Agenda 9/27 DO NOW (3 parts) ◦Prove the following: 1.The length of a square is 5 cm. What is the measure of its width? How can you be sure? 2.What is the sum of the first 5 counting numbers? 3.Solve the equation 2x + 1 = 5. Test corrections – to be turned in during class…graded (worth 10 points) Section 2.1 – An Introduction to Proofs (Notes)

Section 2.1 (Part 1) An Introduction to Proofs Geometry 2010 – 2011

Adding Integers How could you find the sum of the first n odd counting numbers without actually adding them? Complete the table below. Then, see if you can find the sum of the first 100 odd numbers without actually summing them.

Adding Integers Table

Examining Area You are given the figure below, which is built entirely of squares. The area of square C is 64, and the area of square D is 81. Is the overall figure square?

Blaise Pascal (1623 – 1662) French mathematician, physicist, inventor, writer, and philosopher Invented the mechanical calculator ◦Started working on it in his teens ◦Took him 3 years Helped to create two new areas of research ◦Projective geometry (at age 16) ◦Probability theory (applicable in economics and social science) ◦

The BEAUTY of Pascal’s Triangle The triangular array of numbers shown below is commonly known as Pascal’s triangle, named after Blaise Pascal (1623 – 1662). Pascal’s triangle has applications in geometry, algebra, and probability. (a) Can you find the entries for the next row in the triangle? Write the next row. (b) Find the sum of each row. What is the sum of the entries in the nth row? (c) Without actually writing the first 50 rows of Pascal’s triangle, find the sum of the entries in the 50 th row.

Earliest known “Pascal’s Triangle” The illustration above is the earliest known version of “Pascal’s triangle.” It is from a Chinese book printed around 1303 C.E.

What is a proof? A ______ is a convincing argument that something is true. A proof starts with ______, or statements that are accepted as true, and then uses logic to reach a _________.

2.1 (Part 1) Homework Page 83 ◦#7 – 8, 12 – 15, 29 – 32

Agenda – 9/28 DO NOW: ◦The sequence shown below is commonly known as Fibonacci’s sequence, named after Leonardo of Pisa who was known as “Fibonacci” (son of Bonaccio). This sequence has numerous applications in economics, architecture, and even music. Explain how find the next term in the sequence. Then, find the next three terms in the sequence. 0, 1, 1, 2, 3, 5, 8, 13, … Continue working through 2.1

Section 2.1 (Part 2) An Introduction to Proofs Geometry 2010 – 2011

Leonardo of Pisa (c – 1250) Known as Fibonacci Italian mathematician ◦Considered by some “the most talented western mathematician of the Middle Ages” Best known for spreading Hindu-Arabic numeral system and for the Fibonacci sequence (we will be studying this!)

Examine a sequence A __________ is an ordered list of objects (or events)

9.) What number do the sums in the table seem to be approaching? 10.) Because the sequence has infinitely many terms, you cannot add them all with your calculator. However, it is still possible to find the sum of the infinite sequence. Start by drawing a square with an area of 1 square unit. Label the lengths of the sides of the square.

2.1 (Part 2) Homework Page 85 ◦#33 – 45