was created from a two dimensional net that was folded and glued into its three-dimensional form. If necessary, explain that a net is a two-dimensional pattern that can be folded into a three-dimensional shape. Point out that the three-dimensional shape has length, width, and height. Also note its faces, edges, and vertices. Tell your students that they will make three-dimensional figures from two-dimensional nets.
• Distribute copies of Student Guide 26.1 and review the information it contains with your students. Go over the steps for creating their sculptures. If necessary, do some of the steps together as a class.
• Hand out copies of the data sheets your students will need: Data Sheet 26.2: Net for a Cube; Data Sheet 26.3: Net for a Tetrahedron; and Data Sheet 26.4: Net for an Octahedron. (Have extra copies of the nets available in the event that students make cutting mistakes.)
• Remind your students to cut along the solid lines and fold on the dotted lines of their nets. After folding on the dotted lines they should have formed a three-dimensional figure. If the figure does not have three dimensions, the student should double-check his or her steps and may need to start over.
• Show your students the faces, edges, and vertices of each figure. Note that the cube has 6 faces, 12 edges, and 8 vertices. The tetrahedron has 4 faces, 6 edges, and 4 vertices. The octahedron has 8 faces, 12 edges, and 6 vertices.
• Encourage your students to color the faces of their figures to make them more attractive. They should color the faces before they glue them together.
• Instruct your students that after they have glued together the faces of their figures, they are to attach their figures to their Styrofoam bases with pipe cleaners.

Wrap Up

Display the sculptures around the classroom. Review the properties of two-dimensional and three-dimensional fIgures.

Extension

Obtain nets of other platonic solids such as a dodecahedron or an icosahedron and encourage your students to make three-dimensional models of them.

Hands-On Math Projects with Real-Life Applications may have proven my book’s equal if it had included more lessons similar to this “Three-Dimensional Paper Sculpture” lesson. However, because my book uses art to teach many more math topics, it is much better overall at integrating art into math instruction.

The third and final competitor, MathART Projects and Activities, written by Carolyn Ford Brunetto, offers my book the strongest competition. In the process of trying to get my book published by Scholastic, I learned that this book has been a strong seller of theirs for many years. However, Ms. Brunetto’s book is weaker than mine because it states imprecise learning objectives, has less instructional depth, lacks assessments, and frequently relies on project materials not readily available in a classroom.

A supplemental math book’s main purpose is to allow teachers to introduce, reinforce, or expand upon the topics their students are required to learn. A supplemental math book that incorporates art has the added goal of generating student excitement about such topics. Although the lessons in both my book and Ms. Brunetto’s book succeed in increasing student interest in math, Ms. Brunetto’s book is much more limited in its ability teach a variety of specific math concepts.

The beginning of both my book and Ms. Brunetto’s book includes a section that aligns individual lessons with the National Council of Teachers of Mathematics (NCTM) Standards. But while her book only links its lessons to the broadest of the NCTM Standards’ categories (“Mathematics as Communication,” “Number Sense and Numeration,” etc.), my book includes a chart that matches lessons to the more specific sub-standards of each category. (To compare the two sections, see pages 7-8 of Ms. Brunetto’s book and pages v-viii of my book.)

However, it is not enough for any supplemental math book to rely solely on NCTM alignment as evidence of an ability to teach specific math concepts. This is because even the NCTM’s sub-standards are conceptually broad. (For instance, a sub-standard of “Communication” states that students should be able to “organize and consolidate their mathematical thinking through communication.”) In order for supplemental lessons to be useful in the classroom, they should also explicitly state their purpose using precise language.

For instance, let’s compare each book’s most specific lesson guides: my book’s Table of Contents and Ms. Brunetto’s “Projects by NCTM Standards.” (Note: Ms. Brunetto’s table of contents was not selected for the comparison because it provides less specific guidance than her NCTM Standards section.) Below on the left is an excerpt from Ms. Brunetto’s NCTM section, which is her most specific lesson guide but still only references her lessons with the most general of the NCTM’s standards. On the right is an excerpt from my book’s Table of Contents, which states the exact learning objective of each lesson.

Standard 2: Mathematics as Communication
Math Wanted Posters (page 34)

Standard 6: Number Sense and Numeration
Math Wanted Posters (page 34)
Make an Abacus (page 42)
Place Value Snakes (page 46)
Number Collages (page 48)

Standard 7: Concepts of Whole Number Operation
Multiplication Constellations (page 38)
Make an Abacus (page 42)

Standard 8: Whole Number Computation
Number Pattern Connect-the-Dot Puzzles (page 36)
Multiplication Houses (page 40)
Weave a Number Pattern (page 71)

Standard 9: Geometry and Spatial Sense
Stained Glass Windows (page 13)
Nature Symmetry Prints (page 15)
Symmetry Pop-Up Card (page 17)
Amazing Paper Ornaments (page 20)
Copycat Coordinates (page 22)
Graph a Wall Hanging (page 25)
Geometry Sculptures (page 29)
Five Pointed Stars (page 31)
Magic Folding Cubes (page 52)
Terrific Tessellations (page 67)
Number Spirals (page 73)
Fraction Flags (page 82)
Fraction Quilts (page 84)
Symmetry (page 29)
Teaches students how to make symmetrical designs and how to distinguish between vertical symmetry, horizontal symmetry, and rotational symmetry.

Perimeter (page 32)
Teaches students how to determine the perimeter of complex shapes.

Area (page 34)
Teaches students to determine the area of complex shapes.

Place Value (page 37)
Helps students memorize places values, including decimal place values.

Map Scale (page 40)
Teaches students to determine “real life” distances on maps.

Perpendicular and Parallel Lines (page 45)
Teaches students to identify perpendicular and parallel lines.

Angles (page 50)
Teaches students to estimate and compare angle measurements.

Shapes (page 56)
Teaches students to organize 19 shape names into three categories.

How is a teacher supposed to use the index on the left to find a supplemental lesson capable of introducing, reinforcing, or expanding upon a mathematical concept? Nothing is learned about the specific concepts taught in the lessons “Stained Glass Windows” or “Geometry Sculptures” after reading that they align with “Standard 9: Geometry and Spatial Sense.” In contrast, my table of contents is much more informative and useful.

The limitations of Ms. Brunetto’s book are also evidenced by the lessons themselves. For instance, along with its title and materials, each of Ms. Brunetto’s lessons begins with a brief statement that attempts to summarize the purpose of the project. Below are ten examples from ten different lessons:

STAINED GLASS WINDOWS
These translucent designs will shed some light on geometric shapes!

NATURE SYMMETRY PRINTS
This project will help students see the symmetry that is found in so much of nature.

SYMMETRY POP-UP CARDS
These 3-D greeting cards are a fun way for students to explore symmetry—and they’re sure to impress the lucky recipients.

MATH WANTED POSTERS
These funny posters will put everyone on the lookout for math!

NUMBER COLLAGES
Numbers alone are the focus of these works of art!

ONE-METER DESIGNS
You’ll be surprised to see how creative students can be with just one meter!

MAGIC FOLDING CUBES
Students measure and cut three strips of paper, then learn the secret way to fold them into a cube.

NUMBER SPIRALS
Round and round go these graceful patterns of numbers and lines.

FRACTION FLAGS

Just like the index entitled “Projects by NCTM Standards,” these preliminary statements do little to help teachers determine the exact concepts covered by each lesson. They also signify each lesson’s lack of instructional depth and academic rigor.

For instance, my single symmetry lesson teaches more than both of Ms. Brunetto’s symmetry lessons combined. Below, on the left, is my symmetry lesson positioned alongside Ms. Brunetto’s “Symmetry Pop-Up Cards” and “Nature Symmetry Prints.” All three lessons are reproduced in their entirety (except my lesson does not reproduce the black line masters or the assessment).*

Symmetry

Materials

• Symmetry sheets, page 106-108 (1 of each type for every student)
• Markers (1 box per student)
• The completed project prepared by the teacher before the lesson
Introduction

This lesson teaches students how to make symmetrical designs and how to distinguish between vertical symmetry, horizontal symmetry, and rotational symmetry.

Start this lesson by offering students a simple definition of symmetry: “an image that is the same on both sides.” This definition isn’t entirely accurate, since from it one could deduce that the following image is symmetrical, when it actually is not:

However, a more precise definition of symmetry will form in students’ minds once beginning the project.

Completed examples of the three symmetry projects should be hung at the front of the room, along with the corresponding uncolored sheets. The teacher should use these to summarize the three types of symmetry.

An image with vertical symmetry has a vertical line of symmetry that causes the left and right halves of the image to be reflections of one another, but not necessarily the top and bottom halves.

An image with horizontal symmetry has a horizontal line of symmetry that causes the top and bottom halves of the image to be reflections of one another, but not necessarily the left and right halves.

An image with rotational symmetry has both vertical and horizontal lines of symmetry that cause both the left/right halves and the top/bottom halves to be reflections of one another.

The Project

Tell students that they will be using three marker colors to make their first symmetrical design.

Students should start with vertical symmetry (page 106), since this is usually the easiest to grasp visually. While working, it is most important for students to remember that any shape they color on one side they must also find and color on the other side. In other words, “opposite” shapes need to be the same color. Model the coloring of a few “opposites”:

Another way to show students the correct way to color these sheets is to explain that each shape is the same distance away from the line of symmetry as its “opposite.” The teacher can demonstrate this by drawing arrows on the example.

A student is only finished once all the shapes in his or her design are colored.

Horizontal symmetry comes next (page 107). Creating a design with this type of symmetry usually proves slightly more difficult for students. This is probably because, unlike vertical symmetry, horizontal symmetry does not align itself with the natural symmetry of a person’s face and eyes. Still, after making a few mistakes students should get the hang of it.

Finally, when coloring the last sheet (page 108), students are allowed to create designs with two lines of symmetry (rotational symmetry). The teacher may describe rotational symmetry as “finding each shape’s three other opposites,” or “finding for each shape the three others that are like it.” In other words, each shape is part of a group of four:

It’s recommended that you keep the three types of symmetry sheets in an accessible location in the classroom. That way, students that finish other school work early can use their extra time to make new symmetrical designs.

Assessment: Page 167

SYMMETRY POP-UP CARDS

These 3-D greeting cards are a fun way for students to explore symmetry—and they’re sure to impress the lucky recipients.

WHAT YOU NEED
• Construction paper
• Crayons or markers
• Scissors
WHAT TO DO
Student Instructions / Teacher Notes

1) Choose two square or rectangular sheets of construction paper, and fold them in half. Set one aside to be your card. The second sheet will form your pop-up.

2) Draw a pop-up design along the fold of the second sheet. You will only need to draw one half of the design. Include a notched flap along the bottom of the design.

Here are some ideas for pop-up designs:

3) Cut out your pop-up design. You'll notice that it has symmetry--it looks exactly the same on both sides.

4) Place the pop-up along the inside fold of your card. The inside fold of the pop-up should face you. Fold the flaps away from you.

5) Tape the flaps to the card at an angle. When you close the card, the pop-up should fold flat. When you open the card, the pop-up should pop up!

The pop-ups work best when both sides are taped at about a 45-degree angle.

6) Decorate your card and give it to someone you love!

TIPS FOR A SUCCESSFUL PROJECT

Students may need your assistance in taping their pop-ups to their cards at the correct angle.

WANT TO KEEP GOING?

Add more pop-ups to each card. Just be sure to place taller pop-ups behind smaller ones.

Students can add some intricacy to their pop-ups by cutting out other shapes from their centers.

NATURE SYMMETRY PRINTS
This project will help students see the symmetry that is found in so much of nature.

WHAT YOU NEED
construction paper
tissue paper or tracing paper
crayons, colored pencils, or grease pencils (dark colors work best)
large tree leaves

WHAT TO DO
Student Instructions / Teacher Notes

1) Look for a big, flat leaf on the ground. Make sure that the leaf is whole and has no nicks or holes.

Steer students away from leaves that are not flat or have curled points at the end. Also, try to find leaves that have prominent veins. You may want to gather a group of suitable leaves yourself and bring them into class.

2) Lay the leaf on a flat surface so that its veins are facing up. Tape the top and bottom of the leaf to the flat surface. Then tape a sheet of tissue paper over the entire leaf.

3) Rub the crayon firmly across the tissue paper over the leaf. (Don't press too hard, or you'll tear the tissue paper or squish the leaf!) Rub the crayon back and forth in the same direction.

4) When you have rubbed over the whole leaf, remove the tissue paper. Then create a frame for your picture with construction paper.

The frames need not be square--encourage your students to tryout circles and triangles, too. To make frames, have students fold sheets of construction paper rectangles, squares, or circles in half, cut out a smaller shape from the center, and unfold.

5) Hang your symmetry print on a window!

Ask students to find the symmetrical patterns of the leaves. Point out that the veins and edges of one half of the leaf are almost a mirror image of the opposite half.

TIPS FOR A SUCCESSFUL PROJECT

The softer and darker the crayons, the easier it will be to bring up the details of the leaves.

WANT TO KEEP GOING?

Nature's symmetry doesn't stop at leaves--students can also make prints of shells, flowers, plants, and even insects and butterflies.

Have your students look up leaves in an encyclopedia to see more examples of leaf symmetry.

Although all three lessons are fun for students, only my lesson helps students gain a sophisticated understanding of symmetry. Both of Ms. Brunetto’s lessons ignore the concept of lines of symmetry. Rather, they simply explain symmetrical images as looking “exactly the same on both sides.” In contrast, my symmetry lesson teaches students about three different types of symmetry. Furthermore, my lesson explains that images are truly symmetrical only if their matching parts are equidistant from their lines of symmetry.

One can find additional evidence that Ms. Brunetto’s book has less instructional depth by comparing other lessons of hers and mine. If one wishes to make additional comparisons, the chart below lists lessons that teach similar topics.

Ms. Brunetto's book's lack of instructional depth can be also be shown by examining how well her book covers the NCTM standards that it claims to cover. For instance, let’s compare how well both books teach the NCTM’s "Measurement Standard” for grades 3-5. According to Ms. Brunetto's "Projects by NCTM Standards” section, eight of her lessons teach the topics of "Standard 10: Measurement." Similarly, my book's NCTM Standards Chart states that nine of my lessons teach the topics of this same standard. However, the following chart shows that Ms. Brunetto's lessons only superficially teach the NCTM Measurement Standard for grades 3-5, whereas my book teaches the standard more thoroughly. This is true even though my book is written for grades 2-4 and hers for grades 3-5.

Clearly, Ms. Brunetto’s book minimally teaches the academic standards, while my book provides students with a deeper understanding of the topics they are expected to learn.

Finally, whereas all the projects in my book require materials readily available in a typical elementary school, many of the required materials in Brunetto’s lessons are inconvenient for teachers (namely: egg cartons, Velcro tape, adding machine tape, plastic shower curtain liners, tree leaves, sticks, marshmallows, sponges, shoe laces, shoe boxes, drinking straws, and paper-towel tubes).

In summary, my book is more focused, educational, organized, thorough, and convenient for teachers than its main competitors.

* Lessons teaching symmetry were chosen for comparison because both my book and Ms. Brunetto’s book include lessons that explicitly teach this topic. Other topic areas covered by both books are set forth in a chart following the symmetry comparison.