Share Share
Ratios In and Of the World

Life Skills:

  • Communication

Time Frame:
10 class periods that run 45 minutes each.

Group Size:
Small Groups


  

Summary:
This unit is designed to help students explore ratios in various contexts and from different perspectives.

Materials:
Worksheets: Fibonacci sequence for each student Unit Project Sheet/Rubric for students Compass, protracter, straight edge One set for every two students ••• Transparencies ••• Transparency pens ••• 'Middle Rulers' ••• Dice Overhead projector TI-92 Graphing Calculator (with overhead projector if one is available) TI Graph Link and its Software

Background For Teachers:
This unit is designed to be interactive and hands on. It will include daily work as well as a unit project. The unit project is designed to be completed by individuals with help from their families encouraged, but it could easily be turned into small group projects. Students should be allowed 5 to 10 minutes to present their projects.

Intended Learning Outcomes:
Students will use ratios to find unknown measurements of similar objects. Students will explore from at least three different perspectives to gain an appreciation for the diversity of this concept. Students will then use ratios in a hands-on application.

Instructional Procedures:
Use as much of the Fibonacci materials as you have time for, especially the construction of the spiral. Students will explore the relationship between the numbers in the Fibonacci Sequence to develop a ratio. This will be a discovery, co-operative group activity that will link history, Fibonacci and the architecture of Greece, while introducing the basic idea of ratio. Have the students in small groups of four. Each group will choose three angles that they can use to construct a triangle. (The sum of the angles must be 180°) Using large sheets of paper, the groups will split into two subgroups and independently construct triangles using their angles. The groups will rejoin, measure the sides of their triangles, and make conjectures as to the relationship of the sides of the triangles.

The teacher will then give the students angles for triangles to be constructed by the individual groups. Measurements of the sides will be made and a table of these measurements displayed on an overhead or the board. The groups will investigate the relationship of these numbers and write a paragraph on their conclusion and how they arrived at it. (This can also be used as an assignment) Two triangles can be proven congruent by: SSS, SAS, ASA, AAS, RHL (hypotenuese and one leg of a right triangle.) Question for the day: What is needed for two triangles to be similar? For each conjecture, the student groups should show what they did to come to the conclusion, and show that their conjectures do lead to similar triangles. (How do they know if they are similar? Definition of similar: equal angles and proportional sides.) This step introduces pi as a ratio of the circumference of a circle to the diameter of the circle.

Have each student bring a circle to class. Encourage them to bring larger items so they can be measured more easily. Provide string (crochet thread or yarn), scissors, and rulers (not tape measures because you want to emphasize that the circumference is a linear measure).

Working in groups of four, the students will complete the attached chart. They will find the average of their decimal numbers. You then average all of the averages together and ask them what number the class average reminds them of. We then refer back to the Greeks and their fascination for ratios to discuss the origin of pi.

An interesting side demonstration is to take a circle of circumference one, use diameters and chords to create 6 equilateral triangles, find their total area. Then divide it into 12 triangles and find the sum of their areas and watch the area converge to pi. Students will play the Chaos game to gain a further understanding of self-similar and again explore the use of ratio, this time with fractals. Use transparencies for the students pairs, premarked with the vertex points in the same places on each transparency. After demonstrating the game on an overhead, have the students do at least 30 points on their own transparencies.

After they have carefully constructed their transparencies, ask them if they can see anything happening with their triangles. Eventually they come up with the fact that the centers are fairly clear. Then they start finding other 'holes'. Then start putting transparencies on top of each other and Sierpinski's Triangle really comes out. Stack all of the transparencies together on an overhead and then discuss randomness, self-similar, and the idea of ratio with the smaller triangles.

Assignment: construct Sierpinski's triangle by connecting the midpoints of the trianges repeatedly. Color the triangles to emphasize the fractal.

Students may also use the TI-92 program to construct Sierpinski's Triange using random numbers. Set the window to X:[0,1] and Y:[0,1] and type in sierpin() and enter. The Pantheon, a temple that has been called the pinnacle of Roman architecture, has three elementary dimensions on the facade: diameter of the Pantheon's facade; 'clear distance' or space between the columns; and height of the columns. These are in the ratio of 1 to 2 to 9.5. According to Hermogenes, one of the most celebrated architects of the Hellenistic age, this ratio belongs to an ideal facade; thus, we have a numerical recipe for beauty.

A number of student activities can be linked to this introduction:

  • Reconstructing the facade on the football field
  • Construction buildings using their own ratio
  • Linking various similar polygons with the ratio of the sides as was done in step 2
  • Introduce food recipes and use ratios to increase the receipes and then make the recipe
As an alternative assessment, students will design and build a home using ratios.

Web Sites

Author:
NANCY THOMAS

Created Date :
Feb 17 1998 16:20 PM
 8570