THERE'S GOLD IN THEM THAR RATIOS
Grades 7-10

Patterns are everywhere. Patterns help us make sense of our world and help us feel that we have some control over our lives and environment. Pattern recognition helps us create, explain, and predict. In this lesson students will use inductive logic, which is reasoning based on pattern recognition that can lead to predictions. Students will model the initial problem that Fibonacci used to generate his famous sequence and will find his numbers occurring in nature, art, music, architecture and the human body. Students will investigate the golden ratio relationship found in the Fibonacci Sequence, in golden rectangles, and in golden triangles. Students will also find that assumptions based on patterns are not always true. Students will discover that people use inductive logic everyday in their lives, sometimes making incorrect conclusions. By watching segments from a video about an over protective mother, students will discover that false conclusions are made when one's perspective is skewed. This investigation can take one to two days to complete.

Math Vantage #1: Discovering Patterns
Math Vantage #3: Sequences and Ratios
Wishbone #121: The Canine Cure

Students will be able to:
1. draw a model of the bunny problem which generates the Fibonacci Sequence.
2. draw spirals generated from golden rectangles and golden triangles.
3. identify the golden ratio in the human body.
4. find the Fibonacci numbers in nature.
5. investigate situations where false conclusions are made from pattern recognition.
6. analyze why some art is drawn in a ratio other than the golden ratio.
Texas Assessment of Academic Skills (TAAS), Exit Level
Math Objectives:
#3: Demonstrate understanding of geometric properties and relationships.
#12: Express or solve problems using mathematical representation.
#13: Evaluate the reasonableness of a solution to a problem situation.

NCTM Standards for Grades 9-12
Standard 1: Mathematics as Problem Solving
Standard 3: Mathematics as Reasoning
Standard 7: Geometric Form and Synthetic Perspective
Standard 14: Mathematical Structure

Per student:

• ruler
• graph paper
• notebook paper
• pencils

Per group of 4 students:

• tape measure
• calculator
• protractor
• Nautilus shell
• pineapple
• pine cone
• open-faced flowers (wild flowers if possible)
• branches from trees and shrubs, with leaves still attached
• Sunday cartoon section from the newspaper
• clothing advertisements with drawn illustrations of models
• rubber plant (one plant will do for the entire class)

As they enter the room, have students work in groups of four to answer the following questions:
1. What is the first thing a newborn baby notices?
2. What is the most common thing put on baby toys?
Tell them that the answer is the same for both questions.
(a face) Ask them why this is the first pattern that a human notices. Ask students if the eyes are in the middle of the face. (no) Tell students that babies can focus as far away as the mother's face and can see the contrast of the eyes. Ask students what generally happens when the mother holds the baby. (baby is fed, cuddled, warmed) The baby associates a feeling of security with the mother's face; hence, a face is placed on baby toys.
Have each student work with a partner in the group to measure the length of their face from the eyes to the chin in millimeters. This will become the denominator of a fraction. Now have each measure the length of their partner's face from the top of the forehead (hairline) to their chin. This will become the numerator. Ask the students to divide the numerator by the denominator. Have each group of four students average their findings by adding up the decimals (round to the nearest hundredth) and dividing by four. Take a class average. Is the average close to 1.6? If it is, tell them that their faces are in the golden ratio. Tell them that they are to begin an investigation of how this ratio shows up in nature and in some of our most pleasing art and architecture. They will also investigate some situations that lead to false conclusions even though they are determined by a pattern.

Ask students how important patterns are in our lives. Have them list some important patterns in their daily lives. Ask them if they have any certainties they can expect in the future. Accept any reasonable answers. Ask them if they can be sure of anything. Tell students that they will watch a video segment about how patterns are used. To give students a specific responsibility while viewing, ask them to determine how patterns are used to generate a laser light show. Students will list three important uses of pattern recognition. Students will also determine why assumptions made by pattern recognition are not always correct.

BEGIN the Math Vantage video after the narrator points to the planetarium sign when the laser light show begins. PAUSE video after the narrator says, "It might look like a cube cut out of a block." Ask students to "see" the picture both ways. Ask students what this might have to do with making predictions with patterns. (They might not be looking at the entire picture, enough of the pattern to get the big picture, or in the proper orientation.) RESUME the video.
PAUSE at the picture of the old woman that is also a young woman when the narrator says, "...young woman in a hat." Ask students to focus on the picture to see both pictures in one. Again emphasize the importance of viewpoint when making a conclusion. A person could be wrong. RESUME the video. Have students call out the numbers with the narrator when she calls out "10, 20, 30, 40, 50,..." Ask them what it is. (a football field) Who had the correct prediction? STOP the video after she says, "It all depends on how you look at it." Ask students what three things patterns do for us. (create, explain, and predict) Have groups list three things patterns can us help create, three things patterns can help us explain, and three things that patterns can help us predict. Share results with the class.
Tell students they will work in their groups to investigate a famous pattern called the Fibonacci Sequence. Ask students to get out paper and pencil. Place this problem on the overhead. Ask students to copy it on their paper.
How many pairs of rabbits will there be at the beginning of each month if every month each pair produces one new pair that begins to bear young two months after birth?
Ask students if anyone wants to make a guess. Ask students what important information is missing for them to find the number of rabbits? (the time frame that they will be multiplying) Tell them to determine it over the course of one year. Give the groups four or five minutes to discuss the problem.
Tell students that they will now watch a video segment that will help them solve this problem. Tell them they are responsible for determining how the sequence is developed and for generating a flow chart to visually describe the solution to the rabbit problem. Tell students that they must also list three things that have Fibonacci numbers. Students will also explain what is the golden ratio. They will give an example of how the golden ratio is found in the human body.
EJECT the Math Vantage #1 video and INSERT the Math Vantage #3 video, already cued to begin where the narrator is holding a rabbit (after the Math Vantage logo) and just before she says, "What's up, Doc?" BEGIN the video. PAUSE after she says, "There's a pattern here." Tell the students that they are now going to draw a diagram of this problem to make it easier to understand. They will also keep a tally of how many pairs of rabbits are in existence each month. In this model, a circled "A" will represent an adult pair of rabbits that can reproduce when the rabbits are two months old. A circled "Y" will represent a young pair that will not be able to reproduce for two months. Begin with a young pair which cannot reproduce. It cannot reproduce the next month either. However, in March, the pair has its first pair of rabbits.

In April, the young pair becomes an adult pair that is not yet ready to reproduce. However the original adult pair has another pair. Extend the pattern with the students through July.

Is the diagram getting too involved? (Most students will say yes.) Tell the students that you will show them that video segment again. REWIND to the "HUH?" written on the screen and resume the video. PAUSE after she says, "There's a pattern here." Ask students to find a quicker pattern that will generate the correct amount without having to draw the diagram. (Add the two previous entries to find the next in the sequence.) Ask them to share it with the class. RESUME the video. PAUSE after she says, "And there's just not enough carrots to go around." Check for understanding. Did the students come up with the correct pattern?
Have them fill out the chart on the transparency (included with this lesson) to discover the Fibonacci Sequence embedded in the original sequence. Look at the total of rabbits each month. Then compare the number of adults each month separately and the number of young each month. Ask students if they can see it. Have them fill out the chart through December. Ask them why the world is not totally overpopulated with rabbits if they multiply according to the pattern we discovered. (Rabbits do not really multiply this way. Also there are predators and disease.) Point out that this is one pattern that holds true on paper, but does not necessarily hold true in real life. Inductive logic again is not foolproof.
RESUME the video. PAUSE after she says, "...artichokes and pine cones too." Pass out pineapples, pine cones, flowers, and limbs to each group. Have students find the Fibonacci numbers in all the materials. Are there any exceptions? Remind students that the patterns might not be correct in every situation.
RESUME the video. PAUSE when the three ratios are shown next to the rectangles. Have students divide each ratio, changing the ratio to a decimal. Have students round each decimal to the nearest hundred. Ask students what they notice. (all about 1.6) Have the students try larger ratios. Are they about the same? (yes)
RESUME the video. PAUSE after she says, "The result-visual harmony. Where else can we find golden ratios?" Tell students that they will now draw a spiral pattern generated from golden rectangles. Ask students to use their rulers to draw a 1 centimeter by 1 centimeter square in the middle of a sheet of graph paper. The dimensions of this square are 1 by 1 centimeters squared. Have students draw another square attached to the top side of the first square. Ask students for the dimensions of the new rectangle. (1 by 2)

To the left side of the rectangle, draw a 2 by 2 square. Ask students the dimensions. (2 by 3)

To the bottom of the new rectangle, draw a 3 by 3 square. Ask students for the new dimensions. (3 by 5)

To the right side of the triangle, draw a 5 by 5 square. Ask students if they see the pattern of Fibonacci here. What are the new dimensions? (5 by 8)

Ask students to continue the diagram placing the next square on the top, and the next on the left. They will continue until they have a rectangle with dimensions of 13 by 21. Have students draw the spiral as seen in the diagram attached.

Have students compare their picture of the spiral to the Nautilus shell. Ask students to see the similarity in the structure of the shell.
Tell students that a similar spiral pattern can be seen in golden triangles. Golden triangles are isosceles triangles with sides that are consecutive Fibonacci numbers. Have students draw a large isosceles triangle on a piece of notebook paper with congruent sides of 13 centimeters and a base side of 8 centimeters. To help draw this triangle, have students use the protractor to make the base angles 72 degrees. Students will draw the angle bisector of one of the base angles making the new angle 36 degrees. The new triangle formed will also be isosceles. That length of the segment formed in the triangle is the same length as the original base length. This new triangle is also in the golden ratio and has new lengths of 8, 8, and 5 centimeters. Repeat the process for the new isosceles triangle by drawing the angle bisector of one base angle of the new base angle. Continue through four more iterations. Note that each triangle formed is golden. Have students draw the spiral as seen below.





Ask students if they think this pattern will hold with a triangle with legs being 21 centimeters and base being 13 centimeters. [Note: Students should agree that the pattern should hold. Students might note that the spiral becomes bigger as the ratios get larger.] Ask students how they could show that the pattern holds true. (draw it) Remind students that this is only proof for one orientation. Ask the students to compare the spirals formed from golden rectangles and golden triangles. How are they the same and how are they different? (Same: in golden ratio and both form spirals. Different: rectangle pattern spirals out and the triangle pattern spirals in; also different in shape)
Tell students that they can see the Fibonacci numbers in music and in the human body. RESUME the video. PAUSE when the frame of her bent finger is shown surrounded by a rectangle and she says, "In that golden rectangle, there is a golden ratio." Have students in their groups measure their own bent fingers. Ask for the a show of hands of all whose finger fits the ratio.
RESUME the video. STOP after she says, "Try it with
a friend."
Have each member of the group measure themselves with a tape measure from the waist down and from the waist up. Place the waist down measure in the numerator and the waist up measure in the denominator. Use the calculator to get the decimal value. Ask students if measuring the total height to the waist down measure should give the same value. (yes) Have students take those measures and find the decimal value. Ask for a show of hands of the people whose bodies are in the golden ratio.
Have students note that some people do not fit the pattern. Predictions made on patterns can be incorrect. Remind them that the original video stated that if one's perspective is skewed, a correct prediction cannot be made. Tell students that in this video they will see how pattern predicting can lead to false conclusions in a real life situation. This is a video about a dog called Wishbone with very human characteristics. He lives with a young boy. The boy has a friend who spends the night because the friend's house is being painted. The friend's mother believes her son is allergic to paint. As a matter of fact, she thinks he is allergic to everything. To give the students a responsibility while viewing, ask students to watch carefully to determine if they believe that she has correctly diagnosed the condition of her son's health. Students are to determine the pattern the son follows the minute he thinks he is allergic to something.
EJECT the Math Vantage video and INSERT the Wishbone video. BEGIN the Wishbone video when the mother says, "Well, I think that's just about everything." PAUSE after Wishbone says, "Boy, if you look for sickness, that may be all you find." FAST FORWARD and PAUSE at the frame where Wishbone is laying on a chair watching the boys play checkers. While fast forwarding, begin a discussion about Wishbone's comment. How does this comment describe how the mother perceives her child? (She sees him as a sickly child because he was sickly when he was little. She worries about him and tries to protect him.) What item sent with him leads us to think she is too over protective? (a snake bite kit) Ask students how this attitude would lead to misconceptions about her son's health. (Accept any reasonable answer.)
RESUME the video. PAUSE when Wishbone says, "I'll finish his dinner. Joe?" FAST FORWARD to the frame where Wishbone is in the bed with the allergic boy. While fast forwarding, ask students to describe what happened when the over protective mom saw Wishbone in the house. Does she stay true to the pattern we have determined for her? (yes) What immediate change happened to her son as soon as he is told he is allergic to dogs? (starts scratching and sneezing) Is he reacting according to a set pattern? (yes) Ask students to determine why we feel that this is a trained response, not an actual allergy. (He was around Wishbone all day and had no problems.) Tell students that they will now watch a segment where Wishbone proves that the reaction was a trained response.
RESUME video. STOP when Wishbone says, "...nonallergenic pancakes." Begin a discussion on how we use inductive repeated patterns in our daily lives. What patterns do we set to get to school on time? Do we always follow that pattern? What can cause that pattern to change, making it impossible to guarantee that the same thing will happen every day? Discuss how pattern recognition helps us decide who we will consider close friends. Discuss the patterns that lead to success in school and in business. Ask students how studying inductive logic in their math class can help them make good decisions in life.

Patterns make the world less chaotic and more organized. Sometimes patterns are broken deliberately to create something unusual, eye-catching, and interesting. Tell students that they will investigate the golden ratio (or absence of it) in comic strip characters. Give each group of four a section of the Sunday comics from the newspaper. Ask each group to pick 10 characters that they will measure. Three must be animals given human characteristics; three must be normal looking characters that look like real people; and four must be caricatures of real people. Students are to measure in millimeters each character's face to determine if the face is in the golden ratio (from eyes to chin over eyes to top of forehead). Students will next measure in millimeters each character's body to determine if the whole body is the golden ratio (length from the waist down to toe over the waist to the top of the head). Each group will prepare a report to record all findings. The groups will try to determine a pattern of the kind of ratios used most in serious comic strips and the kind of ratios used most in the funny ones. The results will be shared with the class to determine if a predominating pattern is evident. Students will also try to find exceptions to the patterns.
Research indicates that many women in our society are displeased with their physical appearance. The pattern does not seem to hold true for men in our society. Tell students that they are to make an inductive argument that could explain this pattern. Ask students to compare the appearance of women in music videos versus the appearance of men in them. (Women tend to be young, slim, beautiful. Men can be overweight, bald, older.) This is an inductive pattern, so there will be many exceptions.
Have groups look through the newspaper's fashion section. The illustrated models of women are not drawn in the golden ratio. Have students measure the drawings as they did the comic strip characters. Ask students why they are drawn out of proportion. (to emphasize sleekness, elegance, give them a willowy look) Could women look the same as the models in the pictures if they bought those clothes? (no) Why? (because our bodies are in the golden ratio)

Clear your plan first with the local toy store manager. Tell students that they will investigate the golden ratio is children's dolls. They will go to a toy store to measure the ratios in "girl's dolls" and in "boy's dolls". Tell students that dolls are for anyone who wants to play with them, but advertisers market many dolls specifically for girls, such as Barbie, or boys, such as action figures. Students must measure 10 dolls in each of the girl/boy categories. Students are to compare the number of girls' dolls in the golden ratio to the number of boys' dolls in the golden ratio. Everyone must measure the Barbie doll, one of the most popular dolls in America. [Note: The Barbie doll is proportionally impossible as a real human. Young girls may use Barbie as a standard of acceptable beauty and form. This could set the same pattern just as the illustrated fashions, making it impossible to reproduce in real life.] Have students note the differences in the colors used for girl and boy toys. (pastel for girls and primary for boys) Have them compare the names of the girl toys and boy toys. (Boys' toys have much more complex names.) Have each group of students make a poster with their results. They are also to write a short paper on what patterns they found. They must also explain how these differences, if they found them, could affect the kinds of people girls and boys become. Have students comment on how these patterns are good and how they might be harmful.

Architecture: Find the golden ratio in classical architecture such as the Parthenon and the Acropolis.
Art: Find examples of the golden ratio in art
and sculpture.
Mathematics: Derive the golden ratio: 1 + 5
2
Science: Study fractals to see patterns within patterns. Study the patterns in DNA molecules.
Industry: Study the patterns used in a fast food chain to serve food quickly and efficiently. Study assembly lines in factories to determine how patterns speed up production of a product.
Fashion: Study the history of high fashion to determine how changing hem lines and waist lines could change the look of the human body to emphasize or change the golden ratio appearance of the wearer.
Social Studies: Study patterns that repeat themselves over time such as Napoleon's and Hitler's attempt to conquer Russia.
Psychology: Study patterns of behavior that get characterized under certain neurosis.
Medicine: Interview a doctor to determine how he/she uses patterns of symptoms to decide what disease or ailment a patient has.

1995-1996 National Teacher Training Institute / Austin

Master Teacher: Linda Shaub

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