TRANSFORMERS What transformations do smartphones use? Mathalicious 2015 lesson guide Smartphones have to handle some complex calculations to do all the amazing things they do. One of those things is moving and resizing all of the elements on the screen when a user interacts with the device. In this lesson, students identify and categorize the different transformations that occur when a user manipulates a smartphone screen. They also use on- screen coordinates to calculate the results of zooming within an application and to decide whether ponying up for a larger screen is worth it. Primary Objectives Identify and describe geometric transformations that occur on smartphone screens Determine whether given transformations (or a sequence of transformations) produce figures that are congruent, similar, or neither Reason about and discuss how smartphones determine the center of dilation for zooming applications Given various points and scale factors in the coordinate plane, calculate the images of points under dilation Construct arguments about whether larger screens are worth the additional cost, based on dilations Content Standards (CCSS) Mathematical Practices (CCMP) Materials Grade 8 G.2, G.3, G.4 MP.3 Student handout LCD projector Computer speakers Before Beginning This lesson focuses on developing concepts and language surrounding transformations, congruence, and similarity, but assumes no prior knowledge of those terms or ideas. If students already possess the vocabulary to identify particular transformations (e.g., translation ), that s great, but all that s really required is the ability to describe transformations with intuitive language (e.g., sliding ).
Lesson Guide: TRANSFORMERS 2 Preview & Guiding Questions This lesson is all about the geometric transformations that smartphones use, so begin by showing the clip of people interacting with the iphone 5 and have students pay special attention to what s happening on the (small) screens. Students may have varying levels of personal experience with smartphones, so this will either give them the opportunity to observe some user interaction, or to look more closely at something they might take for granted on a daily basis. Once they ve seen the ad, ask students to describe some of the different ways you can interact with a smartphone, and what kinds of changes happen on the screen. Has anyone here ever used an iphone? What about another kind of smartphone? What parts of the device can you interact with on a phone like that? What sorts of gestures change things on the screen? What do you think is the coolest thing that the screen does? Why? Act One In Act One, students look at several examples of transformations that occur on smartphone screens. They describe in their own words what s happening in each scenario, and then they tie those intuitive concepts to more technical language, categorizing each transformation as a rotation, reflection, translation, or dilation. Students then discuss the concepts of congruence and similarity in terms of transformations, and discuss which on- screen scenarios produce congruent or similar results. Finally, students see an example that results in figures that are neither congruent nor similar and discuss the reasons why. Act Two In Act Two, students focus on dilations. They learn about the center of dilation and use examples to hypothesize how smartphones determine the center of dilation when a user zooms in or out. Students learn that a smartphone screen is essentially a coordinate plane, and the phone uses transformations to map pixels from their starting points to their end points when the user changes the screen. They examine this phenomenon in the context of a map application and use coordinate transformations to calculate where given points will end up after zooming in or out. Lastly, students use their knowledge about dilations to discuss whether or not it s worth the additional cost to get a smartphone with a larger screen.
Lesson Guide: TRANSFORMERS 3 Act One: Decept- icons 1 Smartphone operating systems Android and ios use various transformations. Watch the screen recordings of someone using an iphone, and describe what s happening. For each effect, which transformation(s) do you see: rotation, reflection (i.e. flipping), dilation (i.e. scaling), or translation (i.e. shifting)? Description All of the icons slide toward the left. The baseball stadium gets larger in the center of the screen. The entire image rotates 90 clockwise. The video rotates 90 counterclockwise and also gets larger. Transform. Translation Dilation Rotation Rotation and Dilation Explanation & Guiding Questions This opening question has two goals: (1) get students to describe geometric transformations in their own words, and (2) begin to tie mathematical terminology to student intuition. For instance, although the idea of shifting the icons to the left succinctly captures what s happening in the first video, students may not have the technical vocabulary to classify such a change as translation. To help with the move from intuitive to technical vocabulary, the question text includes both. That way students can see a simpler description associated with its appropriate term. Of course it s impossible for the question to include every way students might describe the transformations in the videos, so it might be useful to have students relate other descriptive words to the ones provided. For example, if a student describes the first video as sliding between screens, see whether she can associate that with the synonymous shifting, and finally with translation. In your own words, how would you describe what s happening to the screen/icons in the first video? Could you best describe what s happening as rotating, flipping, scaling, or shifting? So which word rotation, reflection, dilation, or translation best matches what you see? What about in the other videos? Deeper Understanding When the image or video gets larger, why doesn t it look distorted? (It must be getting scaled by the same factor in every direction.) Can you think of an example of when your phone would perform a reflection? (Answers will vary, but some photography apps will let you flip photos horizontally or vertically, resulting in reflections.)
Lesson Guide: TRANSFORMERS 4 2 After a transformation, an object is congruent to the original if it s the same shape and size as before. An object is similar if it s the same shape, but not necessarily the same size. Choose one iphone transformation, and focus on a single screen element (e.g. an app icon, the stadium, etc.). After the transformation, what s different about the element, what s the same, and is it an example of congruence, similarity, or neither? Answers will vary. Sample responses: In the first video everything stays the same size and orientation, but each icon s position moves to the left, so each figure is congruent to its starting figure. In the second video the stadium s position and orientation stay the same, but its size changes, so the ending image is similar to the starting image. In the third video the building s position and size stay the same, but its orientation changes, so the final image is congruent to the original. In the last video, the orientation and size change, but its position stays the same, which means the resulting video is similar to the original video. Explanation & Guiding Questions Congruence and similarity make the idea of geometric sameness precise. They are also intimately bound up with and often defined in terms of transformations. So one way to talk about congruence and similarity is to talk about how shapes or objects are the same and different after a transformation (or a sequence of transformations), compared to before. For instance, in the video showing the image of the U.S. Capitol being rotated, its orientation changes, but everything about its size and fundamental shape remains the same. In the video where the user zooms in on the baseball stadium, the image s orientation and shape are unchanged, but its size increases. The goal is for students to realize that any combination of rotations, translations, and reflections of a figure will lead to a congruent figure, since all of those transformations preserve both shape and size. Dilations, however, produce similar figures, since a dilation doesn t change a figure s fundamental shape, but can change its size. One point that you might want to clarify with students is that every set of congruent figures is also a set of similar figures. After all, if two figures are the same shape and size (congruent), then they must necessarily be the same shape (similar). The converse isn t true, however, since figures can have the same shape, but at various scales. To put it simply: all congruent figures are similar, but not all similar figures are congruent. In each video, what kinds of transformations do you see? What changes about the image? What stays the same? Which transformation(s) will produce figures that are congruent to the original? Why? Which transformations will produce figures that are similar to the original? Why? Can figures be both congruent and similar? Why or why not? Deeper Understanding If you double all the dimensions of an image in a dilation, the scale factor is 2. What scale factor would produce a congruent figure from the original? (1 or - 1)
Lesson Guide: TRANSFORMERS 5 3 Watch a slow- motion recording of someone closing an app. As the screen collapses back into its icon, what transformation(s) do you notice, and do you think this is an example of congruence or similarity? Explain. The full- screen image shrinks in size until it ends up as an icon, which makes it look like a dilation. But it can t actually be a dilation, because the resulting icon is a different shape than the original image (it s much closer to being a square). That means the resulting icon is neither congruent nor similar to the starting image. Explanation & Guiding Questions This question is tricky, because a transformation that starts out looking a lot like a dilation turns out not to be. But it highlights an important point: dilations change a figure s size, but they have to preserve its shape in order to be called dilations. In the first few frames, that s exactly what seems to be happening: the map begins to reduce in size, but it s hard to tell whether it s the same shape or not. By the end of the animation, though, it s clear that we can t be dealing with a dilation, because the original map roughly matches the phone screen s dimensions, while the final icon is nearly square. The key point here is that dilations don t just scale figures, they scale figures uniformly in all directions. In this case, the horizontal and vertical dimensions of the rectangle are being scaled by different factors, so the icon is neither congruent nor similar to the screen at the beginning of the video. What changes about the image as the app closes? What stays the same? What kinds of transformations do you notice? Are the before and after images congruent? Why or why not? Are the before and after images similar? How can you tell? Do you notice anything different about this set of transformations compared to the earlier ones? Deeper Understanding Here we have an example of something that looks like a dilation, but doesn t produce a similar figure. What s different about the scaling that s happening in this case? (The shape is scaled by different horizontal and vertical factors as it shrinks.)
Lesson Guide: TRANSFORMERS 6 Act Two: More Than Meets the i 4 When you zoom in or out, the screen scales around a center of dilation; the object at this point stays in place, while everything around it moves inwards or outwards. Watch as someone zooms in and out using two fingers. In each case, draw the center of dilation. How do you think the smartphone determines where this center is? The phone appears to make the center of dilation the midpoint of the user s fingers. For instance, the tennis ball in the dog s mouth seems to stay in the same place on the screen, so that must be about the center of dilation. Explanation & Guiding Questions Every dilation has a center: the fixed point of the transformation around which everything else changes. So if students can spot the part of the image that appears to stay in place, then they can locate the center of dilation. In the first video, the center of the tennis ball in the dog s mouth doesn t look like it changes position, so that must be the center of dilation. In the second video, the center of the plane appears to stay in place, so that must be the center of dilation. In both cases, the center of dilation looks as though it s halfway between the user s fingers. If students have trouble locating the center of dilation at first, you could start by pointing out some places where it couldn t be. For example, the tip of the dog s tail ends up much closer to the top of the screen than where it started, so it clearly isn t the fixed point. In the plane video, the little patches of grass that end up on the left aren t even on the screen when the video begins, so nothing on that whole side of the screen could be the center. In the first video, what part of the picture seems to stay in the same place? What about the second video? In each video, can you pick out anything that definitely can t be the center of dilation? How do you know? Where does the center of dilation appear to be, relative to the user s fingers? Deeper Understanding Why do you think the phone uses the midpoint of the user s fingers as the center of dilation? (If you think about what would happen if you tried to stretch a little section of tablecloth between your fingers, everything would spread out from the point halfway between them. The phone is probably trying to mimic what happens when you stretch something in real life.)
Lesson Guide: TRANSFORMERS 7 5 A smartphone screen is basically a coordinate plane; each pixel represents a point. Imagine you re zooming in and out on a map using an iphone 5 and an iphone 6. If the center of dilation is at the origin, determine the new coordinates of objects A and B for each scale factor: 0.5x, 2x, and 3x. Based on this, do you think it was smart for Apple to release the larger iphone 6? Screen Resolution: 1136 640 pixels Screen Resolution: 1334 750 pixels Original Coordinate Point A = (204,120) Point B = (- 284,- 16) Scale Factor 0.5x 2x 3x 0.5x 2x 3x New Coordinate (102,60) (408,240) (612,360) (- 142,- 8) (- 568,- 32) (- 852,- 48) On Screen? iphone 5? iphone 6 Answers will vary, but the larger screen obviously means that users will be able to see more of what s going on when they zoom in or out. Explanation & Guiding Questions Earlier in the lesson when students were talking about similar figures, they discussed how images maintain their shape when they are dilated. That means that any scaling must be applied equally in all directions, otherwise things will get distorted. When performing dilations with coordinates, that means each point must be scaled (in this case from the center) by the same factor along the x- axis and the y- axis. Since the center here is (0, 0), that simply means multiplying both coordinates by the scale factor to obtain the new point. Students will see that points on the larger screen stay in frame longer than points on the smaller screen. However, smaller screens may have the edge when it comes to convenience. Which phone students prefer will probably vary, depending on how they weigh these competing factors. Why do dilations preserve the shape of an object? What do you think it means to scale a point equally in all directions? What happens to the point (100, 60) when it gets twice the distance from (0, 0)? What would happen to the map if you only multiplied the x- or y- coordinates by the scale factor? Deeper Understanding How would the situation change if the center of dilation were some other point besides (0, 0)? (You could no longer just multiply the coordinates by the scale factor.)