## Flip-n-Slide: Exploring Transformations through Modeling and Computer Games

• Lesson
6-8,9-12
2

In this lesson, students will explore reflections, translations and rotations. Students participate in a modeling activity where they will learn the rules for translations and reflections. Then students will practice using these transformations, as well as explore the rules for rotations, in the game Flip-n-Slide on Calculation Nation®.

The activities described below will teach students about geometric transformations—translations, reflections, and rotations. Because of the cognitive demand required to discover the rules for clockwise rotations, the activities have been divided into two parts. The rules for translations and reflections are more intuitive and work well with a lesson that is open-ended and exploratory, but rotations are better suited for guided discovery. Consequently, this lesson plan attempts to teach translations and reflections on one day and rotations on the following day, but you may wish to adjust based on the needs of your students.

To deliver this lesson effectively, you must be able to:

• Model reflections over lines, over the x- and y-axes, and about the side of a shape
• Model rotations around the origin at 90°, 180°, and, 270°
• Model translations.

It will be helpful to remember the following rules for clockwise rotations about the origin:

• 90° Rotation: If the original point is (a,b), then the point after rotation will be (b, -a)
• 180° Rotation: If the original point is (a,b), then the point after rotation will be (-a,-b)
• 270° Rotation: If the original point is (a,b), then the point after rotation will be (-b,a)

Begin the class by allowing students some time to play the game Flip-n-Slide. Many students have been exposed to reflections, rotations and translations, but they may not have experienced these geometric movements on a coordinate plane, and they may refer to these transformations as flips, slides and turns rather than their proper names. Once students have had time to play at least one game, review the correct terms and definitions for reflections (flips), translations (slides) and rotations (turns) as a class prior to introducing the coordinate plane.

Next, ask students, “What happens to the coordinates of a shape when it is translated?” Direct students to begin exploring this question using the game Flip-n-Slide. The translation movements are the top six transformation tokens. Three tokens are designated as right or left translations and three are designated as up or down translations. As students explore, encourage them to change the directions (right, left, up, or down) and the number of units the shape will move. If students are having difficulty finding a pattern, ask them to record the coordinates before and after each translation.

Once students have had an opportunity to explore translations individually, have them perform a think-pair-share with a partner. A think-pair-share gives students an opportunity to think about what they have discovered and time to discuss their findings. This discussion can lead to more specific ideas, and students may learn something from their partner. During this time, students should record any of their ideas so they can share them during a whole-class discussion. Designate a recorder within each group.

After the think-pair-share, have the class develop rules for what happens to a shape during a translation. As the rules develop, ask students to check and make sure that they are correct by performing the translation on their own computers.

• When translating up, the y-coordinate increase by the number of units moved.
• For example,if one set of coordinates is at (1,0) and a shape is translated up 8 units, then the new set of coordinates will be (1,8).
• When translating down, the y-coordinate will decrease by the number of units moved.
• For example, if one set of coordinates is at (1,8) and the shape is translated down 3 units, then the new set of coordinates will be (1, 5).
• When translating right, the x-coordinate with increase by the number of units moved.
• For example, if one set of coordinates is at (2,3) and the shape is translated right 4 units, then the new set of coordinates will be (6, 3)
• When translating left, the x-coordinate will decrease by the number of units moved.
• For example, if one set of coordinates is at (0,4) and the shape is translated left 5 units, then the new set of coordinates will be (-5,4).

After students have determined the rules for translations, direct the class to begin exploring the rules for reflection. The question remains the same, “What happens to the coordinates of a shape during a reflection?” You will want students to first move their triangles in the game Flip-n-Slide away from the x-axis and the y-axis. Encourage students to explore what happens when they reflect a shape over the x-axis, y-axis and a side of the shape. If students are having difficulty identifying the pattern, once again ask them to record the coordinates before and after a reflection. Once students have had ample time to explore reflections, ask them to participate in a think-pair-share to develop their understandings. Again, designate a recorder so students can share their ideas in a whole-class discussion.

After students have completed the think-pair-share, work as a class to determine a set of rules for performing a reflection. Allow students to check their games and ensure that the rules being created are correct.

• When reflecting over the x-axis, the x-coordinate stays the same but the y-coordinate is the opposite of the original y-coordinate.
• For example, if the beginning coordinates were (3, 2), after a reflection over the x-axis the new coordinates would be (3, -2).
• When reflecting over the y-axis, the x-coordinate is the opposite of the original x‑coordinate but the y-coordinate stays the same.
• For example, if the beginning coordinates were (4, 5), after a reflection over the y-axis, the new coordinates would be (-4, 5).
• When flipping over a side of the triangle, two sets of coordinates will stay the same (the two endpoints of the side), but the third coordinate will change. The side lengths will not change, and the point will appear on the other side of the figure after the reflection.

At this point, it will be important for students to note that (1) some of the rules described above are specific to triangles and (2) that transformations preserve size and shape; that is, the resulting figure will have the same side lengths and interior angle measures as the original shape.

After these rules have been created, guide students to explore rotations within the game. Give them a few minutes to explore the rotation token without any specific goals.

Distribute the Exploring Rotations Activity Sheet. Ask students to start a new Flip-n-Slide game. At the start of the game, players have the option to adjust their triangles. As a first step, have students move their triangles away from the origin. If you have access to a Smart Board, demonstrate this on your computer screen. Circulate to ensure all students have done this correctly. (Note that there is a time limit for changing the initial size and location of the triangle. If students are not able to do this successfully in the allotted time, have them perform a translation or reflection to move the triangle away from the axes.)

Explain to students that they will only be performing rotations in this activity. If they run out of rotation tokens in their game, they will need to start a new game.

Give students time to complete the Exploring Rotations activity sheet on their own. When the students are working to make conjectures, encourage them to discuss their ideas with their neighbors, as working in pairs or groups will allow them to talk through their ideas. To promote these kinds of discussions, ask questions like, “Jakob, what do you think of Tisha’s idea?” as you circulate and talk to students.

Once all students have completed the activity sheet, conduct a class discussion and encourage students to discuss the conjectures and rules they have developed for this activity. The goal should be to create a standardized set of rules for completing 90°, 180°, and 270° clockwise rotations. Write the students’ formal rules on chart paper and display in the class. The correct rules can be found on the Exploring Rotations Activity Sheet Answer Key.

### Idea for Differentiation

To help struggling students see the results of a transformation, have students draw the figure on graph paper. Then, with a congruent figure drawn on a piece of tracing paper, have students perform the transformation to observe the result.

Assessment Options

1. Students should create a journal entry in which they use words and pictures to explain the rules for performing translations, reflections and rotations. Prompts may be given, such as, “When a geometric figure is translated, the new figure will be…”.
2. Collect the students’ completed Transformations Activity Sheet to check their understanding.

Extensions

1. In this lesson students focused on one transformation at a time. Discuss the value of using multiple transformations to move from one location to the next within the context of the game. Have students explore using multiple transformations as they attempt to win the game.
2. Calculation Nation allows students to play against one another. Have two students compete against each other. (Note that they will need to register, if they have not already done so.) After each move, the player who did not move should describe the path of the other player’s triangle. The player who is moving should write down his path, and the player who is trying to guess should either dictate the path to the first player or write it down. The players can compare their paths to see if they match.
3. Students can examine counterclockwise rotations in Flip-n-Slide in the same way that they examined clockwise rotations. Have them compare counterclockwise rotations and clockwise rotations. They should note that a 90° clockwise rotation is equivalent to performing a 270° counterclockwise rotation, and vice versa.
4. Students can create creative pieces to help them remember the rules of transformations (ex: short story, song, poem, etc.).
5. Give students two consecutive transformations. Then, have students try to find one transformation that gives the same result. They can create a record to see if there are any patterns of which transformations will and won't work.

Questions for Students

1. What is the difference between reflecting across a side of the shape and reflecting across an axis?

[When a shape is reflected about a side, that side stays in the same place. When reflecting across an axis, the shape moves to a different quadrant, but each vertex is the same distance from the axis as it was previously.]

2. What is the difference between rotating 270°, 180°, and 90° about the origin in a clockwise direction?

[If a shape is rotated 90°, it moves one quadrant; if it is rotated 180°, it moves two quadrants; and if it is rotated 270°, it is moved three quadrants.]

3. What happens to each vertex when a 90° clockwise rotation is performed? Does it happen all the time? How do you know?

[If the vertex originally had coordinates (x, y), then its new coordinates will be (y, -x). That is, the x and y values will be switched, and the new y value has the opposite sign. Multiple examples within the game suggest that it does happen all the time.]

4. What happens to each vertex when a 180° clockwise rotation is performed? Does it happen all the time? How do you know?

[If the vertex originally had coordinates (x, y), then its new coordinates will be (-x, -y). That is, the x and y values both become opposites of their original values.]

5. What happens to each vertex when a 270° clockwise rotation is performed? Does it happen all the time? How do you know?

[If the vertex originally had coordinates (x, y), then its new coordinates will be (-y, x). That is, the x and y values switch places and the new x value becomes opposite of its original value.]

6. What would happen to each vertex when a 360º rotation is performed?

[Each vertex would return to its original location. That is, a vertex at (x, y) would return to (x, y).]

7. Translation A is right 3, down 4 and translation B is right 2, up 5. Translation C is the result of translation A followed by translation B. What is the rule for translation C?

[Translation C is right 5, up 1.]

Teacher Reflection

• What observations did you make that show evidence that your students were engaged in the lesson?
• What student misconceptions did you observe for each of the transformations (reflections, rotations, and translations)? How did you address the misconceptions? Which your attempts at addressing the misconceptions seemed most effective to increase student understanding?
• What, if any, appropriate conjectures were students able to make about rotations of a figure about the origin?

### Learning Objectives

Students will:

• Students will accurately perform reflections, translations and rotations.
• Students will develop rules for performing clockwise rotations on a coordinate plane
• Students will be able to verbally describe geometric transformations.

### NCTM Standards and Expectations

• Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.
• Understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices.
• Use various representations to help understand the effects of simple transformations and their compositions.

### Common Core State Standards – Mathematics

Grade 8, Geometry

• CCSS.Math.Content.8.G.A.3
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.