Assignment P : 1, 2-30 even, 42, 44, 45, 46, 48, 50, 56, 58, 63, 72, 73 Da Vinci Code Ratios Challenge Problems

Example 1 In a survey of American families, 150 families had a total of 360 children. What is the ratio of children to families? On average, how many children are there per family? Make sure you pay attention to the order of the wording 360/ children/family

What is a Ratio? ratio A ratio is a comparison of two quantities, usually by division. The ratio of a to b is a:b or Order is important! –Part: Part –Part: Whole –Whole: Part –Units, sometimes important

6.1 Ratio and Proportion Objectives: 1.To recognize and use ratios and proportions to solve problems

Example 2 Find the first 13 terms in the following sequence: 1, 1, 2, 3, 5, 8, … Fibonacci Sequence This is called the Fibonacci Sequence! 13,21,34,55,89,144,233

Foxtrot

Example 3 What happens when you take the ratios of two successive Fibonacci numbers, larger over smaller? What number do you approach? ETC!

The Golden Ratio What happens when you take the ratios of two successive Fibonacci numbers, larger over smaller? What number do you approach?

The Golden Ratio What happens when you take the ratios of two successive Fibonacci numbers, larger over smaller? What number do you approach? It’s the Golden Ratio = …

What’s a Proportion? proportion When two ratios are equal, it’s called a proportion. What’s an example of a proportion? What ratio is equal to ½? Proportions are often used in solving problems involving similar objects.

Solving a Proportion What’s the relationship between the cross products of a proportion? 360  12.4  150 They’re equal!

Solving a Proportion Cross Products Property In a proportion, the product of the extremes equals the product of the means.

Solving a Proportion To solve a proportion involving a variable, simply set the two cross products equal to each other. Then solve! 275  x15  25

Example 4 Solve the proportion. 50x = 1950 x=39

Example 5 Solve the proportion. Show your work in your notebook. x= -6 or 1

More Proportion Properties INTERESTING, huh?

Guided Practice Within your group, take five minutes to work through the following problems. When you are finished, check your answers on the overhead.

Exercise 1 If you work for 2 weeks and earn $380, what will you expect to earn in 15 weeks? $2850 Show your work in your notebook

Exercise 2 Solve for y : Show your work in your notebook y=2

Exercise 3 Which is longer: a yardstick or a meter stick? (Use the conversion factor 1 in. = 2.54 cm) Show work in your notebook. Meter stick Remember: 1 m = 100 cm

Exercise 4 The sides of a rose garden in the shape of a right triangle are in the ratio of 8:15:17. If the perimeter is 60 ft, what is the length of the shortest side? How are you going to do this one? Think about what perimeter means Work it out in your notebook. 12

Pi: The Movie! Darren Aronofsky’s  maybe the first ever math-fi movie. In one scene, Max explains the concept of , except he calls it  ! What a dork!dork In another scene, Max makes a Golden Rectangle. Goes crazy. crazy

The Greeks, Again! The Greeks used the Golden Ratio to do everything from making a pentagram, to constructing a building, to combing their hair.

The Golden Rectangle If you make a rectangle with sides that have the Golden Ratio, you’ve made a sparkly Golden Rectangle.

The Golden Rectangle This happens precisely when the ratio of the long side to the short side is equal to the ratio of the sum of the sides to the long side.

Example 6 Assume that the smaller side of a Golden Rectangle is 1. Use algebra to find the exact value of phi, the Golden Ratio.

Da Vinci Code Robert Langdon in Dan Brown’s The Da Vinci Code teaches his class about phi. He claimed the following were equal to the Golden Ratio. Try them, why don’t you.

Da Vinci Code 1.Tip of head to floor/Belly button to floor 2.Shoulder to fingertips/Elbow to fingertips 3.Hip to floor/Knee to floor

Apophenia? Because we are so prone to finding patterns, we must be careful of apophenia. Apophenia Apophenia (a-poe- FEE-nee-uh) is seeing patterns or connections when there aren’t any.

Assignment P : 1, 2-30 even, 42, 44, 45, 46, 48, 50, 56, 58, 63, 72, 73 Da Vinci Code Ratios Challenge Problems

Extra Credit Opportunities 1.Construct a Golden Rectangle with a compass and straightedge and explain how it demonstrates the Golden Ratio