Students will solve a challenging three-dimensional geometry problem, investigate and discuss a Josef Albers’ artwork, and replicate a Josef Albers design by measuring and drawing.
- We will enhance our mental imagery through a Quick Draw activity
- We will solve the painted cube problem and identify our solutions
- Through close looking and discussion, we will interpret meaning from an abstract artwork
- Participation in class discussions
- Participation in The Painted Cube activity
- Quick Draw worksheet
- The Painted Cube worksheet
- Homage to the Square worksheet
- Document camera/projector
- Art images (linked below in the activities)
- Activity pages (1–6)
- Quick Draw #1–3
- 20 snap cubes per student
- Colored pencils or crayons
North Carolina Curriculum Alignment
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 1).
- 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: The Painted Cube
NOTE: All pages referenced below are linked above in Materials/Resources Needed as “Activity pages.”
1. Project Jon Kuhn’s Crystal Cream as inspiration for the activity. Ask students to describe what they see. If students describe the form as a cube ask them how they know it’s a cube and to define a cube.
2. Organize students in pairs, like ability works best. Provide students with The Painted Cube worksheet (activity page 2) and 20 snap cubes each (or for each pair if you don’t have enough).
3. Ask students to work in pairs to solve the following problem and explain their solution in writing: A large cube is formed from smaller individual cubes. Four of the smaller cubes fit along each edge of the large cube. Imagine if the 4 by 4 by 4 large cube is dipped in paint. How many of the individual cubes will have paint on them?
4. Walk around and observe your students, helping those that need assistance. You can prompt them by asking if they are counting cubes or faces of cubes.
5. After most pairs of students have a solution, have the class come together and ask two pairs to explain in front of the class how they solved the problem. If another pair has a different solution with the correct answer allow them to explain too. Point out that some problems have multiple correct solutions.
Activity Extension #1: Ask students to determine how many individual cubes there are.
Activity Extension #2: Ask students how they could sort the individual cubes into like families if they took the cube apart once the paint dried. What criteria would students use to sort them?
SOLUTION FOR TEACHER REFERENCE: NOT TO BE EXPLAINED TO STUDENTS
The large cube is composed of 64 cubes, four layers of 16 cubes each. Of these 64 cubes, 56 have paint on them.
Solution 1. 16 on the top layer and 16 on the bottom layer are painted. Front and back then have 8 with paint on them. When these cubes (not faces) are considered, there are only 4 cubes in the middle of the sides that have paint on them. Thus 16 + 16 + 8 + 8 + 4 + 4 = 56.
Solution 2. In a 4 by 4 by 4 cube, there is in the middle of this large cube, a 2 by 2 by 2 cube that is not painted. When these 8 cubes are subtracted from the total of 64 cubes the difference is 56 cubes.
There are numerous other ways of solving the problem. Please be open to other methods and avoid imposing or even emphasizing one particular method.
Activity Three: Homage to the Square
NOTE: Read the artist biography of Josef Albers with your students before you begin the lesson. All pages referenced below are linked above in Materials/Resources Needed as “Activity pages.”
1. Project Josef Albers’ Formulation: Articulation, Folio II, Folder 13. Have students look carefully and discuss what they see. This should be a lively discussion. Discussion of color and the interaction of the colors might arise.
2. Provide students with Homage to the Square worksheets (activity pages 3-6). Students can begin by completing questions #1-10. The directions and answers are below. Answers have not been provided for those questions whose responses are subjective. Question #6 may require some class discussion.
- What is a square? A figure with four equal sides and four right angles.
- How many squares do you see in this artwork? Possible responses: 2 sets of four, eight
- What words could you use to describe the colors of these squares?
- Would you want your bedroom painted like this?
- Why or why not?
- Compare the relationship between the widths of the borders at the sides of the squares to the widths of the borders at the bottom of the squares. How are they similar? How are they different? The width of the side borders are twice the size of the width of the bottom border.
- In the space provided on the next page, draw one square inside another so that the distance between the squares at the sides (the side border) is twice the width of the border at the bottom, just like Albers’ squares. Use a ruler and draw carefully. Comment: This is likely to be challenging for the students. They have to measure and draw precisely.
- Measure the distance of the border at the top. Is it related to the other two distances? How? The top border is three times the size of the bottom border.
- Add a third square to your drawing that is outside the second, so that the border at the bottom is equal to the border just above it and the side borders are the same, just like in Albers’ squares.
- Using colored pencils or crayons, fill the squares with colors from the same color family. In the example, this color wheel shows four colors for each color family, but students do not need to limit themselves to just these colors. Any values (lighter or darker versions) or a color can be in a color family. If students don’t have many colors to choose from, they can create multiple values of one color by pressing down lighter or harder when coloring.
3. Take time to review students drawings individually while walking around the room. When most have completed their first drawings, the class can move on to questions #11–13.
- In the space provided on the next page, draw a large square. Draw a second square inside the first one so that the border at the sides is twice the width of the border at the bottom. Comment: It is not easy to draw a square with just a ruler and pencil. They must measure carefully. How will students form right angles? Not easy.
- Try to draw another square inside the second square meeting the same conditions. Can you draw a fourth square inside the third square? Answer depends on the squares they drew and borders used. It is possible that only one square can be drawn or it could be more.
- Using colored pencils or crayons, fill the squares with colors from the same color family.