Divine Proportions: Teaching Ratios

Students sometimes wonder when they’ll use certain information that they learn in school. They may not reap immediate benefits from reading Herman Melville or learning the laws of stoichiometry, but they usually take the teachers at their word that such knowledge is never wasted – it’s all part of the process of becoming educated citizens. In math, however, the benefits of learning ratio and proportion are instantly recognizable. People use ratios every day in all sorts of situations, and are often even unaware that they’re doing so. Ratio and proportion are so prevalent in daily life that their use has become reflexive.

Ratios are a way to show the relationship between two different things. Elementary classes often illustrate the concepts of ratio by comparing the numbers of boys to girls in class, noting the number of right- or left-handed students, and so forth. Older students begin to use ratios in situations that they’ll encounter in daily life, such as figuring out prices per unit, measurement scales, rates of speed, hourly rates of pay, and much more.

Proportion is a way of comparing two ratios; when two ratios are equal, they are said to be in proportion. Some of the most famous ratios are the Fibonucci Sequence, where every number in the sequence (after the second) is the sum of the previous two numbers. In turn, The Golden Ratio (also called the Divine Proportion) is based on the Fibonucci numbers, and reveals a pleasing symmetry based on an object that divides a line at such a place where the sum of the quantities of the larger part is equal to the ratio of the larger part to the smaller part. The Golden Ratio occurs prominently in nature in leaves, seed patterns, the formation of pinecone scales, etc. Leonardo da Vinci’s drawing of the Vitruvian Man uses the Golden Ratio of 1.618033…, and is considered to show the human form in perfect proportion, where the distance from navel to foot is in complete proportion to the distance from navel to head. Moreover, people whose faces conform to the Golden Ratio are considered to be the most beautiful; recent scientific research has confirmed what artists such as da Vinci, Michelangelo, and others figured out long ago.

This week I’ve highlighted three resources on ratios and proportion for a variety of grade levels. As always, I’ll be featuring many more activities, lesson plans, and other resources on the topic throughout the week on our Twitter and Facebook pages, so be sure to check those pages often. Also, please take a look at Peggy’s companion column (linked below) for ideas on teaching ratios in your classroom.

Crazy Putty Ratio
Subjects: Physical Sciences, Measurement, Language Arts
Grade: 3-5
Students mix various ratios of liquid starch and glue to make "Crazy Putty" (their variation of Silly Putty), using knowledge of measurements and ratios. They chart their ratios, make observations, and write summary of activity. This activity was produced by Beacon Learning Center, an online educational resource and professional development center. Beacon Learning Center offers standards-based resources and professional development activities.

Relative Ratios
Subjects: Math, Science
Grade: 6-8
This is an interactive, media-rich lesson in which students examine proportions. By experiment, students will discover that proportions are made up of equal ratios each containing the same building blocks. Using video, students will discover that proportions are direct relationships, and understand how they are used to estimate quantities and find missing numbers. Using the Internet, students will apply their knowledge of direct relationships and proportions to problems in planetary science. Finally, students will demonstrate their ability to use proportions by determining if human bones are a good predictor of height. This lesson utilizes reading, writing, and mathematical skills in an interdisciplinary format. This lesson is offered by Thirteen Ed Online, the educational Web component of WNET, PBS's flagship station in New York. This free service features everything from standards-based lesson plans and classroom activities to a multimedia primer, online mentors, and reviews of curriculum-based Web sites.

I Can Determine the Height of a Rocket!
Subjects: Algebra, Trigonometry
Grade: 9-12
The lesson is intended to give students a fun real-world experience in applying their math skills. They will use trigonometric ratios to calculate heights of tall structures. They will also use the Internet to convert their calculations from standard to metric units and vice versa. This lesson was produced by the Alabama Learning Exchange (ALEX), an award-winning education portal that provides lesson plans, education-related podcasts, best practices, and Alabama professional development activities.