Rigid Transformations

RIGID TRANSFORMATIONS

INTRODUCTION

The objective for this lesson on Rigid Transformations is, the student will utilize tools such as protractors and rulers to explore translations, reflections and rotations of a figure prior to the transformation and after the transformation.

The skills students should have in order to help them in this lesson include, use of protractors, use of rulers, angles, line relationships and parallel lines.

We will have three essential questions that will be guiding our lesson. Number one, what is a rigid transformation? Number two, explain the pattern that exists with the coordinates of the vertices when reflections occur over the y-axis or the x-axis. Number three, describe the options when using a transformation to rotate a figure.

Begin by completing the warm-up on identifying geometric shapes and figures to prepare for the lesson on rigid transformations.

SOLVE PROBLEM – INTRODUCTION

The SOLVE problem for this lesson is, Sean is decorating the nursery for the arrival of the new baby. He is planning the wall decorations using a coordinate grid. He has arranged a rectangular frame with its top left corner at one, two. The frame is one unit tall and two units long on the grid. If he decides to slide the frame three units to the right, what will be the four new vertices of the picture frame?

In Step S, we Study the Problem. First we need to identify where the question is located within the problem and underline the question. The question for this problem is, what will be the four new vertices of the picture frame?

Now that we have identified the question, we need to put this question in our own words in the form of a statement. This problem is asking me to find the four new vertices of the picture frame.

During this lesson we will learn about Rigid Transformations to complete this SOLVE problem at the end of the lesson.

TRANSLATIONS

Throughout this lesson, we will be creating manipulatives to use and they will be stored on the blank page below the SOLVE problem.

Discuss some of the characteristics of the figure on the coordinate plane.

What is the shape of the figure on the coordinate plane? It is a square.

Describe some of the characteristics of a square. A square has four equal sides, two sets of parallel lines, and four ninety degree angles.

Use a ruler to measure the sides of the square to be sure that all sides are equal.

Are all of the measures of the line segments the same length? Yes, they are all a bit longer than one half of an inch.

How can we tell if all angles measure ninety degrees? We can use a protractor to measure each angle to be sure. Measure the three other angles to be sure.

What do you notice about the figure? All of the sides are the same and all four of the angles are right angles.

What can you conclude about this figure? The figure is a square because it has four equal sides and four ninety degree angles.

Let’s work together to identify the coordinates of each corner of the square. Point A is at one, four; Point B is at three, four; Point C is at three, two and Point D is at one, two.

What is another name for these points that represent the meeting point of the two line segments? Vertices

What was the measure of Angle A when we measured the angles of the figure? Ninety degrees or a right angle.

What was the measure of the other three angles that we measured? They were all ninety degreesor right angles.

How could you measure the sides of the square if there was no ruler available? We can also refer to units on the coordinate plane as our official measure. For example, line segment AB measure two units.

What do each of the line segments of the figure measure? Two units because all sides are the same length and we can see that the length of each segment is created with the length of two units.

What is the heading of the last row? Parallel Line Segments

Identify one pair of parallel line segments in the square. Line segment AB is parallel to line segment DC. We can write that with the symbol for parallel line segments.

Identify a second set of parallel line segments in the square. Line segment AD is parallel to line segment BC.

Take a look at the headings of both of the tables.

This lesson will involve manipulating figures by sliding and moving them to explore the end result. Looking at the headings of the tables, what do you think “Pre-image” and “Image” refer to? “Pre” means “before” so we can assume that “pre-image” is referring to the image before any moves are made, while “image” would refer to the figure after the moves are made.

Lay a sticky note over top of figure and trace Square ABCD. Label the square with the letter on the inside of the square, because we will be cutting out the square.

Note: It will be helpful to trace the square so it is on top of the sticky portion of the sticky note. This will help keep it stuck to the page as we work on the activity and keep the manipulatives stored.

Trace the square. Then label the vertices inside the square. Cut the square out, and then place the square over top of square ABCD.

Now slide the traced square to the right two units. Label the vertices inside of the square. Slide the traced square one unit up. Now, mark the vertices of the trace square at the new location after sliding up and to the right. Use a straight-edge to connect the vertices of the new figure. Also label each of the new vertices with the corresponding letters that are on the inside of the traced square.

What do you notice about the way the points are labeled in the second table? Each letter has an apostrophe after it.

What do you think the apostrophe represents? It represents the corresponding vertex after the figure has been moved. This is referred to as “prime.”

For example, the vertex at Point A is moved to the new figure, and Point A in the new figure will be referred to as Point A prime.

Let’s label all the new vertices with the apostrophes such that they will all be prime.

Let’s complete the second column by identifying the coordinates of the new vertices after sliding the figure. Note that the letters are all in the same corresponding positions because the figure has not rotated at all. There was only a slide of the square in its original position.

A Prime is three, five; B Prime is five, five; C Prime is five, three and D Prime is three, three.

What is the measure of Angle A Prime? Ninety degrees

What is the measure of Angle B Prime, Angle C Prime, Angle D Prime? They are all ninety degrees.

Measure the lengths of the line segments of the new figure using units. What are the lengths of the line segments in the new figure? All of the line segments measured two units.

Name one of the sets of parallel line segments of the new figure. Line segment A Prime B Prime is parallel to line segment D Prime C Prime.

What is the other set of parallel line segments? Line segment A Prime D Prime is parallel to line segment B Prime C Prime.

What do you notice about the measure of the line segments? The measure of each line segment is two units in length. There was no change in the measure of the line segments from the first figure to the second figure.

What do you notice about the measure of the angles of the figures? Both figures remain squares and each angle has a measure of ninety degrees. The measures of the angles did not change from the first figure to the second figure.

What do you notice about the x-coordinates from the pre-image to the image? All of the x-coordinates have increased by two. The figure moved two units to the right, which means that all of the points, or vertices, of the square moved two units to the right. To move two units to the right, the x-coordinates increase by two.

What do you notice about the y-coordinates from the pre-image to the image? All of the y-coordinates have increased by one. The figure moved up one unit, which means that all of the points, or vertices, of the square moved up one unit. To move up one unit, the y-coordinates increase by one.

Did the square lose any of its characteristics during the transformation? No

When something keeps its characteristics and does not change, we refer to it as rigid. Therefore, this was a rigid transformation.

How would you describe the transformation that took place from Square ABCD to Square A prime, B prime, C prime, D prime? It appears that Square ABCD slid from its original position into a new position to create the new figure Square A prime, B prime, C prime D prime.

When a person speaks more than one language, he or she sometimes has to slide back and forth between languages to comprehend the meaning of what is being said. We refer to this as translating.

Sliding an image is also referred to as a translation.

REFLECTIONS

What do you notice about the figure? It is the same square that we used as the original figure in the last two translations.

Complete the pre-image chart for the figure.

Place a new sticky note on the coordinate plane so that the bottom left corner of the sticky note should be at the origin. The left side of the note should be aligned with the y-axis and the bottom of the note should align with the x-axis.

Trace Square ABCD with a pen, labeling the vertices on the drawing. It is fine to label the letters of the vertices on the outside of the square, because students will not be cutting this square from the note.

After you have traced the square with a pen, pick up the sticky note by the top of the note and flip it over the x-axis, so that the square is now facing down and the original bottom of the note that was the origin still remains at the origin and aligned with the x-axis.

Now, use a pen or marker to apply ink to the vertices that are facing down. Allow ink to bleed through the sticky note onto the coordinate place. You may need to peek under the sticky note to be sure that the points are recording from the ink bleeding through the note.

Mark the new points with the letter that corresponds to the letter nearest to the newly plotted points and label them as prime vertices, A prime, B prime, C prime and D prime.

What do you notice about the placement of the vertices after the transformation? Vertices D and C are now the top vertices while A and B are now the bottom vertices. Explain why this happened. We flipped the square and didn’t slide it. Connect the new vertices to create the square of the new figure.

Take a moment to measure the new angles and lengths of the new line segments.

What are the new coordinates of A prime? One, negative four

What are the coordinates of B prime? Three, negative four

What are the coordinate of C prime? Three, negative two

What are the coordinates of D prime? One, negative two

What do you notice about the measures of the line segments? The measure of each line segment is two units in length. There was no change in the measure of the line segments from the first figure to the second figure.

What do you notice about the measure of the angles? Both figures remain squares and each angle has a measure of ninety degrees. The measure of the angles did not change from the first figure to the second figure.

What do you notice about the x-coordinates from the pre-image to the image? All of the x-coordinates remain the same.

What do you notice about the y-coordinates from the pre-image to the image? All of the y-coordinates have switched to the opposite sign of the original coordinates.

Did the original square lose any of its characteristics during the transformation? No

Would you consider this transformation a rigid transformation? Yes. Explain. The characteristics of the figure did not change. Even though the shape flipped, it still kept the same angle measures and line segments lengths.

When looking into a mirror, we see our reflections.

This type of transformation is also a reflection that uses the x-axis as the mirror.

When reflecting over the x-axis, we found that the x-coordinates remain the same, while the y-coordinates are the opposites of the original coordinate.

ROTATIONS ABOUT THE ORIGIN

What do you notice about the figure? The figure is the same square that we’ve been using for the previous transformations.

Complete the pre-image chart for the figure.

Take out the sticky note that we used for reflections. If the note is too messy from the ink bleeding, feel free to replace the same square on a new sticky note.

Place a new sticky note on the coordinate plane so that the bottom left corner of the sticky note should be at the origin. The left side of the note should be aligned with the y-axis and the bottom of the note should align with the x-axis.

Trace Square ABCD with a pen, labeling the vertices on the drawing. It is fine to label the letter of the vertices on the outside of the square, because students will not be cutting this square from the note.

Hold your finger on the sticky note at the origin. While pressing the sticky note down at the origin, rotate the sticky note so that the bottom of the note aligned with the x-axis rotates and aligns with the y-axis.

After letting the ink bleed through at the vertices of the traced square, label the vertices with the prime vertices that correspond to the letters closest to the plotted point. Connect the vertices.

With all the transformations thus far, the new parallel line segments always correspond with the original parallel line segments.

Even though we have marked the new vertices with the prime notation, the relationship between the vertices as they relate to the parallel line segments remains the same.

What are the coordinates of A prime? Four, negative one

What are the coordinates of B prime? Four, negative three

What are the coordinates of C prime? Two, negative three

What are the coordinates of D prime? Two, negative one

What do you notice about the measures of the line segments? The measure of each line segment is two units in length. There was no change in the measure of the line segments from the first figure to the second figure.

What do you notice about the measures of the angles? Both figures remain squares and each angle has a measure of ninety degrees. The measure of the angles did not change from the first figure to the second figure.

What do you notice about the x-coordinates from the pre-image to the image? All of the x-coordinates are exactly the same as the original y-coordinates.

What do you notice about the y-coordinates from the pre-image to the image? All of the y-coordinates are the same numerical values as the x-coordinates but opposite signs.

Did the original square lose any of its characteristics during the transformation? No

Would you consider this transformation a rigid transformation? Yes. Explain. The characteristics of the figure did not change. Even though the shape was turned, it still kept the same angle measures and line segment lengths.

What happened to this original figure in this transformation? It was turned.

What type of transformation is this? A rotation

What direction did we rotate the original figure? Clockwise

What was the point around which we rotated? The origin

When we rotated the figure ninety degrees about the origin, the x-coordinates were the same as the original y-coordinates, and the y-coordinates were the opposite values of the original x-coordinates.

If we rotated the sticky note one more time, how many degrees would we have rotated it? One hundred eighty degrees

What about if there was a third rotation? That would be two hundred seventy degrees.

How do you know that each rotation is an additional ninety degrees? Each quadrant is a ninety degree, right angle, and all four ninety degree rotations total three hundred sixty degrees, or the total number of degrees in a full circular rotation.

ROTATIONS ABOUT A POINT

What do you notice about the figure? The figure is the same square that we’ve been using for the previous transformations.