Before How does the painting compare to the original figure? What do you expect will be true of the painted figure if it is painted to scale?

Transcription

1 Dilations LAUNCH (7 MIN) Before How does the painting compare to the original figure? What do you expect will be true of the painted figure if it is painted to scale? During What is the relationship between the longest side lengths of the original figure and the painted figure? What is the relationship between the shortest side lengths of the original figure and the painted figure? After There is humor in the idea of a rectangle posing for a portrait. However, in what ways is it a good choice for this problem? How would the painting be different if the original rectangle were rotated 90 degrees? KEY CONCEPT (3 MIN) In what way are most dilations different from translations? What is the result of a dilation with a scale factor of 1? PART 1 (7 MIN) What relationships should you look for between the original figure and its image? While solving the problem How can you check that the graph you choose makes sense? PART 2 (8 MIN) Kamal Says (Screen 1) Use the Kamal Says button to reaffirm the meaning of dilation in geometry, as opposed to the use of the term to describe a person s pupils. How can you tell which figure is the image and which is the original? Why is it important to know which figure is the image? Kamal Says (Screen 2) Use the Kamal Says button to reinforce that all sides of a dilated figure must change by the same factor. PART 3 (7 MIN) How will the size and shape of the image relate to the original? After solving the problem How do you think the graph would be different if the center of dilation were inside the original figure instead of outside? CLOSE AND CHECK (8 MIN) Can you ever expect a scale factor to be negative? Explain your reasoning. Suppose you know the scale factor relating the side lengths of a figure and its dilated image. How can you find a dilation that would map from the image, back to the original figure? Review the songs Algo sings in this lesson and in the previous topic. Can you write an additional verse connecting translations, reflections, rotations, and dilations?

2 Dilations LESSON OBJECTIVES 1. Use coordinates to describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures. 2. Identify dilations, translations, rotations, and reflections. 3. Explore the properties of dilations. FOCUS QUESTION What effect does an enlargement have on a figure? What effect does a reduction have on a figure? MATH BACKGROUND While studying proportional relationships in Grade 7, students solved problems involving maps and scale drawings. They learned to find segment lengths and draw various figures to create a scale drawing. Students learned to visualize two-dimensional cross sections of three-dimensional figures and describe the characteristics of these figures. They explored transformations in the previous topic, focusing on rigid motions that result in images that are the same size and shape as the original. This lesson extends students work with scale drawings and proportional relationships while introducing them to a new type of transformation: the non-rigid dilation. Students learn how to recognize and describe a dilation. They identify dilations by characteristics such as shape, size, and coordinates, and distinguish them from rigid motions. Students use proportional relationships to compare figures and calculate the scale factor that links a figure to its image. Finally, students graph the effects of a dilation on the coordinate plane. Working with two-dimensional figures on the coordinate plane shows students a connection between geometric concepts and the graphs, proportionality, and linear equations from past lessons. The integration of these concepts provides a preview of the mathematical possibilities students will explore in high school. LAUNCH (7 MIN) Objective: Explain whether a figure is drawn to scale. Students saw scale in Grade 7 and now use the concept to explore dilations. Before starting the Launch, you may want to review what it means for an object to be drawn to scale. Before How does the painting compare to the original figure? [Sample answer: Both are rectangles whose longer sides are horizontal. The image on the canvas is smaller than the original rectangle.] What do you expect will be true of the painted figure if it is painted to scale? [Sample answer: The painted figure will have the same shape as the original but a different size.] During What is the relationship between the longest side lengths of the original figure and the painted figure? What is the relationship between the shortest side lengths of the original figure and the painted figure? [Sample answer: The

3 longest and shortest sides of the original figure are 6 times the length of the longest and shortest sides of the painted figure.] After There is humor in the idea of a rectangle posing for a portrait. However, in what ways is it a good choice for this problem? [Sample answer: It is easy to analyze and compare rectangles on the coordinate plane because you can quickly identify side lengths and angle measures. It is easy to predict how a scaled rectangle should look because the properties of a rectangle are recognizable.] How would the painting be different if the original rectangle were rotated 90 degrees? [Sample answer: The painting would be the size that it is now, but its orientation would be different it would also be rotated 90 degrees.] Solution Notes To solve this problem, students could use ratios to compare each actual length with the corresponding painted length. You know the painted rectangle is to scale if the ratios are equivalent. Discussing such a solution might help students make a connection between ratios, proportionality, and scale factor. Connect Your Learning Move to the Connect Your Learning screen. Talk about how to recognize when a figure is to scale. Listen for students to describe how the painted rectangle in the Launch is scaled down, but you could scale up a figure. See if students can identify a real-life situation in which a figure is scaled up, such as a photograph enlargement. You may want to spend a few moments reviewing the properties of the rigid motions studied in the previous topic. KEY CONCEPT (3 MIN) Teaching Tips for the Key Concept The animation groove tune emphasizes the continuity between dilations and other transformations presented in the topic Congruence. In what way are most dilations different from translations? [Sample answer: Most dilations are not rigid motions because the image is not the same size as the original figure.] What is the result of a dilation with a scale factor of 1? [Sample answer: When the scale factor is 1, the image is the same size as the original figure.] PART 1 (7 MIN) Objective: Identify dilations. Students distinguish dilations from other transformations. They learn that the change in side length must be consistent for all sides in order for the image to be the result of a dilation. Instructional Design Make sure students analyze the graphs that do not show a dilation to determine if they show another transformation. They can draw on the coordinate planes to test theories.

4 What relationships should you look for between the original figure and its image? [Sample answer: You should look for an image that is the same shape as the original but a different size. Each vertex of the image should fall along a ray that starts at the center of dilation and passes through the corresponding vertex of the original triangle.] While solving the problem How can you check that the graph you choose makes sense? [Sample answer: You can compare the lengths of the corresponding sides of the triangles to see if they are equivalent ratios. Or, you can draw a line through corresponding pairs of vertices to see if they all meet at a common center of dilation.] Got It Notes Once students realize that graph III represents a dilation, they can eliminate options A and B. Then they only have to decide whether graph II represents a dilation. Encourage students to explain why each transformation is or is not a dilation as a way of reinforcing the characteristics of dilations. If you show answer choices, consider the following possible student errors: If students confuse dilation with translation, they will likely choose A. If they identify a dilation by the change in only one side length, they will likely choose B or D. PART 2 (8 MIN) Objective: Identify enlargements and reductions of two-dimensional figures in the coordinate plane. The intro explains how dilations in geometry can be described in terms of enlargement or reduction. As students observe the animation of these two types of dilations, they gain a visual understanding of the impact of the scale factor on a dilation. This helps them analyze figures and their images so that they can calculate the scale factor and determine the type of dilation. Kamal Says (Screen 1) Use the Kamal Says button to reaffirm the meaning of dilation in geometry, as opposed to the use of the term to describe a person s pupils. How can you tell which figure is the image and which is the original? [Sample answer: The original figure is PQRS, and the image is P'Q'R'S'. The figure with prime notation is the image.] Why is it important to know which figure is the image? [Sample answer: You need to know which figure is the original and which is the image in order to find the scale factor. If the original figure is enlarged, the scale factor is greater than 1. If the original is reduced, the scale factor is less than 1.] Kamal Says (Screen 2) Use the Kamal Says button to reinforce that all sides of a dilated figure must change by the same factor.

5 Solution Notes Before showing the solution, encourage students to identify a way to check that their answer is correct. Given that this example is a rectangle, they only have to check one other pair of corresponding sides. Differentiated Instruction For struggling students: To help students set up the correct ratios, have them trace over the corresponding sides of the rectangles, using a different color for each pair. Label all side lengths with their measures. Then have students use the same colors to set up the ratios so they can track the flow of information taken from the graph. For advanced students: Have students describe how to use the coordinates of a figure and its image to algebraically find the coordinates of the center of dilation. They can write two equations and solve the system to see where the equations intersect. Error Prevention Remind students to pay close attention to how the figures are named. Although students know what prime notation is, they may not think to use it. However, doing so is important for determining whether the dilation is an enlargement or a reduction. Got It Notes If you show answer choices, consider the following possible student errors: Students who find the scale factor that describes the change from the image to the original will likely choose A. If students choose C or D, they do not understand that an enlargement has a scale factor greater than 1 and a reduction has a scale factor less than 1. PART 3 (7 MIN) Objective: Graph a dilation of a two-dimensional figure in the coordinate plane. In Part 2, students may have seen a connection between the coordinates of the vertices of a dilated figure and its image. Now this relationship is explicitly stated, and students use it to find the coordinates of a figure. Instructional Design You can assign a vertex to three different students and allow each to multiply the coordinates by the scale factor and graph both vertices. Make sure students label each point using the correct notation. How will the size and shape of the image relate to the original? [Sample answer: This dilation is an enlargement because the scale factor is greater than 1, the image will be larger. It will be the same shape as the original.] After solving the problem How do you think the graph would be different if the center of dilation were inside the original figure instead of outside? [Sample answer: If the center of dilation were inside the original figure, the image would surround the original figure.]

6 Got It Notes Here, the center of dilation is a vertex of the original figure. Remind students that in this case, the vertex and its image after the dilation are the same point. If you show answer choices, consider the following possible student errors: Students who choose A multiply the coordinates by a scale factor of 3 instead of 3 4. Students who choose B are restating the coordinates for M. This may be the result of dilating the image about M rather than about the origin. Students who choose D divide each coordinate by 4. In other words, they apply a scale factor of 1 4 instead of 3 4. CLOSE AND CHECK (8 MIN) Focus Question Sample Answer An enlargement is a dilation with a scale factor greater than 1, so an enlargement increases the size of a figure. A reduction is a dilation with a scale factor less than 1, so a reduction decreases the size of a figure. Focus Question Notes Look for answers that describe the result of a scale factor of 1. Students may also address the effects of the location of the point of dilation in relation to the image. Essential Question Connection This lesson addresses the Essential Question about how you can be sure you scale an object correctly. Can you ever expect a scale factor to be negative? Explain your reasoning. [Sample answer: No; a scale factor is the ratio of two lengths, and length is always positive.] Suppose you know the scale factor relating the side lengths of a figure and its dilated image. How can you find a dilation that would map from the image, back to the original figure? [Sample answer: You can find the reciprocal of the scale factor and use that to map back to the original figure.] Review the songs Algo sings in this lesson and in the previous topic. Can you write an additional verse connecting translations, reflections, rotations, and dilations? [See student work.]