Using     Ratios
	Class: Math Theme(s):

Divide any number in the Fibonacci sequence by the number before it, and the answer is always close to 1.61803. This is known as the Golden Ratio. Ratios are pairs of numbers used to make comparisons. Ratios can be written in different ways: 1 to 1.61803 or 1:1.61803, both meaning the same thing: the Golden Ratio.

Like cauliflowers, snails too are touched by nature's golden ratio. Draw a rectangle in the proportions of the golden ratio, then draw consecutively smaller rectangles within it. Join diagonal corners with an arc. The result is a perfect snail shell.

Giovanni Agostino da Lodi Adoration of the Shepherds About 1505 Oil on wood panel Enlarge

Renaissance artists used the Golden Ratio to determine beautiful proportions in painting compositions. The proportion of land to sky in landscapes was figured using the Golden Ratio, and rectangular paintings were planned using the Golden Ratio to determine the sides of the rectangle: 1 to 1.61803. The composition of the Adoration of the Shepherds (Giovanni Agostino da Lodi) conforms to the Golden Ratio, where the width of the painting equals 1 and the height of the painting equals 1.6. If a perfect square is determined using the measurement of the width of the painting, you'll see that the figures occupy the square while the landscape in the background occupies the portion above the square.

The beautiful proportions of the Golden Ratio are still used today. If you measure a credit card, you'll find it is a perfect golden rectangle.

Classroom Time
One to two 40 minute periods

Objectives
The students will:

• Understand the use of ratios in the art of the Renaissance.
• Use ratios to describe proportional situations.
• Successfully mix primary colors to get a secondary color.

Pennsylvania Academic Standards

Materials

• Clear plastic drinking straws, two per student
• Red, yellow and blue tempera paint
• Paper cups, two per student
• Wax paper or plastic palettes, one per student
• Mixing spoons, popsicle sticks or other mixing implements, one per student
• Paintbrushes, one per student
• 8" x 10" heavy watercolor paper, cut in half, one per student
• Pencils
• Rulers
• Permanent markers
• Paper towels
• Water

Optional Resources
Black and white paint could be used by students to mix shades and tints of a color.

Math Lesson

1. Setting the Stage
(Note: The background section of this lesson might serve as a handout to help students in this activity.)

Ratios are valuable to express the proportions used to create all kinds of things. For example, in the Renaissance, artists had to make their own paint by grinding pigments and adding egg and water for tempera paint or oil for oil paint. The proportions of pigment to oil or egg had to be correct to be sure that the paint would dry properly and not crack or flake off.

All of the colors that artists use are mixed from the three primary colors; red, yellow and blue, plus black and white. Mixing red and yellow results in the color orange but what kind of orange depends on the proportions of red to yellow paint. More red than yellow will result in red-orange, while more yellow than red will result in a yellow-orange. Today you will mix ten different variations of a secondary color, and record the ratios of primary colors you used to get each variation.

2. The Renaissance Connection
Instruct students to use pencils and a ruler to draw a grid of ten one-inch squares on their watercolor paper. Make the grid two rows high and five columns wide. Give each student two plastic drinking straws and instruct them to mark one-half inch on the end of each straw using the ruler and a permanent marker.

Each student should select two primary colors to use for their project. For easiest measuring, the paint should be the consistency of heavy cream. Once students have two paper cups of paint, show them how to use the straws to pick up one-half inch of paint by holding the straw in the paint and closing the open end with their finger. Each dip of the straw represents one unit of paint. Show students how to deposit the paint onto wax paper or a mixing palette by releasing their finger at the end of the straw.

Once students understand how to measure their paint, instruct them to mix ten different versions of the secondary color made up by the primary colors they chose. Each version of the secondary color should by painted into a square on their watercolor paper with a paintbrush and the ratio of primaries used to get the secondary should be recorded next to each square. Caution students to carefully rinse their brush in water after each paint application so that the ratios aren't polluted by paint left on the paint brush. It may be helpful for students to see a sample of a completed project.

3. Summary
Have students display their work in the classroom. Were they surprised at all the variation that could be achieved just be mixing two colors? Could they predict how much the colors would change with the different ratios? Were they ever surprised about the color they mixed?

4. Assessment
Did the student successfully mix two primary colors to achieve ten versions of a secondary color? Did the student record the ratios used for each secondary color next to its corresponding square on a grid?

5. Related Activity
Use a good quality reproduction of a Renaissance painting and ask students to match one color in the painting by mixing primary colors. Each attempt can be recorded with ratios, revealing the process of mixing students use to match the color in the painting.

Vocabulary
composition: the arrangement of shapes, forms, colors, areas of light and dark, and other elements in a work of art.

primary colors: red, yellow, and blue. All other colors are mixed from these three primary colors.

proportion: the relationship between two or more parts of a whole.

ratio: the relationship in quantity, amount, or size between two things.

secondary colors: colors formed by mixing together two primary colors. Red and yellow make orange, blue and yellow make green, and red and blue make violet.

Lesson Plans