1.12: Rigid Transformations
Pacing
Day 1 | Day 2 | Day 3 | Day 4 | Day 5 |
---|---|---|---|---|
Exploring Symmetry | Translations | Reflections |
Quiz 1 Start Rotations Investigation 12-1 |
Finish Rotations |
Day 6 | Day 7 | Day 8 | Day 9 | Day 10 |
Compositions of Transformations Investigation 12-2 |
Quiz 2 \begin{align*}*\end{align*}Extension: Tessellating Polygons |
\begin{align*}*\end{align*}Extension Quiz Start Review of Chapter 12 |
Finish Review of Chapter 12 | Chapter 12 Test |
Exploring Symmetry
Goal
This lesson introduces line symmetry and rotational symmetry.
Teaching Strategies
The Know What? for this lesson provides a good starting point for a discussion about symmetry in nature. After defining line symmetry and rotational symmetry, discuss how both are found in nature and the real world. Then, go over the symmetry in the starfish.
Have your students write the alphabet in uppercase letters. Using one colored pencil, show which letters possess horizontal or vertical symmetry by drawing in the line. For example, \begin{align*}B, E\end{align*}, and \begin{align*}K\end{align*} have a line of horizontal symmetry. Using a second color, draw in the vertical lines of symmetry. Have a contest to determine who can write the longest word possessing one type of symmetry. For example, MAXIMUM is a word where all the letters have vertical symmetry. KICKBOXED has a horizontal line of symmetry. You could even do words like TOT. The entire word has vertical symmetry.
When discussing rotational symmetry, some textbooks may refer to the rotations as \begin{align*}n\end{align*}-fold rotational symmetry. This simply means that the \begin{align*}n\end{align*} is the number of times the figure can rotate onto itself. For example, a regular pentagon has 5-fold rotation symmetry, because it can be rotated 5 times of \begin{align*}108^\circ\end{align*} before returning to its original position.
Translations
Goal
The purpose of this lesson is to introduce the concept of translations in the coordinate plane.
Relevant Review
Students should be comfortable with the distance formula and finding the slope between two points.
Teaching Strategies
All the transformations in this chapter are rigid transformations or isometries. Students need to know that these transformations never change the size or shape of the preimage and, therefore, will always create congruent images.
A “translation rule” is not property defined in this lesson. A translation rule is the amount an image is translated (or moved) from the preimage. A rule can only be applied when the translation is in the coordinate plane. The horizontal movement is added or subtracted from \begin{align*}x\end{align*} and the vertical movement is added or subtracted from \begin{align*}y\end{align*}. The horizontal and vertical change will always be the same for every point in a figure. For example, if the translation rule is \begin{align*}(x, y) \rightarrow (x+1, y-2)\end{align*} for a triangle, each vertex of the triangle will be moved to the right one unit and down two units.
Additional Example:
a) If \begin{align*}\Delta ABC\end{align*} is the preimage, find the translation rule for image \begin{align*}\Delta XYZ\end{align*}.
b) If \begin{align*}\Delta XYZ\end{align*} is the preimage, find the translation rule for image \begin{align*}\Delta ABC\end{align*}.
Solution:
a) From \begin{align*}A\end{align*} to \begin{align*}X\end{align*}, the triangle is translated to the left 8 units and down 5 units. \begin{align*}(x, y) \rightarrow (x-8, y-5)\end{align*}.
b) From \begin{align*}X\end{align*} to \begin{align*}A\end{align*}, the triangle is translated to the right 8 units and up 5 units. \begin{align*}(x, y) \rightarrow (x+8, y+5)\end{align*}.
Reflections
Goal
Students will reflect a figure over a given line and find the rules for reflections over vertical and horizontal lines in the coordinate plane.
Teaching Strategies
Using patty paper (or tracing paper), have students draw a small scalene triangle \begin{align*}\Delta ABC\end{align*} on the right side of the paper. Fold the paper so that \begin{align*}\Delta ABC\end{align*} is covered and then trace it. Unfold the patty paper and label the vertices as \begin{align*}A^\prime, B^\prime,\end{align*} and \begin{align*}C^\prime\end{align*}, the images of \begin{align*}A, B\end{align*}, and \begin{align*}C\end{align*}. Darken the fold line; this is the line of reflection. Use a ruler to draw \begin{align*}\overline{AA^\prime}\end{align*}. Mark the intersection of the reflecting line and \begin{align*}\overline{AA^\prime}\end{align*} point \begin{align*}M\end{align*}. Find \begin{align*}AM\end{align*} and \begin{align*}A^\prime M\end{align*}. Ask students what they notice about the distances and how the line \begin{align*}\overline{AA^\prime}\end{align*} intersects the line of reflection.
Students might have a hard time visualizing where a reflection should be placed. Tell students to use the activity described above to help them. They can fold their graph paper on the appropriate line (the \begin{align*}y-\end{align*}axis, for example) and then trace the figure on the other side. Until they get used to using the rules, this can be one way for students to apply a reflection.
One way to help students remember the rules for reflections over vertical or horizontal is that the other coordinate is changed. For a reflection over the \begin{align*}x-\end{align*}axis (or horizontal line), the \begin{align*}x-\end{align*}value will stay the same and the \begin{align*}y-\end{align*}value will change. For a reflection over the \begin{align*}y-\end{align*}axis (or vertical line), the \begin{align*}y-\end{align*}value will stay the same and the \begin{align*}x-\end{align*}value will change. The new coordinates of an image depend on how far away the preimage points are. If a point is 5 units to the left of a line of reflection, then the image will be 5 units to the right of the line of reflection.
The only diagonal lines that we will reflect over in this chapter are \begin{align*}y = x\end{align*} and \begin{align*}y = -x\end{align*}. Again, encourage students to fold their graph paper when starting rotations over these lines and completing homework or class work problems.
Rotations
Goal
In this lesson, students will learn about general rotations and in the coordinate plane. By the end of the lesson, students should be able to apply the rules of rotation for \begin{align*}90^\circ, 180^\circ\end{align*}, and \begin{align*}270^\circ\end{align*}, as well as draw a rotation (using a protractor) of any degree.
Relevant Review
Make sure students remember how to draw and measure an angle, using a protractor. Practice this before starting Investigation 12-1.
Teaching Strategies
Investigation 12-1 should be done individually by each student. The teacher can also lead the students in the activity, on the overhead projector, if desired. Students need to be comfortable rotating a figure around a fixed point. Every student will need a protractor for this activity. After completing the investigation, have students repeat it with another figure of their choosing. Encourage students to pick a figure that has straight sides, such as a quadrilateral or the letters \begin{align*}H, T,\end{align*} or \begin{align*}E\end{align*}.
Unless otherwise stated, rotations are always done in a counterclockwise direction. Tell students this is because the quadrants are numbered in a counterclockwise direction. In the coordinate plane, the origin is always the center of rotation.
After going over the rules for the rotations of \begin{align*}90^\circ, 180^\circ\end{align*}, and \begin{align*}270^\circ\end{align*}, compare the rules learned in the previous lesson to these (reflections over the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} axis and \begin{align*}y = x\end{align*} and \begin{align*}y = -x\end{align*}). At this point, students know seven different reflection and rotation rules that are all very similar.
Reflection over \begin{align*}x-\end{align*}axis: \begin{align*}(x, y) \rightarrow (x, -y)\end{align*}
Reflection over \begin{align*}y-\end{align*}axis: \begin{align*}(x, y) \rightarrow (-x, y)\end{align*}
Reflection over \begin{align*}y = x\end{align*}: \begin{align*}(x, y) \rightarrow (y, x)\end{align*}
Reflection over \begin{align*}y = -x\end{align*}: \begin{align*}(x, y) \rightarrow (-y, -x)\end{align*}
Rotation of \begin{align*}90^\circ\end{align*}: \begin{align*}(x, y) \rightarrow (-y, x)\end{align*}
Rotation of \begin{align*}180^\circ\end{align*}: \begin{align*}(x, y) \rightarrow (-x, -y)\end{align*}
Rotation of \begin{align*}270^\circ\end{align*}: \begin{align*}(x, y) \rightarrow (y, -x)\end{align*}
All of these rules are different, but very similar. It is encouraged that students make flash cards for this rules to help them memorize each one. Ask students which ones have the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values switched and why they think that is. Also, see if they can make a correlation between the reflections over the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} axis and the rotation of \begin{align*}180^\circ\end{align*}. These three are the only rules where there is just a sign change. As it will be seen in the next chapter, a rotation of \begin{align*}180^\circ\end{align*} is a composition of the double reflection over the axes. Students might be able to notice this by looking at the rules now. The reflection over the \begin{align*}x-\end{align*}axis has a negative \begin{align*}y\end{align*} and the reflection over the \begin{align*}y-\end{align*}axis has a negative \begin{align*}x\end{align*}. In the rotation of \begin{align*}180^\circ\end{align*}, both \begin{align*}x\end{align*} and \begin{align*}y\end{align*} are negative.
Composition of Transformations
Goal
This lesson introduces students to the concept of composition. Composition is the process of applying two (or more) operations to an object. In this lesson, we will only apply two transformations to an object and then determine what one transformation this double-translation is the same as.
Teaching Strategies
Students can get easily confused when applying compositions. They may attempt to perform the composition from left to right, as in reading a sentence. Point out to the students they must begin with the object and, according to the order of operations, should perform the operation occurring within the parentheses first. A glide reflection is the only composition where order does not matter.
Have students do Example 6 and see if they can come up with the Reflection over the Axes Theorem on their own. You may need to do an additional example so students see the pattern.
Additional Example: Reflect \begin{align*}\Delta XYZ\end{align*} over the \begin{align*}y-\end{align*}axis and the \begin{align*}x-\end{align*}axis. Find the coordinates of \begin{align*}\Delta X^{\prime \prime} Y^{\prime \prime} Z^ {\prime \prime}\end{align*} and the one transformation this double reflection is the same is.
Solution: The coordinates of \begin{align*}\Delta XYZ\end{align*} and \begin{align*}\Delta X^{\prime \prime}Y^{\prime \prime}Z^{\prime \prime}\end{align*} are:
\begin{align*}& X(3, 5) \rightarrow X^{\prime \prime}(-3, -5)\\ & Y(6, 9) \rightarrow Y^{\prime \prime}(-6, -9)\\ & Z(10, 7) \rightarrow Z^{\prime \prime}(-10, -7)\end{align*}
From these coordinates, we see that a double reflection over the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} axes is the same as a rotation of \begin{align*}180^\circ\end{align*}.
Investigation 12-2 should be a teacher-led activity. You can either have students perform the investigation along with you or you can just do the activity and have student write down the necessary information.
Extension: Tessellating Polygons
Goal
This lesson defines tessellations and shows students how to create a tessellation from regular polygons.
Relevant Review
Review with students the definition of a regular polygon and how many degrees are in a quadrilateral, pentagon, hexagon, octagon, etc.
Teaching Strategies
Take students to the computer lab and let them play with the Tessellation Artist, from the website given in the FlexBook (http://www.mathisfun.com/geometry/tessellation-artist.html). This website does not create true tessellations, but it is fun for students to see if or how their drawing would tessellate.
Students may wonder if shapes other than regular polygons tessellate. You can show them the example below and then see if they can tessellate any quadrilateral. When tessellating this quadrilateral, make sure that every student has a different quadrilateral. When everyone is done, have the students hold up their tessellations or share them with each other.
Additional Example: Tessellate the quadrilateral below.
Solution: To tessellate any image you will need to reflect and rotate the image so that the sides all fit together. First, start by matching up the each side with itself around the quadrilateral.
Now, continue to fill in around the figures with either the original or the rotation.
This is the final tessellation. You can continue to tessellate this shape forever.