<p>This lesson presents translations (slides) as a new sort of transformation. Students broaden their understanding of congruent figures, which have the same size and shape but may have a different orientation. Students practice different transformations while working with tetrominoes. Students will:&nbsp;<br>- perform translations.&nbsp;<br>- identify the rotations, reflections, and translations.&nbsp;<br>- create a pattern that includes all three types of transformations.&nbsp;<br>- work with tetrominoes to understand transformations.</p>

<p>- What strategies might we use to verify symmetry and congruency?&nbsp;<br>- What strategies may we take to continue a sequence?</p>

<p>- Tessellation: Patterns of shapes that fit together without any gaps.<br>- Tetromino: Two-dimensional figures made of four connected congruent squares.<br>- Tiling: Fitting individual tiles together with no gaps or overlaps to fill a flat space like a ceiling, wall, or floor.<br>- Translation: A movement of a figure to a new position without turning or flipping it.<br>- Quadrant: The x- and y- axes divide the coordinate plane into four regions. These regions are called the quadrants.</p>

<p>- student copies of Vocabulary Journal pages (M-5-7-1_Vocabulary Journal)&nbsp;<br>- 4 one-inch square tiles per student (plastic if available or cut out of paper, M-5-7-3_Full Page 1 Inch Grid)&nbsp;<br>- student copies of one-inch grid paper that is 8.5 x 11 inches (M-5-7-3_Full Page 1 Inch Grid)&nbsp;<br>- 2 transparencies of one-inch grid paper, one 8.5 x 11 inches and another 6 x 8 inches (M-5-7-3_Full Page 1 Inch Grid and M-5-7-3_Game Grid 6 x 8)&nbsp;<br>- student copies of Tetromino Game sheet (M-5-7-3_Game Grid 6 x 8), or use the 8.5 x 11-inch grid sheets and have students outline a 6 x 8-inch section (M-5-7-3_Full Page 1 Inch Grid)&nbsp;<br>- a spinner with six equal sections (place a picture of a different tetromino in each of the first five sections; label the sixth section “free choice”)(M-5-7-3_Tetromino Spinner)&nbsp;<br>- student copies of the Lesson 3 Exit Ticket (M-5-7-3_Lesson 3 Exit Ticket and KEY)&nbsp;<br>- index cards (approximately 20 for each student) for vocabulary flashcards&nbsp;<br>- a small sealable plastic bag or an envelope for each student to store their tetrominoes&nbsp;<br>- index cards cut into 2 x 2-inch squares&nbsp;<br>- 9 x 12-inch colored paper or tagboard&nbsp;<br>- glue sticks&nbsp;<br>- tape</p>

<p>- Observations collected during the Partner Translation Activity will help in determining the students' overall level of comprehension.<br>- Observations made during student evaluations of tetromino sets will provide an opportunity to clarify some misunderstandings about transformations.<br>- Student evaluation during Tetromino Cover-Up The game will consist of instructor observations.&nbsp;<br>- Use the Lesson 3 Exit Ticket (M-5-7-3_Lesson 3 Exit Ticket and KEY) to provide further student evaluation.</p>

<p>Scaffolding, Active Engagement, Modeling, and Explicit Instruction&nbsp;<br>W: Display a grid with figures marked on it, similar to the lesson example. Allow students a couple of minutes to describe the transformation they discover. Introduce translations.&nbsp;<br>H: After demonstrating an example, have students work in pairs to perform a translation on a grid and identify it. After several turns, evaluate all of the transformations, as they will all be used in the next activity.&nbsp;<br>E: Describe tetrominoes as two-dimensional figures made of four connected congruent squares. Assign students to work in pairs to discover as many tetromino configurations as possible. Display the five possible results and explain that some of them may be transformations of one another. Allow students to play Tetrominoes Cover-Up game.&nbsp;<br>R: During the game, walk around the room to observe, ask questions, and make suggestions. Discuss with the students some of the strategies they used, including which shapes fit well together and which pieces were difficult to use.&nbsp;<br>E: To measure student comprehension, have them fill out the Lesson 3 Exit Ticket (M-5-7-3_Lesson 3 Exit Ticket and KEY).&nbsp;<br>T: Create a card matching game where students design cards with a vocabulary word on one card and its meaning or depiction on the other card. Students can also design tessellations by combining the transformations studied in this unit.&nbsp;<br>O: In this lesson, students will study about translations and apply the transformations presented in the unit.&nbsp;</p>

<p><strong>Think-Pair-Share: </strong>Draw the figures below or a similar setup on the board or overhead.<br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_49.png" width="298" height="279"><br><br><strong>"Consider the figure marked as the original figure. Spend around 2 minutes on a sheet of paper writing down all of the different transformation of the original that you see. Make your descriptions as clear as possible, and provide the image number."</strong><br><br>Allow students 2 minutes to write their observations. Then, ask them to turn to their partner and share their observations. Go around the classroom and ask each pair to share at least one observation. Be careful to get responses for all three images. The observations should include the following:<br><br>Image 1 is a 90° rotation.<br><br>Image 2 is a reflection on the y-axis, but it is also a slide motion on the x-axis, 11 spaces to the left.<br><br>Image 3 has been diagonally moved to the left; it is the combination of sliding down 5 spaces and moving left 5 spaces.<br><br><strong>"Based on your observations, I see that you recognized the reflection in Image 2, as we saw in Lesson 1. You also noticed the rotation in Image 1, which we covered in Lesson 2. You also mentioned the third type of transformation. When we slide a figure to the right, left, up, down, or a combination of these movements, it is called translation. The main characteristic of this change is a sliding movement."</strong> Identify the slide movements used in the samples on the board.<br><br><strong>"In our lesson today we will look at a variety of situations which include translations (slides), and some will also include reflections and rotations."</strong><br><br><strong>Partner Translation Activity</strong><br><br><strong>"The slide movement can be as simple as going in one direction, as shown in Image 2 of our first example. A figure may be moved horizontally (left to right) or vertically (up or down). What seems to be a diagonal slide is actually two slides—one horizontal and one vertical. Try to find the translation (slide movements) in the figures your partner makes using pattern blocks."</strong><br><br>Give each pair of students a coordinate grid and a set of pattern blocks or pattern cutouts (M-5-7-2_Coordinate Axes.doc, M-5-7-1_Pattern Cutouts 1, M-5-7-1_Pattern Cutouts 2, and/or M-5-7-2_Pattern Cutouts 3). Allow the partners to take turns placing one pattern piece on the coordinate grid, along with two similar pieces that have been translated with either one or two moves (slides). The opposite partner will try to identify the move(s) using phrases such as "the original figure has been translated left 2 and up 4." Show an example on the overhead or on the board to help students comprehend. Walk around the room, observing and assisting as required. Allow students 5-10 minutes to resume their alternate rounds.<br><br><strong>"Now that you are comfortable identifying translations (slides), we are going to work on an activity that incorporates all of our transformations (reflections, rotations, and translations)."</strong><br><br>Review the terms translation (slide), reflection (flip), and rotation (turn).<br><br><strong>Tetromino Try-outs</strong><br><br><strong>"Tetrominoes are two-dimensional figures made of 4 connected congruent squares. You and your partner will try to find the four squares in as many different ways as possible. Trace them on your grid paper as you discover them."&nbsp;</strong><br><br>Display the following rules for arranging the tiles on the board or overhead projector:&nbsp;<br><br>Each square must have a common side.&nbsp;<br><br>Tiles must be set flat. Stacking is not allowed.&nbsp;<br><br>Tiles may not be overlapped.<br><br>Distribute four 1-inch square tiles and an 8.5 × 11-inch sheet of 1-inch grid paper to each student (M-5-7-3_Full Page 1 Inch Grid). If plastic 1-inch squares are not available, 1-inch tiles can be cut from additional sheets of grid paper.<br><br>Ask students to use four squares to find as many two-dimensional tetrominoes as possible. Instruct students to trace the combinations on two sheets of 1-inch grid paper. When pairs of students believe they have found every possible combination of four squares, they should cut them out to create their own set of tetrominoes.&nbsp;<br><br>Discuss the class discoveries with the following questions to guide students' thinking:&nbsp;<br><br><strong>"Do you have all the possible tetromino shapes?"&nbsp;</strong><br><br><strong>"How do you know?"&nbsp;</strong><br><br><strong>"Are some of your tetrominoes the same?"</strong><br><br><strong>"How can you prove it?" </strong>(<i>By turning, flipping, or sliding the tetrominoes and placing them, we can determine whether they are the same or different. They are similar when they fit exactly on top of each other, demonstrating that they are the same size and shape.</i>)&nbsp;<br><br>There are just five actual tetromino patterns, but students should have created many more pieces, including duplicates. At this time, do not tell students that they should only have five. Instead, tell students that they will be looking for duplicate congruent shapes created by rotations, slides, and flips. Demonstrate to their partner how they will prove congruence to start reducing the number of duplicates, using phrases to prove congruence like:<br><br><strong>"I can prove these are congruent by rotating to match these two figures."&nbsp;</strong><br><br><strong>"I can prove these are congruent by flipping one to match the other."&nbsp;</strong><br><br>Allow students 5-8 minutes to narrow down their set to the point so that there is no duplication. Examine each pair's final set. Each student should have five remaining tetrominoes, as shown below. Once each student gets the correct set, move on to the Tetrominoes Cover-up Game. Each player should have their own set of tetrominoes for the game.<br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_50.png" width="240" height="138"><br><br><strong>Tetrominoes Cover-Up Game</strong><br><br>Students play "Tetrominoes Cover-Up" with the tetrominoes they have just created.&nbsp;<br><br>Each pair will need a spinner template and a paperclip to use as the spinner needle when held in place with the tip of a pencil (M-5-7-3_Tetromino Spinner). Students will use tetromino shapes to completely fill the 8 x 6-inch game board grid (M-5-7-3_Game Grid 6 x 8). The goal is to have the least number of uncovered squares left at the end of the game without overlapping any tetrominoes. Students can use flips, slides, and turns to place their tetrominoes.<br><br>Distribute crayons, colored pencils or markers, a spinner, a paper clip, and two game boards to each pair. Once the spinner is spun, the arrow will point to a tetromino figure. That figure must be used by the player during that turn. A player that spin shows "Free Choice" may play any piece he or she chooses. When a piece is used, it should be traced, removed, and its shape color-coded on the board. This allows a player to spin the same piece on the spinner and use it in a different turn. Model how to play the game on the overhead projector or board, then explain the following additional rules:<br><br>Each pair will determine who goes first.&nbsp;<br><br>Player 1 spins the spinner to choose which tetromino piece to play.&nbsp;<br><br>He or she places the tetromino on the game board so that one side touches the bottom or (after the first round) another tetromino.&nbsp;<br><br>They may use slides, flips, or turns to place the chosen tetromino so that the fewest squares will be left uncovered on the game board as the game continues.<br><br>The player then traces and colors the squares covered by the selected tetromino before returning it to his/her pile.&nbsp;<br><br>Player 2 spins the spinner to choose which tetromino would be placed on his/her game board.<br><br>Play continues until there are no more tetrominoes to place on either game board.&nbsp;<br><br>The players' points are calculated by counting the total number of squares not covered on their own game board.&nbsp;<br><br>The winner is the player with the lowest score (fewest squares uncovered).&nbsp;<br><br>Allow students 8 to 10 minutes to play the game.&nbsp;<br><br>As students work on their tetromino sets, go around the room and ask leading questions to keep them moving toward decreasing their sets to the final five. When all students have played the game at least once, have a class discussion about some of the techniques they discovered. The following questions can help to direct the discussion:<br><br><strong>"Do certain shapes fit together well?"&nbsp;</strong><br><br><strong>"How did you decide where to place the tetrominoes?"&nbsp;</strong><br><br><strong>"Was one tetromino shape more difficult to place than the others? Why?"&nbsp;</strong><br><br><strong>"What tetromino shape was the easiest to work with? Why?"</strong>&nbsp;<br><br>If time allows, have each student play the game again using the strategies mentioned. Encourage them to take the game home and play it with someone at home.<br><br>Have each student fill out the Lesson 3 Exit Ticket (M-5-7-3_Lesson 3 Exit Ticket and KEY). Based on the findings of this evaluation and observations from the class activities, identify which, if any, of the instructional strategies listed below should be used for each student.<br><br><strong>Extension:</strong><br><br>Use these tips to personalize this lesson to your students' needs throughout the unit and year.&nbsp;<br><br><strong>Routine:</strong> Discuss the importance of understanding and using the appropriate vocabulary words to convey mathematical ideas clearly. During this lesson, students should record the following terms in their Vocabulary Journals (M-5-7-1_Vocabulary Journal): tessellation, translation. Keep a supply of Vocabulary Journal pages on hand so that students can add them as needed. Bring up examples of translation and tessellation from throughout the school year. Ask students to bring up examples they notice, and encourage them to cut out and bring examples to class. They are likely to see these in patterns in a number of places such as on clothing, household decorations (fabric, wallpaper, floor coverings, posters, pictures, etc.), and on book and magazine covers.<br><br><strong>Vocabulary: Match Me Up Activity</strong><br><br>Students should use index cards to make a matching game. Each new word in the unit should be written on a card. Students should write a definition, picture, or example for each vocabulary phrase on a second card. Each vocabulary word card should have a matching card. Students should properly mix up the cards before placing them face down in a rectangular matrix pattern on the table. On each player's turn, he/she will draw two cards with the purpose of matching a vocabulary word to its mate (definition, graphic, or illustration). If students do not find a match, they replace the cards. If they find a match, they keep the cards and play another turn. Once all of the pairs have been matched, the player with the most cards wins. To make the game more difficult for advanced students, each term may include three cards (term, definition, and image or example). Students would need to find all three cards in one round in order to keep the trio and look for another matching set on their next turn. If a trio is not identified, cards are returned to their original place, and the game continues on to the next player, as in the pairs' version.<br><br><strong>Individual Technology Connection: 20-a-Day</strong><br><br>If computers are available to students, this activity could be used for further practice. This task is appropriate for students who could benefit from more practice with the concepts of rotation and translation, as well as any student who needs a review. If you have the ability to project on a classroom screen from a single computer, these tasks could be used as class practice or review, as well as in a classroom game.<br><br>Practice problems can be found at: <a href="http://www.ixl.com/math/practice/grade-5-reflection-rotation-and-translation">http://www.ixl.com/math/practice/grade-5-reflection-rotation-and-translation</a>&nbsp;<br><br><i>Note to teacher:</i> Users are restricted to 20 questions each day. Additional problems are only available to members.<br><br><strong>Expansion: Tessellation</strong><br><br>Use this activity with students who have mastered the concepts in this lesson. Give each student a square piece of tag board or a part of an index card (about 2 x 2 inches), a full sheet of paper or tag board (approximately 9 × 12 inches), scissors, crayons, and colored pencils or markers.<br><br>Students will cut a unique pattern from the right side of the square card. Students will either slide the cutout across to the left parallel side, flip it, and slide it across to the left side, or rotate it 90° to the top adjacent side before taping it into place.<br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_51.png" width="460" height="140"></p><p>slide to the left &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; rotate to the top &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; flip then slide left<br><br>called a translation &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; called a 90° rotation &nbsp; &nbsp; &nbsp; &nbsp; called a glide reflection<br><br>Students can now use this card as their figure to tessellate. The fundamental region refers to the figure that is used repeatedly to build a tessellation. Students can use their fundamental region and begin tracing it on paper to make a tessellation, or they can create a more complex design using their fundamental region. To create a more complex fundamental region, students can begin with one of the transformations shown above and then cut another unique piece from one of the card's remaining flat sides. They will turn the piece to the final flat side before taping it in place. This more complex figure can then be traced numerous times on a sheet of paper to create a tessellation. The example of a complex fundamental region using translations is provided below. A piece was cut from the right side and then moved to the left. A different piece was cut from the bottom then slid to the top.<br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_52.png" width="159" height="127"><br><br>If a student is having problems, you could suggest creating a very simple fundamental region. Simply make one cutout. Take the cutout design from the bottom right corner and slide it up to the top right corner. This makes the figure much easier to line up, tape, and trace.<br>After taping the figure in place, students should start the tessellation by tracing one copy of the pattern (fundamental region) in the middle of the sheet. It may be straight or at an angle. They will continue to trace the fundamental region until the page is filled. The tessellating piece (fundamental region) will need to be flipped, slid, and/or rotated to fit perfectly next to previously placed parts, like a puzzle, without leaving gaps or overlaps on the page. Students can create M.C. Escher-inspired tessellations that resemble pictures and/or real-world objects. Encourage students to get creative in designing their fundamental regions that looks like a cartoon bird, fish, dog, or tree by adding simple details like eyes, leaves, or feathers and using color. Show examples below. Many more entertaining and creative examples can be discovered online or in books.<br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_53.png" width="489" height="141"><br><br><strong>Technology Option: Interactive Tessellation</strong><br><br>This website can be used to show how to transform a geometric figure and tessellate the resulting figure, or to allow students to experiment with tessellations. <a href="http://www.shodor.org/interactivate/activities/Tessellate/">http://www.shodor.org/interactivate/activities/Tessellate/</a>&nbsp;<br><br>This lesson is exploratory in character. Students analyze and contrast the characteristics of reflection, translation, and rotation. Students work together to identify translations and then use them to create a collection of tetrominoes. Students are asked to use and discuss all three transformations to verify their tetrominoes. The lesson ends with additional exercises that allow students to combine transformations to create real-world designs.</p>