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Analyzing Designs

3-5
1

This lesson encourages students to explore the geometric transformation of rotation, reflection and translation more fully. Students create a design then, using flips, turns, and slides, make a 4-part paper "mini-quilt" square with that design as the basis. This experience focuses students’ attention on both the changes produced by the geometric transformations and on line symmetry.

Choose a quilt block from the Quilt Blocks Activity Sheet. (Alternatively, you can search the Web or some other source for other quilt block patterns.)

Give each child four black‑and‑white copies of the selected quilt square and crayons or markers. Then, ask students to fold one square on its main diagonal. When students color, instruct them to color such that if we were to fold the square back up, like colors would touch. The remaining three quilt squares should be colored the same way.
As students work, circulate the room to ensure that directions are being followed properly (be sure to have extra quilt squares in case students make mistakes). Also, provide each student with a work mat large enough to fit all four quilt squares, such as a poster board.

Ask the children to place one of the colored squares on the top left-hand corner, and then to place an identical square in the same orientation on top of it. Have students slide the identical square to the right (edges touching) and rotate it one half‑turn, or 180 degrees. (From the start, caution students to ensure that the top square is in the same orientation as the square below it. This is often where students make a mistake, which will disrupt the entire design.)

Then, have them place a third square face down (so that no colors/designs are showing) on top of the first square. Ask students to overlay the squares such that like colors are touching. Have students flip it down directly below, edges touching.

Finally, instruct them to place their last quilt square on top of the upper right-hand square (so that like parts are overlapping), and turn it a quarter‑turn (clockwise), and place it directly below.

The result of student designs should be a four‑square quilt with rotational symmetry, as shown below. Encourage students to discuss their resulting designs and the effects of the flips, turns, and slides.

Now give each child a small mirror and ask each to place it on the design in various places to see if the design shows in the mirror what is on the "dark" side of the mirror. (Individual mirrors for each child will aid in the investigation as the children find reflections in the resulting design, but if you do not have enough mirrors for all the students, you may wish to have them work in small groups.) You might ask them to focus on the left half of the quilt, or focus on the right half of the quilt. [They will be the upside‑down copies of each other]. Then compare the squares in the top half of the quilt, then the bottom half. [They will be right‑left copies of each other.] Introducing the term line symmetry, encourage the children to find as many examples of it as they can in the four‑part design they created. They may wish to glue the completed design onto a piece of paper, marking each line of symmetry with a dark crayon line.

When the children are ready, call them together to share their designs and describe how each of the squares in it can be obtained by another square using flips, slides and turns. You may wish to encourage alternate explanations. Then have students discuss the lines of symmetry they found in their designs. You may wish to ask the students to draw a sketch and reflect upon the meaning of line symmetry.

• 4 black and white copies of the selected quilt square for each student
• Crayons or markers
• 6-inch by 6-inch work mat divided into 4 equal parts, one per student
• Small mirrors
• Glue
• Large pieces of paper

Assessment Options

1. Use verbal and/or written explanations to assess the students. At this stage of the unit, it is important for students to know:
• transformation terms
• how to follow a sequence of directions concerning transformations
• find lines of symmetry
2. The guiding questions listed above may provide information that will help you assess the students’ current level of knowledge in this area.
3. You may also wish to keep a copy of the students’ designs and written reflections.
4. Checking to see if students can state how the mathematical skills they have learned in previous lessons enabled them to complete this one will help them develop a value for the mathematics they are learning.

Extensions

1. Samples of other quilt squares in which line symmetry is evident can be seen on various websites. Students can search for such websites and look for examples of line symmetry in quilts.
2. Move on to the last lesson, Planning and Making a Mini-Quilt.

Questions for Students

1. When you made your design what move did you use first? Then what moves?
2. Are any of the blocks turned half way from another block?
3. When figures have a line of symmetry, what is alike between the two sides? Is anything different?
4. Have you ever seen line symmetry before today? Where?

Teacher Reflection

• Which students met all the objectives of this lesson? What extension activities are appropriate for these students? What evidence did you collect to document achievement of the learning targets?
• Which students did not meet the objectives of this lesson? What instructional experiences do they need next? What mathematical ideas need clarification? What evidence did you collect to document lack of achievement of the learning targets?
• What adjustments would you make the next time you teach this lesson?

Parts of a Square

3-5
Students investigate the ways shapes can be divided into equal pieces with one or two cuts. The lesson provides a review of the following vocabulary terms: square, triangle, and rectangle; congruent, one-half, and one-fourth. The other lessons in this unit build on this introductory lesson.

Describing Designs

3-5
Students create designs and describe them to a classmate, using fractional and geometric terms. This lesson provides a context for using the following vocabulary terms: horizontal, vertical, square, rectangle, triangle, diagonal, one-half, and one-fourth.

Exploring Flips and Slides

3-5
This lesson builds on the previous two lessons and encourages students to explore the geometric transformations of reflections and translations. Students create a design. Then, using flips and slides, students make a four-part paper "mini-quilt." While the formal terms are reflection and translation, the more informal terms slide and flip are used at this stage. The experience focuses students’ attention on the changes these geometric transformations make in a student-designed quilt square.

Exploring Turns

3-5
This lesson encourages students to explore the geometric transformation of rotation. Students create a design then, using turns, make a four‑part paper "mini-quilt" with that design as the basis. While the formal term is rotation, the more informal turn is used at this grade band. The experience focuses students’ attention on the changes the geometric transformations make in a student-designed quilt square.

Planning and Making a Mini-Quilt

3-5
Students create a quilt using the three transformations and investigate the ways shapes can be colored to show one-half and one-fourth. They locate lines of symmetry and create a set of directions for making their quilt block.

Learning Objectives

Students will:
• Explore the results of flips, slides, and turns.
• Explore line symmetry and identify it in a design.
• Be able to name the geometric transformations used to create a given design using informal language.

NCTM Standards and Expectations

• Describe location and movement using common language and geometric vocabulary.
• Predict and describe the results of sliding, flipping, and turning two-dimensional shapes.