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| 5 |  Grade 3 / Figure operations: Rotation | |
| Developed by | IPSantarem - Bento Cavadas e Nelson Mestrinho | |
| Grade | 3 | |
| Duration | 3h15 | |
| Energizers |  46 - Movement Ask the students to lie down on the floor (on their backs or tummies) and tell them to demonstrate the sliding movement. Figure 1. Illustration of the sliding movement
Some questions for discussion:
Now, ask the children to lie down on the floor again and give their interpretation of the flipping movement (in a turn the pupils move from back to tummy or from tummy to back). Figure 2. Illustration of the flipping movement Some questions for discussion:
Finally, ask the pupils to lie down on the floor and demonstrate the turning movement. Figure 3. Illustration of the turning movement Some questions for discussion:
Students should be encouraged to use their own words to describe and characterize the different movements. Moves like sliding, flipping and turning constitute rigid motion. They are motions that do not distort shape. Picking something up and moving it around for instance is a rigid motion, but stretching or warping it is not. These intuitive experiments are an excellent way to introduce more advanced geometric concepts, such as translation, reflection and rotation, as fundamental types of isometric transformations. In this lesson, we shall focus on the idea of rotation. (adapted from: National Council of Teachers of Mathematics (1993). Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades K-6. Second-grade book. Reston, VA: NCTM.) | |
| Relaxing Exercises |  41 - Puppets movements Students sit in a circle on the floor and freely illustrate and discuss the different movements of sliding, flipping and turning with puppets. | |
| Objectives | Students will:
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| Preparation | Materials:
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| Introduction | Geometry is a cornerstone subject in school mathematics, both because of its undeniable importance for understanding the space around us and because of its unavoidable historical relevance. Geometric transformations add a perspective of movement to the traditional methods of elementary geometry, and it makes perfect sense to study them from the first years of school. With this activity, students will have the opportunity, starting from an everyday context, to explore the notions associated with the concept of rotation in the plane, in its simplest forms (quarter-turn and half-turn rotations), while at the same time promoting their sense of aesthetics and developing their fine motor skills. | |
| Teaching and learning methods | Exploratory learning from real-life situations. | |
| Interdisciplinary connections | Language – oral and visual communication. Arts – Construction and identification of geometric regularity in patterns Culture | |
| Resource teacher or other specialists activities | Almost all the steps of the lesson are comprehensible to most special needs students who have well-preserved cognitive abilities. The lesson does not comprise written explanations, and the long instructions, which are difficult for the target group’s students (TGSt), are not many. The role of the special needs teacher or the assistant teacher is to help TGSt in case they are not able to cope with some of the tasks or steps. | |
| New vocabulary | Quarter-turn Half-turn Clockwise and anti-clockwise | |
| Work Flow | Phase 1: Presenting the context A quilt is a multi-layered textile, traditionally composed of two or more layers of fabric or fiber. A single piece of fabric can be used for the top of a quilt, but in many cases the top is created from smaller fabric pieces joined, or patchwork, sometimes using scrap pieces of fabric. The pattern and color of these pieces creates the design, and many of them are built as repetitive geometric patterns.  Examples of quilts created with repetitive geometric pattern. By National quilters circle | nationalquilterscircle.com
 By Billvolckening - Own work, CC BY-SA 4.0 https://commons.wikimedia.org/w/index.php?curid=45371912 Check on some exemples of quilt pattern traditions from aroud the world: https://www.marthastewart.com/7836632/traditional-quilts-around-the-world https://www.internationalquiltmuseum.org/ The teacher may show to pupils some local traditional pattern quilts, and discuss about the decorative and utilitarian use of quilts (like bedding, table topping,...) and the process quilting (sustainability by reduction of textile waste, an art form and an hobby,...). If possible, the teacher shows a quilt to students and discuss how the quilt is 10 minutes Phase 2: Creating a quilt square In this phase, teacher shows how to create a paper quilt square: Cut a red 30 cm x 30 cm sheet of construction paper along its diagonals to create four congruent triangles.  Cut a yellow 30 cm x 30 cm sheet of construction paper the same way.
 Glue one yellow triangle and one red triangle onto a blue 30 cm x 30 cm piece of construction paper as shown below:  The teacher needs to have another three copies of this design for steps ahead.Then, the teacher shows students other possible quilt square designs that he/she made previously from construction paper. Some examples that one may consider are:
Teacher discuss with pupils how the pieces of construction paper are glued together to create each design. Organized in pairs, children now can draw on graph paper their own simple design or copy one design of their choice (one design for each pair). Next, the teacher tells pupils that they will construct four identical quilt squares using construction paper, following their own design as drawn. On this stage of the work, teacher should: - Check to ensure that students have drawn a simple design. - Allow children to collect the materials needed to create their four squares (10 cm x 10 cm sheets of coloured construction paper, scissors, glue). - Monitor pupils’ work and collaboration as they build their quilt squares, assisting them, if necessary, to make identical squares. 50 minutes Phase 3: Exploring Rotations When students have finished making their paper quilt squares, the teacher begins by drawing vertical and horizontal axes on the board.
Then, the teacher fixes (with tape or magnets) one of the quilt squares in the upper left quadrant on the board.
Teacher now holds a second quilt square directly superimposed the fixed square. Then, he/she asks the students, “How could I turn this quilt square onto the next (upper right) section?”. A volunteer student can show how to perform the rotation. Then, this second quilt square can be fixed in the upper right quadrant.  It is essential to recognise that, during rotation, there is a point on the square that remains fixed in relation to the movement - the centre of rotation - and the movement can be carried out while placing a finger at the center of rotation, on the vertex of the square positioned at the point where the axes intersect (marked in the image above). Now the teacher ask students, “Why would we call this rotation a quarter turn?”, connecting the quarter turn to the movement of the minute hand on a clock from the upright position to the quarterpast-the-hour position. Then, he/she records “Quarter Turn” next to the square. At this point, the teacher can ask students, "What if instead of turning clockwise we turned anti-clockwise? Where would the square be?" Holding a third quilt square directly on top of the original one, another volunteer student can show how to perform this new rotation, a quarter turn anti-clockwise, keeping the same center of rotation as before. This third quilt square will be fixed in the lower left quadrant, with the corresponding record “Quarter-turn anti-clockwise”, also completing the first record as “Quarter-turn clockwise”.
 To finish the pattern, after superimposing a fourth square on the initial square, the teacher asks a student to turn it round half turn. The quilt square will then occupy the bottom right-hand quadrant.
The teacher can also ask the student if the direction of rotation makes a difference (clockwise or anti-clockwise). At this stage it should be clear that the result of a half-turn is always the same regardless of whether it is taken in the forward or reverse direction. Students can be asked to describe the result of a full turn, to conclude that after a complete rotation the square returns to its original position and record “full turn” next to the original square. The teacher can invite other students to demonstrate quarter turns (both clockwise and anti-clockwise). 40 minutes Phase 4: Students’ quilt patterns In this phase, each pair of students will build their quilt pattern using the quilt squares they built previously and using the process described above, using rotations. Each pair will glue their four quilt squares on a blank sheet of paper, to form the pattern. It's up to the teacher to supervise each pair's work, making sure that the process is correctly implemented and that all groups present a quilt pattern formed from both quarter-turn rotations (clockwise and anti-clockwise directions), as well as half-turn rotation. At the end, the teacher collects the work of all the pairs and, with the help of the students, builds an exhibition so that everyone can enjoy the diversity of patterns constructed, as well as assessing the correctness of the work carried out by the different pairs. 50 minutes Adapted from: Ontario Education (2005). A Guide to Effective Instruction in Mathematics – Kindergarten to Grade 3 – Geometry and Spatial Sense. Ministry of Education. Queens Printer for Ontario. | |
| Reflection | Pairs of students are asked to explain the construction of their pattern. Some questions to ask might be:
Teacher can provide some (e. g. a car making a left or right turn) and ask students to describe the rotation (is it a quarter turn, a half turn or a other type of turn?) 15 minute | |
| Notes | ||
| Digital Resource | ||
