Tumbling Blocks Quilt Algebra

Visual algebra is an educational approach that uses concrete models, diagrams, and pictures to build a deeper, conceptual understanding of algebraic concepts, rather than relying solely on memorization of symbols and rules. By connecting visual representations, like tiles or geometric shapes, to symbolic manipulations and graphs, visual algebra aims to clarify the logic behind algebraic processes and make them more intuitive and accessible.

The tumbling blocks quilt pattern creates an optical illusion of three-dimensional cubes tumbling across the surface, achieved by arranging light, medium, and dark colors of rhombi to create a 3D effect. The physical, or visual block patterns can be used to represent and explore algebraic ideas.

The aim of this project is to explore visual algebra, art and crafts (ex. quilts). Attached is a math-focused handout for teachers with questions and answers.

This project incorporates the CRA Model (Concrete, Representational, and Abstract) that includes three stages:

1. Concrete (C): Hands-On
2. Representational (R): Visual
3. Abstract (A): Symbolic

Tinkercad 3D Design to complete the pattern making task

Shape SVG files (diamond, cube, star)

Physical foam shapes or pattern blocks

Fabric of a Nation: American Quilt Stories (book)

Internet/WiFi for websites such as Cooper-Hewitt

For grades K-8+: Read the book Fabric of a Nation: American Quilt Stories, which features 58 works of textile art that express the personal narratives of their makers and owners and connect to broader stories of global trade, immigration, industry, marginalization, and territorial and cultural expansion.

Students can visit the Cooper-Hewitt website to learn about tumbling blocks, a pattern that "has a long history in the grammar of ornament" that dates as far back as ancient Greece but may have been introduced into quilt making in the late 19th century.

These "readings" can be used as lead-ins for concrete, hands-on projects.

Concrete (C): Hands-On Activity: In the concrete stage, students engage in hands-on learning experiences using physical manipulatives (objects). For example, the following activities reference a "stars and blocks" pattern from 1928.

Students can use physical foam shapes or pattern blocks to create a motif (design) or pattern -- see example image. Note: If you have a 3D printer or laser cutter you can print/cut pieces before classes begin.

and/or

Students can use Tinkercad, a free, online 3D modeling program to create their own designs and patterns.

Instructions for Tinkercad:

1. Launch Tinkercad
2. Click on the blue "Create +" and "3D Design" (lower right side of your window)
3. Import the attached SVG files (diamond, cube, star)
4. Change the color and shades of each imported diamond (light, middle, dark colors)
5. Use the rotate tool: Click on the object to select it, then click and drag one of the three arc rotation handles (bendy arrows) that appear. Use the outer ring for fine-tuning by individual degrees, or move the cursor closer to the center of the ring to snap rotations in 22.5° increments. Hold the Shift key to snap rotations in 45° increments, or input a specific degree value into the box that appears.
6. Rotate the diamond shapes to create a 3D cube
7. Select and bundle the diamond shapes (cube)
8. Shift-click and drag one of the points to re-size the cube, if necessary
9. Click "Duplicate and repeat" button (top left) and create 6 cubes
10. Use the rotate tool to move and reposition the cubes around the star.
11. Select all, duplicate and repeat the group to create a pattern

Tumbling blocks, also known as rhombille tiling, reverse cubes, or cubework, is a popular and very effective optical illusion used in tile and textile design, both in the past and the present. This pattern can also be found in several artworks and other media such as video games. Some examples include:

1. Quilts (see previous step)
2. Clothing (ex. sweater)
3. Sculpture
4. Q*bert games
5. Pottery/ceramics

Representational (R): Visual Activity: In this stage, students transition from physical or digital manipulatives to visual representations. For example, students can practice close-looking that encourages deep, mindful observation of a work of art, focusing on details, visual elements, and the choices an artist makes to convey ideas or meaning. Some examples of questions to ask students include:

1. What specific details do you notice in the artwork?
2. What is the central idea or theme of the artwork?
3. Where and when was this artwork created, and in what historical context?
4. How would you describe this artwork to someone who has never seen it?

You can use the attached slides as examples.

Using visual representations—like close-looking at artworks—can help students understand abstract mathematical ideas. The final stage of the CRA approach is where students understand that math can stand in for the concrete objects and visual representations they’ve encountered earlier.

Abstract (A): Symbolic Representational Activity: In the abstract stage, students work with numbers and symbols to represent algebraic concepts. The term "tumbling blocks" has some applications in visual algebra. Students can use visual algebra to answer the following question:

What is the best way to figure out how many cubes (or clusters) you need to use to create a pattern for a tumbling blocks quilt?

Students can build sequences of cube structures and analyze the pattern's growth. For instance, they might be challenged to build the next structure in a series.

After building a pattern, students can derive algebraic rules that describe the sequence. This can involve writing recursive formulas (e.g., "add 3 blocks to the previous step") and explicit formulas (e.g., "3𝑥+5").

The attached slides provide a few tasks to assist students in answering this question. The slides cover the following:

1. Building and growing patterns
2. Describing patterns
3. Linear expressions
4. Combining linear expressions

As a final activity, students can participate in a "gallery walk" to see and provide feedback on finished projects.

This project is an artifact of Culturally Relevant Math Expressions created with Lesley STEAM Learning Lab at Lesley University and made possible by the generous support of Takeda US CSR & Philanthropy.