Do Now: Write a similarity ratio to answer the question. If you have a vision problem, a magnification system can help you read. You choose a level of magnification. Then you place an image under the viewer. A similar, magnified image appears on the video screen. The video screen pictured is 16 in. wide by 12 in. tall. What is the largest complete video image possible for a block of text that is 6 in. wide by 4 in. tall?

September 13, 2011 Agenda: A. Do now/HW check GPS 7-2 #9 B. Math History: The golden rectangle C. Mini lesson: Properties of Proportion D. Practice Exercises E. Closure/Exit Question F. HW

Similarity in Polygons is not an invention but a pattern from nature. It is found in the golden ratio and the golden rectangle.

The Golden Rectangle A golden rectangle is a rectangle that can be divided into a square and a rectangle that is similar to the original rectangle. A pattern of repeated golden rectangles is shown at the right. Each golden rectangle that is formed is copied and divided again. Each golden rectangle is similar to the original rectangle. In any golden rectangle, the length and width are in the golden ratio, which is about to 1 The golden rectangle is considered pleasing to the human eye. It has appeared in architecture and art since ancient times. It has intrigued artists including Leonardo da Vinci (1452–1519). Da Vinci illustrated The Divine Proportion, a book about the golden rectangle

Supposedly, Pythagoras discovered this ratio. And the ancient Greeks incorporated it into their art and architecture. Apparently, many ancient buildings (including the Parthenon) use golden rectangles. It was thought to be the most pleasing of all rectangles. It was not too thick, not too thin, but just right (Baby Bear rectangles). Because of this, sheets of paper and blank canvases are often somewhat close to being golden rectangles. 8.5x11 is not particularly close to a golden rectangle, by the way. The golden ratio is seen in some surprising areas of mathematics. The ratio of consecutive Fibonacci numbers (1, 1, 2, 3, 5, 8, 13..., each number being the sum of the previous two numbers) approaches the golden ratio, as the sequence gets infinitely long. The sequence is sometimes defined as starting at 0, 1, 1, 2, 3... Zero is the zeroth element of the sequence. See Fibonacci Numbers.Fibonacci Numbers

Here is a 'Fibonacci series'. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,.. If we take the ratio of two successive numbers in this series and divide each by the number before it, we will find the following series of numbers. 1/1 = 1 2/1 = 2 3/2 = 1.5 5/3 = /5 = /8 = /13 = /21 = The ratio seems to be settling down to a particular value, which we call the golden ratio(Phi= ).

DO NOW: Explore the Fibonacci Series Construct a pyramid starting with one toothpick. Write the series as the number of toothpicks for each level. Writing Exercise: Describe the series

An artist plans to paint a picture. He wants the canvas to be a golden rectangle with its longer horizontal sides 30 cm wide. How high should the canvas be?

A map has dimensions 9 in. by 15 in. You want to reduce the map so that it will fit on a 4 in.-by-6 in. index card. What are the dimensions of the largest possible complete map that you can fit on the index card?

Do Now:Design: You want to make a scale drawing of your bedroom to help you arrange your furniture. You decide on a scale of 3 in. = 2 ft. Your bedroom is a 12 ft-by-15 ft rectangle. What should be its dimensions in your scale drawing? Learning Intentions: Today you will learn to: find similarity ratio of similar polygons; apply proportions to real life problems

Vocabulary Check: Ratio Proportion Mean proportional or geometric mean: If the two means of a proportion are equal, either mean is called the mean proportional between the extremes of the proportion.

Mini lesson Practice Exercises using the example from Reteaching 7-2 WS & #3 as counter example. Think, Pair & Share Activity: Do # 1,2,4 Class Discussion Closure HW: Do # 5-9;work on the ws

Memory check: Name some postulates that prove whether the triangles are congruent. September 15,2011 Learning Intentions: Learn to prove that triangles are similar. Do Now: Do # 1 on worksheet

To determine whether two quadrilaterals are congruent, you must check both that their corresponding sides are proportional and that their corresponding angles are congruent. Use investigative approach

AA Similarity Conjecture If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Investigation 1: Is AA a Similarity Shortcut? Draw your own triangle ABC. Use a compass and straightedge to construct triangle DEF, with A D and B E. Are your triangles similar? Explain. Your findings should support this conjecture.