Level H Lesson 23

Rotations and Dilations

In lesson 23, the student will rotate and dilate geometric figures in a four-quadrant graph.

We have three essential questions that will be guiding our lesson. How do the coordinates of a figure change when you rotate a figure 90 degrees clockwise? Number 2, how do the coordinates of a figure change when you rotate a figure 180 degrees clockwise? Number 3; describe how a figure would change if you dilated the figure by a whole number.

The SOLVE problem for our lesson is, Carter’s teacher asked him to dilate a figure on the coordinate plane. The figure is a rectangle. The coordinates of the rectangle are point A (3, 9), point B (3, 3), point C (6, 3 ) and point D (6, 9). What are the coordinates of the dilation if he is supposed to use a scale factor of one third?

We’re gong t study the problem. We’re going to start by underlining the question. What are the coordinates of the dilation if he is supposed to use a scale factor of one third? I’m then going to complete the statement this problem is asked me to find the new coordinates if the figure is dilated one third .

Let’s talk about rotations. When skateboarding, what does it mean if you “do a 180”? It means that you make a half turn. How about a “360”? That means that you make a full turn. These are called rotations. When working with figures on the coordinate plane, you can also rotate figures. Sometimes they are rotated clockwise, or to the right, and sometimes they are rotated counter-clockwise, or to the left. A rotation may also be called a turn.

Think about the coordinate plane. How many degrees are in each quadrant? There is a total of 90 degrees.

We have a coordinate graph here. We’re going to take the trapezoid that you’ve cut out and we’re going to copy it on to the coordinate grid. Point A should be at (4, 2), and point C should be at (0, 4). Go ahead and copy your trapezoid onto the coordinate graph. We have our coordinate grid and now we’ve copied our trapezoid. Our Trapezoid is located in quadrant 1. Remember each of our quadrants has 90 degrees.

Each student should now have a 3 by 5 card. Each corner of our card is a 90 degree angle. We’re going to use this card to rotate the trapezoid 90 degrees clockwise around the origin. We’re now going to place 1 corner of the card at the origin and then line up the side of the card with point A. We’re going to take our card, we’re going to put 1 point or 1 corner at the origin which is (0, 0) and the other one we’re going to line up with point A. I’m going to now take your marker or colored pencil and we’re going to draw a point in the corner where the origin and also on the side of the card where the point A is. We now have drawn a point on our 3 by 5 card, at the origin where point A is. Now I’m going to take my card and I’m going to fold it in half at the corner where I put the origin. I’m going to line up the 2 opposite sides of the card and place a point directly across from point A. We’re now going to take our 3 by 5 card and fold it in half I’m going to line up the opposite 2 sides of the card and place a point directly across from the point that we made A. After we mark our original point, we’re going to fold our card in half, at the corner you put the origin. We’re going to line up the opposite sides of the card and place a dot directly across from where point A is. I’m going to place my card back on the coordinate place so that the first 2 points again line up with A and the origin. The third point you made should be the location of A prime. Because we wanted to rotate our figure 90 degrees. Remember this is a 90 degree angle. I’m going to draw a point directly under the third point and label it A prime. The coordinates of the A prime are (2, negative 4). A prime is now located in quadrant 4.

What happened to our x- and y-coordinates of A to get to the x- and y-coordinates of A prime? Remember that A was at (4, 2). Now A prime is at (2, negative 4). So the x-coordinate became negative, and switched to the y-coordinate. The y-coordinate became the x-coordinate. Again the 4 is now a negative 4 as the y, and the 2 is now a 2 as the x. The x-coordinate moved to the position of the y-coordinate and became negative. The y-coordinate is now the x-coordinate.

We’re now going to list the coordinates, the original coordinates of B, C and D. B is at (2, 2), C is at (0, 4) and D is at (4, 4). We’re going to use the information that we just determined in our rule to find where B prime, C prime and D prime should go. Remember our rule, the x moves into the y position and becomes a negative x and they moves to the x position. So for B prime, B original B was (2, 2), so B prime is (2, negative 2). The original C (0, 4), so now it is (4, 0). Notice that 0 is neither negative or positive. D prime is now going to be (4, negative 4). We’re going to place those points on the trapezoid and then we’re going to check each point.

We’re now going to rotate the trapezoid 180 degrees. You’ll be using your 3 by 5 card again to see where A double prime should go. Since 180 degrees is twice of 90, you can begin the card on the second trapezoid and move it 1 quadrant for 90 degrees from the second trapezoid. That would be the same as a move of 180 from the original trapezoid. We’re going to take our index card again place the point at the corner on our origin, place the point we created on A prime, and then we’re going to do, position the point we made for our first A prime and use that to determine A double prime. A double prime is going to be at (negative 4, negative 2).

We’re now going to use the information to determine a rule for what we just did. The coordinates of A double prime are (negative 4, negative 2). What happened to the x and y-coordinates of A, to get the x and y-coordinates of A double prime? Remember our original A was (4, 2). Our A double prime is ( negative 4, negative 2). Our x-coordinate became a negative x, and our y-coordinate became a negative y.

We’re now going to list the original coordinates for point B, C and D again. We’re going to use our rule to determine our double prime points. B was (2, 2), B double prime is (negative 2, negative 2), C was 0, 4), C double prime is now (0, negative 4), D was at (4, 4), D double prime is now at (negative 4, negative 4).

We’re now going to place the points for the new trapezoid on our coordinate plane. What quadrant is our figure in rotated at 180 degrees? It’s now in the 3rd quadrant.

Let’s rotate the trapezoid 270 degrees. We’re going to be using our 3 by 5 card again to see where A triple prime should be. Because 270 is 3 time 90 we can begin with our card on the third trapezoid and move it one quadrant for a 90 degree change in the third trapezoid. That would be the same as a 270 degree change from the first trapezoid. We’re now going to take our 3 by 5 card, place the point on the origin, place our other point on A double prime, and we find that A triple prime is going to be at (negative 2, 4). What are to coordinates of A triple prime? (negative 2, 4).

What happened to the x- and y-coordinates of A to get the x- and y-coordinates of A triple prime? Remember our original A was at (4, 2). Our A triple prime, or the 270 degree clockwise rotation puts A at (negative 2, 4). Our y-coordinate is now in the x- position as a negative and the x- coordinate is now in the y position.

We’re now going to list our original coordinates B at (2, 2), C at (0, 4) and D at (4, 4). We’re going to use the information of (x, y), is going to rotate and the coordinates will be (negative y, x) value to determine our 270 degree clockwise rotation or our triple prime points. B triple prime will be (negative 2, 2), C triple prime will be (negative 4, 0), D triple prime will be (negative 4, 4).

Go ahead and place the points for the new trapezoid on the coordinate plane. We’re now plotted the points and this is our 270 degree clockwise rotation. This is located in quadrant 2. Is there a trapezoid in each quadrant? And the answer is yes.

What do you think would happen if we graphed using rotation that were counterclockwise? If we did, the trapezoids would be in the same place as they are now, one in each quadrant. However, 90 degrees counterclockwise is the same as 270 degrees clockwise. Our 270 degrees clockwise was in the second quadrant. If we did a 90 degree clockwise it would be in the second quadrant. 180 degrees counterclockwise is the same as 180 degree clockwise. And 270 degree counterclockwise is the same as 90 degree clockwise.

We’re now moving to dilations. In math, dilating a figure means to enlarge or reduce a figure without changing its shape.

We’re going to start out listing the original coordinates of the trapezoid. We had A at (4, 2), B at (2, 2), C at (0, 4) and D at (4, 4). We’re going to dilate our figures using a scale factor of one half. What that means is, we’re going to multiply each coordinate by the scale factor to write the new coordinates. Our x we multiply by one half, and our y we multiply by one half. Remember that our original are listed here below and we’re going to multiply each value by one half. Our new points are A quadruple prime (2, 1), B quadruple prime (1, 1), C quadruple prime (0, 2), and D quadruple prime (2, 2). Let’s go ahead and plot the points on the coordinate graph. We’ve now plotted our points of our dilation of our original figure using the dilation of one half. The next thing we’re going to do is create dilation using a scale factor of 2. (x, y) is going to become (2x, 2Y).to find our new points. Our new points are (8, 4), (4, 4), (0, 8), and (8, 8). In red we’ve now created a dilation of our original figure, that is a dilation of 2.

We’re going to go back and review types of transformation. The first we have is a 90 degrees clockwise which is the same as a 270-degree counterclockwise. It looks like this on the coordinate plane. And our coordinate rules are the x coordinate moves the position of negative x in the y position. And the y coordinate moves to the position of x-coordinate. (2, 1) is now (1, negative 2), (2, 5) is now (5, negative 2), (6, 1) is now (1, negative 6).

When we have a transformation that is 180 degrees it is clockwise and counterclockwise the same. Our rule for that our x- and y-coordinates stay in the same position but they are both now a negative value. So (2, 1) is now (negative 2, negative 1), (2, 5) is now (negative 2, negative 5), and (6, 1) is now (negative 6, negative 1).

The last transformation we’re going to talk about is the 270 degree clockwise which is also the same as the 90 degrees counterclockwise. We’ve plotted that on our coordinate plane and our rule is that the x-coordinate in now in position of the y-coordinate and the y-coordinate is now in the position of the x as a negative value. (2, 1) is now (negative 1, 2), (2, 5) is now (negative 5, 2), and (6, 1) is now (negative 1, 6).

To review dilations, remember we multiply the original points by the scale factor. In this case the dilation is 3. So we multiply each coordinate by 3. (negative 2, 2) is now (negative 6, 6), (2, 2) is now (6, 6), (negative 2, negative 2) is now (negative 6, negative 6), and (2, negative 2) is now (6, negative 6).

Let’s go back to our SOLVE problem from the beginning of the lesson. Carter’s teacher asked him to dilate a figure on the coordinate plane. The figure is a rectangle. The coordinates of the rectangle are (3, 9), (3, 3). (6, 3) and (6, 9). What are the coordinates of the dilation if he is supposed to use a scale factor of one third?

At the beginning of the lesson we did the S step by underlining the question and completing the statement this problem is asking me to find the new coordinates if the figure is dilated one third.

We’re now going to move to the O step where we’re organizing the facts. We first identify the facts. Carter’s teacher asked him to dilate a figure on the coordinate plane, fact. The figure is a rectangle, fact. The coordinates of the rectangle are (3, 9), (3, 3), (6, 3) and (6, 9), fact. We now eliminate the unnecessary facts. A fact is only necessary if it helps us to answer the question. Carter’s teacher asked him to dilate a figure on the coordinate plane. That fact is not necessary. The figure is a rectangle. That also is a unnecessary fact. We do however need to know the coordinates of the rectangle and the fact that we’re dilating by a scale factor of one third. We list our necessary facts.

In the L step we’re going line up a plan. We’re going to choose an operation and we know we’re going to multiply because we know we have a scale factor. Then we write in words our plan of action. Multiply each x- and y-value of the coordinates by the scale factor.

In the V step we’re going to verify you plan with action. We first begin by making an estimate. Because we’re multiplying by a fraction our estimate is going to be that all of the coordinates will be smaller than they were originally. Then we carry out our plan by multiplying each point by a scale factor of one third to get the new point.

We move to the E step where we examine your results. Does your answer make sense? Compare your answer to the question. Yes, because I dilated the figure using the scale factor. Is your answer reasonable? Compare your answer to the estimate. We estimated that all the coordinates were smaller. And yes, all the coordinates are smaller. Is your answer accurate? Go back and check your work. The last part of the E step is to write your answer in a complete sentence. The coordinates of the shape after the dilation will be (1, 3), (1, 1), (2, 1), and (2, 3).

We’re now going to go back and the essential questions from the beginning of the lesson. How do the coordinates of a figure change when you rotate a figure 90 degrees clockwise? The x and y switch places, and the old x is negative.

How do the coordinates of a figure change when you rotate a figure 180 degrees clockwise? The x and y both become negative.

Describe how a figure would change, if you dilated the figure by a whole number. The figure would be he same shape, but a larger size. It would be as many times larger as the scale factor.