360∘ Doesn’t Count - When looking for rotational symmetries students will often list 360∘ rotational symmetry. When a figure is rotated 360∘ the result is not congruent to the original figure, it is the original figure itself. This does not fit the definition of rotational symmetry. This misconception can cause error when counting the numbers of symmetries a figure has or deciding if a figure has symmetry or not. Another important note here is that sometimes rotational symmetry is referred to as point symmetry. The center “point” of the figure is the center of rotation in a figure with rotational symmetry. Students might be thrown by this new term which might appear on a standardized test if it isn’t introduced here.
Review Quadrilateral Classifications - Earlier in the course students learned to classify quadrilaterals. Now would be a good time to break out that Venn diagram. Students will have trouble understanding that some parallelograms have line symmetry if they do not remember that squares and rectangles are types of parallelograms. As the course draws to an end, reviewing helps students retain what they have learned past the final. It is possible to redefine the classes of quadrilaterals based on symmetry. This pursuit will make the student use and combine knowledge in different ways making what they have learned more flexible and useful. You may also want to look at the symmetry of regular polygons.
Applications - Symmetry has numerous applications both in and outside of mathematics. Knowing some of the uses for symmetry will motivate student, especially those who are not inspired by pure mathematics, to spend their time and energy learning this material.
Biology – Most higher level animals have bilateral symmetry, starfish and flowers often have 72∘ rotational symmetry. Naturally formed nonliving structures like honeycomb and crystals have 60∘ rotational symmetry. These patterns are fascinating and can be used for classification and study.
Trigonometry – Many identities of trigonometry are based on the symmetry of a circle. In the next few years of mathematics the students will see how to simplify extremely complex expressions using these identities.
Advertising – Many company logos make use of symmetry. Ask the students to bring in examples of logos with particular types of symmetry and create a class collection. Analyze the trends. Are certain products more appropriately represented by logos that contain a specific type of symmetry? Does the symmetry make the logo more pleasing to the eye or more easily remembered?
Functions – A function can be classified as even or odd based on the symmetry of its graph. Even functions have symmetry around the y−axis, and odd functions have 180∘ rotational symmetry about the origin. Once a function is classified as even or odd, properties and theorems can be applied to it.
Draw - Have students be creative and create their own logos or designs with specific types of symmetry. Using these concepts in many ways will build a deeper understanding and the ability to apply the new knowledge in different situations.
Translation or Transformation - The words translation and transformation look and sound quite similar to students at first. Emphasis their relationship: A translation is just one of the many transformations the students will be learning about in this chapter.
Image vs Pre-image - Students often get these two terms mixed up. Help students focus on the prefix, “pre” which means “before”. The pre-image is the image before a transformation is performed.
Mapping Notation - An ordered pair is use to represent a location on the coordinate plane, and mapping notation indicates how a point is “moved” to create an image of a point. Give students ample opportunity to practice reading and writing with this notation.
The Power of Good Notation - There is a lot going on in these exercises. There are the points that make the preimage, the corresponding points of the image, and the mapping notation used to describe the translation. Good notation is the key to keeping all of this straight. The points of the image should be labeled with capital letters, and the prime marks should be used on the points of the image. In this way it is easy to see where each point has gone. This will be even more important when working with more complex transformations in later sections. Start good habits now.
Use Graph Paper and a Ruler - When making graphs of these translations by hand, insist that the students use graph paper and a ruler. If students try to graph on lined paper, the result is frequently messy and inaccurate. It is beneficial for students to see that the pre-image and image are congruent to reinforce the knowledge that a translation is an isometry. It is also important that students take pride in producing quality work. They will learn so much more when they take the time to do an assignment well, instead of just rushing through the work.
Translations of Sketchpad - Geometers’ Sketchpad uses vectors to translate figures. The program will display the pre-image, vector, and image at the same time. Students can type in the vector and can also drag points on the screen to see how the image moves when the vector is changed. It is a quick and engaging way to explore the relationships. If the students have access to Sketchpad and there is a little class time available, it is a worthwhile activity. You will need to explain how students can write vector from mapping notation in the program.
Rules for Reflections in the Coordinate Plane - Students are likely to have a hard time memorizing these rules. Encourage students to just reflect one point at a time until they notice a pattern. When reflecting over the y−axis, the x changes sign and when reflecting over the x−axis, the y changes sign. Reviewing the quadrants and where x and y are positive/negative may also help. They will then see how the signs change as they cross the particular axes.
Will the Pre-Image and Image Match Up? - Encourage students to visualize whether or not the image and pre-image will “match up” if they fold over the line of reflection. This caheck at the end of the process may help students avoid making careless errors.
Reflections over y=x - This particular reflection is very important for students to be able to recognize for future math courses. The inverse of a function is a reflection of the original function over this line. Understanding the connection between the process of creating this reflection (switching x and y values) and finding an inverse function later in more advanced algebra courses (students will switch the variables in the equation) will help students gain a deeper understanding of both concepts.
Clockwise vs Counterclockwise Rotations - Students sometimes have difficulty keeping these two straight. It is also counter-intuitive to them that a counterclockwise angle is considered positive and a clockwise angle is negative. This text uses all positive angles and indicates clockwise vs counterclockwise but if you decide to use Geometer’s Sketchpad (or other computer programs) to do some additional activities, you will need to explain this concept to students.
Rotating in the Coordinate Plane - There are mapping rules given in the text for the different rotations in the coordinate plane. Sometimes, however, it is helpful to have students attempt to “visualize” these on the coordinate plane. Often students forget the “rules” and cannot complete the assignment or problem on a quiz or test. It may be helpful to do some examples using patty paper. Students can draw a triangle or quadrilateral on the coordinate plane, then trace the figure and the axes onto the patty paper. Now they can rotate the patty paper 90, 180 and 270 degrees using the axes as a reference to see how the figure (and the coordinates of its vertices) change. This technique is particularly beneficial for visual and kinesthetic learners.
Composition of Transformations
Glide Reflections - Students sometimes struggle with the concept that order doesn’t matter here. It is helpful to have students do a problem such as the example below to experience this.
Example: ΔABC has vertices (2, -3), (5, 3) and (6, 0). Translate this figure 5 units left and reflect it over the line y=−2. What are the coordinates of the final image? Now try doing the reflection before the translation. What are the coordinates of the final image?
Answer: The coordinates of the resulting images for both orders is (-3, -1), (0, -7), (1, -4).
Reflections over Parallel Lines - Students may initially struggle with the relationship between this double reflection and the resulting equivalent translation. Have students practice this on the coordinate plane where distances between the vertices of the pre-image and image can be easily calculated to verify the relationship.
Reflections over both Axes - Patty paper can be used again here to help students see that this double reflection is actually a rotation of 180∘ about the origin. Students can also compare the rules for reflections to the rules for rotations to see that the combination of the two reflections results in the rule for the rotation.
Extension: Tessellating Polygons
Review Interior Angles Measures for Polygons - Earlier in the course students learned how to calculate the sum of the measures of interior angles of a convex polygon, and how to divide by the number of angles to find the measures of the interior angle of regular polygons. Now would be a good time to review this lesson. The students will need this knowledge to see which regular polygons will tessellate and the final is fast approaching.
Move Them Around - When learning about regular and semi-regular tessellation it is helpful for students to have a set of regular triangles, squares, pentagons, hexagons, and octagons that they can slide around and fit together. These shapes can be bought from a mathematics education supply company or made with paper. Exploring the relationships in this way gives the students a fuller understanding of the concepts.
Use On-Line Resources - A quick search on tessellations will produce many beautiful, artistic examples like the work of M. C. Escher and cultural examples like Moorish tiling. This bit of research will inspire students and show them how applicable this knowledge is to many areas.
Tessellation Project - A good long-term project is to have the students create their own tessellations. This is an artistic endeavor that will appeal to students that typically struggle with mathematics, and the tessellations make nice decorations for the classroom.