Students will use mental imagery in the Quick Draw activity and be challenged to create a tiling with a given shape. They will use color to explore optical illusion based on an artwork by Josef Albers.

### Objectives

- We will enhance mental imagery skills through the Quick Draw activity.
- We will learn about various geometric shapes and their mathematical names.
- We will use transformational geometry to make a tiling of a given shape.
- We will learn a new geometric shape (concave hexagon) and its various orientations.
- We will explore optical illusion through an artwork that depicts an impossible figure as optical illusion.

### Assessment

- Use performance assessment by learning about each student through his or her responses to Quick Draw and by examining their tiling exercises.
- Quick Draw worksheet
- Tiling worksheets
- Parallel worksheet

### Time Required

1 hour

## Materials/Resources Needed

- Document camera/projector
- Quick Draw #1-3
- Art Images (linked below in the activities)
- Activity pages (8-14)
- Colored pencils or crayons
- Paper for drawing

## North Carolina Curriculum Alignment

After your field trip to the Asheville Art Museum, have your students talk about their visit. Encourage them to discuss artworks they saw, identifying which ones they liked the most/least and why. Ask them to talk about the studio activity and what they created.

## Activity One: Quick Draw

**NOTE:** *All pages referenced below are linked above in Materials/Resources Needed as “Activity pages.”* Quick Draw will be used as a class opener, with the whole class participating. Quick Draw was designed by Dr. Grayson H. Wheatley, Professor Emeritus of Mathematics Education at Florida State University (http://www.mathematicslearning.org). Quick Draw develops spatial sense, encourages the transformation of self-constructed images, and develops geometric intuitions through discussion.

- Provide each student one Quick Draw worksheet (activity page 8).
- Prepare for the activity by telling students that you are going to show them a shape for just three seconds and you want them to study it, building a mental picture of what they see. Avoid the temptation to show it for a longer period of time. It is important that students work from mental imagery rather than copying what they are seeing. Students will draw what they saw in the upper left box on their worksheet. Project Quick Draw #1 for three seconds. Tell students to draw what they saw in box #1 on their worksheet. After a few moments, show the shape again and let students adjust their drawings. If you feel it is necessary, briefly show the shape a third time. This should only be necessary for more complex figures or groups that are struggling. Two times is the norm, and three times is usually sufficient.
- Instruct students to put their pencils down. Show the shape so students can compare their drawing to the actual picture. With the image in view, ask students what they saw, how they drew the shape, and which part they drew first. Talking about mathematics encourages students to reflect on their imaging. Geometric language will be used naturally. You may wish to supply mathematical names for objects such as trapezoids and parallelograms as needed.
- Repeat the same procedure for Quick Draw #2 and #3. Collect papers or go over together as a class.

## Activity Two: Tiling

- Provide each student with four sheets of dot paper (activity pages 9–12). This is an individual activity, but students may want to compare their drawings with others as they are working.
- Project activity page 9 and point to the example of the L shape in the upper right corner. Tell students to draw the same L shape in the middle of their page 9 worksheet. Explain that this is the shape of a tile that will be used to make a pattern for tiling an imaginary floor. Explain that when tiling a floor you don’t want any overlaps, gaps, or holes, so when they draw their tile design the L shapes should fit together. Tell students that the goal is to make arrangements of tiles to form a pattern that can be extended across the whole floor. Tell them not to worry about the edges of the paper; we cut tiles when we get to an edge. Explain they can flip or turn the L tiles as they repeat them.
- Once most students have successfully filled the page with their repeating tiles, project activity page 10 and point to the example of the T shape in the upper right corner. Tell students to draw the T shape in the middle of their page 10 worksheet and repeat the same exercise, filling their paper with T shaped tiles.
- After activity pages 9 and 10 are completed, encourage students to use the same L and T shapes to create different patterns on activity pages 11 and 12. Students can use colored pencils or crayons to visually enhance their patterns. They can outline all their tiles in one color to resemble grout in real tile. Collect papers or go over together as a class.

Expect that some students will have difficulty with this activity. You may be surprised how difficult this is for some students and easy for others. Use as homework and as time permits, come back to this on subsequent days. Most students will have success given sufficient time.

## Activity Three (a): Parallels

- Project activity page 13. Ask students to look carefully at the cube. Discuss the concept of parallel planes. Use any box or cube shape available in your classroom to help illustrate this concept.
- Provide students with activity page 13 and ask them to color one pair of parallel faces (top and bottom, left and right, or front and back) with one color.
- Ask a few volunteers to hold up their work to show the class which parallel faces they colored. As students are sharing, ask the class to identify which pair of parallel faces are colored.
- Ask students to color the remaining two pairs of parallel faces, each pair with their own color. Top/bottom should be one color, left/right should be another color, and front/back should be a third color.
- When students have finished coloring, ask them to complete the questions on activity page 13. Collect papers or go over answers together as a class.

## Activity Three (b): Parallels

- Project
*Formulation: Articulation,**Folio II, Folder 10*by Josef Albers. Tell them to look carefully at this image and make observations. Ask students to describe what they see and if these two-dimensional figures could exist three-dimensionally. The perception of the figures will change with each individual, but neither figure could exist in three-dimensional space. - Provide each student with activity page 14. For each form, ask students to use one color to color any faces that they perceive to be parallel.
- Next, ask students if there are other parallel faces in either form and if so to color those with a second color. Invite students to share their work and discuss the parallel faces as a class. Collect worksheets.