F un E xperiment O n R atios Groups of TWO or THREE Measure your friend's: Height (approximate) Distance from the belly button to the toes (approximate) Divide the 1 st measurement by the 2 nd Approximate your answer to THREE places after the decimal 1 st measurement 2 nd measurement

F un E xperiment O n R atios The Ratio Should Be: …

Experiment !

The Fibonacci Series Leonardo of Pisa ( ), nickname Fibonacci. He made many contributions to mathematics, but is best known of numbers that carries his name: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,... This sequence is constructed by choosing the first two numbers (the "seeds" of the sequence) then assigning the rest by the rule that each number be the sum of the two preceding numbers.

Take the RATIO of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13,..) and divide each by the number before it. 1/1 = 1, 2/1 = 2, 3/2 = 1·5, 5/3 = ?, 8/5 = ?, 13/8 = ?, 21/13 = ? Use your calculator and plot a graph of these ratios and see if anything is happening. You'll have DISCOVERED a fundamental property of this RATIO when you find the limiting value of the new series!

Throughout history, the ratio for length to width of rectangles of has been considered the most pleasing to the eye. This ratio was named the golden ratio by the Greeks. In the world of mathematics, the numeric value is called "phi", named for the Greek sculptor Phidias. The space between the columns form golden rectangles. There are golden rectangles throughout this structure which is found in Athens, Greece. The Golden Ratio

Examples of art and architecture which have employed the golden rectangle. This first example of the Great Pyramid of Giza is believed to be 4,600 years old, which was long before the Greeks. Its dimensions are also based on the Golden Ratio.

Pythagorean Connection

Pythagoras of Samos about 569 BC - about 475 BC

Unpacking

Course 212 –

Course 33 –

Course 33 –

Course 33 –

Algebra 1

Geometry

Pythagorean Connections

Pythagoras of Samos about 569 BC - about 475 BC

Pythagorean Connections

Very Interesting

12 Equal sized Sticks Area 9 Perimeter 12 Area 5 Perimeter 12

The Challenge Area 4 Perimeter 12 Objective:

I should agree I agree Very Interesting

Handout Booklet: Pages 1-2 THIRD GRADE

Handout Booklet: Pages 3 Pages 4- in today’s handout provide a sampling of how Number Sense develops across the grade levels. Your task is to TEACH someone else about the MacMillan math program. List six key points you would include in your presentation.

THIRD GRADE Handout Booklet: Pages 4-

Handout Booklet: Pages 3-4 In Problem Solving Lessons

Handout Booklet: Pages 1-2

Handout Booklet: Pages 9-

Warm Up Fun Activities

You may use calculators 20 minutes

Find the sum of the digits of the number raised to the second power !

Interesting Discovery!!!

Interesting !!!

Interesting Discovery!!! =

Answer 67

Vik Help Me Explain

How Would You Solve The Problem ? Any volunteers ?

Help Me Get The Answer Using Sound Mathematical Reasoning “No Fuzzy Stuff”

6 th Grade by long division

Mathematical Reasoning “No Fuzzy Stuff”

Vik

28x9 28x9

48 x 9 = space fold 4 3 2

28x9 28x9

83 x 9 = space fold 7 4 7

63 x 9 = space fold 5 6 7

85 x 9 = space fold 7 6 5

fingers times 10 3 fingers 2 finger s 6 50

Greater than 5 3 fingers X