## 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.)

 Quilt Blocks
Give each child four black‑and‑white copies of the selected quilt square and crayons or markers. Then ask students to color one of the squares using any colors they wish. Next, ask students to color the other three squares in the same way. Now provide each child with a 6" × 6" work mat which has been divided into four equal parts, as was done in the previous two lessons.

Ask the children to place one of the colored squares in position 1, and then to place an identical square in the same orientation on top of it. Have students slide the identical square to position 2 and rotate it one half‑turn. (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 on top of the square in position 1 and flip it down to position 4. Finally, instruct them to turn a square that is face up in position 2 a half‑turn and place it in position 3.

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 squares in positions 1 and 4 or focus on those in positions 2 and 3. [They will be the upside‑down copies of each other]. Then compare the squares in positions 2 and 4 or 1 and 3. [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 workmat divided into 4 equal parts, one per student
• Small mirrors
• Glue
• Large pieces of paper

Assessments

1. 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

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.

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. It 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. It 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 reflection and translation. Students create a design then, using flips and slides, make a four-part paper "mini-quilt" using that design as the basis. 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.