What comes next?

What comes next? -4, 0, 4, 8… …12, 16

How did we find those answers How did we find those answers? What went through your mind to figure out the pattern? What was told to you in the beginning of the problem?

Inductive Reasoning The process of reasoning that a rule or statement is true because several specific cases are true. Inductive reasoning can be used to formulate a conjecture about something.

Conjecture A statement that is believed to be true. Also known as your hypothesis. What is a hypothesis? Every example must be true for a conjecture to be true. It is very hard to prove a conjecture true.

Counterexample Example that does not support the conjecture. There only has to be ONE counterexample to prove the conjecture false. Usually, it is easier to prove a conjecture false than to prove it true.

Example 1 Formulate a Conjecture Look at the progression of the pattern to formulate a conjecture regarding the number of blocks there will be at the fifth arrangement.

Example 2 a. Testing a Conjecture Michelle made the conjecture, “The expressions 6n +1 and 6n -1 will always result in two prime numbers.” Show that this conjecture is true for n=1, 2, and 3, but not true for n=4.

Example 2 b. Testing a Conjecture Maria looks at the diagram below and conjectures that the number of triangles in the figure is given by the expression 2n + 1. Is this conjecture true for the four steps of the pattern shown below? n = 1 n = 2 n = 3 n = 4

Example 3 A researcher studying crows for several years made the observation that every crow she studied was black. Her research assistant made this conjecture: “All crows are black.” How can this conjecture be tested? Can it be proved?

Your turn to practice! Working in PAIRS, work on the worksheet. What isn’t finished in class is homework!

Perimeter What is the perimeter of a shape? The perimeter is the sum of the side lengths of a closed geometric figure.

Area What is the area of a figure? The area is the size of the region bounded by the figure.

Geoboards! This diagram shows one unit and one square unit. What do we measure with one unit? What do we measure with square units?

Use your geoboard to create a polygon whose vertices are (1,1), (1,4), (5,4), and (5,1) What is the name of this polygon? What is the perimeter of this figure? What is the area of this figure?

Use your geoboard and create a triangle whose vertices are (1,8), (4,8), and (4,4). What is the perimeter? How long is each side? What is the area?

Make your own shape. It can be any shape you want Make your own shape. It can be any shape you want. Switch with a partner and find the area and perimeter of your partner’s shape.

What do you think of when you hear FORMULA?

Mathematical relationship expressed with symbols. Formula Mathematical relationship expressed with symbols.

Area and Perimeter Formulas

Example 1 Find the perimeter of a star with equal side lengths of 5m. Find the perimeter of this figure: 20ft 5ft 5ft 4ft 9ft 17ft 9ft 5ft 17ft 7ft 30ft

Example 2 a. Find the area of a rectangle whose side lengths are 11cm and 4cm. b. Find the area of a triangle with a base of 9in and height of 14in.

Pythagoras http://www.mathopenref.com/pythagoras.html

Pythagorean Theorem The sum of the square of the lengths of the legs, a and b, of a right triangle is equal to the square of the length of the hypotenuse c and is written a2 + b2 = c2. c a b

Example 3 Using the Pythagorean Theorem Find the length of the hypotenuse. Find the area of the triangle. 12cm 5cm 5ft 3ft b

Example 4 Measuring Temperature Different countries use different units to measure the temperature. Much of the world uses degrees Celsius, but a few countries use degrees Fahrenheit. For scientists and travelers, converting between Celsius and Fahrenheit is an important skill. To convert Celsius from Fahrenheit, use this formula: C = 5/9(F – 32) If it is 77oF, what is it in Celcius? If it is 10oC, what is it in Fahrenheit?

HOMEWORK Finish inductive reasoning worksheet and do these problems from the book: pages 50-52 #2, 6, 13-15, 21, 26, 29

(2009). Saxon Geometry: Teacher’s Edition (2009). Saxon Geometry: Teacher’s Edition. United States of America: HMH Supplemental Publishers. Geoboard Activities. (2007). Areas and Perimeters. Retrieved from http://www.jamesrahn.com/geometry/activities/area_and _perimeter.htm