This week I started playing around with the Learning Lab, the Smithsonian online platform that allows you to build collections to use with your students.

Check out the activity I made using the Learning Lab platform called Unpacking Sol LeWitt’s open cubes. The activity allows students to apply what they have learned about drawing 3D shapes and nets. Since making the activity, I have not looked at isometric paper the same. Looking through his variations of open cubes exercised my visualization skills of 3D objects. Noticing which lines were gave the shape more a of 2D feel or 3D feel.

This activity has students make connections between the planning phases, chair design then links to the optical illusions that OK GO! (Smithsonian Ingenuity Award winners) uses in their music video “The Writing’s on the Wall”

Enjoy, and please send your feedback.

Here is the teacher resource for the activity:

# Unpacking Sol LeWitt’s Open Cubes

By Amanda Riske

Students will analyze Sol LeWitt’s variations of the open cube to apply their knowledge of drawing cubes and nets of cubes. Students will extend their knowledge of surface area while observing LeWitt’s Cube without a cube and make generalizations of two formulas. This is an activity for a grade 6 or 7 geometry class. Prerequisite knowledge: volume, surface area and nets of cubes.

# Common Core Standards

**6th grade- Geometry**

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

**7th grade- Geometry **

#### Draw construct, and describe geometrical figures and describe the relationships between them.

#### Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

This activity can be modified for High School students as well.

Geometry HS

Visualize relationships between two-dimensional and three-dimensional objects

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Permutations & Combinations possibilities

# 21st Century skills

Visual Literacy

Visual Literacy is the ability to interpret, use, appreciate, and create images and video using both conventional and 21st century media in ways that advance thinking, decision-making, communication, and learning

Critical Thinking

Higher-Order Thinking and Sound Reasoning include the cognitive processes of analysis, comparison, inference/interpretation, evaluation, and synthesis applied to a range of academic domains and problem-solving contexts

# Activity

### Resources required:

Isometric paper

Dot paper

Rulers

3-colored pencils

*Tinker Toys or Goobi Construction set– if needed

Technology

Video projector and audio

Computer or tablets for students

Geogebra (possibly)

## Part 1: Learning Lab Activity (50-60 minutes)

### Slide 1: Sol LeWitts “Variations of open cubes” (10 minutes)

Using the Learning Lab shared resource Sol LeWitts “Variations of open cubes” have students do the thinking routine Think, Puzzle, Explore for the piece in groups of 2-3.

- What do you
**think**you know about this piece? - What questions or
**puzzles**do you have about Sol LeWitt’s process? - How can you
**explore**this topic further?

Encourage students in groups of 2-3 to look through his organization.

Possible questions that could come out of discussion:

- What other 2D shapes came out of his 3D study?
- Could you do the same study on the net of the shape? Would have the same effect?
- Why does he start at 3 lines?
- Does he include all possibilities of the edges of cube? (possible introduction of combinations and permutations, challenge to find all the possibilities and perform a similar study of another prism, hexagon based prism)
- What other 3D objects could you explore?

### Slide 2- Incomplete Open Cube (10 minutes)

Students will be prompted in the activity to answer the questions about the sculpture:

- Was this variation included in Sol LeWitt’s plans?
- How many edges?
- Is it in the same orientation?
- Draw the 3D of this cube.
- How would you express this cube in a net? Draw your net.
- What are the other orientations could this sculpture have?
- Would that be the same value for other structure from the same edge class?

### Slide 3: Sol LeWitts “Variations of open cubes” (10 minutes)

This slide is repeated so students can easily go back and forth for the next open cube.

Before going to the next example as a class discuss LeWitts parameters to his variations.

Important vocabulary: parallel lines, skew lines, & vertices

Depending on the level of the students you can discuss permutations vs. combinations.

- How many lines are in a cube?
- What does 3/1, 3/2, 3/3 mean?
- You could have more advanced students calculate the amount of combinations of 12 choose 3 or C312and compare with the 3 iterations that LeWitt has recorded.

After then next open cube students discuss why LeWitt only considered sculptures with at least one edge from each axis (x, y, & z). Why did all the edges have to connect in some way?

See Extensions for more on permutations.

### Slide 4 Incomplete Open Cube (10 minutes)

Students will be prompted in the activity to answer the questions about the sculpture:

- Was this variation included in Sol LeWitt’s plans?
- Is it in the same orientation?
- Draw the 3D of this cube.
- How would you express this cube in a net? Draw your net.
- What are the other orientations could this sculpture have?
- Would that be the same value for other structure from the same edge class?

### Slide 5: Chair (5 minutes)

Students will be prompted in the activity to answer the questions about the sculpture:

- Was this variation included in Sol LeWitt’s plans for an open cube?
- Draw the 3D of this cube.
- What are you considering as the edges?
- How would you express this cube in a net? Draw your net.
- What are the other orientations could this chair have?

# Demonstration of knowledge

### Slide 6: Wall Drawing #356 Isometric figure within which are 3’’ wide black lines in three directions. (Cube without a cube) (10-15 minutes)

Demonstration of knowledge

- Students will use isometric paper to sketch the 3D version of the (Cube without a cube).
- Find the volume of the object and justify their answer.
- They will need to determine the side lengths

- Students can use dot paper to draw the net of the Cube without a cube. (if they need to construct using blocks if need be)
- Find the surface area of the object and justify their answer.

### Extension:

Make general formulas for finding the volume and surface of any size of cube without a cube. Define your variables.

## Part 2 (possibly day 2 or a follow up activity) 30 minutes

The music video *The Writing’s on the Wall* by OK GO! (winners of the Smithsonian Ingenuity Award in 2016) is a play between 2D depictions of 3D spaces to create illusion for the audience. Throughout the video the band members are stretching our perception of reality through optical illusions. The illusions are revealed to the audience through changes in angles of the camera. This is a great analogy to the thinking that students develop in math class; an appreciation of perspective taking and how slightly changing approach or the angle we look at a problem can allow us to see the solution. There is also opportunity to partner with the students’ art teacher for students to plan and develop an art work on an optical illusion through math.

- Watch music video The Writing’s on the Wall by OK GO! (5 min)

While watching the video have students jot down examples or plays on Sol LeWitt’s sculptures. After the video write down discuss with the students and collectively the examples (if you can also record the timing). Watch the video a second time together to generate more ideas as the illusions happen quickly.

- Have students pick one of the examples they gave to do the Thinking Routine Parts, People & Interactions in groups of 2-3 students.

Here are a few examples of illusions that involve cubes:

1:01 Isometric paper

1:25 Cubes with black edges

1:47 Cube in center that is 2D with only 3 edges.

3:00 hugging the pole

3:13 Walking up the yellow steps

3:23 Walking on grid

## Extensions for HS:

Putting Sol LeWitt’s variations to a test. Are any of his variations doubled up? How many variations does he not consider because they do not work for his sculpture?

Is there a relationship with the amount of vertices and the amount of edges in each variation? Do they have to be the same?

Students can use combinations formulas or excel to find the possible combinations. Here is an example of an spreadsheet exploring the combinations.

Doing similar variation of work with a prism with a different base.

Possible extensions to IB HL Option Discrete Mathematics with networking.

## Modifications:

If students have a hard time understanding a 2D depiction of a 3D object have them make the 3D object first using Tinker Toys or Goobi Construction set. Then the students can work on the drawing with isometric paper.